HydrologyandWaterResourceSystemsAnalysis

HydrologyandWaterResourceSystemsAnalysis

MARIAA.MIMIKOUNationalTechnicalUniversityofAthens,Greece

EVANGELOSA.BALTASNationalTechnicalUniversityofAthens,Greece

VASSILIOSA.TSIHRINTZISNationalTechnicalUniversityofAthens,Greece

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Names:Mimikou,MariaA.,author.|Baltas,EvangelosA.,author.|Tsihrintzis,VassiliosA.,1959-author.Title:Hydrologyandwaterresourcesystemsanalysis/authors,MariaAMimikou,EvangelosABaltas,VassiliosATsihrintzis.Description:BocaRaton:Taylor&Francis,2016.|Includesbibliographicalreferencesandindex.Identifiers:LCCN2015039103|ISBN9781466581302(alk.paper)Subjects:LCSH:Waterresourcesdevelopment.|Hydrology.|Systemanalysis.Classification:LCCTC405.M492016|DDC551.48--dc23

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To my husband Yiannis and my daughter Alexandra for their supportMariaMimikouTomyfamilyEvangelosBaltasInmemoryofmyparentsAndreasandEvangeliatowhomIoweeverything.To my wife Alexandra and my two sons Andreas and Konstantinos for theirsupport,encouragementandpatienceVassiliosA.Tsihrintzis

1

1.11.21.31.41.51.61.71.8

1.8.11.8.2

1.91.10

2

2.12.22.3

2.3.12.3.22.3.3

2.42.4.12.4.2

Contents

PrefaceAuthors

Introduction

GeneralTheScienceofhydrologyHistoricalevolutionofhydrologyClassificationofhydrologyHydrologicalcycleHydrologicalvariablesandtheirunitsofmeasurementRiverbasinScaleinhydrology

SpatialscaleTimescale

WorldwidedistributionofwaterHydrologicalbalance

References

Precipitationandhydrologicallosses

GeneralFormationofatmosphericprecipitationPrecipitationtypes

RainSnowHail

CoolingmechanismsandtypesofprecipitationCycloniccoolingOrographiccooling

2.4.32.5

2.5.12.5.22.5.32.5.42.5.5

2.5.5.12.5.5.2

2.5.5.3

2.5.62.5.7

2.5.7.12.5.7.22.5.7.32.5.7.42.5.7.52.5.7.62.5.7.7

2.62.72.8

2.8.12.8.22.8.32.8.42.8.5

2.92.9.12.9.22.9.32.9.42.9.5

2.9.5.12.9.5.22.9.5.3

2.10

ConductivecoolingMeasurementofprecipitation

PrecipitationsensorsRaingaugerecordersSnowfallmeasurementMeteorologicalradarJoss–Waldvogeldisdrometers

DescriptionofthedisdrometerRD-80Hydro-meteorologicalparameterswhichcanbederivedbytheRD-69andRD-80disdrometersDerivationofraincharacteristicswiththeuseofaJoss–WaldvogelRD-69disdrometer

InstallationofraingaugesSpacemeasurements

SatellitesInstalledinstrumentsAdvancedprecipitationradarMicrowaveimagerVisibleandinfraredradiationscannerCloudsandtheEarth’sradiantenergysystemLightningimagingsensor

InstallationofnetworkofpointmeasurementdevicesTestofdatahomogeneityandanalysisofdoublecumulativecurvesCompletionofrainfallmeasurements:Adaptationtodifferentaltitudes

MethodofarithmeticmeanMethodofnormalratiosInversedistancemethodCorrelationandregressionCorrectionofrainfallwithaltitude

SurfaceintegrationofarealrainfallfrompointmeasurementsAveragingmethodThiessenmethodIsohyetalcurvemethodOptimuminterpolationmethod(Kriging)Timedistributionsofrainfall

Limited-scalephenomenaMedium-scalephenomenaSynoptic-scalephenomena

Hydrologicallosses

2.10.12.11

2.11.12.11.1.1

2.11.22.11.32.11.4

2.122.12.12.12.2

2.12.2.12.12.2.22.12.2.32.12.2.42.12.2.52.12.2.62.12.2.7

2.12.32.12.3.12.12.3.2

2.132.13.12.13.22.13.32.13.42.13.52.13.62.13.72.13.8

3

3.13.23.3

3.3.13.3.1.1

GeneralEvaporation

WaterbalancemethodsThornthwaite’swaterbalancemethod

EnergybalancemethodsMasstransfermethodsCombinationmethods:Penmanmethod

EvapotranspirationWaterbalancemethodsMethodsforthedeterminationofpotentialevapotranspirationfromclimaticdata

Penman–MonteithmethodThornthwaite’smethodBlaney–CriddlemethodJensen–HaisemethodMakkink’smethodHargreavesmethodPriestley–Taylormethod

MethodsforthedeterminationofactualevapotranspirationTurcmethodCoutagnemethod

InfiltrationrateestimationInfiltrationHotton’smodel(1930)Green–Amptmodel(1911)Huggins–Monkemodel(1966)Holtanmodel(1961)Kostiakovmodel(1932)Philipmodel(1954)SoilConservationServicemethod

References

Runoff

GeneralRiverbasinHydrographs

Characteristicsofthehydrographφindex

3.3.23.3.2.1

3.3.2.2

3.3.33.3.4

3.3.4.13.3.4.23.3.4.3

3.43.4.13.4.23.4.3

3.53.5.13.5.23.5.33.5.43.5.5

3.63.6.13.6.2

3.73.7.1

3.7.1.13.7.1.2

3.7.23.7.33.7.4

3.7.53.7.6

3.7.6.13.7.6.23.7.6.3

3.7.73.7.7.1

HydrographseparationMethodsofbaseflowseparationfromthetotalhydrographHydrographseparationwiththemethodofthelogarithms

CompositehydrographseparationFactorsinfluencingthehydrographshape

ClimaticfactorsTopographicfactorsGeologicalfactors

HydrometryInstallationcriteriaforahydrometricstationMeasurementofwater-levelDischargemeasurementbythemethodofvelocityfield

DischargeestimationusinghydrometricdataPreparationofaratingcurveExtensionoftheratingcurveRemarksontheratingcurvesEstimationofanaveragewater-levelforaspecifictimestepFlowestimationfrommeasurementsofwater-levelmeters/recorders

Rainfall–runoffrelationships:EmpiricalmethodsRationalmethodforestimatingfloodpeaksOtherempiricalmethodsforcalculatingpeakrunoff

Rainfall–runoffrelationships:TheUHBasicassumptionsofUH

PrincipleofproportionalityPrincipleofsuperposition

DerivationoftheUHfromasinglerainfalleventMathematicaldeterminationoftheUHofcompositerainfallDeterminationofUHofacertaindurationfromknownUHofadifferentduration:S-curveEstimationoffloodhydrographusingtheUHSyntheticUHs

Snyder’sUHSCSdimensionlesshydrographmethodTriangularSCShydrographmethod

InstantaneousUHByanS-curve

3.7.7.23.7.7.33.7.7.4

3.7.83.7.93.7.10

3.83.8.1

3.8.1.13.8.2

3.8.2.13.8.2.2

4

4.14.1.14.1.24.1.34.1.4

4.24.3

4.3.14.3.2

4.44.4.14.4.24.4.34.4.44.4.5

4.54.5.14.5.24.5.3

4.5.3.1

4.5.4

ByusingaconvolutionintegralBytheuseofvariousconceptualmodelsRoutingtime-areacurveofbasins

CEH-UK:RevitalizedfloodhydrographShortcomingsoftheUHandtheUH-basedmodelsGeneraloverviewofhydrologicalmodels:Specificrainfall–runoffmodels

CasestudyusingtheHEC-HMShydrologicalmodelGeneralinformationforHEC-HMS

ComponentsofHEC-HMSCasestudy

StudyareaHEC-HMSenvironment

References

Probabilityandstatisticsinhydrology

GeneralconceptsanddefinitionsExperimentsandsamplespacesProbabilityfunctionConditionalprobabilityTotalprobabilityandBayes’theorem

RandomvariableDistributions

NormaldistributionLog-normaldistribution

SomeimportantdiscretedistributionsBernoullitrialsandtheBernoullidistributionBinomialdistributionGeometricdistributionPoissondistributionUniformdistribution

SomeimportantcontinuousdistributionsUniformdistributionExponentialdistributionGammadistribution

Gammadistributionofthreeparameters(PearsontypeIII)

Log-Pearsondistribution

4.64.6.1

4.6.1.14.6.1.2

4.6.24.6.34.6.4

4.74.7.14.7.2

4.84.8.1

5

5.15.25.35.45.55.65.7

5.7.15.7.25.7.3

5.85.8.15.8.2

5.95.9.15.9.25.9.3

5.9.3.15.9.3.25.9.3.3

5.9.45.9.5

StatisticalanalysisofextremesPointfrequencyanalysis

GraphicalmethodMethodoffrequencyfactor

GumbelmaximumdistributionGumbelminimumdistributionWeibulldistribution

TestingofthedistributionsTestX2Kolmogorov–Smirnovtestfortheappropriatenessofadistribution

Intensity–duration–frequencycurvesConstructionoftheidfcurves

References

Groundwaterhydrology

GeneralSoilandaquiferparametersClassificationofaquifersFieldmeasurementsMathematicalproblemofgroundwaterGeneralexpressionofgroundwaterflowAnalyticalsolutionsofsteadyflow

ConfinedaquiferUnconfinedaquiferSemi-confinedaquifer

TheoryofimagesWellnearriverWellnearimpermeableboundaries

Analyticalsolutionsofnon-uniformflowWellhydraulicsProcessingofdrawdowntestsAnalysisofconfinedaquifers

TheismethodCooper–JacobmethodChowmethod

AnalysisofunconfinedaquifersAnisotropicalluvialdeposits

5.105.115.12

5.12.15.12.2

6

6.16.2

6.2.16.2.2

6.36.3.16.3.26.3.36.3.46.3.56.3.66.3.76.3.86.3.9

6.3.10

6.3.11

6.46.4.16.4.2

6.56.5.16.5.26.5.36.5.4

6.66.7

6.7.1

WelllossesAquiferrechargeSalinization

InterfaceofsaltwaterandfreshwaterSaltwaterelevationcone

References

Hydrologicdesign

IntroductionSizingofreservoirs

GeneralSectionsofvolumeandreservoirexploitation

ConventionalmethodofsizingtheactivereservoirvolumeConstructionofacumulativeinflowcurveEstimationoftheactivereservoirvolumeDisadvantagesofRippl’sconventionaldesignSequentpeakanalysismethodNon-conventionalstochasticmethodsofreservoirsizingSyntheticdataofinflowsandtheircumulativecurvesDesignrisk:EstimationofvolumebasedonacceptableriskAdvantagesofthenon-conventionalsizingmethodSensitivityofthereservoirdesignonthemechanismofsyntheticdischargegenerationNon-conventionalmethodofsizingincludingthepersistenceininflowsSizingofareservoirwithaspillwaywiththeuseofthemethodofmaximumshortage

SizingofareservoirinariversitewithoutmeasurementsRegionalhydrologicmodelsRegionalmodelforthederivationofmonthlyinflows

SizingofthedeadreservoirvolumeGeneralSedimentationinareservoir:DefinitionofdeadstorageEstimationofdeadvolumeEstimationofactivevolumelossduetoreservoirsedimentation

Sizingofthereservoir’sfloodvolumeHydrologicdesignoffloodsafety(protection)structures

Spillwayhydrologiedesign

6.7.26.7.36.7.46.7.56.7.6

6.86.8.16.8.26.8.3

6.96.9.16.9.2

6.9.3

6.9.3.16.9.46.9.5

7

7.17.1.17.1.27.1.37.1.47.1.57.1.67.1.77.1.8

7.27.2.17.2.2

7.37.3.1

7.3.2

SpillwaydesignfloodCriteriaofspillwaydesignProbablemaximumflood(PMF)EstimationofthedesignstormandfloodDerivationoffloodhydrographinspecificcases

HydrologicdesignofariverdiversionIntroductionDesigncriteriaofadiversionEstimationofthedesignfloodofdiversion

Hydrologicdesignofotherwaterstructure–specificissuesGeneralConstructionofoperationaldepth–duration–frequencycurvesofrainfallforirrigationneedsDischargepredictioninawaterrecipientofoutflowsfromawastewatertreatmentplant

ConditionsandrequirementsofthemethodModelsofdischargepredictionsofwaterrecipientofsewageDuration–dischargecurvesandtheiruseinthestimationofthehydroenergypotential

References

Urbanhydrologyandstormwatermanagement

IntroductionDefinitionsofurbanhydrologyandstormwatermanagementHistoryDescriptionoftheurbanstormwaterdrainagesystemImpactsofurbanizationFactorsinfluencingurbanrunoffqualityPollutantgenerationprocessesTypesandsourcesofpollutantsImpactstothereceivingwaters

UrbanrunoffquantitycomputationsRationalmethodSCSmethod

UrbanrunoffqualitycomputationsUSEPAmethodforthepredictionofannualunitpollutantloadingsUSGSmethodforthepredictionofmeanannualpollutant

7.3.37.3.47.3.5

7.47.4.17.4.27.4.37.4.47.4.5

8

8.18.2

8.2.18.2.28.2.3

8.38.3.18.3.2

8.48.5

8.5.18.5.1.18.5.1.28.5.1.3

8.5.28.5.3

8.5.3.18.5.3.28.5.3.3

8.68.6.18.6.2

8.6.2.18.6.2.28.6.2.3

quantitySimplemethodofWashington,DCPollutantaccumulationonstreetsurfaceWashoffofaccumulatedpollutants

SurfacerunoffquantityandqualitymanagementGeneralPollutioncontrolatthesourcePollutioncontrolinthesewerandthedrainageditchesPollutioncontrolwithsurfacedetention/retentionandstorageRunofftreatment

References

Sedimenttransportanderosion

IntroductionPropertiesofsediment

SizeandshapeDensity,specificweightandspecificgravityFallvelocity

FlowresistanceChannelflowresistanceFlowresistanceinalluvialstreamsandrivers

IncipientmotionSedimenttransportformulas

BedloadMeyer–PeterandMüllermethodEinsteinmethodEinstein–Brownmethod

SuspendedloadTotalsedimentload

EinsteinmethodAckersandWhitemethodYang’smethod

LanderosionandwatershedsedimentyieldIntroductionUniversalsoillossequation(USLE)

RainfallerosivityfactorRSoilerodibilityfactorKSlopelengthandsteepnessfactorLS

8.6.2.48.6.2.5

8.6.38.6.4

8.6.4.18.6.4.2

VegetativecoverandcroppingmanagementfactorCErosioncontrolpracticesfactorP

ModifieduniversalsoillossequationErosioncontrolmeasures

ErosioncontrolatthesourceStructuralmeasurestocontrolerosion

ReferencesIndex

Preface

The overall goal is to provide students and practitioners with a complete andcomprehensive guidebook on hydrological and water resources issues byfacilitatingtheirunderstandingofbasicandtheoreticalknowledgeandconceptsthrough a significant number of examples. We believe that this book willcontribute to the educationofundergraduate andgraduate studentswhoattendclassesrelatedtohydrologyandwaterresourcesandwillprovidebetterinsightsto scientists, technicians, practitioners and professional engineers regardingintegratedapproachesinhydrologicalprocesses.

Authors

Dr.MariaA.Mimikou isaprofessor in theSchoolofCivilEngineeringat theNationalTechnicalUniversityofAthens(NTUA)anddirectoroftheLaboratoryof Hydrology and Water Resources Management, Athens, Greece. Sheestablished the Center of Hydrology and Informatics (CHI)(www.chi.civil.ntua.gr).Shehasvastexperience(35years)intheareasofwaterresourcesmanagement;waterresourcessystemsplanningandoperation;urban,rural and coastal hydrology; stochastic hydrology; hydrological and waterqualitymodelling;soilerosionandsedimenttransport;climatechangeandlanduse change at the catchment scale; flood forecasting; risk assessment andmappingandwaterscarcityanddroughtanalysis.Shehaswrittenseveralbooksandhasauthoredorco-authoredmorethan80papersinpeer-reviewedscientificjournals and has more than 100 peer-reviewed publications in national andinternationalconferences, inaddition toseveral scientific technical reports.Dr.Mimikou coordinates several undergraduate and postgraduate courses in theSchoolofCivilEngineering(NTUA)andhassupervised90graduatediplomasandpostgraduatetheses.Shehasvastacademicandnon-academicadministrativeexperience.Shehas served as director of theHydrologySectionof thePublicPowerCorporation.Shehasservedasdeanof theSchoolofCivilEngineering(NTUA)anddirectoroftheDepartmentofWaterResourcesandEnvironmentalEngineering. She is experienced in themanagement of research programs andhas been scientifically responsible for and administrator of many Europeancompetitiveresearchprogramsandhasbeenacoordinatorofandscientificallyresponsible for other International competitive programs(http://mimikou.chi.civil.ntua.gr/). Also, she has served as a member of theExternalAdvisoryGroupsinDGResearch,aschairmanandmemberofsteeringand organizing committees in several conferences, as a member of differentEuropeanscientificnetworkslikeEurAqua,EXCIFF,etc.andhascontributedtoposition papers for the Water Framework Directive, Horizon 2020, Water

Science—PolicyInterfacing.Sheestablishedthenationalacademicnetworkonhydrologyandwaterresources‘HYDROMEDON’.

Dr.EvangelosA.BaltasisaprofessorintheSchoolofCivilEngineeringattheNationalTechnicalUniversityofAthens(NTUA),Athens,Greece.Heactivelyparticipated in the establishment of the Center of Hydrology and Informatics(CHI) in Athens, which comprises the NTUA meteorological network, thedatabase of the hydrological information and the experimental basin. He hasmore than 25 years experience in the areas of water resources management,waterresourcessystemsplanningandoperation,hydrometeorology,hydrologicalmodelling, climate change and land use change at the catchment scale, floodforecasting,riskassessmentandmappinganalysis.Hehaswrittenseveralbooksandhasauthoredorco-authoredmorethan70papersinpeer-reviewedscientificjournals and more than 100 peer-reviewed publications in national andinternational conferences, in addition to anumberof technical reports.Hehassupervised more than 60 undergraduate and postgraduate theses and hasextensive academic and non-academic administrative experience. He has alsoofferedengineeringconsultationservicesinthefieldsofhisexpertisetotheEU,Greek ministries, public organizations and private companies in the UnitedStates and Europe. He has been the principal investigator or researcher incompetitive EU (more than 30) and nationally funded programs related tointegratedwaterresourcesmanagement.HealsoservedasasecretarygeneralfortheMinistryofEnvironment,PhysicalPlanningandPublicWorksfrom2006to2009.Duringthatperiod,hewasappointedasaGreekdelegatetoanumberofcouncilsataministeriallevelintheEuropeanUnion,UNESCO,OECDandtheUnitedNation for issuesconcerningenvironmental legislation,climatechange,renewableenergywaterresources,etc.

Dr. Vassilios A. Tsihrintzis is a professor of ecological engineering andtechnology at the School of Rural and Surveying Engineering, NationalTechnical University of Athens, Greece. His research interests concentrate,among others, on water resources engineering and management with anemphasis on urban and agricultural drainage and non-point source pollution,water quality of aquatic systems and pollution control, ecohydrology andecohydraulicsandtheuseofnaturaltreatmentsystems(i.e.constructedwetlandsand stabilization ponds) for runoff and wastewater treatment. His publishedresearch work includes more than 130 papers in peer-reviewed scientificjournals,more than 250 papers in conference proceedings andmore than 100technicalreports.Hehasalsoauthoredorco-authoredbooksandbookchapters

on operations research, urban hydrology and runoff quality management andnatural systems for wastewater and runoff treatment, among others. He hasparticipatedasaprincipalinvestigator/coordinatororateammemberinvariousresearch projects in theUnited States, the EU andGreece.Dr. Tsihrintzis hassupervisedmorethan80undergraduateandpostgraduatethesesand12doctoraldissertations. He regularly teaches engineering hydrology, urban watermanagement, fluid mechanics, groundwater, environmental engineering andnaturalwastewatertreatmentsystems.Hehasalsoservedasaprofessorandthehead of theDepartment ofEnvironmentalEngineering,DemocritusUniversityofThrace,Greece,forseveralyears,andpreviouslywasanassociateprofessorof water resources engineering at Florida International University, Miami,Florida.Dr.Tsihrintzishasextensiveprofessionalexperienceasapracticingcivilandenvironmentalengineerinleadingengineering:consultingfirmsbothintheUnited States (he was a registered professional engineer in California and acertifiedprofessionalhydrologistbytheAmericanInstituteofHydrology)andinGreece, having been involved in several projects related to land development,drainage, urban hydrology, sediment transport and channel design, waterresourcesmanagement,wetlandsrestorationandconstructedwetlands.

Chapter1Introduction

1.1GENERAL

Thisbookcontainsthedescriptionandanalysisofbasicconcepts,aswellas,agreatnumberofapplicationsandsolutionsthatcoverthescientificdisciplinesofengineeringhydrologyandwaterresourcesmanagement.Morespecifically, inChapter2, the formsofprecipitation, theprecipitation-

measuring instruments and their function and installation design, the basicpreprocessing of the point rainfall data (homogenization, addition of missingdata,altitudecorrection)andthemethodsofintegrationfortheestimationofthearealrainfallaredescribed.Also,inthesamechapter,weexplainthetermsandthe estimation methods of evaporation, transpiration and actual potentialevapotranspiration.Finally,thehydrologicallossesonthegroundarepresented,suchasstoragedepression,retentionandinfiltration.Finally,variousmethodsofinfiltrationevaluationarealsopresented.In Chapter 3, we deal with runoff components, the geomorphological and

physiographic characteristics of a river basin and the characteristics and thecomponents of a hydrograph. Moreover, the hydrometry, the rainfall–runoffrelationships,andselectedmodelsarepresented.InChapter4,adetaileddescriptionofgroundwaterisgiven.Particularly,soil

parameters, groundwater aquifers and their classification, the mathematicalproblem and the general solution of groundwater flow, analytical solutions ofsteadyandunsteadyflows,thetheoryofimages,pumpingtests,aquiferrechargeandseawaterintrusionarepresented.InChapter 5, the basic concepts of the probability and statistics theory in

hydrologyaredescribed.Continuousanddiscreteprobabilityfunctions thatareappropriate for use in hydrology and water resources, extreme distributions(floods and droughts) and rainfall intensity–duration–frequency curves are

presented.InChapter6,thehydrologicdesignofhydraulicandotherengineeringworks

is presented. The issues discussed include hydrologic design (sizing) ofreservoirs and safety (flood protection) structures like spillways and riverdiversionworks.Bothdeterministicandstochasticapproachesarepresented.Inaddition,regionalmodelsaredescribedforthedesignofworksatungaugedsitesandotherspecificissuesrelated,forexample,tothedesignofirrigationuptakesandsmallhydroelectricdams.In Chapter 7, issues of urban hydrology and stormwater management are

presented.Thechapterstartswith thedescriptionof theurbandrainagesystemand proceeds with the impacts of urbanization in affecting the shape of therunoffhydrograph.Then,issuesrelatedtourbanrunoffquality,includingfactorsaffectingrunoffquality,thepollutantgenerationprocesses,thetypesandsourcesofpollutantsandtheimpactsonreceivingwatersarediscussed.Thechapterthendescribesmethodstocomputeurbanrunoffquantity,suchastherationalandtheSCSmethods.Then, threemethods for runoffqualitycomputations,aswellasmethodstoevaluatepollutantaccumulationonthestreetsurfaceandwashoffofpollutants,aredescribed.Finally,thelastsectionofthechapterpresentsindetailthedesignofagreatnumberof structuralandnon-structuralbestmanagementpracticestomanageurbanrunoffquantityandquality.Amongtheseareporouspavements,infiltrationtrenches,grassfiltersanddryandwetponds,aswellasmethodstodealwithcombinedseweroverflows.Chapter 8, the last chapter of this book, is an introduction to sediment

transportanderosion.Thechapter first introduces thesedimentproperties,andthenitaddressesflowresistanceissuesinopenchannelsandalluvialstreamsandrivers, bed forms, incipient motion and stable channel design and selectedsediment transport formulas for bed load, suspended load and total load. Itfinallypresentswatershederosionandsedimentyieldandcontrolmeasures.

1.2THESCIENCEOFHYDROLOGY

Hydrology is the science that describes the appearance, circulation anddistribution of earth’s water, as well as the interaction of its physical andchemicalpropertieswiththeenvironmentandhumans.The objective of hydrology is the scientific evaluation of water in various

phases, especially its variation in time and space. In the scienceof hydrology,various complex processes of primary concern are involved, includingevaporation,rainfall,infiltration,transpiration,storageandrunoff.Hydrologyis

closely connected to a wide range of sciences like biology, chemistry,agriculture, geography, geology, oceanography, physics and volcanology, asshown in Figure 1.1. Its connection with these sciences is a physicalconsequence of the close connection of water with the atmosphere and theground. This connection states clearly that hydrology is an interdisciplinarysciencerelevanttoalmosteveryaspectoflife(Singh,1992).

Figure1.1 Relevanceofhydrologytootherscientificareas.

Many techniques and methods with origins from other scientific areas likemathematics, statistics, probability theory, operational research, control theoryandsystemsanalysishavebeenusedtosolvehydrologyproblems.Thefieldstudyofhydrologyincludestheatmosphere(uptoadistanceof15

km),thesurfaceandtheinteriorofthelithosphere(downtoadepthof1km)andthe hydrosphere (oceans). Within the system of these three areas, thehydrologicalcycletakesplace.

1.3HISTORICALEVOLUTIONOFHYDROLOGY

Mankind before many centuries constructed hydraulic projects worth

mentioning, relying mostly on philosophical theories, since the knowledge ofhydrological processes was almost nonexistent, if not conceptually incorrect.The hydrological cycle was known from Plato’s era; nevertheless, a precisetheory was formulated by Marcus Vitruvius during the first century AD(MimikouandBaltas,2012).Theperiodofmodernhydrologyblossomedintheseventeenthcentury,during

which the knowledge of hydrological phenomena started to be based onmeasurements.PerraultmeasuredtherainfallandevaporationoftheSeineRiverbasin (Perrault, 1966; Nace, 1974), Mariotte (Chisholm, 1911) estimated thedischargeofthisriverandlaterexecutedmeasurementsoftheflowvelocityandthewetcrosssection,andfinally,astronomerHalley(Brutsaert,2005)measuredthe evaporation of the Mediterranean Sea, accounting the inputs and outputsfromthatsea.Withthesemeasurements,itbecamepossibletoextractrelativelypreciseconclusionsaboutvarioushydrologicalphenomena.The experimental period in hydrology started in the eighteenth century and

broughtconsequentdevelopmentsinhydraulics.Fordischargemeasurement,thePitottube(Saleh,2002),themouthpieceofBorda(Chanson,2014)andWoltmanwatermeter(Arreguietal.2007)were invented, theequationsofflowinopenchannelconduitsandspillwayswereformulated,andflowinporousmediawasstudiedincludingequationsforpumpedwells.The insufficiency of empirical equations for adequate solutions to practical

problemsbecameobviousduringthebeginningofthetwentiethcentury.Duetothis, through the foundation of special services and scientific organizationsresponsible to look into hydrologymatters, an effortwasmade for systematiccollection of data, research and study of relevant problems. Thus, the rationalmanagementof hydrological phenomena started in the1930s.Sherman (1932)introduced the concept of the unit hydrograph for surface discharge andstatisticalandmathematicalanalysesofdischarges.Finally, in the1950s, theperiodofapplicationof theoreticalmethods to the

studyofhydrologicalproblemsbegan.Thedevelopmentofelectroniccomputersmade feasible the solution of equations arising from the application ofmathematical analysis of hydrological phenomena. Additionally, thedevelopment of complex instruments of high precision and above-groundmeasuring potential (radars, satellites, etc.)made possible today the collectionand assessment of detailed meteorological and hydrological data, and theverificationofappliedmethods.

1.4CLASSIFICATIONOFHYDROLOGY

Dependingonthewayandthetargetoftheapproachonthesubject,thefieldofhydrologyisdividedintovariousbasicsubdisciplines,asshowninFigure1.2.

Figure1.2 Classificationofhydrologyindisciplines.

Awayofdividingthevariousdisciplinesofhydrologyisbasedonthespacein which various hydrological phenomena occur. So, surface hydrology dealswith surfacewater and groundwater hydrology deals with undergroundwater.This distinction exists due to the different kinetic and dynamic behaviours ofwateronthegroundsurfaceandundertheground.Anotherway of distinction is based on themethodological approach of the

hydrological procedures. Hence, deterministic hydrology uses methods andmodelswhoseparametersarecalculatedfromempiricalprocedures(e.g.theunithydrograph method) and statistical hydrology deals with the methods of thetheoryofprobabilitiesandstatistics.Thelatterisdividedintotwodifferentsub-categories: probabilistic hydrology, which analyzes and composes thehydrologicaleventswithouttakingintoconsiderationthetimecontinuityoftheevents,andstochastichydrology,whichtakesintoconsiderationthetimeseriesinthestructureandtheoccurrenceofthehydrologicalevents.The branch of hydrology that targets the understanding of the physical

processes, the causes and mechanisms that evoke them and the naturalphenomenathatareconnectedtothemisknownasphysicalhydrology,whilethediscipline that targets the quantitative estimation and the prediction of thehydrological phenomena is known as engineering hydrology. Thus, theengineering hydrology provides the tools for the study, or in other words thehydrologic design for the functioning and sizing of all hydraulic and otherengineeringworks,forexample,thehydrologicdesignofdamsforthestorageofwater, water supply, irrigation and natural land reclamation networks, floodprotection structures (embankments, levees), urban hydraulic projects (sewagenetworks),roads,bridgescrossingriversandprojectsofgroundwaterrecharge.

1.5HYDROLOGICALCYCLE

Thehydrologicalcycledescribestheperpetualmovementofwaterbetweentheoceans, theatmosphereand the land,accompaniedbychangesamong thewet,gasandliquidphasesofwater.Figure1.3presentsaschematicdescriptionofthehydrologicalcycle.The start of the hydrological cycle can theoretically be placed in the

atmosphere, where the vapour from water evaporation from the land and theoceans,aswellasfromtranspirationfromtreesandvegetation,isgathered.Thevapouriscarriedbywindsandwhenappropriateconditionsdevelop,cloudsareformed.Atalatertime,thevapourformsatmosphericprecipitation(rain,snow,haze)whichreturnstotheearthsurface.Fromthewaterwhichreachestheearthsurface, a part is detained by the vegetation and evaporates, another partinfiltratesintothesoil,andfinally,alastpartrunsasrunoffonthesurfaceandcollects in streams or rivers ending in lakes or seas. From the water whichinfiltrates,apartevaporatesor transpires throughplantsandtherestpercolatesin deeper layers of the ground, recharging the groundwater aquifers, findingwaysout tothesurfaceata later timeat loweraltitude,andendingup,finally,into the sea. Then, through evaporation, the water returns to the atmosphere,completingthehydrologicalcycle.

Figure1.3 Thehydrologicalcycleandtheannualworldhydrologicalbalance.

1.6HYDROLOGICALVARIABLESANDTHEIRUNITSOFMEASUREMENT

The basic hydrological variables are precipitation, evaporation, interception,retention, infiltration and percolation, and surface runoff. Surface runoff endsinto the most important water resources and also creates most hydrologicalhazards(i.e.floods,droughts).Alsoundergroundpercolationisconnectedwiththeexploitationofwaterresourcesandconstitutesamajorsubjectofengineeringhydrology. Atmospheric precipitations are connected to surface runoff andunderground percolation by the cause–effect relation. Evaporation andtranspiration, which are also referred together as evapotranspiration, arehydrological losses,meaning thepartofprecipitation thatdoesnotrunoffandcannot be available for surface or underground storage and exploitation. Inconclusion, runoff (surface and groundwater), precipitation andevapotranspiration are the most characteristic processes and variables of thehydrologicalcycle,andtheirvaluesquantifythestatusofthewaterresourcesinanarea.Table1.1presentsthemostimportantunits(inthemetricsystem)usedinthe

scienceofhydrologyforallmeasuredvariables.Theseunitsaredeterminedbythe precision of the instruments used and by the physical meaning of themeasuredvalue.Themost commonunitofmeasurementofdischarge is cubicmetres per second (m3/s), while also the equivalent depth of water is used,dividedby thesurfaceof theriverbasinusuallymeasured inkm2.Therainfalldepth is measured in mm or cm, indicating the volume of rainfall per unitdrainagearea.Itisunnecessarytoexpressthisparticularvariableinprecisionofmm,sincetheraingaugesandrainmetersdonothavethiskindofprecision.Inanycase,inthemeasurementofthevariablesandexpressionoftheresults,thejudgementofthehydrologicalengineerisincorporated.

Table1.1Somecommonlyusedhydrologicalparametersandtheirunits

Variables Characteristics Units

Rainfall Depth Millimetres(mm)

Intensity Millimetresperhour(mm/h)

Duration Hours(h)

Evaporation Rate Millimetresperday,monthoryear(mm/day,mm/month,mm/year)

Depth Millimetres(mm)

Infiltrationandpercolation Rate Millimetresperhour(mm/h)

Depth Millimetres(mm)

Interception Equivalentdepth Millimetresperstormduration(mm/time)

Retention Equivalentdepth Millimetresperstormduration(mm/time)

Runoff Discharge Cubicmetrespersecond(m3/s)

Volume Cubicmetres(m3)

Equivalentdepth Equivalentmillimetresovertheriverbasin(mm)

1.7RIVERBASIN

It is very important in hydrology to define the entire area that receives therainfall,andtheareawheresurfacerunoffisproducedandendsuptoaspecificlocationorpointof thedrainage system.Thisarea is calleddrainagebasin,orwatershed,orcatchmentareaof the river,or simply riverbasin.Theboundarylinealongatopographicridgeseparatingtwoadjacentdrainagebasinsiscalleddrainagedivide.AsshowninFigure1.4,thesinglepointorlocationatwhichalldrainage from the basin concentrates as outflow is called concentration orcollectionpointoroutletofthebasin(Raghunath,2006;Figure1.4).

1.8SCALEINHYDROLOGY

Depending on a certain hydrological problem, the hydrological cycle and itscomponentsmaybeconsideredatdifferentscalesofspaceandtime.Inthestudyofhydrologicalphenomena, it isnecessary toknowevery typeofhydrologicalvariables in continuous space and time. It is clearly understandable that theextreme complexity of the hydrological phenomena and the vast extent anddepth of time duringwhich they are evolvingmake this approach impossible.The insertion of scale is connected with the separation of a specific spatialterritoryandaspecifictimeperiodinwhichwestudyeachphenomenon.

Figure1.4 Ariverbasinwithitsconcentrationpointandthedrainagedivideline.

1.8.1Spatialscale

Theworldwide scale is the largest spatial scale,while the scale of a drainagebasin is the smallest as far as the hydrological matters are concerned. It isobviousthatthereisnotasingleanduniquedrainagebasinforastream,butateach point, there is a specific drainage (river sub-basin). This is the mostimportantscaleforthescienceofengineeringhydrology,andalltheotherscalesmay be composed of summing different river basins. This approach alsoseparates hydrology from hydraulics; in the latter case, the average scale is apipe/canalorapartofit.Itshouldalsobeunderstandablethatadrainagebasindoes not necessarily coincide with the spatial or district limits that aredetermined for economical or political reasons.Drainage basinsmay have thesizeofasmallparkoreventhesizeofalargeriverbasin.Forexample,thesizeofMississippiriverbasinoccupiesapproximately41%oftheareaoftheUnitedStates. Usually, large basins are divided into smaller sub-basins so that thehydrologicalanalysisiseasier.Thedrainagebasinswhichshowoperationalinteresthavesizesofabouttens

or thousands of km2. However, in research studies regarding mainly theunderstanding of the physical mechanisms which are connected with thehydrologicalprocedures, thedetailedobservationandmeasurement takesplaceinsmallerdrainagebasins(evensmallerthan1km2),theso-calledexperimentalriver basins. Finally, sometimes, the observation of a phenomenon or itsmeasurementtakesplaceinaverysmallareawhichispracticallyrepresentedby

onepointonly,sothenwecanspeakaboutpointobservationormeasurement.

1.8.2Timescale

Allthehydrologicalvariablesintroducetimevariability.Thetrueknowledgeofthe time evolution of a hydrological variable demands its continuous timemonitoring. However, this is usually impossible, either due to the calculatingdifficultiesorbecauseoftheabsenceofdistinctmeasurements.So,thevariablesaremonitoredatdifferent(distinctive)timescalesdependingonthenatureoftheproblem that is confronted. The timescales which are used in hydrology varyfromafractionofanhourtoayearorevenmanyyears.Thetimescalewhichisusedinhydrologicalanalysisdependsonthegoalofthestudyandthenatureofthe examined problem. Usual scales in hydrology are hourly, daily, weekly,monthlyandannualscales.Thechosenscaleisoftendeterminedfromthetimestepoftheavailablehydrologicaldata(e.g.measurementsofriverstageatdailystep,rainfallathourlystep).In conclusion, the timescales ofminutes, hours or even days are proper for

studiesofstorms,floodsorevenfordetailedstudiesorhydrologicalanalysesinanarea.Inothercases,forexample,inwatermanagementstudies,theefficientstandardsaretimescalesofmonthsoryears.

1.9WORLDWIDEDISTRIBUTIONOFWATER

Waterisoneofthebiggestresourcesprovidedbynature,duetoitsnecessityinthe lives of people, animals and plants. The total volume ofwater on earth isestimatedaround1360millionkm3.ThedistributionofwaterisgiveninTable1.2. We can easily observe that the largest amount of water is stored in theoceans(97.2%ofthetotalvolumeofearthwater).Asignificantvolumeofwater(2.15% of the total water volume) is stored in the form of ice glaciers andconsequently is not exploitable. The remaining amount ofwater of the planetcorrespondstoriverandlakewater,biomasswaterandgroundwater.Groundwaterhasavolumeof8.4×106km3;nevertheless,halfofitcannotbe

exploitedbecauseeitherit is locatedsodeepthatpumpingisnoteconomicallyfeasibleor it issaline.Fromtheexploitableamountof theplanet’swater, riverandlakewatercompriseonlyasmallfractionofalmost2%.Theremaining98%oftheexploitablequantityofwaterisgroundwater.Fromthesearguments,itisclear that a minimal quantity of groundwater is crucial for satisfying humanneeds. The recharge of a significant part of groundwater is done via theinfiltrationoftheatmosphericprecipitation,whileontheotherhand,substantial

amountofrunoffendsintotheoceansandseas.

Table1.2Worldwidedistributionofwater

Volume×1000km3 Percentage%

Atmosphericwater 13 0.001

Surfacewater

Oceanwater 1,320,000 97.2

Saltwaterinlakes 104 0.008

Freshwaterinlakes 125 0.009

Freshwaterinrivers 1.25 0.0001

Wateriniceglaciers 29,000 2.15

Biomasswater 50 0.004

Groundwater

Waterintheunsaturatedzone 67 0.005

Wateratadepthupto800m 4,200 0.31

Wateratadepthfrom800to4000m 4,200 0.31

Total 136,000 100

Source:Bouwer,H.,GroundwaterHydrology,McGraw-Hill,NewYork,1978.

1.10HYDROLOGICALBALANCE

The hydrologic or water balance of a drainage basin is the mathematicalquantitativeexpressionofitshydrologicalcycle.Itisexpressedbyequatingthedifference between inputs and outputs in the drainage basin with the rate ofchange of the water storage ΔS in a defined time interval Δt. If the drainagebasin, or a reservoir, is considered as a system, in which all the inputs andoutputsareknownandtheinternalprocessesareunknown(blackbox),thenthehydrologicalbalance(budget)canbeexpressedasfollows:

(1.1)

(1.2)

where and are, respectively, the average inflow and outflow in the timeintervalΔt,whichshouldberelativelysmallfortheestimationofaveragevaluestobemeaningful.Theindices1and2correspondtothestartandendofthetime

interval Δt = t2 − t1 If I andO are continuously changing through time, thenEquation1.2canbewrittenas

(1.3)

For a river basin, the rainfall, the snow, the hail and the other forms ofprecipitation could be considered as inflow. The surface runoff, intermediaterunoff, underground runoff, evaporation, transpiration and percolation are themost common forms of outflow. The storage of the drainage basin has alsovariouscomponents like the surface storage (above thegroundcontainingalsothestorageinwaterstreamsandreservoirs),thesoilstorage(inthevoidsandtherootsofplants),theundergroundstorage(intheaquifers)andinterceptedwater(byvegetation,buildings).Theaforementioned factors areoutlined in thenextbasicequationofhydrologicalbalance:

(1.4)

Accordingtothisequation,thechangeinadrainagebasinstorageΔSisequaltotheamountofwaterfallingintheformofprecipitationP,minustheamountofwater that exits the basin as surface runoff R, percolates underground G,evaporates to the atmosphere E and transpires from the plant leaves T. Forindividual rain incidents, the constituents of evaporationE and transpirationTaresubstantiallylowerthantherestandusuallyareomitted.

Example1.1Arainfallofanintensityof5mm/hhasdroppedonariverbasinofanarea5km2in6h.Attheexitoftheriverbasin,thedischargehasbeenmeasured during this time period at 100,000 m3. Estimate thehydrologicallossofthis6hrainfallevent.

Themeasuredoutflowis

The difference between outflow and inflow gives the hydrologicallosses:

These lossesmostly comprise the amount of water infiltrated in the

soil after subtracting evaporation and transpiration, which, however,arerelativelysmallforatimeintervalof6h.The rateof losses, expressed inunitsof equivalentdepthofwater

perunitareaofriverbasinandtime,isasfollows:

Example1.2Inagivenmonth,a lakeofconstantsurfacearea1,000,000m2hasameaninflowof0.40m3/s,anoutflowof0.34m3/sandanincreaseinstorage of 20,000m3. A rain gauge aside the lake measured a totalrainfalldepthof30mmforthismonth.Ifweassumethatotherlossesfromthelakeareinsignificant,estimatethemonthlyevaporationofthelake.SolutionThe general equation of thewater balancewill be appliedwhich, inthisparticularcase,takesthefollowingform:

whereIistheinflowinthelake=0.40m3/sor1,036,800m3inthemonthOistheoutflowfromthelake=0.34m3/sor881,280m3inthemonthPistheprecipitation=0.030m·1,000,000m2=30,000m3

ΔSistheincreaseinstorage=20,000m3

EistherequestedevaporationByreplacementofthevaluesinthewaterbalanceequation,weget

Evaporationcouldbeexpressedinunitsofverticaldepthofwater(cmormm),bydividingthevolumewiththeareaofthelake:

Example1.3The drainage basin of a river has a total area of 840 km2, and it is

1.2.

3.

4.5.

6.

7.

1.

2.

covered by four rain stationsA,B,C andD. The percentage of theareacoveredbyeachstationis30%,25%,35%and10%,respectively.Duringthehydrologicalyear1998–1999,thesefourstationsrecordedthefollowingmonthlyrainfalldepthsinmm(Table1.3).

Table1.3Monthlydepthsofraininfourrainstations

Estimatethefollowing:ThepointannualrainfalldepthineachstationTheannualrainfalldepthandtheannualvolumeofrainontheriverbasinTheannualvolumeofrunoffdischargeandtheequivalentdepthofdischargeinmm,iftheaverageannualdischargeofrunoffatthebasinoutletis8.3m3/sTheannualrunoffcoefficientThevolumeofhydrological lossesand theequivalentdepthofhydrologicallossesontheriverbasinThe actual evapotranspiration if the hydrological losses frominfiltration and underground percolation are 45% of the totalrainfallThemaximummonthlyrainfalldepthintheriverbasin

SolutionThepointannualdepthof rainfall ineachstationresultsas thesumof12monthlyvaluesofrainfall.TheresultsaregiveninthelastcolumnofTable1.4.For the estimation of the annual rainfall depth, initially wetransform the rainfall point measurements on the surface bymultiplying the point rain of every month with the respectivepercentageofareathateverystationrepresents.Forinstance,inMarch,theaveragemonthlyrainfallisequalto

TheresultsarepresentedinthelastrowofTable1.4

3.

4.

5.

6.

Theannualdepthofrainfallisproducedbythesumofthe12values calculated and is equal to 798 mm. This amountmultiplied by the surface of the river basin gives the annualvolumeofrainfall:

Table1.4Calculationofpointannualandaveragemonthlyvaluesofrainfall

Therunoffvolumeattheoutletoftheriverbasinisdeterminedbymultiplyingdischargebytherespectivetime:

The equivalent depth of rain is the quotient of this volumedividedbytheareaoftheriverbasin:

Theannualrunoffcoefficientistheratiooftherunoffdepth(i.e.runoffvolume)thatexitedtheriverbasintotheannualrainfall.

Inthiscase,C=312/798=0.391.

Thevolumeofthehydrologicallossesis

So,theequivalentdepthofthehydrologicallossesis

The water balance of this river basin is expressed in the

7.

1.

2.

3.

4.

5.

followingequation:

wherePistherainfalldepth=798mmRistherunoffdepth=312mmG+ΔSistheinfiltrationandundergroundstorage=0.45×P=359mmEistheevapotranspiration=798−312−359=127mm

Themaximummonthly depth of rainfall in this river basin isshowninthelastrowofTable1.4andisequalto123.5forthemonthofNovember.

Example1.4Inariverlocation,adamistobeconstructedforthestorageofwaterfor irrigationneedsof thenearby fields.The riverbasinupstreamofthedamhasanareaof818km2andisconsideredimpermeable.Table1.5presentshistoricaldataof (a) theannual surface rainfallPof theriverbasinand(b) theaverageannual runoffdischargeQat thedamlocationfor10hydrologicalyears.Calculatethefollowing:

Thevolumeof runoffdischarge inm3and therespectivedepthofrunoffinmmforeveryyearTheactualevapotranspiration(ET)intheriverbasininmmandinm3foreveryyearThe runoff coefficient for every year and for the given timeperiodThe expected annual volume of water (in mm) that we canexploit if the total losses of the under construction dam (i.e.evaporation, infiltration, overflows) are 18% of the averageannualinflowonthereservoirTheareaof thearable landthatcanbeirrigatedwiththegivenvolumeofwater (in thecaseof irrigationdam) if thepotentialevapotranspiration of the selected crop during the irrigationperiod is 800 mm and the rainfall during the same period isnegligible

Table1.5Dataofaverageannualsurfacerainfalland

1.

2.

averageannualrunoffdischarge

Hydrologicalyear P(mm) Q(m3/s)

1981–1982 1487 23.73

1982–1983 1337 17.01

1983–1984 1357 24.99

1984–1985 1000 15.54

1985–1986 1532 23.10

1986–1987 1182 14.49

1987–1988 1173 13.86

1988–1989 1238 12.39

1989–1990 760 5.67

1990–1991 1477 20.37

SolutionThe discharge volume is given by multiplying the averageannualdischargeoftheriverbytherespectivetime(31,536,000s/yr).ItisseeninthefourthcolumnofTable1.6inm3andinthefifthcolumninmmafterdivisionbytheareaoftheriverbasin.Given that the river basin is impermeable, the only losses arethose of the actual evapotranspiration, which is derived byabstracting the equivalent depth of runoff discharge from thesurfacerainfall:

Thecalculationoftheevaporationforeveryyearinmmandinm3isshownincolumnssixandsevenofTable1.6.

Table1.6Estimationofdischarge,evapotranspirationandrunoffcoefficient

3.

1.

Therunoffcoefficientforeveryyearisdefinedastheratiooftheamountofwaterdischargingfromtheexitof theriverbasin tothetotalamountofwaterthatprecipitatestotheriverbasineachyear,thatis,

Thevaluesoftherunoffcoefficientforeveryyeararecalculatedin the last column of Table 1.6. We remind that the runoffcoefficientisadimensionlessparameter.Therunoffcoefficientforthewholetimeperiodofthesample,i.e. 10 years, is estimated again as the quotient of the totalamountofwaterdischargedtothetotalprecipitationdepth:

Itmustbeunderlinedthat it is incorrect tocalculateCtotas theaverageof10annualvaluesofCfor10years.The average annual inflow in the reservoir is equal to theaveragevalueoftheamountofwaterdischargingintotheriverwherethedamistobeconstructed:

Theexploitableamountofwateris82%ofthequantitygivenor

5.

orbydivisionwiththeareaoftheriverbasin

Letusassume, forsimplicity, that if the irrigationneedsof theplantsareequaltothepotentialevapotranspiration,thentheareaofthearablelandthatcouldbeirrigatedwiththegivenamountofwateris

Example1.5Inariverbasin,areservoirhasbeenconstructed,servingwatersupplyneedsofamajorcity,aswellasirrigationneedsofanearbycultivatedland.Theareaoftheriverbasin,upstreamofthedamlocation,is600km2, while the average area of the lake of the reservoir is 34 km2

(Figure1.5).Based on data of rain gauges and meteorological stations, the

surface rainfall of the river basin and the rainfall and evaporationofthereservoirareestimatedfor10hydrologicalyears,asgiveninTable1.7. In addition,Table1.7presents the absolute altitudeof thewaterlevelofthereservoiratthebeginningofthehydrologicalyearandtheamount of water consumed for water supply and irrigation needs.Finally,basedonmeasurements, theundergroundwater lossesof thereservoirhavebeenestimatedintheorderof0.08m3/s.

Figure1.5Theriverbasin.

1.

2.

3.

1.

Table1.7Waterbalancedataofthereservoir

Estimatetheinflowinthereservoirandtherunoffcoefficientoftheupstreamriverbasinineachhydrologicalyear.Estimatethewaterlevelofthereservoir,on1October2011,ifitis assumed that during the hydrological years 2010–2011, theinflow, rainandevaporationof the lake, aswell as the sumofthe uptakes, were equal to the mean average values of theaforementioned variables for the period of 10 hydrologicalyears.Estimate the mean yearly runoff of the river at location A,upstream of the reservoir, in a sub-basin of 100 km2. Assumethatthesurfacerainfall,upstreamofpositionA,isincreasedby20% in comparison to the whole river basin, while thephysiographic characteristics of these two river basins aresimilar.

SolutionFromthewaterbalance/equilibriumequation,

Table1.8CalculationofΔHandΔV

Hydrologicyear ChangeinwaterlevelΔH(m)

ChangeinvolumeΔV(×106m3)

2000–2001 1.67 56.78

2001–2002 0.65 22.1

2002–2003 1.40 47.6

2003–2004 −3.35 −113.9

2.

2004–2005 −1.10 −37.4

2005–2006 1.73 58.82

2006–2007 2.72 92.48

2007–2008 −2.13 −72.42

2008–2009 2.01 68.34

2009–2010 0.25 8.5

SolvingforVinflow,weget(1.5.1)

whereVoutflow=Evaporation+Watersupply+IrrigationL(losses)=0.08m3/s=0.08·3600·24·365/106=2.52·106m3/year

FortheestimationofΔV,Table1.8isformed,whereΔHisthedifference in the water level of the lake in comparison to thepreviousyear(i.e.ΔH= (2001–2002)−(2000–2001)=52.50–50.83=1.67)andΔV is thedifferenceinvolume(thisvalue iscalculatedbymultiplyingbythemeansurfaceareaofthelake,i.e.1.67m·34·106m2=56.78·106m3).Preservoirisestimatedbymultiplyingtheareaofthelakewiththerainfallofthelake:

In a similar manner, evaporation is estimated by multiplyingevaporationwith the surface area of the lake, i.e. Evaporation(2000–2001)=1364/1000m3·34·106m2=46.4·106m3. By replacing in Equation 1.5.1, theVinflow is calculated foreveryhydrologicalyear.Iftheprecipitationvolumeofthewholeriverbasinisalsoestimated(bymultiplyingtheareaoftheriverbasin with the annual precipitation), the runoff coefficient iscalculated as the ratio of Vinflow to the annual precipitationPriver_basin:C=Vinflow/Priver_basin.TheresultsareshowninTable1.9.Theaveragevaluesfortheperiodof10hydrologicalyearsofthetotalwatersupplyandirrigationneedsarecalculatedas19.6·106m3 and 159.3·106 m3, respectively. Also from Table 1.9, the

3.

average values of Preservoir, evaporation and Vinflow areestimated.Substituting the values inEquation 1.5.1 (solving for ΔV), wearriveatthefollowingresult:

Table1.9EstimationofVinflow,Priver_basinandC

Bydividingwiththeareaofthelake,wehave

Finally,H2010–2011=54.68(2009–2010)+0.385=55.07m.Weassume,accordingtotheexercise,thattherunoffcoefficientofthesub-riverbasinisthesameasthatofthetotalriverbasin.So,

Thus,

REFERENCES

Arregui,F.,CabreraJr,E.,andCobacho,R.,2007,IntegratedWaterMeterManagement,IWAPublishing,UK.

Bouwer,H.,1978,GroundwaterHydrology,McGraw-Hill,NewYork.Brutsaert,W.,2005,Hydrology:AnIntroduction,CambridgeUniversityPress,NewYork.Chanson,H.,2014,AppliedHydrodynamics:AnIntroduction toIdealandRealFluidFlows,CRCPress,

Taylor&FrancisGroup,Leiden,theNetherlands.Chisholm, H., ed., 1911, “Mariotte, Edme”.EncyclopœdiaBritannica, 11th edn. Cambridge University

Press.Mimikou,M.andBaltas,E.,2012,EngineeringHydrology,Papasotiriou,Athens,Greece.Nace, R.L., 1974, Pierre Perrault: The man and his contribution to modern hydrology, Journal of the

AmericanWaterResourcesAssociation,10(4),633–647.doi:10.1111/j.l752–1688.1974.tb05623.x.Perrault,P.,1966,OntheOriginofSprings (Translationof“De l’originedes fontaines” (1674)),Hafner,

NewYork,ASINB0026MBIHE.Raghunath,H.M.,2006,Hydrology:Principles-Analysis-Design,NewAge InternationalPublishers,New

Delhi,India.Saleh, J.M., 2002,FluidFlowHandbook,McGraw-Hill Professional, CambridgeUniversity Press, New

York.Sherman,L.K.,1932,Streamflowfromrainfallbytheunithydrographmethod,EngineeringNews-Record,

108,501–505.Singh,V.P.,1992,ElementaryHydrology,Prentice-Hall,EnglewoodCliffs,NJ.

Chapter2Precipitationandhydrologicallosses

2.1GENERAL

This chapter describes the types of precipitation, the measurementinstrumentation, the basic analysis processes of the point rainfall information(homogeneity, filling used for measurement gaps, altitude adaptation) and themethods for the estimationof average surface rainfall.Moreover, hydrologicallosses are discussed thoroughly,which are defined as the part of precipitationwhichdoesnotendupinastreamafterarainfall.Hydrologicallossesaremainlydividedintoevaporation,transpiration,infiltration,percolationanddetention.The term ‘precipitation’ includes any formofmoisturewhich falls from the

atmosphere to the Earth’s surface. The humidity in the atmosphere is mainlycaused by evaporation fromwet surfaces and transpiration. Large amounts ofwarmwaterandextensivesoilareascoveredwithlushvegetationareexcellentsourcesfortheenrichmentofatmosphericmoisture.The amount of water vapour in the atmosphere is less compared to the

amountsofotheratmosphericgasesbutisimportantforhumanlife.Theamountofwatervapourvarieswithspaceandtime.Thehighestconcentrationsofwatervapour are near the surface of the sea and decreasewith latitude, altitude anddistance from shoreline. Approximately 50% of the atmospheric moisture isfoundinthefirst1500mfromtheEarth’ssurface.

2.2FORMATIONOFATMOSPHERICPRECIPITATION

Precipitation is the main component of the hydrological cycle. It feeds thesurface receptors, renews the stocks of soil moisture and enriches theunderground aquifers. The types of precipitation are rain, snow, hail and

variations (dew,mist, fog),whichareofminor importance inhydrology.Notethat there isnodirect relationshipbetween theamountofwatervapouroveraregionandtheprecipitationofthatregion(Shaw,1994).Thetypeandamountofprecipitation (product of water vapour in the atmosphere) depend on climaticfactorssuchaswind,temperatureandatmosphericpressure,whileairhumidityisnecessarybutnotsufficienttocauseprecipitation(MimikouandBaltas,2012).Figure2.1showstheformationmechanismsofprecipitation.

Figure2.1 Graphicillustrationoftheformationmechanismsofprecipitation.

The precipitated water is the amount of water contained in an atmosphericcolumnof1cm2baseandheightzwhichextendsabovethesoilsurfaceandisgivenby

  (2.1)

whereρwistheabsolutehumidityWisthedepthofwaterincm

TheverticalstructureoftheatmosphereispresentedinFigure2.2.Troposphere: The most important climatic phenomena (clouds, storms,precipitation,winds) occurwithin the troposphere. The entire quantity ofwater vapour is in the troposphere. Temperature decreases with height.Strongverticalcurrentscauseseverestorms.

Stratosphere: Weak vertical currents are developed. Temperature decreaseswithheight,andtheozonecontentincreases.

Mesosphere: There are ionized molecules, and temperature decreases withaltitude.

Thermosphere: The thermosphere constitutes 1% of the mass of the upperatmosphereandischaracterizedbyhightemperatures(230°C–1730°C).

Lower atmosphere: It constitutes 99.9% of themass of the atmosphere andvolumetrically is composed of nitrogen 78.08%, oxygen 20.95%, argon0.93%andwatervapour0%–4%.

Figure2.2 Verticalstructureoftheatmosphere.

2.3PRECIPITATIONTYPES

Therearethreemaintypesofprecipitationasdescribed:rain,snowandhail.

2.3.1Rain

Rainfallvariesgeographically, temporallyandseasonally. It isobvious that thespatio-temporalvariationofrainfall isakeycomponentforallwaterresourcesmanagementandhydrologicalstudies.Forexample,itisimportanttoknowthattheminimumprecipitationcoincideswiththemaximumdemandinaregion.Theamountofraininaregiondifferssignificantlyevenatverysmalldistances.Forexample, records have shown deviating measurements greater than 20% at adistancelessthan8m.Therainconsistsofwaterdropswithadiameterfrom0.5to7mm.Adropof

rainasitfallsinstillairattainsamaximumspeedcalledterminalvelocity.Thisspeed is obtained when the air resistance offsets the weight of the drop. Theterminalvelocity increaseswith thedrop sizeup toadiameterof5.5mmandthenitdecreasesforlargersizesbecausethedropsstarttowiden;thus,resistancein the air increases. Large drops deform easily and break into smaller partsbefore reaching their terminal velocity.Upward air currents affect the averagediameter of the droplets in clouds and thus affect rainfall. Depending on theamount of rain, the event is characterized as light for rainfall depth up to 2.5mm/h,medium fordepth from2.5 to7.5mm/h and intense for depthsgreaterthan7.5mm/h.Arainfallwithdropsofdiametersmallerthan0.5mm,fallingata uniform rate, is referred to as a drizzle andhas an intensity of less than1.0mm/h.Thetotalrainfallwhichreachesthegroundmayundergovariousprocesses.A

part is intercepted by obstacles or covers irregularities of the surface, it istemporarily stored there, and eventually evaporates. Another part covers theneeds of soilmoisture and then recharges the underground reserves,while thefinal part becomes surface runoff (Viessman and Lewis, 1996). The actualdistributionofprecipitationdependsonthetotalamountofrainfall,soilmoistureconditions,topography,landcover,landuse,etc.

2.3.2Snow

Snowisprecipitationintheformoficecrystals.Althoughindividualicecrystalsmay reach the ground surface, usually many ice crystals coalesce and formsnowflakes.Snow is peculiar when compared to other types of precipitation in that it

accumulates and remains on the ground surface for some time beforemeltingand turning into runoff or sublime. Therefore, apart from the study of snowbehaviourasprecipitation,thehydrologistfacestheproblemofdeterminingtheamountofsnowaccumulatedonthegroundandtheconditionswhichdeterminetherateofmeltingandsublimation.

Thedistributionofsnowfalloveranareaismoreuniformthantherespectivedistributionofrain,butbecauseofitsform,snowconcentrationandretentiononthegroundareveryheterogeneous.Especiallyinmountainousregions,thelackof uniformity is more intense due to the combination of strong winds andtopography.Avalanches and unusualmovements of the snow, caused by localturbulentwinds,maycause localaccumulationofsnowwhich ismuchgreaterthantheannualamountofsnowinthearea.Topographicanomalies,gulliesandvarious structures act as traps which accumulate large quantities of snowtransportedbyairfromsurroundingareas.Additionally,vegetationcoveraffectsthe concentration of snow.Within a forest, the concentration of snow on thegroundvaries,dependingon thedensityand typeof treesand thenumberandtypeofforestglades.Themereknowledgeofthethicknessofthesnowinaregionisnotsufficient

to calculate the equivalentwater depth; the density of the snow cover is alsonecessary.Theequivalentwaterdepthisthewaterdepthinmillimetresresultingfrommelting snowwhich has the thickness and characteristics of a particularsnowcover.Thesnowpackisformedbyicecrystalsandhasdifferentshapesanddensities.

As time passes, the snow density changes under the influence of theenvironment.Thechangeinitsstructure,fromfluffy,withatemperaturebelowzeroandlowdensity, tocoarse,granularandliquidwithhighdensity,iscalledsnowmaturation.Themature snowcover is ready togive runoff. In averticalsection, the density of the snow cover may not be homogeneous, since itcomprisesicepartsandsnowlayersofdifferentdensities.Thefactorsaffectingthedensityofasnowcoverareasfollows:

1. Heat exchange due to transportation, liquefaction, radiation and heattransferfromtheground

2. Thewind3. Thetemperatureofthesnowcover4. Thepressureexertedbytheoverlyinglayersofsnow5. Variationsinthewatercontentofthesnowcover6. Infiltrationofthewatercomingfromsnowmelt

Thepropertiesofthesnowlayerwithrespecttorefectionandradiationareveryinterestinginhydrologyastheycanbeusedfor theestimationof theextentofevaporationfromthesnowysurfaceandthemeltingprocess.The snow layer acts as a ‘black body’ with respect to long-wavelength

radiation.Itabsorbsthemajorityoftheincidentlong-wavelengthradiationfrom

the environment, while transmitting radiation back in accordance with theStefan-Boltzmannlaw.

2.3.3Hail

Hail consists of masses of ice crystals of spherical or irregular shape with adiameter greater than 5mm.Under certain conditions, the size of hail is verylargeandhailstonesareformed.Manyhailstonesconsistofalternatinglayersofsolid ice and snow due to successive lifts and falls into the cloud as aconsequenceofhighatmosphericturbulence.Somehailstonesreachthegroundintheformofpureice.Fortheformationofhailstones,itisnecessarythatstrongupdraftsholdtheminthecloud.Upwardvelocitiesintheorderof15m/sarenotuncommon in clouds, and speeds up to 36 m/s have been observed underconditionsof extreme instability.Hailoccursonly in severe storms,usually inlate spring and summer.Another formof hail,whichmay be characterized asmicrohail,consistsofsphericalorconicalparticleswithadiameterof2–5mm.Itusuallyappearswithrain.

2.4COOLINGMECHANISMSANDTYPESOFPRECIPITATION

Cooling of air sufficient to cause a significant amount of precipitation isachievedwhenlarge-scaleupwardmovementstakeplace.Thecharacteristicsofprecipitationareafunctionofthecharacteristicsoftheascendingairandliftingmechanism (Eagleson, 1970). The supply of air with moisture and the liftingmechanisms change seasonally in a region, so does the type of precipitationfallinginthisregion.Themainmechanismsofcoolingandtherespectivetypesofprecipitationarethefollowing.

2.4.1Cycloniccooling

Thiscanbedividedintonon-frontalandfrontalcooling.Thefirstistheresultofconvergence and therefore of the lifting gasmass to lower pressure points. Inthis case, the rainfall intensity is moderate and quite long. Frontal cycloniccooling occurswhen themovement forces the air to rise and penetrate into afrontsurfacebetweentwoairmassesofdifferenttemperatures.Whenthefrontiswarm,therainfallintensityismoderateandthedurationislong,whilewhenthefrontiscold,therainfallintensityishigh,thedurationisshortandthewindsarestrong.

2.4.2Orographiccooling

Airmassesfallonmountainslopes,andtheairexpandsandiscooledtoalowerpressurelevelcorrespondingtothehigherelevation.Windyhillsideshavemanymoreclouds,morerainandlesstemperaturefluctuations.Innon-windyhillsides,theclimateisdrierwithagreatervariationintemperature.

2.4.3Conductivecooling

The heating of the surface of the Earth generates a vertical current transfer,whichinturncausesverticalinstabilityofairandhencealiftingmechanismandcooling.Therainfallinthiscaseisintensiveandofshortduration.Insummary,itcouldbesaidthatanunsaturatedvapourairmassbyvirtueof

one of the three mechanisms rises and cools. The appropriate conditions ofpressureandtemperatureatacertainlevelcausethesaturationofwatervapour.As the cooling continues, it causes condensation and the formation of clouds.Heat is releasedduring condensation,which is offered to themass initiallybytheprocessofevaporationandsublimation.Vapourischangeofgastoliquidorsolid state (for the latter, usually,more coolingof airmass is required).Thus,formedwaterdroplets,snowflakesor iciclesgainsufficientsizearoundseveralcores and precipitate as rain, snow or hail, respectively. These cores arecondensingparticlessuchasseaacids,oxidesofnitrogen,ammoniumsaltsandsoilgranules.Oftentimes, theexistingacids, salts,etc.,create impurities in theprecipitation(characterizedasacidrainwhichdamagesthenaturalenvironmentandthemonuments).

2.5MEASUREMENTOFPRECIPITATION

Themeteorological variable firstmeasured by humanswas probably the rain.TheoldestandlongestperiodofrecordisreportedinEgyptintheNileRiverinabout 980 BC. There are several rainfall measurement and estimationinstruments, such as standard rain gauges, the weather radar and satellites.Today,thereisalargenumberofdifferenttypesofinstrumentsusedtomeasureprecipitation;thosemostlyusedaresummarizedinthefollowing.

2.5.1Precipitationsensors

They are used in the measurement of cumulative rainfall depth andsupplementary snowfall, which are installed at appropriate locations. Theymeasure the total rainfall depth and the equivalentwater depth of snowfall atcertaintimeintervals(usually8,12or24h).Themeasurementisrecordedfromanobserver.Theclassic typeof sensor iscylindrical in shapeandcomprisesa

collector, a funnel and a recipient, as shown in Figure 2.3. The collector is acylindersufficientlytall,withverticalinnerwalls,whichendsupinthefunnel.The angle of inclination of the walls of the funnel is at least 45° in order tominimize thewater losseswhichmay occur during the collision of raindrops.Thefunnel leads to therecipientwhichhasanarrowentranceand isprotectedfromsunlighttominimizewaterlossduetoevaporation.Themost common types are thevolumetric rain gauge and the10-fold rain

gauge. Volumetric rain gaugeswith a known section area of the collector areaccompaniedbyspecificvolumetrictubes.In10-foldraingauges,theareaofthecollectoris10timesgreaterthanthesectionareaoftherecipient,whichmakesthe precipitationwhich reaches the collector 10 times greater in the recipient,leadingtotheincreasedaccuracyofmeasurements.

Figure2.3 Standardrainfallsensor.

Insomecases,thehydrologistisinterestedonlyinseasonalrainfallorinthemeasurementofrainfall inareasnoteasilyaccessible,especiallyduringwinter.Insuchcases,therecipientisformedsothatitiscapableofholdingtheprobableamountofrainfallwhichisexpectedtofallwithinarelativelylongperiod.Therain gauge is suppliedwith an anticoagulant,with preference to ethyl alcohol,when the instrument is locatedmainly inmountainousplaces,where there isapossibilitythatwatercanfreezeinsidetherecipient.Itisalsousefulformeltingthe snow.Additionally, a small amountofoil is added to reduce lossesdue toevaporation.

2.5.2Raingaugerecorders

Raingaugerecordersareinstrumentsusedforthepointmeasurementofrainfall,installed at appropriate locations, collectingmainly rainfall and supplementarysnowfall, recording the change of rainfall depth over time using a simplemechanism,andthus,describingthetemporaldistributionofpointrainfall.SucharaingaugeisshowninFigure2.4.Threemaintypesofraingaugesarecurrently inuse: the tippingbucket, the

gravimetric and the floating type. Rain gauges give continuousmeasurementsevenforveryshortintervals;thus,theyaresuitableforthestudyofvariationsofrainfall intensity. When quipped with recording paper (old type), theirmechanism can be configured so that the drum carrying the recording papermakes a full rotation every day; thus, the rainfall can be measured at 5 minintervals. If the system is configured to change the paper once a week, therainfallcanbemeasuredat30minintervals.However,modernrecordershaveadataloggerandareconnectedtoacomputer.The tipping bucket rain gauge, which is the most advanced technology,

consists of two small buckets mounted side-by-side on a common horizontalaxis,whichmovesrightandleftasoneofthemisfilledwithrainwaterdirectedatthemfromthefunneloftheinstrument.Asonebucketfills,theotherempties.Rainfallequalto0.25mmisessentialforthetipandemptyingofabucket,andtheminimumtimerequiredforatipis2/10ofasecond.Thesepropertiesoftheinstrumentresultinerrorsincasesofverylightrainwhenittakestimetofillthebucket,makingthecollectedwatersubjecttoevaporation,butnotincaseswhentherainisveryintenseandthebucketfillsinlessthan2/10s.Thebucketsareconnected to a recording device, which records the number of tips and thecorrespondingrainfalldepthwithtime.

Figure2.4 Standardraingauge.

The gravimetric type gauge comprises a container which is mounted on aweighing device. As the rainwater is collected in the container, the spring ispresseddownand thecompression is recorded.Thecapacityof the instrumentmayreach1000mmofrain.Thefloating typegaugeconsistsofacontainerwhichcollects therainwater,

and ithasa floatwhich isused tomeasureand recordvariations in thewater-level.To avoid gross errors, a comparison between the amount ofwater collected

andtheamountrecordedshouldbemade.Insteadoftheconventionalrecordingmechanism, current technology enables the conversion of the movingmechanismoftheinstrumentintoadigitalsignal.Thissignalmaybestoredinan electronicdata logger.Furthermore, there is the capabilityof telemetry, i.e.signal transmission (after signal processing) wirelessly (via radio, GPRS, cellphonenetwork,satellite,etc.)orwired(vialandphoneline)toabasestation.

2.5.3Snowfallmeasurement

The snowfall depth is usually measured by snow banks. These are simplehorizontal surfaceswhere snowaccumulates and thedepth ismeasuredwith acommonbar.Aftermeasuring,thebankisclearedofthesnowsothatitisreadyfor the next measurement of snowfall. The equivalent amount of water fromsnow and the corresponding density can be measured at the snow bank if asimpleweighingsystemisprovided,whichmeasurestheweightofsnow.

Thedepth of snow cover ismeasured easily by punching a commonbar orreading a permanently installed stage (level staff), in which the zero levelcorresponds to the ground surface. The equivalentwater bar of snow cover ismeasured by taking a sample of snow through penetration of an appropriatecylindrical snow sampler and then byweighing the collected snow. To obtainrepresentative samples of snow, the average of measurements from about 6pointsalongafixed(permanent) routewithastandard lengthof150–250miscalculated.Anotherclassofmethods formeasuring thesnowfallheight,whichgive the

extentofsnowcover,isbasedonphotogrammetric(withaerialphotographs)orterrestrialobservation(withgroundphotographs).Theextentofsnowcovercanalsobedeterminedfromsatelliteimages.

2.5.4Meteorologicalradar

Conventional rainfall data are generally considered inadequate to describe thespatialandtemporaldistributionofrainfall.Onemethodwhichhasbeengivengreatattentionrecentlyisthemeasurementofrainfallwithdatafromaweatherradar.The radaroutweighs thenetworkof raingaugesbasedon the followingcharacteristics.The total information of the distribution of rainfall in time and space is

concentratedandprocessedseparatelyforeachelement,i.e.problemscausedbyenvironmentalconditionsareovercomebyusingtheradar.Theuseofmeteorological radar ismucheasier thanusinganetworkofrain

gauges.Theraingaugesandtheweatherradararetwodifferentsystemsformeasuring

rainfall.Themeasuringrangeofaraingaugeisverysmallcomparedtothatoftheradar.Anotherreasonfor thedifference in theamountofrainfallmeasuredbythetwoinstrumentsisthattheradarscansfromadistancefromthesurface,whiletheraingaugemeasuresrainfallonthegroundsurface.Thetemporalscaleof the measurements of rain gauges is more than 5 min (1 min for the mostmodern), while the measurements of the radar are almost instantaneous (20s/rotation)sothat,forthecomparisonwiththemeasuredamountsofrainfall,theradardatashouldbecompletedintime.Themeteorological radar utilizes the transmissionof electromagneticwaves

in the atmosphere and their collision with particles of rain. The beam isconcentratedatanangleof1°or2°totheantenna,whichreceivesthepartofthebeamreflectedontherainparticles.Theamountofreturnedenergydependsonthenumberofparticlesof rainpulse inside thevolumeof the radarbeamand

theirsize,composition,relativeposition,shapeandorientation.Thegeneralequationofthemeteorologicalradarissimplifiedtothefollowing

form(AndersonandBurt,1985):

  (2.2)

where

PristhereturnedpowerDisthediameteroftherainparticlesristhedistancefromthetargetC1andC2aretheconstantsoftheradarKisaparameterrelatedtothematerialofthetarget,thetemperatureandthewavelength

The water particles vary in size from 0.2 up to 5 mm. Knowing the sizedistribution, the reflectivity factorZ from allwater particles can be calculated(Collier,1989):

  (2.3)

whereN(D)isthedistributionofthewaterparticlesDisthediameterofthewaterparticlesNiisthenumberofwaterparticleswithadiameterrangingbetweenDandD+dDperunitvolumeoftheatmosphere

The distribution ofwater particles becomes 0whenD <Dmin andD >Dmax,sincethelimitsofintegrationarefrom0to∞.Theunitsoftheradarreflectivityaremm6/m3.Thereflectivitiesstartfrom0.001mm6/m3forfogorsparsecloudsandreachup to50,000,000mm6/m3 for intensehail.Thisdifference led to thedefinitionofanewparameterwhichmeasuresreflectivity:

  (2.4)

whereζisthelogarithmicparametermeasuredindBZandZinmm6/m3.Thisconversiongivesvaluesof ζ from−30dBZfor sparsecloudsup to75

dBZforintensehail.Themeasurementofthereturnedpower(distance,azimuth,altitude, power, speed and other features) at predefined points is importantinformationfortheirdescription.

2.5.5Joss–Waldvogeldisdrometers

The Joss–Waldvogel RD-69 and RD-80 disdrometers are instruments formeasuring raindrop size distributions (DSDs) continuously and automatically.TheRD-80isanenhancedversionofRD-69disdrometer.Theyweredevelopedbecause statistically meaningful samples of raindrops could not be measuredwithoutaprohibitiveamountofwork(DistrometLtd.,2011).Theseinstrumentsusually are combined with weather radars and provide data, such as radarreflectivity values. Moreover, they can be used as stand-alone instrumentsmeasuring rainfall intensity,water liquidcontent,kineticenergy fluxandotherhydro-meteorologicalparameters,aswillbedescribedinthefollowing.Both instruments work under the same principle, that they transform the

verticalmomentumofanimpactingdropintoanelectricpulse,whoseamplitudeis a functionof thedropdiameter.Aconventionalpulseheight analysisyieldsthesizedistributionofraindrops.AnexampleofasystemisshowninFigure2.5. Itconsistsofadisdrometer

RD-80andapersonalcomputer.The sensor is exposed to the raindrops to bemeasured, and alongwith the

processor,producesanelectricpulseforeverydrophitting.Theamplitudeofthepulseisdirectlyrelatedtothediameteroftheraindrop.Intheprocessor,pulsesaredivided into127classesofdropdiameter.Acomputerprogram,developedfor the disdrometer system, can be used to transform the data into a suitableformat.Inordertogetstatisticallymeaningfulsamplesandtoreducetheamountofdata,theprogramreducesthenumberofclassesto20.

Figure2.5 Systemformeasuringdropsizedata.

2.5.5.1DescriptionofthedisdrometerRD-80

Thedisdrometerforraindropsconsistsoftwounits:thesensor,whichisexposedtotherain,andtheprocessorforanalogprocessinganddigitizingofthesensorsignal.Acable,4mlong,isusedtoconnectthetwounits.The sensor transforms the mechanical momentum of drop into an electric

pulse,whoseamplitudeisroughlyproportionaltothemechanicalmomentum.The processor contains circuits to eliminate false signals, mainly due to

acousticnoise,toreducethe90dBdynamicrangeofthesensorsignal.Thesensorconsistsofanelectromechanicalunitandafeedbackamplifier.A

conical styrofoam body is used to transmit the mechanical impulse of animpactingdrop to a setof twomovingcoil systems inmagnetic fields.At theimpact of a drop, the styrofoam body together with the two coils movesdownwardsandavoltageisinducedinthesensingcoil.Thisvoltageisamplifiedand applied to the driving coil, producing a force which counteracts themovement.Therefore,ittakessometimeforthesystemtoreturntoitsoriginalrestingposition,andtherefore,togetreadyforthenextdrop.Theamplitudeofthepulseattheamplifieroutputisameasureofthesizeofthedropthatcausedit(DistrometLtd.,2011).Theprocessorhasthreemainfunctions:

1. Itsuppliespowertothesensor.2. Itprocessesthesignalfromthesensor.3. Itcontainscircuitsfortestingtheperformanceoftheinstrument.

Thesignalprocessingcircuitconsistsoffourparts:

1. Anoiserejectionfilter2. Adynamicrangecompressor3. Asignalrecognitioncircuit4. Anon-linearanalogtodigital(A–D)converter

Thenoiserejectionfilterisanactivebandpassfilter,whosefrequencyresponseisdesignedtogiveanoptimumratiobetweensignalsfromraindropsandsignalsduetoacousticnoiseaffectingthesensor.Thesignalrecognitioncircuitcanseparatethesignalpulsescausedbydrops

hittingthesensorfromthemoreuniformoscillationscausedbyacousticnoise.Ifa pulse caused by a raindrop exceeds the oscillations caused by noise, a gatepasses it to the pulse standardizer, which produces a constant pulse durationwithoutchangingtheamplitudeoftheoriginalpulse.TheRD-80hasthespecificationslistedinTable2.1.

2.5.5.2Hydro-meteorologicalparameterswhichcanbederivedbytheRD-69andRD-80disdrometers

Disdrometers are useful in estimating rain DSDs. A DSD is commonlyrepresented by the function N(D), the concentration of raindrops with thediameterD in a given volume of air. Because of the complicated processesinvolvedin theformationofprecipitation, thefunctionN(D)variesandcannotbe described by a simple function. In many cases, however, a DSD can beapproximated fairly well by an exponential law and the followingparameterizationcanbeusedtocharacterizeit:

  (2.5)

whereN0 is thenumberconcentrationofdropswithdiameter0on theexponentialapproximation

Λisitsslope

Table2.1SpecificationsoftheRD-80disdrometer

Rangeofdropdiameter 0.3–5mm

Samplingarea 50cm2

Accuracy ±5%ofmeasureddropdiameter

Resolution 127sizeclassesdistributedmoreorlessexponentiallyovertherangeofdropdiameters

Outputformat AccordingtoRS-232-Cstandard,7databits,evenparity,1stopbit

Baudrate 9600Baud

Handshake DCDandDTRsignals

Display 8LEDsfor8groupsof16channelseach

Powerrequirements Plug-inpowersupplyincludedindelivery:115/230VAC,5.5VA,50/60Hz(9–18VDC;3.3W,alsopossible)

Operatingtemperaturerange 0°C–40°Cforprocessor

0°C–l50°Cforsensor

Dimensionsofthesensor 10×10×17cmheight

Dimensionsoftheprocessor 12×26×27cmdeep

Weight Sensor,2.9kg;processor,2.2kg

Standardlengthofsensorcable 4m

Table2.2presentsthehydro-meteorologicalparametersthatcanbederived.

2.5.5.3DerivationofraincharacteristicswiththeuseofaJoss–WaldvogelRD-69disdrometer

AJoss–WaldvogelRD-69disdrometerrecordedthedataforarainfalleventon22February2013(Table2.3).Thistypeofdisdrometercanrecordthenumberofraindrops per diameter class per time step (in this example, the time step is 2min), classifying the measured diameters in 20 classes. The lower and upperlimitsofthediametersofeachclassareshowninTable2.3.Thesamplingareaofthedisdrometeris50cm2.Theestimationoftheterminalvelocityusingfourempirical relationships in correlationwith themean raindrop diameter of eachclassispresentedinTable2.4.

Example2.1Findforeveryrelationshipofterminalvelocitythefollowing:

1. Themeandiameterandtherangeofeachclass.2. The number density of drops of the diameter corresponding to

sizeclassiperunitvolumeN(Di)in1/m3mm.3. TheintensityoftherainRinmm/h.4. Thetotalamountofrainandtherespectivecumulativecurvefor

thisrainfallevent.5. ThewaterliquidcontentWging/m3.6. ThekineticenergyfluxEinJ/m2h.7. TheradarreflectivityfactorZinmm6/m3anddBZ.8. LetusassumethattheDSDN(Di)isexpressedapproximatelyby

the exponential relationshipN(D) = N0exp(−ΛD). Estimate theparametersN0(1/m3mm)andΛ(1/mm)ofthisdistribution.

9. FindtheparametersaandbinaZ−RrelationshipoftheformZ=αRβwithunitsofZ(mm6/m3)andofR(mm/h),usingthemethodoflinearregression.

Table 2.2 Hydro-meteorological parameters which can be derived by the RD-69 and RD-80disdrometers

Solution

1. Therelationshipsforthecalculationofthemeandiameterandtherangeofeachclassarethefollowing:

  (2.6)

  (2.7)

whereDiisthemeandiameterclassDimax is the upper limit of each class andDimin is the lowerlimit,respectively

Table2.3Recordeddataofdropnumbersandlowerandupperlimitsoftherespective20classesofdiameters

Table2.4Empiricalrelationshipsoffinalraindropspeedincorrelationwiththemeandiameterofeachclass(V1inm/s,Dincm)

V1=14.2D1/2

V2=9.65ȃ10.3e(−6D)

V3=17.67D0.67

V4=48.54De(−1.95D)

ItisvalidthatDimax=D(i+1)min.Bysubstitutingthevaluesoftheupper and lower limits, the respective mean diameter and

diameterrangeforeachclassarederived.TheresultsareshowninTable2.5.

2. WecalculatetheDSDusingthefollowingrelationship:

  (2.8)

whereNA(Di)isthenumberofhitsperclassiAisthehitareaofthedisdrometercone=50cm2

V(Di)isthefinaldropspeedofeachclassiΔDiistherangeofeachclassi

WesubstitutethevaluesofA=50cm2=50×10−4m2,t=120s,V(Di)⇒ V1(Di) (the same procedure is followed for the otherthreefinaldropspeedrelationshipswhichareomitted),resultinginthevaluesgiveninTable2.5.Thevalueshavebeenmultipliedbythefactor10−3forabetterpresentation.N(Di)isexpressedin1/m3mm.

3. TherainintensityRisgivenusingthefollowingrelationship:

  (2.9)

Allcomponentshavebeenexplainedpreviously.ThecalculationsareshowninTable2.6.Inthelastrow,thesumofeachcolumniscalculated.Figure 2.6 shows the graphical representation of therainintensitywithtime.

4. Thetotalrainfallamountisestimatedbymultiplyingtherainfallintensitywiththerespectivetimeinterval(2min)andsumming:RA=Σ(R×t)inmm.Figure2.7showsthecumulativeplot.

5. Theliquidwatercontentisgivenbythefollowingrelationship:

  (2.10)

TheresultsareshowninTable2.7andFigure2.8.6. Thekineticenergyfluxisgivenbythefollowingrelationship:

  (2.11)

TheresultsareshowninTable2.8andFigure2.9.

Table 2.5 Calculation of mean diameter, class range and drop sizedistributionN(Di)×10

−3

Table2.6CalculationsofrainintensityR

Figure2.6 Rainintensity.

Figure2.7 Cumulativerainfalldepth.

7. TheradarreflectivityZinmm6/m3iscalculatedbythefollowingrelationship:

  (2.12)

ToconvertintodBZ,thenextrelationshipisused:  (2.13)

ThesedatawereusedinpreparingTable2.9andplottingFigures2.10and2.11.

8. N0andΛarecalculatedusingthefollowingrelationships:

  (2.14)

  (2.15)

TheresultsareshowninTable2.10.9. TheestimationofarelationshipofZversusRofthetypeΖ=αRβ

is feasible through logarithmicequation: lnZ= lna +blnR.Thisrelationship is a line of the type y = C1 + C2x in doublelogarithmicaxes (C1= lna,C2=b).Applying linear regression,theresultsarey=5.4224+1.46x,solna=5.4224,a=226.41,b=1.46(Figure2.12).

Table2.7LiquidwatercontentW

Figure2.8 LiquidwatercontentW.

•

•

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2.5.6Installationofraingauges

The main purpose of the rainfall observations is to obtain values which arerepresentative of a region. This objective is difficult to achieve, since theinstrument may affect the measurements, and it is known that the hydro-meteorologicalparametersareaffectedbyenvironmentalparameters.Whethertheyareautographicinstrumentsornot,theymustbeprotectedfrom

theair,becausetheedgesofthecollectorarewellabovethegroundsurface.Thesizeofturbulencevarieswiththevelocityoftheairandtheirregularitiesoftheregion.Theflowfieldofthewindduringarainfallorsnowfallisthemaincauseof heterogeneities and discontinuities in the recording of rainfall: the variousgroundorotherabnormalitiescauselocaldisturbancesinthewindflowlinesandcorrespondingdisordersinthemovementofraindropsorsnow-fakes.Regardingnon-automatic instruments used for measuring precipitation, the water intakeshouldbelessinordertominimizeevaporationlosses.Consequently, the basic rules for the installation of rain gauges are

summarizedasfollows:Installation of the rain gauge in areas sheltered from winds from alldirections,i.e.freefromsevereturbulenceorotherdisorders.The collector should be placed at a height of 1–1.5 m above the groundsurface.The number of trees or shrubs surrounding it should be limited or thedistanceofthesebarriersfromtheinstrumentshouldbeatleasttwicetheirheight.

Table2.8KineticenergyfluxE

Figure2.9 KineticenergyfluxE.

2.5.7Spacemeasurements

2.5.7.1Satellites

Satellitesareclassifieddependingon theirorbitsaroundtheEarth.Thereareavariety of different possible orbits: sun-synchronous,mid-inclination and low-inclinationorbits.Eachoneofthemhasadvantagesanddisadvantages.Thesun-synchronousorbithasanapproximateinclinationof98.6°andhasthe

advantageofproviding samples at the same local timeeachday.Theconstantanglebetweenthesun,thesatelliteandtheobservedspotontheEarthallowsfor

asimplesolararrayandthermaldesign.However,theretrogradeorbit(anorbitofasatelliteorbitingtheEarthinwhichtheprojectionofthesatellite’spositionon theEarth’sequatorialplane revolves in thedirectionopposite to thatof therotation of the Earth) requiresmore launcher capability and therefore ismoreexpensive. Also, in the case of launching multiple satellites, it is difficult toconfigurethelaunchingpatternssothatthesatelliteswillbedistributedthroughdifferentascendingnodesatthesamealtitude.Mid-inclination orbits (35°–70° inclination) provide short revisit intervals

around the inclination latitude. However, due to the 70° constraint, the polarregionsarenotcoveredatallbythesesatellites.Itiseasiertodistributemultiplelaunched satellites to the desired orbits, but they require more complex solararrayandthermaldesignandmayrequireperiodicmanoeuvrestomaintainthedesiredorbit.Finally, for low-inclination orbits (up to 25°–30° inclination), the revisit

intervals around the tropics (and nowhere else) are very satisfactory and thelimited range of sun angles simplifies solar array and thermal design. Theproblemwithusing low-inclinationorbits is that satelliteswithmid-inclinationor sun-synchronous orbits must also complement in order to have coveragebeyond the tropics. Since the mid-inclination and sun-synchronous orbitedsatelliteswillalsocoverthetropics,theresultwillbeperfectcoverageupto30°latitudebutwillhavemanygapsfrom30°latitudeto90°latitude.Theoptimizationprocess,whichinvolvestheorbits’architecture,isbasically

a trade-off between coverage and resolution (see Figures 2.13 and 2.14). Thegreatestcoverageisachievedwithhigh-altitudesatellites(i.e.833km)sincetheswath width will be large. However, this implies poor resolution since theresolution is the swath width divided by the number of beams that theinstrumentsareusing.Sincethenumberofsuccessivebeamsperscanisconstantfor a particular instrument, the resolution is higher when the swath width islower.

Table2.9RadarreflectivityZ

Figure2.10 RadarreflectivityZinmm6/m3.

Figure2.11 RadarreflectivityζindBZ.

Table2.10N0andΛvalues

Time N0(1/m3mm) Lambda(1/mm)

22/2/2013 7:48:00p.m. 2,948.697 2.599

22/2/2013 7:50:00p.m. 3,735.054 2.687

22/2/2013 7:52:00p.m. 4,819.560 2.794

22/2/2013 7:54:00p.m. 2,715.052 2.389

22/2/2013 7:56:00p.m. 4,398.741 2.679

22/2/2013 7:58:00p.m. 10,300.557 2.908

22/2/2013 8:00:00p.m. 11,668.312 3.541

22/2/2013 8:02:00p.m. 21,994.359 4.517

Figure2.12 LinearregressionoftheZ–Rrelationship.

Figure2.13 Swathwidthvs.altitude(at140°sector).

ThesatelliteTropicalRainfallMeasuringMission(TRMM)waslaunchedintoorbitaroundtheEarthon27November1997(Figure2.15).It is theresultofajointeffortbetweentheUnitedStatesandJapan,withthesoleaimofmeasuringprecipitation from space. Due to the uniqueness of the project and the wideacceptance of the products, which are freely available to the scientificcommunity, theoperational lifetimeof the satellitewas extended from3yearsand2months,as itwasoriginallydesigned, to6yearsand2months(Enright,2004).Thisincreasewasmadepossiblebyminimizingtheenergyconsumption.NASAfurtherextendedthedurationof themissionuntil theyear2012,withapotentialriskofuncontrolledre-entryofthesatelliteintotheatmosphere,asthefuelwouldhavebeencompletelyexhaustedby then(NationalAeronauticsandSpaceAdministration,2007).The satellite has a total weight of 3620 kg. The satellite rotates

asynchronously to the sun at an altitude of 350 km above the ground and anangle of 35°. The small angle has the advantage of very densemeasurementsaround the tropics as shown in Figures 2.16 and 2.17. Because of the smallangles of the sun’s rays, the design of solar collectors is greatly simplified(Fotopoulos,2002).Thesecollectorsprovidemostoftheenergyconsumedbyitsinstruments.

Figure2.14 Apertureforfootprintvs.altitude.

Figure2.15 TheorbitofTropicalRainfallMeasuringMissioninadayperiod.

2.5.7.2Installedinstruments

Five different instruments are installed in the TRMM satellite. Three of themaim to assess the precipitation and the other two are used to monitor otherweather parameters. The instruments used to record the precipitation are theprecipitation radar (PR), the microwave imager (TMI) and the visible and

infrared scanner (VIRS). The other two instruments are the clouds and theEarth’sradiantenergysystem(CERES)andthelightningimagingsensor(LIS)(Everett, 2001). A schematic view of the scan geometries of primary rainfallsensorsisgiveninFigure2.18.

Figure2.16 Theorbitsofallsatellitesinadayperiod.

Figure2.17 ThepositionofTropicalRainfallMeasuringMissiononJanuary1,2007,13:40:52.

2.5.7.3Advancedprecipitationradar

ThePRwasthefirstrainradarinspace(Table2.11).Itspurposeisto:providea

3D structure of the rainfall, particularly of its vertical distribution; obtainquantitative rainfall measurements over land and over ocean and improve theoverall precipitation retrieval accuracy. The National Space DevelopmentAgency of Japan (NASDA) has developed the PR in cooperation with theCommunications Research Laboratory, Ministry of Posts andTelecommunications (NASDA, 2001). The measurements taken by thisinstrument,combinedwiththoseofTMI,providetheverticalprofileofrainfall.ThisprofilecanbeusedtoestimatetheemissionoftheEarth’slatentheat.Thewidthoftheswathoftheunitonthegrounddoesnotexceed215km,whiletheverticalresolutionreachesupto250mstartingfromthegroundandreachingaheightofabout20km.Thehorizontalresolutionoftherecordingsrangesfrom4.18to4.42km,dependingonthepositionofthesatellite.

Figure2.18 Schematicviewofthescangeometriesofprimaryrainfallsensors.

Table2.11Majorparametersoftheprecipitationradar

Item Specification

Frequency 13.796,13.802GHz

Sensitivity ≤~0.7mm/h(S/N/pulse≈0dB)

Swathwidth 215km

Observablerange Surfaceto15kmaltitude

Horizontalresolution 4.3km(nadir)

Verticalresolution 0.25km(nadir)

Antenna

Type 128-elementWGplanararray

Beamwidth 0.71°×0.71°

Aperture 2.0×2.0m

Scanangle ±17°(crosstrackscan)

Transmitter/receiver

Type SSPAandLNA(128channels)

Peakpower ≥500W(atantennainput)

Pulsewidth 1.6μs×2channels(transmittedpulse)

PRF 2776Hz

Dynamicrange ≥70dB

Numberofindependentsamples 64

Datarate 93.2kbps

ThePRhasonlyaKuband,whichissufficienttomeasuretherainfallinthetropics.Formid-latitudes,whereweakrainandsnowoccurs,anotherbandhastobeadded to the instrument.TheKabandwill operate simultaneouslywith theKuband.TheKabandwillhaveimprovedaccuracyandwillbeusedtomeasureweakrainfallandsnowfallandtoseparatesnowfromrain.Sincethisisthefirsttime that thisbandwilloperate, itsspecificationswillbebasedonpreliminaryrequirementsas shown inTable2.12,which are subject to change.To achievethesepreliminaryoperational requirementsof theKaband(separationofsnowand ice from rain, accurate estimation of the rain rate and the DSD andcomputation of the effect of non-uniformity of raindrop distribution),informationfrombothbandsisneededatagivenlocationatagiventime.Theradarhastousespecialscanpatternstocombinebothbands(KaandKu)withinasinglesweep.Itwouldbeidealtousethesamescanpatternsforbothbands,sincethetarget

wouldbethesameinbothcasesandnoshiftingalgorithmswouldbenecessary.However, this is not possible because the two bands operate at differentwavelengths and they have different resolutions and swath widths, so thescanningpatternscannotbeidenticalwithinasweep.Also,anexactmatchofthetwo beams is technically impossible. The currently proposed scan pattern isgiven in Figure 2.19. The arrow in Figure 2.19 indicates the direction of thesatellite’s movement. The instrument scans from left to right and the satellitemovesfromthebottomtothetopofthefigure.Theresultisarotatedscanatan

angleΘwithrespect to theplaneperpendicular to thefightpath.Thereare49horizontalKubeams(transparentcircles),yieldingaswathof245km.Thereare25Kabeams(lightcolouredcircles),yieldingaswathof125km.ThenumberofKa beams is subject to change. Finally, the dark coloured circles are KainterlacedbeamsandshouldbeonelessinnumberthanKafootprints ineveryscan.Thisscanpatternshouldnotbeconsideredasfinal.

Table2.12OriginaloperationalrequirementsoftheKaband

Item Specification

Frequency 35.5GHz

Sensitivity 11dBZorbetter

Swathwidth 20–40km

Observablerange Surfaceto15kmaltitude

Horizontalresolution 4.0km(nadir)

Verticalresolution 0.25km(nadir)

Measurablerain

Minimum 0.3mm/h

Maximum 10mm/h(nearsurface)

Figure2.19 SchematicviewofAPRconicalscan.

2.5.7.4Microwaveimager

TheTMI is a nine-channel (frequencies) passivemicrowave radiometer basedupon the special sensor microwave/imager, which has been flying aboard theU.S. Defense Meteorological Satellite Program satellites since 1987 (Table2.13). Itspurpose is toprovidedata related to the intensityof rainfallover theoceans.Theaccuracydecreaseswhenmeasurementsaremadeover land,sincetheheterogeneousemissionfromthelandsurfacecomplicatestheinterpretationof the measurements. The swath of the instrument on the ground is conicalshaped, with a width of 760 km and a horizontal resolution of 6–50 km,dependingonthepositionofthesatellite.The TMI antenna is an offset parabola, with an aperture size of 61 cm

(projectedalong thepropagationdirection)anda focal lengthof50.8cm.TheantennabeamviewstheEarth’ssurfaceatanadirangleof49°,whichresultsinanincidentangleof52.8°attheEarth’ssurface.TheTMIantennarotatesaroundanadiraxisataconstantspeedof31.6rpm.TherotationdrawsacircleontheEarth’ssurface.Only130°of theforwardsectorof thecompletecircle isusedfor measurements. The rest is used for calibrations and other instrumentmaintenancepurposes.

2.5.7.5Visibleandinfraredradiationscanner

Thisinstrumentmeasurestheemittedradiationinfivespectralbands,operatingin the area of visible and infrared radiation. The combined use of themeasurementsofVIRSwiththoseofTMIprovidesamoreaccurateassessmentofprecipitation,whencomparedtoseparatemeasurementsofalltheparameters(NASDA, 2001). That is because the distribution of clouds is estimated byVIRS. The scanning angle of the instrument is ±45° which is translated to afootprintwithawidthof720kmontheground.Thehorizontalanalysisdoesnotexceed2km.

Table2.13TMIinstrumentspecifications

2.5.7.6CloudsandtheEarth’sradiantenergysystem

Thisinstrumentisdesignedtoreduceuncertaintyinthepredictionoflong-termclimate change. The instrumentmeasures the Earth’s radiant energy, which isseparatedfromtheclouds’radiation(NASA,2008).Itisbasedonthedeviationof radiant energy used in physical models of climate prediction and thedeterminationofthebalanceofsurfaceemissivity,whichplaysanimportantroleinatmosphericprocessesandthetransportofenergyfromtheairtotheseaandviceversa.Althoughthenaturalprocesseshavenotbeensimulatedsuccessfully,the measurements of CERES are considered necessary for the success of theresearchinthisscientificfield.

2.5.7.7Lightningimagingsensor

AnLISisanopticaltelescopecombinedwithafilteredimagingsystem,whichrecords not only the lightnings which occur inside the clouds but also thosewhichoccurfromcloudstotheground.InconjunctionwiththemeasurementsofPR, TMI and VIRS, important steps have been taken to link lightning withrainfallandotherpropertiesofstorms(Petersenetal.,2005;PessiandBusinger,2009).

2.6INSTALLATIONOFNETWORKOFPOINTMEASUREMENTDEVICES

Installing a point measuring device, even a simple one such as a rain gauge,meansactuallyinstallingameteringstation.Themeteorologicalinstrumentsofthestations,theirdensityandtheirlocation

of installation are a subject of a special study and depend mainly ongeomorphologicalandclimaticfactorsaswellasontheuseofrainfalldata.Forthisreason,therearenogeneralrulesforthedensityofthestationsandthetypesoftheinstrumentsateachstation.Generally, the denser the network, the more representative the estimated

surface rainfall (Manning, 1997). The heterogeneities of the geomorphologyrequire a dense network. Moreover, the cost of installation, maintenance andeaseofaccessibilityoftheobserverarethefactorswhicharetakenintoaccount.Ingeneral, themeasurementerrorsofrainfall increaseas thesurfacerainfall

increases and decrease with increasing density of the network, the rainfalldurationandthesizeofthearea.Largerdeviationsoccuratthescaleofrainfallevent,thanatmonth,seasonoryearandduringthesummerseason.Insummer,thenetworkshouldbetwotothreetimesdenserthaninwinter(Singh,1992).

•

•

•

•

The adequacy of a network of rain gauges is determined statistically. Theoptimumnumberofraingaugescorrespondingtoaspecifiedpercentageoferrorintheestimationofsurfacerainfallis

  (2.16)

whereNistheoptimumnumberofraingaugesCuisthevariationcoefficientoftherainfallatthemeasuringdevicesεistheallowedpercentageoferror(%)

The standard value of ε is 10%. If this value should decrease, thenmore raingaugesarerequired.If there arem rain gauges in a basin andP1,P2,P3,…,Pm are the rainfall

depthsforaspecifictimeinterval,thenthecoefficientCuis

  (2.17)

wherePisthemeanvalueoftherainfallrecordedintheraingauges

  (2.18)

Sisthestandarddeviation

  (2.19)

The World Meteorological Organization (WMO) has proposed the followingregarding the density of the network in relation to the general hydro-meteorologicalconditions:

One rain gauge per 600–900 km2 in fat areas formildMediterranean andtropicalzonesOne rain gauge per 100–250 km2 in mountainous areas for mildMediterraneanandtropicalzonesOneraingaugeper25km2insemi-mountainousareaswithintensevariationinrainfallOneraingaugeper1,500–10,000km2indryandpolarzones

Ifthepurposeistheestimationofthehydrographorthepeakdischarge,thenthedensity of the network may be different. The densities of rain gauges as afunctionoftheareaofagriculturalbasins,accordingtotheWMO,aregiveninTable2.14.Thenumberandtypeofdevices tobeplacedinabasindependoneconomic, climatic and topographic (accessibility)139 as well as on themethodology of data analysis. The density of a network must be determinedprimarilybasedonthedegreeofirregularityofprecipitationandthepurposetobeserved.

Table2.14Densityofraingaugesdependingontheareaofagriculturalbasins

Basinarea(х1000m2) Radio(km2/station) Minimumnumberofstations

0–120    0.13    1

120–140    0.20    2

400–800    0.25    3

800–2,000    0.40    1per0.4km2

2,000–10,000    1.00    1per1km2

10,000–20,000    2.50    1per2.5km2

>20,000    7.50    1per7.5km2

Table2.15Annualrainfallperraingauge

Example2.2Abasinconsistsofanetworkof six raingauges.Theannual rainfallrecordedfromtheseraingaugesislistedinTable2.15Calculatetheoptimumnumberofraingaugesforthisbasin,with10%errorinthecalculationofthemeansurfacerainfall.SolutionTheaverageoftheannualrainfallfromthesixraingaugesis

 

Thestandarddeviationis

 

ThecoefficientCuis

 

Theoptimumnumberofraingaugestothebasinunderstudyis

 

Thus,onemoreraingaugeshouldbeinstalled.

2.7TestofDataHomogeneityandAnalysisofDoubleCumulativeCurves

Theprobabilityofastormoccurringinaparticularregionisthesameifallthemeteorological conditions remain stable. Such an area is consideredmeteorologicallyhomogeneous.Additionally,aregionishomogeneousifithasthesameannualrainfallandroughlythesamerangeofrain.Thefactorswhichdetermine the annual rainfall and the meteorological homogeneity are thedistance from the coast, the wind direction which determines the direction ofweathersystems,themeanannualtemperature,thealtitudeandthetopography.Ifoneareaismeteorologicallyhomogeneous,thentheprobabilityforastormofthesamedurationanddepthtooccuristhesamethroughouttheregion.Beforeanalyzingtherainfalldataofastation,thequalityandcompletenessof

datashouldbechecked.Thehomogeneity testof thedata includes,apart fromthe logical testing, the test of the data quality. What needs to be checked iswhether the measurements have been obtained under the same conditions.Changingthepositionoftheinstrument,andreplacementoftheinstrumentandtheobserverusuallyleadtonon-homogeneousdata.

Figure2.20 Pointofslopechangeinadoublecumulativecurve.

Thetestingof thehomogeneityofastationdata isfeasible throughmethodssuch as the double cumulative curve, the cumulative deviations and the vonNeumannratio.Themostcommonwaytotestthehomogeneityofrainfalltimeseriesisbyplottingthedoublecumulativecurve(Dingman,1994;Shaw,1994).Thetestofhomogeneityisappliedonlyontheannualtimeseriesofrainfall.Thedoublecumulativecurveisderivedasfollows:considertwoneighbouring

stationsX andY, where the annual rainfall time series is denoted by x and y,respectively.Thecumulativetimeseriesiscalculatedasfollows:

  (2.20)

Eachvalueisequaltothesumoftherainfallofalltheprecedingyears.Thesumiscommonlycalculatedbeginning from themore recentyear.Thepairsof thevalues(syi,sxi)areplottedonadiagram,leadingtothedoublecumulativecurve.Thestandardformofadoublecumulativecurvewithachangeofslopepoint

m is shown in Figure 2.20. In this case, the points of the double cumulativecurvearenotwelldescribedbyasinglestraightline,butbytwolines,formingan angle.This heterogeneity is correctedbyperforming aprocedure, inwhichthe values of the heterogeneous time series are multiplied with a coefficient,suchthatthedoublecumulativecurvebecomesastraightline.Specifically,ifthetime seriesx andy haven values and the double cumulative curve presents abreakpointm,thenthecorrectingcoefficientofthetimeseriesyiscalculatedasfollows:

  (2.21)

Then,thevaluesfromym+1toynaremultipliedwithλ,and the timeseriesy iscorrected.

Example2.3The annual rainfall of station X and the average rainfall of 10neighbouring stations are given in Table 2.16. Determine thehomogeneityofthemeasurementsofStationX.Inwhichyeardidthechangeoccur?CalculatetheaverageannualrainfallatstationXforaperiod of 30 years with and without the homogeneity of themeasurements.

Table2.16AnnualrainfallofstationXandaveragevaluesofthe10neighbouringstations

SolutionItisassumedthatthemostrecentlyrecordedvaluesaremoreaccuratethan older values,whichwill be corrected.Therefore, the values areclassified in reverse chronological order, as shown in Table 2.17(columns4,5and6).Then, the cumulative values of the two stations are calculated in

Table 2.18 (columns 7 and 8), and a double cumulative curve is

plotted,asshowninFigure2.21.Achangeintheslopeofthedoublecumulativecurve isobserved in1963–1964.Therefore, thevaluesofstationX will be corrected from the year 1950–1951 up to the year1962–1963.Applyinglinearregressionbetweenthedataofcolumns7and8,the

slopeofthecurvefortheyears1979–1980~1963–1964isλ1=1.0192andfortheyears1963–1964~1950–1951,itisλ2=1.2669.Theratioof the twoslopes ism=λ1/λ2=0.8045.Thevalues incolumn5aremultiplied with this value from the years 1950–1951 to the years1962–1963,inordertogetthecorrectedvaluesforstationX,whicharegiveninTable2.18(column9).ThenewcumulativevaluesofStationXarepresentedincolumn10.Thenewdoublecumulativecurve(basedondataofcolumns10and

11ofTable2.19)hasauniformslope,asshowninFigure2.22.The average annual rainfalls at station X before and after the

homogeneityofthemeasurementsare328and297mm,respectively.

2.8COMPLETIONOFRAINFALLMEASUREMENTS:ADAPTATIONTODIFFERENTALTITUDES

Themainsourcesoferrorinmeasurementsofrainfallare

1. Instrumentationerrors2. Improperpositioningoftheraingauge3. Humanerrors

Table2.17Rankofvaluesinreversechronologicalorder

The rainfall data are usually incomplete. The lack of datamay involve a fewdaystoseveralyears.Popularmethodsofdatacompletionarethemethodofthe

arithmetic average, the method of the normal ratios, the method of inversedistances, the contour method, the Lagrange method, interpolation and theKrigingfamilyofmethods.Data completion is essential to anyhydrological study and is basedondata

fromadjacentrainfallstations.Besides,thetotaloperatingperiodofaraingaugestationmaybeshort,whileotherstationswithalongeroperatingperiodmaybein theneighbourhoodof the station. In this case, an expansionof the station’sdatacouldbeachieved,usingthedataoftheneighbouringstations(Maidment,1993). The general methodology for completion and expansion is the same;however, the case of expansion normally concerns longer periods thancompletionandneedsmoreattention.

Table2.18Cumulativevalues

(7) (8)

Cumulative

Year StationX Averagevalue10StationX

1979–1980  340    350

1978–1979  610    600

1977–1978  890    860

1976–1977  1190    1210

1975–1976  1530    1540

1974–1975  190    1880

1973–1974  2130    2160

1972–1973  2430    2450

1971–1972  2680    2710

1970–1971  3070    3060

1969–1970  3390    3390

1968–1969  3680    3720

1967–1968  3960    3950

1966–1967  4310    4230

1965–1966  4670    4570

1964–1965  4930    4820

1963–1964  5130    5040 Thissetofvaluesconstitutesthepointofslopechange.

1962–1963  5470    5280

1961–1962  5880    5540

1960–1961  6460    5940

1959–1960  6810    6220

1958–1959  7180    5670

1957–1958  7540    6740

1956–1957  7830    7000

1955–1956  8180    7300

1954–1955  8430    7530

1953–1954  8700    7790

1952–1953  9120    8150

1951–1952  9360    8360

1950–1951  9830    8650

Figure2.21 Doublecumulativecurve.

Table2.19Newcumulativevalues

(9) (10) (11)

Newvalues Cumulative

Year StationX Newcumulativevalues(stationX)

Averagevalue10Station

1979–1980  340  340    350

1978–1979  270  610    600

1977–1978  280  890    860

1976–1977  300  1190    1210

1975–1976  340  1530    1540

1974–1975  370  1900    1880

1973–1974  230  2130    2160

1972–1973  300  2430    2450

1971–1972  250  2680    2710

1970–1971  390  3070    3060

1969–1970  320  3390    3390

1968–1969  290  3680    3720

1967–1968  280  3960    3950

1966–1967  350  4310    4230

1965–1966  360  4670    4570

1964–1965  260  4930    4820

1963–1964  200  5130    5040

1962–1963  274  5404    5280

1961–1962  330  5733    5540

1960–1961  467  6200    5940

1959–1960  282  6482    6220

1958–1959  298  6779    6480

1957–1958  290  7069    6740

1956–1957  233  7302    7000

1955–1956  282  7584    7300

1954–1955  201  7785    7530

1953–1954  217  8002    7790

1952–1953  338  8340    8150

1951–1952  193  8533    8360

1950–1951  378  8911    8650

Figure2.22 Newdoublecumulativecurve.

The methods which will be studied in the context of this course are thearithmeticmean,thenormalratios,theinversedistanceandthelinearregression.

2.8.1Methodofarithmeticmean

Thismethodusestheaveragevalueofthemeasurementsofthestations,whichsurrounds the stationwithmissingdata.Thismethod is usedwhen the annualmeasurementsof the stationwithmissingdatadiverge less than10%from the

averageannualrainfalldepth:

  (2.22)

whereαiis1/NNisthenumberofthestationsPiistherainfallrecordedonstationiPiistheaveragerainfallofthestations

2.8.2Methodofnormalratios

Thismethodisverysimpleandwidelyused,andusesaweightingcoefficientforeach station. Three stations surrounding the station whose distance from thestationwithmissing data is almost the same are usually selected.The generalrelationshipisthefollowing:  (2.23)

wherePistherainfalldepthNistheannualrainfalldepthXistheindexwhichdenotesthestationwithmissingdataTheindicesi=1,2,3,…,mdenotethesurroundingstations,mintotal.

2.8.3Inversedistancemethod

Thismethod involves the surrounding stations in relation to the distance fromthestationwithmissingdata.Thesedistancesarecalculatedassuming that theoriginoftheaxesisthestationwithmissingdata.Therainfalliscalculatedusingthefollowingequations:

  (2.24)

and

  (2.25)

wherekisthenumberofthestationsPyistheestimatedrainfalldepthatthestationwithmissingdataPiistherainfalldepthateachstationdiisthedistanceofeachstationfromthestationwithmissingdatabistheexponentwithatypicalvalueof2wiistheweightingcoefficientofeachstationi

2.8.4Correlationandregression

Additionally,thecompletionofrainfallmeasurementsmaybeachievedbasedonthelinearcorrelationofthemeasurementsofthestationwithmissingdatawithotherbasestationsifthedegreeoflinearcorrelationbetweenthetwostationsishigh.Correlationimpliesthedeterminationoftherelationshipbetweentwoormore

random variables. For example, if X and Y are the random variables whichrepresent the precipitation in two adjacent stations, then the variables areassociatedwiththerelationship:

  (2.26)

Thecorrelationofthevariablesconsistsofdeterminingtherelationshipf.Regression is the correlation which is based on the least squares method,

whichpractically,isidenticalwiththecorrelationsincetheleastsquaresmethodis used in almost all cases. Themost common one is the linear regression, inwhichthefunctionfislinear.All time series, independent and dependent, should have a common

measurementperiodwithoutmissingdata,inordertocarryouttheregression.Thesimplelinearregressionwillbedescribedhere.Accordingtothismethod,

thevalueforcompletiony=Pyisestimatedfromthecorrespondingvaluex=PxoftheneighbouringstationX(fortheperiodwithlackofdataatstationY)basedonthelinearrelationship:

  (2.27)

wherea and b parameters which are estimated so as tominimize the sum ofsquare errors of the estimate. If xi and yi are simultaneous measurements at

stationsXandY,respectively,atthetimeperiodI(usuallyyearormonth),thenthefollowingholds:

  (2.28)

  (2.29)

wherexandyaretheaveragevaluesofxiandyi,respectively,i.e.

  (2.30)

and n is the (common for xi and yi) sample time period (Koutsoyiannis andXanthopoulos,1997).Thedegreeofsuitabilityof themethodfor thespecificdata iscalculatedby

theestimateofthecorrelationcoefficientr:

  (2.31)

Thecloserthecorrelationcoefficienttotheunit,themoresuitablethemethodis.Usually, the following is required for the application of the linear regressionmethod:

  (2.32)

Insimplerterms,itisrequiredthatr≥0.7.

Example2.4The raingauge stationXwasout of operationduring a stormevent.Themeasurementsof rainfall,whichwere recordedat three adjacentstations(meteorologicallysimilar)A,BandC,were117,99and124mm.Theannual rainfallat thestationsX,A,BandC are958,1130,985and1220mm,respectively.CalculatetherainfallatstationX.SolutionThedataoftheexercisearesummarizedinTable2.20.TherainfallatstationXisdeterminedusingthenormalratiomethod

asfollows:

  

  

  

Table2.20Annualrainfalldata

Station Rainfalldepth(mm) Station Annualrainfall(mm)

A    117  X    958

B    99  A    1130

C    124  B    985

 C    1220

Example2.5Calculate therainfallatstationA inFigure2.23using themethodofinversedistances.ThedistancesoffiveadjacentstationsfromstationA and the corresponding recorded rainfall depths are given in Table2.21.SolutionAccordingtothemethodofinversedistances,therainfallatstationAisgivenbytherelationship

 

wherePiistherainfalldepthateachneighbouringstationandwiisthecoefficientateachstationwhich isafunctionof thedistancedi fromstationA,where

 

k=5.TheproductPi·wiiscalculatedasgiveninTable2.22.

Figure2.23 Raingaugestations.

Table2.21DistancesoftheneighbouringstationsfromstationA

Station Xi−X0(km) Yi−Y0(km) Rainfall(mm)

1   1.3   0.8   25

2   0.6   1.1   35

3   0.7   0.4   14

4   0.6   1.3   23

5   1.2   0.9   19

Table2.22Procedureoftheinversedistancemethod

Therequestedrainfalldepthis

 

Example2.6Theannualrainfallinmillimetresattwostationsfortheperiod1970–1971 to1989–1990 isgiven inTable2.23. Ithasgaps insomeyearsduetoproblemsinoneofthetwostations.Calculatethemissingdatafor station 2 by applying the statistical method of simple linearregression.SolutionThe linear regression between the rainfall depths of the two stations(without taking into account theyears forwhich there arenodata atstation2)givesthestraightline: 

withacoefficientoflinearcorrelationr=0.752.

Table2.23Annualrainfalldepthofthestations

Hydrologicalyear Station1 Station2

1970–1971 1145.8 1354.7

1971–1972 1311.5 1466.2

1972–1973 1078.5

1973–1974 1278.5 1455.8

1974–1975 1262.8 1422.7

1975–1976 1088.6 1469.2

1976–1977 1175.6 1366.7

1977–1978 963.4 1368.9

1978–1979 1066.8

1979–1980 985.2 1128.6

1980–1981 1066.9 1208.6

1981–1982 1306.4

1982–1983 1285.6 1502.3

1983–1984 1139.6 1402.5

1984–1985 1297.5 1544.3

1985–1986 958.3

1986–1987 1144.2 1499.8

1987–1988 978.3 1275.4

1988–1989 1145.2 1399.8

1989–1990 955.3 1255.9

Table2.24Completionofvaluesusinglinearregression

Hydrologicalyear Station1(x) Station2 Station2(y)

1970–1971  1145.8  1354.7

1971–1972  1311.5  1466.2

1972–1973  1078.5 —  1340.55

1973–1974  1278.5  1455.8

1974–1975  1262.8  1422.7

1975–1976  1088.6  1469.2

1976–1977  1175.6  1366.7

1977–1978  963.4  1368.9

1978–1979  1066.8 —  1332.42

1979–1980  985.2  1128.6

1980–1981  1066.9  1208.6

1981–1982  1306.4 —  1498.91

1982–1983  1285.6  1502.3

1983–1984  1139.6  1402.5

1984–1985  1297.5  1544.3

1985–1986  958.3 —  1257.02

1986–1987  1144.2  1499.8

1987–1988  978.3  1275.4

1988–1989  1145.2  1399.8

1989–1990  955.3  1255.9

The completion of the requested values is done by applying the lastrelationshipfortherespectivevaluesofx.TheresultsarepresentedinTable2.24.

2.8.5Correctionofrainfallwithaltitude

Useful for thesupplementof the rainfalldataofa station is the fact thatpointrainfallincreaseswithincreasingaltitude.Actually,thiscanbegenerallyappliedtoanyrainfalldataprocessingforstationsatdifferentaltitudes.Theaverageincreaseintheannualrainfallatastation,per100mincreaseof

altitude, iscalled rainfallgradient.The rainfallgradient isusuallyobtained foreachregionfromthegraphoftheaverageannualrainfallheightsofthestationsinrelationtothealtitudeofthestations.

Oftentimes,thelinearregressionlinewhichisplottedforthedeterminationofthe rainfall gradient has a low degree of linear correlation. In this case, thepotential exception of some stationsmay be considered or the regionmay besplit into subregions, for which there is a straight regression line withsatisfactorycoefficientoflinearcorrelation.

2.9SURFACEINTEGRATIONOFAREALRAINFALLFROMPOINTMEASUREMENTS

The rainfall measurements taken by rain gauges are point measurements, andthus, represent the point where precipitation was measured. In most cases,however,suchasintheassessmentofthewaterbalance,thesurfacerainfallisofparticularimportance,representingtheentirewatershedunderconsideration.Forthis reason, a network of rain gauges is installed at the watershed, whosepositionsshouldbesuchastodescribeasbestaspossiblethespatialvariabilityofrainfall.Then,thepointmeasurementsareusedforthecalculationofsurfacerainfall,usingsurfaceintegrationmethods.Therearenumerousmethodswhichhavebeendevelopedandusedtoestimate

theaveragesurfacerainfall.Thesecanbedividedintodirectintegrationmethodsandmethodsofarealadjustment(KoutsoyiannisandXanthopoulos,1997).Thedirectintegrationmethodscalculatethearealrainfalldirectlyfromthevaluesofthepointrainfall.Thebest-knownmethodswhichbelongtothiscategoryaretheaveragingmethod,theThiessenmethod,thetwo-axismethodofBethlahmyandtheoptimalintegrationmethod(Kriging).Onthecontrary,thesurfaceadaptationmethodsfirstestimate thegeographicvariabilityof rainfall in theregionunderstudy and based on this, they calculate the surface rainfall. Methods of thiscategoryarethemethodofisohyetalcurves,themethodoflinearinterpolation,themethodofinversedistance,themethodofmultisquareinterpolation,theleastsquares method with polynomials, the Lagrange polynomial method, theadaptation method of splines and the optimal interpolation method (Kriging).The methods of areal adaptation are further divided into interpolation andsmoothing methods. The averaging method, the Thiessen method and theisohyetalmethodarepresentednext.Regardlessofwhichmethodisused,thereliabilityofthefinalresultdepends

primarily on the density of the point information: the integration is moresuccessful when the density of the network of the rainfall stations is dense.Unfortunately, networks are usually not quite dense and in somemountainousinaccessibleareas,thereisascarcityofstations.

2.9.1Averagingmethod

This is thesimplestmethod,according towhich theweightsofall stationsaretaken to be equal, i.e.wi = 1/k. The method may be used for the first roughestimatesofrainfallduetoitssimplicity.Theaccuracyistolerableonlywhenthearea is relatively fat, the stations are evenly distributed and the rainfall depthdoes not vary greatly fromone station to another. The surface precipitation isgivenbythefollowingequation:

  (2.33)

2.9.2Thiessenmethod

According to this classicalmethod, the total areaA is divided into geometriczonesAi,oneforeachstation,sothat

  (2.34)

Theweightingcoefficientdependsontheareaofthestation’szone,i.e.

  (2.35)

Figure 2.24 Thiessen polygons for the PiniosBasin in Thessaly,Greece. (FromMimikou,M.,NationalBankofHydrologyandMeteorologyData(NDBHMI),FinalReport,NTUA,Athens,Greece,2000.)

Thegeometriczonesaredeterminedinawaythatthedistanceofeachpointofthezoneofstationitostationj islowerthanitisfromanyotherstationinthearea.This principle leads directly to a simple geometric construction of zonesbased on the segments perpendicular to the line segments connecting the twostations.ThiscreatestheThiessenpolygons(Figure2.24).Despiteitsprolongeduse, themethod remainswidespread today because of ease of implementationandreliableestimates.Theestimatesofthemethodarebetterwhenthenetworkof rainfall stations is dense and the timescale of the study is long (e.g. theestimates are more accurate at hyperannual scale than the estimates at stormevent scale). The altitude adjustment of the areal rainfall of the method isconductedthroughanaltitudecorrectioncoefficient,basedonthelinearrelation

betweentherainfallandtheelevation.

2.9.3Isohyetalcurvemethod

A rainfall isohyetal curve is defined by the group of pointswhere the rainfalldepthhasaspecificvalue.Dependingontherangeofrainfall,isohyetalcurvesareplottedwithanintervalDP.Theexactplotofanisohyetalcurvedependsontheavailabledataandtheexperienceofthehydrologist.Aftertheplottingofthecurves,theareasbetweensuccessivecurvescorrespondingtorainfalldepthsPiandPi−1arecalculated.Thearealmeanrainfalloftheregioncanbecomputedasfollows:

  (2.36)

Therainfall isohyetalcurvesfortheregionofThessalyarepresentedinFigure2.25,whicharebasedon thehyperannual rainfallvalues (from1/10/1955)andintervalof100mm(Mimikou,2000).

Figure 2.25 Rainfall isohyetal curves in the region of Thessaly. (FromMimikou,M., National Bank ofHydrologyandMeteorologyData(NDBHMI),FinalReport,NTUA,Athens,Greece,2000.)

Figure2.26Stations’locations.

Example2.7The monthly rainfall data and the respective altitudes for 10 raingauges in a sub-basin of Pinios River in Thessaly (Figure 2.26) areshowninTable2.25.Calculate theaverage surface rainfallusing themethodsofarithmeticmean,Thiessenandisohyetalcurves.Moreover,an adjustment of the average surface rainfall calculated by theThiessen method with respect to the average elevation of the basinshouldbemade.Theareaof thebasin isequal to2940km2,and theaverageelevationofthebasinisequalto532m.Solution

1. MethodofarithmeticmeanThe surface rainfall is equal to the average of the pointmeasurementsofthe10stations:

 

Table2.25Altitudeandmonthlyrainfallatstations

α/α   Station Altitude(m) Rainfall(mm)

1 MOUZAKI  226  74

2 TAVROPOS  220  73

3 AGIOFILLO  581  92

4 MALAKASIO  842  101

5 MEGALIKERASIA  500  83

6 METEORA  596  87

7 PALAIOCHORI 1050  110

8 STOURNAREIKA  860  103

9 TRIKALA  116  82

10 FARKADONA  87  70

Figure2.27DrawingtheThiessenpolygons.

1. ThiessenmethodThe design of Thiessen polygons is made by drawing linesperpendicularandatmidpointtotheonesconnectingthestationsonebyone,asshowninFigure2.27.Thesepolygonsaredefineduniquelyforaparticularnetworkofraingauges. The procedure for the calculation of the surface rainfall issummarizedinTable2.26.Thepercentagewisequaltotheratioof the Thiessen polygon of the station to the total area of thebasin.Thesurfacerainfallisestimatedas86mm.

2. IsohyetalcurvemethodTherainfallisohyetalcurveshavebeendrawnat5mmintervalinFigure2.28.Tenintervalsareformed. Thedrawingoftherainfallisohyetalcurvesisnotunique,asin

the Thiessenmethod, but depends on the interpolationmethod.Thesimplestwayisthegraphicdrawingoftheisohyetalcurves,which involves the discretion and experience of the designer,while in geographical information systems, the commonly usedmethods are the inverse distance or the nearest neighbour. Theprocedure for the calculation of the surface rainfall is given inTable2.27. Thesurfacerainfallisestimatedat87mm.

Table2.26Thiessenmethod

Figure2.28Drawingofrainfallisohyetalcurves.

Table2.27Procedureoftheisohyetalcurvemethod

1. RainfalladaptationtothemeanaltitudeofthebasinThesurfacerainfallcalculatedbythemethodofThiessenignoresthe actualmean altitude of the basin. It only takes into accountthe altitude of the stations. Therefore, in a basin where moststationshavebeeninstalledinthelowlands,theThiessenmethodwould underestimate the true surface rainfall and vice versa.Therefore, a correctionof the surface rainfall resulting from theThiessen method would be required based on the actual meanaltitudeof thebasin,which, in this example, is equal to532m.The correction involves assessing the rainfall gradient and thedifference between the weighted mean altitudes of the stationsandtheactualmeanaltitudeofthebasin.

Figure2.29Linearregressionbetweentherainfallandthealtitude.

Therainfallof thestationsandtheiraltitudearelinearlycorrelatedtoestimatethemonthlyrainfallgradient,asshowninFigure2.29.Theslope of the line formed (λ= 0.039) is themonthly rainfall gradientand denotes the increase in the monthly amount of rainfall withaltitude. The rainfall gradient is usually expressed in millimetres ofrainfall per 100 m of altitude change, i.e. here it is equal to 3.86mm/100m.Theweightedmeanaltitudeofthestationsmaybederivedusingthe

ThiessencoefficientswhichwerecalculatedinthepreviousstepusingtheprocedurepresentedinTable2.28.Theadjustedsurfacerainfallisasfollows:

 

wherePt=86mm,therainfallwhichresultedfromtheThiessenmethodλ=0.039(mm/m),therainfallgradientΔh=HKBsurface−Hweighted=532−449=83m

Table2.28Calculationoftheweightedaltitudeofthestations

Station Thiessenpercentagew

AltitudeH(m) w×H

1  0.107  226 24.29

2  0.052  220 11.45

3  0.073  581 42.69

4  0.072  842 60.61

5  0.081  500 40.48

6  0.198  596 118.00

7  0.066  1050 69.30

8  0.060  860 51.78

9  0.187  116 21.70

10  0.103  87 8.94

Total  1 449

Theadjustedsurfacerainfallis 

The difference in the estimated rainfall is 3 mm in relation to therespectiveamount,whichresultedfromtheThiessenpolygonmethod.

2.9.4Optimuminterpolationmethod(Kriging)

Interpolationisamethodofconstructingnewdatapointswithintherangeofadiscretesetofknowndatapointsbyminimizingtheexpectederrorfluctuations.Krigingisageostatisticalestimatorwhichderivesthevalueofarandomfieldatanunobservedlocationfrompointmeasurements(e.g.rainfall).Itisusedinthespatialestimationofrainfallinareaswithoutraingaugemeasurements.Krigingactuallycomprisesagroupofmethodsnamedafter theminingengineerD.G.Krigewhopioneeredintheirdevelopment.Krigingisbasedontheprinciplethatthevalueatanunknownpointshouldbetheaverageoftheknownvaluesofitsneighbours,whichisweightedbytheneighbours’distancetotheunknownpoint.Acharacteristicofthesemethodsisthatanimportantroleplaysafunctioncalledthevariogram(orsometimesthesemivariogram)whichquantifiestheextenttowhich the difference between values tends to become significant as the timebetweenobservationsbecomeslonger,assumingthat thisdoesinfactoccur.Inthecaseof raingauges,pairsofgaugesare selected fromawiderarea, takinginto account that gauges are positioned in places with similar climatecharacteristics.Plotting consists of values of distances between the two stations in the

horizontalaxisanddifferencesofmeasurementsDij=0.5(yi−yj)2intheverticalaxis.It appears to be a tendency as measurements become far apart and the

differencebetween themgrows.Aplotof this type is called avariogram.The

variogramitselfisacurvethroughthedatathatgivesthemeanvalueofDijasafunctionofthedistance.Anexperimentalorempiricalvariogramisobtainedbysmoothingthedatatosomeextenttohighlightthetrend.Amodelvariogramisobtainedbyfittingasuitablemathematicalfunctiontothedata,withanumberofstandardfunctionsbeingusedforthispurpose.Typically, a model variogram looks like the one in Figure 2.30. Even two

pointswhichareclosetogethermaytendtohavedifferentvalues,sothereisanuggeteffect;theexpectedvalueof0.5(yi−yj)2isgreaterthanzeroevenwithh=0.Themaximumheightofthecurveiscalledthe‘sill’.Thisisthemaximumvalueof0.5(yi−yj)2whichappliesfortwopointswhicharefarapartinthestudyarea. Finally, the range of influence is the distance at which two points haveindependent values. This is defined as the point inwhich the curve’s value isequal to 95% of the difference between the nugget and the sill. There is anumber of standardmathematicalmodels for variograms.One is theGaussianmodelwiththefollowingequation:

  (2.37)

wherecisthenuggeteffectSisthesillaistherangeofinfluence

Figure2.30Atypicalvariogramshowingthesill,thenuggeteffectandtherangeofinfluence.

Hereareothermodelswhichareoftenconsidered:

1. Thesphericalmodel

(2.38)

1. Theexponentialmodel

(2.39)

1. Thepowermodel

(2.40)

Forallofthesemodels,cisthenuggeteffect.Thesphericalandtheexponentialmodels alsohavea sill atS, but thepowermodel increaseswithout limit ashincreases.Inappliedhydrology, themodelswhichareusuallypreferredaretheGaussianandtheexponentialmodel,basedonX2test.Supposethatinthestudy

area, samplevaluesy1,y2,…,yn are known at n locations, and it is desired toestimatethevaluey0atanotherlocation.Forsimplicity,assumethattherearenounderlying trends in thevaluesofY.Then,Krigingestimatesy0usinga linearcombinationoftheknownvalues:

  (2.41)

wheretheweightsa1,a2,…,an for theseknownvaluesareselectedso that theestimatorfory0isunbiased,withtheminimumpossiblevarianceforpredictionerrors.Theequationsfordeterminingtheweightstobeusedinthisequationaresomewhat complicated. They are a function of the assumed model of thevariogram.Somevariationsofthemethodareasfollows:OrdinarysimpleKriging:Thisisthemostcommonformandisbasedonthefollowing assumptions: (1) The variable is normally distributed, (2) theestimateisunbiased,and(3)thereisacontinuityoftheseconddegree,with(d1)whenthelocalmeanisknown(simple)and(d2)whenthelocalmeanisunknown(ordinary).TheuseofordinaryKrigingforrainfallestimationislimiteddue toparticular assumptions.Rainfall is not normallydistributedduetointermittence(largenumbersofzerorainfall);theestimatedweightsandvariances are also independentof thedatavalues.These assumptionsmaycauseKrigingtooverestimateintheno-rainandlowrainfallsituationsandtounderestimateinhighrainfallsituations.

Indicator Kriging: The indicator approach is one overcoming both theselimitations and therefore of obtaining better estimates of rain areas. Anindicator function is a binary variable representing zero and non-zerorainfall amounts. So, the departure from normality in rainfall should begreatly reduced when rain and no-rain intervals are separated. Also, theproblem of data independence of the Kriging variance is reduced sinceindicatorKrigingdependsonthedatavalues.

Neighbourhood Kriging: If the local mean and variance are constantthroughout the region (cases of permanency and isotropic field), inmostapplications,thedatacontainvariationsonalocalscale.Forthisreason,thenearestpointsorthosecontainedinthesurroundingareaareinvolvedintheestimationofthevalueinanunknownarea.

UniversalKriging:Thisisappliedwhenthedatacontaintrends.Cokriging: The estimation with normal Kriging is improved significantlywhenthevariablebeingtestedisconnectedwithanothervariableforwhich

therearemeasurements.ExamplesoftheuseoftheKrigingmethodareshowninFigures2.31and2.32.Figure2.31depicts thepiezometricmapproduced through theKrigingmethodin ArcView, while Figure 2.32 depicts the mean areal rainfall for westernGreece.

2.9.5Timedistributionsofrainfall

Fordesignpurposes,amatterofgreatimportanceisthedefinitionofspatialandtimedistributionsof thedesign storm,whose importance in hydrologic designwill be discussed thoroughly in Chapter 6. The categorization of the spatialdistributionofrainisdoneaccordingtoscales,discussednext.

2.9.5.1Limited-scalephenomena

Inthislimitedscale,mostlyconvectiverainsaredominantincelltype(cellsofcompactedvapours,which aremovingwith thehelpof thewind).These cellshaveanaveragelengthfrom1toafewkilometresandmeanraindurationfrom30minto1h.Forapointobserver,thistypeofrainhasashortdurationoffewminutes. These clouds follow a random movement which tends to beindependentofthewind’saveragespeed.

2.9.5.2Medium-scalephenomena

Theaforementionedsmallcellscanbegatheredingroupsandseparatedfromthemediumcellswithaveragecharacteristicdimensionsfrom8to50km.

Figure2.31ThepiezometricmapforCentralGreecebyusingtheordinaryKrigingmethod.

2.9.5.3Synoptic-scalephenomena

Stormsystemswhichcometogetherwithweatherfrontsandintensebarometriclowpressurescancoverusuallyanareaofhundredsofkilometresandarebigenough tobepresented in continentalweathermaps.These systems are calledsynoptic-scalephenomena.Takingforgrantedtheclimatologicconditionsofanarea,itisverylikelyfor

phenomenaofthesamescaletoshowsimilardistributionsiftheyarenormalizedfor theirdurationand size.Basedon this fact, a timedistributionof rainswasconductedandexpressedbythepercentagecumulativecurve.Thiscurveshowsthecumulativedepthofraindividedintosections,wherethepercentageofrainduration is plotted on the x-axis and the percentage of total rainfall depth isplottedonthey-axis.Therainiscategorizedintofourstandardtypeswiththeirrespective group of curves and according to their respective quartiles (first,second, third or fourth), when the rain height reaches its peak, with an extraparameterwhichistheleveloftrustofoccurrenceforeachcurveinsideacertainquartile.Thesedistributionshavebeenderivedaftertheexperimentalanalysisofmanystorms(Huff,1970),andtherespectivelevelsoftrustshowthepercentage

ofraineventswhichcorrespondtoeachcurve.Ingeneral,convectiveandfrontalrains tend to reach their peak in the beginning of the rain (a first quartiledistribution), while stratiform rains usually reach their peak during the thirdquartile(Laurenson,1960;Figure2.33).

Figure2.32 MeanannualrainfallforwesternGreeceusingtheordinaryKrigingmethod.

2.10HYDROLOGICALLOSSES

2.10.1General

Waterfromprecipitationisdelayedorretainedinmanydifferentwaysduringitscoursetoastream.First,manyobstaclesgetinitsway,liketreesandplants.Thisis called interception. The excess water fills the ground cavities (detention),while a film of water is also retained in the surface. The latter is known assurfacedetention.Partofthewaterreturnstotheatmospherebyevaporationandplant transpiration,while the restenters thegroundby infiltration.Figure2.34

shows a qualitative view of the distribution of quantities of water held byinterception,retention,infiltrationandsurfacerunoff.Infiltration is an important process in the hydrological cycle because it

determinesthepercentageofwaterreachingastreamandalsothepercentageofwaterenteringthegroundrechargingtheaquifers.infiltrationratehastimeandspatialvariations,dependingnotonlyon thephysicalpropertiesof thesoilbutalsoonvegetationand rainfall intensityand timedistribution. In the followingparagraphs, infiltration analysis and the respective empirical relationships aredescribed.

Figure 2.33 Rainfall time distributions. (a) First quarter distribution and (b) second quarter distribution.Rainfall time distributions. (c) Third quarter distribution and (d) fourth quarter distribution.(FromHuff,F.A.,WaterResour.Res.,6,447,1970.)

Figure2.34 Thedistributionofinterception,retention,infiltrationandsurfacerunoff.

2.11EVAPORATION

Evaporation is the process of water transportation from the Earth and watersurfacestotheatmosphereintheformofwatervapour.Thisprocessismainlydependent on solar radiation, temperature, water vapour pressure, wind speedandtypeoftheevaporationsurface.Solarradiationisamajorfactoraffectingevaporation,becauseitsuppliesthe

energyrequiredforthechangeinthewaterphase.Moreover,forconstantwatertemperature, evaporationdependson (i) thewind speed and (ii) thedifferencebetweenthesaturationpressureofwatervapouratthewatertemperatureandthepartial pressure of water vapour of the overlaying air. All these factors areinterdependent,andthus,theeffectofeachcannotbeestimatedaccurately.Evaporation, as it depends on solar radiation, varies with latitude, season,

altitude, timeofdayandcloudcover.Theevaporationofwaterfromasurfacealso depends on the available amount of water. Therefore, the potential forevaporation is unlimited from water surfaces, while it varies at soil surfaces,whereitrangesfromunrestrictedwhenthegroundissaturatedtonearlyzerofordrysoil.Evaporationisusuallymeasuredasthemassofwaterperunitareaandtime.

Alternatively, it ismeasured as the equivalentwater depth inmillimetres at acertainpointoftime.

2.11.1Waterbalancemethods

The water balance for the estimation of evaporation in a watershed can bewrittenasfollows(Sakkas,1985):  (2.42)

whereEistheevaporationIistheinputPistheprecipitationOistheoutflowOsistheinfiltrationΔsisthechangeinstorage

The infiltrationOs can be measured or calculated directly, and the degree ofaccuracy of this quantity affects the accuracy of measurement of the actualamountofevaporation.Theinflow,theoutflow,theprecipitationandthechangein storage can be measured with sufficient accuracy. This method, used forassessing long-termevaporation, canbeusedasa reference forcomparing theefficiencyofothermethods.

2.11.1.1Thornthwaite’swaterbalancemethod

Thornthwaitedevelopedawaterbalancemethodfortheestimationofavailablewatercontentinsoil,whichisdescribedbythefollowingequations:

 

(2.43)

whereAnandAn−1aretheavailablewatercontentsinthesoilforthetimeintervalsnandn−1,respectively

PnistheprecipitationAETnistheactualevapotranspirationPETnisthepotentialevapotranspirationAWCisthewatercapacityofthesoilROnistherunoff

Thismethod’sprerequisiteisthestartofcalculationatthebeginningofthewetseasonandthataniterativeapproachbeusedtocorrectfor thefirstcycle.Thedisadvantagesofthemethodaretheuseofaveragevaluesinmonthlyintervals,theassumptionthatallrainfallisevaporatedandthesoilmoisturedryingcurvebeingapproximatedbyasimplisticexponentialrelation.

2.11.2Energybalancemethods

ThetotalnetradiantenergyattheEarth’ssurfaceis  (2.44)

ThisquantityofenergyisconvertedintolatentheatΛ,i.e.energyspentfortheevaporation of water, and sensible heat,H, i.e. the quantity of energy that isinduced by the water body to the atmosphere. Neglecting the smaller energylosses (e.g. conduction to the ground, biochemical processes and temporarystorage), the equation (Koutsoyiannis and Xanthopoulos, 1997) can beapproximatelystatedas  (2.45)

Themost essential parameter in energyprocesses is the latent heatΛ, since itprovides the energy required for evaporation.TheBowen ratio connects latentheatwithsensibleheataccordingtothefollowingequation:

  (2.46)

Combiningtheseequations,theevaporationcanbecalculatedastheratioofthelatentheatΛtothelatentheatforwatervaporizationλ:

  (2.47)

whereEt=dh/dtistheevaporationinkgofwaterperdayandm2ofareaλisthelatentheatforwatervaporization(kJ/kg)Rnisthenetradiationtothemassofwater(kJ/m2d)BistheBowenratio[dimensionlessparameter]

2.11.3Masstransfermethods

These methods consider evaporation as mass transfer or diffusion of watervapour from higher to lower concentrations.As a diffusion process, it can bedescribedbytheFickdiffusionlaw,whichin thecaseofevaporationtakes thefollowingform:

  (2.48)

whereEaistheevaporationrateeisthepartialpressureofwatervapourMisthetransfercoefficientzisthealtitude

Basedonthisequation,complexequationshavebeendeveloped,whichcalculatethe evaporation rate, using the logarithmic profile of wind speed near thesurface, which depends on the roughness of the ground. In practice, theempirically determined relationship of Dalton has been used since 1802.According to this relationship, the evaporation rate is given by the followingequation:

  (2.49)

whereD=es−e(hPa)isthedeficitofwatervapourF(u)isthewindspeedfunction

••

••

•

The wind speed function F(u) contains the wind speed and is given by thefollowinglinearequation(Penman):

  (2.50)

whereu is thewind speed in (m/s)measured at an altitude of 2m above theground.ThelatterrelationshipisoneofthemanyapproachesformulatedfortheestimationofthelinearrelationshipofF(u)withthewindspeedu.

2.11.4Combinationmethods:Penmanmethod

The combination methods for assessing evaporation integrate energy balancewithmasstransfermethods.ThefirstcombinationalmethodforcalculatingtheevaporationfromawatersurfacewasgivenbyPenmanin1948.Later,Penmanequation was modified by Monteith (1965), in order to computeevapotranspiration from soil surfaces. The meteorological data used by thePenman method are measurements of temperature, relative humidity, relativesunshineandwindspeedataheightof2mabovetheground.According to Penman, evaporation from a water surface is given by the

followingequation:

  (2.51)

whereγisthepsychrometriccoefficient(hPa/°C)∆ is the slope of the saturationwater vapour curve (hPa/°C) given by the

relationship ,whereTisthemeanairtemperature(°C)and es is the saturated vapour pressure (hPa) estimated from

(hPa)Rnisthenettotalradiantenergy(kJ/m2day)Rn=Sn−Ln(kJ/m2/day),whereSnistheshort-wavenetradiationandLnisthelong-wavenetradiation.Theshort-wavenetradiationcanbedescribed

by , where S0 is the solar radiationreachingφthe latitudeof thesite,n themeandailysunshine(h),n= totalmonthlysunshine(h/daysofthemonth)andNthemeandayduration(h)Ln, the long-wave net radiation, is given by the expression

•

•

•

, where σ is Stefan–Boltzmann constant (σ = 4.9 × 10−6 [kJ/m2/K4/d]), TK is the meantemperatureinK(TK=T+273)andUisthemeanrelativehumidity(%)λisthelatentheatofevaporation(kJ/kg),whereλ=2501−2.361·T(kJ/kg)andTin°CF(u) is the function ofwind speed (kg/hPam2 d)whereF(u) is given byEquation2.50Disthewatervapoursaturationdeficit(hPa),D=es−esU/100(hPa)

The ratios Δ/(Δ + γ) and γ/(Δ + γ) function as weighing coefficients for theparticipationoftheenergytermandthemasstransferterm,respectively, intheevaporationfromwatersurfaceandaggregateonetotheunit.Usingtheunitsreportedintheearlierequations,evaporationisgiveninkgof

water per square meter and day. Dividing by the water density (kg/m3), theevaporationisgiveninm/day.Usually,thefinalresultsaregiveninmm/dayormm/month.

2.12Evapotranspiration

Evapotranspiration includesboth transpirationfromvegetationandevaporationfromwater surfaces, soil, snow, iceandvegetation.Thereare severalmethodsfor assessing evapotranspiration, some of which are accurate and reliable andothersprovidesimpleapproaches.Theselectionofthemethoddependsmainlyon the type of the surface and the purpose of the study. Consequently, theselectionofthemethoddeterminesthescalesoftimeandspaceandtheprecisionrequirements.Otherequallyimportantfactorsarethecostandtechniqueofeachmethod.

2.12.1Waterbalancemethods

Thesemethodsarerelatedtothewaterbalanceequationandcontributedirectlyor indirectly to the determination of the evapotranspiration. In most studiesbased on the water balance equation, evapotranspiration is calculated as theresidualtermoftheequationwhiletheothercomponentsareeithermeasuredorcalculated.Ifthereisnoirrigation,evapotranspirationisgivenbytheequation  (2.52)

wherePistheprecipitation

ΔSWisthechangeofthewatercontentofthesoilROisthesurfacerunoffDisthedeepinfiltration

Thisequationcanbeappliedatanyscale.The disadvantages of this method are the low-level accuracy of the

measurements and the difficulties in assessing theET during periods of rain.Evenwiththeuseofcarefulmeasurements,itisdifficulttodetectchangesinsoilwaterwithprecisiongreaterthan2mm.Whenthismethodisappliedoverlargeareas, the main problem is the lack of good average spatial values of itscomponents,duetothevariabilityofrainfalloverlargeareasandthediversityoftopography and the soils beneath the area. The errors associated with thismethodmakeitsuseinsufficientonadailybasis.

2.12.2Methodsforthedeterminationofpotentialevapotranspirationfromclimaticdata

2.12.2.1Penman–Monteithmethod

The Penman method assumes that water vapour is at saturation at the watersurface. Therefore, it cannot be used in the case of transpiration from plantsurfaces.Toaddressthisweakness,Monteithintroducedthesurfaceresistanceoffoliage in evaporation. The Penman–Monteith method is suitable for theestimationofevapotranspirationfromsoilsurfacesandisgivenbythefollowingequation:

  (2.53)

where

Thisisthemodifiedtermofpsychrometriccoefficientwhichtakesintoaccounttheresistanceoffoliage.Windspeeduisgivenin(m/s):

ThisisthemodifiedwindspeedequationwithTin(°C)anduin(m/s).Therestof the variables are calculated just as in the Penman method, except of the

reflectancecoefficienta(albedo),whichismodified,takinggreatervalues,thuschangingthenetradiationRn.

2.12.2.2Thornthwaite’smethod

Thornthwaite(1948),exceptoftheestimationofavailablewatercontentinsoildescribedearlier in thischapter,createdanequationwhichcanbeusedfor theestimationofmonthlyevapotranspirationonlimitedwateravailabilitybasedontheaveragemonthlytemperature,whichhasthefollowingform:

  (2.54)

whereEpisthepotentialevapotranspirationinmm/monthtiistheaveragemonthlytemperaturein°CμisthenumberofdaysNisthemeanastronomicaldurationofthedayJistheannualtemperatureindicatoraisanempiricalparameterwhichdependsonJ(a=0.016·J+0.5)

Thetemperatureindicator,J,isgivenbythefollowingequation(KoutsoyiannisandXanthopoulos,1997):

  (2.55)

The monthly temperature indicator ji is a function of the average monthlytemperature,accordingtothefollowingequation:

  (2.56)

Themonthlypercentages(%)ofdayhourswithrespecttothedayhoursoftheyeararegiveninTable2.29.

Table2.29Monthlypercentagesofdayhourstothedayhoursoftheyear

2.12.2.3Blaney–Criddlemethod

Blaney and Criddle (1962) developed an empirical formula which connectsevapotranspirationwithmeanair temperatureandmeanpercentageofdaylighthoursperday.Evapotranspirationdirectlydependsonthesumofproductsoftheaverage monthly temperatures and the percentage of hours of daylight in amonth,inanactivelygrowingcropwithsufficientsoilmoisture,accordingtothefollowingrelationship:

  (2.57)

whereETisthemonthlypotentialevapotranspirationinmmkisanempiricalfactor,referringtoaparticularcrop(cropfactor)Tistheaveragemonthlyairtemperaturein°Cpisthepercentageofdaylighthourspermonth

piscalculatedasfollows:

  (2.58)

whereNistheaverageastronomicaldurationofthedayinhoursμisthenumberofdaysofeverymonth

Values for k are given in Table 2.30 for a variety of crops, as estimated by

BlaneyandCriddle.Thismethodwasinitiallydesignedfortheestimationoftheseasonalwaterneedsduringthegrowingperiodofeachcrop.InTable2.31, thevaluesofmonthlykaregivenfordifferentcrops.FromthecomparisonofTables2.30and2.31,onecanconcludethatthereareconsiderabledifferencesbetweenseasonal and monthly crop factors k. This is due to the difference in thedevelopment of the root system and the aboveground part of the plant,depending on the growing phase. For this reason, for the estimation of themonthlywaterneeds,theuseofthemonthlycropfactorsissuggested.

Table2.30Valuesofseasonalcropfactork(Blaney–Criddlemethod)

Crop Growingdurationinmonths k

Clover   Betweenfrost 0.80–0.85

Corn   4 0.75–0.85

Cotton   7 0.60–0.70

Cereals   3 0.75–0.85

Citrus   12 0.45–0.55

Deciduousfruittrees   Betweenfrost 0.60–0.70

Grassmeadow   Betweenfrost 0.75–0.85

Potato   3–5 0.65–0.75

Rise   3–5 1.00–1.10

Sugarbeet   6 0.65–0.75

Tomato   4 0.65–0.70

Vegetables   2–4 0.60–0.70

Table2.31Monthlycropfactorsofirrigatedcrops(Blaney–Criddlemethod)

Note:Thesymbol“//”meansthesameasabove.

Remark:Smallervaluesofk are applied in coastal areas and largervalues areappliedindryclimates.

2.12.2.4Jensen–Haisemethod

By analyzing 3000measurements of evapotranspiration collected by a certainprocedure for a period of 35 years, Jensen and Haise (1963) developed thefollowingequation:  (2.59)

whereRsisthesolarradiationinly/dayCTisatemperatureconstantequalto0.025Tx is the x ordinate of the temperature curve in zero value, equal to 3 iftemperatureisgivenin°C

Thesecoefficientsareconstantforacertainarea.CT isgivenby thefollowingequation:

  (2.60)

Intheserelationships,e2ande1are,respectively,thesaturationpointofvapourin themaximum andminimum temperature during the warmestmonth of theyear,withC2=7.6°C.Cisdefinedasfollows:

  (2.61)

and

2.12.2.5Makkink’smethod

Makkink’s method (1957) is based on the theory that much of the energyconsumed for evapotranspiration almost entirely comes from two sources:radiationenergyandenergyofairwhichiswarmerthanthesurface.Thesetwoenergysourcesareactuallytransformedintosolarenergy.So,evapotranspirationiscorrelatedwithsolarradiationandmoreoverisdependentintenselyonshort-wave radiation in a linear manner. The dependence of evapotranspiration onradiation is not constant throughout the year as the climate and surfaceconditionschange.Itmustbenotedthatindryareas,theenergytransferonthehorizontallevelanditsdownwardtransferformanimportantpercentageoftherespective evapotranspiration in subtropical and semiarid areas. In these cases,even though the transferred heat comes from the sun, it leads to a non-linearcorrelationofETpandshort-waveradiationRs.Manyof themethodsbasedonsolarradiationhaveembeddedatermdependentontemperature.Forthisreason,Makkink (1957)has suggested the following relationship for the estimationofETp(mm/day)fromsolarradiationmeasurements:

  (2.62)

where Rs is converted into equal units of vaporized water. Makkink’srelationshiphasofferedsatisfyingresultsindryareas.Accordingtothemodifiedformforgrass,theequationofET0inmm/dayis

  (2.63)

where

Rsisthesolarradiationinequivalentevaporationinmm/dayWisacoefficientwhichdependsontemperatureandgeographicallatitudeofthearea

C is a factorwhich depends on average humidity andwind conditions onlyduringdaytime

2.12.2.6Hargreavesmethod

Hargreaves(1974)developedamethodforETpestimationwhichissimpleandneedsminimumclimatedata.Thismethodcanbeexpressed in the formofanequation:

  (2.64)

whereETpisestimatedinmm/monthMFisamonthlyfactordependentongeographicallatitudeandgivenintablesTaistheaveragemonthlytemperature(in°C)CH is a correction factor for the relative humidity RH which applies onlywhentheaveragerelevanthumidityexceeds64%

CHisestimatedbythefollowingformula:  (2.65)

For mean relative humidity less than 64%, CH is equal to unit. Hargreavescomparedhismethodwithdatacollectedbylysimetersinvariousareasaroundthe world and developed equations (from the various linear regressions)connectingactualETwithETpfordifferentclimaticconditions.

2.12.2.7Priestley–Taylormethod

The Priestley–Taylor model (Priestley and Taylor, 1972) is a modification ofPenman’stheoreticalequation(UniversityofWaterloo,2001).Usedinareasoflow moisture stress, the two equations have produced estimates which differapproximately by 5% (Shuttleworth and Calder, 1979). An empiricalapproximation of the Penman combination equation ismade by the Priestley–Taylor model to eliminate the need for input data other than radiation. Theadequacy of the assumptions made in the Priestley–Taylor equation has beenvalidated by a review of 30water balance studies inwhich itwas commonly

found that in vegetated areas with no water deficit or very small deficits,approximately95%oftheannualevaporativedemandwassuppliedbyradiation(Stagnittietal.,1989).Thereasonisthatunderidealconditions,evapotranspirationwouldeventually

reacharateofequilibriumforanairmassmovingacrossavegetationlayerwithan abundant supply of water. The air mass would become saturated and theactual rate of evapotranspiration (AET)would be equal to thePenman rate ofpotential evapotranspiration. Under these conditions, evapotranspiration isreferred to as equilibrium potential evapotranspiration (PETeq). The masstransfer term in the Penman combination equation approaches zero and theradiationtermsdominate.PriestleyandTaylor(1972)foundthattheAETfromwell-wateredvegetationwasgenerallyhigherthantheequilibriumpotentialrateandcouldbeestimatedbymultiplyingthePETeqbyafactor(α)equalto1.26:

  (2.66)

whereKnistheshort-waveradiationLnisthelong-waveradiations(Ta)istheslopeofthesaturationvapourpressureversustemperaturecurveγisthepsychrometricconstantρwisthemassdensityofwaterλvisthelatentheatofvaporization

Althoughthevalueofαmayvarythroughouttheday(Munro,1979),thereisageneral agreement that a daily average value of 1.26 is applicable for humidclimates(StewartandRouse,1976;DeBruinandKeijman,1979;Shuttleworthand Calder, 1979) and temperate hardwood swamps (Munro, 1979). Morton(1983) notes that the value of 1.26, estimated by Priestley and Taylor, wasestimatedusingdatafrombothmoistvegetationandwatersurfaces.Mortonhasrecommended that the value should be increased slightly to 1.32 for estimatesfromvegetatedareasasaresultof theincreaseinsurfaceroughness(BrutsaertandStricker,1979;Morton,1983).Generally,thecoefficientαforanexpansivesaturatedsurfaceisusuallygreaterthan1.0.Thismeansthatthetrueequilibriumpotential evapotranspiration rarely occurs; there is always some component ofadvectionenergywhichincreasestheactualevapotranspiration.Highervaluesofα, ranging up to 1.74, have been recommended for estimating potentialevapotranspirationinmorearidregions(ASCE,1990).

Theαcoefficientmayalsohaveaseasonalvariation(DeBruinandKeijman,1979), depending on the climate being modelled. The study by DeBruin andKeijman(1979)indicatedavariationinαwithminimumvaluesoccurringduringthemid-summerwhenradiationinputswereattheirpeakandmaximumduringthespringandautumn(wintervalueswerenotdetermined),butincomparisontoadvective effects, radiation inputs were large. The equation was successfullyapplied not only for open water bodies but also for vegetated regions. Thesatisfactory performance of the equation is probably due to the fact that theincomingsolar radiationhas some influenceonboth thephysiological and themeteorologicalcontrolsofevapotranspiration.Avalueof1.26hasbeenusedforαthroughout.TemporalvariationsinαassuggestedbyresearchersareemulatedbytheconversionfactorsusedinthecalculationofAETfromthePETwhichisdescribedinthefollowing.EstimatesofPETusingthePriestley–Taylorequationhavebeenadjustedasa

functionofthedifferenceinalbedoatthesitewheremeasurementsofradiationhave beenmade (albe) and the land classeswith differing albedo (alb). In theadjustment,itisassumedthatthegroundheatflux(whichshouldbeincludedinthenettotalradiationdataifitisavailable)contributes5%oftheoverallenergy.The remaining 95%of the potential evapotranspiration estimate is scaled as afunctionofthedifferenceinalbedo:

  (2.67)

Example2.8In a close distance to a natural lake, a meteorological station isinstalled (the station is sited in latitude 38°N). Based on themeasurements of a year, the monthly values of four variables areestimatedaccordingtoFigure2.35.Thefollowingareasked:

1. Estimate the evaporation of the lake according to the PenmanmethodforJuly.Thealbedoofthewetsurfaceisestimatedata=0.05.Alsogivenareγ=0.67hPa/°C,psychrometricconstant;σ=4.9×10−6kJ/m2/K4/d,Stefan–Boltzmannconstant;andρ=1000kg/m3,thedensityofwater.Julyhas14.4hmeandaydurationforlatitude 38°N, and the solar radiation which reaches theatmosphere from the outer space has the value S0 = 40,731.0kJ/m2/d.

2. Estimate the potential evapotranspiration on the soil using thePenman–MonteithmethodforJuly.Soilalbedoisestimatedata=0.25.

3. In the particular area, an irrigation network is about to bedeveloped with a water consumption coefficient of Kc = 0.80.Estimate the potential evapotranspiration of this crop using theBlaney–Criddlemethod.

4. Comment on the differences between the results of the variousmethods.

Solution

1. Thelatitudeofφ=38°hascosφ=0.79.ThetableinFigure2.36is filled using Equation 2.51 and the subsequent analyticalexpressionsofeachterm.Thelightcolouredcellsarefilledwithgivenvalues,whilethedarkonesareestimated(inalogicalorderfrom top to bottom). Evaporation inmm/d units is derived bymultiplicationwiththedensityofwater,whileinmm/month,itisobtainedbymultiplicationwiththedaysofthemonth.

2. The potential evapotranspiration on the soil is estimated by themodifiedEquation2.53andthefollowingexpressionsofγ′andF′(u). The results are presented in Figure 2.37, where the valueswhicharedifferentfromsection(1)of theexamplearegiven indarkgray.

3. In this section, Equations 2.57 and 2.58 of the Blaney–Criddlemethodareused,andtheresultsarepresentedinFigure2.38.

4. Although temperature-based methods like Blaney–Criddle areuseful when data of other meteorological parameters areunavailable, the estimates produced are generally less reliablethanthosewhichtakeotherclimaticfactorsintoaccountlikethePenmanandPenman–Monteithmethods.

Month TemperatureT(°C) RelativehumidityU(%)

Windspeedu2at2m(m/s)

Monthlysunshineduration(h)

October 17.6 60.4 1.9 169.1

November 11.7 66.4 1.9 125.2

December 8.4 67.9 2.3 117.7

January 6.3 66.6 2.5 119.8

February 7.6 64.4 2.6 111.2

March 11.4 62.2 2.6 153.1

April 15.3 57.9 2.8 208.4

May 20.9 54.2 2.4 262.3

June 26.1 48.3 27 315.2

July 28.9 44.8 2.5 338.2

August 27.9 47.2 2.6 305.2

September 23.5 51.4 2.4 240.2

Figure2.35Meteorologicaldata.

Month July

Days  31

Temperature(°C)  28.9

RelativehumidityU(%)  44.8

Windspeedu2(m/s)  2.5

Monthlysunshine(h)  338.2

Y(hPa/°C)  0.67

TK(K)  301.9

es(hPa)  39.84

Δ(hPa/°C)  2.30

N(h)  14.4

n(h)  10.91

S0(kJ/m2/d)  40731.0

σ(kJ/m2/K4/d)  4.9×10−6

Albedoa  0.05

ρ(kg/m3)  1000

cosϕ  0.79

Sn(kJ/m2/d)  24966.13

Ln(kJ/m2/d)  5721.57

Rn(kJ/m2/d)  19244.56

 0.61

F(u)(kg/hPa/m2/d)

D(hPa)  21.99

Figure2.36Penmanmethod.

2.12.3Methodsforthedeterminationofactualevapotranspiration

Forthedeterminationofactualevapotranspirationforarelativelysmallduration,thesoil–plant–atmospheresystem is taken intoaccount.For long timeperiods,theaverageactualevapotranspirationcanbeevaluatedwithsufficientaccuracybased on themean values of rainfall and air temperature. The average actualevapotranspirationofanareadependsonmanyfactorsliketotalrainfall,rainfalldistribution, plant cover, geological features and meteorological conditionsaffecting evaporation and transpiration. In an area which is geologically andclimatically homogeneous, the annual actual evapotranspiration can beexpressedasafunctionofannualrainfallandannualtemperature.TheTurcandCoutagnemethodsweredevelopedonthisprinciple(Sakkas,1985).

Month July

Days  31

Temperature(°C)  28.9

RelativehumidityU(%)  44.8

Windspeedu2(m/s)  2.5

Monthlysunshine(h)  338.2

Y′(hPa/°C)  1.22

TK(K)  301.9

es(hPa)  39.84

Δ(hPa/°C)  2.30

N(h)  14.4

n(h)  10.91

S0(kJ/m2/d)  4,0731.0

σ(kJ/m2/K4/d)  4.9×10−6

Albedoa  0.05

ρ(kg/m3)  1,000

cosϕ  0.79

Sn(kJ/m2/d)  1,9710.10

Ln(kJ/m2/d)  5,721.57

Rn(kJ/m2/d)  13,987.35

F(u)(kg/hPa/m2/d)  0.74

D(hPa)  21.99

λ(kJ/kg)  2,432.77

E(kg/m2/d)  9.40

E(mm/d)  9.40

E(mm/month)  291.42

Figure2.37Penman–Monteithmethod.

Month July

Days 31 Temperature(°C) 28.9

Kc 0.8

N(h) 14.4

P(hourspercentage/daysonthemonth/daysoftheyear) 10.19

E(mm/month) 173.87

Figure2.38Blaney–Criddlemethod.

2.12.3.1Turcmethod

Turc (1961)analyzeddatacollected from254watersheds, sitedworldwide;hecorrelatedevaporationwithrainfallandtemperature,accordingtothefollowingrelationship:

  (2.68)

whereEistheannualevaporationorevapotranspirationinmmPistheannualrainfallinmm,ITorL=300+25T+0.05T2Tistheaverageannualtemperaturein°C

Turc has also formulated another equation which embodies the effect of soilhumidityvarianceasfollows:

  (2.69)

whereEistheevaporationinmmforaperiodof10daysE10 is the estimated evaporation (for a period of 10 days) from plain soil,assumingthatthereisnoprecipitationorthatitislessthan10mm

Kisacropfactorgivenby

  (2.70)

where100Misthefinalperformanceofdriedmassinkg/ha,10Gisthevalueofgrowingperiodindaysandc isafactorfor thecrop.IT is theabilityofairevaporation:  (2.71)

Inthisequation,Tistheaverageairtemperatureforthe10-dayperiodin°CandRsistheincidentsolarradiationincal/cm2·day.For the climatic conditions of western Europe, Turc estimated the

evapotranspirationinmm/dayfor10-dayperiodsasfollows:

  (2.72)

forrelativehumidity(RH)>50%and

 (2.73)

forrelativehumidity(RH)<50%,withTtheaveragetemperaturein°CandRsthesolarradiationinly/day.

2.12.3.2Coutagnemethod

TheCoutagnemethodisbasedonthesameassumptionsoftheTurcmethodforthe estimation of the average annual value of the actual evapotranspiration(Sakkas,1985).TheempiricalformulagivenbyCoutagneis

  (2.74)

whereEandPhavebeendefinedbeforeandIisatemperaturefunction:  (2.75)

whereT stands for the average annual temperature in °C.These equations arevalidonlyforthedepthofprecipitationwithinthelimits:

  (2.76)

If the rainfall depth is lower than I/8, then the lack of flow is equal to therainfall,andthereoccursnoflow.Therefore,

  (2.77)

IftherainfalldepthisgreaterthanI/2,thenthelackofflowisindependentoftherainanditcanbedeterminedbythefollowingequation:

  (2.78)

TheresultsoftheequationsusingboththeTurcandCoutagnemethodsaregivenbypreconstructedchartstodemonstratethesimplicityoftheprocedure.

2.13INFILTRATIONRATEESTIMATION

2.13.1Infiltration

Infiltration is aprocessofwatermovementbeneath the surfaceof theground.Although it is different from percolation, very often, these two processescoincide.In the first stage, water accumulates on the ground and then moves

downwardsintothesoil,mostlybygravity,andalsoduetomolecularforcesandcapillarypotential.Capillarypotentialiscausedbythepresenceofinfinitepipelinesintheground

indifferentdirections.Inanunsaturatedground,thesepipelineshavelittleornoquantityofwater,andthecapillarypotential isgreat.Onthecontrary, inmoistground,thecapillarypotentialislow.Forthisreason,afterrain,theinfiltrationcapacityislower.At the beginning of a rainfall, the downward direction of the capillary

potentialandgravityisthesame,whichhelpsthewatertorechargetheaquifersfromafewmetrestotensofmetresbelowthesurface,asdescribedinChapter5.Upon ground saturation, the capillary potential diminishes. The methods ofinfiltrationestimation(Horton,etc.)arediscussedinthissection,andtheknownφ-indexmethodisgiveninChapter3.Bytheendoftherainfall,theuppergroundmoisturestartstodecreasedueto

evaporation and transpiration, and the capillary potential reverses andcontributestothemovementofwaterupwards.Thisgroundzonevariesfromafewcentimetrestoafewmetresabovetheaquifer,andthehydrostaticpressureinthiszoneislowerthantheatmosphericpressure.Theimportanceofcapillarypotential for plant survival is easily understood in intense drought periods. Indifferentcases,watercouldonlyexistatgreatdepthsintheground.The role of infiltration is important in a storm event. When intense rain

exceeds the infiltration capacity of the ground, part of the water quantityreachingthegroundbecomessurfacerunoff.Also,watersufficiencyintheupperground layers defines evapotranspiration, while water quantity in the aquifersuppliesstreamsduringthedryseason.Infiltrationalsoplaysanimportantrolein seasonal distribution ofwater supply. If an impermeable layer exists in theground,infiltrationisminimum,butontheotherhand,watercanstaylongerinthegroundandsustainbaseflowinastream.So, it iscritical todeterminetheinfiltration rate.The factorswhich influence itare theextentand typeofplantcover,thesaturationandtypeofground,thetemperature,therainintensityandwater quality. The measurement of infiltration rate is possible throughinfiltrometersandrespectivehydrographanalysis.Theinfiltrationrateestimationvariesincomplexity:fromsimpleobservations

of different categories of vegetation to the use of differential equations. Theinfiltration process is complicated, so many empirical relationships based onsaturatedflowequationshavebeenproposed.

2.13.2Hotton’smodel(1930)

Horton’sequationisasfollows:

  (2.79)

wherefpistheinfiltrationrate(mmorcm/h)intimetkistheconstantdependingmostlyonsoilandvegetationfcisthefinalconstantrateofinfiltrationcapacityf0istheinitialrateofinfiltrationcapacity

Horton’s equation is shown inFigure2.39 in conjunctionwith a rain episode.Theareaunderthecurverepresentsthewaterinfiltrationcapacityofthegroundatanytimet.Using thisgraph, f0andk canbeevaluatedbychoosing twopairsofvalues

andsolvinga2×2systemofequationswithsuccessionalapproaches.

Figure2.39 Horton’sinfiltrationcurveandtheraingraph.

2.13.3Green–Amptmodel(1911)

Thiswasproposedoriginally in1911andaftervariousmodificationshasbeenfound to have immense applications in simulationmodels like the stormwatermanagementmodel.In Figure 2.40, the assumptions of the model are shown: the water moves

downwardsinalinearfront,andthegroundiscoveredwithathinlayerofwater.KsisthefactorofinfiltrationcapacityinthesaturatedzoneandunderDarcy’s

law:

  (2.80)

whereListhedistancefromthesurfaceSisthecapillarypotentialinthewaterfront

IfFistheaccumulatedinfiltrationdepthandIMDistheinitiallackofmoisture,usingtheequation(2.80),

  (2.81)

Consideringfp=dF/dt

  (2.82)

andthenbyintegrating

  (2.83)

Figure2.40 Green–Amptmodel.

This form ofGreen–Amptmodel is easier in simulationmodelling due to theconnectionbetweenaggregatedinfiltrationandtime.Thevaluesofθs(θs−θi=IMD,whereθsistheporouscoefficientandθiistheinitialcontentofwater),SandKscanbetakenbyUSDA,dependingonthesoilcharacteristics(Maidment,1993).

2.13.4Huggins–Monkemodel(1966)

Many researchers omitted the variable of time. An example is the followingequationbyHugginsandMonke:

  (2.84)

whereAandParethecoefficientsS is the storage capacity over the impermeable ground layer (Tp minusmoisture)

FisthetotalinfiltrationwatervolumeTpisthetotalporosityofthesoilovertheimpermeablegroundlayer

Thecoefficientsareevaluatedbyinfiltrometerexperiments.Fmustbecalculatedforeverytimestep,whileift=0,F=0.

2.13.5Holtanmodel(1961)

Thismodelisgivenbythefollowingequation:

  (2.85)

wherefistheinfiltrationcapacity(in./h)α is the infiltration capacity [(in./h)/in.1.4] of the available storage volume(coefficientrelevanttoporoussurface)

Fpistheavailablestoragecapacityinthesurfacelayer(first6in.oflayer,ininches)

fcistheultimateinfiltrationrate(in./h)nisthecoefficientusuallyequalto1.4

2.13.6Kostiakovmodel(1932)

Kostiakovhasdevelopedthefollowingrelationship:

  (2.86)

Thetotalinfiltrationisobtainedby  (2.87)

wherefistheinfiltrationcapacity(in./h)Fistheaccumulatedinfiltration(in.)aandbarethecoefficients

Itcanalsobeexpressedinthelogarithmicform:

  (2.88)

Parametersaandbareestimatedusingadouble logarithmicpaperwith linearregressionamongaccumulatedinfiltrationandtime.

2.13.7Philipmodel(1954)

Philiphasproposedthefollowingequation:  (2.89)

Therefore,theaccumulatedinfiltrationis

  (2.90)

wherefistheinfiltrationcapacity(in./h)Fistheaccumulatedinfiltration(in.)sisthesoilstoragecapacityAistheconstantrelevanttohydrauliccharacteristicsoftheground

2.13.8SoilConservationServicemethod(1972)

SoilConservationService(SCS)usesanempiricalprocess,involvingCNcurvenumber,notdirectlyestimating infiltrationbutwithan indirectevaluationof it(seealsoChapter7).Thismethodestimatestheactiveprecipitationdepth,thatis,theprecipitation

whichgivesdirectrunoffaccordingtothefollowingrelationship:

  (2.91)

whereheistheactiveprecipitationdepthhisthetotalprecipitationdepth

SisaparameterconnectedtoCNcurvenumberaccordingtotherelationship:

 (2.92)

The SCSmethod uses empiricalmatrices of direct runoff in conjunctionwithsoiltypes.ACNcurvenumber(1–100)iscomputed.Thecategorizationofsoiltypesisasfollows:

A. Stronginfiltration(sandorgravel)B. Mediuminfiltration(sandyclay)C. Lowinfiltration(clay)D. Verylowinfiltration(clayoverimpermeableformation)

Table2.32CurvenumbersforthehydrologicalstatusII

AweightedCNcurvenumbercanbeestimatedbydifferentpercentagesofsoiltypesandlanduses.CNcurvenumbersaredividedaccordingtosoilmoisture(antecedentmoisturecondition,AMC)indifferentsoilstatuses:AMCI:DrysoilsAMCII:AveragemoistureofawetseasonAMCIII:Intenseprecipitationinthelast5days;soilinsaturation

ValuesofCNforAMCIIaregiveninTable2.32.ThoseforAMCIor IIIarederivedfromTable2.32usingtheAMCIIvalues.

REFERENCESAmericanSocietyofCivilEngineers,Committeeon IrrigationWaterRequirementsof the Irrigation and

DrainageDivision of theASCE, 1990,Evapotranspiration and IrrigationWater Requirements: AManual,ASCE,NewYork,332pp.

Anderson,M.G.andBurt,T.P.,1985,HydrologicalForecasting,Vol.372,Chichester,Wiley.Brutsaert, W. and Stricker, H., 1979, An advection-aridity approach to estimate actual regional

evapotranspiration,WaterResourcesResearch,15(2),443–450.Collier,C.G.,1989,ApplicationsofWeatherRadarSystems,EllisHorwood,Wiley.De Bruin, H.A.R. and Keijman, J.Q., 1979, The Priestley-Taylor evaporationmodel applied to a large,

shallowlakeintheNetherlands,JournalofAppliedMeteorology,18,898–903.Dingman,S.L.,1994,PhysicalHydrology,PrenticeHall,Inc.,NewJersey.DistrometLtd.,2011,Basel,Switzerland.http://www.distromet.com/,lastaccessedJanuary5,2016.Eagleson,P.S.,1970,DynamicHydrology,McGraw-Hill,NewYork.Enright, L., 2004, Low-cost re-architecturing ofNASA’sTRMMmission control center,Proceedings of

GroundSystemArchitecturesWorkshops,ManhattanBeach,CA.Everett,D.,May2001,GPMsatellites,orbitsandcoverage,GoddardSpaceFlightCenter,Greenbelt,MD.Fotopoulos,F.,May2002,Simulationof thesamplingpropertiesof theglobalprecipitationmission,MSc

thesis,MassachusettsInstituteofTechnology,Cambridge,MA.Green,W.H.,andAmpt,G.A.,1911,StudiesonSoilPhyics.TheJournalofAgriculturalScience,4(01),1–

24.Holtan,H.N.,1961,Conceptforinfiltrationestimatesinwatershedengineering.U.S.Dept.Agr.,Agr.Res.

Serv.,ARS41–51,25pp.Huff,F.A.,1970,Timedistributioncharacteristicsofrainfallrates,WaterResourcesResearch,6,447–454.Huggins,L.F.,andMonke,E.J.,1966,Themathematicalsimulationofthehydrologyofsmallwatersheds.

Tech.Rep.No.1,PurdueWaterResourcesReasearchCentre,Lafayette.Kostiakov,A.N.,1932,Onthedynamicsofthecoefficientofwater-percolationinsoilsandonthenecessity

forstudyingitfromadynamicpointofviewforpurposesofamelioration.Trans,6,17–21.Koutsoyiannis,D.andXanthopoulos,T.,1997,EngineeringHydrology,NTUA,Athens,Greece(inGreek).Laurenson,E.M.,1960,TemporalpatternofSydneystorms,Seminalrain,Paper7/4,Sydney,Australia.Maidment,D.R.,1993,HandbookofHydrology,McGraw-Hill,NewYork.Manning,J.C.,1997,AppliedPrinciplesofHydrology,Vol.276,UpperSaddleRiver,NJ:PrenticeHall.Mimikou, M., 2000, National Data Bank of Hydrological and Meteorological Information (NDBHMI),

Finalreport,NTUA,Athens,Greece.Mimikou,M.andBaltas,E.,2012,EngineeringHydrology,Papasotiriou,Athens,Greece(inGreek).Morton,F.I.,1983,Operationalestimatesofarealevapotranspirationand theirsignificance to thescience

andpracticeofhydrology,JournalofHydrology,66,1–76.Munro,D.S., 1979,Daytimeenergy exchange and evaporation fromawooded swamp,WaterResources

Research,15(5),1259–1265.NationalAeronauticsandSpaceAdministration,2007,TRMMturnsTen,http://www.eorc.jaxa.jp/TRMM/-

documents/data_use/text/handbook_e.pdf,lastaccessedJanuary5,2016.National Space Development Agency of Japan, February2001, TRMM Data Users Handbook, Earth

ObservationCenter,Saitama,Japan.Pessi, A. and Businger, S., 2009, Relationships between lightning, precipitation, and hydrometeor

characteristicsovertheNorthPacificOcean,JournalofAppliedMeteorologyandClimatology,48,833–848.

Petersen,W.A.,Christian,H.J.,andRutledge,S.A.,2005,TRMMobservationsof theglobal relationshipbetweenicewatercontentandlightning,Geophys.Res.Lett,32(1–4),L14819,4pages.

Philip,J.R.,1954,Aninfiltrationequationwithphysicalsignificance.SoilScience,77(2),153–158.Sakkas,J.,1985,EngineeringHydrology-SurfaceWaterHydrology,DUTH,HanthiGreece(inGreek).Shaw,E.M.,1994,HydrologyinPractice,3rdEdition,ChapmanandHall,London,U.K.,569pp.Shuttleworth,W.J. andCalder, I.R., 1979,Has the Priestley-Taylor equation any relevance to the forest

evaporation?,JournalofAppliedMeteorology,18,639–646.Singh,V.P.,1992,ElementaryHydrology,PearsonCollegeDivision.SoilConservationService,1972,SCSnationalengineeringhandbook,section4:hydrology.TheService.

UnitedStates.Stagnitti, F., Parlange, J.Y., and Rose, C.W., 1989, Hydrology of a small wet catchment,Hydrological

Processes,3,137–150.Stewart,R.B.andRouse,W.R.,1976,Asimplemethodfordeterminingtheevaporationfromshallowlakes

andponds,WaterResourcesResearch,12(4),623–662.UniversityofWaterloo,2001,Watfloodmodel.http://www.civil.uwaterloo.ca/watflood/manual/manualstar-

t.htm,lastaccessedJanuary5,2016.ViessmanW.,JrandLewis,G.L.,1996,IntroductiontoHydrology,Fourthedition,HarperCollinsCollege

Publishers,NewYork.

Chapter3Runoff

3.1GENERAL

Partof theamountofwater that reaches thegroundasprecipitation isheldbythe canopyof plants covering theground.The amount ofwater intercepted inthiswayisnotconstantbutisdependentontheplanttypeandthepercentageofcover in the area and the characteristics of the precipitation. Some amount ofwater is retained by soil irregularities; part of this quantity goes back into theatmospherethroughtheprocessofevaporationandtranspiration.Theremainingmovesonthegroundorpercolatesintothesoil.Onepartoftheinfiltratedwatermoveslaterally,immediatelybeneaththesurface,andreappearsdownstreamonthesurfaceorstreambanks(interflow).Theremainingmovesintodeeperlayersandenrichestheaquifers,becomingpartofgroundwater;thegroundwater,againmoving sideways, can reach the bed of a stream or even move outside theboundaries of the basin.Above-ground runoff starts almost immediatelywhenprecipitationintensityexceedsinfiltrationandsurfacedetentioncapacitiesor isdelayed and starts in accordance with the upper soil layer saturation levelprovided the precipitation continues unabated. The water moving above andbelow the ground surface following any of these paths forms the runoff. Inparticular,thepartofthewatermovingonthesurfacemakesthesurfacerunoff,which combines with interflow to form the direct runoff. The water movingundergroundasgroundwateralsoflowstostreambeds,formingthemainrunofforbaseflow.

3.2RIVERBASIN

Allpointscoveringthegroundsurfacereceivingrainfallandthroughwhichthe

1.

2.

water runoffs supplying the stream at a specific point are considered as thedrainagebasinofthisstreamorriverat thatpoint.Innature, theboundariesoftheareawhichcontributesgroundwatertoastreammaynotbeidenticaltothoseof the regionwhich contributes surface runoff. For small basins, groundwatercanmove from one catchment to the neighbouring one or even further away.This causes someuncertainties in defining the overall boundaries of basin.Toovercometheseuncertainties,theareacontributingwaterasdirectrunoffintoastreamhasbeendefinedasthestreamorriverbasin.Theboundarywhichseparatesariverbasinfromanadjacentoneiscalledthe

ridge. The basin boundary follows the ridge around the basin and crosses thestreamonlyattheoutlet.Often,itisnecessaryforpracticalreasonstodividealarge basin into smaller ones, which are called sub-basins and are defined byinternalbasins(BedientandHuber,1992).Figure3.1showsthewayofplottingthedrainagenetworkonatopographicmap.

Figure3.1 Wayofplottingthedrainagenetworkonatopographicmap.

Basinshavecertaincharacteristicsthatmodulatetheirhydrologicalbehaviourtoalargeextent.Thesefeaturesarethefollowing:

The drainage surface of the basin: This is the plan view of the areaboundedby thewatershedboundary; it isdefinedby the area calculatedfromtopographicmaps,anditisexpressedusuallyinsquarekilometresorhectares.Streamorder:Theorderofstreamsisacharacteristicwhichreflectstheirdegree of branching in the basin. Horton (1945) describes that streamswhicharesmallandhavenobranchesas thefirst-orderstreams,streamswhichhavebranchesasthesecond-orderstreamsandstreamswhichhave

3.

4.

branchesofthesecondorderasthird-orderstreams.Thus,theorderofthemain stream shows the extent of branching streamswithin a catchment.Other classifications of streams are known by Shreve and Strahler. Theclassification by Strahler is similar to that by Horton, whereas in theclassification by Shreve, the streams are summed each time and, as aresult, the final main stream has a great order. Figure 3.2 shows theclassificationofstreamsbyShreveandStrahler.

Figure3.2 Streamorderby(a)Horton(1945)andStrahlerand(b)Shreve.

Thedrainagenetworkdensity:This isaparameterof thebasin,which iscalculated,ifthetotallengthofstreamsisdividedbytheareaofthebasin.Itisexpressedinthefollowingform:

  (3.1)

whereL is usually expressed in kilometres andAd in square kilometres.The drainage network density shows the length to be travelled on thegroundsurfacebythewateruptoacertainstream.Generally,ariverbasinisconsideredtohavealowcoefficientofhydrographicnetwork,whichisproportionaltothedrainageofthebasin,ifDd≤0.5,whileitisconsideredtohaveahighcoefficientofhydrographicnetwork,implyingaverygooddrainageifDd≥3.Thehypsometric curve:Hypsometric curve is the distribution of surfaceelevationofabasin. It refers to the relationshipbetweenanaltitudeandthe percentage of the surface of the basin which has a value above orbelow it. It is usually presented in charts derived in geographicinformation systems (GISs) (Heywood and Carver, 1998), as shown inFigure 3.3. These charts are useful because they give a picture of theaverage ground slope of the basin and they can set the basis for

5.

6.

comparisons between catchments. Also, the hypsometric curve is animportant guide in the design and installation of a network ofhydrometeorologicalstations.

The calculation of the average elevation of the basin is based on thehypsometriccurveusingthefollowingequation:

  (3.2)

wheretherangeofvaluesofthehypsometriccurvehasbeendividedintointervalsoflengthΔLpandtheZvaluescorrespondingtotheedgesoftheintervalΔLrareZrandZr+1.

Figure 3.3 Hypsometric curve using GIS methods. (From National Data Bank of Hydrological andMeteorologicalInformation(NDBHMI),Athens,Greece,http://ndbhmi.chi.civil.ntua.gr/.)

Thealignmentandlongitudinalprofileofthestream:Thealignmentofthestream is the path that the stream follows on a map. The longitudinalsectionof themainstreamof thebasin isagraphofelevationalong thelength (alignment) of this stream. Based on this, the slope of the mainstream may be defined in many ways. One of them is to divide thedifference of the elevation between the highest and lowest points of thestreambythetotallength.Thisslope,ingeneral,isnotveryrepresentativebecauseofthesignificantslopevariationalongthestream.Morerealisticand useful are nearly uniform slope values obtained for various reachesalongthestream,basedonitslongitudinalsection.Averageslopeofthebasin:Thisparameteriscalculatedwithaprocedure

7.

whichbeginsbytheconstructionofagridoverthecatchment.Then,thenumber of contour lines which intersect each horizontal grid line iscountedinonedirection,andthisnumberisdividedbythetotallengthofthelinesinthisdirection(Figure3.4).Ifhisthecontourinterval,Nisthenumber of intersections of the contour lines with the grid lines in onedirection andL is the length of all lines in this direction, the slopeS isgivenby

   (3.3)

The surface of the streams of the river basin: It is obvious that such acalculation is very complex, and therefore, some simplifications areneeded.Ifweassumethatthewidthofthemainstreamattheoutletofthebasinorthewidthofanystreamatthepointofitsconfluencewithhigher-order stream is b and the width at the beginning of the streams ispracticallyzero,thenthewetsurfaceAWofeachstreammaybeestimatedbyarelationofthefollowingform:

   (3.4)

whereListhelengthofthestream.Ithasbeenobservedthatinmostriverbasins,thetotalsurfaceofthestreamsdoesnotexceedapercentageof5%ofthebasinarea.

Figure3.4 Gridoverthecatchmentareatodeterminetheslopeofthebasin.

3.3HYDROGRAPHS

Thedischargeofariverbasinataspecificlocationisdefinedasthechangeinthe volume of water per unit time, measured in m3/s. The variation of thedischarge of a stream (which is a result of the rainfall–runoff process in thebasin)withrespecttotimeiscalled‘hydrograph’.Thebasiccomponents(parts)ofahydrographarethesurfacerunoff,theintermediaterunoff(orinterflow)andthebaseflow.Figure3.5showsthesethreecomponentsofatypicalhydrograph.The surface runoff (component) includes the water flowing over the groundsurface.The intermediate runoff (component) includes thewaterwhichmoveslaterally beneath the ground surface in the unsaturated zone and returns aftersome travelled distance back to the ground surface or directly to the streambanks.Thebaseflow(component) includesthegroundwaterdrainagefromthesaturated zone.The surface and intermediate runoff components constitute thedirect runoff which is related to a rainfall event. The simple sketch given inFigure3.6showsthefactorswhichmodulatethedischargeofastream.

3.3.1Characteristicsofthehydrograph

Theshapeofahydrograph,causedbya relativelyshortdurationrainfalleventwhichcoversthewholecatchmentarea,typicallyfollowsageneralpattern.Thispatternisbellshaped,asshowninFigure3.5,atthebeginningthereisaperiodof rise,whichmeans thatduring this time, the streamdischarge increases to amaximum value (peak flow), and then, a period follows where the dischargecontinuously decreases, in which it can even reach zero, depending on theexistenceornotofbaseflow.Insmallcatchments,wherethecontributionoftheintermediate and baseflow in stream discharge is usually limited or thegroundwater table is low(e.g.aridareas), thehydrographisformedessentiallyfromsurfacerunoffonly.

Figure3.5 Thethreemainrunoffcomponentswhichformahydrograph.

Figure3.6 Diagramshowingthecomponentswhichmodulatethestreamdischarge.

Figure3.7 Typicalformofhydrographcausedbyanindividualrainfallevent.

Figure3.7showsatypicalhydrograph(dischargeofwaterfromabasinwithtime)ofasinglefloodevent,incombinationwiththehyetograph,whichcausedtheflood(rainintensityovertime–whichispresentedataninverteddirectionofordinates, just as shown in the figure, but using the same timescale). ThehyetographshowsthatattimetA′aneventofrainbegins,andatalatertimetA,after an initial deficit, the excess rainfall begins, which is converted to directrunoff.TherainendsintimetNH.TheexcessrainfalleitherendsatthesametimetHorinaprevioustimetM,iftheintensityattheendoftheeventisshort.At time tA ≡ tB where the excess rainfall begins, the discharge of the river

basin begins to rise at an increasing rate, until the discharge reaches itsmaximumvalueattimetC.ThebranchBCofthehydrographiscalledtherising

limb,pointCiscalledthepeakofthehydrographandthedischargeattimetCiscalledthepeakflowdischarge.Therisinglimbofahydrographbeginswiththesurfacerunoffandendsatthepointwheretherateofthehydrographisreduced.Theshapeofthiscurvedependsonthecharacteristicsofthebasin,formingthetravel timeofwater on the surface and streams, andon theduration, intensityandhomogeneityoftherain.Itsinitialsectionisconcavebecauseonlyapartofthebasincontributesandalsoatthisstage,theinfiltration,theretentionofwaterinsoilirregularities,theevaporationandthewaterretentionfromvegetationareproportionatelyhigherthanthatinthenextstages.The(maximum)peak,causedbyarainfallwithgivenintensityandduration,

occurswhenallpartsofthebasincontributewater,i.e.boththepartofthebasinneartheoutletisstillcontributingwaterandthewaterfromthefurthestpointofthebasinhas reached theoutlet. Inorder for this tooccur, thedurationof therainfallmustbeequaltoorgreaterthanthetimeofconcentrationofthebasin.Then, thedecreaseof thedischarge follows in time, as shown in the falling

limbCD.Thedescending limb includes the remainingpart of thehydrograph,constitutingthewithdrawalofwaterwhichistemporarilystoredatthesurfaceofthebasinaftertheactualcompletionoftherainfallevent.Theshapeofthiscurveis independent of fluctuations in the intensity of the rainfallwhich caused therunoffandsoil infiltration,dependingalmostexclusivelyon thecharacteristicsof the stream bed. At time tD, the direct runoff ceases, but the baseflow(presenting only contribution from groundwater) continues flowing in thestream,formingtheDEbranch.Thisbranch, i.e. thebaseflow, isnotrelatedtothespecificrainfallevent.Thedirectrunoffoccursatthetime(tB,tD),whilethebaseflowiscontinuous.

Theduration tb = tD−tB is known as flood duration or hydrograph base time.Othertypicaldurationsofthehydrographarethetimeoftherisinglimb,ta=tc−tB,andthelagtime(moreprecisely,thepeaklagtime),tL=tc−tS,wheretSisthe time corresponding to the centroid S of the active rainfall graph. Moreclearly,thelagtimeisdefinedasthetimedifferencebetweenthecentroidofthedirectrunoffhydrographandtheexcessrainfallgraph,i.e. ,wheretKisthe time corresponding to the centroidK of the direct runoff hydrograph. Forbetter understanding, time is also referred to as centroid lag time. Forconvenience, thefactor tL isusedhere insteadof ,withoutbeing regardedasthe characteristic of the basin, as it also depends on the form of the rainfallgraph.Anotherapproximatelyinalterabletimeperiod,whichisalsocharacteristicof

the basin, is the time of concentration or confluence time. The time ofconcentration is defined as the time which is required from the water whichcontributestodirectrunofftoreachfromthehydrologicallymostdistantpointofthebasintotheoutletofthebasin.ThetimeofconcentrationisshowninFigure3.7asthetimedistancefromtheendofactiverainuntiltheendofdirectrunoff,i.e.tc=tD−tM.In order to evaluate the aforementioned characteristic time durations, it is

necessarytoseparateonthegivenhydrographthetwocomponentsoftherunoff,base and direct flow.To do this, the following have to be determined: (1) thestartingtimeofdirectrunoff,i.e. thepointB; (2) theendtimeofdirectrunoff,i.e.thepointD;and(3)thevariationofthebaseflowduringtime.The separation of excess rainfall and losses (mostly infiltration) in a given

hyetograph follows the separation of direct and baseflow in the floodhydrograph. After separating the direct runoff from the baseflow, the volume[L3]ofdirectrunoffcanbecalculatedasfollows:

  (3.5)

whereQd(t)isthedirectrunoffattimet,whichresultsfromthedifferenceofthetotal dischargeQ(t)minus thebaseflowdischargeQb(t).Asmentioned earlier,the direct runoff hydrograph constitutes a transformation of the active rainfallgraph.Thevolume[L3]ofwateroftheexcessrainfallgraphis

  (3.6)

whereie(t)istheintensityoftheactiverain[L/T]heisthecumulativeverticaldepthofexcessrainfall[L]Aistheareaofthebasin[L2]

Massconservationrequiresthat  (3.7)

whichimpliesthat

  (3.8)

Therefore, after determining the depth he, the rainfall excess graph can besubsequentlyestablished.

3.3.1.1φindex

Infiltrationindicatorsoftenassumethattheinfiltrationiscarriedoutatafixedoraveragerateduringthestorm,somethingnottrue.Typically,thesemethodstendtounderestimatetheinitialinfiltrationratesandoverestimatethefinalones.Thebestapplication isduring largestormsandmoist (i.e. fullysaturated)soils, i.e.stormswhereinfiltrationratesmaybeconsidereduniform.Themostwell-known indicator is theφ index, according towhich the total

volume of losses during the storm is distributed evenly throughout the event(Figure3.8).Therefore,thevolumeofprecipitationoverthelineoftheindexisequal to therunoff.Avariantof theφindex,whichdoesnot takeintoaccountthe soil storageand retention, is theWindex. Initialwater quantities are oftendeductedfromthefirststagesofthestormsoastoexcludeinitialretention.In order to determine the φ index for a given storm, the observed runoff is

computed based on themeasured direct runoff hydrograph usingEquation3.5and then by subtracting this volume from the volume of the total measuredprecipitation. This resulting volume of losses (which include interception,depressionstorageandinfiltration)isdistributedevenlyalongthehyetograph,asshowninFigure3.8.Theuseoftheφindexforcalculatingthevolumeofdirectrunofffromagiven

stormprofileisthereverseprocess.Theφindexisdefinedforaspecificstormandisnotgenerallyapplicabletootherstorms,andifitisnotassociatedwiththecharacteristicparametersofthebasin,ithaslowvalue.

Figure3.8 Theφindex.

Example3.1Calculatetheφindexforabasinwithanareaof37,297km2.Arainfalleventresultedin therunoffof32mm.Thetemporalevolutionof therainfallintensityisshowninTable3.1.SolutionThe φ index will be calculated by a trial-and-error procedure. It isobserved that the time step of the rainfall is not constant. The totalrainfalliscalculatedforeachtimestepbymultiplyingthevolumewiththe corresponding period. Let us assume an initial value of φ = 3.0mm/h.For thisvalue, the losses are calculatedat each time stepandsubtractedfromtherainfall,resultingintheamountofrainflowingonthesurface(netrainfall).Oneshouldbecarefulofthetimestepswhenrainfallislessthanthelosses;thedifferenceinthiscaseshouldbesetto0.0,sincethereisnophysicalmeaningforthesurfacerunofftohavenegativevalues.TheprocedureispresentedinTable3.2.Iftheincrementalrunoffdepthsaresummed,thetotalsurfacerunoff

results to 40.78mm, which is greater than the amount of the givenrunoff (32 mm). Therefore, the assumed value of φ should beincreasedinthenexttrial.Withsuccessivetrials,itwasfoundthatthevalueofφ is equal to3.85mm/h (Table3.3),making surface runoffalmost equal to the given one; therefore, the actual value of the φindexis3.85mm/h.

Table3.1Temporalevolutionofthe15December1967rainfallevent

Time(min) Intensity(mm/h) Time(min) Intensity(mm/h)

0–20 3 152–213 2

20–50 7 213–300 3.8

50–62 3.2 300–343 9

62–121 12 343–405

121–132 6 405–422 0.2

132–140 21 422–720 7

140–148 3 720–805 4

148–152 6

Table3.2Runoffcalculationwithφindex:firsttrial

Table3.3Runoffcalculationusingtheφindex:lasttrial

Lasttrial

φ=0.59(mm/h) Losses(mm) Surfacerunoff(mm)

3.85 1.50 0.00

3.85 2.25 1.58

3.85 0.90 0.00

3.85 4.43 8.01

3.85 0.83 0.39

3.85 0.60 2.29

3.85 0.60 0.00

3.85 0.30 0.14

3.85 4.58 0.00

3.85 6.53 0.00

3.85 3.23 3.69

3.85 4.65 0.00

3.85 1.28 0.00

3.85 22.35 15.65

3.85 6.38 0.21

Sum 31.96

3.3.2Hydrographseparation

During the process of hydrological analysis, it is sometimes necessary toseparateahydrographfromitscomponents.Thisseparationisnotaneasytaskbecause it is difficult to identify and separate with accuracy the differentcomponents, especially in thecaseof surface runoff and intermediate flow.Toseparate thehydrograph, variousmethodshavebeendevised,which, however,aretoadegreearbitraryandsubjective.

3.3.2.1Methodsofbaseflowseparationfromthetotalhydrograph

Withthesemethods,thehydrographisdividedintodirectflow(surfacerunoff)andbaseflow,andtheseparationprocessisbasedonidentifyingthefallinglimbofthehydrograph,thepointwherethedirectrunoffends.Thedescendinglimbafterthispointonlyrepresentsessentiallythebaseflowcurve.Basedonthis,itisassumed that the baseflow curve has the same slope as a point just below thepeakofthehydrograph.So,if thiscurveisextendedbackwardsfromthepointwherethedirectrunoffends(pointB)toapointbelowthepeak(pointC), thisextension(straightlineBC)canbeassumeddividingthepartofthehydrographfromitspeaktopointCintodirectflowandbaseflowunderneath.ThenpointCis linkedwithastraightlinewiththepointwhichstartsthedirectrunoffat thebeginning of the hydrograph (point A), so the separation is completed. ThisprocedureisgivenschematicallyinFigure3.9andrepresentedbythelineACB.The difficulty in applying this method, similarly to the other two methods

whichfollow,istodeterminethepointwherethedirectrunoffends,i.e.pointB.Onewaytodeterminethisisbasedontheexaminationofseveralhydrographsofthecatchmentareafromwhichthispointcanbeidentifiedastheslopechangingpointofthedescendingcurve.ThedefinitionofpointB isessentiallybasedonthe fact that the base time of the direct flow hydrograph is limited and theamount of the baseflow is not comparatively large.The definition can also beaidedby examining topographic andgeological characteristics of thebasin. Inaddition to these, several empirical relationships for the definition of pointBhavebeenoccasionallyproposed (Linsleyet al.,1982).Oneof thosegivenbyLinsleyetal.(1949)hasthefollowingform:  (3.9)

whereNisthenumberofdaysafterthepeakflowthedirectrunoffstopsAistheareaofthebasininkm2

Another way to separate the hydrograph into two parts is just by connectingpointsAandBwithastraight line, i.e. thepointswhichindicate thebeginningandtheendofdirectrunoff.ThisprocessisshowninFigure3.9bythelineAB.ThethirdmethodofseparationistoextendthecurveatthebeginningofrunofffrompointAuntil itcrossestheverticalfromthepeaktoapointD.Then, thispointisconnectedtopointB.ThisprocessispresentedinFigure3.9bythelineADB.

Figure3.9 Hydrographseparationindirectrunoffandbaseflowwiththreedifferentgraphicalmethods.

Ithas tobenotedthatall these threemethodsare toadegreearbitrary,so it

cannot be distinguished whether or not a certain one is advantageous. Whatshouldbeclarified is thatduringahydrologicalanalysis,allhydrographsmustbeseparatedusingthesamemethod,regardlessofwhichisused.

3.3.2.2Hydrographseparationwiththemethodofthelogarithms

Thedescendinglimb,aspreviouslymentioned,representstherecessionofwaterwhichisstoredonthebasinsurfaceaftertherainstops.Theshapeofthiscurveis independent of fluctuations in the intensity of rainfall and resulting surfacerunoff and soil infiltration, but is dependent almost exclusively on thecharacteristics of the basin and the stream bed (i.e. slope, roughness, cross-sectional shape). Horner and Flynt (1936) and Barnes (1939) found that thedescending limb can be described by the following equation (Papazafiriou,1983):

  (3.10)

whereQ1andQ2arethedischargesattimest1andt2,respectivelyKisaconstant∆tistheintervalbetweenthetimest1andt2

This equation, if drawn on semi-logarithmic paper for constant K, gives astraight line. However, it is observed that the valueK is not constant for theentirelengthofthedescentandthiscurve,ifdrawnonsemi-logarithmicpaper,doesnotrepresentastraightline,butadistinctpointofslopebreakisseen.Thisslope break is attributed to the interruption of direct runoff contribution. ItshouldbenotedthatmanyapproacheshavebeentriedandalmostallleadtotheconclusionthataconstantKisnotsufficienttodescribetheentirecurve.The procedure followed for the separation is, then, the following: First, the

descendinglimbfromthepeakandbelowisplottedonsemi-logarithmicpaperas shown in Figure 3.10. Then, the two straight segments are separated. Thebreakpointiswherethedirectrunoffceasestocontribute,definedinthegraphbytimetD.

Figure3.10 Descendinglimbofhydrographplottedonsemi-logarithmicpaper.

This hydrograph separation method, although strictly empirical, producesmore consistent results and should be preferred to the previously mentionedmethods,whenamoredetailedanalysisisrequired.

Example3.2A rainfall event with intensity I = 12.2 mm/h and duration t = 1 hoccurs in a catchment area of 50 km2. The following values ofdischargesweremeasuredattheexitofthebasin(Table3.4):

1. Plottherainfallgraph(hyetograph)andthefloodhydrograph.2. Separatethebaseflowfromthedirectfloodrunoff.3. Calculatethetotalfloodvolumeatthebasinoutlet.4. Determinethedirectrunoffatthebasinoutlet.

SolutionThehyetographandthefloodhydrographarepresentedinFigures3.11through3.13.In order to separate the baseflow from the surface runoff, the

logarithmofthedischargevalueswithtimeisdrawnandthepointofslopeshiftisdefined(inthisexample,thepointisalsoobviousintheflood hydrograph). The slope changes at time T = 6 h, so it isconsideredthatthistimerepresentstheendofthedirectfloodrunoff.It is considered that the baseflowvaries linearly between the start

point (T = 0 h) and the completion of flood runoff (T = 6 h), thuscalculatingthesetofbaseflowvaluesQb.ThefloodrunoffQfwillbe

thedifferenceofthebaseflowfromthetotalrunoff,asshowninTable3.5.The total floodvolume isderivedbycalculating theareaenclosed

bythecurveofthetotalflowandthetimeaxis.Inthiscase,itcanbecalculated by the trapezoid method approximately equal to 871,200m3.Byremovingthevolumeofbaseflow,thefloodvolumeisderivedanditisequalto588,960m3(Table3.5).

Table3.4Dischargesatthebasinoutlet

Figure3.11 Hyetographoftherainfallevent.

Figure3.12 Floodhydrographoftheobservedrunoff.

Figure3.13 Separationofbaseflowfromthefloodrunoff.

Table3.5Calculationofthefloodrunoff

3.3.3Compositehydrographseparation

Themethodspresentedearlierinvolvetheseparationofhydrographswhicharecausedbyindividualrains,i.e.hydrographshavingapeakandthecorrespondingascendinganddescendingcurves.However,duringarainfallevent,itispossibletohavesomeintervalswithnorain.Thisresultsinacompositehydrograph,i.e.a hydrographwith two ormore peaks and intermediate incomplete ascending

and descending curves. Such a hydrograph, caused by two successive rainfallevents,isshowninFigure3.14.The separation of a composite hydrograph into direct hydrograph and

baseflowismorecomplexthaninthesimplecase;first,thedurationofthedirectrunoffafterthepeakshouldbecalculated.Thistimeperiodcanbeestimatedbyone of the methods mentioned, among which is Equation 3.9. A generalizedseparation process in such hydrographs is meaningless, because each case isunique. Instead of this,we present the separation of the hydrograph shown inFigure3.14.After the estimationof the timeN from thepeakuntil the endof thedirect

runoff,which is expressed byEquation3.9, the descending limbAC after thefirstpeakextendstothepointDwhichislocatedNtimefromthepeakA.Thisway, the composite hydrograph is analyzed in two simple oneswhich can beseparated by any of the aforementioned methods, whereas there will be anoverlapintimebetweenthetwopeaks.Forthisperiod,theseparationwouldbeappropriate as best fits. In the case of Figure 3.14, the separation of the twosimplehydrographsismadeusingthethirdmethodpreviouslymentioned,andasa dividing line between the direct runoff and the baseflow of the compositehydrographistheIHDKFline.Insteadof thismethod,anyothermethodcouldbeusedfromthosewehave

mentionedwithbetterfit;e.g.theonebasedonplottingthedescendinglimbonsemi-logarithmic paper. In this case, the descending limb BE is plotted on alogarithmic paper and the pointF, which is highlighted, indicates the end ofdirect runoff(this is thesecondfromthesideofpeakBpointofslopeshiftofsemi-logarithmic line).With this procedure, both the shape of the descendinglimbandthetimeN’indaysthatthedirectrunofflasts(timebetweenpointsBandF)aredetermined,withoutusingEquation3.9.Then,thedescendinglimbisdrawn after the first peak A, based on the form found by analyzing the BEsegmentof the curve, thereby resulting in two simplehydrographs.Fromhereon,theprocedureofthepreviousparagraphfollows.

Figure3.14 Compositehydrographseparationintodirectflowandbaseflow.

3.3.4Factorsinfluencingthehydrographshape

The time distribution of runoff, which is expressed by the shape of thehydrograph, is affected by climatic factors as well as topographical andgeological features of the basin (Papazafiriou, 1983).Generally, it canbe saidthattheascendinglimbofhydrographisformedprimarilybythecharacteristicsofraincausingrunoff,whilethedescendinglimbisalmostindependentofraincharacteristics.

3.3.4.1Climaticfactors

Themainclimaticfactorsaffectingthevolumeanddistributionofrunoff,i.e.theshapeofthehydrograph,are

1. Theintensity,durationandtimedistributionofrainfall2. Therainfalldistributionoverthecatchment3. Thedirectionwhichtherainmoves4. Theformoftheprecipitation5. Thetypeofrain

3.3.4.1.1Intensity,durationandtimedistributionofrainfall

Thesethreecharacteristicsofraindefinethevolumeoftherunoff,thedurationofthesurfacerunoffandtheheightofthehydrographpeak.In the case of rain with steady intensity, the duration determines, to some

extent,themagnitudeofthepeakandthedurationofsurfacerunoff.Foragivenrainfalleventduration,anincreaseintheintensitywouldincreasethepeakandthevolumeofrunoff,provided,ofcourse,thattherainfallintensityexceedstheinfiltration capacity of the soil. The effect of such an increase in the intensitydoes not affect the duration of the surface runoff. Finally, variations in theintensityofraincanaffecttheshapeofthehydrographofsmallcatchmentareas,whileinlargerbasins,thiseffectisverylimited.Generally, a rain of high intensity and short duration causes high peak,

ascending and descending hydrograph limbs with relatively steep slopes andlimiteddurationofsurfacerunoff(shortertimebaseofhydrograph).Therainofthe same total depth but with low intensity and long duration causes lowhydrographpeak,ascendinganddescendinglimbswithsmallslopesandlongerdurationofsurfacerunoff.ThesecasesareshowngraphicallyinFigure3.15.

3.3.4.1.2RaindistributioninthecatchmentIf the catchment area is large enough, thedistributionof a rain that falls on itmaynotbeuniform.Inotherwords,morerainmayfallinonepartofthebasinwhencomparedtoanotherpart.Ifthishappens,thereisanimpactontheshapeofthehydrograph.Specifically,ifmostoftherainfallfallsatthepartofthebasinwhichisnear

its exit, the hydrographwill normally have a strong peak and steep ascendinganddescendinglimbs,andifitfallsatadistantpartofthebasin,thenthepeakofthe hydrographwould be lower andmore flattened.These cases are shown inFigure3.16.

3.3.4.1.3RainfalldirectionThedirectioninwhichtherainmovesinrelationtothemainorientationofthenetworkofstreamsofthehydrologicalbasincanalsoaffectthemagnitudeofthepeak of the hydrograph and the duration of surface runoff. The effect of thisfactorismoreintenseinelongatedbasins.Insuchbasins,ifthedirectionoftherainisfromtheoutlettotheupperpart,thehydrographwillhavealowerpeakandalongerdurationofsurfacerunoff,comparedtowhentheraindirectionisfromthemostdistantendofthebasintotheoutlet.Thesecasesarequalitatively

showninFigure3.17.

Figure 3.15 Hydrographs caused by rains of equal depths but of different intensities and durations. (a)Shortduration,highintensityand(b)longduration,lowintensity.

Figure3.16 Effectoftheraindistributionontheshapeofthehydrograph.(a)Mostoftherainfallhasfallenneartheoutletofthebasinand(b)mostoftherainfallhasfallenawayfromtheoutletofthebasin.

Figure 3.17 Effect of the rainfall direction on the shape of hydrograph. (a) Rainfall direction from theoutlettotheupperpartand(b)rainfalldirectionfromthemostdistantendofthebasintotheoutlet.

3.3.4.1.4FormsofprecipitationUnliketherainfall,whichhasadirecteffectonthehydrographshape,theeffectofsnowfallisnotrelatedtotheamountandthedurationbuttothetimeandrateofsnowmeltinglayercoveringthecatchment.Therunoffcausedbysnowmeltisarelativelyslowprocess.Thisisduetotheunevendistributionofthesnowlayerthicknessoverthebasinandthedailytemperaturevariationswhichresultintheaveragerateofmelting,whichcannot,inmostcases,exceedthesoilinfiltrationrate.Thus,inthesecases,mostdirectflowisderivedfromintermediaterunoff,resulting in hydrographs having low peak but very long duration. These, ofcourse,are relevant tobasinswhichhave formedasnow layerand thenormalmeltingperiod is in spring. If the snow fallsoccasionallyona riverbasinandmeltsratherquickly,theshapeofthehydrographissimilartothatcausedbytherainfall.Asfor its final form, itwillbea functionofsnowpack thicknessandmeltingrate.Finally,insomecases,thesnowmeltingisduetothefallofwarm

rain,causinghighandfairlyprolongedrunoff.Suchincidentssometimescauseveryseriousflooding.

3.3.4.1.5TypeofrainfallThetypeofrainfallplaysamajorroleinshapingthepeakofthehydrographinrelation to the size of the catchment area. Rainswhich are caused by intenseupdrafts (spring and summer storms) produce the highest flow peaks in smallbasins,while their impact isnotsignificant in largebasinsbecause theupwardrainfalls are limited in extent. On the contrary, rains caused by widespreadfrontal systems combined with orographic factors give high and prolongedpeaks.

3.3.4.2Topographicfactors

Thedirectrunoffhydrographofacatchmentreflectsthecombinedeffectofallphysical characteristics of the basin. Some of the main features which exertsubstantialinfluenceonshapingthedirectrunoffofarivercanbethefollowing:

1. Thesizeandshapeofthebasin2. Thedistributionanddensityofstreams3. Theslopeofthebanksofthebasin4. Theslopeofthemainwatercourses5. Theterrain6. Thepercentageandtypeofvegetationcover

3.3.4.2.1SizeandshapeofthebasinThemaineffectofthebasinsizeontheshapeofthedirectrunoffhydrographisrelated to the duration of the hydrograph. So, regardless of other factors, aparticularrainfallingona largebasinwillgivehydrographof longerduration,whiletherunoffperunitareaatitspeakwillbesmaller.The shape of the basinmodulates the rate atwhichwater reaches themain

streams, and it is an important factorwhich affects the ascending limb of thehydrographanditspeak.Thus,ifthebasinshapeissuchthatthestreamshaveashortlengthandconvergetowardstheoutlet,theascendinglimboftheproducedhydrographwillbesteeperand thepeakwillbehigher,compared towhen themajorpartofthebasinisrelativelydistantfromtheexit.Inthesecondcase,the

basinsdohaveanoblongshape.TheeffectofthebasinshapeondirectrunoffisgivenqualitativelyinFigure3.18.

3.3.4.2.2DistributionanddensityofstreamsThedenserandmoreuniformthenetworkofthestreamsinabasin,thesmallerthewaterpathsoverthebasinsurfaceandtheshorterthetimethewatertakestoreachthebasinoutlet(timeofconcentration).Theoppositejustholdsinthecaseof a basin with sparse network of streams. Therefore, in the first case, theascending limb of the hydrograph would be steeper, the peak higher and,generally, the direct runoff greater because the ponding time of water on thegroundisshorter,andhence,lesswaterinfiltrates.

3.3.4.2.3SlopeofthebanksofthecatchmentTheslopeof thebasincarriesasubstantial impactonsurfacerunoffbecauseitdeterminesthecontacttimeoftherainwatertothegroundsurface,thusaffectinginfiltration, and also it is the most important factor affecting the time ofconcentration since it regulates the water velocity over the surface. Accuratecalculation of this velocity cannot be achieved because of the many factorswhichinfluenceit.Generally,itcanbesaidthattheflowofwateronthesurfacecanbeexpressedbyanequation suchas theone formulatedbyButler (1957).Thisequationisthefollowing(Papazafiriou,1983):

  (3.11)

whereqistheflow(discharge)perunitwidthofareaDistheflowdeptha, b, and c are the constants which depend on the Reynolds number, thedistributionofraindropsandtheroughnessofthesoilsurface

Sisthechannelslope

Figure 3.18 Hydrographs produced by the same rainfall in two basinswith the same area but differentshapes.

This equation shows that the flow rate increases according to the slope. Thegreatertheslope,thesmalleristhetimebaseofthehydrograph,thegreaterthepeak,andthesteepertheascendinganddescendinglimbsofthehydrograph.

3.3.4.2.4SlopeofthemainwatercourseWhenthewater reaches themainstreamofabasin, the timerequired toreachtheoutletdependsonthelengthandtheslopeofthestream.Generally,theflowvelocity in open channels can be calculated by an equation of the followingform:  (3.12)

whereVistheflowvelocityCisaparameterdescribingtheroughnessofthechannelwallsRisthehydraulicradiusSisthechannelslopemandnaretheexponentsoftheRandS

When the Manning formula is used, m and n have the values 2/3 and 1/2,respectively.ThetimetrequiredforwatertotravelapathoflengthLis

  (3.13)

Thus,thetraveltimeisinverselyproportionaltotheslopeofthestream,i.e.thistimedecreaseswithincreasingslope.Theeffectofthisfactoronthehydrographis associated with the time of the peak, the time base of hydrograph and theshapeofthedescendinglimbwhichbecomessteeperastheslopeincreases.

3.3.4.2.5TerrainofthecatchmentareaThe rainfall which falls on the ground surface, before moving to the surfacestreams,mustfirstfillanykindofterritorialirregularities.Itis,therefore,naturalthat the number and the form of soil cavities, together with the slope of thebasin,willaffecttheshapeofthehydrograph.Inbrief,itcanbesaidthatbasinswithlittlesoilcavitiesandlargebankslopesderivehydrographswithshorttimebases,steepascendinganddescendinglimbsandhighpeaks.Incontrast,basinswith a large number of soil cavities give hydrographswith lower peaks and alongtimebases.

3.3.4.2.6PercentageandspeciesofvegetationcoverThe vegetation affects the runoff in twoways: one is direct and refers to theretention of rainwater from vegetation, which then evaporates back into theatmosphere,andtheotherisduetotranspiration.Catchmentareas,whichhavealarge percentage of vegetation coverage, apart from trees with dense foliage,giveverysmoothhydrographswithlongtimebasesandlowpeaks.Inaddition,thevegetation increases thebasinandchannel roughness, retards the flowandincreasestimeofconcentrationandinfiltration.

3.3.4.3Geologicalfactors

Geological factors affecting the shape of the hydrograph are those whichregulate the flow of water to the streams, either in the form of intermediaterunofforintheformofgroundwaterflow(baseflow).Thesefactorsrelatetothefollowingcharacteristics:

1. Surfacesoillayers2. Underlyinggeologicalformations

3.3.4.3.1SurfacesoillayersThegroundsurfacecondition, incombinationwith the textureandstructureoftheunderlyinglayers,determinestheinfiltrationrateofwater,directlyaffectingthesurfacerunoffvolume.Also,thelayoutofthesoillayersaffectstheverticaland lateral rate of movement of water which has infiltrated into the ground.Thus, if the surface soil is sufficiently permeable but it is underlain by animpermeable layer, a significant percentage of infiltratedwaterwill bemovedlaterally, thus making the intermediate runoff an important factor in the finalshape of the hydrograph. The hydrograph will have a low peak and a longduration.Ifthesurfacelayerofthebasinconsistsoflowpermeabilitysoils,mostoftherainfallwillbecomesurfacerunoffcollectedinsurfacestreams,resultinginahydrographwithahighpeakanda short timebase.Finally, if all the soillayers are relatively permeable, most of the water will be deeply infiltrated,resultinginlowpeakandgenerallylowvolumeofdirectrunoff.

3.3.4.3.2DeepergeologicalformationsThe type and the arrangement of geological formations beneath a river basinaffectthebaseflow.Dependingonthelayoutoftheseformations,theareawhichcontributeswatertothestreamsofabasincanbegreaterorlessthanthesurfaceareaofthebasin,asdefinedbythewatershed.Moreover,thislayoutcancontrolthelocationofthegroundwater-levelinsuchawaysoastoprovidethestreamscontinuously with water, in which case the baseflow is continuous andsignificant;inothercases,thegroundwaterispermanentlybelowthestreambed;insuchcases,thestreamenriches(recharges)theundergroundlayerswithwater,andthebaseflowisminimalordoesnotoccur.So far, the influence of the various factors on the hydrograph shape was

examined separately. Innature,however, it is common forone factor to eithercanceltheeffectsofanotherfactor,orfactorstohaveasynergeticeffect,i.e.thefinalhydrographmaybedependentonthecumulativeeffectofmanyfactorsastheyactindividuallyorincombination.

3.4HYDROMETRY

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The purpose of hydrometry is the measurement and the evaluation of flowparameters, i.e.water-level,velocityandriverdischarge.Hydrometry isaverycomplex and costly process, which requires skilled personnel for both thefieldwork and the storing and processing of collected data at the office. Thefieldwork consists mainly of measuring the water-level and performing watermeasurements inorder to estimate the flowdischargeof the river.Thesedata,after verification and required processing, are stored in the archives of theresponsible regulatory ormonitoring authorities (e.g. theHydroelectric PowerCorporation, theMinistryof theWaterResourcesand theEnvironment)whichhaveestablishedthemonitoringlocations.Thedataarefinallyprocessedinorderto estimate themaximum or average discharge of the river at a specific timeperiod.

3.4.1Installationcriteriaforahydrometricstation

A fully equipped hydrometric station should be placed at an appropriate crosssection of the watercourse. It includes instruments used for water-level anddischargemeasurement.Oneinstrumentneededinsuchstationisthewater-levelmeter. Often, the station has more than one water-level meters. Appropriateplacestoinstallthesemetersarethevariousstructuresthatcrossthestream,i.e.bridges.Theseshouldbecheckedfrequentlyinordertoobserveanymovementor deviation from the correct position. In places where we are interested incontinuousmonitoringof the level, the station includes awater-level recorder,whichcontinuouslymonitorsthevariationofthelevelintime,thusallowingfora detailed evaluation of the temporal variation. In addition, the station mayinclude hydrometric instruments to measure river flow velocity for theassessment of river discharge, including both regular and rare events, such asfloods.The following are the criteria of a site for the installation of a hydrometricstation:

Theentireflowshouldbeinasinglebranch.Theriverbedshouldhaverelativelystableanduniformgeometry.The river reach should be relatively straight, which helps avoiding theinfluenceoftheflowfromdownstreamobstacles.Thereshouldbereducedpresenceofchannelerosionordeposition.The change in water-level should have a good relation to the dischargechange.The station should be easily accessible, even during floods, in order tomaintainitandmostlycarryoutthemeasurements.

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Itshouldbefavourablesitefortheconstructionofreservoirs,bridges,etc.According to the recommendations of theWorldMeteorologicalOrganization,the minimum density of hydrometric stations in the Mediterranean countriesshouldbe:

Inlowland/plainregions,1stationper1000–2500km2

Inmountainousregions,1stationper300–1000km2

Thehydrological stations should be distributed in altitude zoneswith a heightdifferenceofabout500mbyzone.

3.4.2Measurementofwater-level

The most common measuring instruments of the water-level, as previouslymentioned,arethelevelmetersandrecorders.Thewater-levelmeterisasinglevertical rulerwith an imprinted scale in centimetres. Its zero refers to a fixedreferencealtitude.Incaseswhenthetotalflowoftherivercannotbedescribedbyasinglewater-levelmeter, thenmoremetersare installed in thesamecrosssection(right,left,middle)orevenupstreamanddownstream.Thewater-levelmeasurement isdonemanuallybyanobservereverydayat

8:00 a.m.,while in the case of floods, the levelmeasurement is usually donemore frequently and during the flood event. The use of thewater-levelmeterentails the risk that significant level changes which may occur in the timeintervalbetweentwoconsecutiveobservationsmaynotberecorded.Moreover,inmanycases, it isnecessary tomakeobservationsatpositionswhicharenoteasilyaccessiblebytheobserver.Toovercomethesedifficulties,autorecordinginstruments, calledwater-level recorders, are used.A typical arrangement of awater-levelrecorderconsistsofthefollowingcomponents(Papazafiriou,1983),asshowninFigure3.19:

A vertical stilling well of circular or rectangular cross section made ofsteelorreinforcedconcrete,whosebottomissetatleast50cmbelowthelowerpointof the streambedandwhich is locatednear thebankof thestream.Two or three tubes of relatively small diameter, horizontally levelled,connectingthestreamwiththewell.Thesetubeshaveasmalldiameterinorder to damp any variations and disturbances of the streamwater-leveldue to surface waves and/or turbulence. A practical design rule is thatwhenthelengthofthetubesislessthan5m,theratioofthetubediameterto thewelldiameter is1:50,and for lengthsexceeding5m, this ratio is1:25.Since the pipes may be clogged by sediments, a purification system is

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usuallynecessary,whichconsistsofasmallwaterreservoirlocatedabovethegroundsurface,whichisconnectedbypipingtothehorizontaltubes.The reservoir, from time to time, is flushing water into the tubes. Thisprocedureisusuallysufficientforcleaning.The water-level recording mechanism is placed in the well outside themaximum expected water and consists of a float in the well suspendedthrough a cable and connected to a counterweight.This system, using asuitablestructure,transferslevelchangesintothesystemlogger.Anothertype of awater-level recorder has been used, instead of a rotating drumwhichhasaperforatingmechanismwhichtransfersthelevelchangesinaspecific tape, for direct data transfer to a computer. However, these areold-typerecorders; todaytheloggerstoresmeasurementsinmemoryandtransfersthemtoapersonalcomputer.

Figure3.19 Typicalwater-levelrecorderinastillingwell.

Atypicalinstallationofawater-levelrecorderisshowninFigure3.19.

3.4.3Dischargemeasurementbythemethodofvelocityfield

To estimate the discharge of a stream, several methods are used, such asmeasurementbyflowmeterofthevelocityfield;thedilutionmethod;floats;orestimates using empirical relationships.A very usualmethod is themethod ofvelocity field, using a current meter. Note that the velocity profile in a river

section is not uniform, but thegreater velocity value appears at themaximumdepth and the zero velocity at the cross-section solid boundary, as shown inFigure 3.20. The higher velocity is noted just below the water surface. Thevertical velocity profile at the location ofmaximumdepth is shown in Figure3.21.Tocalculatethedischargeataparticularlocationoftheriverwiththemethod

of thefieldofvelocities, it isabsolutelyessential toknowthegeometryof thewetsection.

Figure3.20 Velocitiesdistributionprofileinariversection.

Figure3.21 Verticalvelocityprofileatthemaximumdepthofthecrosssection.

Figure3.22 Thecrosssectionofariver.

This is determinedbydividing the examined cross section intoN subsections,usually one every 1–2m, but depending on the width of the section and thedesired accuracy, as shown in Figure 3.22. An empirical rule is applied toseparatetheriverintosubsectionsbydividingitintopartswhichallownomorethan10%ofthetotaldischargeineachofthem.Todefinethegeometryofeachsubsectioni,theheightofitstwoverticalsides

ismeasured (di-1 and di), as well as the distance between them, which is thedifferenceofthehorizontaldistanceoftheverticals(bi‒bi-1),fromafixedpointusedtoassessthemeasurementsononebankshore.Withthehelpofacurrentmeter, themeanvelocityvi isestimated foreachsubsection i, vertically in themiddleof the interval (bi‒bi-1).Thecurrentmeterusually isequippedwithapropeller,andwhenimmersedinthecrosssectionofthestreamatthatpoint,itrotatesundertheinfluenceoftheflow(Figure3.23).Today,therearecommonlyusedelectromagneticcurrentmeterswhichhavenopropeller,buttheiroperationisbasedonthealterationbytheflowofanelectromagneticfieldaroundthetipoftheprobe.

Figure3.23 Typicalconfigurationofacurrentmeter.

Theaccessandthemeasurementofthevelocitycanbedoneinvariouswaysdepending on the flow conditions. In shallow streams of low flow rate, theprocess is done with wading and the lifting of the current meter is donemanually,whileindeepfloatingstreams,aproperlyequippedboatcanbeusedorthecurrentmetercanbesuspendedfromabridge.Incaseswherelargeflowvelocitiesaredeveloped,anoverheadwiringwithpulleysmustbepermanentlyinstalled.Thisallowstheaccesstoanypointofthesectionfromthebankofthestream. The necessary condition for the results to be valid is the correctorientation,horizontalandparalleltotheflowdirection,oftheaxisofrotationofthecurrentmeter.Forthisreason,currentmetersareequippedwithfinstoself-aligntotheflowdirection,whileaweightissuspendedbelowthecurrentmetertokeepitinplace,resistingtheflowpressureforce.Thevelocity at eachpointof thevertical is a linear functionof the rotation

frequencyofthepropellerat thatpoint(i.e. theangularvelocityoftherotatingpropelleristransformedtolinearvelocityoftheflow).A question which arises at this point is on the proper depth to which the

current meter should be immersed so that the measured point velocity, ui, isrepresentative of the velocity of the particular vertical and thus of thecorresponding part and subsection, since it is known that the velocity varieswithinthesection.Accordingtothelogarithmiclaw,itcanbecomputedthatthemeasuredpointvelocityismorerepresentativeoftheaverage,ifmeasurementismade at a distance from the surface equal to 60% of the flow depth at theparticular subsection. Therefore, if only one measurement is made in thesubsection, thecurrentmeter shouldbesubmerged toadepthequal to60%ofthe total depth from the surface. In practice, this is donewhen the subsectionflow depth is less than 0.6 m. For larger depths, more than one velocity

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measurementsarerequired,usuallytwo(fordepthsupto3m).Thesearetakenatdistancesfromthesurfaceequalto20%and80%ofthedepth,respectively.Inthiscase,theaveragevelocityineachvertical(andhenceineachsubsection)isestimated adequately by the average of these two values, as shown in thefollowingequation:

  (3.14)

Whenthewaterdepthismorethan3m,threevelocitypointmeasurementsareperformedtoaccuratelyapproximatethemeansubsectionvelocity.Theseareatdistances from the surface equal to 20%, 60% and 80%of the depth, and theaveragevelocityisgivenbythefollowingequation:

  (3.15)

Then,ameasurementdatasheetisfilledonsitebytheobserverwiththedataofwater measurement and the total flow of the river, the date and time ofmeasurement, aswell as the corresponding level of the riverwhichmay varyduringthemeasurement.Theaverageflowqiofeverysegmentresultsfromtheequationofcontinuity,

as the product of the average speed i of the segment and the intersectionAi,namely,

  (3.16)

andthetotalflowattheparticularcrosssectionisgivenby

  (3.17)

3.5DISCHARGEESTIMATIONUSINGHYDROMETRICDATA

Toestimatetheaverageflowoftheriveratagivenperiod,watermeasurementsat regular intervals are required (e.g. weekly, fortnightly). However, thefrequency of measurements will never be the required one due to the specialdifficulties and the significant cost. The procedure for the estimation of theaverage flow rate for a shorter time interval (e.g. hourly, weekly) is to use aratingcurve,i.e.acurverelatingdischargetowater-level,whichcanbepreparedbasedonthefollowingsteps:

Generationofaratingcurveatthissectionoftheriver

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Estimationoftheaverageleveloftheriveratthispointforthesametimeinterval(e.g.hourly,daily)ExtensionoftheratingcurveConnectionbetweentheratingcurvesandtheextension

Itisobviousthatinordertoproperlyassesstheaveragelevel(especiallywhenthe time step is less than one day), the station must necessarily have anautomaticwater-levelrecorder.

3.5.1Preparationofaratingcurve

The measurements of discharge and water-level at one station are plotted,formingtheratingcurveatthatposition.Thiscurvehasusuallyparabolicshape,andsometimes, itpresentsvariationsdependingontheshapeandvariabilityofthe section. Inmost streams, the computed rating curvemay be changed overtime,asaresultofthechangeinthecrosssectionandtheslope,duetoerosionor sedimentation of the bed. This fact shows that the curve discharge versusstageisnotpermanent,butitshouldberevisedfromtimetotime.Therefore,astationhasa setofcurves,eachonevalid fora specificperiodof time,whichvariesandmaybefromseveralmonthstoseveralyears.

Figure3.24 Ratingcurvesinnormalchart.(FromNationalDataBankofHydrologicalandMeteorologicalInformation(NDBHMI),Athens,Greece,http://ndbhmi.chi.civil.ntua.gr/.)

Thefirststepforthepreparationoftheratingcurveistotakeandhomogenize

all the measurements collected at different time periods. The next step is togroup themeasurementswhich present the same behaviour and they could berepresentedbythesameratingcurve.Thisisaverytediousprocessthatrequiresconsiderable experience. After grouping, the identification of curves follows(one for each subset). In each time period, the theoretical model describes aspecific curve whose validity is defined by the start and the end of the timeperiod.TypicalratingcurvesforaparticularriverarepresentedinFigure3.24.Foreachgroupofpointsdefinedbytheuser,adifferentmodelforcalculating

each curvemay be used.The rating curve can have various forms; one is thepowerequation:

  (3.18)

whereQisthedischargehisthestagekandbarestation-specificempiricalcoefficientsaisthedistancebetweenzeroaltitudeofwater-level[meters]andaltitudeofzeroflowofthesection

Otherequationswhichcandescribetheratingcurvearepolynomials:

  (3.19)

orsecond-degreepolynomiallogarithms

  (3.20)

whereQisthedischargeHisthewater-levelA0,A1andA2arethecoefficientsofthemodels

3.5.2Extensionoftheratingcurve

A major problem often encountered when using rating curves is thatmeasurements,particularlyathighflows(e.g.floods),aremissing,sotheratingcurve is generatedwithout using such data. So, the question is what to do toestimatethedischargeduringeventsofhighflows.Forthis,theextensionoftherating curve is required, which is usually done using the following Manningequation.ItmustbepointedoutthattheManningequationisvalidonlyinthecaseof

normal flow. For that reason, it only approximates real streamflow conditions

anditsusemustbethoughtful.

  (3.21)

whereQisthedischargeAisthecross-sectionalareaRisthehydraulicradiusJisthefrictionslopenistheManningroughnesscoefficient

To define the constant term J1/2/n, discharge data of the highest watermeasurements are used. This fixed term is calculated by fitting a regressionequation between the discharge and the term (AR2/3), which can be calculatedfromthegeometryoftherivercrosssectionatthepositionoftheratingcurve;obviously, the cross section has to be surveyed. Alternatively, instead of theManningequation,theChézyformulacanbeusedtoextendtheratingcurve:

  (3.22)

whereQisthedischargeCistheChézyroughnesscoefficient

In this case, based on measuring the pairs (Q, AR1/2), the term CJ0.5 can becomputedusinglinearregression.Thewayof linkingbetweentheratingcurveanditsextensionconcerns thewayoffit.Thefitof thecurve isastraight lineconnectingtheedgeofeachcurveofthegroupwithapointoftheextensiontoalwayssatisfytherelationDQ/DH>0.

3.5.3Remarksontheratingcurves

Due to the variability of the curves, systematic discharge measurements arerequired throughout the period the station operates without any interruption.Otherwise, it will not be possible to identify changes in the rating curve.However,theassumptionholdsthatthechangeinthecurveusuallytakesplaceduring a significant flood event (due to the erosive power of water); thiscontributes to the theory that the transitionfromone to thenextcurveconcurswith the highest water-level ever recorded by the water-level meters at thestationintheperiodbetweenthetwoflowmeasurementsfromtwoconsecutivesubsets.

Foragivencrosssectionofariverandacertaintimeperiod,withnochangein profile and characteristics of the channel, there is a uniquematch betweenstageanddischarge.However,amoredetailedstudyof the issuebasedon theprinciplesofhydraulicsshowsthatthisistrueonlyundertheconditionthattheflow is steady. This condition is not valid during flood events, which arecharacterizedbyhighflowvariationsovertime.Inthecaseofunsteadyflow,theratingcurvehasdifferentrisingandrecession

limbs;noneofthemconcurwiththecurveofsteadyuniformflow,astypicallyshowninFigure3.25. It isobserved thatat the initialstageof thefloodevent,thedischargeincreasessignificantlybutthewater-levelincreasesatalowerratecompared to the uniform flow rating curve. The opposite is true during floodrecession,i.e. thedischargedeclinesatafasterratethanthecorrespondingrateofwater-level reduction. Itcan, thus,besaid that the leveldoesnotaccuratelymonitorthechangeintheflow,butasshowninFigure3.25,forthesamevalueofwater-level,theflowisgreaterintherisingphaseoffloodandsmallerintherecession.Usually,whencalculatingthetimeseriesofdischargefromthecorresponding

stagetimeseries,thisphenomenonisignoredduetothecomplexityandlackofdata.So,inthecaseoffloodevents,auniqueratingcurveapplies,whichcreatesa source of error on the estimation of flood hydrographs. The error is moresignificantwhentheslopeofthebasinismild.

Figure3.25 Loopratingcurveinunsteadyflowconditions.

3.5.4Estimationofanaveragewater-levelforaspecifictimestep

An importantpartof theprocessingof the recordedwater-level information isrelatedtotheconsolidationofthemeasurementsfromthewater-levelmeterandthewater-level recorder, ifboth instrumentsexistat thehydrometric station. Ithas been observed that simultaneous measurements from the water-levelrecorderand thewater-levelmetermaybedifferent.Generally, thewater-levelmeterdatacomprise timeseriesatadailystep,recordedonasuitableformbythe observer, while water-level recorder data arise from decoding of the tapewithatimestepofhalfor1h.Normally,simultaneousmeasurementsofwater-levelgaugesandwater-level

recordersshouldbe identical. If thishappened, then thedataof thewater-levelgauge would be practically useless for the periods when both instrumentsoperate in parallel, because the time series of water-level recorders would bemore accurate and of greater time discretization. This is difficult to occurbecausethepositionsofthetwoinstrumentsusuallydifferfromeachotherandthere are involved errors in the measurements. In the case of the water-levelgauge,thereisonlyanerrorinthereadingoftheleveloverthestage,whileinthecaseofwater-levelrecorders,theerrorsourcesaremore(e.g.defectsinthesensor, recording mechanism, clock mechanism, improper positioning of thefilm). For these reasons, the level of the water gauge is considered moreaccurate, while for the time series of water-level recorders, corrections areneededbasedonthemeasurementsofwater-levelgauges.Anotherreasonforthemeasurements of water-level gauges to be considered correct is that in theplottingoftheratingcurve,measurementsofwater-levelgaugesareonlyused.Thecorrectionofwater-level recordermeasurements isasimpleprocess.At

times ti where there are available simultaneous measurements of water-levelgauges and recorders, the measurements of water-level gauges are taken intoaccount, while the interim values available from water-level recorders arecorrectedbylinearequationinfunctionwithtime.Toestimate the average level for a specifiedperiod, theoperationofwater-

level recorders is necessary. Initially, the reduction of the measurements ofwater-level recorders is basedon themeasurements ofwater-levelmeters, andthen the average of thewater-levelmeasurements is obtained. For example, iftherecordingtimestepwas1/2h,thenthedailyaveragelevelisderivedastheaverageof48consecutivevalues,whileiftherecordingtimestepwas1h,thenthedaily average level isderivedas the averageof24values.Toestimate theweeklyormonthlyaveragewater-level,theaveragevalueofthecorrespondingdailyvaluesistakenintoaccount.

3.5.5Flowestimationfrommeasurementsofwater-levelmeters/recorders

After having retrieved and processed all the data from thewater-level gaugesand the water-level recorders and having derived the rating curve groups, thecalculationof discharges follows in time steps similar to the steps used in theestimate of the average water-level (half hour, hourly, daily, etc.). For thispurpose,theappropriateratingcurveisused.Ifho(t)isthelevelrecordedattimetfromthewater-levelgaugeorrecorder,thentheratingcurvegroupthatappliesat time t is used for the conversion of level to a corresponding discharge.However, if at the same time t a flow measurement was carried out (withrecorded level hp (t) and discharge Qp(t)), then the application of the ratingequation will not give the metered dischargeQp(t), but a different discharge,sincethecurvedoesnotpassexactlythroughallthemeasuredpoints(hp,Qp).In order to remove this deficiency, the Stout correction can be used. This

correction isdonebyapplying thereverseprocess, i.e. forall timepointswithflow measurements, the estimated stage is calculated from the rating curve.Then,thedifferencebetweentheestimatedstageandthemeasuredstageofthewater-levelmeteriscalculatedforeachofthesetimepoints.Therefore,aseriesofnon-zerowater-leveldifferences∆h isdeveloped,and it is assumed that thesize∆hbetweentimestiandti+1varieslinearly.

Figure3.26 Estimationofdischargetimeserieswithastepofhalfanhourwithandwithoutcorrectionof

the level. (From National Data Bank of Hydrological and Meteorological Information(NDBHMI),Athens,Greece,http://ndbhmi.chi.civil.ntua.gr/.)

Havingcalculatednowthelevelcorrectionsateachtimet,forwhichthereisastagemeasurement,thecorrectedlevelserieshd(hd=hs—∆h) is formed,andthus, when applying the rating curve, the corrected series of dischargeestimationsisacquired.Figure3.26showsthedischargetimeseriesinhalf-hourtimestepswithandwithoutcorrectionofthelevel(Stoutcorrection).Inordertoestimate the average daily discharge, the level is converted in high temporalresolution (e.g. every half hour) to corresponding discharges, and then again,these values are converted into daily, weekly, monthly, yearly, etc. Thecomputing process of assessing the discharge with correction and withoutcorrectionofthelevelisgiveninthefollowingexample.

Example3.3

Inahydrometric station, systematicmeasurementsof river stageanddischargehavebeenmade,asshowninTable3.6,foraperiodwithoutanysignificantchangesinthegeometryandthecharacteristicsof theriver section.Based on thesemeasurements, plot the rating curve oftheriverfortherelevantperiod.UsingthisratingcurveandthedataofmeasuredflowandaveragedailylevelsfortheperiodshowninTable3.7,calculate theaveragedailydischargeseriesof thestation for theaforementionedperiod.

Solution

SettinguptheratingcurveToplottheratingcurvefromthedataofthewatergauge,itisessentialto initially calculate the logarithms of the values of stage anddischarge,asshowninTable3.8.Thisisdonebecausethedischargeisrelatedtothelevelthroughanequationoftheformof

Table3.6Measurementsofwater-leveldischarge

A\A Date Stage(m) Discharge(m3/s)

1 20/3/1984 6.50 670.20

2 29/4/1984 5.70 510.30

3 27/5/1984 5.10 420.20

4 23/6/1984 4.60 370.70

5 20/7/1984 3.90 286.45

6 25/8/1984 3.70 278.20

7 20/9/1984 4.00 302.30

8 15/10/1984 4.42 319.79

9 13/11/1984 5.33 455.63

10 29/11/1984 6.10 582.98

11 23/12/1984 6.46 650.90

12 12/1/1985 7.19 820.70

13 30/1/1985 7.50 905.60

14 27/2/1985 7.74 984.84

IS 4/3/1985 8.05 1098.04

16 10/3/1985 8.47 1296.04

17 28/4/1985 5.20 440.20

Table3.7Water-levelmeasurements

A\A Date Stage(m)

1 27/2/1985 7.85

2 28/2/1985 7.89

3 1/3/1985 7.94

4 2/3/1985 7.99

5 3/3/1985 8.06

6 4/3/1985 8.09

7 5/3/1985 8.16

8 6/3/1985 8.23

9 7/3/1985 8.29

10 8/3/1985 8.32

11 9/3/1985 8.38

12 10/3/1985 8.41

Equation3.18,i.e.Q=A(h—ho)n;thus,thelogarithmsofQandharelinkedthroughalinearequation.Linear regression is used to estimate the coefficients between the

logarithms of the discharge (y) and of stage (x), resulting in thefollowingequation:

EstimationofaveragedailydischargeThe estimation of daily dischargeswill bemadewith the use of theratingcurve.Ifaneffortismadetoapplytheequationofthepreviousquestion, it is observed that for 3 days, water gauging data areavailable(27/2,4/3and10/3)andtheresultingdischargesdifferfromthe corresponding measured ones. To avoid this phenomenon, theStoutcorrectionisapplied.Withtheuseofthismethod,themeasuredlevels will be corrected, and when using the rating curve, thecalculateddischarge for thedays thatmeasurementsexistwillbe thesameas thatof themeasuredones.Theprocedure is shown inTable3.9.

Table3.8Processofcalculatingthestage-dischargecurve

Table3.9Stoutcorrection

InTable3.9,inthecolumnofwatermetering,themeasuredvaluesofstage and discharge for the three dateswith availablemeasurementsareplaced.Qest is thedischargederivedbyapplying theratingcurveforthedataofthewater-levelgauge.hestisthelevelresultingforthemeasureddischarge,fromtheratingcurve,i.e.,

Δh is the difference between the measured stage and the hest. Inplaces where there are no values of best, Δh is obtained by linearinterpolation between the known values. hcor is calculated bycomputingthedifferenceofstageandΔh (in thiscase, thevaluesarenegative and added).Finally,Qcor is obtained by applying the ratingcurveforalevelequaltohcor.Itisobservedthatforthethreedatesforwhichdischargedataareavailable,thisisalmostequaltothecorrectedcalculatedvalue.

3.6RAINFALL-RUNOFFRELATIONSHIPS:EMPIRICALMETHODS

Oneofthemostimportantconcernsoffloodhydrographanalysisistoestimatethe (flood) peak flow. The early methods investigated for this purpose wereempirical, but now more sophisticated methods are increasingly used mostlyrelatedtothetheoryofunithydrograph(UH)andfrequencyanalysis.Otherkeyelements in theanalysisof rainfall-runoffare theflooddischargevolumefora

singleeventorforaspecifiedperiodoftimeandtherisetimeandthedurationofthefloodhydrograph.Thechoiceofthemethodtouseforthisanalysisineachcaseisbasedoncertaincriteria(Papazafiriou,1983):Theavailable data: If there is a long-term hydrological observation record,statisticalanalysisproceduresmaybeused,butthesuccessofsuchmethodsislimitedwhenthereislackofavailabledata.

Thebasinareaandothercharacteristics:Thesecharacteristicsformtheshapeof the hydrograph in general and in particular the size and the time ofoccurrence of the peak. This part analyzes some representative empiricalmethodsoffloodpeakassessment.

Theintendedpurpose:Thepurposeistodesignahydraulicprojectbasedonfloodpeakortotalfloodvolume.Thisparagraphreferstotheestimationoffloodpeaksbyusingempiricalmethodsdescribedinthefollowing.

3.6.1Rationalmethodforestimatingfloodpeaks

Thismethodisusedtoestimatetherunoffpeakofrelativelysmallbasins(<35km2). It is based on the principle that, for a rain with uniform intensity anddistributionoverthearea, themaximumrunoffoccurswhenthewaterfromallpartsofthebasinreachestheoutlet(seealsoChapter7).Therunoffconstitutesacertainpercentageoftherainintensitywhichproducesit.Therationalmethodisexpressedbythefollowingequation:

(3.23)

whereQpisthepeakrunoffinm3/sCisadimensionlessparameterknownastherunoffcoefficientIistherainfallintensityinmm/hAistheareaofthecatchmentinkm2

For the proper use of the method, it is necessary to clarify the limits of itsapplication. At first, the method requires the rainfall intensity and spatialdistributiontobeuniformthroughoutthecatchment.Thisconditionisrarelymetinnatureandthisassumptionmayonlybevalidforsmallcatchments.Anotherfactor which should be taken into account is the estimation of the rainfallintensity.Toget themaximumpeak, thewatermustreachtheexitof thebasinfromallpoints,i.e.therainfalldurationwithuniformintensityshouldbeatleast

equaltothetimeofconcentrationtcofthebasin.Thissuggeststhatthemethodcannot be applied for rainfall duration less than tc. But if the rain duration isgreaterthantc,itcannotbeusedintheequation.Accordingto theprocess, thepeakrunoffperunitareaofabasincausedby

therainofuniformintensityandofunlimiteddurationwillbe

(3.24)

This suggests that the runoff factor C represents the ratio qp. The runoffcoefficientCshouldbeselectedbasedonthefollowingfactors:(1)thelanduseofthebasin,(2)theextentanddensityofvegetationcover,(3)theslopeofthebasinand(4)themoisturecontentofthesoilduringthestartoftherain.Thesefactors suggest that the runoff coefficient is not constant even in the samecatchment area, as it is a function of themoisture content of the soil and therainfall intensity. Since these factors are difficult to evaluate, C is usuallyselected from tables taking into account all other factors, apart from the two.Such values given by theU.S.ArmyCorps of Engineers (USACE; 1948) arelistedinTable3.10(seealsoTable7.3).Asalreadymentionedearlier, the rationalmethodcanbeappliedfor rainfall

eventswhoseduration is equal to or greater than the concentration time. It is,therefore,necessarytohavewaystocalculatethistime.Inthecaseofhydrologicalbasins,wheretheriverpathlengthsarerelatively

large and the surfaces are uneven, several empirical relationships have beendevised for the calculation of the concentration time. The following are theindicative equations devised by Kirpich (1940), Giandotti and the U.S. SoilConservationService(SCS):TheKirpichequationis

(3.25)

wheretcistheconcentrationtime(min)Listhemaximumpathlengthofwateroverthecatchment(m)S is the slopeequal to the ratioH/L,whereH is the difference between thehighestpointofthebasinandtheoutlet

Table3.10IndicativevaluesofthecoefficientC(rainfallforthereturnperiodof5–10years)

Typeofbasin RunoffcoefficientC

Permeablesoiltypes(sandy) 0.10–0.20a

Mediumpermeabilitysoiltypes 0.30–0.40a

Lowpermeabilitysoiltypes 0.40–0.50a

Urbanareas 0.70–0.90a

Industrialareas 0.50–0.90a

Forests 0.10–0.25

Roads(asphalt,concrete) 0.70–0.95

Rooftops 0.75–0.95

aThelowvaluesconcernwoodlands,whilethehighestvaluesconcernruralareas.

TheGiandottiequationis

(3.26)

whereAisthecatchmentareainkm2

ListhedistanceofthemainstreamtotheoutletofthebasininkmH is the difference between the average elevation of the basin from theelevationattheoutletofthebasininm

TheSCSequationis

(3.27)

wheretcisthetimeofconcentrationofthebasininhListhelengthofthemainstreaminmH is theelevationdifferencebetween thefarthestpointand theoutletof thebasininm

Example3.4A bridge is planned to be constructed at the outlet of a basin, andtherefore,itisnecessarytoestimatethemaximumpeakflood.Atthis

location,thereisnowatergauge.Thebasinupstreamofthislocationis62 km2, themaximum length of themain stream is 10 km, and thedistance from the basin outlet to the nearest place in the river basincentroidis3.75km.Ithasalsobeenestimatedthattheaveragealtitudeof the basin is 252 m, the basin outlet altitude is 121 m, and thealtitude at the farthest point of the basin is 310m.The basin runoffcoefficientisequalto0.2.Calculate the maximum flood peak making use of the rational

methodwhentheconcentrationtimeiscalculatedbytheGiandottiandSCSmethods.TheidfcurveofthebasinforareturnperiodofT=20yearsish=28.6t0.416,wherehinmmandtinhSolution1.GiandottiTheconcentrationtimeisgivenby

whereHav andHout are the average basin and outlet altitude in m,respectively.Therainfallresultsfromtheidfcurves

andthecorrespondingintensitywillbe

Thepeakdischargeiscalculated,usingtherationalmethod:

2.SCSTheconcentrationtimeisgivenby

whereListhelengthofthemainstreaminfeetH is thealtitudedifferenceof thefarthestpointof thebasinand

theelevationattheoutletofthebasin

Fromtheidfcurves,thefollowingrainfallisobtained:

andthecorrespondingintensitywillbe

Thepeakdischargeiscalculatedwiththeaidofrationalmethodasfollows:

3.6.2Otherempiricalmethodsforcalculatingpeakrunoff

Apart from the rational method, several other empirical methods have beeninvestigatedtoestimatefloodpeaks,whichareusedincaseswherehydrologicaldata for an area are inadequate. The difficulties encountered in theimplementationofthesemethodsarenotduetothefactthattheseareempiricalbuttothelackofknowledgeofthespecificconditionsunderwhicheachcanbeapplied.The most common and simplest methods use the catchment area as a

parameterandhavethefollowinggeneralforms(Papazafiriou,1983):

(3.28)

(3.29)

whereQpisthedischargeduringthepeakAisthecatchmentareaC,a,b,d,nandmareempiricalcoefficientswhichmustbeevaluatedforeachparticularcase

AnequationwhichwasdevelopedbytheU.S.SCS(1957)uses,apartfromthebasinareaA,theamountofexcessrainfallPrandtherisetimetp,asfollows:

(3.30)

whereQp isinm3/swhenA is inkm2,Pr inmmand tp inh.Finally,Kinnison(1945)developedanequationwhich,apartfromthebasinareaA,alsousestheaveragebasinelevationaboveitsoutletpointh,theaveragepathneededtomakethe water reach the outlet Lm and the percentage of the basin a covered bynaturalorartificialponds.Theequationisexpressedasfollows:

(3.31)

whereQp is inm3/swhenA is inkm2,Lm in kmanda is thepercentage.Theequationmaybeusedonly in the casewherepart of thebasin is occupiedbylakes;otherwise,a=0andQpgetsaninfinitevalue.Itshouldalsobeclarifiedthattheequationprovidesthemaximumpeakofthebasin.

3.7RAINFALL-RUNOFFRELATIONSHIPS:THEUH

ThecomputationofafloodhydrographforanyrainfallcanbebasedontheUHtheory,whichwasfirst introducedinhydrologicalanalysisbySherman(1932).AccordingtoSherman,aUHisthehydrographcausedbyaunitrainfallexcess(i.e. in the metric system equal to 1.0 cm), which is evenly distributedthroughoutthecatchmentareaandhasauniformintensity.Ingeneral,theUHisarunoffhydrographcausedbyarainfallexcessequalto10mmandofadefiniteduration.

3.7.1BasicassumptionsofUH

ThetheoryoftheUHisbasedonthefollowingassumptions:

1. In a particular catchment area, rainfalls of equal duration, which causerunoff,derivedirectrunoffhydrographsofthesametimebaseregardlessofrainfallintensity.

2. Atagivendrainagearea, thedirect runoffcausedbya specific rainfall isindependentofthepreviousornextrainfalls.

3. Thebasincharacteristicsremainunchangedwithtime.

The aforementioned conditions are validated approximately only in natural

catchments (Wilson, 1990). Regarding the first criterion, it is easy to observethatthecapacityofstreamsincreasesasthewater-levelrises.Thus,forrainfallswiththesameduration,themorethewaterstoredinthestreams,thegreatertheintensityoftherain.Thestoredwaterwillcontinuedrainingafterthestopoftherainfall,resultinginaprolongeddirectrunoff.Therefore,itisexpectedthattherewillbesomevariationintherunofftimedependingontherainfallintensity,andthedefinitionofasingletimebaseshouldbearesultofassumptions.Regardingthesecondcondition,rainfalleventsprecedingthefloodaffectthe

entirehydrographandsubstantiallythebaseflow.Forthisreason,theUHcannotbeappliedtothetotalrunoffbutonlytothedirectrunoff.Buteveninthiscase,thiscriterionisambiguoustobeapplied,sincethedirectrunoffdependsonthesoilmoisturelevelandthefillingofthesoilcavitiesbeforethestormevent.Finally, the third condition can be considered valid because there are no

significantchangesinthestateofhydrologicbasinswithinreasonableperiodsoftime, and there is no human intervention with construction of projects oralteration of vegetation cover in the basin. After such interventions, it isexpectedthatthehydrologicalbehaviourofthebasinwillchangeaccordingly,sotheUHs tobe investigatedwillbedifferent from those investigatedbefore theintervention.Theoretically, an infinite number of UHs could be investigated due to

variationsinthedurationanddistributionofrainfallevents.However,itisonlynecessarytodeterminethedurationofrainfallforeachriverbasin,forwhichtheUH will be investigated in conjunction with the direct runoff, which will berepresentative for thisbasin. In thepast, therehavebeenvariousproposals forestimatingthedurationofthetypicalrainfall.Sherman(1949)recommendsthefollowing: for basins with an area larger than 2500 km2, the typical rainfalldurationshouldbe from12 to24h; forbasins rangingbetween250and2500km2, the duration from6 to 12 h; for basins ranging from50 to 250 km2, theduration from 2 to 6 h and for smaller basins, the duration must be selectedbetween the 1/3 or 1/4 of the time of concentration of the catchment area.Linsleyetal.(1949)concludedthatthedurationofthetypicalrainfallshouldbeabout1/4ofthetimelagofthebasin:thishasbeendefinedasthetimeintervalbetween the centroid of the excess rainfall and the peak of the hydrograph.Moreover,theUSACE(1948)proposedthatinbasinswithanarealessthan250km2,thetypicaldurationoftherainfallshouldbehalfofthetimelag.Exceptforselectingthemostappropriaterainfalldurationforeachbasin,itis

usefultoinvestigateUHsofotherdurationslessorgreaterthanthetypicalones.For classification purposes, the investigated UHs are characterized by thedurationoftheexcessrainfallfromwhichtheyoriginate.Thus,forexample,a6

h UH is the direct runoff hydrograph resulting from a rainfall excess of 6 hduration(anddepth1cm).The UH theory is based on two basic principles: the principle of

proportionalityandtheprincipleofsuperposition.

3.7.1.1Principleofproportionality

Accordingtotheprincipleofproportionality, tworainfallexcessesof thesameduration but with different intensities derive hydrographs with the same timebase,butwithordinatesproportionaltotheintensities,i.e.arainfallexcesswithdouble intensity would result in a hydrograph with double discharges. Theprinciple of proportionality is demonstrated in Figure 3.27. This principle isdependent upon the linearity of the basin, where the flood volume is directlyproportional to the rainfall volume (double rainfall provides double runoffvolume).

3.7.1.2Principleofsuperposition

Accordingtotheprincipleofsuperposition,thetotalhydrographresultingfromindividual rainfall events, is thehydrographwithordinates at agiven time thesumoftheordinatesatthattimeoftheindividualhydrographsfromeachevent.The principle of superposition is demonstrated in Figure3.28,where the totalhydrographis thesumof individualhydrographscorrespondingtothreeeventsof rainfall excess.Thestartof the individualhydrographs,whichare summed,concurswiththestartoftherespectiveexcessrainfallevents.

3.7.2DerivationoftheUHfromasinglerainfallevent

The necessary data for the derivation of a river basin UH are simultaneousobservationsofrainfallanddischargeforaperiodoftime,usuallyseveralyears.From these time series, 4–5 intense rainfall events of the same duration areselected, as uniformly distributed as possible over the catchment and withuniform intensity in time. From the measured hydrographs of these rainfallevents,UHsarederived,whichareusedtoextractarepresentativeaverageUHofthebasin.

Figure3.27 Theprincipleofproportionalityasappliedtotheunithydrograph.

Figure3.28 Theprincipleofsuperpositionasappliedtotheunithydrograph.

With the assumption that a rainfall with considerable intensity, evenlydistributedoverthecatchmentandintime,andwithrepresentativedurationforthebasinisselected,inaccordancetowhathasbeensaidearlier,theprocedurefortheconstructionoftheUHfollowsthefollowingsteps:

1. Thetotalhydrographisseparatedintodirectrunoffandbaseflow.2. Thedirectrunoffhydrographisplotted.3. Thetotalvolumeofdirectrunoff(m3)andtherainfallexcessdepth(mm)is

computed.4. Basedontheprincipleofproportionality,theordinatesofthedirectrunoff

hydrograph are divided by the corresponding rainfall excess depth,calculated in thepreviousstep.Thevalues resulting fromthisprocessare

theordinatesofthedesiredUH.5. The duration of the rainfall excess, which is equal in magnitude to the

knowndirectrunoff,isdetermined.

Table3.11FloodhydrographattheoutletofbasinA

Table3.12ComputationofUH

Time(h) Directrunoff UH

0 0 0

1 119 99.17

2 418 348.33

3 510 425.00

4 481 400.83

5 344 286.67

6 236 196.67

7 106 88.33

8 44 36.67

9 5 4.17

10 0 0

Example3.5The rainfall excess of 12mmand of 1h duration resulted in a floodhydrograph at the outlet of basin A, which is shown in Table 3.11whichfollows.Findthe1hdurationUHofthebasin.SolutionFirst,thedirectfloodhydrographwillbecalculatedbytheseparationof the baseflow,which, in this case, seems constant and equal to 98m3/s. Since the desired direct hydrograph and UH have the sameduration,thelatterwillresultfromtheprincipleofproportionalitybysimplymultiplying thedischargesof thedirect runoffhydrographbytheratioofrainfallexcessdepths:

TheprocessofcalculatingtheUHisshowninTable3.12.

3.7.3MathematicaldeterminationoftheUHofcompositerainfall

The linear basin response at any precipitation is proportional to the UH, asreferredearlier.Thisratiocanbegeneralizedincompositestormsandnotonlyinaspecificuniformevent(Jones,1997).Figure3.29 shows a composite storm,which extends, for example, in three

equaltimeperiodsof1heach.TheUHcorrespondingtoatimeperiodofexcessrainfallcanbeusedforthecalculationofthefloodhydrograph.BasedonFigure3.29, it is obvious that the following equations apply for the ordinates of thehydrograph(principlesofproportionalityandsuperposition):

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

Figure3.29 Hydrographofcompositestorm,asderivedusingthe(a)excessrainfall,(b)unithydrographand(c)floodhydrograph.

This systemof equationsmust be solved each time in order to estimate theUH. The general form of these equations can be expressed in the followingmatrixform(Bras,1990):

(3.38)

where

(3.39)

Itshouldbenotedthatthenumberoftheordinateskofthefloodhydrographiscalculatedby

(3.40)

wherenisthenumberoftheUHordinatesjisthenumberoftheexcessrainfallordinates

Thegeneralformoftheequationwhichdescribesthemethodofcalculatingthehydrographis

(3.41)

Thisisalsoknownasconvolutionintegral.

Example3.6Excess rainfall of 2 h duration (shown inTable3.13) resulted at theoutletofBasinA in thefloodhydrographthat isgiveninTable3.14.Thebaseflowis96.23m3/s.Derivethe1-hourUH.

Table3.13Analysisofrainfall

Time(h) 1 2

Rainfall(mm/h) 30 7.5

Table3.14Floodhydrographattheoutletofthebasin

SolutionFirst,thedirectrunoffiscalculatedbyabstractingthebaseflow.Time(h) Floodhydrograph(m3/s) Directrunoff

(m3/s)

0 96.23 0

1 342.28 246.05

2 311.29 215.06

3 248.65 152.42

4 174.84 78.61

5 110.07 13.84

6 96.23 0

The direct flood hydrograph results from the superposition of tworainfall events.Suppose that the requiredordinatesofUHareU0 (atthe time t = 0) to U6 (for t = 6 h). Applying the proportionalityprinciple and the principle of superposition, the following system ofequationsisdetermined:t(h) UH Displacement Floodhydrograph

0 U0 3·U0=0

1 U1 U0 3·U1+0.75·U0=246.05

2 U2 U1 3·U2+0.75·U1=215.06

3 U3 U2 3·U3+0.75·U2=152.42

4 U4 U3 3·U4+0.75·U3=78.61

5 U5 U4 3·U5+0.75·U4=13.84

6 U6 U5 3·U6+0.75·U5=0

Solving the system of equations (line by line) gives the requiredordinatesofUH:

3.7.4DeterminationofUHofacertaindurationfromknownUHofadifferentduration:S-curve

The calculation of a UH with a duration that is an integer multiple of thedurationofaknownUH is relatively simple.Torexample, fromaUHof6h,

anotherUHof12hshouldbecalculated.TheUHof6hisaddedtoitselfwitha6 h time lag. The resulting hydrograph will last 12 h but will have a runoffvolume equivalent to 2.0 cm depth, since each UH of duration of 6 h has avolume equivalent to 1.0 cm depth. If the ordinates of this hydrograph aredividedby2, the resultinghydrographwill have equivalent volumeof 1.0 cmandwillbethebasinUHof12hduration.Thismethodisknownasthelaggedhydrographmethod.So,itisverysimpletoderiveUHswithdurationamultipleofthedurationof

otherknownUHs.DifficultiesarisewhenitisnecessarytoshortenorextendtheUHdurationbyafraction.Inthiscase,theS-curveisused.The S-curve is a hydrograph resulting at the basin outlet from infinite

successive storms of a given duration. According to the UH theory, the S-hydrograph is thesumof theordinatesof infinitenumberofUHsof thegivenduration,asshowninFigure3.30.TheS-curveofD-hourdurationisobtainedbyadding the corresponding UHs ofD-hours, each one lagged by timeD. It isunderstoodthatalltheseUHsareequalandeachoneresultsfromastormwithduration i = 1/D (cm/h) (Figure3.30). The generatedS-shaped curve takes itsmaximumvalueandas therainfallexcesscontinues, itbecomesparallel to theaxisoftimes,afterthesumofT/DUHs,whereTisthetimebaseofeachUHofD-hours. Since, asmentioned, the rainfall excess intensity of the hypotheticalstormofuniformvolumeshouldbe1/D,theS-curveismaximizedandstabilizedat the rateQ = 1/D, in units of discharge per surface area, orA/D in units ofdischarge,whereAisthesurfaceareaofthebasin.

Figure3.30 CalculationofS-curvefromtheunithydrograph.

WhentheS-curveofD-hourdurationisavailable,itcanbeusedtoderivetheUH of any other duration t, using the linearity property, according to thefollowingprocedure:

1. LagtheS-curvebythours.2. Subtract the ordinates of the lagged curve from the originalS-curve. The

resultisahydrographofrainfallexcessintensityt/D.3. The resulting hydrograph from the previous step is converted to unit

volumebymultiplyingitsordinatesbyD/t.

Thisprocess,showninFigure3.31,resultsinaUHofthours.DuringtheprocessofcalculatinganS-curveofagivenUH,anasymmetrical

fluctuation of values around themean value (at the partwhere the S-curve issupposedtobeastraightlineparalleltothetimeaxis)isusuallyobserved,whichbecomesmoreintenseasitapproachestheend.Thisvariationismainlyduetothe lackofprecision in thechoiceof theUHduration: the realdurationof theUH differs somewhat from that used in the calculations. In this case, anormalization of the S curve is needed in order to be adjusted to the normalvalues.

Figure3.31 TheuseoftheS-curvetoderiveaunithydrographofdifferentdurations.

Example3.7DerivetheS-curveanduseittoderivetheUHof3hbasedontheUHof1hshowninTable3.15.SolutionTheS-curveofthoursisthesumofinfiniteUHlaggedbyth.Inthiscase,whiletheS-curvebasetimeis1h,itiscalculatedmoresimplybysumming up at each time interval the ordinates of all theaforementionedUH.Then,theS-curveisshiftedbytimeequaltothebase time of the requested UH, i.e. 3 h. Finally, the difference iscalculatedateachtimeinterval(Table3.16;Figure3.32).ThedesiredUHisderivedbytheimplementationofthemultiplicativeproperty,asfollows:

Table3.15UHordinatesof1h

T(h) UHof1h

0 0.00

101.63

2 344.86

3 434.83

4 405.67

5 295.72

6 198.25

7 93.3

8 43.32

9 5.83

10 0

Table3.16ProcessofcalculationofUHof3h

Figure3.32 TheS-curveandtheunithydrographof3h.

Example3.8BasedontheUHof2hshowninTable3.17andusingthemethodoftheS-curve,calculatetheUHof3hofthesamebasin.SolutionThe2hUHisthemeasuredrunoffattheoutletofthebasinresultingfromastormthatlasted2handhadexcessintensity5mm/h.TheS-curve,whichcorrespondstotheUH(curveS-2h),istherunoffattheoutletof thebasin, resultingfromahypotheticalrainfalleventwhichhasinfinitedurationandintensity5mm/h.Itis,thus,clearthattheS-curve is calculated as the sum of infinite events similar to thoseproducing the 2 hUH, shifted by 2 h relative to each other. In thiscase, 3–4 aggregations are sufficient to stabilize the S-curve at themaximum value. The algorithmic process of shifting the individualeventsandsummingthem,inordertoconstructtheS-curve,isshowninTable3.18.Forthecalculationof3hUH,theS-curveshouldfirstbeshiftedby

3hand theordinateshave tobesubtractedfromtheordinatesof theinitialS-curve.Thissubtractionproducesan intermediatehydrographcorrespondingtotherainfallexcessof3handintensity5mm/h,equaltotheintensityoftheS-curve.ThisdirectrunoffhydrographofrainfallexcesshasthesamedurationwiththedesiredUH(3h)butadifferentintensity (5 mm/h for the intermediate hydrograph and 10/3 = 3.33mm/hforthe3hUH).Multiplyingthedirecthydrographordinatesby3.33/5 = 2/3 gives the requested UH results. The correspondingalgorithmicprocedureispresentedinTable3.19.

Table3.17UHof2h

Table3.18ConstructionofS-curveofUH2h

Table3.19Derivation3hUHfromthe2hS-curve

t(h) S−2h S−2hshift/Dt Difference UH3h

0 0 0 0

1 50 50 33.3

2 120 120 79.92

3 145 0 145 96.57

4 170 50 120 79.92

5 170 120 50 33.3

6 170 145 25 16.65

7 170 170 0 0

3.7.5EstimationoffloodhydrographusingtheUH

Once theUHof a basin is calculated (from simultaneous observations of rainandrunoff),thenitcanbeusedforthederivationofdirectrunoffforanystormevent, so that, the existing runoff data of a catchment can be used to coverperiods for which there are observations of rainfall but not runoff data. It isnecessaryat thispoint tohighlight the restrictionswhichaffect thehydrologicanalysisbasedontheUH.TheUH, asnoted earlier, isdeterminedon thebasisof aparticular rainfall.

The distribution of this rainfall over the catchment affects the shape of thehydrograph.Thus, theuniformityof thedistributionof the rainfall imposes anupperlimitonthesizeofthebasin,towhichthismethodcouldbeapplied.Whatshouldbe theupper limitof thissize isaquestion thatuntilnowhasnotbeenanswered. Ingeneral, itcanbesaid that for theappropriatechoice, the typeofprecipitation should be taken into account, such as the land use, the averageslopeandtheshapeofthebasin.Thus,ifinaregionorographicrainfallsprevail,the maximum extent of catchments should be less than that of another areawhichisdominatedbyfrontalorcyclonicrains.Indicatively,itcanbesaidthatinthefirstcase,theareaofthecatchmentshouldnotexceed5,000km2,whileinthesecond,thelimitcanreach10,000–12,000km2.If theobservationsof the rainfallare recorded in24h, theareaof thebasin

shouldbelargeenoughsothatthetimeofconcentrationofthewaterisgreaterthan24h.Inthiscase,theareaofthebasin,ontheaverage,mustbeatleastinthemagnitude of 2000 km2, and in certain conditions (basin shaped long andnarrow,smallslopesofembankments,streamslopes,etc.),thethresholdcanbeloweredto1000km2.The hydrologic analysis based on the UH can be applied only when the

characteristicsof the streamsof thebasindonot changewith time.Moreover,thiscanbeappliedtoareaswhichhavenosignificantabilityofretainingwateron their surface, because the ratio between the rate and the volume of runoff,requiredbythetheoryoftheUH,impliesalinearequationbetweenthestreamdischarge and the stored water on the surface of the basin. This condition isviolatedwheninsidethebasin,thereareartificialornaturallakeswhichcanholdasubstantialvolumeofwaterorwhen,intheplainsofthecatchment,thewaterofthemainstreamisfloodingtherespectiveareas,sotheeffectisthesameasinthecaseofanartificiallakestoringwater.Finally, several problems arise when part or all the runoff water is derived

from snowmelting. In such cases, significant attention is required, andbeforeapplyingthemethod, it isnecessarytosolvesomespecificproblemsregardingtheareaandtherateofmelting,thedistributionratesofrunoffderivedfromrainandsnowmelting,etc.If the conditions for the implementation of the UH are defined, the

investigated hydrograph for a catchment area can be used to produce directrunoffhydrographsofthisbasinforanyrainfall.The construction of such hydrographs can be done by the following

procedure:

1. Calculatetherainfallexcesswithoneoftheknownmethods(Φ,W,etc.).

2. Calculate the direct runoff hydrograph for each period having excessrainfall,bymultiplyingwiththedepthofexcessrainfallofthatperiodtheordinates of the appropriateUH, i.e. theUH of the same duration to theperiod.

3. The ordinates of the direct runoff hydrographs of each period aresuperimposedlinearly,aftermakingthepropertimeshift.

Example3.9DeterminethetotalfloodhydrographcausedbythefollowingrainfallexcessbasedontheUHofExample3.5.

Rainfallexcess

T(h) 1 2 3 4

I(mm/h) 0.4 1.2 1.9 1.6

Table3.20Calculationoffloodhydrograph

SolutionApplyingtheprincipleofproportionalityforeachindividualrainfallof1hduration,itisobservedthateachindividualrainfalleventisshiftedin time from the previous one by 1 h. The direct runoff hydrographresults by horizontal sum while for the total flood hydrograph, thebaseflowisaddedtoeachordinate,accordingtoTable3.20.

3.7.6SyntheticUHs

The UH method is one of the most popular for flood analysis; however, it

requiresstreamflowdatainordertoderivetheUH.Becausemanystreamshavenohistoricalrecords,researchershavedevelopedprocedurestoderivesyntheticUHs.TheprocedureistodeterminethreecharacteristicsofthesyntheticUH—thepeaktime,thepeakflowandthebasetime.Withthesethreecharacteristics,asyntheticUHcanbecreated.TheshapeofthesyntheticUHisformulatedsuchthat the area under the curve equals 1 cm of runoff. It is important in thesemethodstocalibratetheparametersamongthepeakflowandthelagtime.

3.7.6.1Snyder’sUH

Themostwell-knownandusedmethodforsyntheticUHderivationisSnyder’smethod,whichwasdevelopedbasedon theanalysisofmanyrainfallevents intheAppalachianMountaininNorthAmerica.Thismethod defines the lag time tp, the peakQp, the time base T and the

widthsoftheUHW50andW75atdischargesthatcorrespondto50%and75%ofthepeak(McCuen,1998).TheequationsusedtoderiveSnyder’sUHare

(3.42)

whereLcaisthedistancebetweentheoutletandthecentroidofthebasin,measuredalongthemainchannelofthebasin(mi)

Listhelengthofthemainstreamfromtheoutlettotheheadwater(mi)Ct is a constant representing topographical and soil characteristics of thewatershednormallyrangingbetween1.8and2.2.Forsteepbasins,Cttendstothelowervalue.

Cpisacoefficientthatdependsontheappliedunitsystemintheequationandbybasincharacteristics(rangingbetween0.56and0.69)

Aistheareaofthebasin(mi2).Tisthebasetimeofthehydrographwithaminimumvalueof3days(Figure

3.33)

Itisnotedthattpisthelagtime(timebetweenthepeakofthehydrographandthecentroidofexcessrainfall).

Figure3.33 Snyder’ssyntheticunithydrograph.

tRisconnectedwithtpthroughtheequation

(3.43)

IftherequiredUHoftR′>tR,thentR,isadjustedbytheequation

(3.44)

ThisadjustedvaluetR′mustbereplacedinEquations3.42 inorder tocalculatethevaluesof andT′.The widths of the UH in 50% and 75% of the peak, W50 and W75, are

estimatedbythefollowingequations:

(3.45)

whereqp=Qp/AisthepeakdischargedividedbytheareaofthebasinA.Aftercalculationof theaforementionedvalues,sevenpointsof thesynthetic

UHgrapharedetermined(includingthefirstandthe lastof it),andbyjoiningthemthefinalhydrographisplotted.

Example3.10Atariverbasinoutletwithsmallinfiltrationrates,abridgeisplannedtobeconstructed.Estimatethepeakdischargeofthesteam.Dischargemeasurementsinthispositionarenotavailable.Thetotalbasinareais38km2, themaximummainstream length is 15km, and thedistancebetween the outlet and the nearest point of the main stream at thecentreofthebasinis4.25km.UseSnyder’ssynthetichydrographmethodtocalculatetheUHof3

hrainfallduration.CtandCpvaluesare2and0.65,respectively.Solution1ft=0.3048m1mi=1.609kmtpisgivenby

withLca=4.25km=2.641miL=15km=9.322miCt=2

ThebasetimeofexcessrainfallinhisSince the requested UH has a time of ( = 3 h)tR, adjusted iscalculated:

The peak discharge of theUH of tR, divided permi2 is given by

So,thefinalpeakdischargeis

whereA=14.67mi2,theareaofthebasinThegraphwidthsin50%and75%ofthepeakare,respectively,

Theseordinatesextendby1/3totheleftand2/3totherightofthexordinateofthepeak(Figure3.33).Finally,thetimebaseofthehydrographis

3.7.6.2SCSdimensionlesshydrographmethod

ThisisanempiricalmethodbasedontheanalysisofalargenumberofUHsbytheSoil(NaturalRecoursesnow)ConservationService(1971).They-axisisinunitsh/hpandforthevolume-timediagramtheunitsareVa/V(Figure3.34).The volume of the rising limb of the hydrograph includes 37.5%

approximatelyofthetotalvolume.

3.7.6.3TriangularSCShydrographmethod

SCSalsoproposedtheuseofatriangularhydrographwiththesamepercentageofvolumeontherisinglimb(37.5%)(Figure3.35).tbisgivenbytheempiricalrelationshipssuchastb=0.1021(L/S0.5)0.968ortb=

0.1L/S0.5forsimplicity.ThismethodisasimpleversionofSCSdimensionlesshydrographmethod.

Figure3.34 Theunithydrograph(UH)attimet/peakUHandaccumulatedvolume/totalvolume.

Figure3.35 ThetriangularSoilConservationServicehydrograph.

3.7.7InstantaneousUH

Inorder to remove the restrictionof uniformdistributionof rainfall in time, acasewhichisdifficulttomatchwiththecollectedactualdataofexcessrainfall(ER),forthederivationofaUHoftr-time,amoreadvancedtool,whichistheinstantaneousUH(IUH),hastobeused.IUH is adirect runoffhydrographproducedbyaunit rainfall (Pnet=1cm)

instantaneously over a catchment. Therefore, the IUH is independent of thedirect runoff (DR) duration, and it is considered an impulse response of thesystem.ForthederivationofIUH,fourmethodsareapplicable.

3.7.7.1ByanS-curve

IfanS-curveisproducedthroughthesuccessivelaggingofaUHbyatimestepandthenthegivenUHsaresummedup,inordertoproduceS-curveordinates,thenextmethodisimplied:thecurveislaggedbyasmalltimeof .Then,the

UGOat time t canbe expressed as ,whereSt is anordinateintheS-curveatanygiventimetandSt�,isanordinateafterthelagby

(Figure3.36).AscanbeseeninFigure3.1, throughdeferentialanalysiswhen →0, then

theordinateofaIUHU(0,t)=dSt/dt=slopeofS-curve.ThemethodisapproximatesincetheS-curvederivedfromtherainfall-runoff

dataisnotexact.Nevertheless,itisusefulfortheoreticalpurposes.

3.7.7.2Byusingaconvolutionintegral

Convolutionintegral(orDuhamelintegral)isthedefinedfunction:

(3.46)

ThismethodisbasedontheinfinitedivisionofaUGininfinitesimalelementsofPnet.

Figure3.36 TheS-curveandthelaggedS-curve.

3.7.7.3Bytheuseofvariousconceptualmodels

OnemodelwhichiscommonlyapplicableistheoneproposedbyNash(1957).The concept under this model is that a drainage basin consists of a series oflinearreservoirs.Linealreservoirsarefictitiousreservoirsinwhichthestorageisdirectlyproportionaltotheoutflow:

(3.47)

Also, the inflow and the outflowof a system can be connected through theprincipleofcontinuityas

(3.48)

UsingtheconditionO=0,att=0,itisconcludedthat

(3.49)

TwoparametersinNashmodelhavesignificantimportance:(1)n,consideredashapeparameter, isameasureof thecatchmentchannelstorage,whichdefinestheshapeofIUH,and(2)Kisascaleparameter(smallK-valuerepresentslowerpeak time of the runoff hydrograph and higher K-value the opposite). It isconsideredthatbyusingalargernumberofreservoirs(increasingn),thestorage

time (k) decreases reversely. Therefore, it is possible to have different sets ofthesetwoparameterswhichgivesimilarresults.

3.7.7.4Routingtime-areacurveofbasins

Theprinciplewhichunderliesthismethodisthedivisionofacatchmentintoaseriesofsub-areas,eachcontributinginflowintodrainagechannels(whichhavestorage) due to a flash storm. These sub-areas are called isochroneswhen therainfallinginanysub-areahasthesametimeoftraveltotheoutflowpoint.Clark(1945)usedtime-areadiagrams(TADs)forthecalculationofIUH.Two

parameters other than TADs are necessary for this method: the time ofconcentrationfor thebasin tcandK a storagecoefficient.Clark’smethod isasfollows(Figure3.37):AnyordinatefollowingO2,providedthefirstO1,isgivenbytherelationship

wheretistheΔtcbetweensuccessiveisochronesKisthestoragecoefficientIistheinflowtotheisochronesub-area

Kcanbedeterminedfromanobservedhydrographasfollows.

3.7.8CEH-UK:Revitalizedfloodhydrograph

TheRevitalizedFloodStudiesReport(ReFSR)of2005,presentedbytheCentreofEcologyandHydrology(CEH,UK)asareplacementofanoldermethodfromFloodEngineeringHandbook—1999 (FEH), introducedanewapproach in thedevelopment of event-based rainfall-runoff models concerning hydrologicdesign.Thismethodwasappliedsinceananalysisofalargenumberofobservedflood events (1488) and model parameter estimation from 143 gaugedcatchmentsintheUnitedKingdomwerecarriedout.Themodelparameterswerelinked to basin characteristics usingmultivariate linear regression providing auseful tool applying the rainfall-runoff model at ungauged catchmentsthroughouttheUnitedKingdom.

Figure3.37 Therelationofthedischargeinthefallinglimb.

The components which shape the revitalized flood hydrograph (ReFH) areloss,routingandbaseflowmodelsbasedonPDMMoore(1985)model,timetopeak tpparameteradopted from the triangular-shaped IUHand linear reservoirconcept, respectively. This method is implemented both in gauged and inungauged catchments but with different methodologies in each one. Thisparagraphwillfocusontheungaugedsitesroutingmodelandtheestimationoftpparameter.TheReFHmodelusesthetriangularIUHconceptforroutingthedirectrunoff

butgoesfurtherwithanewshape,i.e.twisted/bendedtriangleform,asshowninFigure 3.38. The new shape is described by a timescale factor tp and twodimensionless parameters Qp and Qk, affecting the height and thetwisting/bending,respectively,oftheIUH.Qk isamultipliertotheQcordinateofanon-bendedtriangularIUH.In the ungauged sites, the optimal combination of catchment descriptors

adoptedbyHoughton-Carr (1999) helped to predict time to peakbyusing theFEHmethodology.Asaresult,fortheReFHroutingmodel,

(3.50)

whereDPLBARisthemeandrainagepathlength(km)DPSBARisthemeandrainagepathslope(m/km)PROPWETistheproportionoftimewhensoilmoisturedeficit(mm)wasless

thanorequalto6mmURBTEXT1990istheextentofurbanandsuburbanlandcover(year1990)

Figure3.38 Instantaneousdimensionlesshydrographadoptedinrevitalizedfoodhydrograph.

FromthenewgeometryofthesuggestedUH,itisacquiredthat

(3.51)

where tbt = 2(tp/Qp) ensuring unit area under the non-bended triangular UH.Otherusefulequationscompletingthisnewapproachare

(3.52)

whereQp = 0.65 andQk = 0.8, which are designated as average values afterunsuccessfulattempts to relate theseparameters.Due to thenewapproach, theReFHhasalowerpeakfowvalueandlongerbasetimethanthetriangularIUH.In order to convert the dimensionless IUH into required units of m3/s/mm, ascalingfactorisapplied.ThisfactorisequaltoAREA/(3.6tp),whereAREAisinkm2andtpinh.

3.7.9ShortcomingsoftheUHandtheUH-basedmodels

An important restriction in the use of the UH and consequently in UH-based

modelsisthatitisaprerequisitetobasinlinearity,orinotherwords,thevolumeoftherunoffhastobeproportionaltotheraindepthfallingoverthebasin.Thiscase is invalid in non-linear basins where depending on the extent of non-linearity,therunoffintheexitofthebasinisnotproportionaltotheprecipitationover the basin. In order to overcome this issue, a number of rainfall–runoffmodelsinasemi-distributedorfullydistributedmodehavebeendevelopedforthecalculationoftherunoffhydrographsinnon-linearbasinsaswell.

3.7.10Generaloverviewofhydrologicalmodels:Specificrainfall–runoffmodels

Hydrological models refer to a wide range of mathematical transformationsusing fielddataand logicalassumptionsaboutphysicalprocesses.Thegoalofthese models is to make quantitative estimates of different hydrologicalvariables.Suchmodelsare,forexample,waterbalancemodelsusingsimulationtools which represent the physical processes at the scale of a river basin orrainfall-runoffmodels estimating,e.g. the flood peak, flood volume and floodduration.Thehydrologicalmodelsarecategorizedaccordingtothefieldofapplication,

thetimeresolution,thespatialscaleandthemathematicalstructure.The time resolution of a hydrological model depends on the specific

application of themodel.Watermanagementmodels need a daily ormonthlyscale,whilefloodmodelsadoptfinerscalesand,insomecases,anhourorlesstimestep.Accordingtothespatialscale,themodelscanbedividedasfollows:

1. Lumped models are the models whose parameters are the same over thestudyarea.

2. Semi-distributed models are the models which can be applied withdifferentiation in their parameters in river basin which are divided intodiscrete areas corresponding to natural sub-basins with the samehydrologicalandgeomorphologicalcharacteristicsforeachone.

3. Semi-lumped models are an intermediate form between the lumped andsemi-distributedschematization,whichconsidersdiscretespatialareaswithdifferent characteristics but common parameters for the entire modellingarea.

4. Distributed models are the models which can be applied in a fullydiscretizedarea throughGISapplications(using, forexample, themethodofgrids)withdifferentsetofcharacteristicsandparametersforeachgrid.

Dependingontheparametricormathematicalstructure,thehydrologicalmodels

canbecategorizedasfollows:

1. Physicalmodelsaredetailedmodelswhichusealargenumberofequationswithlimitedapplicationatariverbasinscaleduetohighuncertainty.

2. Conceptual models with parametric relations which represent thehydrologicalprocessesonariverbasin.

3. Deterministicwithdeterministicdescriptionofthevariablesandprocessesandstatistical(probabilistic–stochastic)modelswhichsimulatethenaturalbehaviour of the observed measurements by using statistical tools. Theprobabilisticmodels use the probability theory, while the stochastic onesinclude a stochastic component and moreover the time evolution of theprocesses(timeseries).

4. Black-box models which only take into account water balance equationsand consider only the input—output of the models without taking intoaccounttheintermediatesystems(e.g.thecharacteristicsofthebasin).

Inthewebsitehttp://www.hydrologicmodels.tamu.edu/models.htm,asubstantialeffort has been undertaken to list, according to pre-defined categories, all theavailable commercial software of hydrological models, accompanied with therespective links. In the category of the distributedmodels,MIKESHE,StormWaterManagementModelandCentralValleyGroundwaterandSurfaceWaterModel are themost famous, while that of the lumpedmodels,MIKE 11 RR,HydrologicalModel and Forecasting System (WATFLOOD) and SCS-CN arecommonlyused.AmonthlywaterbalancemodelwithmanyapplicationsistheWater Balance Simulation Model. In the category of GIS applications inhydrology and hydraulics, the frequently used models are BASINS, HEC-GeoRAS,HEC-GeoHMS and Soil andWater Assessment Tool, among others.An example of global hydrology model is theGlobal Hydrologic EvaluationModel, while a stochastic model is the Stochastic Analysis, Modelling andSimulation.In the following, the HEC-HMS model is presented as a case study. This

model is a UH-based conceptual rainfall–runoff model and is applied to anexperimentalsemi-urbanbasininthegreaterAthensarea.

3.8CASESTUDYUSINGTHEHEC-HMSHYDROLOGICALMODEL

3.8.1GeneralinformationforHEC-HMS

HEC-HMS is a product of theHydrologicEngineeringCenter of theUSACE.HEC-HMSisanopen-sourcesoftwarewhichcanbefreelydownloadedfromthewebsite of USACE (http://www.hec.usace.army.mil/software/hec-hms/). HEC-HMS is designed to simulate the precipitation–runoff processes of watershedsystems. It is applicable in a wide range of geographic areas for solving thewidestpossiblerangeofproblems.Thisincludeslargeriverbasinwatersupplyandfloodhydrologyandsmallurbanornaturalwatershedrunoff.Hydrographsproducedbytheprogramareuseddirectlyorinconjunctionwithothersoftware.Thismodelofferstheoptiontoimportageographicalfilewiththestudyarea,

its sub-basins and their properties, created in GIS environment. One of theoptionswhichcanbeused is theextensionHEC-GeoHMS.Particularly, it isageospatial hydrology toolkit, which can be downloaded from the website ofUSACEaswell.Theprogramallows theuser tovisualize spatial information,documentwatershedcharacteristics,performspatialanalysisanddelineatesub-basinsandstreams.Generally,itcreateshydrologicalinputsforHEC-HMS.

3.8.1.1ComponentsofHEC-HMS

The analysis is based on formulation of models for the computation of thefollowing hydrological parameters: hydrological loss, direct runoff, baseflowand channel flow routing. Once themethods for thosemodels are selected, ameteorologicalmodelandacontrolspecificationmanageraredesigned.Then,alist of representative events is drawn, and these events are simulated by themodel.Someeventsneedtobeusedforcalibration,andtherestofthemcanbeused for validation purposes. In order to compute each one of theaforementionedmodels, theusermaychoosebetweenawiderangeofrelevantoptions.Themaincriteria for the selectionof theappropriatemethod foreachmodelaretheavailabilityofdatarequiredbyeachmethodandtheexperienceofthe modeller, based on which the modeller may exclude some methods forspecific applications. In the following sections, indicative methods for eachmodelofeachelement(basinelements,reachelements)areincluded.

3.8.1.1.1Basinelements3.8.1.1.1.1HYDROLOGICALLOSS

Hydrologicallossesincludeinteraliainfiltration,subsurfacelossandretention.AlltheseprocessesandtheirinteractionaresimulatinginaunifiedwaybyHEC-HMS through a loss method which needs to be defined for every sub-basin.

HEC-HMS provides 12 different loss methods. The Deficit and constant lossmethod, which uses a single soil layer to account for continuous changes inmoisture content and is combined with a meteorological model for theevapotranspiration estimation. The Exponential loss method which representsincrementalinfiltrationasalogarithmicallydecreasingfunctionofaccumulatedinfiltration.There isanoptionfor increased initial infiltration,whenthesoil isparticularly dry before the arrival of a storm. However, it is not suitable forcontinuous simulation. TheGreen and Ampt loss method assumes the soil isinitiallyatuniformmoisturecontentandinfiltrationtakesplacewiththepistondisplacement.Thismethodautomaticallyaccountsforpondingonthesurface.Incorrespondence with the above, the Gridded defcit constant loss methodimplements thedeficitconstantmethodonagridcellbygridcellbasis,whereeach grid cell receives separate precipitation and potential evapotranspirationfrom the meteorological model. The gridded Green and Ampt loss methodimplements theGreenandAmptmethodonagridcellbygridcellbasis.TheGriddedSCScurvenumberlossmethodimplementstheSCScurvenumberlossmethod on a grid cell by grid cell basis, aswell as, thegridded soilmoistureaccountingwith the soilmoistureaccountingmethod.The initial and constantlossmethod is simplier andmoreappropriate forwatersheds that lackdetailedsoilinformation.TheSCScurvenumberlosscomputesincrementalprecipitationduringa stormby recalculating the infiltrationvolumeat theendofeach timeinterval. The Smith–Parlange loss approximates Richard’s equation forinfiltration into soil by assuming thewetting front can be representedwith anexponential scalingof thesaturatedconductivity.Thesoilmoistureaccountinglossmethoduses three layers to represent thedynamicsofwatermovement inthe soil and isusuallyused in conjunctionwith a canopyand surfacemethod.Some of the methods are designed for simulating events, while others areintendedforcontinuoussimulation.3.8.1.1.1.2DIRECTRUNOFF

Thecalculationsfordirectsurfacerunoffareperformedusinga transformationmethod.HEC-HMSprovidessevendifferenttransformmethods.TheClarkUHtransform, which is a syntheticUHmethod, in the sence that, the user is notrequired to develop a UH through the analysis of the earlier observedhydrographs.Thekinematicwavetransformisaconceptualmodelthatincludesone or two representative planes (pervious-impervious planes) and is appliedeither inurbanareas,or inundevelopedregions.TheModClark transform isalinear, quasi-distributed transform that is based on the Clark conceptual UH.This represents the sub-basin as a collection of grid cells. The SCS UH

transform,itusesobserveddatacollectedinsmall,agriculturalwatersheds,thatare generalized as dimensionless hydrographs and a best approximatehydrographisdevelopedforgeneralapplication.TheSnyderUHtransformisasyntheticUHmethod,wheretheoriginaldataonlysupportedthecomputationofthepeakflowasaresultofaunitofprecipitation.TheUser-specifiedS-graphtransformmethod is not synthetic and uses a summationUH to represent theresponseofasub-basintoaunitprecipitation.Thes-graphisdefinedintermsofpercentage of unit flowversus percentage of time lag.TheUser-specifiedUHtransform is,also,notsyntheticandaseparateUHmustbecomputedforeachsub-basin.Usually, theseUHsaredeveloped frommultiple stormobservationswhenprecipitationandflowhavebeenmeasuredatthesametimeinterval.3.8.1.1.1.3BASEFLOW

Baseflowmethod isused for theestimationof thesubsurfaceprocesses.HEC-HMSprovidesfourdifferentmethods.TheBoundedrecessionbaseflowmethodisintendedprimarilyforreal-timeforecastingoperationandisverysimilartotherecessionmethod.However, themonthlybasefow limits canbe specified.Thebaseflowiscomputedaccordingtotherecessionmethodologyand themonthlylimitsareimposed.Afterastormevent,thismethoddoesnotresetthebaseflow.Theconstantmonthlybaseflowdoesnotconservemasswithinthesub-basinandisintendedprimarilyforcontinuoussimulationinsub-basinswherethebaseflowis approximated by a constant flow for each month. The linear reservoirbaseflowusesalinearreservoirtomodeltherecessionofbaseflowafterastormevent,whileitconservesmasswithinthesub-basin.Infiltrationcomputedbythelossmethod is connected as the inflow to the linear reservoir. Thenon-linearBoussinesqbaseflowisdesignedtoapproximatethetypicalbehaviourobservedinwatershedswhenchannelflowrecedesafteranevent.Thismethodisintendedprimarilyforeventsimulation.Itmaybeusedforanycontinuoussimulation,asit has the ability to automatically reset after each storm event. The recessionbaseflow is designed to approximate the typical behaviour observed inwatersheds when channel flow recedes exponentially after an event. It isintended primarily for event simulation, but, because of its ability toautomatically reset after each storm event, it may be used for continuoussimulation.

3.8.1.1.2Reachelements3.8.1.1.2.1CHANNELFLOWROUTING

Channel flow routing is applied to reach elements that represent streamsegments. HEC-HMS provides six different methods to define channel flowrouting.Eachofthesemethodsincludedinthemodelprovidesadifferentlevelof detail and not all methods are equally adept at representing a particularstream.Thekinematicwaveroutingmethodapproximatesthefullunsteadyfowequationsbyignoringinertialandpressureforces.Thismethodisbestsuitedtofairly steep streams, as it is assumed that the energy slope is equal to the bedslope.Thelagroutingmethodrepresentsthetranslationoffloodwaves.Itdoesnot include any representation of attenuation or diffusion processes and it issuitable for short stream segmentswith a predictable travel time that doesnotvarywithflowdepth.ThemodifiedPulsrouting isoftencalledstorageroutingorlevelpoolroutingandusesconservationofmassandarelationshipbetweenstorageanddischarge to route flow through the stream reach.TheMuskingumroutingusesasimpleconservationofmassapproach torouteflowthroughthestreamreach.TheMuskingum-Cungeroutingisbasedonthecombinationoftheconservation of mass and the diffusion representation of the conservation ofmomentum.Itrepresentsattenuationoffloodwavesandcanbeusedinreacheswith a small slope. The Straddle stagger routing, which is the last one, usesempiricalrepresentationsoftranslationandattenuationprocessestoroutewaterthroughareach.3.8.1.1.2.2LOSS/GAINMETHOD

It is also applied to reach elements, and it represents the modelling ofinteractions with the subsurface. Particularly, it represents losses from thechannel, additions to the channel from groundwater or bidirectional watermovementsdependingonthespecific implementationofamethod.HEC-HMSprovidestwodifferentmethods.TheConstantloss/gainmethod,whichusesanempiricalrelationshiptocalculatechannellossusingafixedflowratereductionand a ratio of the flow, and the Percolation loss/gain method, which uses aconstantinfiltrationrateincombinationwiththeinundatedareainthereachtocomputechannelloss.

3.8.1.1.3MeteorologicalmodelThe meteorological model is one of the main components of HEC-HMS. Itspurpose is to prepare meteorologic boundary conditions for sub-basins. Itincludesprecipitation,evapotranspirationandsnowmelt.Sixdifferenthistoricaland synthetic precipitationmethods are included. There aremanymethods todefinethemeteorologicalmodel.

3.8.1.1.3.1PRECIPITATION

Therearemanyprecipitationmethodstochoosefromortheusercanchoosetohave no precipitation at all. If the basin model contains sub-basins, theprecipitationmethod is necessary. In the casewhen the basinmodel does notcontain sub-basins, the precipitation method cannot be used. HEC-HMSprovides seven differentmethods. The frequency stormmethod is designed toproduce a synthetic storm from statistical precipitation data. However, thismethod uses the same parameter data for all sub-basins. The gauge weightsmethod, such as the Thiessen polygons method, is designed to work withrecording andnon-recordingprecipitationgauges.Thegriddedprecipitation isbased on the ModClark gridded transform, but it can be used with othertransformmethods. The most common use of this method is to utilize radar-basedprecipitation.Theinversedistanceisdesignedforapplicationinreal-timeforecastingsystemsandcanuserecordinggaugesthatreportatregularintervalslike15minor1h.TheSCSstormimplementsthedesignstormdevelopedbytheNaturalResourcesConservationService.Thismethodwasdevelopedmainlyforagricultural applications,but it canbeused forother applicationsaswell.Thespecified hyetograph gives the opportunity to user to specify the exact timeseriestousethehyetographatsub-basinsanditisusefulwhenprecipitationdataare processed externally by the program and importedwithout alteration. Thestandard project storm is no longer frequently used, but it is included forprojectswhereitisstillnecessary.3.8.1.1.3.2EVAPOTRANSPIRATION

Fortheevapotransirationfactor,theusercanuseeitheroneofthethreedifferentevapotranspirationmethodsproposedbyHEC-HMS,ornoevapotranspirationinthecasethatbasinmodeldoesnotincludesub-basins.The availablemethods are: thegriddedPriestley–Taylor which can be used

when ModClark transform method is selected, themonthly average which isusedwhenmeasuredpanevaporationdataareavailableandwhichusesseparateparameterdataforeachsub-basininthemeteorologicalmodel.3.8.1.1.3.3SNOWMELT

For the snowmelt factor, the user may choose to either use one of the twodifferentmethodsproposedbyHEC-HMS,ornottomodelsnowmeltatall.Generally, using the temperature determines whether the precipitation

previouslycomputedwas liquid rainor frozen snow.Thegridded temperatureindexisdesignedtoworkwithModClarktransformmethodandisthesameastheregulartemperatureindexmethod.Themaindifferenceisthattheequations

for simulating the snowpack are computed separately for each grid cell withseparate precipitation and temperature boundary conditions. The temperatureindex is an extensionof thedegree-dayapproach tomodelling a snowpack.Atypical approach to thedegreeday is to have a fixed amount of snowmelt foreachdegreeabovefreezing.

3.8.1.1.4ControlspecificationmanagerThefinalstepisthecreationofasimulationtimewindow,whichisreferredtoas‘control specification manager’. In this window, the time span and the timeintervalareset.Thereisthepossibilitytohaveseveralsimulationtimewindowsbyaddingnewonesorcopyingthealreadyexistingones.3.8.1.1.4.1SIMULATIONRUNS

Simulation run is themaincomponentwhichcancompute results.Each run iscomposed of one basin model, one meteorological model and one controlspecification. Results can be visualized as graphs, summary tables and timeseriestables.Theformatoftheresultswillbeshownintheexamplewhichwillbepresentedinthefollowing.3.8.1.1.4.2OPTIMIZATIONTRIALS

Having completed the simulation runs, themodel gives the opportunity of anautomatic optimization. For this process, an observed outflow in a particularposition is necessary. Particularly, to complete optimization trial and acquireresults,theusershouldchooseasimulationrunandanobservedoutflow.Fortheoptimizationtrial,theusershoulddefinetheobjectivefunction,themethodthatminimizes the objective function and the controlling search tolerance. Manyparameterscanbeoptimizedatthesametime.3.8.1.1.4.3OBJECTIVEFUNCTION

The objective function measures the goodness of fit between the computedoutflow and observed streamflow at the selected element. Seven differentfunctionsareprovided:

1. Peak-weightedRMSerror2. Percenterrorpeak3. Percenterrorvolume4. RMSlogerror5. Sumabsoluteresiduals6. Sumsquaredresiduals

7. Time-weightederror

3.8.1.1.4.4SEARCHMETHOD

Two search methods are available for minimizing the objective function andfindingoptimalparametervalues.Thefirstoneistheunivariategradientmethodwhich evaluates and adjusts one parameter at a time while keeping the otherparameters constant. The other one is theNelder–Meadmethod which uses adownhillsimplextoevaluateallparameterssimultaneouslyanddeterminewhichparametertoadjust.3.8.1.1.4.5CONTROLLINGSEARCHTOLERANCE

TwomethodsareprovidedforcontrollingthesearchprocesswiththeunivariategradientorNelder–Meadmethods.The tolerancedetermines thechange in theobjectivefunctionvaluewhichwillterminatethesearch.

3.8.2Casestudy

3.8.2.1Studyarea

TheareawhichisusedfortheimplementationofthehydrologicmodelisRafinaBasin.ItisaperiurbanareainthegreatersoutheastMesogeiaregionineasternAttica, Greece. The area covers 126 km2 and geographically extends east ofYmittosMountain to the coastline ofEvoikosGulf. Themean altitude of thisregion isapproximately227m,with themaximumvaluebeing909mand theminimum0m.Regardingthegroundslope,itrangesfrom0%to37.8%andthemeanvalueisestimatedas7.5%(Figure3.39).

Figure3.39 TheDEMofAtticaregionandtheboundariesofthestudyarea.

3.8.2.2HEC-HMSenvironment

3.8.2.2.1BasinmodelThebasinmodelinthisstudywascreatedthroughtheuseofHEC-GeoHMSinArcGIS9.3environment(Figure3.40).

3.8.2.2.2Definitionofhydrologicalparameters3.8.2.2.2.1HYDROLOGICALLOSS

The method adopted in this study was SCS curve number loss. For theapplicationof thismethod, it is necessary to defne someparameters.The firstoneistheinitialabstraction.Thisparameterisoptional.Incasetheuserleavesit‘blank’,itisautomaticallycalculatedas0.2timesthepotentialretention,whichiscalculatedfromthecurvenumber.Thesecondparameteristhecurvenumber(CN) which quantifies the impact of soil properties and land use on the sub-basin.Thelastoneisimperviousness.ThispercentagecanbeadjustedtotheCNparameterorcanbecalculatedseparately.Inthecurrentstudy,thepercentageofimperviousnesswasincludedintheestimationofCNparameter.

Figure3.40 Basinmodelwiththesub-basinsandtheelements.

3.8.2.2.2.2DIRECTRUNOFF

Themethod adopted for the estimationof direct runoffwasSnyder’ssyntheticUH.Todefinethismethod, lagtime(h)andpeakingcoefficientarecomputed.

Thevalueofpeakingcoefficientrangesfrom0.4to0.8,anditisestimatedusingthebestjudgementoftheuseraccordingtothewatershedphysicalfeatures.3.8.2.2.2.3BASEFLOW

Themethodwhichwas adopted in the current study for the estimation of thecontribution of baseflow to the sub-basin outflow was recession baseflowmethod, amethod intended for event simulation.At first, the initial dischargetype needs to be defined. There are two options: initial discharge and initialdischargeperarea.Thefirstwasselected.Theotherparameterswhichhavetobeestimatedforthisoptionaretheinitialdischarge,whichistheinitialbaseflowasadischarge, the recessionconstantwhichdescribes the rate atwhichbaseflowrecedesandthethresholdtypewhichisthemethoddetermininghowtoresetthebaseflowduringtheevent.3.8.2.2.2.4CHANNELFLOWROUTING

Channel flow routing is applied to reach elements. In the current study, theMuskingummethodwas applied. For thismethod, the travel time through thereach(MuskingumK[h]),theweightingbetweeninflowandoutflowinfluence(MuskingumX)andthenumberofsubreachesneedtobeimported.RegardingtheparameterMuskingumX,itrangesfrom0to0.5.Thevalue0correspondstomaximumattenuation,whilethevalue0.5correspondstonoattenuation.3.8.2.2.2.5ESTIMATIONOFINITIALVALUESOFPARAMETERS

TheestimatedvaluesforalltheearliermethodsarepresentedinTable3.21.

3.8.2.2.3MeteorologicalmodelIn this study,aprecipitationmethodwas selected, as thebasinmodelcontainssub-basins.Particularly,thegaugeweightsmethodwasadopted.TheseweightswerecalculatedusingThiessenpolygons.ThisprocedurewasperformedinGISenvironmentusingArcMap9.3.

3.8.2.2.4ControlspecificationmanagerIn thissection, thestartand theendof thesimulationrunweredefined,whichwere the same as the start and the end of the rainfall event that was used.Particularly, the rainfall eventof3February2011was selected.The formatofthecontrolspecificationisclearinFigure3.41.

Table3.21Initialestimatedparameters

Figure3.41 Formatofthecontrolspecificationmanager.

3.8.2.2.5SimulationrunandresultsHavingestimatedall theaforementionedparameters, the simulation runwas

accomplished. The results which are presented in the following deal with thepositionsofRafinaandDrafi(Figure3.42),inwhichthereweremeasurements.

3.8.2.2.6ResultsAfloodhydrographhasoccurred ineachof thepositionsofRafinaandDrafi.ObservingFigures3.43and3.44,thegraycolourrepresentsthesimulatedresult,whiletheblackcolourrepresentstheobservedone.

Figure3.42 Positioninwhichtheresultsarepresented.

Figure3.43 ComputedandobservedfloodhydrographatDrafistation.

3.8.2.2.6.1RAFINAPOSITION

Observing theFigure3.44, the continuous line represents the simulated result,whilethelinewithpointsrepresentstheobservedone.

3.8.2.2.7Calibrationofmodel

Togetthebestresults,calibrationisanecessaryprocess.TheparameterwhichwascalibratedinthisstudyistheCNparameterandtheparametersofSnyder’sUH.Particularly,withregardtoSnyder’sUH,areductioninthelagtime(Ct)isimplemented to bring the peaks closer and to increase the peaking coefficient(Cp).Finally,areductioninthevaluesoftheCNparameterwasrealizedinordertogetbetterresults.TheexactvaluesofthecalibratedparametersarepresentedinTables3.22and3.23.

Figure3.44 ObservedandsimulatedfloodhydrographatRafinastation.

Table3.22Calibratedvalues(Ct,Cp)

Snyder’sUH

Lagtime(h) Cp

W15020 2 0.7

W17990 2 0.6

W15460 0.8 0.8

W16890 2 0.6

W16340 3 0.7

Table3.23CalibratedCNvalues

Loss(SCScurvenumber)

CN

W15020 40

W17990 40

W15460 45

W16890 55

W16340 40

3.8.2.2.8ResultsaftercalibrationFor Drafi, the difference between the observed and the simulated flow issignificantlysmallerandflowpeaksarealsocaptured.ForRafina,thesimulatedpeakflowandthedischargevolumeareslightlybigger thantheobservedones(Figures3.45and3.46).

Figure3.45 Drafposition.

Figure3.46 Rafinaposition.

REFERENCES

Barnes,B.S.,1939,Thestructureofdischargerecessioncurves,TransactionsoftheAmericanGeophysicalUnion,20,721–725.

Bedient,P.B.andHuber,W.C.,1992,HydrologyandFloodplainAnalysis,2ndedn.PublishedbyAddison-Wesley.

Bras,R.L.,1990,AnIntroductiontoHydrologicScience,PublishedbyAddison-Wesley.Butler,S.S.,1957,EngineeringHydrology,Prentice-Hall,EnglewoodCliffs,NJ.Clark, C.O., 1945, Storage and the unit hydrograph, Transactions of the American Society of Civil

Engineers,110,1419–1446,PaperNo.2261.Giandotti,M.,1934,Previsionedellepieneedellemagredeicorsid’acqua.MinisteroLL.PP.,Memoriee

studiidrografici,Vol.8,Rep.No.2,ServizioIdrograficoItaliano,Rome,Italy(inItalian).Horner, W.W. and Flynt, F.L., 1936, Relations between rainfall and runoff from small urban areas,

TransactionsoftheAmericanSocietyofCivilEngineers,101,140–206.Horton,R.E.,1945,Erosionaldevelopmentofstreamsandtheirdrainagebasins:Hydrophysicalapproachto

quantitativemorphology,BulletinoftheGeologicalSocietyofAmerica,56,275–370.Jones,G.V.,1997,Asynopticclimatologicalassessmentofviticulturalphenology,Dissertation,Department

ofEnvironmentalSciences,UniversityofVirginia,Charlottesville,VA,394pp.Kinnison,H.B.andColby,B.R.,1945,Floodformulasbasedondrainagebasincharacteristics,Proceedings

oftheAmericanSocietyoftheCivilEngineers,69,849–876.Kirpich,Z.P.,1940,Timeofconcentrationinsmallagriculturalwatersheds,CivilEngineering,10(6),362.Kjeldsen, Thomas Rodding. 2007. The revitalised FSR/FEH rainfall-runoff method. Wallingford,

NERC/CentreforEcology&Hydrology,68pp,ISBN0903741157.(FloodEstimationHandbook,SupplementaryReportNo.1).

Linsley,R.,Kohler,M.,andPaulhus,J.,1982,HydrologyforEngineers.3rdEdn,McGraw-Hill,NewYork.Linsley,R.K., Jr.,Kohler,M.A., andPaulhus, J.L.H.,1949,AppliedHydrology,McGraw-HillBookCo.,

NewYork,689pp.McCuen,R.H.,1998,HydrologicAnalysisandDesign,2ndedn.PublishedbyPrentice-Hall,Inc.,NJ.Moore,R.J., 1985,The probability-distributed principle and runoff production at point and basin scales,

HydrologicalSciencesJournal,30,273–297.Mimikou,M.,Baltas,E.,Zobanakis,G.,Michaelides,S.,Spanides,N.,andGikas,A.,2001,NationalData

Bank of Hydrological and Meteorological Information (NDBHMI), Athens, Greece,http://www.ndbhmi.chi.civil.ntua.gr/.

Nash,J.E.,1957,Theformoftheinstantaneousunithydrograph,IAHSAISHPublications,42,114–118.Papazafriou,Z.,1983,SurfaceHydrology,AristotleUniversity ofThessaloniki,Thessaloniki,Greece (in

Greek).SCS, 1957/1971, Hydrology, Engineering Handbook, Supplement A, Section 4, U.S. Department of

Agriculture,Washington,DC.SCS,1986,Urbanhydrology for smallwatersheds,TechnicalReleaseTR-55,SoilConservationService,

HydrologyUnit,U.S.DepartmentofAgriculture,Washington,DC.Sherman,L.K.,1932,Streamflowfromrainfallbytheunit-graphmethod,EngineeringNews-Record,108,

501–505.Sherman, L.K., 1949, The unit hydrograph method, inHydrology, pt. 9 of Physics of the Earth, O.E.

Meinzer(ed.),DoverPublications,NewYork,pp.514–525.Wilson,E.M.,1990,EngineeringHydrology.PublishedbyPalgraveMacMillan,U.K.

Chapter4Probabilityandstatisticsinhydrology

4.1GENERALCONCEPTSANDDEFINITIONS

The design of hydraulic structures often requires hydrological informationgovernedbythelawsofprobabilityandstatistics.Thisresultsinthenecessityofusingthestatisticalandprobabilisticanalysesofhydrologicaldata.Deterministichydrologydealswithcausesandassesseshydrologicalvariablesandprocesseswith complete certainty; thus, it cannot cover the random component of thehydrological variables, something significantly impacting both simulation andforecast. This has led to the development of the field of statistical hydrology.This branch of hydrology, which deals with the analysis of hydrologicalvariables, is divided into probabilistic hydrology and stochastic hydrology.Probabilistic hydrology analyzes and synthesizes hydrological events withouttakingintoaccounttheirtemporalsequence,i.e.theprobabilityPofthevariableto have a specific value at the spatiotemporal point of zero. The theory ofprobability provides the framework for modelling processes that we cannotdetermineprecisely.Stochastichydrologytakes intoaccount the timesequenceand considers the probability, previously reported equal to ε, where ε is apositive number between 0 and 1. In hydrological variables, there are twocategoriesofuncertainty: (1)natural randomnessand (2) impreciseknowledgeofreality.Each feature which presents variability is called a variable. Variables are

divided into continuous, when their values are the result of continuousmeasurement, forexamplerainfalldepth inmm,anddiscontinuousordiscrete,when the values are enumerated, for example number of rainfall days. Thevaluesofthevariablesarethedata.Withtheterm‘population’,weimplyasetofvaluesofavariablewhichwewanttostudy.Apopulationisfinitewhenwecancount all the members, while it is infinite when its members cannot be

•

•

•

•

enumerated.Inaninfnitepopulationoreveninafinitepopulation,technicalandeconomic reasons make in most cases the study of the entire populationimpossible. Therefore, we must limit ourselves to studying a part of thepopulation,calledsample.Asampleiscalledrandomwheneachmemberofthepopulationhasthesameprobabilityofbeingincludedtherein.Somebasicconceptsofprobabilisticanalysisareasfollows:

Experiment: Conditions underwhichwe observe a random variable, forexampletherainydaysofamonthandthedailyrainfalldepthOutcomeorsamplepoint:Theresultofsuchobservation,forexample10daysofrainfallpermonthand30mmofrainfallonthe10thdaySample space: The concentration of all possible outcomes of anexperiment,forexampletherainfalldaysofamonthS1=(0,1,2,3,4,…,31)andtherainfalldepthgreaterthanzeroS2=(x/x≥0)Event:Theconcentrationofpossibleoutcomesorthesubsetofthesamplespace,forexampletheeventE1=8orfewerdaysofrainfallinthemonthandtheeventoftherainfallbetween5and15mm,E2=(x/5≤x≤15)

The main task of statistics is to draw conclusions from the sample for thepopulation in a way which allows the calculation of the uncertainty of theconclusions drawn. This uncertainty is quantified through the theory ofprobabilitywhichplaysacentralroleinstatistics.TheclassicaldefinitionofprobabilitywasformulatedbyLaplaceasfollows:TheprobabilityofaneventAistheratioofthenumbermofcasesfavourable

toittothenumbernofequallypossiblecases.TheprobabilityofAisdefinedasP(A)=m/n.

4.1.1Experimentsandsamplespaces

Weareofteninterestedinsetsofpointsofthesamplespace.Eachsetiscalledincidentoreventand isasubsetof thebasicset.Thebasicpropertiesof thesesetsaregiveninthefollowing.IntersectionoftwoeventsAandBisthesetofallobjectswhicharemembers

ofbothAandB.ItisdenotedbyA∩B(Figure4.1).TheunionofthesetsAandBisthesetofallobjectswhichareamemberofA

orBorboth.ItisdenotedbyA∪B(Figure4.2).ThedifferenceofsetBfromsetAisthesetofallmembersofBwhicharenot

membersofA(Figure4.3).TwoeventsAandBof thesamesamplespaceΩarecalled incompatibleor

mutuallyexclusiveiftherealizationofoneexcludestheother.Therelationship

forincompatibleeventsis

where∅istheemptyset.Theeventwhichissymbolizedthiswayisconsideredimpossible.

Figure4.1 IntersectionofAandB(A∩B).

Figure4.2 UnionofthesetsAandB.

Figure4.3 DifferenceofsetBfromsetA.

A set of events [A1,…,Am] is considered to include anymutually exclusiveevents,ifAi∩Aj=∅foreveryi,j.TwoeventsAandBofthesamesamplespaceΩarecalledcomplementaryif

therealizationofoneexcludestherealizationoftheotherandtheirsum(union)givesacertainevent,theentiresamplespaceΩ.ThecomplementaryeventofAisdenotedbyAC.

Asetofevents[A1,…,Am]issaidtoexhaustthesamplespaceif .Properties

•••

Commutative:A∪B=B∪A,A∩B=B∩ACombinational:A∪(B∪C)=(A∪B)∪C,A∩(B∩C)=(A∩B)∩CDistributive:A∩(B∪C)=(A∩B)∪(A∩C),A∪(B∩C)=(A∪B)∩(A∪C)

Furthermore,

Thecommutative,combinationalanddistributivepropertiesdonotapplytothedifference:

4.1.2Probabilityfunction

Aprobability functionP[·] is a functionwhich gives a value (real number) ineachevent,satisfyingthefollowingrelations:

IfA1,A2,…,ANisasetofmutuallyexcludedevents,thenthefollowingrelationapplies:

Thefollowingarethepropertiesofaprobabilityfunction:

ForanytwoeventsAandB,thefollowingapplies:

IfA⊂BthenP[A]≤P[B]

(Thesymbol⊂denotesthattheeventAisasubsetofeventB.)

4.1.3Conditionalprobability

Theconditionalprobabilityof aneventA is theprobability that theeventwilloccur given the knowledge that an event B ≠∅ has already occurred. ThisprobabilityiswrittenasP[A|B]

ItisnotdefinedifP[B]=0.The conditional probability is obtained by applying the general rule ofprobabilitymultiplicationasfollows:

The eventsA and B are independent when the probability of event A is notaffectedbytheprobabilityofeventB.ThenP[A|B]=P[A]andP[B|A]=P[B],

soP[A∩B]=P[A]P[B].Thepropertiesofconditionalprobabilityare

IfA1⊂A2,then

4.1.4TotalprobabilityandBayes’theorem

IfB1,B2,…BMisasetofmutuallyexcludedeventsofthesamplespace,thenforeacheventA,withP[A]>0,thefollowingrelationapplies:

Bayes’theoremIfB1,B2,…BMisasetofmutuallyexcludedeventsofthesamplespace,thenforeacheventA,withP[A]>0,thefollowingrelationapplies:

Example4.1A company is supplied with three types of automatic telemetricequipmentformeasuringrainfalldepth,typeA,typeBandtypeC,at50%,30%and20%,respectively.Iftheelectronicequipmentoftypes

1.2.3.

1.

2.

3.

1.2.

1.

2.

A,BandCarefaultyby2%,3%and10%,respectively,calculatethefollowing:

TheprobabilitythattheequipmentisoftypeAandgoodTheprobabilitythattheequipmentisfaultyTheprobability that theequipment isof typeA given that it isfaulty

SolutionThe probability that the equipment is in good condition (K)giventhatitisoftypeAiscomplementarytotheprobabilitythattheequipmentisdefective(E)andoftypeA,i.e.P[K|A]=1−P[E|A]=1−0.02=0.98.Thus,theprobabilitythat the equipment is of type A and good is P(A ∩ K) =P(A)*P(K|A)=0.5*0.98=0.49.The probability that the equipment is faulty is given by therelationship

Theprobability that theequipment isof typeA given that it isfaultyisgivenby

Example4.2Two automatic rain gauges A and B operate independently. If thepercentageofoperatingtimeis0.85forAand0.70forB,whatistheprobabilitythatatagivenpointoftime?

BothofthemoperateNoneofthemoperates

SolutionTheprobabilitythatbothofthemoperateis

Theprobabilitythatnoneofthemoperateis

1.

2.

whereTheprobabilityofoperationfailureofAis

TheprobabilityofoperationfailureofBis

4.2RANDOMVARIABLE

Each outcome of a random experiment is usually characterized by a numericvalue.Thecharacterizationofeachoutcomewithanumberisequivalenttothedefinitionofafunctionwhichtakesacertainnumericalvalueforeachoutcomeoftheexperiment.Thatfunctioniscalledrandomvariable.Thefunctionoftherandomvariableisalsoarandomvariable.IfXisarandomvariableandf(x)isafunctionofX,thenthefollowingapplies:

Mean(expected)valueE[X]orμxoftherandomvariableXThisisgivenbythefollowingrelationship:

E[X]defnesthecentreofweightoffX(·)(Figure4.4).OthermeasurementsofthecentraltendencyFor(median)med(X):

AssumingthattheNvaluesarerankedinorder,themedianisthevalueon

3.

eithersideofwhichthereisanequalnumberofvalues.IfthenumberNiseven, themedian is taken as the average of the twomiddle values.Themostprobablemode,mod(X),isgivenas

Themostprobablemeanofaparameteristhevaluewhichhasthegreatestfrequency(Figure4.5).

Figure4.4 Mean(expected)value.

Figure4.5 Medianandmostprobablemode.

ExpectedvalueofafunctionofrandomvariableIfg(·)isarealfunctionoftherandomvariableX,then

4.a.b.c.

5.

6.

PropertiesofthemeanE[C]=CforeveryconstantCE[Cg(X)]=CE[g(X)]foreveryconstantCandrealfunctiong(·)E[C1g1(X)+C2g2(X)]=C1E[g1(X)]+C2E[g2(X)](meanislinear)

Variance,Var[X]

The varianceVar[X] is themoment of inertia of the probability densityfunction (pdf) around the axis which passes through the mean. It is ameasureoftheopeningofthepdf(Figure4.6).

Figure4.6 Variance,Var[X]<Var[Y].

Standarddeviation

7.

8.a.b.c.d.

9.

10.

VariationcoefficientThe coefficient of variation is a dimensionless parameter of dispersion,whichisdefinedastheratioofstandarddeviationtothemeanandisgivenbyVx=σx/μx.Variationproperties

Var[c]=0foreveryconstantcVar[cX]=c2Var[X]Var[a+bX]=b2Var[X]Var[X]=E[X2]−(E[X])2

Proof

Asymmetryαx

αxisameasureofthesymmetryofthepdf(Figure4.7).Coefficientofasymmetry:γx=αx/σx3

KurtosisKx:

Kxisameasureofthekurtosisofthepdf(Figure4.8).

11.a.

b.

12.

13.

Figure4.7 Asymmetry

Figure4.8 Kurtosis.

Generalfunctionsofstatisticalmoments:Centralmoments(aroundthemean)

(r=2,dispersion;r=3,asymmetry)Momentsaroundtheoriginoftheaxes

(r=1,meanvalue)Functionforthecreationofmomentsmx(t):

Standarderror:One of the most important theorems in statistics is the central limittheorem, according towhich themean value of n independent variableswithmeanmandstandarddeviationσalsofollowsthenormaldistribution

14.

withmeanmandstandarddeviation or inthecaseofsample.Theterm iscalledstandarderror.Frequencydistributions:In the science of hydrology, depending on the time step used for dataanalysis, theamountofdatamaybetoolarge; thus, theirmanagementisdifficult.Insuchcases,thedataaredividedintoclasses.Theselectionofspaceinclassesaffectstheappearanceofthehistogram.Severalscientistshave derived various relationships on the number of classes. Spiegel(1961)proposedthatthenumberofclassesshouldbefrom5to20,SteelandTorrie(1960)suggestedthatspaceshouldbebetween¼and½ofthestandarddeviationofthedata,andSturges(1926)presentedtheequationm=1+3.3logN,wheremisthenumberofclassesandNthenumberofdata.Afterdeterminingthenumberofclassesandtheirrespectivelimits,thenumberofvaluesineachclassiscalculated.Thisnumberiscalledtheabsolutefrequencyoftheclass.

Relativefrequencyisthepercentageratiooftheabsolutefrequencyoftheclasstotheabsolutefrequencyofallclasses.Cumulativefrequencyisthesumofthefrequenciesofallclassesuptothelimit,whilerelativecumulativefrequencyisthesumof therelativefrequenciesup to the limit.Basedon thesereasons, therespectivepolygonsoffrequencyordiagramsgivinginformationonthedatacanbe generated. The histograms are used for the graphic representation offrequency distributions of continuous data. The same applies for frequencypolygons.

4.3DISTRIBUTIONS

From the plethora of theoretical distributions available in the literature, thischapterfocusesonthosewhichfindapplicationsinhydrology.Thedistributionsdescribedinthefollowingaredividedintodiscreteorcontinuous,dependingonwhether thevariable isdiscreteorcontinuous.Thestudyandunderstandingofthe theoretical distributions are necessary to approximate the empiricaldistributionsofhistoricaldata.

4.3.1Normaldistribution

Itisthemostimportantdistributionfromthetheoreticalandpracticalpointsofview.Itarisesincaseswherearandomvariableisthesumofalargenumberofindependent random variables, each contributing a small amount in the total

score (central limit theorem). The normal distribution is characterized by asymmetricpdfintheshapeofabell(alsoknownasGaussiandistribution).Ithastwo parameters, themean μ and standard deviation σ. The pdf of the normaldistributionisgivenbytherelationship

Themeanvalueof thenormaldistribution isdenotedasE[X]=μx themod is

,andthevarianceisdenotedasf(x)→0asx→±∞andf(x)ismaximumatx=μ.Therefore,f(x)issymmetric

aroundμ.Thecentralmomentrofthedistributionisgivenby

Thedistributionfunctionofthenormaldistributionisgivenbytherelationship

ThevaluesofF(x)representtheareaunderthecurveofthenormaldistribution.Tofacilitatethecalculationofthevaluesofpdfandthedistributionfunction,thestandardnormal distributionvariableZ = (x−μ)/σ is used,which follows thenormaldistributionandhasazeromeanandastandarddeviationof1.ItappearsthatthepdfforthevariableZisgivenby

andthecumulativedistributionfunctionis

ThevalueofZistakenfromtables,whereitisrelatedtothevaluesofF(Z),and

thenxiscalculatedfromtherelationship

foranyvalueofmean(μ)andstandarddeviation(σ).Table4.1showsF(Z)inrelationtoZfor0≤Z≤3.09andTable4.2for−3.09

≤Z≤−0.10.In a specially designed graph, known as a normal distribution graph or a

Gaussiangraph, thedistribution function is a straight line.Specifically, in thisgraph, the pairs of points are designed by plotting the value of the standardnormal variable Z and the respective value of the variable x. The chart isdesignedinsuchawaythatbyjoiningthesepointsastraightlineisobtained.AnormaldistributiondiagramisgiveninFigure4.9.All thesecalculationsconcernasampleof thepopulation.Thus, theaverage

valuecorrespondstotheaveragevalueoftherepresentativesample.Inpractice,itisusefultosetlimitsaroundthemeanofthesample,withinwhichisthemeanof thepopulationwithsomeprobability.These limitsareknownasconfidencelimitsandforthenormaldistributionaregivenbythefollowingequations:

whereZ(1+α)/2isthevariableofthestandardizednormaldistributionforconfidencelevelα%σTisthestandarddeviationofX(T)σisthestandarddeviationofthesampleNisthenumberofobservationsofthesample

Table4.1CumulativedistributionfunctionF(Z)for0≤Z≤3.09

Thenormaldistribution is themostwidelyuseddistribution. It isused for theanalysisofvariance, thehypothesis testing, theestimationof randomerrorsofhydrologicalmeasurements, thecomparisonofdistributionsand thegenerationof random numbers. A random variable is expected to follow a normaldistributionifitisthesumofindependenteffects.

Remark 1: The tables of normal distribution calculate the distribution of thestandardizednormalvariableZ=(X−μx)/σx,whichhaszeromeanvalueandunitstandarddeviation.Then,thefollowingapplies:

Table4.2CumulativedistributionfunctionF(Z)for−3.09≤Z≤−0.10

Remark2:FX(x)=1−FX(−x)symmetrical

Remark3:IfX1~N(mx1,σx1)andX2~N(mx2,σx2)aretwoindependentnormal

randomvariables,thenX=X1+X2isalsonormalwith .

Figure4.9 Normaldistributiondiagram.

4.3.2Lognormaldistribution

Alognormaldistributionoccurswhenarandomvariableistheproductofmany

randomvariables,e.g.insedimenttransport,wherethevolumeofparticlesisthemultipliereffectofmanyconflictssuchasthefailureofstructuresormachines:

Fromthecentrallimittheorem,giventhatthenumberNisrelativelylarge,lnXwillbenormallydistributed.Letμ1nX,σ1nX themeanandstandarddeviationoflnX.Then,thepdfofthelognormaldistributionis

or

The variation coefficient Vx and asymmetryCs are given from the followingrelations:

Theconfidencelimitsforconfidencelevelαaregivenasfollows:

whereZ(1+α)/2isthevariableofthestandardnormaldistributionforconfidencelevelα%σTisthestandarddeviationofX(T)σisthestandarddeviationofthelogarithmsofthesampleNisthenumberofobservationsofthesample

Remark1:IfXfollowslognormaldistribution,thenthesameappliesforY=aXbwhere

Remark2:If followlognormaldistributionandare independent, then thesameappliesfor and

Remark3:Usingthetablesofnormaldistribution(Tables4.1and4.2),

where fU and FU are the pdf and the cumulative probability function of thestandardnormalvariable.

4.4SOMEIMPORTANTDISCRETEDISTRIBUTIONS

4.4.1BernoullitrialsandtheBernoullidistribution

Bernoullidistributiongivestheprobabilityofanevent.Thevariablexisdefinedasfollows:

Thedistributionisgivenbythefollowingrelationship:

where

4.4.2Binomialdistribution

LetusdefineNBernoullivariables [X1,X2,…,XN]which takeon independentvalues.IfX=X1+X2+…+XN,then

where0≤p≤1andNisaninteger.Theparametersaregivenbythefollowingrelationships:

4.4.3Geometricdistribution

Let[X1,X2,…,XN]beNBernoullivariablesofprobabilityp.IfX=thenumberofexperiments,whereaneventoccursforthefirsttime,then

where0≤p≤1and

4.4.4Poissondistribution

LetpbetheprobabilityoftheoccurrenceofastorminaperiodΔtj,ΣΔtj=t.Theassumptionsarethefollowing:(1)theprobabilityoftheoccurrenceofmorethanonestormisnegligible,(2)stormsareindependenteventsand(3)thedurationofthe storm is very short compared to the time periods. Let x be the number ofstormswhich occurred during the time t of an experiment in which an eventoccursforthefirsttime.Then,fromthebinomialdistribution,

where0≤p≤1andNisanintegerE[X]=NpistheexpectednumberofstormsintimetIfΔtj→0,thenE[X]=Np=v.

Inthiscase,f(x)followsthePoissondistribution

Theparameterv = λt,where λ is themean, is equal to themean value of theeventsduringtheperiodt.Then,

Also, if X1, X2 are two variables which follow the Poisson distribution withparameters v1, v2, then the sum X = X1 + X2 also follows the Poisson withparameterv=v1+v2.

4.4.5Uniformdistribution

Itisthesimplestdistribution.Allresultshavethesameprobabilityofoccurrenceandaredefinedasfollows:

Theparametersofthedistributionaregivenbythefollowingrelationships:

4.5SOMEIMPORTANTCONTINUOUSDISTRIBUTIONS

4.5.1Uniformdistribution

Thepdfoftheuniformdistributionis

4.5.2Exponentialdistribution

Thepdfoftheexponentialdistributionis

whereλ>0and

4.5.3Gammadistribution

IfX1,…,XKareKindependent,exponentiallydistributedrandomvariables,with

fXi(xi;λ)and ,thenthepdfofgammadistributionis

Generally,

where

The limits of confidence for confidence level a% are given by the followingrelationship:

whereZ(1+α)/2isthevariableofthenormaldistributionforconfidencelevelα%

xisthemeanofthesamplesisthestandarddeviationofX(T)k(T)=Z(1−1/T)CisthevariabilitycoefficientNisthesizeofthesample

Agammadistributionpaper for each function cannot bemanufactured, exceptforthecasewhentheparameterkhasaspecificvalue.

Remark:IfX1,X2areindependentgammavariables,withparameters(κ1,λ),(κ2,λ),thenX=X1+X2isalsoagammavariablewithparameters(κ1+κ2,λ).

4.5.3.1Gammadistributionofthreeparameters(PearsontypeIII)

Addingaconstantc to the two-parametergammadistribution, the result is thethree-parametergammadistribution,whichiswidelyusedinhydrology.Thepdfiswrittenasfollows:

where

Thedistributionfunctionis ,andtheparametersareestimatedasfollows:

Theestimationoftheconfidencelimitsisacomplexprocedureandthegammadistributionpapercanlinearizeeverygammadistributionfunctionforagivenk.

4.5.4Log-Pearsondistribution

IthasbeenwidelyusedintheUnitedStatessince1967aftertheformationoftheWaterResourcesCouncilforadoptionastheofficialfloodfrequencydistributiontoallservices. If therandomvariableY= lnX follows thedistributionPearsontypeIII,thenXfollowsthedistributionlog-PearsontypeIII,withpdf:

where

The distribution function is given by the relationship , and theparametersareestimatedasfollows:

Theestimationoftheconfidencelimitsisacomplexprocedure,andthegammadistributionpapercanlinearizeeverygammadistributionfunctionforagivenk.

1.

2.

3.

4.6STATISTICALANALYSISOFEXTREMES

ConsiderasampleofN independentandidenticaldistributedrandomvariablesY1,…,YN.IfXisdefinedasX=max{Y1,…,YN},then

Consequently, the distribution of X depends onN and the parent distributionFY(·)andfY(·).UndercertainconditionsandforavarietyofparentdistributionswhenNbecomessufficiently large, theextremevaluedistribution fx(·) tends tocharacterizethefollowinglimitdistributions:

Type I: The paternal distribution is not limited in the direction of thespecificlimitandhasfinitemoments(e.g.exponential,normal,lognormalandgamma).TypeII:TheparentdistributionisthesameasintypeI,butallmomentsarenotnecessarilyfinite.TypeIII:Theparentdistributionis limitedtothedirectionofthedesirededge(e.g.exponential,lognormalandgamma).

Type II has no important applications in hydrology. Types I and III are oftenusedforthesimulationofmaximumorminimumrunoff,rainfalldepth,etc.

4.6.1Pointfrequencyanalysis

Frequency analysis is carried out to determine the frequency of hydrologicalevents. The hydrological data for frequency analysis should be processedaccording to the purpose of analysis, the length of the recording period, thecompletenessofdataandthehomogeneity.Theperiodlengthshouldbegreaterthan25years,sotheresultsofthedistributionareacceptable.Themissingdatashould be estimated using regional analysis or by correlation with otherhydrological data in the region. The hydrological data are presented inchronological order. These data constitute the full series. The frequency

distributions may be combined with data using two methods: the graphicalmethodandthemethodoffrequencyfactors.

4.6.1.1Graphicalmethod

This method involves the combination of hypothetical probability distributiondataandobserveddata.Thedataareallocatedinascendingordescendingorderof size. In descending ranking, the unit is given to the highest value.The lastvalue is set to n, wheren is the number of data. This classification gives anestimate of the probability of exceedance, i.e. the probability of a value to beequaltoorgreaterthanthevalueoftheparameter.Iftheclassificationisdoneinascendingorder,thenwehaveanestimateoftheprobabilityofnon-exceedance,i.e. the probability of a value to be equal to or less than the given value.Thepointdataaredesignedwiththehelpofanequation.Oneofthemostcommoninhydrologyis

wherePm is the probability of exceedance of them point data of the sample,whichisrankedindescendingorder.Thereturnperiodofthempointdatais

Theobserveddataandtheirexceedanceprobabilitiesaredesigneddependingonthehypotheticaldistributionofprobability.

4.6.1.2Methodoffrequencyfactor

Chow(1964)proposedtheuseofafrequencyfactor inhydrologicalfrequencyanalysis.IfahydrologicalvariableXisdesignedchronologically,thenanxvaluewillconsistoftwoparts:theaveragexandthedifferenceΔxfromtheaverage,i.e.

The value of Δx may be positive or negative and may be expressed bymultiplyingthestandarddeviationSwiththefrequencyfactorK.Inthiscase,

whereKdependsonthereturnperiodTandtheprobabilitydistributionofx.Kisthe number of standard deviations above and below the average. Fordistributionsoftwoparameters,thevalueofKvarieswiththeprobabilityorthereturnperiodT.Forasymmetricdistributions, thevalueofK changeswith theasymmetry coefficient and is sensitive to the length of the recording period(Singh,1992).

4.6.2Gumbelmaximumdistribution

Consider a set ofN observations of a random variable, whereN is relativelylarge. If the series is divided into n subsamples of sizem, soN = nm, eachsample contains a large and a small value, which are the extremes. Gumbel(1958)showedthatthenhighervaluesofthesamplesasymptoticallyfollowanextremevaluedistribution(TypeI).Thepdfandcumulativedistributionfunctionaregivenbythefollowingrelations:

where−∞<x<∞,−∞<u<∞,α>0

Gumbel(1954)wasthefirstwhousedtheextremevaluetheoryfortheanalysisof flood frequency. The distribution is often referred to as the Gumbeldistributionorthedoublenegativeexponentialdistribution.Itisoneofthemostwidelyuseddistributionsfortheanalysisoffloodfrequency,maximumvaluesofrainfall,etc.:

In line with the confidence limits of the normal distribution, the confidencelimitsoftheGumbeldistributionaregivenbythefollowingrelations:

Remark1:ThetablesgivethedistributionofthestandardvariableW=(X−u)a.Thefollowingapplies:

TheprobabilityofexceedanceF1istheinverseofthereturnperiodT,whiletheprobabilityofnon-exceedanceis1−F1.TheGumbelgraphpaperisconstructedbasedontherelation(W,T).DifferentvaluesofTareselected,suchas1.01,1.5,2, 5, 10, 20, 50, 100, 200 and 250, and the corresponding values ofW arecalculated.Theplottingofthepair(X,T)ofthemaximumvaluesofavariableontheGumbelgraphpaperisatestofhowwelltheGumbeldistributionfitsonthedata.However, basedon a relation, there is a correspondencebetween themaximumvaluexandthereturnperiodT.Differentmethodssuchasthemethodoffrequencycoefficients,maximumlikelihoodandmaximumentropyhavebeenapplied in the estimationof theGumbelparameters.Theonepresented in thischapterandwidelyusedisthemethodofmoments.

4.6.3Gumbelminimumdistribution

Consider a set ofN observations of a random variable, whereN is relativelylarge. If the series is divided into n subsamples of sizem, soN = nm, eachsample contains a large and a small value, which are the extremes. Gumbel(1958)showedthatthenminimumvaluesofthesamplesasymptoticallyfollowan extreme value distribution (Type I). The pdf and cumulative distributionfunctionaregivenbythefollowingrelations:

where

In line with the confidence limits of the normal distribution, the confidencelimitsoftheGumbeldistributionaregivenbythefollowingrelations:

with

Remark1:ThetablesgivethedistributionofthestandardvariableW=(X−u)a.Thefollowingapplies:

4.6.4Weibulldistribution

A Type III distribution in its full form is a three-parameter distribution. Itdescribesthedistributionofindependentvariablesintherangeofvalues(c,+∞)withdistributionfunction

whereρandaarepositiveconstants.

TheminimumdistributionfunctionofthetypeIIIextremevaluedistributionis

In the special case where c = 0, we obtain the simplified, two-parameterdistribution known as the Weibull distribution. The Weibull distribution iswidelyapplicableinhydrology,sinceitdescribesthedistributionoflowflowsinstreams.ThepdfoftheWeibulldistributionis

ThedistributionfunctionoftheWeibulldistributionis

Themeanandthestandarddeviationare

Theestimationof theparametersof thedistribution isbasedon themethodofmoments:

where theparametera can be estimated only by the numerical solution of therespectivecorrespondingequation.In the case when the lower bound on the parent distribution is not zero, a

displacementparameterisaddedandthedistributionfunctionbecomes

Byusingthetransformation,

Tablesofe−ycanbeusedtocalculatetheprobabilityP.ThemeanandthestandarddeviationbecomeHaan,(1985)

Throughthetransformationofyandtheuseofequationsofmeanandvariance,thefollowingequationsareobtained(Haan,1985):

where

The distribution function of the Weibull distribution is a straight line whendrawnon thespecialgraphof theWeibulldistribution.Thepairson thegrapharedesignedbyplottingthevalueofthequantityln[−ln(1−F)]onthehorizontalaxisandthevalueoflnXontheverticalaxis.

4.7TESTINGOFTHEDISTRIBUTIONS

1.2.

3.

4.

Thesuccessfulstatisticalanalysisofthehydrologicaldatadependsonthefitofthedistributiontotheempiricaldistributionofthesample.Thequestionishowwell the theoretical distribution is fitted to the empirical distribution. Thefollowingdescribes two important tests for the adjustment of the sample data,whicharewidespreadinhydrology:theX2andtheKolmogorov-Smirnovtests.

4.7.1TestX2

X2testissuitableforthetestofthefitofadistributionofapdffX(x;Θ1,…,Θρ)innpoints.Itissummarizedinthefollowingsteps:

Theparameters{Θ1,…,Θρ}ofthepdfarecalculated.The sample data are classified from the larger sample (m = 1) to thesmallest(m=N).kdifferentclasses((x0,x1],…,(xk−1,xk])aredefinedsuchthattheexpectednumberofpointsofeachclassisgreaterorequaltoSandapproximatelythesameforallclasses,wheretheexpectednumberofpointsofclassi(xi−1,xi]is

andfX(.;…)istheselectedandcalculatedpdf.The actual number of points ni in each class i is measured, and thefollowingstatisticaltermiscalculated:

isdistributedaccordingtothedistributionX2withv=k−1−pdegreesoffreedom,wherepisthenumberofparameterscalculatedfromthedata.

If ,thedistributionisappropriate.If ,thedistributionisnotappropriate.If , the conclusion cannot be drawn regarding the

appropriateness of the distribution. The test X2 should be repeated after thecollectionofmoredata.

4.7.2Kolmogorov-Smirnovtestfortheappropriatenessofadistribution

TotestthefitofthedistributionofapdffX(x;Θ1,…,Θρ)foragroupofnpoints,in accordance with the Kolmogorov-Smirnov test, the difference between thecumulativedistributionfunctionFX(Xi)and theobservedcumulativehistogramF*(X)iscalculated.TheobservedcumulativehistogramisgivenbyF*(Xi)=i/n,where i is the biggest observed value from a sample of size n. Based on thesampledata,thestatisticalparameterDiscalculated:

ThedistributionisacceptableifD<c,wherethevaluesofparametercaregivenintablesdependingonthesamplesizeandthedesiredsignificancelevel,a.Thesignificance level a is defined as the probability of making type I error, i.e.rejecting thenullhypothesiswhile it iscorrect. It isdescribed indetail in test-relatedstatisticalbooks(Eadieetal.,1971;HollanderandWolfe,1973).

Example4.3CalculatethemeanandthestandarddeviationoftherandomvariableY:

whereX1,X2,…,XNarerandomvariables,whichareindependentandhavethesamemean,μ,andvariance,σ2.

SolutionCalculationofthemeanµY:

CalculationofthemeanμY:

1.2.

1.

Thus,thestandarddeviationoftherandomvariableYisσY=σ/N1/2.

Example4.4ForthestandardvariableU=(x−mx)/σχ,provethat

E[(X−mx)/σχ]=0andVar[(X−mx)/σχ]=1ThecorrelationcoefficientbetweenthevariablesXandZisthecovariancebetweentherespectivestandardvariables[(X−mx)/σχ]and[(Z−mZ)/σZ]

Solution

E[(X−mx)/σχ]=0Proof:

Theexpectedvalueofthemeanisthemeanμ=E(X).Thus,

Proof:

Thus,

1.

2.

2.

Proof:Weknowthat

and

whereE((X−mx)/σχ=0,E((Z−mZ)/σχ=0;thus,

Example4.5A flood control reservoir was designed for theN-year flood, i.e. itscapacitywillbeexceededbytheN-yearorgreaterflood.ThesizeoftheN-yearfloodisdefinedasthatwhichhastheprobability1/Ntobeexceededeachyear.Weassumethat thesuccessiveannualfloodsareindependent.

Whatistheprobabilitythatafloodequaltoorgreaterthanthe50-yearfloodwilloccurin50years?Whatistheprobabilitythatthreefloodsequaltoorgreaterthanthe50-yearfloodwilloccurin50years?

3.

4.

5.

What is the probability that one or more floods equal to orgreaterthanthe50-yearfloodwilloccurin50years?If a company constructs 20 independent systems – i.e. in 20differentareas–designedforthefloodof500years,whatisthedistributionofthenumberofsystemsthatwillfailatleastonceinthefirst50yearsaftertheconstruction?In1958,thesizeofthe50-yearfloodwasestimated.Inthenext10years, twofloodswerefound tobegreater than thissize. Iftheinitialassessmentiscorrect,whatistheprobabilityofsucharemark?

SolutionThe random variable is the annual flood and follows a discretedistribution. For the calculation of probabilities, the binomialdistributionisused:

wherexisthenumberoffloodsequaltoorgreaterthanthefloodwithareturnperiodof50years

Nisthenumberofyearsforwhichtheprobabilityofoccurrenceorexceedanceofthe50-yearfloodisexamined

p=1/Tistheprobabilityofoccurrenceorexceedanceofthe50-yearfloodateachtimestep(1year)

1.

N(years) T(years) p=1/T

x

50 50 0.02 1

Thus,theprobabilityofafloodoccurrenceequaltoorgreaterthanthe50-yearfloodinatimeperiodof50yearsis

2.

N(years) T(years) p= x

1/T

50 50 0.02 3

Thus,theprobabilityofoccurrenceofthreefloodsequaltoorgreaterthanthe50-yearfloodinatimeperiodof50yearsis

3.Theprobabilityofoccurrenceofoneormorefloodsequaltoorgreaterthanthe50-yearfloodinatimeperiodof50yearsisP(X≥1)=1−P(X=0).Thus,theprobabilityofoccurrenceofnofloodequaltoorgreaterthanthe50-yearfloodcanbecalculated.

N(years) T(years) p=1/T

x

50 50 0.02 0

and

4. Initially, the probability of failure of each system j isexamined. The system is considered to fail in the case ofoccurrenceorexceedanceofthe500-yearfloodatleastonceina50-yeartimeperiod.xisthenumberoffloodsequaltoorgreaterthanthe500-yearflood.

N(years) T(years) p=1/T

x

50 500 0.002 0

Thus,

Thus,theprobabilityofthefailureofanysystemjis

Theprobabilityoffailureofnone(X=0)to20(X=20)systemsina50-yeartimeperiodwithafailureprobabilityofeachsystemp=0.095iscalculatedasfollows:

wherexisthenumberofsystemswhichfailNisthetotalnumberofsystemsp=1/Tistheprobabilityoffailureofeachsystemjina50-yeartimeperiod

N(systems) p=1/T

20 0.095

and

Thus,theprobabilityoffailureofthesystemsforx=0,1,2,…,20isgiveninTable4.3.WeobservethattheprobabilityoffailureofXsystemsdecreasesasXincreasesfrom1to20.

5.Inthenext10years,thefloodwiththe50-yearreturnperiodwasexceededtwotimes.Thus,

N(years) T(years) p=1/T

x

10 50 0.02 2

and

1.

2.

3.

Table4.3Probabilityoffailure

X P(X)

0 0.136

1 0.285

2 0.284

3 0.179

4 0.080

5 0.027

6 0.007

7 0.001

8 2.5E−049 3.5E−0510 4.1E−0611 3.9E−0712 3.1E−0813 2.0E−0914 1.0E−1015 4.4E−1216 1.4E−1317 3.5E−1518 6.2E−1719 6.8E−1920 3.6E−21

Example4.6

Themean annual and themaximum daily runoff are given in Table4.4.Thevalueswere estimated at a river cross section for a 30-yeartimeperiod.

Calculate the statistical characteristics of both samples (mean,standard deviation, coefficients of variation, skewness andkurtosis,maximumandminimumvalues).Fitthenormaldistribution(Gaussian)tothesampleofthemeanannualdischarges.Design the sample and the theoretical distributionon anormaldistributiongraph.

4.

5.

6.

7.

8.

9.

1.

2.

Estimate the values of the mean annual discharges, whichcorrespondtoreturnperiodsof10,50and200years,basedonthenormaldistribution.If 75% of themean annual discharge is sufficient tomeet thewaterneedsofanadjacentcity,findtheprobabilityoffailureofthecompletecoverageofthecity’swaterneedsduringayear.AdjustthedistributionsofGumbelandlognormaltothesampleof themaximumdailydischarge.Check the appropriatenessoftheGumbeldistributionwiththetestX2.Design the sample of themax daily runoff and the theoreticalGumbeldistributiononaGumbeldistributiongraph.Estimate the maximum daily runoff values corresponding toreturn periods of 10, 20 and 1000 years, using bothfitdistributions.Calculatethe95%confidencelimitsofGaussiandistributionforthevaluesofquestion4andGumbeldistributionforthevaluesofquestion8.

SolutionThestatisticalcharacteristicsofbothsamplesaregiveninTable4.5.Fitofnormaldistribution

Thestatisticalcharacteristicsofthesampleofmeanannualdischarge,which are necessary for normal distribution, are the mean and thestandarddeviation(calculatedinquestion1).Havingcalculatedthesecharacteristics, the sample is ranked in descending order and thesamplevaluesarenumbered.ThenthereturnperiodiscalculatedusingtheWeibullrelationship(T=(N+1)/m),whereNisthenumberofthesample values and m is the rank. Then, the probability of non-exceedanceFiscalculated(fromF=1–1/T).ForeachvalueofF,thestandardvariableZiscalculated(fromTable4.6).BasedonthevaluesofZ,X is calculated (fromX =μ+Z*σ)which corresponds to eachprobability.TheresultsareshowninTable4.6.

Table4.4Meanannualandmaximumdailyrunoff

Table4.5Statisticalcharacteristicsofbothsamples

Meanannualdischarge(m3/s)

Maximumdailydischarge(m3/s)

Mean 24.71 379.07

Standarddeviation 6.63 198.58

Variationcoefficient 0.27 0.52

Skewnesscoefficient 0.07 0.76

Kurtosiscoefficient −0.38 0.42

Min 10.60 85.00

Max 37.20 904.00

Table4.6Resultsofthefitofnormaldistribution

3. Inordertocheckthefitofthenormaldistributiontothesampleof mean annual discharge, the sample and the theoreticaldistribution are designed on normal distribution graph. Thevalues of the distribution function F are marked on thehorizontal axis of the graph, and the values of discharge aremarkedontheverticalaxis.Thepoints(F,X)ofthetableoftheprecedingquestionare joinedbya straight line.Thecloser the

4.

5.

6.

straight lineis tothelinejoiningthepoints(F,Q) (whereQ isthe time series with the descending values), the morerepresentativeisthenormaldistributionforthesample.Themean(µ=24.71m3/s)andthestandarddeviation(σ=6.63m3/s)of thesampleofmeanannualdischargeswerecalculatedin question 1. The probability of non-exceedance is calculatedforreturnperiodsT=10,50and200yearsusingtherelationF=1–1/T,and then thestandardvariableZ isbasedon tabulardata.TherespectivevalueofthemeanannualrunoffisobtainedbyapplyingtherelationshipX=µ+Z*σforeachZ,giventhatµandaareknownfromquestion1.Thus,weobtainTable4.7.Suppose that the water needs satisfy the normal distribution.Then,thestandardizedvariableZisZ=(18.54–24.71)/6.63=™0.932. The value of non-exceedance probability, F, whichcorrespondstoZisequalto17.57%,asperTable4.8.Thus,theprobability of failure of the complete coverage of the city’swaterdemandduringanygivenyear reflects theprobabilityofexceedanceandisgivenbyF′=1−F=0.8243.FitoftheGumbeldistribution.

The statistical characteristics of the sample of maximum dailydischarges,whichareusefulfortheGumbeldistribution,arethemean,standarddeviationcalculatedinquestion1andtheparametersaandc,whicharegivenasmean=379.07m3/s.Standarddeviation=198.58m3/s,a=0.01andc=289.70,respectively.Once the statistical characteristics of the sample have been

calculated, in order tofit the Gumbel distribution to the sample, thesampledatavaluesarerankedindescendingorderandthevaluesarenumbered.Then,thereturnperiodfromtheWeibullrelationship(T=(N+1)/m)iscalculated,whereNisthenumberofthesamplevalues)andthenon-exceedanceprobability,F=1–1/T,iscalculated.ForeachvalueofF,thequantityln[ln(T)−ln(T−1)]iscalculated

and then the respective value ofX directly from the equation of thetheoreticalGumbel distribution.The process has been tabulated, andtheresultsaregiveninTable4.8.

Fitofthelognormaldistribution

The statistical characteristics of the sample of maximum daily

dischargeswhichareofinterestforthelognormaldistributionarethemean, standard deviation of the sample calculated in question 1 andthemeanandstandarddeviationofthetimeseriesoflogarithmsofthesample:

Mean(sample)μx=379.07m3/sStandarddeviation(sample)σx=198.58m3/sMean(samplelogarithms)μy=5.79m3/sStandarddeviation(samplelogarithms)σy=0.57m3/s

Table 4.7 Probability of non-exceedance

Returnperiod,T

Probabilityofnotbeing

exceeded,F

Standardvariable,

Z

Meanannualrunoff(m3/s)

10 0.9 1.2816 33.21

50 0.98 2.0537 38.33

200 0.995 2.5758 41.79

Table4.8ResultsofthetheoreticalGumbeldistribution

Once the statistical characteristics of the sample are calculated forthefit of the lognormal distribution to the sample, the samples areranked in descending order and the values are numbered. Then, thereturnperiodfromtheWeibullrelationshipT=(N+1)/miscalculated,whereN is thenumberofthesamplevalues,andthenon-exceedanceprobability,F=1−1/T,iscalculated.The standard variable Z is calculated for each value of F (from

Table4.9), and then, the respectivevalueofXwhich corresponds toeachprobabilityiscalculatedfromtheequation

Theprocesshasbeentabulated,andtheresultsarepresentedinTable4.9.

TestX2

ThefirststepforthetestX2isthecalculationoftheparametersofthedistribution to be adjusted. The Gumbel distribution has twoparameters (r = 2), and the parameter values were calculated at thebeginning of question 6. Then, the sample is divided into k classes(intervals)ofequalprobability.Thedivisionof thesampleintok=5classeswaschosen.

Table4.9Resultsofthetheoreticallognormaldistribution

Thedegreeoffreedomofthedistributionisv=k−r−1=5−2−1=2.Xisdetermined,whichcorrespondstothecumulativeprobabilityof

eachclassandthelimitsoftheclasses.

7.

Theexpected(theoretical)numberofobservationsiscalculatedforeachclass,Ei=N*pi,whereNisthesizeofthesample(N=30).The valuesNi of the sample, which are within the limits of each

class,aremeasured,andthestatisticalparameterDiscalculated:D=∑[(Ni−Ei)2/Ei].Thus,weobtainTable4.10.The value of parameter D is compared with the value from the

tablesofX2forthespecifcdegreeoffreedom(v=2)andsignificancelevelα.Theusualvaluesofαwereconsidered:1%,5%and10%.

Forv=2andα=1%,itresultsfromthetablesofX2thatX2=9.2Forv=2andα=5%,itresultsfromthetablesofX2thatX2=6Forv=2andα=10%,itresultsfromthetablesofX2thatX2=4.6

Table4.10TestX2

Limitsofclasses

Expectedprobability,

pi

Ei Ni (Ni−

Ei)2/Ei

0<X<215.988

0.2 6 6 0

215.988<X<303.246

0.2 6 6 0

303.246<X<393.756

0.2 6 6 0

393.756<X<522.047

0.2 6 6 0

522.047<X<1

0.2 6 6 0

D 0

TheresultisthatD<X2fortheusualsignificancelevelsforwhichthesamplewastested.Thus, thesample isconsidered to follow theGumbeldistribution fortheusualsignificancelevels.

To check the adjustment of the Gumbel distribution to the

8.

sample of daily maximum discharges, the sample and thetheoreticaldistributionaredesignedontheGumbeldistributiongraph.ThevaluesofXaremarkedon theverticalaxisand thevalues ofF on the horizontal axis, which results in a straightline. Then, the points are plotted with the values of thedescendingorderof the table inquestion6on theverticalaxisand the values of F on the horizontal axis. The shorter thedistance of these points from the straight line, the morerepresentativeistheGumbeldistributionforthesample.Gumbeldistribution

Given the values of the return periods and the parameters of theGumbel distribution, which were calculated in question 6, the non-exceedanceprobabilityF(=1–1/T) iscalculated,andtherespectivedischarge value is based on the relevant equations. The results areshowninTable4.11.

Lognormaldistribution

First,thenon-exceedanceprobabilityF(=1−1/T)iscalculatedusingthe given return periods and the parameters of the lognormaldistribution,ascalculatedinquestion6.Then,therespectivevalueofthe standard variableZ is calculated, and the discharge is calculatedbasedontherelationshipX=eZ*σy+µy.

TheresultsareshowninTable4.12.

Table 4.11 Non-exceedanceprobability using the Gumbeldistribution

Returnperiod,T

Probability,F

Discharge(m3/s)

10 0.9 638.290

20 0.95 749.793

1000 0.999 1359.651

Table 4.12 Non-exceedanceprobability using thelognormaldistribution

Return Probability Standard Discharge

9.

period,T

ofnon-exceedance,

F

variable,Z

(m3/s)

10 0.9 1.2816 684.63

20 0.95 1.6449 843.63

1000 0.999 3.0902 1936.37

Table4.13Non-exceedanceprobabilityusingthenormaldistribution

Table4.14Calculationofthelimitsforα=95%

NormaldistributionFor the return periods of question 4, the probability of non-exceedance (F = 1 – 1/T), the respective standard variableZFand the parameters δ and ST are calculated. For a = 95%, thevalueofZ(1+α)/2=1.96(fromTable4.13)iscalculated.

ThecalculationsareshowninTable4.13.

Gumbeldistribution

Forthereturnperiodsofquestion4,thenon-exceedanceprobabilityF(F=1−1/T) and theparametersK(T)=−0.45−0.7797*ln[−ln(1−1/T)],δandSTarecalculated.Forα=95%,thevalueofZ(1+α)/2=1.96(fromTable4.14).

1.

2.

ThecalculationsareshowninTable4.14.

Example4.7

Theminimumannualmonthlydischargesonariverarefoundtohavean average of 150m3/s, a standard deviation of 70m3/s and a skewcoefficientof1.4.UsingboththeWeibulldistributionandtheGumbelldistribution, calculate the probability of an annual minimum flow,whichislessthan100m3/s.

Solution

TheWeibulldistributionBy solving the equations given in theWeibull distribution,wehavea=1.266,A(α)=0.098andB(α)=1.36

where

TheGumbeldistributionUsingthemethodofmoments,wecalculatethetwoparametersαandu

where

4.8INTENSITY–DURATION–FREQUENCYCURVES

Severalmajorwaterresourceprojectssuchasstormsewernetworks,reservoirsandfloodprotectionworksaredesignedbasedonananalysisofrainfallintensity(i),duration(d) and frequency (f). Sometimes, rainfall intensity is replaced bytherainfalldepthandthefrequencybythereturnperiod.Therainfallintensityisusually the average intensity during a rainfall.A high-intensity rainfall occurslessfrequentlythanalow-intensityrainfall.Theannualhighestrainfalldepthh(mm)andtheintensityi(mm/h),observed

atarainfallmonitoringstation,foraparticularrainfalldurationt,duringnyearsofmeasurements are ordered in descending order ofmagnitude. The result ofthisanalysisistheconstructionoftherelationship(i,t,T)or(h,t,T)betweentherainfallintensity(ordepth)andthedurationandreturnperiod,whichisknownastheidfcurve.idfcurvesarenecessaryforthehydrologicdesignofnumeroushydrologicalworks,suchastherainwatersewagenetwork,thedesignofawaterretentionbasinoraspillway.Basedonthem,therainfalldepthandintensitycanbeeasilycalculatedforeveryrainfalldurationandreturnperiod,andgenerally,oneofthethreevariables(hori,t,T)canbeestimatedwhentheothertwoareknown.Forexample,undersuchestimation,theresultcanbethedesignstormofaninfrastructurewithaspecificreturnperiodandduration,whosestandardsaremost often used as design criteria (the researcher decides for the scope). Thedesignstormproperlydistributed(duringitsduration)intimecanbeusedwiththeresponsefunctionofthebasin(e.g.theunithydrograph,ifthebasinislinear)atthesiteoftheinfrastructure(spillway),andinthiswaythedesignfloodfortheinfrastructure is derived for a specific return period and risk, as explained inChapter6.Theprocess, following thealreadyknownfrequencyanalysisof theordered

seriesofmaximarainfalldepthshiorrainfallintensitiesiiforacertaindurationt1, consists of repeating the frequency analysis ofmaxima hi or ii for severaldifferentdurationst2,t3,…,trdurations.Fromtheproperprobabilitydistribution(e.g.Gumbel)foreveryt1,t2,…,trdurations,thevalueofh1,h2,…,hrori1, i2,…,iroftherainfalldepthorintensityforaspecificreturnperiodTisestimated.Theanalytical idf relationships fora specified returnperiodT between rainfalldepthhandrainfall intensity iwithduration tareusuallyoneof thefollowingforms:

Simplifiedexponential:

wherekandmareconstants.Theserelationshipsarethesimplestonesbecausetheycaneasilybecome linearusing logarithmic transformation,and thevaluesofkandmarecalculatedusingtheleastsquaresmethod.

Hyperbolicform:

wherebisanadditionalconstant.

Mixedform:

These relationships have better fexibility and generality as they contain thecorrection parameter b, which corrects the timescale as the scattering pointsaroundtheidfcurveareoptimized.Theacquisitionofrelationships(hori,t)fordifferentreturnperiodsleadstoidffamilyofcurves(hori,t,T)oftheform

where α is an additional parameter. In another approach, the idf curves areparallelindoublelogarithmicpaperwithTparameterandhoriand(t+b)axes,followingtheformlogi=log(kTα)—mlog(t+b).The setting of this equation begins with the calculation of the correction

parameterb so that the scatteringpoints around the idf curves is optimized (itcanberesolvedanalyticallyaswellasgraphicallyindoublelogarithmicpaper).Then, for everyT duration of the analysis and by the use of the least squaresmethod,AT = log(kTα) andm can be calculated. Finally, based on the pair ofvalues(AT,logT)whicharealreadyknown,kandαareestimatedwiththeuseoftheleastsquaresmethodandtheanalyticalequationAT=logk+αlogT.Therefore,allfourparameters(b,m,k,α)usedintheanalyticalexpressionof

theidfcurvesforaspecificsiteofamonitoringstationarecalculated.Usually,idfcurvesareusefulwhentheyrepresentawholebasinandnotaspecificsite.Inthat case, for every t1, t2,…, tr durations of the analyzed rainfalls, the annualmaximum average values for thewhole basinmust be calculated (e.g. by the

1.

Thiessenmethod)followedbyananalysisusingtheaveragevaluesofthebasin.Theacquisitionofprimarydata,meaningtheserieshiorii foreveryrainfall

duration time, is in great importance for the frequency analysis of the rainfallmaxima. Particularly, in order to record rainfall durations andmultiples, evenafter24h,itisnecessaryaraingaugetobeinstalledattheanalysissiteinorderto continuously record the rainfall. In recent years, fully automatic telemetricmonitoringstationswithahightemporalresolutionarebeingusedwidely,fromwhichitiseasytoderivetherainfallmaximadepthatdifferentdurations.Fromthegauge’stape(usuallyweeklyordaily)andforeveryyeartheperiodt1,t2,…,tr,ofhourswiththehighestannualdepthischosen.Therainfallbeingselectedmustbecontinuousorwithminorgapssothatitisnotconsideredasadifferentevent.Therefore,everymaximumshouldbereferredtoasinglemeteorologicaleventandnottotwoconsecutiveevents.Also,specialattentionshouldbegiveninthecasewherethemaximahioriiofatduration,butaverageforabasin,iscalculated.First,themaximumvalueforaspecificmonitoringstationispointedout(usuallyfortheonewiththehighestrainfalldepth).Then,thevaluesoftheother stations, which are situated at the same basin and for the same rainfalleventforthespecificduration,arebeingpointedout.Finally,theaveragevalueis determined using thesemethods, e.g. the Thiessenmethod and theKrigingmethod.Theaveragehighestvalueofthebasinforthespecificdurationmustbeverified by estimating the average highest value of the basin for this durationstarting from the highest annual point value of the same duration in anotherstation situated in the basin or by depleting practically the probability of theexistenceofanotherhighestvalueinthesameyear.Finally, theconstructed idf curves inmanycountrieshavebeenencountered

massively for extensive geographical areas. So, ready-to-use maps have beenestablishedforeachregion,so idfcurves foranypointcanbeeasily retrieved,without the need of primary historical data. The U.S. Weather Bureauestablishedsuchmaps in1961(Viessmanetal.,1989;Wanielistaetal.,1997).Thesemapsillustratethecurvesofequalprecipitationofmaximarainfalldepthsfordifferentrainfalldurationsanddifferentreturnperiods.

4.8.1Constructionoftheidfcurves

The precipitation curves may be designed using the frequency analysis ofrecordedrainfalldata.Thefollowingarethestepsfortheirdesign:

Therainfalldurationisselected,suchas5,10,20,30and60minor2,6,12and24h.

2.

3.4.

5.6.

7.

8.

1.

Fortheselectedduration,theannualmaximumrainfallintensityorrainfalldepthiscalculatedforeachyearofrecords.Aproperfrequencycurveisadjustedtothevaluesgivenearlier.Fromthefrequencycurve,weobtainthevaluesofrainfallintensityfortheselectedreturnperiods(5,10,20,30,50,80and100years).Steps2and3arerepeatedfordifferentdurations.Thedataareredistributedasrainfall intensityversusdurationforvariousreturnperiodsorfrequencies.Foraselectedfrequency,thevaluesofrainfallintensityareplacedontheverticalaxisandthedurationonthehorizontalaxisofalogarithmicgraph.Thesameisrepeatedforotherfrequencies,resultinginagroupofcurves(Figure4.10).

Figure 4.10 Precipitation curves. (From Singh, V.P., Elementary Hydrology, Prentice Hall, EnglewoodCliffs,NJ,1992.)

Example4.8Themaximumannualrainfalldepthfordurationsof1,2,6,12,24and48haregiveninTable4.15.ThevalueshavebeenrecordedwiththeuseofaraingaugeinAttica,Greece.

Assuming that the values of maximum rainfall depth of alldurations follow the Gumbel distribution, calculate theprecipitation curves for return periods T = 5, 20, 50 and 100years.

2.

3.

4.

1.

Theconstructionofafloodprotectionprojecttoanearbywaterstream is studied. The upstream basinwith an area of 60 km2

andaveragealtitudeof350mishilly,withmoderatevegetationandasmallpercentageof impermeablesurfaces.The lengthofthemainstreamis9.65km,andthealtitudeatthepositionoftheproject is 120 m. Estimate the runoff coefficient and theconcentrationtimeofthebasin.Estimatethepeakdischargeofthestreamforreturnperiodsof5and100years,applyingtherationalmethod.Calculate the risk of the project for the aforementioned returnperiods,giventhattheusefullifetimeis50years.

Solution

Based on the values of rainfall depth, the respective values ofrainfall intensity are calculated for each rainfall duration. Foreach time series, the following are calculated: mean, standarddeviation and the values of the parameters a and c of theGumbel distribution. The results of the calculations arepresentedinTable4.16.

Thenon-exceedanceprobabilityF(F=1–1/T)iscalculatedforreturnperiodsof5,20,50and100yearsandtheparameterln[-ln(1−1/T)].Then,thevaluesofXarecalculatedforeachTandrainfall duration based on the statistical characteristics of thesample. The results of the calculations are presented in Table4.17.

For the calculation of the precipitation curves for the givenreturnperiods, theparametersa andb of the equation i =aDb

willbeestimated.Ifthevaluesofrainfallintensityareplottedonnormalgraphinrelationto theduration(1,2,6,12,24and48h), then Graph 1 is obtained, while if the values of rainfallintensity are plotted on a logarithmic graph in relation to theduration,thenGraph2isobtained(Figures4.11and4.12).

Table 4.15 Maximum rainfalldepth

The quantities lni and lnD are related linearly through anequation of the form lni = c1 + c2lnD. Thus, for thedetermination of the precipitation curve, the calculation of theparameters c1 and c2 is sufficient for the calculation of theparametersaandb. For each return period, T, the logarithms of intensity andduration of Table 4.17 are calculated. lni and lnD are linearlyrelated (lni = lna + b*lnD), where lna(=c1) = intercept andb(=c2) = slope, while a = elna = eintercept. The calculations areshowninTable4.18.

Table 4.16 Rainfall intensityforeachrainfallduration

Thus, the precipitation curves are given by the followingequations:

ForT=5years→i=19.69*D−0.660

ForT=20years→i=28.88*D−0.661

ForT=50years→i=34.70*D−0.661

ForT=100years→i=39.07*D−0.661

Figure4.11Graph1–rainfallintensitiesversusduration.

2.

Figure4.12Graph2–rainfallintensitiesversusdurationonalogarithmicgraph.

Table4.17Rainfallintensityforeachreturnperiodandrainfallduration

Table4.18Interceptandslopeforeachreturnperiod

The concentration time of the basin is calculated based on theequationofGiandotti:

For a hilly area, the runoff coefficient assumes thevalueC1=0.24.Thepercentageofimpermeableareasislow;thus,C2=0.05.Formoderatevegetationconditions,therunoffcoefficientisC3=0.07.Due to lack of data for the hydrographic network, an averagevalueisobtainedforcoefficientC4=0.08.Thus,thecompositerunoffcoefficientofthebasinisestimated

3.

4.

asfollows:

Fromquestion1, forT=5years, theprecipitationcurve is i=19.69*D−0.660.Replacingd=tc=3.75h,theresultisasfollows:i=19.69*3.75−0.660∼8.2mm/h=2.3*10−6m/sTherefore,bysubstitutingthesevaluesintherationalformula,itfollowsthat

Similarly, for T = 100 years, the precipitation curve is i =39.07*D−0.661.Replacingd=tc=3.75h,weget

The runoff coefficient increases by 25% and is equal to 0.55,andbyreplacingintherationalmethod,itfollowsthat

TheriskisgivenbytheequationR=1−(1−1/T)n,wherenisthe useful lifetime of the project. Forn = 50 years andT = 5years, theriskisR=99.999%,whileforn=50yearsandT=100years,theriskisR=39.499%.

REFERENCESChow,V.T.,1964,HandbookofAppliedHydrology,McGraw-HillNewYork,NY.Eadie,W.T.,Drijard,D.,James,F.E.,Roos,M.,andSadoulet,B.,1971,StatisticalMethodsinExperimental

Physics.Amsterdam:North-Holland.pp.269–271.ISBN0-444-10117-9.Gumbel, E.J., 1954, Statistical Theory of Extreme Values and Some Practical Applications: A Series of

Lectures,Vol.33,Washington:USGovernmentPrintingOffice.Gumbel,E.J.,1958,StatisticsofExtremes,ColumbiaUniversityPress,NewYork.Haan,C.,1985,StatisticalMethodsinHydrology,TheIowaStateUniversityPress,Ames,IA.Hollander,M.andWolfe,D.A.,1973,NonparametricStatisticalMethods,WileyandSons,NewYork.Mimikou,M.andBaltas,E.,2012,TechnicalHydrology.Papasotiriou,Athens,Greece(inGreek).Singh,V.P.,1992,ElementaryHydrology,PrenticeHall,EnglewoodCliffs,NJ.Steel, R.G.D. and Torrie, J.H., 1960,Principles and Procedures of Statistics, with Special Reference to

BiologicalSciences,McGraw-Hill,NewYork.Sturges, H.A., 1926, The choice of a class interval, Journal of the American Statistical Association,

21(153),65–66.Viessman,W.Jr.andLewis,G.L.,1996,IntroductiontoHydrology,fourthedition,HarperCollinsCollege

Publishers,NewYork.

Wanielista,M.P.,Kersten,R.,andEaglin,R.,1997,Hydrology,JohnWileyandSons,NewYork,567pp.

Chapter5Groundwaterhydrology

5.1GENERAL

Themain topics discussed in this chapter are hydrogeological parameters, theclassification of aquifers, the principles of groundwater movement, thehydraulics of water wells (steady and unsteady flow) and the assessmentmethodsofhydrogeologicalparametersofconfinedandunconfinedaquifers.Wateravailable innatureand thatusedbyhumanscanbedistinguished into

surfacewaterandgroundwater. Surfacewater is thewater found in lakes andrivers, while groundwater is the one stored ormoving under the ground. Theparticular characteristics which distinguish groundwater from surface waterresourcescanbesummarizedasfollows(Latinopoulos,1986):

1. Spatialdistribution:Surfacewatercanbefoundlocally(lakes)orfollowingaparticular route (rivers),whilegroundwateroccupiesmuch larger areas.As far as the exploitation facilities are concerned, surfacewater, inmostcases,demandsmoreexpensivetransportationsystems,whilegroundwatercanmeetthelocaldemandeasilywithjustdirectpumping.

2. Temporal variability: Groundwater presents much slower variability inmovement,while insurfacewaters, thevariability isobvious.Asa result,surfacewaterreservoirsareusuallylargeandcanmeetthedemandsspreadoverdifferenttimeperiods.

3. Facilities and operational cost: Surface water collection projects have arelatively high construction cost (dams, reservoirs, pipelines, etc.), and alowoperationalcost.Onthecontrary,thecostofgroundwaterexploitationfacilities(drilling,pumping,etc.) isquiteinsignificant,but theoperationalandmaintenancecostsareimportant,especiallyincaseswherepumpingisfromdeepaquifers.

4. Waterquality:Thisisaveryimportantissueasfarastheexploitationand

managementofwaterresourcesareconcerned.Groundwaterislessexposedtopollutionthansurfacewaters.However,restoration/clean-upprocedures,incaseofpollution,areextremelydifficultandexpensive.

Finally, it should be mentioned that groundwater aquifers can serve multiplepurposes,suchasthefollowing(Bear,1979):

1. Actaswatersupplysources:Thisis,ofcourse,themostimportantfunction.Duetotherefillingofinventoriesbyprecipitation,undergroundwatersareconsideredasrenewableresources.

2. Act as reservoir tanks: In particular, groundwater aquifers, due to theirability to refill their inventories andbecauseof their large area, can storeextremelylargeamounts

3. ofwater.Thestoragecapacityof these layerscanbegreatlyenhancedbytheartificialrechargetechnique.

4. Actaspipelines:Thisfunctioncanbeactivatedonlybyhumanintervention(e.g.byalterationoflocalhydraulicconditions).

5. Act as filters: Using different artificial recharge techniques, surfacewastewaterscanbefilteredinthesoilforpartialortotalpurification.

6. Actassurfacewaterflowregulators:Thisfunctioncanbeaccomplishedinrivers and in wells by regulating the underground water level (e.g. bypumping)of the aquiferswhichhave ahydraulic connectionwith surfacewaters.

Thesereasonsmaketheimportanceofundergroundwaterintheexploitationandmanagementofwaterresourcesquiteobvious.Inthischapter,basicknowledgeand information on this topic is presented, which includes the theory, themethods and the techniques required to address the most common problems,wheregroundwater flow isdominant.However, thevery importantproblemofgroundwaterpollutionandremediationtechniquesisnotpresented.

5.2SOILANDAQUIFERPARAMETERS

Theporositynofasoil(orarock)isthepropertythatexpressesthevolumeofvoidsorporespresentinthetotalvolume,expressedastheratioofthevolumeofvoidsUntothetotalsoilvolumeU:

(5.1)

VoidsratioeistheratioofthevoidstothetotalvolumeofsolidsUs,definedbythefollowingequation(Figure5.1):

(5.2)

Figure5.1Verticalsectionofsoil.

Therefore, porosity and voids ratio are related by default, according to thefollowingrelationship:

(5.3)

ThespecificyieldoreffectiveporositySyofasoilor rockrefers to theratioofthevolumeUy,whichupon saturation canmove through the interstices of themediumduetogravity,tothetotalvolume,thatis,

(5.4)

Thevalueoftheeffectiveporositygenerallyissmallerthantheoverallporosityvalue.Thisisduetotheeffectofcapillaryforces,whicharestrongerincohesivesoils;asaresult,aportionofthetotalvolumeofthewatercannotbedrainedbygravityandremainsinthesoil.So,afterdraining,avolumeofwaterUrremainsin theaquifer and isknownas specifc retentionSr,whichcanbemeasuredasfollows:

(5.5)

Fromtheserelationsandthedefnitionoftherespectivequantities,itisobvious

that

(5.6)

Directly linked to the effective porosity is the aquifers’ property of storagecapacity. The exact definition of storage capacity differs in confined andunconfined aquifers (which are described in the next section). In the case ofconfined aquifers, storage capacity, S, is defined as the volume of water ΔU,removed(oradded),fromaunithorizontalareaA,duetotheunitdecrease(orincrease) Δφ of the hydraulic head. The following expression describes thestoragecapacity:

(5.7)

It is a dimensionless parameter. It is obvious that in confined aquifers, thestoragecapacitydependsonthecompressibilityofwaterandtheelasticityoftherock’ssolidfracturewhichenclosesit.Thehigherthevalueofstoragecapacity,the greater potential to store or abstract water in this reference volume and,therefore,thegreaterpotentialforexploitationoftheaquifer.Thestoragecapacityofanunconfinedaquiferissimilarlydefined.Inthiscase

notthehydraulicheadbutthefreewatersurfaceisdecreasing(orincreasing)byΔh, from an abstraction (or replenishment) of a water volume ΔU in a unithorizontalareaA.Thelevelofthefreesurfacewilldrawdownfurther,evenbyΔh.Similartoexpression5.7,thestoragecapacitycanbedefinedas

(5.8)

In confined aquifers, the removal ofwater is causedby the compressibility ofsoilgrains and the fluid,whereas inunconfinedaquifers, thedecrease in levelreveals removal orwater transfer bygravity, from the volumeof cavities of aparticular area to another. In other words the storage capacity of unconfinedaquiferscoincideswith theeffectiveporosity,andofcourse,asamagnitude ismuch higher than the storage capacity of comparable geological formationsunderpressurizedconditions(Aftias,1992).

5.3CLASSIFICATIONOFAQUIFERS

Waterflowintheaquifersiscommonlyreferredtoasflowinporousmedia.Theterm‘porousmedia’referstoallsoilsandrocksthatconsistofasolidfracture,inthe form of a solid granule assembly separated and surrounded by gaps, i.e.pores or cracks. However, when the fractures have large dimensions, flowbehaviour changes and it is treated as a special class phenomenon (flow infracturedmedia).Thebasiccriterionforthegeneralclassificationoftheaquifersistheposition

ofthemaximumwaterlevelinsoil,asitcanbeseeninFigure5.2.Consideringarandomverticalsectionofthesoil,twodifferentzonescanberecognized,wherewatermovement isquitedifferent: (1)aerationzone (orunsaturatedzone)and(2)saturationzone(orsaturatedzone)(Figure5.3).Iftheupperlimitofthezoneof saturation exists, this is called phreatic aquifer (or phreatic zone), thesynonymoustermofwater table(ofundergroundwater) isused.Becausemostof the problems related to groundwater resource management refer to watervolumes which move or are stored in the saturation zone, the followingdescription is limited to the particular water movement mechanisms and therelatedphenomenathatareobservedonlyinthiszone.

Figure5.2Verticalsoilstructure.

Figure5.3Classificationofaquifers:(a)aconfinedaquiferwithanartesianandanon-flowingwelland(b)anunconfinedaquifershowingalsothesub-caseofaperchedwatertable.

Therefore, a conventional classification of the aquifers is made taking intoaccountthegeologicalstructureandthelocalhydraulicconditionsaswell.Soanaquiferisreferredtoasconfinedorpressurized, if it islimitedfromaboveandbelow by impervious geological formations,which are identified as aquitards.Thepressurizedaquifershavethecharacteristicthat ifawell isdrilledthroughthem, thewater level in thewellwillbehigher than theaquitardand, insomecases, may even reach the surface of the ground. So if the well is properlyconstructed(concerningfilterplacement),thewaterlevelwithinthisobservationwell,orpiezometerasitisusuallycalled,willindicatethehydraulicheadatthespecificpoint.Asaresult,apiezometricsurface isdefinedbythe levelsof thepiezometersdrilledatdifferentpointsalongthehorizontalareaofthespecified

aquifer.Aconfinedaquiferiscalledartesian,whenthelevelofitspiezometricsurface

ishigher than the soil surface. In this case, if awell isopened, thewaterwillflow out of it freely because of its high pressure, without pumping (artesianwell).The secondmajor category relates to those aquiferswhere,while the lower

limitisanimpermeablelayer,theupperlimitisthefreesurfaceofgroundwater.The main characteristic of these aquifers, which are called unconfined orphreaticaquifers, is that they are recharged directly by infiltratingwater fromthesoilsurface.Phreaticaquiferscanalsobeclassified into theonesrestrictedbysomeoverlyingimpermeablelayer,whosepositionishigherthanthepositionof the free surface. This indicates that under conditions of intense waterabstraction from a confined aquifer (e.g. intense pumping), it is possible tocreatefavourableconditionsfortheappearanceofafreesurface.Asubcategoryof phreatic aquifers are the perched aquifers (Figure 5.2). In this case, waterusuallyinsmallervolumescanbefoundabovethewaterlevelofanaquiferduetoacurvedimpermeablelayer.Therearealsosituationswherethelayeraboveorbeloworevenbothlayers,

whichdelineateaconfinedaquifer,arenottotallyimpervious(semi-permeable).Despite their high resistance to water movement, in the case of large areaaquifers,theamountofwaterwhichwillgointothemainaquiferisimportant.Inthis case, the aquifer is said to be pressurized with leakage (leaky). It is, ofcourse, difficult to assess from a spot investigationwhether a layer should beregardedassemi-permeableornot;however,usingthismethod,thepracticeistocharacterize small thickness layers with low infiltration capacity (lowpermeability) in relation to the main aquifer. Respectively, with the confinedaquifer, there are phreatic aquifers with leakage, where certainly the semi-permeablelayerformstheirlowerlimit.

5.4FIELDMEASUREMENTS

Thedistributionandmovementofgroundwatercanbeinvestigatedandanalysedeitheronsiteorbytheoreticalmethods.Butanytheoreticalstudyrequiressomehydraulicdatafromthefield,andthemostbasiconeisthefreesurfaceandthehydraulicheadsmeasuredinthepiezometersandobservationwells.A piezometer consists of a perforated pipe at the lower end, which is

positionedverticallybyopeninguptheaquiferunderstudy.Pressureismeasuredby the piezometer, i.e. the hydraulic head at a particular point of a confined

aquifer,exactlyattheplacewheretheendisperforated.Inadditiontoexploratorywells,whichaimtoinvestigate thegeologicaland

hydrogeologicalcharacteristicsofthedifferenttypesofsoilandrockformations,therearetwoothertypesofwells,withregardtothehydrologyofgroundwaterflow, which are made for observation and exploitation of groundwater,respectively.An observation well consists of a perforated end, along the whole aquifer

thickness, which, in contradiction to the piezometer, is not sealed from thesurrounding aquifer. So at least in theory, this well does not cause anyobstructiontogroundwaterflow.It isnormalthatapipeofthiskindcannotbeused to measure the hydraulic head in a confined aquifer, since the resultingvaluewouldnotbeneitherthehydraulicheadnorthefreesurfacelevelbuttheircombination.Finally, theexploitationwell isusuallyconstructed forpumpingeither from

anunconfinedorfromaconfinedaquifer.Itisalsousedformeasuringthewaterlevel or the hydraulic head. In the second case, for the confined aquifer, themeasurementof thehydraulichead is ameanvaluealong the thicknessof theperforatedpartoftheaquifer.Thevaluestakenbypiezometersandexploitationwellsaregenerallydifferentandareequalonlyincaseswherethereisnoflowintheparticularaquifer.

5.5MATHEMATICALPROBLEMOFGROUNDWATER

Thewatermovementinaquifersdependsontheirhydrodynamiccharacteristicsand local flow conditions. The fundamental law of groundwater hydraulics,whichwasexpressedbytheFrenchengineerDarcyin1856,andbearshisname,refers to the following expression with regard to water movement in porousmedia.Itsgeneralformcanbewrittenasfollows(Hermance,1999):

(5.9)

whereQisthedischarge[L3/T]K isaporousmediumandfluidparameterwhichisameasureoftheporousmediumpermeabilityandiscalledhydraulicconductivity[L/T]

Aisthecross-sectionalareaoftheaquiferthroughwhichtheflowpasses[L2]J is the hydraulic gradient of the free or piezometric surface [L/T], whichequalsΔh/L(Figure5.4)

Thehydraulicconductivity isexpressedinvelocityunitsanditsvaluedepends

onthefluidandtheporousmedium.The equations ofmotion (Darcy law) in three dimensionsxi (i = 1,2,3) that

characterize a general anisotropic and heterogeneous medium of an artesiansystemcanbewrittenasfollows:

(5.10)

Figure5.4Darcy’slaw.

Intheseequations,thecomponentsofdischargeQinthreedimensionshavebeenreplacedbythecomponentsofthespecificdischargeorDarcyvelocity,q=Q/A,whichbydefaultequalsq=nV,whereVistherealorintrinsicflowvelocityandnisthesoilporosity;qhasvelocityunits.In addition to these parameters, two more parameters of the aquifer are

necessary to define the mathematical problem of the groundwater flow: (1)transmissivity T, which defines the ability of an aquifer to transfer water andequals to thehydraulic conductivity integratedover thevertical thicknessbofthe aquifer, and (2) storage capacity (or storativity) S which, as alreadymentioned,expressestheamountofwaterdrainedperunitareaoftheaquifer,asaresultofaunitchangeofthehydraulichead.Theflowequationforaquifers,initsgeneralform,isbasedontheprincipleof

conservation of mass of the fluid (water) in the porous medium (soil).CombiningtheexpressionbasedonthisprinciplewithDarcy’s law,weendupwith the following general differential equation applicable to a heterogeneousandanisotropicmedium(Bear,1979):

(5.11)

wheretistimeSsaparametercalledspecificcapacityoftheporousmedium(usuallySs=S/b,whereSisthestoragecapacityandbisthemeanthicknessoftheaquifer)

As alreadymentioned,most flow problems are solved as horizontal bivariate,usingthehypothesisofthehydraulicapproach.However,therearespecialcaseswhere it is required that the solution of the problem, mainly for unconfinedaquifers,beonaverticalplane.Forthisreason,thedescriptionofmathematicalmodelswhichsolvethevarioustypesofmathematicalproblemsisdividedintothesetwocategories.

5.6GENERALEXPRESSIONOFGROUNDWATERFLOW

This analysis can be generalized easily for 3D flow, considering a differentialreferencevolumedxdydz.Followingthesamepatternofcalculations,weendupwiththefollowingequation,respectivetothecontinuityequation(Gupta,1989):

(5.12)

where,besidesthe3Dapproachoftheflowdomain,thetermWisadded,whichis thevolumetric fluxperunitvolume(L/T),expressing theexternal infloworoutflow of water (recharge of leakage). This equation generally applies tohomogeneous,heterogeneous, isotropicandanisotropicmedia.Foran isotropic(homogeneousorheterogeneous)medium,theaforementionedexpressioncanbewrittenasfollows:

(5.13)

andforahomogeneousandisotropicmedium,itiswrittenas

(5.14)

InthecaseofuniformflowwithouttherechargingtermW,inwhichthereisnochangeintheflowdomainwithtime,theaforementionedequationsaremuch

simpler,astherighthandsideequalstozero.So,theequationofuniformflowinaheterogeneousanisotropicmediumiswrittenasfollows:

(5.15)

Theequationofuniformflowforhomogeneousandisotropicmediumiswrittenasfollows:

(5.16)

ThelastexpressionisknownastheLaplaceequation.Thegoverningequationofundergroundwaterdescribesthewatermovement

inbothconfinedandunconfinedaquifers. In thecaseofconfinedaquifersandwithouttakingintoaccountthetermW, thegeneralequationforananisotropicandheterogeneousmediumismodifiedasfollows:

(5.17)

Foranisotropicandhomogeneousmedium,theequationisasfollows:

(5.18)

In the case of unconfined aquifers, assuming that the flow is horizontal(Dupuitapproximation)andwithouttakingintoaccountthetermW,thegeneralequationforananisotropicandheterogeneousmediumismodifiedasfollows:

(5.19)

The last expression is known as the Boussinesq expression (Singh, 1992).Linearizationof this equation canbemadewhen the change in thedrawdownlevelisrelativelysmallinrelationtothedepthofthewater,wherethedepthhreplacesbythedepthboftheaquifer.Forahomogeneousaquifer,theequationismodifiedasfollows:

(5.20)

Theintegrationofthemathematicalproblemismadebyaddingtheinitialand

boundaryconditions.Theassignationofapiezometric(orhydraulic)headatthebeginningofthephenomenonisdefinedasaninitialconditionforanarbitrarilydefinedtimet=0,withthegeneralformh=f(x,y,0),whichisaknownfunctionateachpointx,yofthehorizontalflowfield.Three types of boundary conditions, usually used for groundwater flow

problems, are as follows: (1) conditions of a known head, h = f1(x,y,t); (2)conditionsofknowndischarge,Q′n= f2(x,y,t),whereQ′n is theperunit lengthperpendicular to the boundary curve infiltrated discharge; and (3) semi-permeable boundary conditions. The most common application of knownconditionreferstotheboundariesoftheaquiferwherethereishydrauliccontactwithsurfacewaters (lakes, rivers,seas), thecorrespondingconditionofknowndischargeisusedforimpermeableboundaries(noinfiltrateddischarge)andtheboundary condition of semi-permeable boundary applies to cases of partiallybounded riverbed or lake due to the deposition of fine-grained material thatreducesthehydrauliccommunicationbetweentheaquiferandthesurfacewaterbody.From this discussion, it is clear that the management of confined aquifers

where the boundaries are defined geometrically is simpler. For unconfinedaquifers, thefactthattheupperlimitoftheflow(freesurfaceboundary)isnotfixedgeometricallybutisdeterminedbythezeropressureconditionintroducesconsiderable difficulty. Very often, flow problems in phreatic aquifers aresimplifiedbytheuseoftheso-calledDupuitapproximation,accordingtowhichthe flowcanbe considered substantially horizontal (assuming the slopeof thephreatic horizon is small). The consequences of this hypothesis is that (1) thevertical component of the specific discharge is zero, (2) the horizontalcomponentsof thespecificdischargearefixedineachvertical lineand(3) thehydraulicheadineachverticallineisfixed.

5.7ANALYTICALSOLUTIONSOFSTEADYFLOW

The groundwater flow, which is described by the Laplace equation, can besolved based on the theory of partial differential equations. For conditions ofactualflow,thesolutionmustsatisfytheboundaryconditionswithregardtothepiezometric heads. The porous medium is assumed as homogeneous andisotropicinallcases.

5.7.1Confinedaquifer

The flow in a well, which fully penetrates in a homogeneous and isotropic

aquifer,isradiallysymmetrical.Theradiusoftheboreholeismeasuredfromthecentreof theborehole (Figure5.5).Foran isotropicandhomogeneousaquifer,theequationdescribingthegroundwaterflowis(FreezeandCherry,1979)

(5.21)

Figure5.5Confinedaquifer.

Byusingpolarcoordinates,

Theequationoftheundergroundflowbecomes:

(5.22)

or

(5.23)

Thismeansthatthefollowingtermisaconstant:

(5.24)

whereC1isaconstant.ThedischargecanbeobtainedusingDarcy’slaw,andanareaA,whichisthe

lateralsurfaceofacylinderofradiusrandheightb,accordingtothefollowing

equation:

(5.25)

which,afterintegration,resultsinthefollowingequation:

(5.26)

ApplyingEquation5.26atdistancesR(wherethechangeinthepiezometriclineisminimal, practically zero, and the head isH) and r, and by subtracting bymembers,weget

(5.27)

whereHisthehydraulicheadatdistanceR(Figure5.5)[L]histhehydraulicheadatanydistancer[L]Qisthedischarge[L3/T]bisthethicknessoftheaquifer[L]bKisthetransmissivityT[L2/T]

Equation5.27isknownastheThiemequation,whichcouldalsobeextractedbytheintegrationofDarcy’sequation:

(5.28)

(5.29)

Example5.1An aquiferwith a thickness of 40m is overlaid by an impermeablelayerof25mthickness.Afteratrialpumpingwithdischarge0.2m3/sfor a long time from a borehole with 0.4 m diameter, the waterdrawdownsattwoobservationwellslocatedatdistances20and80mwere5and2m,respectively.Estimatethehydraulicconductivityandthedrawdowninthepumpingborehole.SolutionIfh1andh2arethehydraulicheadsonthetwoobservationwellsand

r1andr2aretherespectiveradii,thenusingEquation5.27,

The difference in hydraulic heads is the negative value of thedifferenceofdrawdownsfortherespectivepositions.Thedrawdownsw intheboreholeiscalculatedaccordingly(hw the

hydraulicheadintheborehole):

Example5.2Apumpingwelloperateswithasteadydischargerateof62.8m3/hatthecentreofaconfinedaquiferwithashapethatcanbeapproximatedbyacylinderwithathicknessb=10mandradiusR=578m,understeady flow. The drawdown of the piezometric surface in twoobservationwellslocated50and136mapartfromthepumpingwellis2 and 1m, respectively. The pumping stops after 8 h of continuousoperation, and in a short span of time a new state of equilibrium isachievedwithpiezometric surface reducedby48cm from the initialsurface.Thefollowingvariablesaretobeestimated:

1. Thetransmissivity(T)oftheconfinedaquifer2. The drawdown of the piezometric surface 300m from thewell

whileitisinoperation3. Thespecificcapacity(Ss)oftheconfinedaquifer

Solution

1. Forsteadyflowcondition,theThiemequation(5.27)isvalid,so

2. AgainfromtheThiemequation

3. Afterreachingtheequilibriumstate,wehaveΔh=0.48m.Theabstractedvolumeis

Thespecificcapacityoftheaquiferis

whereAistheareaoftheaquifer.

5.7.2Unconfinedaquifer

The analysis of an unconfined aquifer is made by using the Dupuitapproximation,whereitisconsideredthat(1)theflowishorizontaland(2)theflowvelocityisproportionaltothehydraulicslope.Abasicdifferencebetweentheflowofaconfinedandanunconfinedaquiferisthatintheunconfinedflowcase, the aquifer thickness changes and, as a result, the section that provideswater to the pumping borehole changes. The equations mentioned in thepreviousparagraphnowbecome

(5.30)

sincethedischargedoesnotdependonthethicknessoftheaquiferbbutonthedepthofthewaterh.Thelastonetakestheform

(5.31)

Itssolutionresultsin

(5.32)

Applying again the last expression for distances r (with level h) andR (withlevelH),andbysubtractingbymembers,gives

(5.33)

or

(5.34)

ThisequationcouldalsobeextractedbyintegratingDarcy’sequation:

(5.35)

(5.36)

AllmentionedparametersarepresentedinFigure5.6.

Example5.3A borehole of 30 cm diameter has its lower end 60 m below thepiezometricsurface.After16hofpumpingatadischargerateof0.85m3/s, the piezometric surface is stabilized 15 m below the initialpiezometricsurface,whereasatanobservationwellthatislocatedatadistanceof400m from theborehole, the leveldrawdown is2.60m.Calculatethehydraulicconductivityoftheaquifer.SolutionTocalculatethehydraulicconductivityK,theequationthatrelatesthelevelHatdistanceR fromtheboreholeand the level in theborehole(5.34)canbeused:

Figure5.6Unconfinedaquifer.

5.7.3Semi-confinedaquifer

As semi-confined aquifer is called the one that is restricted between a lowerimpermeable layer and an upper semi-permeable layer. Above the semi-permeable layer, there is an unconfined aquifer. It is assumed that at thebeginning, the hydraulic head of the semi-confined aquifer coincideswith theleveloftheunconfinedaquifer,ascanbeseeninFigure5.7.Afterpumpingfromthe semi-confined aquifer, a piezometric difference develops between theunconfined and the confined aquifer, resulting in a flow through the semi-permeable layer.The flow is assumed to be horizontal in the confined aquiferand vertical in the semi-permeable layer. In the general groundwater flowequation,atermisaddedwhichrepresentsleakagefromtheunconfinedaquifertothesemi-confinedone:

(5.37)

whereq′isthevelocityK′isthehydraulicconductivityb′istheaquiferthickness

Figure5.7Semi-confinedaquifer.

Therefore,theequationforthesemi-confinedaquiferinto1Dformisasfollows:

(5.38)

Usingtheleveldrawdowns=H0‒handsubstitutinghinEquation5.38,

or

(5.39)

where

(5.40)

Thesolutionofthelastequationisasfollows:

(5.41)

wheresistheleveldrawdownΓ0(r/B)isthefirst-classmodifiedBesselfunction

K0(r/B)isthesecond-classmodifiedBesselfunctionC1,C2areconstantsresultingfromtheboundaryconditions

Thefinalequationforainfiniteareasemi-confinedaquiferisasfollows:

(5.42)

TheK0(r/B)valuesfordifferentr/baretabulated.Forr/b<0.05,Equation5.42iswrittenasfollows:

(5.43)

Example5.4An aquifer is located between an impermeable layer and a semi-permeable one of 6 m thickness, having a hydraulic conductivitycoefficientof2×10−8m/s.Themeanthicknessoftheaquiferis150mand the hydraulic conductivity coefficient is 1.5 × 10−3 m/s.Groundwater is pumped from the aquifer at a discharge rate of 0.20m3/sthroughaboreholeof10in.indiameter.Calculatethedrawdownatadistanceof1500mfromtheboreholeandintheboreholeitself.SolutionAtfirst,thetransmissivityofthesemi-permeablelayeriscalculated:

AndthentheconstantB(Equation5.40)

For the calculation of the level drawdown, first the ratio r/B isestimated:

Therefore,thefollowingsimplifiedequationcanbeused:

Forthecalculationoftheleveldrawdownatthe1500mdistance,theratioisagainestimated:

In thiscase the simplifiedequation isnotvalidand the leveldrop isgivenbythegeneralequation:

withthevaluesoftheBesselfunction takenfromtables.

5.8THEORYOFIMAGES

The application of the general equation of groundwater flow requires that theaquifer boundaries are infinite. However, all the aquifers are surrounded byeitherimpermeablelayersorsteadysupplyboundaries,suchaslakesandrivers.In the case where pumping wells are located close to such boundaries, theequationsforainfiniteaquiferarenotapplicable;toaddressthesecases,theuseof the theory of images is necessary,where boreholes – images – are used inordertocreatetheconditionsofainfiniteaquifer.Inthischapter,twocasesareexamined,oneclosetoariverandtheotherclosetoanimpermeablelayer.

5.8.1Wellnearriver

Along the river, there is a constant hydraulic head equal to the level at thesurface of the river. Therefore, the cone of the underground water leveldrawdown approaching the river, shouldmatch the river surface. This can beaccomplishedifamirrorwellwhichenrichestheaquiferissituatedontheothersideof theriver,at thesamedistancefromtheinitialwell,asshowninFigure5.8. The recharge rate of the mirror well will be the same as the pumpingborehole,sothattheleveldrawdown,becauseoftheabstractioninthepumpingborehole, equals the level rise due to recharge, with the two levels to beassimilatedalongtheriver,asshowninFigure5.9.Letusassumethat thedistancefromthepumpingborehole to theriver isa.

ThedistancesofapointIwithcoordinates(x,y)fromthepumpingandrechargeboreholesare

(5.44)

Theleveldrawdownduetotheactualboreholeis

(5.45)

whilethemirrorboreholecreatesalevelriseequalto

(5.46)

The final level drop to the point (x,y) will be equal to the sum of the leveldrawdowncausedbytheactualboreholeandthelevelriseduetothepresenceofwatercourses,i.e.

(5.47)

whereαisthehorizontaldistanceoftheriverfromtheboreholex,yarethecoordinatesofthepointwheretheleveldropoccurs

Figure5.8Distancesofthepumpingandenrichmentboreholefromtheriver.

Figure5.9Combinationofleveldrawdownofthepumpingandenrichmentboreholes.

Example5.5A0.30mdiameterboreholeisdrilledintoanaquiferof40mthicknesswith a hydraulic conductivityof 30m/day. If a drawdownof2m isobserved in the borehole due to continuous pumping, calculate thefollowing:

1. Thepumpingrateiftheboreholeissituatedatadistanceof80mfromariver

2. Thepumpingrateiftheboreholeissituatedatadistanceof4000mfromariver

Solution

1. Thedrawdownintheboreholeisgivenbythefollowingequation:

wherex,yarethecoordinatesofthewell,i.e.,x=−80+0.15andy = 0, while a is the distance from the river: a = 80 m. Byreplacingthevalues,weget

2. As incase1, the respective inputsare:a=4000m,x=−4000+0.15,y=0,sowehave

5.8.2Wellnearimpermeableboundaries

Inthiscase,thereisnoflowbeyondtheimpermeablelayer.Ifamirrorboreholeisplacedontheothersideofthelayerandpumpswiththesamedischarge,thenthedrawdownwillmeetintheimpermeableboundary,asitcanbeseeninFigure5.10. If the radius of influence isR, then the drawdown for each borehole isgivenbythefollowingequation:

(5.48)

wherer1,r2arethedistancesasdefinedRistheinfluenceradius

Example5.6A borehole of 0.30 m diameter is drilled into an aquifer of 40 mthicknesswithahydraulicconductivityof30m/day.Ifadrawdownof1.5m is observed in the borehole due to continuous pumping, thencalculate the pumping rate if the borehole is situated at a horizontaldistanceof100mfromanimpermeablelayerandtheinfluenceradiusis1.0km.

Figure5.10Combinationofleveldrawdownofthepumpingboreholes.

SolutionEquation5.48isappliedforr1=0.15m,equaltotheboreholeradius,andr2=2×100=200m, the sameas thedistance from themirrorborehole:

5.9ANALYTICALSOLUTIONSOFNON-UNIFORMFLOW

Inproblemsofnon-uniformflow,thewaterlevelinunconfinedaquifers,orthehydraulic head in confined aquifers, is not stable but changes with time. Inresolvingsuchproblems,theparameteroftimeisinvolved,aswillbediscussedinthefollowingparagraphs.

5.9.1Wellhydraulics

A standard solution of the non-uniform flow equation regards the flow fieldaround the borehole. The equations arising have a direct and extensiveapplicationinassessingthecharacteristicsofanaquifer.

Consider the confined horizontal aquifer of similar shape, which has aconstant thickness b and infinite extent (in both horizontal dimensions). Theaquifer is considered to be of homogeneous and isotropic medium, oftransmissivityTandstoragecapacityS,withnorechargeorleakagebetweentheneighbouring layers.At a certain location in the aquifer, a vertical borehole isopened.Waterstorageintheboreholeisconsiderednegligible.Attimet=0,thehydraulicheadintheaquiferisconstant,equaltoh0.Atthesamemoment,waterispumpedfromtheboreholeataconstantdischargeQ.Duetothecircularsymmetry,theflowequationmaybewrittensimplyusing

polarcoordinates:

(5.49)

where the zero recharging termW is omitted. In the case of level drawdown,Equation5.49becomes

(5.50)

Theis (1935) has shown that the solution of this differential equation for theboundaryandinitialconditionsdescribedearlierisasfollows:

(5.51)

where

(5.52)

where

(5.53)

Intheseequationsristhedistancefromthecentreofthepumpingborehole[L]tisthetime[T]Sisthestoragecapacity[dimensionless]Tisthetransmissivity[L2/T]s is the hydraulic head drawdown at time t and at a distance r from theborehole[L]

The parameter W(u) is known in mathematical terminology as exponentialintegral and in technical terminology as the well function. Equation 5.51 isreferredtointheliteratureastheTheisequation.TheintegralofEquation5.52has the following alternative expansion in infinite sequences, used forcomputationalpurposes(thefirstforsmallvaluesofuandthesecondforhighervalues):

(5.54)

(5.55)

Alternatively, the values of the well function can be obtained from Table 5.1(Chow, 1964). A plot of well function values in the range 10−7 to 10−1 ispresentedinFigure5.11.

Table5.1ValuesofthewellfunctionW(u)forvariousvaluesofparameteru

Source:Chow,V.T.,HandbookofAppliedHydrology,McGraw-Hill,NewYork,1964.

Figure5.11RelationshipbetweenW(u)andu.

These equations are used to find the hydraulic properties of the dischargecapacityandthestoragecapacityofanaquifer.Thevaluesrequiredtosolvetheequationswehaveseenarederivedfrompumpingtests,whereaboreholepumpswith a constant discharge over a period of several hours to several days. Theleveldrawdownisrecordedatspecifictimesinobservationboreholes,whicharelocated at different distances from the pumping borehole. An importantadvantageoftheTheisequationisthatitcanbeusedforsmallertimeintervals,i.e. before achieving equilibrium. Several procedures have been proposed forsolvingtheseequations.ThemostessentialonesaretheTheis,Cooper-JacobandChowmethods.

Example5.7Inanaquiferwith35.0mthickness,hydraulicconductivityK=16.0m/dayandstoragecapacityS=0.00550,awellispumpedatarateof1500m3/day.Estimatethedrawdownatadistanceof30.0mfromthewell,after5daysofcontinuouspumping.SolutionThetransmissivityTisequaltoT=K×b=16×35=560m2/dayTheparameteru

FromTable5.1foru=0.00044⇒W(u)=7.16,therefore

Sothedrawdownis1.52matadistanceof30.0mfromthewellafter

5daysofpumping.Example5.8Awellispumpedatarateof1000m3/dayfromanaquiferwithS=2×10−4andT=150m2/day.Findthedrawdownatadistanceof5mfromthewellafter2hofpumpingandalsoatadistanceof500mafter3daysofpumping.SolutionWith substitution inEquation5.53 for r = 5.0m and t = 2 h (=2/24days):

FromTable5.1:W(u)=8.63

whichisthedrawdownatadistanceof5m,after2h.Similarlyforr=500andt=3days,

FromTable5.1:W(u)=3.04.Therefore,

Example5.9Usingthepreviousexample,estimatetheradiusofinfluenceafter2hand3daysofcontinuouspumping,respectively.SolutionWeassumethedrawdownintheradiusofinfluenceis2cm,whichispracticallyzero(verysmall).

FromTable5.1wegetu=2.31

Fort=2h,weget

Similarlyfor3days,

5.9.2Processingofdrawdowntests

Drawdownandrecoverytestsareaimedatassessingthebasicparametersoftheaquifers in the vicinity of a borehole. Pumping tests are one such way ofassessinganddeterminingtheparametersofcontrolledexperimentsratherthanhistorical data of the aquifer. The parameters of the aquifer, which can beassessed,arethetransmissivityTandthestoragecapacityS.Dependingonthenatureofthetestandthedatacollected,itispossibletoalso

evaluate other parameters, e.g. the leakage factor λ and information about theboundaryconditions.Duringtestinginaborehole,pumpingshouldbeataconstantdischargerate

Qwwhilechangesindrawdownintheboreholebeingpumpedarerecordedovertime (drawdown test). The same recording is done in one or more standingneighbouring boreholes (interference test) if they have the possibility of levelmeasurement.The aforementioned parameters are estimated frommeasured data,with the

useoftheappropriateequationsandregressionanalysismethods.Thechoiceofthe appropriate equation is perhaps the most difficult step in a pumping test,since this equation depends on the boundary conditions of the aquifer, ongeology, on leakage between aquifers, etc.Here, assumptions should bemadebasedonpreviousgeologicalinformation,sincethereisnosystematictheoryfortheselectionoftheappropriateequation.It is generally assumed that the aquifers behave as a representative aquifer,

since the flow in these layers is practically horizontal. It is also assumed thatimperviousformationsbetweenpreviousonesaren’tinsertedintothehydrauliccalculations.Thedrawdown test is an experiment bywhich characteristics of the aquifer

are specified in twophases: theabstractionphaseand the recoveryphase.Thepumping begins with constant dischargeQ, so that the level in the borehole

beginstodrawdown.Theleveldrawdownsisrecordedasfunctionofthetimetfromthestartofpumping.Whenthelevelstopsfalling(thishappenswhentheaquiferreachesequilibrium),pumpingstopsandthelevelbeginstoriseagain.

5.9.3Analysisofconfinedaquifers

5.9.3.1Theismethod

TheequationusedintheTheismethodis

(5.56)

Bytakingthelogarithmofbothsides,

(5.57)

with

(5.58)

Takingthelogarithmofbothsidesofthelastequationfollowedbylogarithmicanalysisgives

(5.59)

wheresistheleveldrawdown[m]risthedistancefromthecentreoftheborehole[m]Sisthestoragecapacity[dimensionless]Tisthetransmissivity[L2/T]Qisthedischarge[L3/T]W(u)isthewellfunction

For constant dischargeQ, the procedure of the Theismethod is based on thefollowingsteps:

1. Preparealog-loggraph,withY-axisthewellfunctionW(u)andX-axistheparameteru.TheparametersW(u)anduareassociatedunivocallyandthecorrespondingdiagramisintheformpresentedinFigure5.11.

2. Fromthepumpingdata,preparealog-loggraphoftheleveldrawdownwithr2/t quantity.This graph is known as a data graph.The data are acquiredfrompumpingwherethedischargeremainsconstant.Theleveldrawdownmayberecordedinanobservationboreholewhichislocatedatadistancerfromthepumpingboreholeatseveraltimes.Also,arecordcanbetakenatthe same time from a number of boreholeswhich are spaced at differentdistances from the initial pumping borehole. This is called drawdownanalysisbydistance.However, inbothcircumstances, thetermr2/tcanbecalculatedandplottedinrelationtotheleveldrawdown.

3. Thedatagraph isplaced tooverlap thegraphof thecurve, and is shiftedhorizontallyorverticallykeeping theaxesof the twographsparalleluntilthedatagraphcoincideswiththecurveofW(u)andu.

4. Anarbitrarypointisselectedwhichmaynotbeoverthecurve.Thispointisusuallychosensothatthecoordinatesaremultiplesof10.Thenthevaluesofsandr2/tofthespecificpointarerecorded.

5. The discharge capacity and the storage capacity are calculated from thefollowingequations:

(5.60)

(5.61)

TheprocedureoftheTheismethodispresentedinFigure5.12.

5.9.3.2Cooper-Jacobmethod

CooperandJacobhaveshown thatwhen theparameterubecomesverysmall,thefunctionW(u)canbeapproximatedbythefirsttwotermsofEquation5.54.Therefore,accordingtoCooperandJacob,theequationsbecome

(5.62)

Figure5.12Overlappingdiagramsandpointselection.

ThedischargecapacityTcanbecalculatedbasedontheCooper–Jacobequation:

(5.63)

or

(5.64)

or

(5.65)

wherer is thedistancebetween thepointwhere the level ismeasuredand thepointwherepumpingislocated.In the phase of pumping, the drawdown s is recorded at various time

instances.If thepairofvalues(t,s)areplottedonsemi-logarithmicgraph, thentheyarearrangedapproximately inastraight line.Thus,performingregressionbetweenthevariableslntandsresultsinastraightlineofthefollowingform:

(5.66)

AndfromEquation5.65itresultsinto

(5.67)

The first of these equations provides directly the discharge capacity. For thestoragecapacity,thefactorofdistancerenters into thefunction.Normally, thelevelshouldbemeasuredatsomedistancerfromtheborehole,butusually,thereisonlylevelmeasurementinthepositionofthedrilling.Inthatcase,thedistancerbecomesequaltotheboreholeradius.Supposethatatamomentt=t1,pumpingfromtheaquiferstopsandthewater

levelstartsgraduallytoreturntotheinitiallevels,asshowninFigure5.13.Thedischarge capacity for the recovery phase is calculated by the followingequation:

(5.68)

Thisappliesforlongtimeintervals:

(5.69)

Duringthereturntests,theboreholewhichinitiallywaspumpedwithdischargeQstopstopumpattimet1andtheremainingleveldrawdowns ismeasuredintheboreholeoratanotherobservationstation(drillingorwell)locatedcloseto

the borehole. According toHantush (1964), when , the followingrelationholds:

(5.70)

Thatis,theleveldrawdownisproportionaltoln(t/[t−t1]),sodataarearrangedinastraightline.

Figure5.13Leveldrawdowndiagramduringpumpingandrecoveryperiod.

However,themethodologyisagainidentical.Astraightlinefitsbetweenthevariablessandln(t/[t−t1]),andtheslopeofthelineisQ/4πT′,fromwhichT′canbecomputed.So,thisprocedureresultsintwodischargecapacities,oneforthepumpingandoneforthereturnperiod.Asalreadymentioned,thelastequationappliesonlyinthecaseoflongtimeintervals;therefore,itmaybedefinedbytheuserthattheregressionwillonlyconsiderthehighpointsofthediagram.Also,ifbending is presented,more regressions can bemade and generate differentT,dependingontimet.

5.9.3.3Chowmethod

TheChowmethodisacombinationofTheisandCooper–Jacobmethodsandisdefinedbythefollowingequations:

(5.71)

(5.72)

(5.73)

ThecorrelationbetweenthemagnitudesF(u),u,W(u)isdefinedbythegraphinFigure5.14.

Ithastobementionedthattheearlierequationsareappliedinthecaseswhenthe discharge Qw is constant during the pumping period and the aquifer isconfinedwithpracticallyverylargedimensions.Inthecaseofpumpingtests,thedischargevaluefluctuatesat thebeginningbutrapidlystabilizes.Also, inmostofthecases,measuringthedischargeaccuratelywasnotpossible,duetolackorto non-continuous hydrometric stations, so the estimation of the pumpingdischargewas calculated empirically (e.g. using containers of known capacityandclockingoffillingtime).Also,thelimitsandthesupplyoftheaquiferwereunknown.

Figure5.14ChowdiagramgoverningtherelationsbetweentheparametersF(u),uandW(u).

Example5.10Aboreholeisconstructedforwaterpumpingfromaconfinedaquifer.The pumping rate is 134m3/h. In an observationwell located 70maway,thefollowingdrawdownswereobserved,whichcanbeseeninTable5.2.Determinethetransmissivityoftheaquiferandthestoragecapacity,

usingtheCooper–Jacob,TheisandChowmethods.

Solution1.Cooper–JacobmethodUsingthesimpleequationofCooper–Jacob,wecorrelatelinearlythe

logarithms of time with the level drawdown. The equation gets thefollowingform:

where

Table5.2Leveldrawdownatanobservationborehole

Timesincethestartofpumping(min)

Leveldrawdown(m)

1 0.18

2 0.34

5 0.52

10 0.65

15 0.73

20 0.80

25 0.84

30 0.89

35 0.92

40 0.96

45 0.97

50 0.99

55 1.01

60 1.03

90 1.11

120 1.18

150 1.23

180 1.26

Data are converted to the same units and the logarithms of time arecalculatedaccordingtoTable5.3:

Subsequently,theslopeandtheordinateofthelinewhichcorrelateslinearlythelogarithmsoftimewiththeleveldrawdowniscalculated,accordingtoFigure5.15.From the linear regression, we derive the transmissivity of the

aquifer T, and the storage capacity S, according to the followingrelations:

Table5.3Datameasurementsofwaterleveldropvs.time

t(min) t(s) ln(t) s(m)

1 60 4.094 0.19

2 120 4.787 0.33

5 300 5.704 0.52

10 600 6.397 0.66

15 900 6.802 0.74

20 1,200 7.090 0.80

25 1,500 7.313 0.85

30 1,800 7.496 0.89

35 2,100 7.650 0.92

40 2,400 7.783 0.95

45 2,700 7.901 0.97

50 3,000 8.006 0.99

55 3,300 8.102 1.01

60 3,600 8.189 1.03

90 5,400 8.594 1.11

120 7,200 8.882 1.17

150 9,000 9.105 1.22

180 10,800 9.287 1.25

Figure5.15Correlationoflogarithmsoftimewiththeleveldrawdown.

2.TheismethodAccordingtotheTheismethod,themagnituder2/t isestimatedat thebeginningofeverytimeinterval,asshowninTable5.4,andthenitisdrawn in a double logarithmic diagram in relation to the leveldrawdowns,asshowninFigure5.16.

Table5.4Conversionofdatatoappropriateunits

t(min) s(m) r2/t(m2/min)

1 0.19 4900.00

2 0.33 2450.00

5 0.52 980.00

10 0.66 490.00

15 0.74 326.67

20 0.80 245.00

25 0.85 196.00

30 0.89 163.33

35 0.92 140.00

40 0.95 122.50

45 0.97 108.89

50 0.99 98.00

55 1.01 89.09

60 1.03 81.67

90 1.11 54.44

120 1.17 40.83

150 1.22 32.67

180 1.25 27.22

Figure5.16Diagramofleveldrawdowninrelationtor2/t.

Figure5.17Diagramoverlapping.

Then, thediagram inFigure5.16 is placedon the diagramwhich correlatesthewell functionWwith the parameteru and then is shifted horizontally andvertically,untilthetwocurvescoincide.ThisprocedureisshowninFigure5.17anditrequiresthatthetwodiagramshavethesamelogarithmicaxes,i.e.thesizeofeachlogstepoftheaxisisthesameinbothdiagrams.Asthetwocurvesoverlap,apointisselectedandthecorrelationbetweenthe

parameters in both systems of axes is found. In this case, a point with thefollowingattributes(coordinates)isselected:

1. u=0.12. r2/t=9800m2/min3. W(u)=24. s=0.52m

Thedesiredcharacteristicsof theaquiferarederivedby replacing thesevalueswithTheis’expressions.Thetransmissivityisderivedbythefollowingequation:

Andthestoragecapacityisequalto

3.ChowmethodUsingtheChowmethod,thefollowingprocedureisfollowed:

a. The level–time drawdown data have already been presented in semi-logarithmicpaperinFigure5.15.Inthisdiagram,apoint isselected,suchas the onewith coordinates s = 0.52m and t = 5.0min. From the samediagram, for a time step on the logarithmic axis, a difference of leveldrawdownΔs = 0.205 m arises (same as the slope of the straight line).Therefore

b. FromFigure5.14andforFu=2.54,oneobtainsu=0.005andW(u)=4.8.c. FromEquation5.60,thetransmissivityoftheaquiferiscalculated:

d. Finally,fromEquation5.61,thestoragecapacityis

Itshouldbenoticedthattheresultsfromthethreemethodsdiffersignificantly.

Example5.11Awellwith0.50mdiameterisconstructedforwaterpumpingfromaconfinedaquifer.Thepumpingrateis105m3/handtheaquifers’ thicknessis20m.ThetransmissivityTis47.573m2/h,whilethestoragecapacitySequals7.44×10−5.

1. After6hofconstantpumping,estimatethedrawdowninthepumpingwellandinanobservationwellthatislocated200maway.

2. In a certainmoment between the 6th and the 16th hour (where the dailyoperation of the pump stops), the system attains a state of equilibrium(steady flow)where the piezometric cone is stabilized. Find themomentwhen the steady flow begins and the drawdown is 100 and 200m apartfromthewell,ifthedrawdowninthewellis3.35m.

Solution

1. FromtheCooper–Jacobmethod:Thedrawdowninthewell(r=0.25m)after6his

Similarly,thedrawdown200mfromthewell(r=200m)after6his

2. Letusassumethats1isthedrawdownatposition1located100mfromthewell,s2isthedrawdownatposition2located200mfromthewellandswisthedrawdowninthewell.Whenthesystementersthestateofequilibrium,thefollowingequationbecomesvalid:

Similarly

tcanbeestimatedbytheCopper–Jacobrelationshipasfollows:

Position2givesthesameresults:

5.9.4Analysisofunconfinedaquifers

The equations described here are not fully applicable to the case of anunconfinedaquiferduetothefollowingreasons:

1. Theleveldrawdownoftheaquifer2. Theverticalflowbeingclosetotheborehole3. Flowdelayduetogravity

Inthecasewhentheleveldrawdownissmallinrelationtothethicknessoftheaquifer,theeffectoftheverticalflowandtheleveldrawdowncanbeconsideredasnegligible.AccordingtoHantush(1964),theverticaleffectisnotimportantiftimeequalsto

(5.74)

wheretisthetime[t]bisthethicknessoftheaquifer[L]Syisthespecificyield[dimensionless]Kzistheverticalhydraulicconductivity[L/T]

AccordingtoStallman(1971),theflowdelayappliesforaperiodoftime:

(5.75)

Theobservedleveldrawdownarecorrectedbyusingthefollowingequation:

(5.76)

wheres′isthecorrectedleveldrawdown[L]sistheobservedleveldrawdown[L]bisthethicknessoftheaquifer[L]

Thevalueofthespecificyieldcanbecorrectedbyusingthefollowingequation:

(5.77)

whereSyisthemodifiedspecificyield[dimensionless]Styistheestimatedspecificyield[dimensionless]isthecorrectedleveldrawdownattheendofpumping[L]

Example5.12A pumping borehole penetrates a phreatic aquifer of thickness 20mandpumpsatadischargerateof92m3/h.Therecordeddrawdownintime, at an observation borehole located 12 m from the pumpingborehole, isgiven inTable5.5.Theparametersof theaquifershouldbespecified.

SolutionThedataareconvertedtotheappropriateunitsandtheleveldrawdowniscorrectedaccordingtothefollowingequation:

whereb=20misthethicknessoftheaquifer.TheresultsaregiveninTable5.6.

Table5.5Drawdownatanobservationboreholeintime

Timesincethepumpingstarted(min)

Leveldrawdown(m)

30 0.23

60 0.38

90 0.47

120 0.54

150 0.59

180 0.63

360 0.79

720 0.95

1440 1.11

Table5.6Correctedlevelvalues

t(min) t(s) ln(t) s(m) s′(m)

30 1,800 7.50 0.23 0.23

60 3,600 8.19 0.38 0.38

90 5,400 8.59 0.47 0.46

120 7,200 8.88 0.54 0.53

150 9,000 9.10 0.59 0.58

180 10,800 9.29 0.63 0.62

360 21,600 9.98 0.79 0.77

720 43,200 10.67 0.95 0.93

1440 86,400 11.37 1.11 1.08

ThentheCooper–Jacobmethodisapplied,withlinearregressionfor

thepairsofvaluesofthelogarithmsoftimeandthecorrectedvalueofthe level drawdown (see also Example 5.10). The results of theregressionarepresentedinthefollowinggraph(Figure5.18).Asusual,theparametersoftheaquiferarecalculatedasfollows:

Thesevaluesarevalidgiventhenextassumptionsregardingtime:

or 5.62 h, in order toneglectverticalflow

or 0.62 h, in order toneglectthedelayofflow

Figure5.18Linearcorrelationofthevariablesln(t)ands′.

Overall,thevaluesofS,Tapplyfortimet>5.62h.Finally,thevalueofthespecificyieldisadjustedaccordingtotherelation

where is the corrected value of the level at the end ofpumping. In phreatic (or unconfined) aquifers, the specific yield SycoincideswiththestoragecapacityS.

5.9.5Anisotropicalluvialdeposits

A common situation in alluvial deposits is the layering of materials withdifferentKvalues.Thisphenomenoniscalledanisotropyanditistheruleratherthan the exception in alluvial deposits. Anisotropy is caused also whenindividualparticleswithashapedifferentthansphericalaredepositedwiththeirfat side in a horizontal direction.Moreover,when deposition takes place on ariverbed, fat particles tend to be tilted slightly upward in the direction of theflow.Thisarrangementiscalledimbricationandisobservedingraveldeposits.Watermolecules flowing through imbricatedmaterial find lesser resistance inthehorizontalratherthanintheverticaldirectionduetoimbrication.Also,watertravelling throughananisotropicaquiferconsistingof separatehorizontal sandandgravellayerswillfindmoreresistanceintheverticaldirectionwhereallthewater has to move through both sand and gravel layers than the horizontal

direction where most of the water moves through just the gravel layers.Consequently,thehydraulicconductivityKzintheverticaldirectionwillbelessthanKx in the horizontal direction. It is not unusual to findKz values that areonlyone-fifthorone-tenthofKx(Bouwer,1978).ThecalculationofKxandKzvaluesinanisotropicalluvialdepositsisexplainedinthenextexample.

Example5.13Determine the total hydraulic conductivityKz in a vertical directionandthetotalhydraulicconductivityKx inahorizontaldirectioninanalluvialdepositsystemwhichconsistsof threehorizontal layers.Theheightzi and the hydraulic conductivityKi of each layer is given inFigure5.19.

Figure5.19Theheightandhydraulicconductivityofeachlayer.

Solution

1. If there is horizontal flow through the system, the hydraulicgradient J is the same in each layer (if J were not the same,pressure head differences would exist along the interfacesbetweenthelayers,whichisimpossibleinhorizontalflow).Kxistheaveragehydraulicconductivityofthemediuminahorizontaldirection,ZisthethicknessoftheentiresystemandZ1,Z2,Z3arethe thicknesses of each layer, respectively. The flow qi in eachlayer per unit thickness of the system can be expressed asfollows:

(5.78)

Summing theq values for each layer to get the total horizontal

flowqxperunitthickness,weget

2. If thereisaverticalflowthroughthesystemofFigure5.19, theflowqperunithorizontalareacanbeexpressedforthetoplayerasq =K1(ΔH1/z1),whereΔH1 is the total head loss in the firstlayer.SolvingtheequationforΔH1yields

(5.79)

Sinceq is thesameforall layers, thetotalheadlossΔHtcanbecalculatedasthesumoftheheadlossesineachlayer:

5.10WELLLOSSES

Thetotalleveldrawdownonawellconsistsoftwocomponents.Oneistheleveldrawdownsaobservedattheouterpartoftheboreholeandthesecondswasthewater moves through the filters at the pump. The second level drawdown isknownaslossesduetoboreholeshapingandisgivenbythefollowingequation:

(5.80)

whereCistheconstant,<0.5min2/m5,foragoodwelldesignnistheexponentofthedischarge,usually2

Thetotallossesofaboreholeforanequilibriumstate(uniformflowconditions)are

(5.81)

Butforanon-equilibriumstate(non-uniformflowconditions),totallossesare

(5.82)

Thewellefficiencyisgivenbythefollowingequation(Figure5.20):

(5.83)

or

(5.84)

Thespecificcapacityofawellisthewellefficiencyforaunitleveldrawdown:

(5.85)

Figure5.20Welllosses.

Example5.14Apumpingboreholeofdiameter0.4mpenetratesaconfinedaquifer.Theobserveddrawdownattheboreholeaftera12hpumpingwith1.6m3/mindischargeis7.1m.Thetransmissivityandthestoragecapacityof theaquiferare560m2/dayand7×10−3, respectively. (1)Thewelllosses,(2)thewellefficiencyand(3)thespecificcapacityofthewellmustbespecified.Solution

1.a.

b.

1. The well losses for the non-equilibrium state are calculated asfollows:

2. The well efficiency is the ratio of losses during movementthroughthesoiltototallosses:

3. The specific capacity of the well is calculated as the ratio ofdischargetothetotallosses:

5.11AQUIFERRECHARGE

Recharge is called the replenishment of groundwater resources from surfacewater resources. Rainfall is a natural way of enriching a basin. The artificialrechargeofanaquifer isdonebyavarietyofmethodswhen it isnecessary toincreasegroundwater resources.Thereare fivestandard techniquesofartificialrecharge; the first one is further divided into four subcategories (Mimikou&Baltas,2012):

SurfacewaterrechargeBasinmethodWater is diverted or pumped from streams, filling one or a series ofsmallnaturalbasinstoslowlyinfiltrateandreachtheaquifer.Thewatersupplyrateisshowntobedirectlyproportionaltothedifferenceinlevelbetweenthewatersupplyandthegroundwaterlevelanddecreaseswithtimeduetofillingofsoilvoids.Stream–CanalmethodThismethodenhancesthealreadysignificantsupplyoftheaquiferfrom

c.

d.

2.

3.

4.

5.

streams, by riverbed modifications and construction of dams andembankments.TrenchmethodSimilar to the stream–canal method, a series of shallow trenches areconstructedinafatbedwiththewaterdivertedtothemfromstreams.FloodingmethodItisthesimplestandmosteconomicmethodofrecharge.Itissimilartothebasinmethod,withtheexceptionthattheextentofsupplyisnaturalwith artificial embankments at its ends where the terrain is notconducivetoflooding.

SupplypitsThe excavation of a supply pit is a very effective method of artificialrechargeinthecaseofapermeablesoillayerwithashallowdepthbeneaththeimpermeablesurface.Oftenthebottomofthepitiscoveredbygravel.RechargewellsThese are equivalent to the abstraction wells but in reverse operation,since they are supplied with water from the surface. Around these anelevation cone is createdwhich is the image of a depression cone. Theformulasofthepumpingboreholesarealsoappliedtothesupplywells.InducedsupplyThis isdone inahydraulicallyconnectedsystemofaquifer–watercoursewhen water is pumped from a nearby well. In this case, the increasedpumpingfromthewellcreatesadditionalhydraulicgradientintheaquiferfromthewatercourseand,therefore,betterenrichment.WastewaterdisposalUsing wastewater that has undergone secondary treatment for aquiferrechargenotonlyenrichestheaquiferbutalsoleadstofurthertreatmentofwastewater.

Thesemethodsareselectedonacase-by-casebasis.

5.12SALINIZATION

Themixingof saltwater and freshwater is a commonproblemof groundwaterqualitydeteriorationofcoastalaquifersmainlyduetotheentryofseawater.

5.12.1Interfaceofsaltwaterandfreshwater

InFigure5.21,asectionofacoastalaquifer ispresented.Ateachpointof theinterface, thepressureof theoverlying freshwaterbalances thepressureof the

underlyingsaltwater,andthefollowingequationapplies:

(5.86)

or

(5.87)

whereρfisthefreshwaterdensity[M/L3]ρsisthesaltwaterdensity[M/L3]Zisthedepthofinterfaceateachpoint[L]histhehydraulicheadoverthesurfaceoftheseaateachpoint[L]

Figure5.21Balanceoffreshwaterandsaltwaterinanunconfinedcoastalaquifer.

The last equation is called the Ghyben–Herzberg expression and ignores thewatermovement. For typical values, if density ρs = 1.025 g/cm3 and ρf = 1.0g/cm3,theequationgetsthefollowingform:

(5.88)

That is, the saltwater is found at a depth equal to 40 times the height offreshwater,asobserved inpractice,or if thefreshwaterdepthdeclinesby1m,thesaltwaterprogressesinlandby40m.

5.12.2Saltwaterelevationcone

In thecasewherea layerof saltwater isbelow the freshwaterzoneandawellpumpsonly from the freshwater layer, there is an elevationof the interfaceofsaltwater and freshwater, as can be seen in Figure 5.22. This phenomenon is

knownaselevationcone.After its stabilization, the phenomenon may be described by the following

equation:

(5.89)

whereKisthehydraulicconductivity[L/T]disthedepthoftheinitialsaltwater–freshwaterinterfacebeneaththeboreholebottom[L]

When the elevation becomes critical (e.g.Z/d = 0.3−0.5), saltwater enters theborehole, polluting the supply. Consequently, the maximum discharge for theelevationtostayunderthecriticallimitcanbecalculatedbysubstitutingwithZ=0.5dinthelastexpression:

(5.90)

Figure5.22Saltwaterelevationconebeneathapumpingborehole.

Example5.15Inadeepaquifer, good-qualitywater lies at adepthof125munderwhich there is brackish water of specific weight 1.026 g/cm3. Apumping borehole pumps water at a depth of 80 m. Define the

maximumpumpingratetoavoidpumpingofsaltwater.Thehydraulicconductivityis800m/day.

SolutionThedepthof the initialsaltwater–freshwater interfacefromthedepthoftheboreholeisd=125−80=45m.Consequently,themaximumpumpingrateis

REFERENCESAftias,E.,1992,WaterSupply,NTUA,Athens,Greece(inGreek).Bear,J.,1979,GroundwaterHydraulics,McGraw-Hill,NewYork.Bouwer,H.,1978,GroundwaterHydrology,Vol.480,McGraw-Hill,NewYork.Chow,V.T.,1964,HandbookofAppliedHydrology,McGraw-Hill,NewYork.Freeze,A.R.andCherry,J.A.,1979,Groundwater,EnglewoodCliffs,NJ:Prentice-Hall.Gupta,R.S.,1989,HydrologyandHydraulicSystems,Vol.1110,EnglewoodCliffs,NJ:PrenticeHall.Hantush,M.S.,1964,HydraulicsofWells,inAdvancesinHydroscience,Vol.1,V.T.Chow(ed.),Academic

Press,Inc.,NewYork.Hermance,J.F.,1999,AMathematicalPrimeronGroundwaterFlow,NJ:PrenticeHall.Latinopoulos,P.,1986,HydraulicsofGroundWater,AUTH,Thessaloniki,Greece(inGreek).Mimikou,M.andBaltas,E.,2012,EngineeringHydrology,Papasotiriou,Athens,Greece(inGreek).Singh,V.P.,1992,ElementaryHydrology,NJ:PrenticeHall.Stallman, R.W., 1971,Aquifer-test design observation and data analysis, techniques of water resources

investigations,ChapterB1,inBook3,ApplicationsofHydraulics,USGS,Washington,DC.

Chapter6Hydrologicdesign

6.1INTRODUCTION

Hydrologicdesignmainlyreferstotheappropriatesizingofastructureinordertofulfilitsspecifiedpurposerelatedtowater(e.g.protectionfromfoods)andisconsidered a fundamental design preceding any other type of design (e.g.hydraulic).Ingeneral,hydrologicdesignisanecessaryelementinthedesignofa wide range of engineering works, hydraulic or not (e.g. reservoirs, bridges,roads,buildings,networks).Inthischapter,wewillfocusonlyonthehydrologicdesignofhydraulicworksthatarecategorizedas(1)reservoirsthatstorewater,(2) food safety (protection) structures, (3) networks (water supply, irrigation,etc.)and(4)worksthatimprovethewaterquality(watertreatmentplants,etc.).Morespecifically,wewillfocuson(1)and(2).Thedominantpositionamonghydraulicworks isoccupiedbydams, storing

riverwaterinreservoirsandaffectingthehydrographoveraperiodoftime.Thesizing of reservoirs is an important part of the hydrologic design. Substantialimportanceisalsoaccordedtothehydrologicsizingoffoodsafety(protection)structures, such as the spillway in a dam that provides protection from foodovertopping and the diversion of a river at a construction site (during theconstruction phase). In this chapter, we will discuss in detail the hydrologicdesign of a reservoir, a spillway, a river’s diversion tunnel and other specificdesigntopics.Bothdeterministicandstochasticapproachesinhydrologicdesignareexaminedthroughoutandthedifferencesarealsopresented.

6.2SIZINGOFRESERVOIRS

6.2.1General

Areservoirisanartificiallakecreatedbehindadamthatstoreswaterinflowingfromadrainagebasin.Themostcommonexampleofareservoirisariverdam(smallorlarge)thatisconstructedacrossariver.Also,therearereservoirsthatarecreatedundergroundorinartificialcavitiesoftheearth,wheretheextractionofwaterispossibleonlybymechanicalmeans(i.e.pumping).Waterstoragehasonlyonecommonpurpose:flowcontrol,bothspatiallyand

temporally. Indetail, a reservoirchanges thedistribution, in time,of thewatersurplusandshortage(quantitiesaboveandbelowareferencelevel,respectively,i.e.thewaterdemandortheaverageflow)oftheriver,ataparticularlocationofthe river.For instance, the irrigationneeds that are greater during the summerperiodnormallycannotbesatisfiedbythenaturalflowofariver,whichislowinthe summer. So, there is a need of a reservoir construction, in order to retainwaterduringtheseasonofhighriverflowandsupplythiswaterinthesummerseason, through proper flow management. Usually, the reservoirs do notexclusively serve a single purpose but serve multiple needs, i.e. water forirrigation, hydropower, water supply and recreation. In many cases, thesepurposes are antagonistic to one another: for example, in the case of energyproductionandirrigationpurposesduringthesummerseason.Nevertheless, themajority of reservoirs that have been created by big river

damsaremeantprimarily forpowergeneration. Insuchprojects,hydroelectricpower production units are installed at the bottom of the dam.High-diameterwater pipes (penstocks) drawwater from the reservoir and pass it through theunit. The water rotates the turbines of the power generation unit, whichtransforms the kinetic energy into electric energy. Then the voltage of theelectric current is stepped up, so that it can be transported,with small energylosses,across longdistancesuntil it reachesdistributioncentres.Theproducedelectric energy is proportional to the water discharge flowing through theturbinesandtheheightofthewaterfall.Single-purposehydropowerplantsalsoprovidewater forotherneeds (e.g. irrigation),but this factdoesnotalter theircharacterizationassingle-purposedams, if thesesecondaryabstractionsarenotprogrammedandtakenintoaccountintotheoptimizationprocessoftheproject,atleastduringthedesignphase.Fromthehydrologicdesignpointofviewofareservoir,whatmattersmostis

the reliability of the design. In order to have a reliable design, the storagevolumesizemustbecompatiblewiththetimeseriescharacteristics(mainlythestatistical) of the reservoir inflows. A tolerable risk must be chosen, for theestimated storage volume to satisfy the production of a defined output (e.g.primaryenergy)orgenerallytosatisfythedemand.

6.2.2Sectionsofvolumeandreservoirexploitation

Thetotalstoragevolumeofareservoircomprisesthefollowing:

1. The volume from the river bottom until theminimum level of operation,whichisdefinedasthelevelofwaterabstraction.Thiswaterlevelmustnotfall lower than thispoint,or the reservoirwillhaveoperationalproblems.This volume is called dead volume Vd. In this part of the reservoir,sedimentsaredepositedduetotheslowdownoftheflowvelocitytonearlyzero.

2. Thevolumebetweenminimumandmaximum (normal) operational level,which is called active volume Va. This is the volume used for watersubtraction (i.e. for hydroelectric production and other uses) and can bemanaged in time. Themaximum (normal) operation level determines thehydroelectricpotentialoftheproject,sincethenormalenergyproductionisapplicablebetweentheseminimumandmaximumwaterlevels(Figure6.1).

3. Thevolumebetween themaximumoperation leveland the spillway levelwhichiscalledfoodvolumeVf.Thisvolumeisusedforthecontainmentofthedesign foodof the spillway.The spillway level is themaximum levelthat the water is predicted to reach during the design food (this is themaximumfoodthatthespillwaycanroutesafely).Abovethislevel,afreeboard is added up to the crest level of the dam. In many hydroelectricprojects, where the flow is controlled by flow gates, the maximum foodleveldoesnotstaypermanentlyemptywaitingforanextremelyrare foodevent (for which the common return period is 10,000 years or even lessfrequent), but it canbe exploited for energyproduction, at least partially.So, the maximum operation level and the active volume may not beconstant, but they increase seasonally especially when the risk ofoccurrenceofthedesignfoodisextremelysmall.

Figure6.1Characteristicstoragevolumesinareservoir.

The permanent withdrawal Qt from the reservoir, for various uses can beexpressedasapercentage(α)ofthemeanvalueofdischargeoftheriverĪ(e.g.theannualmeanvalueofinflowsIt)foranumberofNyearsas

(6.1)

The percentage α is called the efficiency coefficient of the reservoir. If α isrelativelylow,e.g.30%,thenthereservoiriscalledlow-efficiencyreservoir.Ifαishigh,e.g.95%,thentheefficiencyofthereservoirishigh:alargeproportionofwatervolumeiscontrolledforvarioususes.Themaximumwaterefficiencylevelis100%(idealreservoir);inthatcase,theentireriverflowcanbestored,controlledandusedinthereservoir.Inthefollowing,thesizingmethodsforallthreevolumecomponentswillbepresented.

6.3CONVENTIONALMETHODOFSIZINGTHEACTIVERESERVOIRVOLUME

One of the most common sizing methods of a reservoir is the Rippl method(Rippl,1883),whichisbasedontheanalysisofthefluctuationofthehistoricalinflow volumes of the reservoir. The method is simple, but it has manydisadvantages, which will be discussed later here. This method creates manytechnological problems, especiallywith regard to the reliability of the design.The problem addressed by the Rippl method is explained in two alternativescenarios(Schultz,1976):

1. Anobserved timeseriesof inflows to the reservoirand the timeseriesofabstractions(consumption)aregiven.Thequestionistofindthenecessaryvolumeofthereservoirtosatisfythedemand.

2. An observed time series of inflows to the reservoir and an existing (asdesigned)maximumvolumeofthereservoiraregiven.Whatisaskedisthemaximum possible abstraction (demand for consumption) from thereservoir.

Atthispoint,onlythefirstscenarioisdiscussed.

6.3.1Constructionofacumulativeinflowcurve

TheRipplmethodusesasimplecumulativecurveof thehistoricalvolumesofinflowsinthereservoir.SoifIT isthetimeseriesofinflowdischargesintothereservoir(m3/s),thecumulativecurveisdefinedas

(6.2)

whereTisthetimebetween0,…,t.Forpracticalreasons,thecontinuouswatergraphIT,T=0,…,t,isexpressedonamonthlybasisandthecumulativecurveis

(6.3)

whereidenotesmonthΔtistheperiodoftime(1month)Nisthenumberofthemonthlyvaluesofthetimeseries(N=12×n,nisthenumberofyearsoftheobservedtimeseries)

The time series should be carefully selected to represent a significant droughtperiod.The Rippl method can be applied to any reservoir independent of the

efficiency coefficient α. However, for practical reasons, in order to find themaximum fluctuation of inflows in comparison with the demand curve, weassume that α = 100% (ideal reservoir) and that the demandQt is constant intimeandequaltothemeaninflowĪ:

(6.4)

Figure6.2presentstheconstructionofacumulativeinflowscurve(α=100%),plusthedemandcurvewhereQt=Ī.

6.3.2Estimationoftheactivereservoirvolume

Aftertheconstructionofthecumulativecurveofinflowsandthedemandcurve,thenecessaryvolumeofthereservoircanbeestimatedbytheRipplmethodinthenextstep: fromthe twopointsof thecumulativecurve thatdiffer themostfromthedemandcurve(maximumsurplusandmaximumshortage),twoparallellinestothedemandcurvearedrawnthatosculatetothecumulativecurve.TheverticaldistancebetweenthesetwolinesiscalledthemaximumrangeRt,swheretdenotesthestartpointoftheinflowcurvesandsdenotesthetimeperiodofthedatausedtoestimatetherange.Theanalyticalexpressionoftherange,asshowninFigure6.2,isthefollowing:

Figure6.2Cumulativecurveofinflowvolumesinareservoir.

(6.5)

In the aforementioned equation, the range is expressed as the algebraicdifference of the maximum surplus and the maximum shortage. The activevolume of the reservoir based on this method is Rt,s. If the reservoir that isdesignedthroughthismethodisfullwhenitsoperationstarts,thentheoretically,itwillnevergetempty,andthedemandwillbealwayssatisfied,aslongasthetime series of inflows that was used for the design is reproduced during theoperationofthereservoir(Chowetal.,1964).

6.3.3DisadvantagesofRippl’sconventionaldesign

During the application of the Rippl method for the sizing of a reservoir, weshould not oversee a critical point: in order to verify the assumption that thereservoirwithvolumeRt,swillneveremptyandwillalwayssatisfythedemand,theotherassumptionmustalsobeconsidered,namely,thetimeseriesofinflowswillbe reproduced in the futureduring theoperationphaseof theproject.Theoccurrence of this event has zero probability. In other words, there is asubstantiallynon-zeroprobabilitythatthereservoirwillnotsatisfythedemand,andalso,thereisanon-zerochancethatthereservoirwillbeempty.Another disadvantage of the method is that the volume Rt,s is influenced

directlybythedurationsofthetimeseriesthatisusedfortheestimation.Ifweassumethatthefollowingequationholds:

(6.6)

ThevolumeRt,sisanexponentialfunctionofs(withexponentthecoefficientH)(Hurst et al., 1965). The departure of the exponentH from 0.5 is the Hurstphenomenon,which shows quantitatively the persistence of the time series It,given that St,s is the standard deviation from time t + 1 to t + s andK is aconstant.Forexample,ifsomeonechangesthedurationsofthetimeseries,Rt,swillalsochange.Thedependenceofthereservoirvolumeonthedurationofthetime series is an important disadvantage, given the fact that long observationperiods (something desirable) lead to huge and expensive reservoirs (which isundesirable).Finally, another disadvantage is the inefficiency of the method to provide,

alongwiththeestimatedvolume,theprobabilityofthisvolumetobeinsufficienttomeetthedemand(sinceitisobviousthatthereisahighprobabilityforthistohappen).Alldisadvantagesofthisconventionalmethodcanbesignificantlydiminished

byusinganon-conventionalmethod:thecumulativecurveofRipplisappliedtoa large number of synthetic time series of inflows, which are generated by astochastic model of simulation of the historical series, as explained in thefollowing. The non-conventional method of reservoir sizing satisfies bothnecessary conditions that can make the Rippl method technically correct andreliable;e.g.itcanprovidethecapabilityofdeterminationoffailureprobabilitytosatisfythedemandandseparatestheestimationofthereservoirvolumefromthedurationoftheobservedhydrologicdata.

6.3.4Sequentpeakanalysismethod

Sequent peak analysis method is an alternative conventional method forreservoir sizing. Itderives the largestdeficitvolumeofadischargeserieswithrespect to a certain threshold level (usually zero).Thismethodcanbe appliedeither for constant or varying demands and it is more suitable for longobservationperiods.Theinflowsequenceisassumedtorepeatandanalgorithmis carried out, over more than one cycle if necessary. The algorithm can beexpressedmathematicallyinthefollowingform:

(6.7)

whereQtistheinflowRtistheoutflow,inacertaintimestept

Atthebeginningoftheprocess,weassumethatK0=0.IfthevalueofKtiszero,afterthefirstcycleofcalculations,thentheprocess

ends: thecriticalvalueiscontainedin thiscycle.TheprocessalsoendsafterapointwhereinthevaluesKtofeachtimestepareidenticaltothecorrespondingonesofthepreviouscycle.Themethodisexplainedinthenextexample:

Example6.1:ReservoirsizingwithsequentpeaksanalysismethodEstimatethereservoirsizewithinflowsanddemandsasseeninTable6.1, for a period of 6 years. Total inflows are equal to total demand(=24units).

SolutionTable6.2isfilledforthefirstcycleofthealgorithm.Asitisobserved,duringtheendofthefirstcycle,thevalueofKtis

notzero,sothenextcycleisrunning(Table6.3).Weperformalsoathirdcycle(Table6.4):

Table6.1Inflowsanddemands

Table6.2Firstcycleofcalculations

t Qt Rt K(t‒1) Kt=K(t‒2)+Rt‒Qt1 3 5 0 2

2 6 4 2 0

3 8 6 0 0

4 4 4 0 0

5 2 3 0 1

6 1 2 1 2

Table6.3Secondcycleofcalculations

t Qt Rt K(t‒1) Kt=K(t‒1)+Rt‒Qt1 3 5 2 4

1 6 4 4 2

3 8 6 2 0

4 4 4 0 0

5 2 3 0 1

6 1 2 1 2

Table6.4Thirdcycleofcalculations

t Qt Rt K(t‒1) Kt=K(t‒1)+Rt‒Qt1 3 5 2 4

2 6 4 4 2

3 8 6 2 0

4 4 4 0 0

5 2 3 0 1

6 1 2 1 2

Bycomparisonofthesecondandthirdcycles,itisobviousthatallvaluesofKt respectivelymatch,meaningtheendof theprocess.Themaximum volume of the reservoir is themaximum value ofKt andequalsfourunits(inthefirstyearofthesecondcyclecoincidingwith

thesameyearinthethirdcycle).

6.3.5Non-conventionalstochasticmethodsofreservoirsizing

Thebasicnotionbehindthesemethodsistheobservationofthefactthateveryset of time series of natural inflows in the reservoir is only one case ofpracticallyinfinitepossiblescenariosofavariable’svalue(inthiscase,theriverdischarge),which is actually a stochasticvariable (Yevjevich, 1967;Mimikou,1985).So,insteadofdependingononlyoneobservedtimeseriesthatgivesonlyasingleanswer,thereservoirvolumeinadeterministicway(theprobabilityofthe scenario to occur is equal to unit), a number of synthetic time series ofinflows can be used, which are statistically equivalent to the historical timeseries, having the same probability of occurrence.With thismethod,we get arange of volume for each time series by using the cumulative curve method(Rippl).Theresultingvolumevalueshaveaprobabilitydistributionfunctionthatcanbecalculated,whichdeterminesarelationbetweenvolumeandprobabilityof exceedance and actually determines a relation of volume and the risk ofdesign. As far as the selection of the model for the stochastic component isconcerned,wemayhaveautoregressive(AR)modelsusingtheMarkovprocess,movingaverage(MA)models,autoregressivemovingaverage(ARMA)models,autoregressive integrated moving average (ARIMA) models, fast fractionalGaussiannoise(FFGN)models,etc.(Reddy,1997).Ashortdescriptiontothesemodelsfollows.

AR(p),autoregressivemodels

TheMarkovprocessconsidersthatthevalueofanevent(i.e.streamflow)atonetimeiscorrelatedwiththevalueoftheeventatanearlierperiod,i.e.aserialorautocorrelationexists in the time series, indicating theorderofmemory (p)ofthe process. In a first-order Markov autocorrelation process AR(1), thiscorrelationexistsintwosuccessivevaluesoftheevent.Thefirst-orderMarkovmodel,whichconstitutestheclassicapproachinsynthetichydrology,statesthatthe value of a variable x in one period is dependent on the value of x in aprecedingtimeperiodplusarandomcomponent.Thus,thesyntheticflowsforastreamrepresenta sequenceofvalues,witheachvalueconsistingof twoparts(Gupta,1989):

(6.8)

wherexiistheflowatithtime

diisthedeterministicpartoftheflowatithtimeeiistherandompartatithtime

Thevaluesxiarelinkedtothehistoricaldatabyensuringthattheybelongtothesame frequency distribution and possess similar statistical properties (mean,deviation, skewness)as thehistorical series.ThegeneralMarkovprocedureofdatasynthesiscomprisesthefollowing:

1. Determiningstatisticalparametersfromtheanalysisofthehistoricalrecord2. Identifyingthefrequencydistributionofthehistoricaldata3. Generating random numbers of the same distribution and statistical

characteristics4. Formingthedeterministicpartbyconsideringthememoryandpersistence

(influenceofpreviousflows)andcombiningitwiththerandompart

After the execution of the aforementioned procedure, we have the followingequationforflowxi:

(6.9)

where(thefirsttwocomponentsarethedeterministicpartwhilethethirdistherandompart)xiisthestreamflowattheithtimeisthemeanrecordedflowr1isthelag1serialorautocorrelationcoefficientti istherandomvariatefromanappropriatedistributionwithameanofzeroandvarianceofunity

iistheithpositioninseriesfrom1toNyears

MA(q)movingaveragemodels

The other category known as moving average MA(q) models considers astochastic component (streamflow event) to be a constituent of a number ofrandomvariates,withthecurrentvariateassignedaweightofunityandrandomvariatesgeneratedatantecedenttimesmultipliedbytheassignedfactors.Thegeneralequationsare

(6.10)

where f1,…fqareweightcoefficients. isa linearcombinationofq+1whitenoise variables andwe say that it isq-correlated,meaning that and areuncorrelatedforalllagsτ>q.Soqistheorderofcorrelation(Reddy,1997).Theautovarianceequationis

(6.11)

Andfinally

(6.12)

Itisvalidthat andE(at-Kat)=0.Thevarianceis

(6.13)

Theautocorrelationfunctionis

(6.14)

Ifq=1,wehavetheMA(1)model:

(6.15)

(6.16)

(6.17)

Respectivelyifq=2,MA(2)is

(6.18)

(6.19)

(6.20)

(6.21)

(6.22)

ARMA(p,q)models

MA(q)modelsaregenerallyinappropriatefordirectapplicationtohydrology.AnARMAmodel,however,combinesanautoregressivemodelAR(p)andamovingaveragemodelMA(q)toproduceamixedmodelknownasanARMA(p,q)model.ARMA(p,q) consists of two polynomials of the orderp andq, respectively, asfollows(Gupta,1989):

(6.23)

whereαiistheithvariableofthesequence(stochasticcomponent)ofzeromeanandunitvariance

φp,1φp,2,…areautoregressiveparametersorweightsni,istherandomnumberatithtimefq,1,fq,2,…aretherandomvariateweights

Estimationof theparametersφand f isnotastraightforwardprocedure. In theclassofmixedmodels,thesimplestistheARMA(1,1)model,givenby

(6.24)

ARIMA(p,d,q)models

If thevalueofparameterφ1,1 isclosetoits limits−1and1, thenon-stationarybehaviourofthehistoricalhydrologicsequenceisindicated.Thenon-stationarityisaccountedforbymeansofthedth-orderdifferenceoperator,whichrepresentssuccessivedifferenceofd termsof stochasticvariables (i.e.αi values).This isknown as an autoregressive-integrated moving average ARIMA(p,d,q) model(Reddy,1997).

FFGNmodels

TheFFGNmodelisavariationinthefamilyofFGNmodelswiththeadvantage

of smaller computational time required. FFG type I, FFG type II and filteredFGNmodelsarethemostcharacteristicmodelsofthisfamilyexceptFFGN.AnFGNtypeIImodel,forexample,isgeneratedby(Reddy,1997)

(6.25)

wherehisHurst’scoefficientMisthememoryoftheprocesszuareindependentstandardnormalvariates

TheFFGNmodelhas twoadditivecomponents ,where takes intoaccountthelow-frequencyeffectsand takesintoaccount thehigh-frequencyeffects.Thecomponent isgivenby(Reddy,1997)

(6.26)

whereXt(GM/i,B)isastandardAR(1)normalprocesswithρ1equalto(-B)-Iandwith standard normal independent variates as a random component. TheparameterB takes a suggested value of 2–4.Wi areweights. The parameter ρbetween the range of 15 and 20 is found to be sufficient in practice. Thecomponent isaseparateAR(1)normalprocesswithzeromean.

6.3.6Syntheticdataofinflowsandtheircumulativecurves

Thesynthetictimeseriesofdischargesaregeneratedbyastochasticsimulationmodelsuchastheonespreviouslydescribed.Theproducedgroupoftimeseriesofthephysicalvariableshasthesamestatisticalcharacteristicsintherangeofaselectedorderwiththehistoricaltimeseriesandcanhaveanydesiredduration(usually must cover the economic life of the project, i.e. 50–100 years).Practically,wecanproducean infinitenumberof timeseriesof this typefromthe simulation, by changing the stochastic component (thewhite noise) of thesimulation (MonteCarlo technique). Every single produced time series has atleast theoretically the same probability of occurrence in the future as thehistoricaltimeseries.Inthecaseoftherunoffprocess,dependingonthegroundwaterstorageand

permeabilitycharacteristicsofthebasin,aconsiderableamountofrainfallmaybecomegroundwaterandittakesalongertimeforthisgroundwatertoappearas

runoffinthestream.Thisisespeciallytrueinthecaseoflargebasins.Sorainfallof a previous yearmay contribute to the runoff of the next year. This storageeffect,variouslyknownasthememoryoftheprocessorthepersistence,istrulyreflectedinthefirst-orderARmodelasthevalueofthevariableatanytimethatis partly dependent on the previous variable. Therefore, theAR(1) model is acommonchoice todescribeannual flowseries. If thememoryextends for twotimeperiods,perhapsAR(2)mayhavetobetried,somethingveryrare.The historical characteristics that are embedded in a synthetic time series

depend on the type of simulation used. For instance, the Markov type ofsimulations keeps the first and second order of statistical moments and thecharacteristicsofautoregressionofthehistoricaltimeseries.AMarkovchainisa stochastic process with theMarkov property (thememoryless property of astochasticprocess).The term‘Markovchain’refers to thesequenceofrandomvariablesthataprocessmovesthrough,withtheMarkovpropertydefiningserialdependenceonlybetweenadjacentperiods(asina‘chain’).Itcanthusbeusedfordescribingsystemslikesometypesofstreamrunoffsthatfollowachainoflinkedevents,wherewhathappensnextdependsonlyonthecurrentstateofthesystem.Onthecontrary,othertypesofsimulations,likeFFGNmodels,keep,inaddition,thestatisticalpersistence.At this point, given the fact that the produced group of time series has a

usually long duration, in order to cover the lifetime of the project, itmust beunderlinedthatthedesignerdoesnotperformahydrologicalprognosis(whichisactuallyapredictionofthefuture).Thesimulatedsynthetictimeseriesrepresentonly statistically equivalent time series with the observed one. After theproductionofmsynthetictimeseries(e.g.100monthlytimeseriesof100yearsduration),whichhavebeengeneratedbyastochasticsimulationmodel,synthetictime series are plotted for every synthetic time series (i = 1,2,…, m). TheprocedurethenisdoneaccordingtoEquation6.3or6.4forthetotaldurationofthecumulativecurve.

6.3.7Designrisk:Estimationofvolumebasedonacceptablerisk

Fromeverycumulativecurveofasynthetictimeseries,themaximumrange,andconsequentlyastoragevolumeVi, iscalculatedby theRipplmethodassumingthat there is full efficiency (a = 100%) and a constant discharge equal to theaverageinflow.VolumesVi=1,2,…,m,haveacumulativecurveofprobabilitydistributionfunctionFvi(v)thatcaneasilybefound.Thisdistributioncorrelatesthevolumesofstoragewiththeprobabilityofexceedancej(%)accordingtothefollowingrelationship:

(6.27)

Theprobability j(%) is defined as the risk of designwith volumeVi = v andrepresentstheprobabilitythatthespecifiedvolumevdoesnotmeetthedemand.By defining in this way an acceptable risk of design, the volumeVi = v thatcorresponds to the analytical or graphical expression of the distributionFvi(v)canbeinstantlycalculated.ThesecalculationsareshowninagraphicalwayinFigure6.3forasumofm=100volumesofVibetweenthevaluesof3and10×109m3.Itisobviousthatifsomeonechoosesarelativelysmallriskj=20%,that is,

100×0.20=20years,theeconomiclifeoftheproject,thevolumewillnotmeetthedemand,and thevolumemustbe7.65×109m3.On theother hand, if j=90%,thenthevolumeis3.20×109m3.It was found (Schultz, 1976) that if non-conventional sizing based on a

historical time series ofn years is repeated using only a part of the historicaltimeseries(i.e.from15years,only10isused),theresultingFvi(v)ispracticallythe same. This presumes that the stochastic method is independent of theduration of the historical time series, in contradiction to the totally dependentconventional method. This fact is an important improvement in the area ofhydrologicdesignofareservoir.

Figure6.3Determinationofdesignriskinthestoragevolumeofareservoir.

6.3.8Advantagesofthenon-conventionalsizingmethod

From theprevious analysis, it is obvious that theuseof thenon-conventional,stochasticmethodof reservoir sizing cannowsolve theproblems that exist inthe conventional method. The basic assumption of the conventional method,whichwastheimpossiblereproductionofthehistoricaltimeseriesinthefuture,is no longer needed and it is replaced by a more logical assumption that anumber of statistically equivalent time series can occur, intending to cover allpossiblescenariosthatnaturecanproduce.Additionally,insteadofdeterministicdesign with only one answer for the required volume, the non-conventionalmethodcreatesavolume-riskdistributioncurve,asseeninFigure6.3.Animmediateresultistheabilitytochoosetheacceptablerisk,orviceversa,

with the volume given (that is, in a number of cases, dictated by technical oreconomicalrestrictions)toknowtheparticularriskofthisvolume.Inbothcases,the qualitative result is the reliability enhancement of the design. Finally, theindependenceof the stochasticmethod from the timedurationof theobservedtime series of the discharge solves the problem of dependence of risk (which

must be selected by technical, economic or safety criteria) on the duration ofobservations (which is irrelevant to the decision criteria and, definitely, isdependentonthealgorithmused).

6.3.9Sensitivityofthereservoirdesignonthemechanismofsyntheticdischargegeneration

Apoint thatneedsextraattention,regardingthestochasticmethodofsizing, isthat the anticipated statistical equivalenceof the synthetic time serieswith thehistoricalonesmustbecompatiblewiththeneedsandthespecialrequirementsofthedesign.It isknown thateverystochastic simulationmodel,dependingon themodel

(infrastructure) capabilities, keeps in the synthetic time series a group ofhistorical statistical features. Also there are some special requirements in thereservoir design to keep some historical statistical features. In order tostandardize theserequirements,wemustconductasensitivityanalysis, relativetothedesignoftheinflowgenerationmechanism.Ithasbeenfound(WallisandMatalas,1972) that thestoragecapabilityofa

reservoirisdirectlyrelatedtothegenerationmechanismoftheinflows,inotherwords, to the chosen stochastic simulation model. More particularly, it wasfound that the minimum storage capacityVm of a reservoir that must meet aspecified demand is related to the efficiency coefficient α, the value of theautocorrelation coefficient of first-order r1 and the Hurst coefficientH of theinflows.For relatively low-efficiency coefficients (α < 0.80),Vm dependsmainly on

the autocorrelation coefficient r1. For this reason, all stochastic models thatsustain this statistical characteristic, i.e. Markov models and models withmoving averages, can be used. For high-efficiency coefficients (α ≥ 0.80),Vmdependsmostlyon theHurst coefficientH,which expresses thepersistenceofthe time series. In these cases, the former models that do not retain thepersistenceofthehistoricaltimeseriesareimproperanddifferentmodelsmustbe used, such as FFGN that retain (apart from the first and second order ofmomentandautocorrelation)thestatisticalfeatureofpersistence.

Example6.2:EstimationofstoragevolumeusingRipplandstochasticAR(1)methodsAtariversite,whereadamisplanedtobeconstructed,wehavethemonthly inflows It for a period of 20 years, given inTable 6.5. The

lifetimeoftheprojectis50years.ThetimeseriesofoutflowsfromthereservoirQt is assumed tobe constant andequal to the annualmeanvalueofinflows.Thefollowingareasked

1. Thesizingof thereservoirusing theconventionalRipplmethodfor10and20yearsoftimeseriesdata

2. Thesizingofthereservoirwiththenon-conventionalmethodforrisk = 20% (without keeping the statistical persistence) for thewholetimeseriesof20years

3. Thesizingofthereservoirwiththenon-conventionalmethodforrisk=20%(withoutkeepingthestatisticalpersistence)usingonlythefirst10yearsofthetimeseries

4. The value of risk that corresponds to the volume that wasestimated with the conventional method, with the use of theresultsofthenon-conventionalmethod

5. Comparison of the results of the non-conventional methodbetween thecaseof20-year timeseriesand thecaseof10-yeartimeseries

Remarks

Thetimeseriesofinflowsisnotconstantandasafirststepneedstobestabilized.Forthenon-conventionalmethodofsizing,themodelAR(1)willbeimplemented.At least 40 synthetic time series of 50-year length will beconstructed.The random part of the time series of inflows is assumed tofollowtheGaussiandistribution.

Solution

1. In order to calculate the storage volume with the conventionalRipplmethod,wecreate thecumulativecurveof inflowvolumefor the time seriesof20years.The range that is createdby themaximum surplus andmaximum shortage is the needed storagevolume,asitisseeninFigure6.4.Inthisexample,themaximumsurplus for a time series of 20 years is 0.72 × 109m3, and the

maximumshortageis0.47×109m3.Intotal,thestoragevolumeis1.19×109m3.

Table6.5Monthlyinflowstothereservoir(m3/s)

Figure6.4CumulativeinflowsusingtheRipplmethodfora20-yeartimeseries.

Figure6.5CumulativeinflowswiththeRipplmethodfora10-yeartimeseries.

Respectivelyforonlythefirst10years,weplotFigure6.5.In this example, the maximum surplus for a time series of 10yearsis0.44×109m3,andthemaximumshortageis0.34×109m3.Intotal,thestoragevolumeis0.78×109m3.

2. In order to calculate the active volume of the reservoir with agiven riskusing thenon-conventionalmethod,wemust initiallyproduce synthetic time series statistically equivalent with theactual time series of inflows to the reservoir, and then, proceedwiththeestimationofthevolumeofagivenrisk.Thesearethestepsthatwemustfollow:

Productionof40synthetictimeseriesof50-yearduration.Estimation of themaximum range and subsequently of thestoragevolumeforeveryoneofthesetimeseries(convertedtovolumeunits),usingtheRipplmethodSortingofthesevolumesindescendingorderCorrelationforeachoneofthesevolumesoftheprobabilityP=m/(N+1)wheremistheorderandNistheirnumberPlottingofacumulativecurveofprobabilitydistributionofvolumes,where thevolume isplotted in thex-axis and therespectiveprobabilityofsurpass(risk)J=P%isplotted intheiny-axisSelection from thediagramof the respectivevalue forR=20%

ProcedureofsynthetictimeseriesofinflowsThestepsthatwemustfollowtoproducesynthetictimeseriesofinflowsarethefollowing:a. Creationofadatafileofactualvaluesofinflows(m3/s).b. Estimationofthestatisticalcharacteristicsofthetimeseries

of inflows like the mean, the variance, the standarddeviation, the skewness coefficient and the kurtosiscoefficient.IfN is thenumberofthemonthlyvaluesofthetime series, then these coefficients are calculated by thefollowingequations:

(6.28)

(6.29)

(6.30)

(6.31)

(6.32)

c. Estimation of the autocorrelation coefficients of the timeseriesforhalfofitslengthandretainingtheautocorrelationcoefficientsofthefirstandsecondorder.Theautocorrelationcoefficient of the time series, with itself shifted for everytimestep(coefficientofTorder),is

(6.33)

whereTisthetimestepoftheshiftistheaverageofthetimeseriesintimetistheaverageofthetimeseriesintimet+T

d. Stabilizationofthetimeseriesofthemonthlyoutflows.Thestabilization is possible through the abstraction from eachvalueoftheaveragevalueandthedivisionoftheresultwith

the standard deviation of the respective month. In thefollowing, we can see the relationship that is used for thestabilization:

(6.34)

whereX1(I,J)isthestabilizedvalueofthedischargeX(I,J)isthemonthlyoutflowinm3/sJisthemonthIistheyearM(J)istheaveragevalueofmonthJSD(J)isthestandarddeviationofmonthJ

e. Estimation of the statistical characteristics of the stabilizedtimeseries.

f. Estimation of the autocorrelation coefficients of thestabilizedtimeseriesforhalfof its lengthandretainingtheautocorrelationcoefficientsofthefirstandsecondorder.

g. ApplicationofthemodelAR(1)fortheestimationoftheF1= R(1) coefficient where R(1) is the coefficient ofautocorrelationofthefirstorderofthestabilizedtimeseriesand of the theoretical deviationC1 given by the followingrelationship:

(6.35)

whereC2isthevarianceofthestabilizedtimeseries.h. Production using a computer of random normal numbers

withaverage0andstandarddeviation1.Calculatethevaluesof synthetic stabilized time series of outflows using thefollowingrelationships:

(6.36)

whereC1andF1arethetheoreticaldeviationandthecoefficientofthemodelAR(1)NR(I)istherandomnumber

i. Estimation of the statistical characteristics of the synthetictime series. The values of average and variance mustcoincidewiththerespectivevaluesoftheactualtimeseriesofinflows.

j. Estimationoftheautocorrelationcoefficientofthefirstorderof the synthetic time series which must coincide with the

respectivevalueoftheactualtimeseries.k. Productionof40synthetic timeseries,of50-year length in

thesameprocedure.Inthisexample,aftertheapplicationofthe procedure mentioned earlier, the following results areproduced(Tables6.6through6.8):

Finally,weplotthenextfigureandselectthevalueofvolumeforrisk20%,whichis2.2×109m3.

Table 6.6 Statistical characteristics for 20 years for anon-stabilizedtimeseries

Mean 24.27

Variance 442.00

Standarddeviation 21.02

Skewcoefficient 1.05

Kurtosiscoefficient 4.05

R(1) 0.63

R(2) 0.33

Table6.7Statisticalcharacteristicsfor20yearsfor thestabilizedtimeseries

Mean 1.71E–8

Variance 0.95

Standarddeviation 0.98

Skewnesscoefficient 0.91

Kurtosiscoefficient 3.88

R(1) 0.42

R(2) 0.18

Table6.8Statisticalcharacteristicsfor20yearsfor themodelAR(1)

F1 0.42

C1 0.89

Mean 25.23

Variance 455.731

Standarddeviation 21.35

Skewnesscoefficient 0.76

Kurtosiscoefficient 2.62

R(1) 0.66

R(2) 0.35

3. Iftheprocessisrepeatedusingonlythedataofthefirst10years,itgives(Tables6.9through6.11).Fromtheaforementioneddiagram,thevolumevalueforrisk20%isselected,whichis1.85×109m3.

4. Inthecaseoftheuseofa20-yeartimeseries,thevolumeof1.19×109m3,derivedfromtheRipplmethod,correspondstoariskof100% (Figure 6.6), while with the use of 10 years of data andvolume0.78×109m3,theriskisalso=100%(Figure6.7).

5. If20yearsofdatasetisused,thefinalvolumeforriskof20%is2.2×109m3,while in theothercase(10yearsofdata)with thesamerisk,thevolumeis1.85×109m3.Thedifferenceis16%andjustifiesthefactthat thelengthofthetimeseriesisunimportantconsidering the use of stochastic models, like AR(1), for theestimationofthestoragereservoirvolume.

Table 6.9 Statistical characteristics for 10 years for anon-stabilizedtimeseries

Mean 25.81

Variance 500.19

Standarddeviation 22.37

Skewnesscoefficient 1.09

Kurtosiscoefficient 4.39

R(1) 0.65

R(2) 0.34

Table 6.10 Statistical characteristics for 10 years forthestabilizedtimeseries

Mean 1.52E–07

Variance 0.91

Standarddeviation 0.95

Skewnesscoefficient 0.44

Kurtosiscoefficient 2.98

R(1) 0.46

R(2) 0.27

Table 6.11 Statistical characteristics for 10 years forthemodelAR(1)

F1 0.46

C1 0.85

Mean 25.82

Variance 489.69

Standarddeviation 22.13

Skewnesscoefficient 0.89

Kurtosiscoefficient 3.23

R(1) 0.65

R(2) 0.31

6.3.10Non-conventionalmethodofsizingincludingthepersistenceininflows

In the cases of design with high-efficiency coefficient, more than 80%, theretention of persistence of the historical time series of reservoir inflows isachievedbypropermodels,liketheFFGNmodel(Reddy,1997).Persistence is a long-term statistical feature of the discharge time series;

according to its definition, periods of high discharges tend to follow otherperiods of high discharges,while the same phenomenon tends to exist also inperiodsoflowdischarges.Persistence characterizes the fluctuation of discharges.Hurst (1951), during

his study on long-term reservoirs, defined that the maximum range of thecumulative fluctuations from the average time series of inflows of s years isequal to the reservoir storage capacity, which is needed to satisfy a yearlydemandequaltotheaverageinflowofsyears.

Figure6.6 Probabilitycurveofvolumesfor20years.

Figure6.7 Probabilitycurveofvolumesfor10years.

Hurstfound(Equation6.6)thattherangedividedbythestandarddeviationofthe inflows varies exponentially with the duration of the time series with anexponentH (H > 0.50)which expresses the persistence of the time series. Inotherwords,foraspecifieddurationofhistoricaldata,thesizingofareservoir

withtheuseofsynthetictimeseriesthatareproducedbymodelsnottakingintoaccountstatisticalpersistence,willleadtosmallerreservoirvolumeinrelationtomodels with statistical persistence, for a given value of risk j%. The use ofmodelswithpersistence leads to increase indesign risk for agivenvolume incomparisonwithmodelswithoutpersistence.This results in theuseofmodelswith statistical persistence that gives more conservative and reliable reservoirdesigninhighexploitationsystems(Mimikou,1985).

6.3.11Sizingofareservoirwithaspillwaywiththeuseofthemethodofmaximumshortage

The determination of the reservoir volume with the method of the maximumrangeoftheinflowcumulativecurvesfromthedemandcurveisnotsuitableinmany actual cases of reservoirs with overflow structures. Gomide (1975)introducedtheterm‘maximumcumulativeshortage’orjust‘shortage’,whichisdefinedinthefollowingtext.Assume that the reservoir is initially full andV0=0corresponds to the full

storage capacity. In this case, overflow will occur when the water balanceequationfor thereservoirVt=Vt−1+It−Qt (It the inflow,Qt theoutflow) ispositive(evapotranspirationandotherminorpartsoftheequationareomittedforsimplicity reasons).AssumingVt = 0 and for time equal to t + 1, the storagechangesaccordingto thebalanceequation.InFigure6.8, it isobvious that theneededstoragecapacityisequaltoDt,sexpressedanalyticallybythefollowingequations(Bayazit,1982):

(6.37)

(6.38)

ThestatisticalvariablesofshortageDt,sandrangeRt,s(whicharebothstochasticvariables if estimated for a number of synthetic time series) are completelydifferentevenif thereservoirdoesnotoverflow.TheaverageofDt,sissmallerthan the average Rt,s but the reverse occurs in case of variances of the twovariables.

Figure6.8 GraphicaldefinitionofshortageDt,s.

TheasymptoticequationsofFellow(1951)forindividualinflowsinthecaseofcompletemanagementofthereservoirareasfollows:

(6.39)

Gomide(1975)hasderivedsimilarequations:

(6.40)

wheresisthedurationofthetimeseriesEdenotestheexpectedvalue

Intheexaminedliterature(Gomide,1975;Pegram,1980),significantefforthasbeen put on the statistical behaviour of the shortage, especially in the case ofcompletemanagement.FigureshavebeenpublishedforthemeanvalueandthevarianceoftheDt,sdistribution.Nevertheless,thereareafewstudiesdevotedtothepartialmanagementofthe

reservoir(thecaseofE(It)>E(Qt)).Thereasonfor this is that it isdifficult todeduce analytical equationswhen the inflows are linearly correlated (Bayazit,1982). In Figure 6.9, we can see the diagram for the cumulative distributionfunctionF(Dt,s) ofDt,s, for a group of autocorrelation coefficients r1, of firstorder of inflows, whereDt,s have been estimated by synthetic time series ofMarkovmodelsAR(1),fora50-yearlifetimeofaproject.

Figure6.9 CumulativecurveofshortageDt,sprobabilities,fortotalexploitationandvariousvaluesofr1.(FromBayazit,M.,J.Hydrol.,58,1,1982.)

ThesizingofthereservoirthatisbasedontheshortageDt,sfollowsthesameprocedureasthenon-conventionalmethoddescribedfortherangeRt,s.Initially,manysynthetictimeseriesofinflowshavebeendeducedbyaproperstochasticsimulationmodel(MarkovmodelinthecaseofpersistencerejectingandFFGNwhenkeepingofpersistenceistargeted).Foreverytimeseries,weestimatetheshortage Dt,s and then we construct the cumulative probability distributionF(Dt,s),which,accordingtoFigure6.9,dependsontherangeofautocorrelationof the time series of inflows. Figure 6.9 covers a wide variety of cases; forannualdischarges,r1isalmostineverycasesmallerthan0.5andthelifetimeofprojectsisusually50years.Finally,bychoosinganacceptableriskj%,wecanestimatetherespectivevolumeequaltotheshortageDt,sthatcorrespondstothisriskfromthedistribution.Forexample,inFigure6.9forj=20%andr1=0.20,wehaveD50=V50=12×109m3.

6.4SIZINGOFARESERVOIRINARIVERSITEWITHOUTMEASUREMENTS

Veryoftendamsandreservoirsaretobeconstructedinremoteareaswithoutanynearbyflowmeasurementstation,withthenearestonelocatedatalargedistancealong the same river or, even worse, in another river. The remoteness of theposition sometimes prohibits the construction of a measurement station evenafter the decision for the project construction is taken. In this case, thehydrologicdesignmuststartfromapriorlevel.Thetimeseriesofdischargesatthepositionofthedamiscalculatedbytransferringthehydrologicinformationfromnearbystations.Thistransferisfeasiblethroughrainfall-runoffmodelsthatare calibrated at existing measurement locations. Then, given the assumptionthattheyarealsovalidforthedamposition,theyareapplied,usingasinputdatatherainfallon thedamposition(acquiredbyraingaugeswhicharedistributednationwide).Nevertheless,thistransfermighthavemanyproblems,suchasthepossibility

that the equation of rainfall-runoff is not valid in the second position, or thedistance between the two positions is significant, so that the calibratedparameters of the model differ. One method that is often used in hydrologicinformation transfer, especially in relatively extended but hydrologicallyhomogenousareas,isthedevelopmentofregionalrainfall-runoffmodels.

6.4.1Regionalhydrologicmodels

Regionalhydrologicmodelsareempiricalrelationsdeveloped(usuallywiththemethodofsingleormultipleregressionanalysis)betweenadependentvariable(i.e.discharge),whichlackstheobservationsataspecificoutletofariverbasin,and several independent physical variables of the basin (i.e. rainfall, basindrainage area, river length, slope), which can be estimated from archives andmaps.Themodelparameters(proportionallytothecorrelationlevel)expressthegeographical variation of the examined dependent variable within ahydrologically homogenous area. The latter is an area where the hydro-meteorologicalconditionsareverysimilar.Aregionalmodelcanbecalibratedwithdatafromsiteswheremeasurements

ofthedependentvariable(i.e.discharge)exist.Thesesitesshouldbedistributedinthehydrologicallyhomogenousarea.Thisway,thetransferofthehydrologicdata (i.e. discharges) to ungauged sites within the same area occurs, inaccordance to the laws of geographical variation of the parameters of theempirical equationsdeveloped.Many regionalmodelshavebeendeveloped in

many cases, for instance, a regional model to predict sediment transport atpositionswithoutmeasurements(Mimikou,1982),fortheestimationofaveragemonthlyinflows,(MimikouandRao,1983),fortheestimationofannuallossesandtheunithydrograph(Mimikou,1984),fortheestimationofthedesignflood(Mimikou, 1984) and for the estimation of the discharge duration curves(MimikouandKaemaki,1985).Amonthly inflow regionalmodel isdescribedforsizingofareservoiratsiteswithoutdischargemeasurements.

6.4.2Regionalmodelforthederivationofmonthlyinflows

A simple regionalmodel of rainfall-runoff has been developed (Mimikou andRao, 1983),which is valid for both linear and nonlinear basins.Themodel isexpressedbythefollowingequation:

(6.41)

whereQp,tisthemonthlydischarge(m3/s)onyearp(p=1,2,…,N)andmontht(t=1,2,…,12)

Pp,tistherespectivemonthlyaveragesurfacerainfallontheareaofinterestkandnareparametersofthemodelat,iarethemodel’scoefficients

kandndeterminethestructureofthemodeland,morespecifically,theorderandthe memory, respectively. The order k expresses the monthly rainfall-runoffrelationandisequaltotheorderoftherelation(linearornonlinear).Toestimatek,onemuststudythemonthlyrainfall-runoffrelationshipoftheareaofinterest.In the case of a linear basin, k is equal to 1,while in a nonlinear basin, it ispolynomial:

(6.42)

kisequaltotheorderofthispolynomialequation.Inthecasewhenthevarianceofthepoints(Qp,t,Pp,t)inthemonthlycorrelationforaparticularbasinishigh,then themodel needs touse thememoryparametern.Then, the correlation isrestartedwithdischargeQp,tandtheaveragePp,tduringmonthst,t−1,…,t−n,wherenisthememoryofthecorrelationofrainfall–runoff.Thisphenomenonisactually present during the periodwhen runoff is coming from snowmelting,whichprecipitatedduring thepreviousnmonths.With theuseofparametern,

wecancorrectthecorrelationforthesemonths(springperiod).Wecanseethatkandnarecharacteristicsofthebasinforeverymonth.Parameterk,nevertheless,is more logical to be considered a constant characteristic of the basin and toexpressthelinearityornotofthebasin.Themodelwascalibratedintwobasins,withsizes217and640km2.Fromthe

monthlycorrelations,onebasinwasfoundlinear(k=1)andtheothernonlinear(k=2).FromthesamerelationshipswhentherunoffcoefficientCt=Qt/Ptwasfoundforaparticularmonthtobegreaterthan1(April,May,June),theuseofnwas necessary (n = 2). After that, all coefficients at,i, i = 0,1,…, k, werecalculatedwiththemethodofleastsquares.Regionalrelationshipsthatcorrelatethesecoefficientswith themonthlyrunoffcoefficientsunder thesamediagramweredeveloped. Itwas found that the coefficients forma line for i = 0, an -Scurvefori=1andaparaboliccurvefori=2,asshowninFigure6.10.The monthly runoff coefficients ct for all basins were plotted on the same

figure,dividedbythebasinarea(km2),alongwiththevaluesofmonthlyrainfall(divided also by the basin area). Six different curves were found that varythroughout the year as shown in Figure 6.11. The model works at locationswithoutmeasurements through the following:byusing thedataof theaveragemonthly rainfall divided by the area of the basin, we estimate the runoffcoefficientCt,foreverymonth,andthenwemultiplybytheareaofthebasinatthepointofinterest,asshowninFigure6.11.

Figure6.10 Graphofregionalvarianceofcoefficientsat,i.

Then,usingFigure6.10,weestimateat,i,i=0,1,…,k.Atthispoint,kmustbeestimatedindirectly.Thefollowingistheproposedsolutionfortheestimationofk:forbasinssmallerthan217km2,k=1;forbiggerthan640km2,k=2;andforintermediatebasinareas,theparameterkdependsonthedesigner’s judgement.ThenparameterofmemoryisequaltozeroformonthswithCt<1,whileforCt> 1, n = k.With all parameters estimated, we finally use Equation 6.25 andestimatethedischargeQp,t,foreverymonthlyrainfalldataPp,t.

6.5SIZINGOFTHEDEADRESERVOIRVOLUME

6.5.1General

Inordertodesignthedeadvolumeofthereservoir,wemustestimatethevolumeofsedimentstransportedintothereservoir.Allreservoirsthatareconstructedonnatural river flows are subject to sedimentation. The sediments that enter thereservoirwiththeriverflow,duetothereductionoftheflowvelocity,depositatthebottom:thecoarser-grainsedimentfirstfollowedbythefinerones.Againstthebackdropofsedimentation,thebasicproblemsforthedesignare

1. Theestimationoftheannualrateofsedimentationandthetotalvolumeofdepositedsedimentthroughouttheprojectlifetime

2. Theestimationofthetimeperiodthatthesedimentswillstarttoaffectthebasicfunctionsoftheproject

Figure6.11 Curvesofmonthlyvarianceofrunoffcoefficienttorainfall

Inthecasewhenthisisestimatedtohappenduringtheeconomiclifetimeoftheproject, an additional volume should be provided to accommodate depositedsediments, and design changes should bemade, like the position of thewateroutletinthedam.

6.5.2Sedimentationinareservoir:Definitionofdeadstorage

The life cycle of solids, startingwith the generation of sediments through theerosion process, and the transport through the water bodies, is affected byclimatic, topographical and other factors (see alsoChapter 8). The amount ofsedimentsthatreachareservoircanbeestimatedbyvariousmethods.Afterthattheestimationof thepropervolume(deadvolume)of the reservoir is thenexttask.Thedeadvolumeisusuallydefinedbyvarioustechnicalcharacteristicsofthe project, like the point of water abstraction. The estimated dead volumeshouldbelargerthantheestimatedsedimentation.Definitelythelatterisalowerlimitofthedeadvolume.

Table6.12Reservoiroperationtypes

Reservoiroperationalcharacteristics Type

Sedimentsarealwaysoralmostalwayssunktothebottomofthereservoir

I

Medium-tolarge-levelfluctuations II

Emptyreservoirs(i.e.floodprotection) III

Reservoirswithsediments(bottomsedimentonly)

IV

Some technical terminologies regarding thesedimentationofa reservoirareasfollows:Storagecapacity: It is defined as the quotient of the amount of transportedsolids that deposit to the total transported solids. It depends on the flowvelocity and the falling velocity of the particles in the reservoir. Theparticles’ fallingvelocitydependson theirsize, theviscosityof thewater,itschemicalcomposition,etc.

Sedimentindicator: It isdefinedas thequotientof theperiodofretention totheaveragevelocitythroughoutthereservoir.

Specific weight of the deposited sediments: The average inflow of solids(ton/day)mustbeconvertedtoanequivalentofvolumewiththeuseofanestimated specific weight of solids. Many factors (Lara and Pemberton,1963)affectthespecificweightofsolids:

1. Theoperationofthereservoir2. Thetextureandsizeoftheparticles3. Thesedimentationrate

4. Thecurrentsandtheflorainthereservoir

Among these factors, the most important is the first (1). According to theoperation-mode, reservoirs are categorized into four main types, as shown inTable6.12.

6.5.3Estimationofdeadvolume

After the evaluation of the reservoir type by the designer according to Table6.12, based on the operation characteristics, the initial unit weight of thesedimentsisestimatedusingthefollowingequation:

(6.43)

whereγistheinitialunitweightinlb/ft3Pc,Pm,Psarethepercentagesofclay(size<0.004mm),silt(0.004mm<size<0.0625mm)andsand(>0.0625until2.0mm),while thecoefficientsofclay,siltandsand,Wc,Wm,Ws,respectively,areestimatedbythereservoirtypeusingTable6.13

TheinitialunitweightγincreasesintimeandbecomesWafterTyears:

(6.44)

Table6.13EstimationofcoefficientsWc,Wm,Ws

 Clay   Silt   Sand Reservoirtype Wc Wm Ws

I 26 70 97

II 35 71 97

III 40 72 97

IV 60 73 97

Kisaconstantthatdependsonthegradationofthesedimentsandthereservoirtype:

(6.45)

ThecoefficientsFc,Fm,FsaregiveninTable6.14.Equation 6.44 is valid for T years. Actually every additional year, extra

compactiontakesplaceandweendupinincreasedcompactioninthelifetime.Miller(1953)haspresentedananalyticalexpressionfortheapproximationofthe

integrationoftheaverageunitweightWafterTyearsofoperation:

(6.46)

For example, in a reservoir of type I, with the following proportions ofsediments:40%clay,30%siltand30%sand,wehaveWc=26,Wm=70,Ws=97andtheinitialunitweightis60.5lb/ft3.ThevalueofKfromEquation6.45andTable6.14isK=8.11.TheunitweightisshowninTable6.15.ThelastcolumnofTable6.15definesthevolumeofthesedimentsfor10,20,

50and100yearsinthereservoirusingtherelationST=(G/WT)×T,whereGistheaveragesolidstransferintothereservoir.Forthepreviousexample,G=1.2×106m3/year.Inotherwords,ifthelifetimeoftheprojectis50years,thenthedeadvolumemustbe53×106m3orbiggerifotherrestrictionsareapplied,liketheselectionoftheabstractionpoint.

Table6.14EstimationofK

K

Reservoirtype Clay Silt Sand

I 0 5.7 16.0

II 0 1.8 8.4

III 0 0 0

IVa – – –aTypeIVonlyhasabottomsediment.

Table6.15AverageunitweightafterTyears

T(years) WT(lb/ft3) WT(ton/m

3) VolumeofsedimentsST(10

6m3)

10 65.99 1.055 11

20 68.08 1.088 22

50 71.04 1.136 53

100 73.36 1.173 102

6.5.4Estimationofactivevolumelossduetoreservoirsedimentation

Aftertheestimationofthetotalvolumeofsedimentsthatdepositatthebottomofthereservoir,especiallywhenthedeadvolumeisdeterminedbyit,onemust

determine the distribution of sediments in time. This distribution leads to theestimationofpossiblelossofactivevolume,incasethesurfaceofthesedimentsis extremely rough with formations that enter the active volume (above theminimum level) and reduce the storage capacity of the reservoir. It is alsopossiblethatdepositedsedimentscreatetechnicalproblemslikethecoveringofthereservoiroutlet.Themethodthat isusedfor theestimationofthesedimentdistribution is the empirical method of reduced surface (Borland and Miller,1960;Lara,1962).Astudy in30reservoirs in theUnitedStatesshowed thatapredefined relationshipexistsamong the reservoir shapeand thepercentageofthedepositedmaterialatdifferentdepths.Theshapeofthereservoir isdefinedby the relationshipofdepthandstoragecapacityby theclassification inTable6.16,wherem is thereverseof theslopeof thecurverelatingstorage todepth(horizontalaxisisdepthandverticalaxisisstoragecapacity)plottedinadoublelogarithmic paper. Analytically, it can be estimated as the slope in therelationship:logV=a+mlogH,whereHisthedepthandVisthestorage.Figure6.12presents the empirical relation (%)of thedepthof the reservoir

and thevolumeofdeposited sediment andalso the shapesof the reservoirsoftypesIandIVwhicharethemostcommon.SometimesthetypeofthereservoirwhichwasclassifiedbasedonTable6.12canbechanged.Forexample, if thereservoirbelongstotypeIII(Table6.16)withregardtotheshape(i.e.betweenhills) but the level is lowered frequently, or the sediment is mostly clay, weshould classify it as type IV (Table 6.12), because a large percentage of thesedimentisdepositednearthebank,acharacteristicoftypeIVinTable6.12.Thebasicequationthatthemethodusesis

(6.47)

whereSisthetotalvolumeofsedimentsthataredepositedinthereservoir0istheinitiallevelofthedamYoisthezerolevelinthedamaftertheinflowofsedimentsAistheareaofthereservoirdyisthedifferentialdepthHmisthetotaldepthofthereservoiratthenormallevelK is the proportionality constant for coefficient transformation of relativeareasofsedimentstoabsoluteareasforagivenreservoir

Aisthecoefficientofrelativesurfaceofsediments

Table6.16Typesofreservoirshapes

Reservoirtype  Classification m I Lake 3.5–4.5

II Floodingplane 2.5–3.5

III Betweenhills 1.5–2.5

IV Canyon 1.0–1.5

Figure6.12 Shapetypesofreservoirs(Mimikou,1994).

Equation6.47,afterintegration,becomes

(6.48)

whereU0istherelativevolumeofthereservoiratthenewzerolevela0isthecoefficientofrelativesurfaceofthereservoiratthenewzerolevelV0isthetotalvolumeofthereservoiratthenewzerolevelHmistheinitialtotaldepthofthereservoir(fromthemaximumnormallevel)A0isthetotalareaofthereservoiratthenewzerolevel

Wedefine

(6.49)

and

(6.50)

whereP is the relative depth, meaning a percentage of reservoir depth that ismeasuredfromtheriverbank

VPHisthetotalvolumeofthereservoiratdepthPHAPHisthetotalareaofthereservoiratdepthPH

FromEquations6.49and6.50,hp is equal toh′p,whenwehave thenewzerolevelYo.After evaluationof data collected in theUnitedStates in a large number of

reservoirsofeverytype,dimensionlesscurvesweredeveloped:forthestorageofsediments which are presented in Figure 6.12; for the relative surface of thereservoira,basedontherelativedepthP,inFigure6.13;andfortherelationofhpbasedontherelativedepthP,inFigure6.14.

Figure6.13 DimensionlesscurvesofrelativesurfaceαtorelativedepthP(Mimikou,1994).

Figure6.14 DimensionlesscurvesofhptorelativedepthP(Mimikou,1994).

Example6.3:EstimationofdeadreservoirvolumeThemethodologyfor thedistributionofsedimentsatdifferentdepthsin a reservoir for instance, inSections6.5.2 and6.5.3, for a 50-yearlifetime project, withST = 53 × 106m3 volume of sediments, is thefollowing:

Wedeterminetheinitialdataontheprojectlifetime:ST=53×106

m3,themaximumdepthofthereservoirHmatthenormallevelinourexampleis100m,thelowernormaloperationalis400mandthelevelofthebottomis300m,abovemeansealevel.Alsothedepth-volume and depth-surface relations of the reservoir aregiveninTable6.17.

Table 6.17 Relations between reservoir depth and storagevolume,andreservoirdepthandsurface

Level(m) DepthH(m) Surface(km2) Volume(106m3)

300 0 0.000 0.0

310 10 0.025 0.23

320 20 0.110 0.93

330 30 0.230 2.63

340 40 0.380 5.68

350 50 1.050 16.00

360 60 2.180 26.00

370 70 3.050 60.00

380 80 5.350 115.00

390 90 10.000 215.00

400 100 17.000 350.00

410 110 23.200 530.00

Table6.18Calculationsoftheh′prelationship

ThelogarithmicrelationbetweendepthHandstoragevolumeVwiththedataofTable6.17forH>50misasfollows(withhighcorrelationcoefficient):

Thecorrelationhasaslopem=4.64thatisneartheupperlimitofmfor type I reservoirs (Table6.16). Ifweusealldata fromTable6.17(notonlyforH>50),theaverageslopeism≈3.4,implyingreservoirbetweentypesIandII.Finally,wedecidetoclassifyitastypeIbasedon the general characteristics. We estimate the function h′p fromEquation 6.50, for this particular reservoir, as shown in Table 6.18.ThevolumeVPHandtheareasAPHfromTable6.17areused.Theactualfunctionh′pofthereservoirisplottedinthediagram(h,

hp)ofFigure6.14.Thesectionof (h,h′p)with the theoretical (h,hp)curve, for the type of the reservoir (here is type I), gives the new(relative)zerolevel(whereh=h′p)−P0=0.473–andthenewzerolevelisP0Hm=47.3m,andy0=300+47.3=347.0m

ThenwehavethefollowingestimationofthesedimentdistributionpresentedinTable6.19:

Columns1,2and3ofTable6.19areusedfromTable6.17.Column4givestherelativedepthPforeverylevel.Column5gives the relativesurfaceofPwith theuseofFigure6.17.Column6(sedimentarea)isderivedwhenwemultiplycolumn5with the constant K = APOHm/a0 = 0.50, where APOHm is thereservoir surface at the new zero level, anda0 is the respectivecoefficientofrelativesurface.Column7(sedimentvolume)hasbeenderivedfromcolumn6as(Ai+Ai+1)/2×(Hi−Hi−1).Column8isthecumulativesumofcolumn7.IfthefinalsumisdifferentfromST,thenwecorrectKwithamultiplyingfactorK1= K(S/S1). In this example, S1 = 51.7 < ST, so S1 = 0.50 ×(53/51.7)=0.512.Theprocesshasbeenrepeatedfromcolumn6.Thecorrectedvaluesareinparentheses.Column 9 gives the absolute cumulative sediment volume. Itstarts fromvalueS and is reduced in every level by abstractingtherespectiveamountofcolumn8.Column10givestherevisedareaofthereservoirasadifferenceofcolumns2and6.Column 11 gives the revised values of the active volume as adifferenceofcolumns3and9.Wenoticeareductionoftheactivevolume(in50yearsofthereservoirlifetime)from100%to15%whenmovingfromthelowertothehigherlevels.

Table6.19Estimationofsedimentdistributioninareservoir

6.6SIZINGOFTHERESERVOIR’SFLOODVOLUME

Thethirdcomponentofthevolumeofareservoir,asdiscussedpreviously,isthefloodvolume.Thesizingofthefloodvolumeisaprocessthatisstronglyrelatedto the hydrologic design of the spillway. The reason is that the spillwaymustprotect thedamfromitsdesignfloodwhichwhenroutedthroughthereservoirwillincreasethemaximumoperationlevel.Thus,thereservoirfloodvolumeisdetermined. Design methods differ according to the adopted definition of thedesignfloodasitwillbedescribedinthefollowing.Example6.4dealswiththeestimationofthedesignfloodofaspillwayincombinationwithacasestudyoffloodroutinginsideareservoir.Moreover, the reservoir storage can be expressed as a function of water

surface elevation and of the surface area of the reservoir (storage-water level-surfacecurve),whichisderivedbasedonlyontopographicmaps.Thus,storagecan be estimated in every time step based on the inflow hydrograph by usingappropriateroutingmethods.Ingeneral,thefloodroutingmethod(PulsMethod)andtheriverrouting(MuskingumMethod)aredescribedinChapter7.

6.7HYDROLOGICDESIGNOFFLOODSAFETY(PROTECTION)STRUCTURES

The hydrologic design of flood protection structures is concerned with theestimationofthedesignfloodofthestructure.Thisistheflooduptowhichthestructureprotectsdownstreamlifeandproperties.Thedesignfloodisproducedbytheoccurrenceofthedesignstormoverthebasin.Thedesignfloodisthefloodthatisexpectedtooccurintheprojectsitearea

atanexceedancefrequencyequalto1/T,whereTisthereturnperiod.RiskJistheprobabilityofoccurrenceofthedesignfloodwithfrequencyequalto1/T(orgreater),at leastonce in theselected timeofn yearsperiod (i.e. the economiclife time of the structure). The risk J, assumingn years of project lifetime, isequalto

(6.51)

The risk should be technically and economically acceptable, in relation to theintended purpose and the safety level that this construction will provide. Forinstance,aspillwayofadammustprovidesafetysothatthestoredwaterdoesnotovertopthecrestofthedamthroughouttheeconomiclifetimeoftheproject,

while a diversion of a river at a dam construction site must be designed toprovidefloodprotectiononlyduringtheconstructionperiodoftheproject.ItismorepropertodefineJasthe‘hydrologierisk’,sincethetotalriskofa

projectdependsonmanyotherfactorssuchasthestructuralandtheseismicrisk.For example, in the case of an earth dam, the hydrologie risk (for failure byovertopping)isapproximately25%ofthetotalriskoftheproject.TheselectionofJdefines thedesignflood,withfrequency1/T,of the structure.ThecriteriaforselectionofJareusuallybotheconomicandtechnical.ItisobviousthatasJdecreases resulting in more safety to the structure, the bigger and moreexpensive the project becomes. For this reason, in projects of paramountimportance,atechno-economicanalysisiscarriedouttooptimizetherelationofconstructioncosttothecostoftheconsequencesfrompartialortotalfailureoftheproject.Fortheestimationofthedesignfloodinthebasinofinterest,theestimationof

thetotaldepthhofthedesignstormisrequired.ThiscanbeestimatedusingtheidfcurvesdescribedinChapter5,whichrelatedepth,durationandfrequencyofstorm(h,t,T)or alternatively, intensity,durationand frequencyof storm (i,t,T).These curves are developed for a particular basin from frequency analysis ofannual maximum rainfall depths. As input data are used, the duration of thedesignstormanditsfrequency1/T,anddescribednextarethedesigncriteriaoftheproject.Thereafter,a rainfall–runoffmodel isused,and thedesign floodhydrograph

(producedbythedesignstorm),showingthepeakoftheflood,isestimated.Theestimationofthedesignflood,withtheuseoftheunithydrograph,issimple.Inthe example that is presented next, the estimation method of a design floodapplyingaunithydrographisexplainedindetail.In the case of nonlinear basins, where the application of unit hydrograph

generateserrorsinthehydrologicesign,theunithydrographshouldnotbeusedfor the estimation of the design flood. In this case, a method of nonlinearestimationoffloodpeaksmustbeimplemented,asexplainedinthefollowing.Finally, in the casewhere the design takes place at a sitewithout discharge

measurements,theestimationmethodofthedesignfloodmustfollowdifferentapproaches.Inthiscase,aregionalmodeltransferringhydrologicdatafromotherplaceswheremeasurementsareavailable isneeded,asdescribedpreviously inthechapter.Thiscouldbeafloodregionalmodelfromanearbylocationinthesame river, taking into consideration thewater contribution from intermediatebasins(betweenthemeasurementsiteandthepointofinterest).Alternatively,itcouldbea floodregionalmodel fromothernearby locationsof themajorareawith hydrologically homogeneous characteristics around the design site

(MimikouandRao,1983).Safety structures are levees, rainwater networks, spillways, river diversion

tunnels,etc.Inthischapter,wewillfocusonthehydrologiedesignofspillwaysand river diversion tunnels. The procedure followed for the estimation of thedesign flood is almost identical for all flood protectionworks, but the designcriteria(returnperiodT,durationandtimedistributionofthedesignstorm)varyinaccordancewiththeparticularimportanceofthestructureunderstudy.

6.7.1Spillwayhydrologiedesign

Aspillway is themain safety feature for adambuilt in the routeof a river. Itprovidessafetyagainstovertoppingwhenthiscanhappen,i.e.anearthdam,andgenerally, inallhydraulicprojects, thespillwaycontrolstheoutflowofafloodrunoffandthecontainmentofthefloodinsidethereservoir,fortheprotectionofallstructuresdownstreamofthedam,likeapowerstation,orthefoundationofthedam.Spillways, especially on earth dams, are expensive structures and in many

casesareconsideredthecrucialfactorfortheeconomicfeasibilityofthistypeofwater resourcesprojects. Inextremecases, thecostof thespillwaycanexceedeventhecostofthedamitself,andinnormalcases,aspillwaycosts40%–80%ofthetotalcostof thedam.Nevertheless,despitethisenormousburdenonthetotalcostoftheproject,untilnow,allattemptstosurpassitsnecessity, i.e., theconstructionofdams resistant to total overtopping,or improvements indesignand construction, are at early stages. A basic obstacle to the achievement ofeconomicdesignseemstobethelackofcontemporarydesigncriteria,basedonup-to-dateknowledgeandexperiencethatcomplieswiththedesigncriteriaandsafetyfactors,adoptedbythesocietyandusedinotherfieldsandtechnologicalareaslikehighwaysafety.Thispersistenceofthedesigncriteriaofaspillwayatveryconservativelevelsiscontrarytothenewperspectivesandpossibilitiesinhydrologiedesigngivenbytheimprovementsofthemethodsoffloodanalysis,anareawhereimportantstepshavebeenmadeduringthelastyears.The types of spillways generally applicable to dams are the following

(Nicolaou,1971;Moutafis,2003):

1. Withfreeflowfrominsideofthedam.Itconsistsofacrestandachannelwith side-walls across the external side of the dam, ending in an energystillingbasinorajumpstructure

2. Anopenchanneloffreeflowthat issitedat theedgesof thedam.Italsoendsinanenergydissipationbasin

3. Atunnelthroughthedamslopinginthefirststageandhorizontallayerthatendsalsoinanenergydissipationbasin

4. A side canal along the crest of the dam that ends up in an inclined openchannelortunnel

5. Afunnel-shapedspillwaywithatunnelintheabutmentsofthedam6. Abell-mouthspillwayinsidethedam7. Afreefallinsidethedam

A spillway is a structure of major importance especially with regard to earthdams.Types(2)–(5)aresuitableforearthdams,becausethefailureofaspillwayinearthdamsisconsideredasatotalfailureoftheproject,resultinginfloodingof the downstream area. For other dam types, like arch or gravity dams, aspillway is considered more like a structure that keeps the flood inside thereservoir, discharging the floodwater in away that prevents possible damagesdownstream.The regulationof discharges froma spillway is achievedby the additionof

floodgates forwater level control (above the crest of the spillway), inside thereservoir, especially when we want to exploit a part of the flood volume forusefulpurposes.Inthiscase,weaddfloodgatesonthecrestthatopenandclosepartiallyor completely,dependingon the regulation schedule. In this case, thespillway iscalledacontrolledspillway.On thecontrary, the spillwayswithoutfloodgateinstallation,fromthecrestlevelandabove,releaseanyfloodvolume,waitingfor thedesignflood.VF is the floodvolumeestimatedby thedesignerforthecontainmentoftheflood.From the point of view of the hydrologic design of a spillway, the design

floodmustbeestimated.Thisisanextremelyrareflood(offrequency1/10,000or less) with a respectively low risk. Finally, it is worth mentioning that aspillway,apartfromasafetyconstruction,inmanycaseshasalsootherusesofecological importance:anexample is theopeningof the floodgatesof ‘BonnetCarre’spillwayintheMississippiriver;thewaterdivertedtoPontchartrainlakewith the purpose of the lake enrichment with water and nutritious elements(Laneetal.,2001).

6.7.2Spillwaydesignflood

The design flood of the spillway is an extremely rare flood from which thespillway protects the dam. The more rare the flood is (or the smaller thefrequencyorthebiggerthereturnperiodare),thebiggerthevolumeoftheflood,themorecostlythespillwayandthebiggertherequiredfloodvolumeofthedam

are.Asamatteroffact,thedesignfrequencyisdirectlyassociatedwiththecostof thesafetyconstruction.Adecrease in theselected frequency(or increaseofthereturnperiod)ofdesignleadstoanincreaseintheproject’ssafetyandalsoan increase in theprojectbudgetandviceversa.Nevertheless, the selectionofthedesignfrequencyofaspillwayisbasedmainlyonsafetycriteria.

6.7.3Criteriaofspillwaydesign

As spillway design criteria,we consider the hydrologic parameters,which aredecidedand selectedby thedesigneron thebasisof technical andeconomicalparameters of the project. Hydrologic design criteria of a spillway are thefrequency or the return period of the design flood and the duration anddistributionofthedesignstorm.Frequencyorreturnperiodofthedesignflood:Asmentionedbefore,costisa

functionofthereturnperiodandsafety.Evenifanoptimalsolutioncanbefoundafter a feasibility study among safety and budgeting of the project, the designfrequencyisselectedmainlybysafetyrestrictions.Inmanycases,mostlyintheUnitedStates,theprobablemaximumflood(PMF)atthesiteoftheprojectisthedesign flood of a spillway. Theoretically, its frequency cannot be exceeded.Recently,acriticismhasbeenraisedagainstthispracticebymanydesigners,intheUnitedStates,forbeingveryconservativeandunrealistic.Variousstatisticalstudies have proven that a frequency of occurrence with a size class of 10−4,leading topracticallyzero risk,during theeconomic life timeof theproject isacceptableforthespillwaydesign(Benson,1973).Thedurationandthedistributionofdesignstorm:Thedesignflood,atthesite

ofthedam,isusuallyestimatedbytheunithydrographmethodusingthedesignstormover thebasin.Thedesignstorm isestimated fromthe idfcurvesof thebasinbasedonthefrequencyanddurationofthestorm.Thefrequencyisequaltothefrequencyofthedesignflood(thismatchdoesnotapplyforhigherdesignfrequencies as it will be mentioned later), and its duration is also a designcriterionselectedonthebasisoftechnicalandeconomic-specificfeaturesoftheproject. Small durations give small flood volumes and vice versa. So, if theproject has the containment of the flood through the reservoir as an importantobjective,thenhighfrequenciesmustbeselected.Inthecasewhenthepeakofthefloodisimportant,thenwemustfocusonallhydrologievariablesrelativetothe peak (i.e. the time distribution of storm), and through a trial and errorprocedure,wemustfindthedurationofthestormwhichmaximizesthepeak.Forthedurationofthedesignstorm,therearealsoothersuggestions,suchas

theoneproposedbytheInstituteofHydrologyofEnglandin1978.Theduration

is estimated through climatic and morphologic features of the basin. Theempiricalformulaproposedis

(6.52)

whereDrepresentsthedurationofthedesignstorm(h)SAARistheaverageannualprecipitationoverthebasin(mm)Tpisthetimetothepeakfromthestartoftheunithydrographattheprojectlocation(h)

Thelastfactorisknown(Mimikou,1984)tobecorrelatedtothemorphologicalfeatures of the basin (area, stream length, etc.). In the case of the design of aspillway,wherewehavetoestimatetheadditionaloutflowoftheflood(afterthecontainment),itissuggestedthatthedurationDmustbeincreasedbyatimelagTrinTp:

(6.53)

Thetimedistributionofthestormcouldbeselectedfromthefourstandardtypesand their respective group of curves (level of confidence), according to therespectivequarter(first,second,thirdorfourth)whenthestormdepthreachesitspeak,asdiscussed inChapter2 (Huff,1970).Theselectionof the firstquarterstormfitsbettertheneedsofastormsewagenetwork,whereasthefourthquarterstormismoreappropriateforthespillwayofalargedam.

6.7.4Probablemaximumflood(PMF)

ThePMF,inthedesignofadamspillway,isusuallyestimatedbytheprobablemaximumprecipitation (PMP)which is used in this case as the design storm.The definition of PMP as per the National Weather Service (AMS, 1959),mentioned in the international literature (Shaw, 1994), is the theoreticallymaximumdepthof storm for agivenduration that isphysicallypossible for agiventimespotintheyear.PMFisextractedbyusingPMPofaknownduration,usingtheunithydrographatthepointofinterest.ThebasicadvantagefortheuseofPMFistheprovisionofmaximumpossible

safety.Nevertheless,thereisalackofrationalityconsideringhydrologierisktobe actually zero, and issues arise with impacts on the design, as far as thedefinition and the conditions of application of PMP are concerned. ThedefinitionofPMP is quite abstract and can lead to errors, like the assumptionthat rainfall events can be used in basins with different climatic and

morphological features, the unjustified estimation of themaximum volume ofstorm(precipitatedwater)atthepointofinterest,andtheomissionofuseinthePMPofantecedenthumidityconditionsofthebasin.For the reasonsmentioned, the use of PMF has been abandoned because it

leads to over-conservative estimations and has been replaced by a flood withfrequency 10−4. Thus, the probable maximum flood of the project has afrequencyof10−4.

6.7.5Estimationofthedesignstormandflood

Theestimationofthetotaldepthhofthedesignstormofaspillwayforthebasinin the position of the dam is possible through the idf curves that have beendeveloped for thebasin, through frequencyanalysisof theextremestorms.Asentry data,we use the duration of the storm and frequency 1/T,which are, asmentionedbefore,thedesigncriteriaofthespillway.Totaldepthh(mm)ofthestorm must be distributed in time according to one of the four types ofdistributions.One usual distribution adopted for peakmaximization is the second quarter

distributionwith50%levelofconfidence(seeExample6.4).Ifthemethodusedfortheestimationofthedesignfloodisbasedontheunithydrograph,thenthetime distribution relies on the base of the unit hydrograph (i.e. 2 h). The unithydrograph ismultiplied, asmentioned,by thedepthof theexcess rainfall foreverytimestepoftherainfalleventandfinallysummedup.Tothesevalues,thebasic flowof the stream is addedand the total design floodof the spillway iscalculated.Additionally,whentheunithydrographhasarelativelylongduration(i.e. longer than 6 h), it should be converted into one of smaller durationaccording toaknownmethod,e.g. theS-curve (Linsley et al., 1975;MimikouandBaltas,2012).Theprocedureisdescribedinthefollowingexample.Based on the fact of the extreme rareness of this phenomenon, rain losses

(infiltration,percolation)areconsideredverysmall (equalor less than1mm/hontheaverage).Thetimedistributionoftherainlossesishomogeneousduringthe rainfall, or (more correctly) follows an exponential curve like Horton’s(1933),whichbetter simulates themechanismof infiltration in theground.Byabstracting the rain losses, anetdesign rainfallof2hdistribution isacquired.Thisisadesignrainfalloffrequency10−4,withasecondquarterdistributionfor50%levelofconfidence(Figure6.15).

Figure6.15 Designstormofadamspillway.

6.7.6Derivationoffloodhydrographinspecificcases

The theory of unit hydrograph and the application of it, for the estimation offlood hydrographs, have been analysed in Chapter 3. The unit hydrographusuallyisnotderivedbyasinglefloodeventbutbyusingagroupofrecordedstorm events that give an average result, as long as they do not differsubstantiallyfromeachother.Thewayofderivingaunithydrographwhenthenetrainfallisknownisrelativelyeasy,basedonthetheory.The derivation of unit hydrographs at sites without measurements is

accomplishedby severalmethods suchasSnyder,SCS,etc. (Mimikou, 1994).Animportantproblemwhenapplyingtheseempiricalmethodsistheestimationof the parameters of the relationships of peak discharge and time durationbetween the centre of the storm volume and the flood peak. The use ofparameters applicable to other areas is prohibited, because it has been proventhattheyareextremelysitespecific.We must also mention that linear methods, like the unit hydrograph, are

inappropriate for nonlinear basins, where the runoff volumes are not directlyproportional to storm volumes. Researchers (Rogers and Zia, 1982;Mimikou,1983c)haveproventhatthereisananalyticalcriterioninordertoclassifybasinsas linear or nonlinear.This is the slopeof a groupof curves of standardpeakdischarges distribution, which are plotted in a double logarithmic paper,correlating thepeakdischargesQp and the respective stormvolumesV. It has

beenproven(Mimikou,1983c)thattheslopemofthecurve

(6.54)

isanecessaryandsufficientcriterionoflinearitytest.Ifthebasinislinear,thentheslopeis1.Ifmislessthan1,thebasinisnonlinearandthevaluedepictsthedegreeofnonlinearityofthebasin(i.e.closeto1,itisrelativelynonlinear,whilenearzero,itisextremelynonlinear).Thesameresearchhasalsoshowedthatthepercentageofnonlinearbasinsincreaseswiththeincreaseoftheareasizeofthebasinsexamined.Also,itwasfoundthattheuseoflinearmethods,liketheunithydrograph,in

nonlinear water basins, results in significant mistakes in design, either ofunderestimationorover-estimationof thedesign flood in theorderof60%.Aquestionraisedconcernsthepropermethodtobeappliedifthebasinisfoundtobe nonlinear, a test that is obligatory before the hydrologic design. A simplemethod is proposed (Mimikou, 1983) for the prediction of peak discharges ofselected recurrence periods, applicable to linear and nonlinear basins. Thevolumeofdirect flooddischargeVn caneasilybe estimated for agiven returnperiodT from idfcurvesoranyothermethod,bysubtracting from the rainfallvolume discharge, of the same period T and known duration t, a properlyestimated percentage of losses for the particular basin (based on previoushistorical data). Additionally, based on historical data, an analysis of therelationsof total andnet rainfall for hydrographs that havebeenderived fromstorm events of the same duration with the one selected for the hydrologicdesigncanbeconducted.Inthisway,theaveragepercentageofincreaseofthedirect runoff due to basic runoff is estimated. This increase is added to thepreviously estimated net volumeVn of flood discharge to give the total floodvolume V. After that, we can estimate the peak discharge Qp of a givenfrequencyfromEquation6.54.

Example6.4:Estimationof thedesign floodof a spillwayand floodroutingthroughthereservoirusingthestorageindicationmethod(seealsoSection6.2.4.5)InasmallbasincoveringanareaofAkm2,asmallreservoirisabouttobeconstructedforirrigationpurposes.Theestimatedlifetimeoftheproject is 50 years and the risk 5‰. Idf curves on the area areestimatedusingTable6.20,with themaximumvaluesofstormdepthfortherespectivedurationofthestorm(12,24,36,48and72h),foratimeperiodof18years.

Table6.20Estimationofstormcurves

Table6.21Designstorm

The duration of the design storm is equal to t = 4 h. The designstormfollowsthedistributiongiveninTable6.21.Fortheestimationofthetotalhourlystorm,intermediatevaluesare

needed,whichwillbeobtainedthroughlinearregression.The2hunithydrographofthebasinhasbeenconstructedandispresentedinTable6.22.Estimatethedesignfloodofthespillway.Totallossesareconsidered

constantforthewholedurationofthestormandequalto3mm/h.

SolutionInTable6.23,thedepthofthestormisshowninmmfordurationsof

12,24,36,48and72h.Theequationoftheidfcurveis

For each storm duration, we have a = 1.282/Sx and c = x − 0.45Sx(Table6.24).From equation X = c − ln{ln (T) − ln (T −1)}la, where X = i

(intensityofstorm),andforreturnperiodsT=10,20,50,100,1,000,10,000 years, we estimate the maximum intensity of storm for thegivenstormdurations(Table6.51).

Table6.22Unithydrographof2hofthebasin

Time(h) Discharge(m3/s)

1 0

2 8.1

3 51.0

4 90.0

5 122.6

6 143.1

7 124.3

8 106.3

9 91.7

10 77.1

11 65.1

12 54.9

13 44.6

14 36.9

15 28.3

16 22.3

17 12.3

18 12.0

19 8.6

20 4.3

21 2.1

22 0

Table6.23Stormdepth(mm)fordurationsof12,24,36,48and72h

Table6.24Estimationofmeanvalue,Sx,aandc

Table6.25Calculationofmaximumintensityofstorm

Table6.26Calculationofy=logiandx=−logt

ForthecalculationofslopemandtheconstanttermAToftheline,

withb=0,thatis,logi=AT+m×(−logt),y=logiandX=−logt,asshowninTable6.26.Inthesameway,slopeaandtheconstanttermlogKofthelineAT

=logK+a×logT,withy=ATandX=logT,arecalculated(Table6.27).Inconclusion,theidfcurveequationis

From this equation and for t = 4 h, the intensity of the storm iscalculated.Bymultiplyingwitht,wehavethevalueofhofstorm.ForthespillwayT=10,000years,i=21.11mm/handh=84mm.For thedistributionof the totaldepthofstorm,weuseTable6.28.

Weareinterestedintheactiverainfall;sofromthevaluesofdepthsofrain,weabstracttherainfalllossesas(φ=3mm/h.Fortheestimationofthedesignflood,weassumethat thebasinis

linear,sotheunithydrographcanbeapplied(Table6.29).

DesignfloodQiareevaluatedbasedontheprincipleofproportionality

Table6.27CalculationofaandK

Table6.28Distributionofthetotalstormdepth

whereQistheordinaresoftheunirhydrographhiistheactiverainfalldepthofeveryhour

Qiisthedischargeforeveryhour(Table6.30)

FloodroutingthoughthereservoirInordertocalculatethefloodroutingThroughthereservoir,weneedanexrracurveinaddiriontotheinflowdischargeversustimecurveofthedesignstorm.Thiscurveisthe((2S/Δt)+O)curveversusQ.Thequantity((2S/Δt)+O) iscalledstorage indicationand it iscalculatedbysolvingthemassbalanceequationofthereservoir:I−Q=ΔS/Δt,where I is the inflow to the reservoir,Q is the discharge from thereservoirandΔS/Δtisthechangeofstorageinsidethereservoir.

UsingmathematicalTransformations,wehavethissolution:

(6.55)

where1and2areindicatorsfortwoconsecutivetimesteps.The curve of ((2S/Δt) + O) versus Q is nonlinear and can be

estimated by the characteristics of the spillway and the reservoir. Arelationshipthatisusedtoestimatethedischargefromaspillwayis

(6.56)

whereQisthedischargeListhelengthofarectangularspillwayHistherelativeheightabovethespillway’screstcisaparameter,usuallyequalto3

Inthisexample,forsimplereasons,thecurveof((2S/Δt)+O)versusQhasbeenestimatedandisgiveninTable6.31:The estimation of flood routing through the reservoir is shown

analyticallyinTable6.32.

Column2isthesumoftwoconsecutivevaluesofcolumn1.Forn=1(firsttimestep)inline1,wehaveI(n)+I(n+1)=2S(n+1)/Δt+Q(n+1).ThroughTable6.31withlinearinterpolation,weestimatevaluesofcolumn5usingcolumn4.Column6isderivedbysubstitutionofthevalueofQin2S(n+1)/Δt+Q(n+1),incolumn4.

Column3 in line1has zerovalue,but in line2,weestimate itusingcolumns5and6fromthepreviousline1.Column 4 is the sum of columns 2 and 3 in every line (Figure6.16).

Table6.29BasedurationoftheunithydrographD=22h

Table6.30Estimationofthedesignflood

Table6.31 versusQcurve

6.8HYDROLOGICDESIGNOFARIVERDIVERSION

6.8.1Introduction

Many times,during theconstructionperiodofhydraulicprojects,especiallyofconsiderablesize,thecompletionoftheprojectisimpossibleinonedryseason(summer). For this reason, a diversion of the river is necessary around theconstructionsite.Themainpurposeofthediversionistoallowtheconstructiontotakeplaceunhinderedinadryenvironmentbutalsotoprotecttheprojectanditsresources(personnel,machinery,etc.)fromaflood.Themaintechnicalphaseofadiversionusuallyconsistsoftheconstructionof

an open channel, or closed tunnel (depending on the topographical features)where the river water will inflow at its upstream end and outflow fardownstream from the project site. The design the diversion channel is placednearoneriverbankattheconstructionsite.Specialimportancemustbegiventothe simplicityof thedesign, inorder to cutdown the cost, since thediversionchannel is an auxiliary temporary construction that stops operating after thecompletion of the project. However, in many cases, especially in dams, thediversion channelmay have a supplementary use after the construction of thedam,e.g.serveasthespillway.Nevertheless,thesimplicityofthestructuremustnotendanger thesafetyof theconstructionsiteand thewholeproject.Anotherauxiliarytechnicalworkthatusuallysupportsadiversionisasmallembankment(pre-dam)constructedupstreamof themainproject. Inside thereservoirof thepre-dam,riverwateraccumulates,andatasecondphase,itisdischargedthroughthe diversion channel. The hydrologic design of the diversion consists of theevaluationofthedesignfloodwhichhasafrequencyofoccurrencegreaterthanthespillwaydue to the limited timeof therequiredprotection(1–2yearsforatypical dam). The frequency of occurrence ranges from 1/10 to 1/100 years,depending on the tolerable risk accepted by the designer and the cost of thediversionineachcase.Forexample,theriskJofafloodequaltoorgreaterthanthefrequency1/T=1/10atleastonceduringthe2yearsofconstructionisJ={1− (1 − 1/T)2} = {1 − (1 − 1/10)2} = 19%. If the diversion design flood hadfrequency1/100,thenJwouldbesignificantlysmaller:J={1−(1−1/100)2}=1.99%,resultinginanincreaseinthediversion’sconstructioncost.

Table6.32Estimationofthefloodroutingusingstorageindicationmethod

Figure6.16 Floodroutingthroughthereservoir.

6.8.2Designcriteriaofadiversion

Thedesigncriteriaofariverdiversionareexactlythesameasthatofaspillwaybut with quantitative differences. These criteria are the frequency, the returnperiodofthedesignfloodandthedurationofthedesignflood,asdescribedinthefollowing.Thefrequencyandthereturnperiodofthedesignflood.Duetotherelatively

smalldurationofthedemandedprotectiontimeoftheprojectbythediversion,the frequency of the design flood is chosen much higher than the one of aspillway. A common choice is between 10 and 100 years for less or moreconservativedesigns,respectively.ThecorrespondingrangeofriskJforausualconstructionperiodof2yearsis19%and1.99%,respectively.Thefinalchoiceofthedesignfrequencyinthisrangeisbasedonthedesigner’sexperience,thehydrologicresponseoftheriver(i.e.oftenfloodsneedgreaterprotectionandasmaller design frequency) and the tolerable diversion cost. At this point, it isworth mentioning that, for the range of frequencies for which a diversion isdesigned, there isamismatchbetween the frequenciesof thedesign floodandthedesignstorm(BritishInstituteofHydrology,1978).Inparticular,thepairsoffrequenciesof thedesign stormand thedesign flood, resulting from the stormareshowninTable6.33.Thismeansthatastormwithsmallerfrequencyisrequiredforthederivation

ofa floodpeakofaknownfrequency.Thismismatch isgettingsmalleras thereturnperiodincreases,anddiminishesintherangeof500–1000years.

The design storm duration and distribution. These parameters are chosenbasedon technicalandhydrauliccharacteristicsof thespecifiedproject. In thecaseofdams,usuallyasmallerdurationischosencomparedtothedesignstormofthespillway.Rarely,therearecaseswhereinthesamedurationischosenforthe diversion and the spillway. The distribution is chosen according to thespecificgeographicalandclimaticdataoftheareaofdesign.

Table6.33Pairsoffrequencyofthedesignstormandthefrequencyofthedesignflood

6.8.3Estimationofthedesignfloodofdiversion

Fortheestimationofthedesignfloodofthediversion,firstofall,theestimationof the corresponding design storm over the river basin at the position of theproject isrequired.Thedepthhof thedesignstormhasbeencalculatedwithaknownfrequency1/TorareturnperiodT.Forexample,fromTable6.33,afloodwithfrequency1/50musthaveafrequencyofthedesignstorm1/81andareturnperiod of 81 years; for the duration t, the idf curves (h, t,T) that have beenconstructed for the basin are used. Then, we apply one of the four useddistributions.Definitelythepercentageoflossesforthecaseofdiversionwillbehigherthantheoneestimatedforthedesignstormofaspillway,duetothefactthatitis,inthefirstcase,amuchmorefrequentphenomenon.Inthesameway,aconversionofthedesignstormtoanappropriateunithydrographtakesplace,attheprojectposition,sothedesignfloodofthediversioncanbecalculated.For the case of nonlinear basins, but also for those where there are no

discharge measurements available at the construction site, models for theestimationofthedesignfloodofthediversionareapplied,liketheonesappliedtothedesignfloodofspillways.

Example6.5:EstimationofthedesignfloodofariverdiversionIt has been estimated by statistical analysis that the idf curves of aparticularbasinhavethefollowinganalyticalexpression:

(6.57)

Attheoutletofthisbasinwheretherearegaugingstationsofdischargeandwaterleveloftheriver,adamisabouttobeconstructed,soit isessential to estimate the design flood of the diversion.Based on the

technical, hydraulic, economic and the other characteristics of theproject,weselectadesignfrequencywithadiversionvalue1/50.Alsowechoosefor thedurationof thedesignstormthevalue48h.Fromthe idf curves, we come up with the following intensity and stormdepthforthedesignstorm(Table6.34):Notice that frequency 1/81 for the design storm of diversion

correspondstothedesiredfrequency1/50ofthedesignflood.Afterthat,thestormdepthhasbeendistributedin48hwith2htime

step,basedonthedistributionofsecondquarter,withconfidencelevel50%.Inthenextstep,wedeductthelossesequalto35%ofthetotalrainfall, respectively, for the design storm of the diversion. ThedistributeddesignstormandthelossesareshowninFigure6.17.Net48 h storm is implemented next on a 2 h unit hydrograph that wasderived fromstationdataon thebasinoutlet,and is shown inFigure6.17.Thefloodthat isproducedfor thediversion isshowninFigure6.17. In the total flood, the baseflow has been added, as shown inFigure6.17.

Table6.34Intensityandstormdepth

Frequency Intensityi(mm/h) Stormdepthh(mm)

Designstormofthediversion

1/81 2.84 136.32

Figure6.17 Designfloodofariverdiversion.

6.9HYDROLOGICDESIGNOFOTHERWATERSTRUCTURE–SPECIFICISSUES

6.9.1General

Sofar,fromthehydrologicdesignpointofview,theinterestwasparticularlyinreservoirsandtheircomponents.However, therearealsootherwaterresourcesprojects, like irrigation and water supply networks and wastewater treatmentplants (WWTPs).The social benefits of theseprojects areof great importanceduring the effort of exploitation of a country’s water resources, especially indeveloping countries. The necessity of construction of projects of this typethroughout a country, rural and urban, creates an apparent need for optimumdesignandoperation,soastocovereffectivelythesocialneedsforwater,withtheguaranteedquantityandwithoutcostoverruns.In the effort for optimum design and operation of hydraulic development

projects, the maximization of hydrologic information for each system that isdesignedhasasignificantpart.Thisisaccomplishedbycertaintechniqueslikemodelsandstatisticalanalysesthatenhancethequalityandquantityofhistorical

data (like discharges, rainfall depths) measured at the point of interest. Theenhancement of these data is usually based on a proper process of optimizingrecordedfilesinaformatwhichallowsustotakedirectlymoreinformationforthehydrologicdesignofaproject.Forinstance,inanirrigationsystem,thebasicgoal of the hydrologic design is to forecast the water quantities, properlyadjustedintimeandspace.Ifweassumethatthewatercomesfromamultiplepurposereservoir,theimportanceofexistenceofrainfallinformationintheareaisobviousforthesystemdesign,inordertooptimizetheabstractionsfromthereservoir,whicharenaturally supplementedby the rainwater in this area.Thisprocedurehas asdirect result on theoptimumprogrammingof abstractionsofwater and the saving of increased abstractions from the reservoir. This istranslatedfromahydrologicpointofviewtotheabilityofreliablepredictionofrainfall over an area in a way that everyone can estimate directly, at everymoment,therainwaterofagivendurationthatwillprecipitate,withatolerablerisk of failure of that prediction. In the following sections, we deal with thisproblem, but also with other ones relevant to the optimization of hydrologicinformationduringthedesignphaseofvarioushydrologicprojects.

6.9.2Constructionofoperationaldepth–duration–frequencycurvesofrainfallforirrigationneeds

We present here the prediction of rainfall water over an agricultural area forirrigationalneedsthroughtheconstructionofmonthlydepth–duration–frequencyrainfallcurvesoverthebasin(GabrielandNeumann,1962;Mimikou,1983).Duetothecomplexityofthisissue,firstofallthetheoryoftherainfalldepth

modeloveraperiodonndaysisanalysed.Letusassumethat

Thenumberofrainydaysis

(6.58)

Wesetxv,v=1,2,…,n,thedailydepthofrainfall,inthevrainyday.Thetotaldepthof rainfall in aperiodofndays,S(n), is calculatedwith the help of thenextrelationship:

(6.59)

ThefunctionofS(n)is

ByusingthepartofthesampleareaforwhichS(n)≤x,weget

(6.61)

If we assume that the information of the number of rainy days does notnecessarily mean that the quantity of rain is also known, we transform theaforementionedrelationshipintothefollowing:

(6.62)

P[Nn=v] is calculated throughaMarkovchain relationship (Mimikou,1994),

while the other factor can be calculated by a frequency analysis(usually log-normal distribution) of the total rainfall amount for the specificperiod ofn days, for the years that data exist (Gabriel and Neumann, 1962).Thus,wecancorrelatetherainfalldepthS(n)withtheprobabilityP[S(n)≤x]forvarious periods of n days, where n is not the rain duration of course but theperiodthatconcernsanirrigationprojectforinstance.Thismeansthatforeachmonthofanalysisduringtheirrigationperiod,wehavetherelationshipbetweenthe rainfall depthS(n) and total depth in n dayswith the probability that thisdepth is lessorequal toacertainvaluex.The resultsof such researchcanbepresentedintheformofcurvesthatareusefulfortheplanningandmanagementofwatersystems,asdescribedinExample6.5.If wewant to constructmonthly curves of rainfall, we use as raw data the

averagedailyarealrainfalldepthsofthebasinforthemonthsthatareimportantfor irrigation, i.e. May, June, July and August. For every month and every

durationn,weplottherelationshipsoffrequencyandrainfalldepth.Ifwedefineas risk j the probability of the irrigationwater prediction through these curvesthatdonotsatisfythegivenneeds(i.e.theirrigationofagivenareaandacropthatisdependentonthiswater),thenwehave

(6.63)

Sotheidfcurvesofdepth-durationandfrequencycanprovidedirectlyalsotheriskofeachpredictionofirrigationwaterforagivendurationofndaysineverymonthofirrigation.Throughthesecurves,wecanestimatethefollowing:

1. Therainfalldepthoftheareaofconcernforacertaindurationofndaysofanirrigationperiodandanestimatedacceptedriskj

2. Theriskjforacertainwaterdepthxduringacertainperiodofndaysofanirrigation period (supplementary to the water abstractions from thereservoir)

It has been proven (Mimikou, 1983) that if the idf curves (Figure 6.19) areplotted on a double-axis logarithmic paper for every frequencyP[S(n) ≤ x]=japart(withthedurationinhorizontalaxisandtherainfalldepthinverticalaxis),thereare straight lines (Figure6.20), parallel to eachother, following thenextanalyticalrelationship:

(6.64)

wherek,λ,mareconstantsdependingonclimaticfactors(seeFigure6.20).Thestochasticcurvesofdepth-durationandfrequencyofEquation6.62aresimilartothe idf curves (h, t,T) that have been presented earlier.Nevertheless, they aretotallydifferent,sinceintheidfcurves,tisthedurationoftherainfall,buthere,n is a period of days independent of single rainfall events. Also, the rainfalldepthsinregularidfcurvesaredealtlikeextremeevents(maximum)throughouta year, but from the stochastic point of view, they are dealt like a constantphysical process. This results in a stochastic prediction of depth–duration–frequency(orrisk)ofraintobeoperationallybetterthantheclassicalidfcurves,since someone can adjust thedurationn of the curves to any desired durationappropriateforthedesignbutalsototheoperationalneedsofthewatersystem.Thisway,wecanavoidtheindividualanalysisofextremerainfallsofalimitedduration (usually2–3daysatmost) that ismandatorywhenusingclassical idfcurves.

6.9.3Dischargepredictioninawaterrecipientofoutflowsfromawastewater

treatmentplant

The prediction of discharges of a river that accepts sewage of an urban orindustrial area, after biological treatment, offers a crucial information. Thereason is the potential of lowering the outflow quality of theWWTP, whichreducestheenergyconsumption,duringhighdischargesoftheriver,duetotheself-cleaningabilityoftherecipient.

6.9.3.1Conditionsandrequirementsofthemethod

Letusassumea siteA ina riverwith industrial anddomestic sewageoutflowfromanearbyarea,afterthetreatmentinaWWTP.AflowmeasurementstationislocatedatsiteB(basinoutlet)andrainfall-measuringgaugesarealsoinstalledinthebasin,asshowninFigure6.18.

Figure6.18 Generallayoutofthebasin.

ThemethodconcernsndaysoftheyearwhenthedischargeatsiteAandthenearbysiteBarehighenough.Inthiscase,waterindicators(dissolvedoxygen,concentration of ammonium nitrogen, etc.) that define the quality of water

treatment(%BODremovaletc.)arehighenoughthatallowthemoderationofthetreatmentplantoperation,loweringthelevelsofthedesiredefficiency,duetothe enhanced self-cleaning ability of the river. So, initially an estimate of athreshold river discharge was made, which, when exceeded, the plant canoperate with a lower efficiency. For example, in a treatment plant that canprovidesecondandthirddegreeoftreatment,itispossiblethesedaystooperateonly the second degree of treatment, with substantial economic benefits inenergysaving.Anindicativefactorfortheselectionofthiscrucialdischargeistheduration(indays)withgreaterorequalactualdischargesinanaverageyear.ThisisobtainedfromthecurveofdailydischargesatsiteB.Thisdurationmustbe long enough so that the reduction of full capacity of the plant operation ismeaningfulfromatechnicalandeconomicalpointofview.Usually,weproperlycombine the two indicative elements (criteria), which are the substantialmoderationof theplantoperationand thedurationofaverageannual time thatexceedsthecrucialdischarge,fortheoptimumselectionofit.The requirements of the predictionmodel at siteB aremainly regarding its

operational features. Itmusthave theabilityof real-time implementation,withdirectwarningof dischargesgreater or equal to the selected crucial discharge.Also itmustbeequippedwithrelatively longprediction time, inorder tohavetheabilitytoactivatethesystemfortheexploitationofasuddenriseoftheriverdischarge.Thislastrequirementcomplicatesthesolutionoftheproblem,giventhefactthatinhydrologicsimulationmodels,theincreaseofthepredictiontimeis against the credibility of the prediction itself. This problem is confrontedpartially by the use of an automated correction mechanism (updatingmechanism),whichduringthepredictiontimecorrectsthefinalpredictionwithupdateddatainsertedinthemodelinrealtime.

6.9.4Modelsofdischargepredictionsofwaterrecipientofsewage

Thedemandforlong-termpredictionsrestrictssubstantiallythenumberofreal-timepredictionmodels that canbe selected.Somodels like theonepresentedpreviously,whenthepredictiontimeislimitedtothedurationoftheroutingoftheriver,areinappropriateforthiscase.Theselectionmustbeamongmodelsofonevariable(e.g.theriverdischarge)orwithtwoormorevariables.Inthiscase,a combination of two ormoremodels is required in order to obtain the entryvariablesintheinitialmodel(i.e.arainfallpredictionmodel).The single-variable models of the river discharge predict the future values

using mainly the correlation of the successive measurements of the variable,usingthememorycharacteristicofthevariable’stimeseries.Basedonthis,itis

obviousthatthemostpropersingle-variablemodelsfordischargepredictionsinseasonal(mostcommonlydaily)basisisthestochasticmodelsoftheARMA(p,q)category.Basicadvantagesofthesemodelsarethefastprocessofcalibration,theirsimplicityandtherequirementofminimumfielddata(onlythetimeseriesofdischargesinsiteBofthepreviousfigureisneeded).Abasicdisadvantageistheincreaseoftheaveragesquareerroroftheprediction,whenthetimeoftheprediction increases. It has been proven (Quimbo, 1967) that the mostappropriate stochasticmodel for simulation, and therefore prediction, of dailydischargesistheself-correlationmodelofsecond-orderAR(2).In thecategoryofmodelswith twovariables (rainfall–runoff), someonecan

usedeterministicandstochasticmodelsaswell.Forbetterfunctionalcapabilityin real time, and also for automation (of input and output data), ‘black-box’modelsaresuperior.Calibrationofthesemodelsisbasedontherespectivetimeseries of discharges at siteB and rainfall (through thedaily data of thebasin)fromtherainfallgaugesinFigure6.18.Forthehydrologicuseofthesemodels,wemustfollowanempirical(andnotautomated)processofcalibration,inorderto take into account additional basin information like snowfall and precedentmoisture. In order to use both models for prediction in real time, we mustcombineparallelsingle-variablemodels(stochastic)ofrainfallprediction,whichtaketheroleofinputdata.So,withastochasticmodel,i.e.AR(2),wepredictLtimeunitsaheadofthevalueofrain(averagedailyrainfall)intime(t+L),andthenwecomputethedischargeatsiteB,andconsequentlyatthesiteofinterestA at time (t + L). The models of rainfall–runoff have increased credibilitycomparedtosingle-variablemodels,becausetheybetterdescribetheresponseofthe basin. Nevertheless, more effort for calibration and implementation isneeded.Anevenbetterdescriptionof thebasin’sbalancemechanism isprovidedby

holistic models of the basin that incorporate, except rainfall, storage andevaporation. These models must be connected in parallel with three otherstochasticsingle-variablemodelsforpredictionofLtimeunitsaheadofrainfallvaluesof storageandevaporationof thebasin.Thesevaluesareusedas inputdatainthemodelfordischargepredictionatsiteBfortime(t+L).Withcriteriamainlythefielddata,whichareprovidedforthebasinandalso

the computing capacity, and the existing means of hydrologic data transfer(systemautomation)fortheimplementationofthemodel,wecanselectoneoftheaforementionedmodelsforthepredictionofthedischargeatsiteB(Figure6.18). If, for example, only discharge data or rainfall and discharge data areavailable,thereisalimitationtosingle-variablemodelsortwo-variablemodels,respectively.Alsoanassumeddeficiencyofcomputationalmeansandscarcityof

thehydrologicdatatransfersystemcouldprohibitcredibleimplementationofaholisticmodeleventhoughthefielddataexist(foritscalibration).Aftertheselection,calibrationandimplementationofthepredictionmodelin

real time(connectionwith thesystemofdirectdata transferofentrydata),wemust assure that a credible notification system towards the treatment plantoperatorsexists.Eachoneofthemodelsthathavebeendescribedmustprovidetheprediction in real time tof thedischargeL timeunits ahead (i.e.10days).Exactlyattimet,thesystemofdirectnotificationoftheplantmustautomaticallybeactivated(withasoundoranelectronicmessageonapcscreen),inordertopredictwhichdischargewillexceedthecrucialvalueduringtime(t+L).Then,properadjustmentsmustbedoneintheplant’soperation.Inthecasewhenthemodelhasacorrectingmechanism,andnewreal-timedataenterthepcintime(t+1),(t+2),…,(t+L−1),anextrachange,i.e.thecancellationofthepredictionthat the dischargewill exceed the crucial value in time (t +L), must also betransmittedautomaticallytotheplant.

Example6.6:OptimizationofirrigationuptakesfromareservoirAmountainous basin of a river, with an area of 2744 km2, is to beirrigatedbyanearbymultipurposereservoir.Sincethereisscarcityintheuptakesfromthereservoir,apredictionmodelhastobedeveloped,with acceptable risk in the anticipated amount of rainwater over thebasin, during the irrigation period (May until August), in order tooptimize the irrigation programming, from the reservoir, for watersavingpurposes(Ross,1970).Forthisreason,astochasticmodelhasbeenused.The raw inputdatagive theaveragedaily rainfallsof thebasin(from18raingaugesusingtheThiessenaggregationmethod)foraperiodof20years. In thisanalysis, rainyday isdefinedas thedaywith at least 0.25mmof rainfall occurring;otherwise theday is dry(Chin,1977).The simulation of the daily events of rain for 20 years for the

months May, June, July and August has been performed for everymonthlysub-periodoftheirrigationperiod.Thus,we have computed for everymonth the initial probabilities,

thetransitionprobabilitiesandtheprobabilitytocopewith(v=1,2,…,n)rainydaysinanyperiodofndays(n=1,2,…,30)in1month:P[Nn=v].Afterthat,theproperprobabilitydistributionhasbeendefinedasthecumulativeamountofrainfall,

(6.65)

forv = 1,2,…, 30,with selection between a bi-parametrical gamma,tri-parametricalgammaandlog-normaldistribution,thelastonebeingbetteraccordingtothex2testcriterion.Sotheprobabilitieshavebeencalculated:

(6.66)

foreveryvandxvaluesbetween5and100mm(with5mmstep).Thiscombinedprobabilityacquiresthetotalprobabilitydistribution,fortheamount of rain S(n), in a period of n days with risk j equal to theprobability. In this way, for every month of this period, the depth–duration–risk curves j (x, y, j) have been plotted, as seen in Figure6.19. After this step, the lines have been plotted in double-axislogarithmic paper in Figure 6.20, based on the previous equation,while the parameters k, λ,m, for May, June, July and August areshowninTable6.35.

Figure6.19 Depth–duration–frequencycurves(risk)ofrainforirrigationpurposes.

Figure 6.20 Logarithmic curves of depth–duration–frequency (risk) of rain for irrigationpurposes

Table6.35Parametersofthestochasticcurvesoffrequencyofrain

Months

Parameters May June July August

k 1.08×10−2 2.8×10−3 1.9×10−3 0.83×10−3

λ 1.33 1.60 1.62 1.65

m 1.02 0.93 0.96 1.00

From Figures 6.19 and 6.20, the expected rainfall depth over thebasinisestimatedforagivenperiodofndayswithstandardriskj,i.e.foraperiodof20days,inMay,withrisk20%(failureprobability),theexpected rainfall depth is 10 mm. Also the risk can be found for acertainperiod,i.e.for20mmofrainin20daysinMay,riskis36%.

6.9.5Duration–dischargecurvesandtheiruseinthestimationofthehydroenergypotential

The duration–discharge curve in a river position is a very useful tool for thehydrologic design of hydroenergy installations and other hydraulicworks likeflood protection, irrigation networks and works for sustaining water quality.Especiallyinthedevelopmentofhydroelectricprojects,itisveryusefulduringthe preliminary phase of the design for the estimation of the hydroelectricpotential of the position,mainly in projects with a small reservoir or withoutone.Forprojectswithlargereservoirs,thesecurvesarenotusefulenoughgiventhefactthattheydonotprovidetheinformationofatimeseriesofdischargeofthatcertainposition.ThecurvedisplaysthedischargeQinrelationtothepercentageoftimeDthat

a discharge is equal or surpasses a certain value. In the x-axis, there is thepercentageoftimeandinthey-axisthedischargeQ.Wecanhavecurvesonadaily, monthly, etc. basis. In Figure 6.21, a duration–discharge curve isdisplayed.

Figure6.21 Duration–dischargecurve.

Figure6.22 Duration–dischargecurveforadjustedflowrateat20m3/s.

The duration–discharge curve provides the necessary data for sizing therequiredstorageinordertomaintainaspecificdischargevalue.Forexample,ifthenaturalflowis8m3/sfor the100%oftimeandthedesireddischargeis20m3/sforthe100%oftime,thenthefollowingstepsareapplied(Figure6.22):

1. TheparallelfrompointE(valueof20m3/s)tothehorizontalaxisisdrawn.TheintersectionoftheparallelwiththecurveispointA.

2. ThesectionAEDZisdrawnontheleftsideofthecurve,insuchawaythatthe areas of sectionsAEDZandABCon the other side of the curve, areequal.

3. The new curve is created with downward parallel transposition of thepreviouscurvefrompointZtopointHandhastheshapeofUHAC.

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Borland,W.M.andMiller,C.R.,1960,Distributionof sediment in large reservoirs,Trans.Am. Soc.Civ.Engs.,125(1),166–180.

Chin,E.H., 1977,Modeling daily precipitation occurrence processwithMarkov chain,WaterResourcesResearch,13(6),949–956.

Chow,V.T.,Maidment,D.R.andLarry,W.,1964,HandbookofAppliedHydrology,McGraw-HillCo.,New

York.Fellow,W.,1951,Theasymptoticdistributionoftherangeofsumsofindependentrandomvariables,The

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QuarterlyJournaloftheRoyalMeteorologicalSociety,88,90–95.Gomide,F.L.S.,1975,RangeandDeficitAnalysisUsingMarkovChains,Hydrologypapers,ColoradoState

University,FortCollins,CO.Gupta,R.S.,1989,HydrologyandHydraulicSystems,PrenticeHallEnglewoodCliffs,NewJersey.Horton, R.E., 1933, Determination of infiltration capacity for large drainage basins,Eos, Transactions,

AmericanGeophysicalUnion,18,371–385.Huff,F.A.,1970,Spatialdistributionofrainfallrates,WaterResour.Res.,6(1),254–260,DOI:http://dx.do-

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London,U.K.,145pp.Institute of Hydrology, 1978, Methods of Flood Estimation: A Guide to the Flood Studies Report,

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Carre spillway to divert Mississippi river water into lake Pontchartrain, Louisiana, EcologicalEngineering,17(4),411–422.

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Lara,S.M.,1962,RevisionoftheProceduretoComputeSedimentDistributioninLargeReservoirs,BureauofReclamation,Denver,Colo.,USA.

Linsley,R.K.,Kohler,M.A.andPaulhus,J.L.H.,1975,HydrologyforEngineers,McGrawHill,NewYork.Miller, C.R., 1953, Determination of the Unit Weight of Sediment for Use in Sediment Volume

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Greece,HydrologicalSciencesJournal,27(3/9),369–383.Mimikou,M.,1983a,Monthlyrainfallfrequencycurvesduringirrigationperiods,InternationalJournalof

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Chapter7Urbanhydrologyandstormwatermanagement

7.1INTRODUCTION

7.1.1Definitionsofurbanhydrologyandstormwatermanagement

Asignificantpartof engineeringhydrology isdedicated to the so-calledurbanhydrologyorurbanstormwaterhydrology.Thesetwotermsaresynonymousandare closely related to various other terms used, such as urban drainage, urbanstorm drainage, urban stormwater management and urban surface watermanagement.Urban hydrology deals with the hydrological cycle within the urban

environment.Aspartofengineeringhydrology,itstudiesthefollowing:The precipitation–losses–runoff processes within urban or urbanizingwatersheds, emphasizing on the generation of stormwater runoff and itstransportation both on the ground and street surface and in the sewerconveyancesystem,aswellasthroughitsappurtenancesThequalityofthestormwaterrunoff,itsfateanditsimpactsonreceivingwatersThe technical aspects of the planning, design, construction, operation,maintenanceandeffectivenessevaluationofvariousmeasures tomanagestormwaterrunoffquantityandquality

In this sense, urban hydrology uses various hydrological methods for runoffquantitycomputations(therationalmethodand theSCScurvenumbermethodare the most commonly used); some of them are specifically used in urbanwatersheds.Urban stormwater management is a more comprehensive term than urban

hydrology. It also involves the implementation of variousmeasures to resolvequantityandqualityproblemsrelatedtopopulationgrowthinurbanareas,urban

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expansion and/or urbanization of natural areas. Although urban stormwatermanagementdoesnotnecessarily involveconstructionof specific structures, itmost often does. Typical designs include specific cross sections for the streetsurfaceconveyancesystems;curbs,guttersandcatchbasininlets;stormsewersandtheirappurtenances;openchannels;detentionandretentionbasinswiththeirinlet/outlet structures; and special structures such as energy dissipators. Eitherspecific modifications of existing structures or specific designs are made forcontrollingwaterqualityofurbanstormwaterrunoff.Finally,urbanhydrologyisessentialinprovidingtechnicalsupportinimplementingnon-structuralmeasuresin urban stormwater management, such as flooding area delineation andawareness/educationofthepublic.AccordingtoWalesh(1989),theobjectivesofurbanstormwatermanagement

includethefollowing:Theprotectionoflifeandpublichealth,particularlyrelatedtolesseningoffloodinganddrainageofstandingwatersThe protection of damage to public and private property resulting fromfloodingTheminimizationofdisruptionofhumanactivities,asaresultoffloodingTheprotectionofreceivingwaterqualityimpairment(bothofsurfaceandgroundwater) from pollutants carried by runoff (non–point sourcepollution)Theenhancementofaestheticsand,generally,qualityoflifeinurbanareasby incorporating natural features into the surface drainage system (e.g.opengrassed swales andchannels,grass filters,ponds,wetlands),whichaddrecreational,aestheticandecologicalvalues.

7.1.2History

Eventhoughadvancedmasonryorclayceramicurbanstormandsanitarysewershave been used 4000 years ago in ancient Greece, e.g. during the Minoancivilization (Angelakiset al.,2005), sanitary seweruse inwesternEuropeandtheUnitedStateswasnotcommonuntilthemid-nineteenthcentury;wastewaterwas then freelydischargedonto thegroundsurface,even thoughstormsewerswereinoperation.Thispracticeresultedintheoccurrenceofinfectiousdiseases,such as typhoid fever, dysentery and cholera. The medical doctors could notaddress the problem, since this was mostly due to infection from consumingcontaminatedwater.Theengineersprovidedthesolutiontotheproblem,aroundthe middle of the nineteenth century, by constructing combined sewers. Thispractice continued until early in the twentieth century, when combined sewer

overflow(CSO)andresultingenvironmentalproblemsledtotheconstructionofseparate sewer lines. Nevertheless, old cities or parts of modern cities stilloperate combined sewers even today; attempts aremade, however, to resolveCSOproblems.Eventhoughseparatesewerlinesandalsowastewatertreatmentplants (WWTPs) have been constructed, at least in developed countries,pollutionofsurfacewaterbodiescontinuestobeaproblem.Thishasidentifiednon–pointsourcepollution,frombothagriculturalandurbanareas,asthemaincause of water quality degradation of aquatic systems and the main problemcurrentlyaddressedinurbanstormwatermanagement(Walesh,1989).Theevolutionofurban stormwatermanagement isdepicted inTable7.1. In

short, the table presents early human activities to drain wetlands and usefloodplainsforagriculturalandurbandevelopment,resultinginthereductionofthe natural flood dissipation features of streams and rivers. Later activitiesinclude the construction of major dams for flood control and hydropowergeneration,whichprovidedfloodprotectionofmajorrivers,eventhoughthis,inmost cases, caused significant erosion and ecological impacts in downstreamreaches and the coastal environment.Further urbangrowth,with street pavingand curb and gutter construction, resulted in the shortening of time ofconcentrationand the increase inhydrographpeaks.Asa result, local floodingproblems occurred, leading to the necessity for enlargement of sewer linesand/or the use of detention basins. In the 1990s, stormwater-related qualityproblems of receiving waters were identified, leading to the necessity forstormwater quality management. Today, the tendency is to promote ‘natural’floodandpollutioncontrolmeasures,whichalsoincluderecreationalvalues,andpresent more aesthetic and ecological methods in urban storm watermanagement.

Table7.1Historicaldevelopmentinurbanstormwatermanagement

Milestone Indicativetimescale Impact-result

Naturalenvironment—wetlands:villagesandsmallcities

150yearsago Limited

Drainageofwetlands—agriculturalandurbandevelopmentonfloodplains

100yearsago Reductionofnaturalflooddissipation

Damconstructionforfoodcontrolandhydropower

Floodprotection—erosionandecologicalimpacts

Urbangrowth—pavementofroads,streetandgutterdrainage,combinedsewers,separatestormsewers

1970s Shorteningoftimeofconcentration,increaseofpeaks,localflooding—qualityproblems

Enlargementofsewers—useof Stormwaterquantity

1.

2.

3.

4.

5.

detentionbasins management—qualityproblems

Qualityproblemsemerge 1990s Stormwaterqualitymanagement

Useofmore‘natural’systems—propermaintenance

Today Effectivestormwatermanagement

Source:AdaptedfromDeboandReese,2002.

7.1.3Descriptionoftheurbanstormwaterdrainagesystem

Theurbanstormwaterdrainagesystemaimsto• Provide the means for enhancing water infiltration and minimizingrunoffquantity•Effectivelyandsafelycarrythestormwaterrunoffquantity•Providestorageforexcessrunofftoavoidlocalflooding•Providemeansofretainingorminimizingpollutantscarriedbyrunofftoenterreceivingwaters

Toaddress these fouraims, the stormwaterdrainage systemcomprisesvariouscomponentscategorizedintofivemajorcategories:

Those where runoff is produced, such as rooftops, parking lots andperviousareas.Thoseprovidingincreasedinfiltrationofrainwater:lessimperviousareasresult in less runoff volume and lower peaks. As a consequence, thedrainage system costs less and fewer pollutants reach receiving waters.Incorporatingmoregrassedareas in theurbanenvironment, constructingsurface drainage systems (e.g. swales) and building permeable hardsurface(e.g.porouspavements,infiltrationtrenches)allleadtoincreaseininfiltration.Those used to safely convey runoff, such as open swales and ditches,culverts, street cross sections, inlets andcatchbasins, and storm sewers.Regardingthesewersystem,asmentioned,threetypesoflinesmayexistintheurbanenvironment:separatestormsewers(theyonlycarryrunoff)and sanitary sewers (they carrywastewater), which is the preferred andcurrent practice, and combined sewers (they carry both wastewater andrunoff).Those used to store runoff, such as detention and retention basins andstoragetanks.Those used to control sediments and other pollutants carried by runoff,such as sedimentation basins, erosion control structures, dry and wetbasinsandconstructedwetlands.

7.1.4Impactsofurbanization

Urbanizationofanarea,i.e.thetransformationofanaturalareaintoacity,hasamajorimpactonboththequantityandqualityofrunoffifmeasuresarenottaken(NovotnyandOlem,1994).Regardingthequantityofrunoff,thecommoneffectis the increase inboththevolumeandthepeaksofrunoffhydrographs,withaconsequent increase of water stages in drainage channels, increased risk offloodingandexpansionofinundationareas.Increasesinvolume(andpeak)arearesult of changing land cover by paving roads and constructing buildings andparkinglots,i.e.byincreasingtheimperviousareaoftheurbanbasincomparedtothenaturalbasin,somethingthatdecreasesinfiltrationandotherlosses.Thisisdemonstrated in the hydrographs of Figure 7.1a. Two of the hydrographscorrespond to an undeveloped (natural or agricultural) watershed (A) and anurbanizedwatershedwithasurfacedrainagesystemusingopenswalesorgrasssideditches(B).Thefigureshowsthatboththevolumeofrunoff(areaunderthehydrograph curve) and the peak are increased due to reduced infiltration.However, the time of the peak and the time base of the two hydrographs arecomparable(becausethetimeofconcentrationinthetwocasesiscomparable).Afurtherincreaseinpeak(hydrographC;Figure7.1a) isaresultofshorteningthetimeofconcentrationintheurbanizedcase,andthisismainlyanoutcomeofconcentratingrunoffinconcreteopenchannelsand/orstormsewers,wherewaterflowswithincreasedvelocitiescomparedtooverlandflowandflowinswalesornatural channels. All these can be easily understood by also comparing thehydrographsofFigure7.1b,whichcanalsobeusedtounderstandtheeffectonhydrographpeakandshapeintwourbanizedwatersheds.Thetwohydrographsnow correspond to two hypotheticalwatershedsA andBwhere allwatershedcharacteristicsarethesameexceptonethatisdifferent,assummarizedinTable7.2.Inallcases,therainfallisconsideredthesame,andtheshapeandareaofthetwowatershedsarethesame.

Figure 7.1 (a) Impact of urbanization: (A) undeveloped, (B) urbanized with surface drainage throughswalesorsideditches,(C)urbanizedwithdrainageinconcretesewersandopenchannels;(b)comparison of the resulting hydrographs from the same rainfall in two similar watersheds

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differinginonlyonecharacteristic,assummarizedinTable7.2.

Table7.2ComparisonoftwohypotheticalwatershedsofFigure7.1b

Featureofcomparison WatershedA WatershedB

Urbanversusnatural Urbanized,drainageonpavedroads,instormsewersand/orlinedchannels

Naturaloragricultural

Urbanwatershed

Totalimperviousarea High Low

Hydraulicallyconnectedimperviousarea High Low

Watershedslope Steep Mild

Stormsewerorchannelslope Steep Mild

Stormsewerorchannelroughness Low High

Similartorunoffquantity,urbanizationhassignificantadverseeffectsonthequalityofthewaterbodiesreceivingtherunoff(NovotnyandOlem,1994).Thisis a result of changes in pollutant composition and content in the atmosphereabovetheurbanarea,hydrologyanddrainage,soils,landusesandanthropogenicactivities, among others. Pollutants come from various sources such as theindustry,municipalwastewater,solidwastesandairpollutantemissions.Duringthe construction, relatively high amounts of solids are produced from erosionandearthmovement(estimatedat100tons/ha/year).Thesesolidsendintowaterbodies causing turbidity and possibly pollution (theymay also carry adsorbedcompounds)and,consequently,adverseecologicalimpacts(NovotnyandOlem,1994).In addition to changing the shape of hydrographs and the resulting

drainage/floodingproblems,pollutionistheothermajorimpactofurbanization.Stormwater runoff pollution is of the non–point source or diffuse type. Thecharacteristicsof this typeofpollutionmake itdifficult toaddress,because itssources are widespread over the drainage area, and pollutant loads cannot beeasily quantified. More specifically, non–point source pollution has thefollowingcharacteristics(NovotnyandOlem,1994):

It enters the aquatic systems at various points, at intermittent timeintervals,closelyrelatedtothevariationtometeorologicalfactors.Thepollutantsareproducedinawideareaandarecarriedthroughsurfacerunoffforsometimebeforereachingthereceivingsurfacewaterbodyorgroundwater.The pollution is hard to control at the point it enters the water body,

•

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•

because of the large number of entry points and the significantly highvolumes/rates of runoff, which would require very large and expensivestructures.Treatment of runoff analogous to wastewater treatment is a veryuncommon option. Non–point source pollution is controlled throughmeasuresappliedthroughoutthewatershed,aimingtominimizepollutantsat the source and/or retain them in special structures. These measures,called best management practices (BMPs), are based onmanaging landuse, anthropogenic activities and surface runoff, and either involveconstructionofafacilityorareofthenon-structuraltype.RestorationofthereceivingwatersqualityisamuchmoreexpensiveandprobablyacompletelyunrealisticoptioncomparedtotheuseofBMPs.Non–point source pollution depends on meteorology, extrememeteorological or climatic events, geographic and geologic conditions,and varies from place to place and throughout time. Therefore,implementationofcontrolmethodsissitespecificandcannotdirectlybetransferredfromoneareatoanother.

Main urban non–point source pollutants include the suspended sediments,nutrients,heavymetalsandvarioustoxicsubstances.Thesepollutantsareresultsoftrafficandotheranthropogenicactivitiesandwetordrydepositionfromtheatmosphere;theyareaccumulatedonthegroundsurface,washedoffbyrainfalland transported by runoff. The quantities of pollutants also depend on thedrainage system. For example, it is estimated that the pollutant load fromresidentialareaswithnaturaldrainagewithoutsewers(e.g.usinggrassedswales)issignificantlylessthanthatfromsimilarwatershedswiththesamelanduseandlandcoverbutwithdrainagebysewers(NovotnyandOlem,1994).Asmentioned,theurbandrainagecanbeseparate(i.e.runoffandwastewater

flowinseparatelines)orcombinedsewers(thetwoflowinthesameline).Thefirst case is the modern and preferred practice, but there are still combinedsewers in operation in various cities of the world. Combined sewers operatenormallyduringdryweather,carryingsewageandnuisancewatertotheWWTP;in this case, no environmental problem occurs because wastewater is treated.However,duringwetweather(whentheextrasewercollectssurfacerunoffandwastewater),occurrenceoftwodistinctphasesispossible:first,atthebeginningofsurfacerunofforwhenrainfallintensityanddurationaresmall,thecombinedflowrateissmallandmixeddomesticsewageandrunoffendupintheWWTPandcanbetreated.Thus,thereisnoenvironmentalprobleminthiscase.Second,

inthecaseofhigh-intensityrainfall, thedischargemayexceedthedesignflowrate (capacity) of theWWTP; in this case, a hydraulic control structure (flowdivider) prevents entry to the treatment plant, leading flow into the receivingwaters. As a result, mixed runoff andwastewater are transferred directly (i.e.withouttreatment)inthereceivingwaterbody,aphenomenoncalled‘combinedseweroverflow’ (CSO).CSOcanbequite dangerous for ecosystems, fish andhumans, sincehighpollutant andmicrobialquantities andconcentrations enterthe water body in relatively short time. In developed countries, during CSO,beaches close to bathing, and fish and mollusc consumption is forbidden.Research has shown that nitrogen and phosphorus loadings from urban areaswithcombinedsewerscanbedoublecomparedtothosefromsimilarareaswithseparatesewers(NovotnyandOlem,1994).

Figure7.2 (a)Percentoftransportedpollutantsasafunctionofsurfacerunoffdepth;(b)hydrographandpollutographqualitativelyindicatingfirst-flushphenomenon.

USEPA (1974) first proposed that pollutants in surface runoff are nottransporteduniformlyduringthehydrograph.Thelargestportionofpollutantsistransported during the first 13–25 mm of runoff, as schematically shown inFigure 7.2. This phenomenon is called first flush. Further research (e.g. Ellis,1986) concluded that this phenomenon is not present in every rainfall event;however,itisaconceptuallyrationalphenomenonandisusedinrunoffqualitymanagement.

7.1.5Factorsinfluencingurbanrunoffquality

Several factors influence the quality of surface runoff including (Chui et al.,1982;TsihrintzisandHamid,1997a)rainfallquantityandqualitycharacteristicssuchasdepth, intensity,duration,numberofdayswithno rainandconstituent

concentrations, traffic volume, land use and cover, geologic characteristics,geographic location, streetmaintenancepractices (e.g. sweeping, flushing) andthetypeandgeometryofthedrainagesystem(i.e.surfacedrainage,separateorcombinedsewers).Thenumberofdrydaysbeforetherainfalldeterminespollutantaccumulation

on the surface (Huber andDickinson, 1988). The rainfall intensity affects thevolume and discharge of surface runoff, which then determine the rate ofwashoff, the concentrationof the accumulatedpollutants and their transport tothe receivingaquatic system. In thecaseofhighway runoff, accumulationandwashoffoftraffic-relatedpollutantsisafunctionoftrafficvolume.Landuseandlandcoverare themost important factorsdetermining the typeofpollutants insurface runoff. Human activities in the watershed, e.g. land development andconstruction,contribute toadditionalpollutant loadings in runoff (Guptaetal.,1981;BoydandGardner,1990).Thepercentimperviousnessofthewatershedisan important factoraffecting runoffvolumeandpollutantquantities (Griffinetal., 1980). Geographic, geological andmeteorological conditions of the studyareashouldalsobetakenintoconsideration.

7.1.6Pollutantgenerationprocesses

Theprimaryprocessesleadingtoproductionofpollutantsandqualityproblemsofreceivingwatersarethefollowing(BoydandGardner,1990;TsihrintzisandHamid,1997a):

Atmosphericscrubbing:Atmosphericparticulatesfromvarioussources(i.e.dust,soot,aerosolsandemittedgases)comedownwithprecipitationandbecomepartoftherunoff(Randalletal.,1981).

Scouranderosion:Theyresultfromrainfalldropimpactoverexposedlandsurfaces,whichloosensthesoilstructure,andsubsequententrainmentandtransportbyrunoff.Inaddition,pollutantsadsorbedonsoilgrains(e.g.fertilizersandpesticides)becomepartoftherunoff,endinginthereceivingwaters.

Surfacewashoff:Stormwaterwashesoffalltheaccumulatedpollutantsintothereceivingwater.Inurbanareas,especiallyonroadsandhighways,thewashoffvolumeisrelativelyhighbecauseoflargerimpervioussurfacesuchasparkinglots,streetsandsidewalks.

Deposition:Sedimentsandpollutantserodedfrompervioussurfacesandcarriedbyrunoffmaydepositbyadheringtothesurroundingsurfaces,settlingoutandthroughprocessesthatremovethemfromthemainrunoffstream.

Transportandtransformation:Thetransportprocessmovespollutantsbetweenlocationsandthetransformationprocessalterspollutantcharacteristicsasaresultofphysical,chemicaland/orbiologicalmechanisms.Typicaltransportprocessesincludeadvectionanddiffusion,andtypicaltransformationprocessesincludebiodegradationandphotolysis(Ferrara,1986).

7.1.7Typesandsourcesofpollutants

The following pollutants are typically found in stormwater runoff (TsihrintzisandHamid,1997a):

Suspendedsolids:Streetdustanderodedsedimentsmaymakethewaterturbidorcloudy.Sedimentparticlesmaydepositinthebottom,smotheringincubatingfisheggs,fillingspacesbetweenrocksusedforcoveranddisturbingaquatichabitat.Inaddition,pollutantstendtoadheretosedimentparticles,particularlytofine-sizedones.WallerandHart(1986)reportedthatsolidloadsinurbanstormwaterrunoffaremuchhigherthanloadsfoundintreatedsewage.

Heavymetals:Vehiclesareamajorsourceofcontributionofmetalstotheenvironment.Concentrationofheavymetalsinstormwaterrunoffisamultipleofthatinsanitarysewage.Norman(1991)measuredseveralmetalsintheOhioriver,suchaslead,zinc,copper,chromium,arsenic,cadmium,nickel,antimonyandselenium.Certainheavymetals(e.g.copper,cadmium,leadandzinc)aremoresolubleinwaterthanothersandmaycausetoxiceffectsatconcentrationsexceedingthresholdvalues.Industrialandcommerciallandusescontributemostheavymetalsinrunoff.Theseincludeleadandcopper,whicharepredominantlyassociatedwithparticulates,andthus,theirconcentrationrelateswelltothatofsuspendedsolids.Itisestimatedthat40%–75%ofmetalsarederivedfromhighwayrunoffsources(Hunteretal.,1979).Themosttoxiccompoundsoriginatefromtraffic.Sourcesincludeleadfromtheuseofleadedfuel,leadoxideandzincfromtirewearandcopper,and

chromiumandnickelfromwearoftheplating,bearings,brakeliningsandothermovingpartsofavehicle.

Chlorides:Saltsareappliedduringtheearlyphaseofsnowfalltopreventbondingofsnowwiththepavement.TypicalsaltquantitiesintheUnitedStatesrangefrom180to550kgpertwo-lanestreetpermile(Hoffmanetal.,1981).Repeatedapplicationsattheseratesovertheentirewinterseasonendinhighamountsofchlorides,whicharewashedoffintotheaquaticsystems.Streamscrossingsaltedhighwayshavebeenfoundtocontainhigherchlorideconcentrationsdownstreamofthecrossingthanupstream,withpeaklevelsoccurringbetweenwinterandearlyspring.

Othersconstituents:Oils,greaseandotherrelatedhydrocarbonsarefrequentlyfoundinhighwayrunoff.Itisestimatedthat4.2×109LofautomobileandindustriallubricantsarelosttotheenvironmentannuallyintheUnitedStates,eitherdirectlybydisposaltosewersandapplicationtolandorindirectlybyspillageorleakingfromvehicles(Hunteretal.,1979;Bomboietal.,1990).Petroleum-derivedhydrocarbonsareregularlyreleasedintotheenvironmentinproportionsrelatedtosurroundingurbanizationanddevelopment(Stenstrometal.,1984).Parkinglotsarebelievedtohavethehighestloadingfactor,approximately25timeshigherthanthatofresidentialareas.Hydrocarbonsareemittedfromvehicularexhaustsystems,aresuspendedintheatmosphere,getscrubbedbyrainfallandbecomepartoftherunoff.Theprimaryhydrocarbonloadings(particulatesandlubricantoils)inrunofforiginatefromvehicleexhaustsystems.Unresolvedcomplexmixtures(UCMs),constitutingapproximately80%ofthetotalcontentofaliphatichydrocarbonsinurbanrunoff,havealsobeenfoundinmostsamples(BoyerandLaitinen,1975;BomboiandHernandez,1991).

Itisestimatedthatsome50%ofsolidsand70%ofthetotalpolycyclicaromatichydrocarbons (PAHs) entering aquatic systems originate from highway runoff(Ellis, 1986). Increased PAH andn-alkane loadings in urban runoff correlatedwell with increased urbanization and traffic density (Bomboi and Hernandez,1991;HewittandRashed,1992).The annual BOD5 loading into a water body from urban runoff is

approximately equivalent to that of a secondaryWWTPeffluent (Gupta et al.,

1981).ThecommonsourcesofBOD5areusuallyvegetation,litterandgarbage,and animal waste. Typical mean BOD5 concentrations of 12 mg/L wereobservedinstormwaterdischargesfromtheNationwideUrbanRunoffProgram(NURP)studies(USEPA,1983).

7.1.8Impactstothereceivingwaters

The impact of urban runoff pollutants on thewater quality of aquatic systemsdependson itspriorwaterqualityand the ratesatwhich thesepollutantsenterthesystem.Loadingsenteringinrelativelyshortperiodsoftime(shockloadings)usuallybringdramaticchangesinwaterquality(TsihrintzisandHamid,1997a).Thesechangesmaybecomepermanentifshockloadingsarefrequentduringtheyear.As these pollutants travel along a river, theymay settle down at variousdistances from theoriginalpointof introduction,andslowlystart affecting thelocalenvironment.Toxicsubstances thatdissolveslowly, i.e.heavymetalsandPCBs,usuallyshowsuchcharacteristics(Gjessingetal.,1984).Thevariouscompoundshavedifferent typesof impactson thewaterquality

of the aquatic system (Tsihrintzis and Hamid, 1997a). For example, leadbioaccumulates on the bottom and may retard fish growth and reducephotosynthesis.Atcertainconcentrations,zincandcopperaretoxictofishandaquatic micro-invertebrates. Cadmium and chromium have shown mutagenicandcarcinogeniceffects.Biochemicaloxygen–demandingmaterial candepletethe oxygen level in the aquatic ecosystem, resulting in decrease in the fishpopulation. Excess nutrients, such as nitrogen and phosphorus, cause algalbloomswhichblocksunlightandconsumeoxygenastheydecompose.Oilandgreasedisposedof fromvehicles is toxic toaquaticorganismsandaffects fishreproduction.Totalsuspendedsolids inwater increase turbidity level,affectingfishsurvival(PittandBozeman,1980;Ferrara,1986;Schueler,1987).

7.2URBANRUNOFFQUANTITYCOMPUTATIONS

Typicalmethodscommonlyused inurban runoffhydrologic computations andsewersizingaretherationalmethod,theSCScurvenumbermethodandtheunithydrographmethod.Theemphasiswillbegivenheretothefirsttwo,sincetheyarethemostcommonlyused.TheunithydrographmethodhasbeenpresentedinSection3.7.

7.2.1Rationalmethod

Therationalmethodisbasedonthefollowingformula:

(7.1)

whereQpistheflowpeakofthehydrograph(m3/s)CisthesurfacerunoffcoeffcientCfisthecorrectiontothesurfacerunoffcoeffcientaccountingforthedesignreturnperiod

Iistherainfallintensity(mm/h)Aisthesurfaceareaofthedrainagebasin(ha)(1/360)isaunitconversionfactor.

ThesurfacerunoffcoeffcientCisconceptuallydefnedastheratiooftotalrunoffover total rainfall for a given event. Therefore, it only assumes values less orequal to 1.0. The value ofC depends on various factors, such as the percentimperviousness of the watershed, the soil cover, the land use, the slope, theantecedent rainfall, the depression storage, the soil type, the soilmoisture, theshape of thewatershed and the rainfall intensity, among others.However, themain factors that determine the C-value in practice are the first three, i.e.imperviousness,coverandlanduse.Toguidethehydrologist,severalauthoritieshaveproducedtablescontainingtypicalvaluesofrunoffcoefficient(Table7.3);thesevaluesarevalidforstormeventsuptoareturnperiodof10years.TheCfcoefficient is used to correct the C values taken from such tables for lessfrequentstormevents.Itassumesthevalues1.00,1.10,1.20and1.25forthe1-to 10-, 25-, 50-and 100-years return periods, respectively. However, to havephysicalmeaning,theproductCf·Cshouldalwaysbelessorequalto1.0.

Table7.3TypicalvaluesofsurfacerunoffcoefficientC

SurfacerunoffcoeffcientC

Landusedescription Low High

Commercial

  Urbancentre–downtown 0.70 0.95

  Neighbourhoodareas 0.50 0.70

Residential

  Singlefamily 0.30 0.50

  Multiunits,detached 0.40 0.60

  Multiunits,detached 0.60 0.75

  Suburban 0.25 0.40

  Condominium,apartments 0.50 0.70

Industrial

  Light 0.50 0.80

  Heavy 0.60 0.90

  Parks,cemeteries 0.10 0.25

  Constructionsites,railroadyards 0.20 0.35

  Natural–undeveloped 0.10 0.30

Streets

  Asphalt,concrete 0.70 0.95

  Brick,stone 0.70 0.80

Roof

Lawns,sandysubsoil

  Flat,<2% 0.05 0.10

  Mildslope,2%–7% 0.10 0.15

  Steepslope,>7% 0.15 0.20

Lawns,compactedsubsoil

  Flat,<2% 0.13 0.17

  Mildslope,2%–7% 0.18 0.22

  Steepslope,>7% 0.25 0.35

Source:ASCE,Designandconstructionofsanitarysewers,ManualofPracticeNo.37,1970.

Note:ThevaluesofCarevalid for returnperiodsof less than10years.For larger returnperiods,Cfisgreaterthan1.0andisusedforcorrectionofC.

When using the rational formula, the rainfall intensity is obtained from idfcurves (seeSection 4.8),where the intensity is a function of both the rainfalldurationandthereturnperiod.TypicalsuchcurvesareshowninFigure7.3.Thedesigndurationisselectedasequaltothetimeofconcentrationattheparticularpointofthedrainagebasinwherethecalculationofthepeakdischargeismade.Thisassumptionresultsinmaximumintensityanditisadequateforcalculatingthepeakdischarge.However, itmaynotbeconservative fordesigningstoragestructures.Thetimeofconcentrationofanurbandrainagebasinisqualitativelydefined

asthetimeneededforadropofrainwatertomovefromthe‘hydraulically’mostremotepointofthebasintothepointofconcentration(i.e.thepointofinterestwhereapeakestimateisattempted).

1

Figure7.3 Typicalidfcurve.

Thistimecomprisesthreedistincttimes:thetimeofoverlandflow(To),thetimeofstreetflow(Ts)andthetimeofconduitflow,i.e. insewersoropenchannels(TL),asfollows:

(7.2)

Overlandflowisdefinedasflowonarelativelyflatplainsurface,characterizedbysmalldepthandlargewidth,andrelativelyparallelstreamlines.Inanaturalwatershed,itoccursattheuppermountaintopnearlyflatareas,wheregulliesandsmall channels havenot been formed. In anurban area, it occurs on rooftops,parking lots and similar flat paved or unpaved surfaces. The overland time ofconcentration To is calculated from empirical formulas developed for thispurpose (Gupta, 1989).Themost commonones are the following (in allTo iscomputedinhours):

ThefollowingistheKerby(1959)equation:

(7.3)

whereListhelengthofflowtravelpath(km),andtheequationisvalidforL<0.4kmHistheelevationdifferencebetweentheupperandlowerpointsalongthetravelpathofoverlandflow(m)δisthecoefficientofsoilcover,whichassumesthefollowingvalues:

δvalue Description

0.02 Smoothpavement

2

3

4

5

0.10 Barecompactedsoil

0.30 Roughbaresoilorsparsegrass

0.40 Mediumdensegrass

0.80 Densegrass,rowees

ThefollowingistheIzzard(1944)equation:

(7.4)

whereIistherainfallintensity(mm/h),andEquation7.4isvalidforIL<3.8Cistherunoffcoefficientεisthecoefficientofsoilcover,whichassumesthefollowingvalues:

εvalue Description

0.007 Smoothasphalt

0.012 Concretepavement

0.017 Tarandgravelpavement

0.046 Sparsegrass

0.060 Densegrass

ThefollowingistheBransbyWilliams(1922)equation:

(7.5)

whereAisthesurfaceareaoftheoverlandflowarea(km2).ThefollowingistheFederalAviationAgency(1970)equation:

(7.6)

The following is the Manning kinematic wave equation (Overton andMeadows,1976;Gupta,1989):

(7.7)

whereNistheManningroughnesscoefficientforoverlandflow.Valuesforthiscoefficient are presented in Table 7.4. As seen, these values are significantlyhighcomparedtorespectivevaluesforopenchannelflow(seefollowingTable7.5);S is the slope along the path of overland flow (m/m); andP2-year is the

depth of the rainfallwith duration 24 h and return period 2 years (mm).L inEquation7.7isexpressedin(m)andislimitedtolessthanabout90m.In the case when the equation used contains the rainfall intensity I (e.g.

Izzard’sequation,Equation7.4),forthecomputationofTo,onehastoalsousethe idf diagram (Figure 7.3) in a trial-and-error procedure, according to thefollowingsteps:(1)OneassumesavalueforTo;(2)fromtheidfdiagram,foragivenreturnperiod,andusingtheassumedTovalueforduration,onecangetavaluefortheintensityI;(3)usingthisIvalueinEquation7.4,onegetsavalueforTo;and(4)thisnewTovalueiscomparedtotheoneassumedinthefirststep,andifnotdifferent,onestops;otherwisetheprocedureisrepeatedfromthefirststep.

Table7.4ValuesofManningroughnesscoefficientNforoverlandflow

Nvalue

Condition Min Normal Max

Concrete 0.010 0.011 0.013

Asphalt 0.010 0.012 0.015

Baresand 0.010 0.010 0.016

Gravel 0.012 0.012 0.030

Bareclayloam 0.012 0.012 0.033

Turf 0.39 0.45 0.63

Shortgrass 0.10 0.15 0.20

Densegrass 0.17 0.24 0.30

Trees,forest 0.30 0.45 0.48

Source:Wanielista,M.etal.,Hydrology:WaterQuantityandQualityControl, 2ndedn., JohnWiley&Sons,NewYork,1997.

Table7.5ValuesofManningroughnesscoefficientnforconduitsandchannels

nvalue

Conduitmaterial Min Normal Max

Brass,smooth 0.009 0.010 0.013

Steel,welded 0.010 0.012 0.014

Steel,riveted 0.013 0.016 0.017

Castiron,coated 0.010 0.013 0.014

Castiron,uncoated 0.011 0.014 0.016

Wroughtiron,black 0.012 0.014 0.015

Wroughtiron,galvanized

0.013 0.016 0.017

Corrugatedmetal,smallcorrugations

0.020 0.022 0.025

Corrugatedmetal,largecorrugations

0.030 0.032 0.035

Smoothwall,spiralaluminium

0.010 0.012 0.014

Concretepipe,straight 0.010 0.012 0.013

Concretepipewithcurves

0.011 0.013 0.014

Stormsewer,straight 0.013 0.015 0.017

Sanitarysewer 0.012 0.013 0.016

Source:Wanielista,M.etal.,Hydrology:WaterQuantityandQualityControl, 2ndedn., JohnWiley&Sons,NewYork,1997.

TraveltimesonstreetTsandinconduitTLarecomputedusingtheManningequation:

(7.8)

or

(7.9)

Thesearecombinedwiththeequation

(7.10)

toget

(7.11)

Qpisthepeakdischargeflowingintheconduit(m3/s)Visthemeanvelocityinthestreetorconduit(m/s)Listhelengthoftravelinthestreetorconduit(m)Sistheslopeofthestreetorconduit(m/m)AL is theflowsectionarea in thestreetorconduitperpendicular to theflowdirection(m2)

RisthehydraulicradiusdefinedasthequotientoftheflowsectionareaoverthewettedperimeterP(=AL/P)(m)

The wetted perimeter is defined as the length of the perimeter of the streetsection or conduit section which is in contact with water; n is the Manningroughnesscoefficientforopenchannelflow.Valuesfornaretakenfromtablesasafunctionoftheconduitmaterial(Table7.5).TheuseofManningequation is straightforward,basedongeometry, for the

casesofnormalconduitsections,suchasrectangularandtrapezoidal.Figure7.4presents geometric definitions for these two sections. The needed geometryequations to apply Manning equation for these two cases are summarized inTable7.6,whereBisthewidth(m)ofthebaseofthesection,distheflowdepth(m)andz1andz2aretheslopes(i.e.ratioofhorizontaltoverticalunits)ofthesides of the trapezoidal section.The triangular section is a special case of thetrapezoidalsectionwhereB=0.Particularlyforthecaseofcircularpipes,thegeometryismorecumbersome

and the solution of Manning equation can be aided by the use of Table 7.7,whered(m)isthedepthofflowandD(m)isthepipediameter.Theuseofthistableisexplainedinthefollowing.TheManningEquation7.8canbewrittenasfollows:

(7.12)

Figure7.4 Commongeometriesforchannelsections.(a,b)Rectangularsectionand(c)trapezoidalsection.

Table7.6Geometryequationsuseful for thesolutionofManningequationfor rectangularandtrapezoidalchannelsections

Section FlowareaAL(m2) WettedperimeterP(m)

Rectangular–open(Figure7.4a) AL=Bd P=B+2d

Rectangular–closed(Figure7.4b) AL=Bd P=2B+2d

Trapezoidal(Figure7.4c)

Note:ParameterdefinitionsinFigure7.4.

When the flowQp, theManning roughness coefficient n, the slope S and thediameterD are known, then the right-hand side (RHS) of Equation 7.12 isknown, and thus, the left-hand side (LHS), i.e. parameter (ALR2/3)/D8/3, can becomputed.Table7.7canthenbeusedtocomputebothparametersd/DandR/D,fromwhichonecangettheflowdepthdandthehydraulicradiusR.Similarly,ifdorRisknown,thetablecanbeusedtocomputeanyunknownparameter.Finally, the Manning equation can be used in any channel section, for

example,anirregularsection(Figure7.5).Inthiscase,thecrosssectionshouldbe defined by the known coordinates (x,y) of the various points defining thesection.To applyManning equation, one has to create a tablewhere the flowareaAL(m2),wettedperimeterP(m),hydraulicradiusR(m)andflowQ(m3/s)are computed as function of the vertical coordinate y (m) assuming variousdepthsd(m)fromy=0uptotheupperlimitofthecrosssection(fullsection).This procedure creates a rating curve for the section, where the flow rateQ(m3/s) is related to various water depths d; then, the solution of Manningequation is straightforward through theuseof the tableor rating curve. In thecase of the cross section of Figure7.5, and assuming a slopeS = 0.0001 andManningroughnessn=0.050,onegetsthevaluesshowninTable7.8.In addition to the computation of the peak discharge, the rational method

providesthepossibilitytoderiveahydrograph,whichisrequiredwhenthereisneed to compute a pollutograph, or to design storage and/or pollution controlstructures.The rationalmethod hydrograph has a triangular shapewith a timebaseequalto2TcandaheightequaltoQp.

7.2.2SCSmethod

Thismethodwasdevelopedby theSoilConservationService (SCS;currently:NaturalResourcesConservation Service orNRCS) of theU.S.Department of

Agriculture (USDA-SCS, 1985; USDA-NRCS, 2015) and is one of the mostwidely used hydrological methods (Ponce and Hawkins, 1996). According tothismethod,surfacerunoff iscomputedfor fourhydrologicsoilcategories (A,B,CandD)basedontheirdrainageproperties(soilhydrologieclassification),asdescribedinTable7.9.Regarding texture, soils are divided in threemain categories: sand, silt and

clay.SCSprovidesamoredetailedclassificationin11categories,accordingtothepercentcontentinsand,siltandclayinthesoilsample,asshowninFigure7.6.Table7.10presents soil categoriesand their correspondence tohydrologiesoilclassificationandminimuminfiltrationrates.

Table7.7SolutionaidofManningequationforcircularpipes

Figure7.5 Typicalirregularopenchannelsection;(x,y)coordinatesarealsoshown.

Table7.8Computedflowsasafunctionofdepth(ratingcurve)for thecrosssectionofFigure7.5,forS=0.0001andn=0.050

Depth Flowarea Wettedperimeter Hydraulicradius Discharge

d(m) AL(m2) P(m) R(m) Q(m3/s)

0.0 0.000 0.000 0.000 0.000

0.5 1.875 7.575 0.248 0.148

1.0 6.667 11.865 0.562 0.908

1.5 13.542 16.155 0.838 2.408

2.0 22.500 20.445 1.101 4.797

2.5 37.500 35.495 1.056 7.780

3.0 55.500 37.731 1.471 14.357

3.5 74.500 39.967 1.864 22.568

4.0 94.500 42.203 2.239 32.348

4.5 115.500 44.439 2.599 43.667

5.0 137.500 46.675 2.946 56.513

Table7.9HydrologicsoilclassificationaccordingtoSCS

Category Description

A Soilscharacterizedbylowpossibilitytoproducesurfacerunoffandhighpermeability.Mostlysandwithgravelandverylowcontentsoffinesand

B Soilscharacterizedbyadequatelylowpossibilitytoproducesurfacerunoffandabovemeanpermeability.MostlysandysoilsbutfinerthanthoseofcategoryA

C Soilscharacterizedbyadequatelyhighpossibilitytoproducesurfacerunoffandbelowmeanpermeability.Soilswithsignificantclaycontent

D Soilscharacterizedbysignificantlyhighpossibilitytoproducesurfacerunoffandbelowmeanpermeability.Almostimpermeableclaysoils

Source:U.S.DepartmentofAgriculture,SoilConservationService(USDA-SCS),NationalEngineeringHandbook,Section4,Hydrology,USDA-SCS,Washington,DC,1985;U.S.DepartmentofAgriculture,NaturalResourcesConservationService(USDA-NRCS),NationalEngineeringHandbook,2012,http://w-ww.nrcs.usda.gov/wps/portal/nrcs/detailfull/national/water/?cid=stelprdb1043063.

A combination of the SCS method (USDA-SCS, 1985) is now presentedcombinedwiththeSantaBarbaraUnitHydrograph(SBUH)method(Stubchaer,1975),aspresentedbyTsihrintzisandSidan(1998).TheSCSequationcomputesthe cumulative surface runoff depth as a function of time during the rainfallevent,accordingtothefollowingequation:

(7.13)

Figure 7.6 SCS soil classification. (FromU.S. Department of Agriculture (USDA), Soil surveymanual#18,U.S.DepartmentofAgriculture(USDA),Washington,DC,1951.)

Table7.10SCSsoiltextureclassification

Texturecategory Minimuminfiltrationrate(mm/h)

SCShydrologicsoilcategory

Sand     210.1      A

Loamysand     61.2      A

Sandyloam     25.9      B

Loam     13.2      B

Siltyloam      6.9      C

Sandyclayloam      4.3      C

Clayloam      2.3      D

Siltyclayloam      1.5      D

Sandyclay      1.3      D

Siltyclay      1.0      D

Clay      0.5      D

Source:Wanielista,M.etal.,Hydrology:WaterQuantityandQualityControl, 2ndedn., JohnWiley&Sons,NewYork,1997.

and

(7.14)

where∑R(t)isthecumulativesurfacerunoffdepthasafunctionoftimet fromthebeginningoftherainfall(mm)

∑P(t) is the cumulative rainfall depth as a function of time, i.e. the rainfallmasscurve(mm)

Sistheultimatestoragecapacityormaximumretentionofthewatershedandthesoil(mm)

The termIa = λS represents the initial abstraction, i.e. interception, depressionstorage and infiltration occurring before the beginning of runoff (mm), andparameterλistheratiooftheinitialabstractionIatothemaximumretentionS,i.e. the portion of the total loss that is due to initial abstraction. Since thenumerator of Equation 7.13 is always positive, this equation would predict arunoffdepthevenwhentherainfalldepthislessthantheinitialabstraction,i.e.∑P(t)≤λS.Equation7.14thenstatesthatthishasnophysicalmeaning,i.e.thereisnorunoffwhentherainfalldepthislessthantheinitialabstraction.Atypicalvalueforparameterλis0.2(USDA-SCS,1985).Thisvalueresulted

frommeasureddataofIaandSinexperimentalwatersheds(USDA-SCS,1985).However, when plotted, this empirical data showed significant spreading, andactually,50%ofthedatapointsfellwithintherange0.095≤λ≤0.38.Severalother studies (e.g.McKurk et al., 1980; Cazier andHawkins, 1984; Bosznay,1989;PonceandHawkins,1996)haveshownthatλcanactuallyvarybetween0and0.3.The maximum retention S in Equation 7.13 is expressed by the following

empiricalequation:

(7.15)

whereSinmmandCNisthecurvenumber,anempiricalparameterthatdependsonthesoiltype,thelanduse,thesurfaceconditionandtheantecedentmoisturecondition(USDA-SCS,1985;Wanielista,1990;PonceandHawkins,1996).CNvariesbetween0(foratheoreticalinfinitelyabstractingwatershed)and100(foran impermeable watershed). Values for this parameter are provided in Tables7.11through7.14.

Table7.11CNvaluesforurbanlandusesa

Table7.12CNvaluesforagriculturalwatershedsa

Tables 7.11 through 7.13 are used to select values that closely describe theconditions of the watershed. Table 7.14 is used to adjust the selected valuesaccordingto theantecedentmoistureconditions(AMCs)whicharedeterminedaccordingtotheguidelineonthefootnoteofthetable.NormalconditionisAMCII,withAMCIbeingthedryconditionandAMCIIIthewetcase.Equations 7.13 and 7.15 can be combined into the following equation to

computerunoff:

(7.16)

Table7.13CNvaluesforpermeablesurfacesinurbanareas

Hydrologicsoilgroup

Description A B C D

Bareground 77 86 91 94

Gardensorrowcrop 72 81 88 91

Grassingoodcondition(coverexceeding75%oftheperviousarea) 39 61 74 80

Grassinfaircondition(cover50%–75%oftheperviousarea) 49 69 79 84

Grassinpoorcondition(coverlessthan50%oftheperviousarea) 68 79 86 89

Woods 36 60 73 79

Source:U.S.DepartmentofAgriculture,SoilConservationService(USDA-SCS),NationalEngineeringHandbook,Section4,Hydrology,USDA-SCS,Washington,DC,1972.

Table7.14AdjustmentofCNvalueaccordingtoantecedentsoilmoistureconditions

Antecedentmoisturecondition(AMC)

AMCII(normal)   AMCI(dry)   AMCIII(wet)100    100    100

95    87    98

90    78    96

85    70    94

80    63    91

75    57    88

70    51    85

65    45    82

60    40    78

55    35    74

50    31    70

45    26    65

40    22    60

35    18    55

30    15    50

Source:U.S.DepartmentofAgriculture,SoilConservationService(USDA-SCS),NationalEngineeringHandbook,Section4,Hydrology,Washington,DC,1972.

Note:AMC I:Relativelydry soils, rainfall in thepreceding5days<12.5mm.AMC II:Normal case,rainfall in thepreceding5daysbetween12.5and38mm.AMCIII:Relativelywetsoils, rainfall in thepreceding5days>38mm.

and

(7.17)

This equation shows that runoff is a function of both CN and λ Theinstantaneous(incremental)runoffdepthR(t)(mm)atanytimet(i.e.foratimeperiodΔt) canbe computed from the cumulativeone (Equations7.13, 7.14 or7.16,7.17)using

(7.18)

and the instantaneous discharge (hydrograph) Qt(t) (m3/s) is given by thefollowingequation:

(7.19)

whereAisthewatershedsurfacearea(ha)Δtisthetimestepofthehyetograph(h)

The final hydrograph is produced after routing the instantaneous hydrographthrough the drainage basin. One appropriate method (Tsihrintzis and Sidan,1998)istheSBUHmethodwhichwasfirstpresentedbyStubchaer(1975).ItispresentedhereinamodifiedformwheretheSCSmethodisusedtocomputetherunoff depth, as presented in Equations 7.13 through 7.19. The instantaneoushydrographroutingisdoneusingthefollowingequation(Stubchaer,1975):

(7.20)

whereQf(t)istheroutedhydrographthroughtheurbanwatershed(m3/s)Kistheroutingconstant,which,accordingtoStubchaer(1975),isdefinedas

(7.21)

whereTc is the time of concentration defined using themethods presented inprevioussections.

7.3UrbanRunoffQualityComputations

7.3.1USEPAmethodforthepredictionofannualunitpollutantloadings

Severalauthorshaveproducedannualunitloadingsofpollutantsperunitareaofthe urban watershed. Marsalek (1978) provides values for BOD, nitrogen,phosphorus suspended solids and seven metals for urban areas with bothseparateandcombinedsewers,andforfourlanduses,leadingtoverylow,low,medium or high pollutant production rates. Sonzogni et al. (1980) provideannualunitloadingsforsuspendedsolids,nitrogenphosphorusandfourmetalsforfourspecificlandusesandonegeneralurbanlanduse.Similar work was presented by Heaney et al. (1977) for separate and

combinedsewersand isusedbyUSEPA(1979) inamethod tocomputemeanannual pollutant loadings in surface runoff. The method uses the followingequation:

(7.22)

wheremxistheannualunitloadingofpollutantX(kg/ha/year)βxisthepollutantXloadingcoefficient,takenfromTable7.15Pyearisthemeanannualrainfalldepth(mm)φisthepopulationdensitycoefficient(Equations7.23through7.25)σisthemechanicalstreetsweepingcoefficient(Equations7.26and7.27)

The population density coefficient is given by the following equations. Forresidentialareas,

(7.23)

wherePd is the population density (capita/ha). For commercial/industrial landuses,

(7.24)

andforotherurbanareas(e.g.parks,cemeteries,educationallanduses),(7.25)

The mechanical street sweeping coefficient σ depends on the frequency ofsweeping,i.e.thetimeintervalTs(days)betweenstreetsweepings.ForTs>20days,

(7.26)

andforTs≤20days,

(7.27)

Table7.15Loadingcoefficientβx(Equation7.22)

Source:U.S.EnvironmentalProtectionAgency(USEPA),1978NeedsSurvey–ContinuousStormwaterPollutionSimulationSystem,User’sManual,EPA430/9-79-004,U.S.EnvironmentalProtectionAgency,Washington,DC,1979.

In addition to the mean annual unit loading, the method can also predict themean annual concentrationCx of pollutantX. To do so, one needs the meanannual runoff depth, for which Heaney et al. (1977) propose the followingequation:

(7.28)

whereRyearisthemeanannualsurfacerunoffdepth(mm)Iisthepercentimperviousnessofthearea(%)Ds is the depression storage (mm), which can be estimated using knownmethodsor,alternatively,thefollowingequation:

(7.29)

7.3.2USGSmethodforthepredictionofmeanannualpollutantquantity

TheUSGSmethodisbasedonthedevelopmentofregressionequationsfromthestatisticalanalysisof2813rainfalleventsin173stationsin30urbanareasintheUnitedStates(DriverandTasker,1990),resultinginthefollowingequation:

(7.30)

whereMxistheannualquantityofpollutantX(kg/year)Nisthemeanannualnumberofrainfallsevents(arainfalleventisdefinedastheonehavingaminimumdepthof1.3mm;eventsareseparatedbya6hperiodofnorain);ce,a,b,c,d,e,fcoefficientsdependingonthepollutanttype(takenfromTable7.16)

Aisthesurfaceareaoftheurbanwatershed(km2)Iisthepercentimperviousness(%)Pyearisthemeanannualrainfalldepth(mm)TisthemeanminimumtemperatureofJanuary(°C)y is the landusecoefficient (y=1when thecommercialand industrial landusescovermorethan75%oftheurbanwatershed;y=0foranyothercase)

Table7.17providesupperandlowerlimitsforparametersA,I,PyearandT;themethodisapplicablewhentheseparametersarewithintheselimits.

7.3.3SimplemethodofWashington,DC

ThismethodwaspresentedbySchueler(1987)inthemanualofmethodsforthemanagement of surface runoff and pollution control ofWashington, DC. Themethodcalculatesthequantityofapollutantforayearoracertaintimeperiod,accordingtothefollowingequation:

(7.31)

Table7.16Valuesofcoefficientsa,b,c,d,e,fandceofEquation7.30

Source: Driver, N.E. and Tasker, G.D., Techniques for estimation of storm-runoff loads, volumes, andselectedconstituentconcentrationsinurbanwatershedsintheUnitedStates,U.S.GeologicalSurveyWaterSupplyPaper2363,U.S.DepartmentofInterior,Washington,DC,1990.

Table7.17UpperandlowerlimitsofparametersA,I,PyearandT forwhichEquation7.30 isvalid

Pollutant A(km2) 1(%) Pyear(mm) T(°C)

COD 0.049–1.831 4–100 212.9–1574.8 −16to14.8

Suspendedsolids 0.049–1.831 4–100 212.9–1254.3 −16to10.1

Dissolvedsolids 0.052–1.166 19–99  260.1–955.3 −11.4to2.1

Totalnitrogen 0.049–2.150 4–100 300.5–1574.8 −16to14.8

Ammonia 0.049–1.831 4–100 212.9–1574.8 −16to14.8

Totalphosphorus 0.049–2.150 4–100 212.9–1574.8 −16to14.8

Dissolvedphosphorus

0.052–1.831 5–99 212.9–1173.0 −11.8to2.1

Cu 0.036–2.150 6–99 212.9–1574.8 −9.3to14.8

Pd 0.049–2.150 4–100 212.9–1574.8 −16to14.8

Zn 0.049–2.150 13–100 212.9–1574.8 −11.4to14.8

Source: Driver, N.E. and Tasker, G.D., Techniques for estimation of storm-runoff loads, volumes, andselectedconstituentconcentrationsinurbanwatershedsintheUnitedStates,U.S.GeologicalSurveyWaterSupplyPaper2363,U.S.DepartmentofInterior,Washington,DC,1990.

whereMxisthequantityofpollutantXforacertaintimeperiod(kg)Pistherainfalldepthforthegiventimeperiod(mm)(whenthecalculationisforthemeanannualpollutantquantity,thenthemeanannualrainfalldepthPyearisused)

p is theannualpercentageofrainfallevents thatresult insurfacerunoff(%)

(inthecasewhenthecalculationisforasinglerainfallevent,thisparameterassumesthevalue1.0)

Cxisthemeanpollutantconcentrationinsurfacerunoff(mg/L),whichresulteither from measurements in the area or by using Table 7.18, whichcontainstypicalvalues

Aisthesurfaceareaoftheurbanwatershed(ha)(themethodisvalidforareaslessthan260ha)

C is thesurfacerunoffcoefficient for thestudyarea,which,aspresented, istakenfromtables(e.g.Table7.3)orcomputedusingthefollowingequationasafunctionoftheimperviousnessI(%)oftheurbanwatershed:

(7.32)

Table7.18Typicalpollutantconcentrationinurbanareas

Source:Schueler,T.R.,ControllingUrbanRunoff:APracticalManualforPlanningandDesigningUrbanBMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.

Figure7.7 Sedimenteventmeanconcentrationinmg/Lasafunctionoftheurbanareasizeinacres(1acre= 0.4047 ha). (From Schueler, T.R., Controlling Urban Runoff: A Practical Manual forPlanning and Designing Urban BMPs, Washington MetropolitanWater Resources PlanningBoard,Washington,DC,July1987.)

Table7.19CriteriaforusingeachcurveinFigure7.7(Schueler,1987)

Sedimenteventmeanconcentration(EMC)

Criterion Low Moderate High

Channelstabilityconditioninerosion

Vegetatedswalesorsewers

Intermediatesituation Openchannels,cutbanksalternatingwithchannelsandbars,fallentrees

Capacityofsedimentstorageinthechannel

Smalldepositsinstormdrains,stabilizedlanduse

Intermediatesituation Largesiltorclaydeposits,evidenceofrecentorongoingconstruction,waterbecomesmurkyafterdisturbingbottom

Flowvelocityinthestream

Lowslope,lowimperviousness

Intermediatesituation Highslope,highwatershedimperviousness

The concentration of suspended solids in surface runoff cannot be computedusing Equation 7.31 (Schueler, 1987). The following method is used instead,whichhasresultedfromstatisticalanalysesofdatafrom25urbanwatershedsinthe area ofWashington,DC,which varied in area from2 ha to 405km2. The

meanconcentrationsof suspended solids for each rainfall eventwere found inthesewatershedsandthegraphinFigure7.7wascreated,whichcontainsthreecurves (for low,medium and high suspended solids concentration) and showsthat themean concentration depends onwatershed size (Schueler, 1987). Theselection of curve in Figure 7.7 is in accordance with the description of thedrainagesystemconditioninTable7.19.

7.3.4Pollutantaccumulationonstreetsurface

In the previous sections, methods were presented to compute mean pollutantconcentrations in runoff.Since the time intervalof computationwasannualoreventbased,anassumptionwasmadethatpollutantproductionfollowsalineartrend.However,studieshaveshownthatpollutantproductionandaccumulationon the street surface is not a linear function. Actually, it has been found thatpollutantaccumulationreachesanequilibriumquantitywithtime,whichneitherincreasesnordecreases,unlessitrainsorthestreet ismechanicallyswept.Theexplanation of the equilibrium is that the rate of accumulation reaches after atimetherateofremovalbecauseofwindandtraffictransport.Variouspollutantaccumulation equations have been proposed. For example, the following areused in the USEPA Storm Water Management Model (SWMM) (Huber andDickinson,1988):Linear(b=1)orhyperbolic(b≠1)

(7.33)

Exponential

(7.34)

Michaelis-Menton

(7.35)

whereM(t)isthemassofpollutantonthestreetsurfaceattimeta,bareempiricalcoefficients,whicharesitespecific

ToapplyEquations7.33through7.35,onehastoknowthevaluesofcoefficientsaandb.Thesecanbederivedataspecificsitebycurvefittingoncollecteddatawith timeor throughcalibrationof a stormwaterqualitymodel that uses them(TsihrintzisandHamid,1997a,b,1998).AlternativelytoEquations7.33 through7.35, thefollowingmethodcanalso

be used to predict pollutant accumulation. It is based on mass conservation(USEPA, 1979; Novotny and Olem, 1994). During any time interval Δt, thechange of mass of a pollutant on the street surface is equal to the mass ofpollutant that enters the areaminus themass that escapes from the area. Thepollutants that enter the area come from (1) solids, tree leaves, etc.; (2) dry(without rain)deposition (settlement)ofpollutantsandsolidssuspended in theatmosphere; and (3) deposition from traffic as a result of pollutants fromautomobile emissions andproductsof frictionbetween thevariousmechanicalparts of automobiles. Pollutants escape from the area as a result of turbulencecreated by traffic and wind. Based on these, the following equation wasproposedbyUSEPA(1977):

(7.36)

whereM(t)isthepollutantquantityattimet(kg/ha)M(t−Δt)isthepollutantquantityattimet−Δt(kg/ha)t and Δt are the time and time interval (Δt is usually taken as 1 day),respectively

misthemeanrateofpollutantaccumulationduringΔt(kg/ha/timeintervalΔt)ξisthepercentofpollutantescapeduringthetimeintervalΔt(%)

Whenξ=0,thereisnopollutantescape,theequationislinearandaccumulationisunlimited.Whenξ≠0,theequationisnon-linear.USEPA(1979)proposestoseparateparameterξ into twocomponents:escapedue tonaturalandchemicalreasonsandescapedue tohumanmanagementpractices(e.g.streetsweeping).Therefore,wegetthefollowingequation:

(7.37)

whereξisthetotalpercentofpollutantescape(%)ξd is thepercentofpollutantescapeduetonaturalandchemicalphenomena(%),withvalues for typicalpollutants6.67%forBODandTKN,and0.0forsuspendedsolidsandlead

ξristhepercentofpollutantescapeduetohumanmanagementpractices(%)

Thisparametercanbeusedtoquantifytheeffectofthesepracticesonpollutantaccumulation. In thecaseof street sweeping, ξr represents theeffectivenessofsweepinginpollutantremoval(whichisdiscussedinthefollowingparagraph).

NovotnyandOlem(1994)presentthefollowingequationforξd:

(7.38)

whereξd,dayisdailypercentescapeofdustandsolids(%)whichusuallyrangesbetween20%and40%.ForΔt<1day,aproportionofξd,day isused;H is thecurbheight(cm),usuallyrangingbetween15and18cm;V is thetrafficspeed(km/h);andWisthewindspeed(km/h).Theparameterm inEquation7.36 iseithercalibratedwithmeasurements in

the urban watershed or it is calculated based on the methods presented inprevious sections (73.1 through 73.3). The preferred method is the USEPAmethod (Heaney et al., 1977), presented in Section 7.3.1, but any methodprovidingmeanannualpollutantquantitiescanbeused,asfollows:

(7.39)

whereΔtisthetimeinterval(days)mxistheannualunitloadingofpollutantX(kg/ha/year)

7.3.5Washoffofaccumulatedpollutants

Washoff of accumulated pollutants by surface runoff can be expressed byequations similar to Equations 7.33 through 7.35. Mostly an exponentialequation is used, something based on the hypothesis thatwashoff is a processthatfollowsfirst-orderkineticswithtime,describedbythefollowingdifferentialequation(NovotnyandOlem,1994):

(7.40)

whereM(t)isthequantityofpollutantremainingonthestreetsurfaceasafunctionoftimet(kg)

kisthewashoffcoefficient(mm−1)risthesurfacerunoffrateorintensity(mm/h)

ThesolutionofEquation7.40isoftheform

(7.41)

whereM(t) andM(t − Δt) are the pollutant masses remaining on the street

surfaceattimes tand(t−Δt) (kg).FromEquation7.41, thepollutantquantityΔM(t)thatiswashedoffduringthetimeintervalΔtcanbecalculatedasfollows:

(7.42)

whereΔRistheincrementalsurfacerunoffdepthforthetimeintervalΔt (mm).The Hydrologic Engineering Center of the U.S. Army (USCOE, 1977) foundthatinthecaseofsolids,notallofthemareavailableforwashoffforallrunoffdepths,basedonEquation7.42.This is due to differences in size, texture andspecificweight.TocorrectEquation7.42,theyproposedtouseacoefficientofavailabilityγontheRHSpartofthisequation.Then,Equation7.42becomes

(7.43)

where

(7.44)

and

(7.45)

Whenusingthecoefficientofavailabilityγ,Equation7.41becomes

(7.46)

NovotnyandOlem(1994)hascalculatedthevalueofthewashoffcoefficientkas 0.19 mm−1 where r and t are expressed in mm/h and h, respectively.According to USEPA (1979), the washoff coefficient k is a function of bothwatershedimperviousnessandslope,andvariesfrom0.06mm−1forflatpervioussurfacesto0.18mm−1forinclinedimpervioussurfaces.USEPA(1979)providesa graph to compute the appropriate value of k for watersheds served withseparatestormsewers.Sincemostpollutantsareproducedandwashedofffromimperviousinclinedsurfaces,theproposedvaluebyNovotnyandOlem(1994)(0.19mm−1)isnearlythesameasthemaximumvaluegivenbyUSEPA(1979)(0.18mm−1).Ithastobenoticedthathighervaluesshouldbeusedforareaswithcombinedsewers.ThevolumeΔVofsurfacerunoffduringthetimeintervalΔtcanbecomputed

fromthefollowingequation:(7.47)

whereΔVisthesurfacerunoffvolume(m3)Aisthesurfaceareaofthewatershed(ha)10isaunitconversionfactor

Therefore, theconcentrationofpollutantXoriginatingfromwashoff insurfacerunoffcanbecomputedfromthefollowingequation:

(7.48)

whereCx(t)istheconcentrationofpollutantX(mg/L)ΔMx(t)resultsfromEquations7.42and7.43

Equations 7.42 through 7.48 can be used to produce a loadograph and apollutographforthewashed-offpollutants.

7.4SurfaceRunoffQuantityAndQualityManagement

7.4.1General

Becauseofitsnature,themanagementofnonpointsourcepollutionfromurbanareas,ingeneral,doesnotfollowthetraditionalmethodsusedinthetreatmentofpoint source pollution. Several methods have been developed, with somerequiring construction (structural methods) and others not (non-structuralmethods). These methods are called best management practices’ (BMPs),becausethegoalisnottoachievetreatmentsothattheeffluentisbelowacertainconcentrationfor thepollutants(somethingimpossiblesince theconcentrationsvary with time, magnitude of rainfall event, antecedent dry days and severalother factors), but to achieve retention of pollutants to the maximum extentpossible. This implies minimization or elimination of pollutants entering theaquaticsystem.BMPscanbedescribedasmeasuresthatslow,retainand/orabsorbpollutants

producedfromnonpointsourcesandassociatedwithsurfacerunoff(Mandelker,1989). BMP selection criteria depend on climatic, geographic and economicfactors. Beale (1992) pointed out three main options to be considered in thecontrolofurbanrunoff:toreduceflowsenteringthedrainagesystem,toincreasethe capacity of the drainage system and to attenuate the flow entering thedrainagesystem.AccordingtoSchueleretal.(1992a),aneffectiveBMPsystemdesign comprises six basic components, namely runoff attenuation, runoffconveyance, runoff pre-treatment, runoff treatment, system maintenance andsecondaryimpactmitigation.

Runoffattenuation:MostBMPsindirectlycontrolpollutionbyminimizingthevolumeofrunoff.Therunoffvolumecanbeprimarilycontainedbyminimizingeithertotalbasinimperviousnessorthe‘hydraulicallyconnectedimperviousarea’(HCIA)ofthewatershed,anddelineatingandprotectingtheenvironmentalreserveareas.RunoffattenuationservestoenhancetheperformanceoftheremainingcomponentsoftheBMPsystem.TheHCIAofthewatershedistheonethatdirectlydrainsintothedrainagesystemandisnotisolatedfromit.ThisisdepictedinFigure7.8.ThestreetsurfaceandareasdirectlydraininginthestreetformtheHCIA.RoofssurroundedbyimperviousareasarepartofthetotalimperviousareaofthewatershedbutnotoftheHCIA.

Runoffconveyance:RunoffshouldbesafelytransportedtotheBMP.Swales,exfiltrationdrainsandparallelpipesystemsaresomeofthesemethods.

Runoffpre-treatment:Toassurelongevityandproperoperation,someBMPsrequirepre-treatment.Forexample,insomeBMPs,coarsesedimentsshouldbecapturedortrappedbeforeenteringtheBMPtopreservestoragevolumeand/orpreventclogging.Examplesofsomepre-treatmentmethodsincludestillingbasins,grassfilterstripsandfilterclothbarriersforinfiltrationsystems.Forindustrialortransportationapplications,waterqualityinletsorsettlingbasinsshouldbeusedtoremoveoilywastespriortoenteringintotheBMPsystem.

Runofftreatment:ThisistheheartoftheBMPsystem.Ingeneral,therearefourbasictreatmentoptionsavailablewhichincludefiltration,detention,retentionandinfiltration.AneffectiveBMPsystemmayusetwo,threeorevenallfouroftheseoptions.

Systemmaintenance:Itisnecessarytomaintainthelong-termperformanceoftheBMPsystem.Sincethesystemaccumulatessignificantquantitiesofsedimentandpossiblytoxicpollutants,thedisposalorcontainmentoftheseresidualsshouldbecarefullyplanned.ThedesignoftheBMPmustincludeeasyandpermanentaccess,aswellasenvironmentallyandeconomicallysoundremovalprocedures.

Secondaryimpactmitigation:ThisdetermineswhethertheBMP

systemcreatesanysecondaryimpactsonthedownstreamcommunity.Forexample,secondaryimpactsfrompondsincludethedischargeofhypoxicand/orthermallyenrichedwaterdownstream,theremovalofdownstreamripariancoverandpossiblyfillingoralterationofwetlandareasadjacenttothepond.Forinfiltrationsystems,theyincludepossibleriskofgroundwatercontaminationandsystemfailure.

Figure7.8 Definitionofhydraulicallyconnectedimperviousarea.

BMPs can be either structural or non-structural. The first kind requires theconstructionofastructurewheresurfacerunoffiscollected,treatedorretained.Thenon-structuralBMPsincludepracticesandpolicieswhichreducepollutionusuallyat the source,e.g. environmental education, reductionof fertilizeduse,mechanical street sweeping, regular maintenance of the sewer system andreductionofsaltusefordeicing.Toeffectivelyreducenonpointsourcepollution,thestructuralBMPsarecombinedwithnon-structuralones,andinmostcases,acombinationofvariousBMPsisusedinanurbanwatershed.SeveralgovernmentagenciesintheUnitedStatesandvariousauthorsprovide

manuals and guidelines for effectiveBMPdescription, design, installation andmaintenance. These include, among others, the Maryland Department of theEnvironment (1984,1987);MarylandDepartmentofNaturalResources (1985,1987);Schueler(1987);CityofSeattle(1989);KingCounty(1990);MinnesotaPollution Control Agency (1992); Urbonas and Roesner (1986); Roesner,UrbonasandSonnen(1988);Torno(1989);StahreandUrbonas(1990);Novotnyand Olem (1994); Metropolitan Washington Council of Governments (1992);Tsihrintzis and Hamid (1997a). Table 7.20 presents main BMPs by dividingthemintofourmaincategoriesandvarioussub-categories.

Table7.20VariouscategoriesandtypesofBMPs

No. Category Description

1 Pollutioncontrolatthesource

Environmentaleducationandawareness

2 Reductionofpollutantsfromatmosphericscrubbing(drydeposition)

Reductionofpollutantsendingtotheatmosphere

3 Streetsandimpermeablesurfaces

Streetandimpermeablesurfaceflushing

4 Streetsweeping

5 Permeablesurfaces Erosioncontrolmeasures

6 Coveringwithgrass,mulchandspecialmeshes

7 Controlofuseofchemicals(e.g.fertilizers,pesticides)

8 Hydrologicalmodificationofthesurfacedrainagesystem(increaseofpermeability,increaseofstorage,reductionofimpermeability)

Infiltration Porouspavements

9 Infiltrationtrenches

10 Infiltrationinlets

11 Increaseofsurfaceretention

Retentionontheroof

12 Rainwaterharvesting

13 Reductionofhydraulicallyconnectedimperviousareas

Grassfilters

14 Environmentalcorridors,protectionzonesandvegetationdevelopment

Grassswales

15 Pollutioncontrolinthesewerandthedrainageditches

Beforetheentrancetothesewer

Oil/greaseseparators

16 Waterqualityinlets

17 Grids

18 Inthesewerorchannel Channelstabilizationstructures

19 Vegetatedcanals

20 Riprap,gabionsandothermethodsofchannelprotection

21 Storage,retention/detentioninthestormsewer

22 Stormsewerwashing

23 Controlofillicitsanitarysewerconnections

24 Pollutioncontrolwithsurfacedetention/retentionandstorage

Floodcontrol–drybasins

Detentionbasins Inseries

25 Inparallel

26 Waterqualitycontrol Basinforinfiltrationoffirstflush(inparallelwiththestormsewer)

27 Extendeddrybasin

28 Wetbasin

29 Wetlands

30 Detentionbasinsforcombinedsewer Inseries

31 Inparallel

32 Insidetheaquaticsystem

33 Runofftreatment

Source: Adapted from Novotny, V. and Olem, H., Water Quality – Prevention, Identification, andManagementofDiffusePollution,VanNostrandReinhold,NewYork,1994.

7.4.2Pollutioncontrolatthesource

The control of the source of pollution, i.e. the reduction or minimization orelimination of pollutant production, is the most important action towards thereductionofurbannonpoint sourcepollution.TheBMPs in this category (No.1ȓ14;Table7.20)aremostlyapplicable in thecaseof separate sewer systems;theirapplicabilityincombinedsewersislimited(NovotnyandOlem,1994).InthismainBMPcategory,asTable7.20shows,therearefivemainsub-categoriesofBMPsandseveralothersub-categories.BMP1–Environmental educationandawareness: It is avery importantnon-structuralBMP.Itmaycomprise,amongothers,solidwastecontrolprogrammeson the streets, public environmental awareness and education programmes,presentations in schools by experts, organization of environmental excursionsand cleaning campaigns of aquatic systems, advertisements in mass media,productionanddistributionofpamphlets,installationofwarningsignsatstormsewer inlets, promotion of programmes for used oil recycling, promotion ofregularmaintenance of cars, promotion of use of public transportationmedia,invitationofthepopulationtoparticipateinenvironmentalprogrammesthroughecologicalorganizationsandapplicationofapolicyimposingfines.BMP2–Reductionofpollutants fromatmosphericscrubbing(drydeposition):This implies the control of atmospheric pollution resulting mostly from

emissionsfromtheindustryandthetraffic.Examplesincludetheuseofnaturalgasinsteadofotherfossilfuelsintheindustryandtheuseofrenewableenergysources.BMP3–Streetandimpermeablesurfaceflushing:ThisBMPandthefollowingone aremostly used for aesthetic reasons, i.e. to remove solids and dust fromimpervious surfaces. Street flushing is more common in Europe while streetsweepingismorecommonintheUnitedStatesandisnowexpandinginEurope.Fromtheaestheticpointofview,thereisnopreferredmethod.However,ifoneconsidersthefactthatsomepollutantsarealsowashedoffduringflushing,thispracticeshouldbeappliedonlyincombinedsewersystems,whenpollutantsaretransported to theWWTP and not directly to the receiving waters, impactingwaterquality(NovotnyandOlem,1994).Forstreetflushing,specialvehiclesareused,equippedwithawatertankandspecialorifices.BMP4–Streetsweeping:Streetsweepingismoreappropriateforseparatesewersystemsbecauseaccumulatedpollutantsareremovedbeforebeingwashedoffbyrain (Novotny and Olem, 1994). Therefore, they are not discharged into thereceivingwaters.Forstreet sweeping, specialvehiclesareused.Thereare twotypes,oneequippedwith tworotatingbrushes (mechanical type)and theotherwith brushes and vacuum (vacuum type). The second type is generally moreeffective; however, the effectiveness is usually less than 50% in all cases,particularly for fine particles of diameter less than 3.2 mm. Other factorsaffectingremovalisthespeedofthevehicle,thenumberoftimesitpassesfromthe area, the size of the particle and the content of particles per street length(Clark and Cobbins, 1963; Sartor and Boyd, 1972; Pitt, 1979; Novotny andOlem,1994).BMP5–Erosioncontrolmeasures:BMPsinthefourthsub-categoryofcontrolof pollution at the source comprise pollutants from permeable surfaces.Mainpollutants are transported sediments, a result of erosion, but also chemicalsubstances adsorbed on sediments. It has to be emphasized that erosion is aprocess that may produce significant amount of pollutants from bare soils inurban areas, particularly when construction activities take place, and needsattention.Thefollowingmethodisoftenusedinurbanareas tocontrolerosionfromsmallsurfaces.BMP 6 – Covering with grass, mulch and special meshes: Erosion issignificantly reducedwhen the bare soil is seededwith grass or coveredwithmulch. Mulch results from tree trimming and branch cutting, straw or fromvegetation remnants fromvariousagricultural activities.Whenappliedonbaresoil,mulchmay reduce raindrop impact, increase roughnessand reduce runoff

velocity,increasedepressionstorageandreducerunoffvolumeandincreasesoilmoisture, enhancing vegetation growth. It is effective in reducing erosion onrelatively mild slopes up to 15%, while its ability is enhanced when usedtogetherwithvariousmetal,plasticorpapermeshesornetsorgrassseeding.BMP7–Controlofuseofchemicals:Underthismeasure,fertilizerandpesticideuse in lawns and parks in the urban landscape should be reduced to theabsolutely minimum. The advice of an expert on the minimum chemicalrequirementsisnecessary.Inaddition,alternativestochemicalsshouldbeused,such as the use of compost, reuse of wastewater for irrigation and use ofwastewatersludge.All theseofcourseshouldbeusedwithcaution.Finally, inareaswith intensivesnowfall in thewinter,saltsandsandshouldbeusedwithcaution.BMP 8 – Porous pavements: This BMP belongs to the general category ofhydrological changes, and more specifically, it is one of the four methods ofincreasing infiltration. The main aim is to reduce the volume of runoff and,subsequently,toreducepollutantloadingsinseparatesewersystemsoroverflowfrequency in combined sewer systems.Porouspavements are the firstBMP inthis category. They are constructed so as to allow runoff to infiltrate throughthemtothesubsoil.Thisisachievedeitherbynotusingfinematerialinasphaltconstructionwhichmakesitpermeable,byspecialconstructionofconcreteorbyusing special paving stones which allow water to percolate. In general, theinfiltrationcapabilityofporouspavementsissignificantlyhigh,evenexceedingtheintensityofveryinfrequentstorms(NovotnyandOlem,1994).Porous pavements are mainly used in open parking areas or in secondary

streets.Theyareeffectiveinareaswithmildslopeandquitepermeablesubsoil,wherethewatertableandtheunderlyingrockaredeep.Intermsofconstructioneffortandcost, theyarecomparabletotraditionalpavements.Theiradvantagesover traditional pavements include the following: They reduce or even caneliminate runoff, they result in smaller sewer size downstream and theycontributetogroundwaterrecharge.Theirmaindisadvantageisthehighriskofclogging,withahighcostforrepairanddangerofflooding.Forthisreason,caremustbeexercisedsothatnosolidsandsedimentsendintotheporouspavement.In addition,maintenance is amust, for example, regular sweeping. Finally, tominimizetheriskofflooding,anescaperouteshouldbedesignedintheareaoftheporouspavementforwatertooverflowincaseofclogging.Regardingdrainage,therearetwotypesofporouspavements:totalinfiltration

andpartialinfiltration.ThefirsttypeisdepictedschematicallyinFigure7.9.Thefunction of this structure is the following: due to the significantly high

permeabilityoftheporouspavement,rainfallorrunoffwaterentersfastfromthegroundsurfacethroughthepavementtotheundergroundreservoir.Thiscontainsrockofdiameter3.8–7.6cmanditsthicknessisadesignparameter.Therockisseparatedfromtheupperporouspavementandthenaturalsubsoilbeneathbyagravelfilterand/orageotextilematerial(Figure7.9).Waterisstoredinthevoidsoftherockandleavesthestructurethroughseepagetothesubsoil.Toreducetheriskofsedimentsentering the topof theporouspavement,agrassstripcanbedesignedtosurroundtheporouspavement,asshownintheplanviewofFigure7.9.Finally,anoverflowpathisprovidedatthelowerpointofthepavementtoallowforfloodwatertoescapeincaseofpavementclogging.Theundergroundreservoircanbedesignedtoholdthehydrographofthedesignflood.Thepartial infiltration system is depicted inFigure7.10. It is similar to the

totalinfiltrationsysteminfunction;theonlydifferenceisthatitisequippedwithanoverflowpipewhichallowsfortheundergroundreservoirtooperateuptoacertain level. This pipe connects to the downstream storm sewer and conveysanyexcesswaterintoit.Usually,storageintheundergroundreservoirisforthefirstflushuptothe2-yearrainfallevent.

Figure7.9 Typicalverticalsectionandplanviewofatotalinfiltrationporouspavement.

Incaseswheretherearesedimentsand/oroilintherunoffenteringtheporouspavement, pre-treatment may be needed, as mentioned. This may comprise agrassstripand,possibly,asandfilter(seealsofollowingBMP)surroundingtheporous pavement. A way to do this is depicted in the schematic diagram ofFigure7.11,where one sees a section of the grass strip, a smallweir and thesand/gravelfilter.Theoperationofthisstructureisself-explanatory.To size the depth of the underground reservoir, the main parameter is the

volume of the voids of the rock. This volume can be determined from theporosity,whichisdefinedastheratioofthevoidsvolumetothetotalvolumeofthe rock. Typical porosities for variousmaterial textures can be found in anygeotechnicalengineeringorgroundwaterhydraulicstextbook(i.e.0.25–0.40forgravel).Itisrecommendedthattheporosityofthematerialisalwaysdeterminedinsitubeforeconstructionbystandardexperimentalprocedures.

Figure7.10 Typicalverticalsectionandplanviewofapartialinfiltrationporouspavement.

Figure7.11 Pre-treatmentbeforetheporouspavement.

Theporouspavementsareveryeffectiveinremovingdissolvedpollutantsorfinesolids.Theundergroundreservoir ismainlyusedfor runoffstorage,whilepollutant retention takes place only after infiltration in the subsoil. Maintreatment processes taking place in porous pavements include the following(Schueler,1987):(1)sorptionofpollutants,whichtakesplaceintheupper30cmof the soil under the underground reservoir and concerns mostly dissolvedconstituents(e.g.orthophosphate,zinc);(2)trappingofsolidparticles,inwhichcase the soil acts as a filter; and (3) biological reactions by various aerobicbacteriaunderthereservoir,whichareenhancedasthedrydaysbetweenrainfalleventsincrease,i.e.floodingislessfrequent.Pollutant removal of porous pavements is satisfying for metals, solids and

nitrogen,and toa lesserdegreeforphosphorus.According toSchueler (1987),long-termremovalpercentagesrange82%–95%forsedimentsandsolids,65%fortotalphosphorus,80%–85%fortotalnitrogen,82%forCOD,99%forzincand98%forlead.Several criteriahave tobe followedandconditions shouldbeapplied in the

design of porous pavements and other infiltration systems. These aresummarizedinthefollowingTable7.21.BMP 9 – Infiltration trenches: Similar to porous pavements, the infiltrationtrenches remove dissolved pollutants and solid particles effectively. A mainproblem, similar to all infiltration systems, is clogging; therefore, their useshouldbeavoidedinareaswherecoarsesolidsareaproblem.Alternatively, insuchcases,protectivegrassstripscanbeusedtofilterrunoffbeforereachingthetrench. To use a trench, aminimum infiltration capacity of the subsoil shouldexist;therefore,subsoilsofSCShydrologicclassificationD(Table7.10)shouldbeavoided.Therearetwotypesoftrenches,namely,surfaceandsubsurface,dependingon

how runoff enters the trench.Typical cross sections andplanviewsof surface

andsubsurfacetrenchesarepresentedinFigures7.12and7.13,respectively.Themaindifferencebetweenthetwotypesisthatoneisopentotheairattheupperpartandrunoffentersfromthere(Figure7.12),whileotherkindisundergroundand runoff enters either directly through a sewer that ends into the trench orthroughaperforatedpipethatemptiesacollectionboxorawaterqualityinletoranoil/greaseseparator,wherethesewerends(Figure7.13;seealsothefollowingBMPs). Similar to porous pavements, infiltration trenches can be of total orpartial infiltration,dependingoncollectingtheentiredesignstormoraportionofit.Inthecaseofsurfacetrench,overflowcanoccurfromtheupperpart,whilethe subsurface trench should be equipped with a perforated pipe that collectsoverflow and leads it to the downstream sewer, as discussed in porouspavements. In several cases, the trench isdesignedonly for thevolumeof thefirstflush(i.e.13–25mmofrunoff).

Table7.21Rules, criteria and conditions for optimumdesign and best performance of porouspavementsandotherinfiltrationsystems

No. Forbestperformance

1 Mostoftheannualrunoffvolumeshouldbeinfiltratedtothesubsoil.

2 Largerinsurfaceandlessdeepsystemsarepreferred.

3 Theundergroundreservoirshouldemptythroughinfiltrationatmostin3days(preferablyin2days),somethingthatallowsforthedevelopmentofaerobicbacteria.Forfasterdrainage,onlysoilsofSCShydrologicclassificationA–Cshouldbeused(Tables7.9and7.10).SoilsofcategoryD(Table7.10)arenotappropriate.Aminimumrunoffretentiontimeof6–12hintheundergroundreservoirshouldalsobeallowed.

4 Regularsweeping(withvacuumvehicles)shouldtakeplace.High-pressureflushingshouldtakeplaceatleastfourtimesayear.

5 Areaslopeforporouspavementsandsurfacetrenchesshouldbelessthan5%.Forsubsurfacetrenches,slopescanbeupto20%.

6 Aminimumdistanceof0.6–1.2mshouldexistbetweenthebottomoftheundergroundreservoirandtheunderlyingbedrockand/orthehigheststageofthewatertable.

7 Tominimizecontaminationrisk,aminimumdistanceshouldbemaintainedbetweenpotablewaterwellsandinfiltrationstructures.Infiltrationstructuresshouldbelocateddownstreamofgroundwaterflowdirection.

8 Toavoidbuildingfoundationsettlementproblems,aminimumdistanceof3mshouldbemaintainedbetweeninfiltrationstructuresandbuildingfoundations.

9 Anobservationwell(Figures7.9and7.10)shouldbeplacedintheundergroundreservoir.Thisallowsforcheckingdrainagetimeandclogging,andalsosamplingforchemicalanalyses.

10 Toavoiddamageandmalfunction,theexistenceoftheinfiltrationstructureshouldbeidentifiedbyplacingsignsonthesurface.Thesizeshouldidentifythetypeofthestructureandshouldprohibitsurfacepavinganddischargeofsand,oilandotherpollutants.

11 Porouspavementsarerecommendedforparkingareasandsecondaryroads.Recommendeddrainageareassizesare0.1–4ha.Forinfiltrationtrenches,recommendedareasareupto2ha.

Source:Schueler,T.R.,ControllingUrbanRunoff:APracticalManualforPlanningandDesigningUrbanBMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.A particular type of trench can be used to drain specifically roofs or

impermeable surfaces, to offer direct infiltration of runoff from such areas.AschematicpresentationofsuchasystemisshowninFigure7.14.Themain advantages of infiltration trenches include reduction of both peak

andvolumeofrunoff(usuallydesignstormwithareturnperiodof2–10yearsisused) and recharge of groundwater (it is expected that 60%–90% of annualrunoff may end into groundwater when these systems are used). Similar toporous pavements, pollutant removal only includes dissolved constituents andfine solid particles with processes similar to those mentioned for porouspavements.RemovalshavebeenpresentedbySchueler(1987).Theyvaryfrom75%to99%forsedimentsandsolids,50%to75%fortotalphosphorus,45%to70%fortotalnitrogen,75%to99%fortracemetalsand70%to90%forBOD(Schueler, 1987). Lower removals correspond to infiltration of first flush (13mm)andhigheronestototal infiltration.DesigncriteriaandrulespresentedinTable7.21forporouspavementsalsoapplyforinfiltrationtrenches.BMP10–Infiltrationinlets:Therearenostandarddesignsforthesesystems,buta typical design is presented in Figure 7.15. They are made of reinforcedconcrete.Theirbaseisopentoallowforrunoffinfiltration.Becauseofthesmallstorage capacity, they are usually designed for first flush (i.e. 13–25 cm ofrunoff),while the restof runoffoverflows throughaweir to thesewer (Figure7.15).Thedesign/sizingofthesystemcomprisesoffourparts:(1)theinlet,(2)theundergroundreservoir,(3)theweirand(4)thesewer.

Figure7.12 Typicalsectionandplanviewofasurfaceinfiltrationtrench.(AdaptedfromSchueler,T.R.,Controlling Urban Runoff: A Practical Manual for Planning and Designing Urban BMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

BMP11–Retentionontheroof:ThisBMPaimstoreducetherunoffvolume.Itisappliedonflatroofs,wherethegutterinletsareraisedabovetherooffloortohavewaterretainedontheroofandeventuallyevaporated.Ausualretentionisforthefirstflush,i.e.13cmofrunoff,whichinthiscasecoincideswithrainfall.For rainfalleventsexceeding thisdepth,excesswateroverflows to thegutters.ThisBMPiseffectiveinreducingrunoffvolume;intermsofrunoffquality,theimprovementisminor,sinceonlypollutantsfromatmosphericscrubbingand/orwetdepositionsareretained.Furthermore,toapplythisBMP,theroofshouldbeproperly designed/checked to sustain additional loads frompondingwater andavoidleakageintothebuilding.BMP 12 – Rainwater harvesting: Rainwater harvesting is the collection andstorage of rainwater for potable or non-potable in-house uses (Gikas and

Tsihrintzis, 2012; Tsihrintzis and Baltas, 2014); it is a relatively inexpensivemethodtoreducetheconsumptionofpotablewatersuppliedbypublicsources,and may contribute to the reduction of runoff volumes and peaks. Untreatedharvested rainwater can be used for non-potable uses, such as toilet flushing,washingclothes,otherhouseholdusesandgardenirrigation,whilepotableusesare also common in several countries (e.g. Australia), but they may requireappropriate treatment of harvested rainwater, depending on its quality(Tsihrintzis andBaltas, 2014).A layout similar to the one presented inFigure7.14 is used for a rainwater-harvesting system; however, the trench should bereplaced by a collection tankwhich receives and stores the rainwater.Variousdescriptions and sizing techniques of a rainwater-harvesting tank have beenpresented(TsihrintzisandBaltas,2014).Thequalityofharvestedrainwaterandhealth risk assessment have been studied by various authors (Gikas andTsihrintzis,201).

Figure7.13 Typicalsectionofasubsurfaceinfiltrationtrench.(AdaptedfromSchueler,T.R.,ControllingUrban Runoff: A Practical Manual for Planning and Designing Urban BMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

Figure7.14 Typicalinfiltrationtrenchforaroof.(AdaptedfromSchueler,T.R.,ControllingUrbanRunoff:APracticalManualforPlanningandDesigningUrbanBMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

Figure7.15 Atypicaldesignofaninfiltrationinlet.

BMP 13 – Grass filters: Grass filters were presented with porous pavements(Figures7.9to7.11)andinfiltrationtrenches(Figure7.12),wheretheyareusedforrunoffpre-treatment,i.e.retentionofsolidstoavoidclogging.Theycanalsobeusedtoreducethehydraulicallyconnectedimperviousareaoralongstreamsto retainmostly solids and, possibly, other pollutants from reaching thewater(Figure7.16).They are flat slightly sloping surfaces, where flow is shallow and evenly

distributed perpendicular to the flow path, i.e. overland flow. Their goodperformance is based on this property. Performance getsworstwhen rills andgutters are formed due to erosion, and flow concentrates in certain paths. Toavoid erosion, a certain type of erosion-resistant grass is used, e.g. Bermudagrass.Inadditiontograss,filterscancontaintreesandbushesand,generally,bebetterdesignedtofitthelocallandscape.

Figure7.16 Grassfilteralongastream.

Theirfunctioncanbesummarizedasfollows(NovotnyandOlem,1994):(1)They reduce the runoff velocity, resulting in an increase in the time ofconcentrationandareductionofpeak;(2)theyreducetheimperviousnessoftheurbanwatershed, resulting in a reduction of runoff volume and an increase ofinfiltration; and (3) they remove solids, organic matter and metals. Theirtreatment performance depends on several parameters, including filter lengthalong the flow path, slope, watershed area, runoff velocity, solid particlediameterandsoil infiltrationcapacity.The twomainparametersare the lengthandtheslope.Anabsolutelyminimumlengthis7m,whilelengthsof20–100mare often used (Figure 7.16). The slope should not exceed 5% for pollutantremoval (NovotnyandOlem,1994),butevenhigherslopesshouldbecoveredbygrassforerosionprotection;(4)dissolvedconstituentsareremovedthroughinfiltration,which ismoreeffectivewhenslope ismildand thereare treesandbushes; (5)maintenance is recommendedwhich includesgrass cutting (two tothreetimesayear)andrepairoferodedareas;(6)grassfiltersalongstreamscanbepartofenvironmentalcorridorswhichincreasebiodiversityandaestheticsofthelandscape,andcansupportparkandrecreationalactivities(walking,jogging,biking, nature observation, etc.) in the urban area. To function as wildlifecorridors,thelengthofthegrassfilter(Figure7.16)shouldincreasesignificantly(200–1000m).BMP14–Grass swales:Grass swalescanbeplaced: (1) along street sides insuburbanareas,whenthereisareaavailabletoreplacecurbs;(2)alonghighwaysat the right side to collect runoff; and (3) along highways at the middle toseparate the two directions of traffic (Figure 7.17). Swales usually are small,open,shallowcanalsofnearlytriangular,trapezoidal(Figure7.17)orparabolicsectionofverymildbankslope.

Theyoffer(1)reductionofrunoffpeakandvolumethroughtwomechanisms,reductionofvelocityand,thus,increaseoftimeofconcentration,andinfiltrationofpartofrunofftothesubsoil.Theinfiltratingvolumecanbeincreasedifsmallcheckdamsorweirsareplacedatseverallocationsacrosstheswaletoallowforwaterpondingandincreaseofdepthupstreamof thecheckdam(Figure7.17);(2) in terms of water quality improvement, similarly to grass filters, theirfunction is restricted to reducing solid particles and the portion of dissolvedpollutantscontainedintheinfiltratingwater;and(3)theyareamoreeconomicsolution, in terms of construction, compared to standard curb and gutter andseweroption.Designrules/stepsforgrassswalesincludethefollowing(NovotnyandOlem,

1994): (1) The slope along the swale should be very mild, preferably nearlyhorizontal.Inanycase,themaximumallowedslopeshouldbe5%.(2)Thebankslopeshouldbemilderthan3:1horizontaltovertical(Figure7.17).(3)Thegrassspeciesusedshouldberesistanttoerosion.(4)Swalesshouldbeusedonlywithsubsoils of SCS hydrologic category A and B (Table 7.9). (5) To increaseinfiltration, check dams or weirs can be used. These can be either concrete,wooden or made of earth. For safety reasons and to avoid mosquitodevelopment,thepondingdurationshouldnotexceed24handthedepthshouldbelessthan50cm.(6)Thedesignvelocityshouldbelessthan0.9m/stoavoiderosion. (7)Maintenance is necessary and should include grass cutting once ayear, repair of bare areas by replanting,minimization or elimination of use offertilizers and collection of garbage. (8) To increase ecological and aestheticvalue,treesandbushescanbeplanted.

Figure7.17 Aschematicplanviewandtypicalcrosssectionofhighwayswales.

7.4.3Pollutioncontrolinthesewerandthedrainageditches

BMP 15 – Oil/grease separators: They are placed at the beginning of stormsewers or at inlets. They are concrete boxes used to separate oil and greasefloating on runoff and solids carried by runoff (Schueler, 1987). They aregenerally small in volume (usually the effective volume is about 11 m3);therefore, theydonotprovidestorageandconsequent reductionofhydrographpeak.Theyretainrelativelylargesolidparticles(i.e.sand),oilandgrease,whilefineparticlesanddissolvedconstituentspassdownstreamwithrunoff.Forthesereasons, they are mostly used for pre-treatment or in combination with otherBMPs(e.g.Figure7.13).Theyareplaced inareaswherehydrocarbonsandoilderivativesfromvehiclesareexpected,e.g.parkinglots,gasstationsandstreets.The drainage area to one oil/grease separator should not exceed 0.4 ha.Maintenanceismandatory(recommendedevery6months)andrequiresremovalof collected sediments and pollutants which may be toxic. If these are notremoved, theyget resuspendedandcarriedby therunoffofan intenserainfall.

To remove solids, special vehicles with vacuum suction are used. Therefore,manholes are needed at the top of the structure.The oil/grease is removedbyspecial materials which can absorb it without absorbing water. This materialneedsregularreplacement.A typical design is shown in Figure 7.18. The structure comprises of three

compartments (Schueler, 1987).The first one retains solidswhich deposit, thesecond one retains floating oil/grease and the third one connects to thedownstreamstormsewer.Thefirstandsecondcompartmentsareseparatedbyavertical wall and communicate through two circular openings, 15 cm indiameter,andplacedatabout1.2mfromthebottomofthestructure.Agrid(13mm in size) is placed in front of the twoopenings to hold floating solids andprotect theorifices fromclogging.Theoil andgrease float andare retained inbothcompartments.Totrapit, thesecondcompartmentcommunicateswiththethirdthroughapipe,whichhasaT-shapetoconveywateronlyfromthebottom(30cmabovethebottom).Thetwoverticalwallsbetweencompartmentsdonotmeet the ceiling of the structure but allow for a 10 cm space for overflow. Insomecases,holes(15cmindiameter)canalsobedrilledatthebottomtodrainthestructuretothesubsoil.However,thesearenotusuallyeffective,sincetheyquicklyclogwithsediments.BMP16–Waterquality inlets:These sewer inlets are specificallydesigned toretaincoarsesolidparticlesandsolidwaste,andoil/grease(Schueler,1987).Inaddition,theycontrolodoursfromsewerstoreachtheatmosphere.Thereisnostandard design, but it depends on local practice; therefore, Figure 7.19 onlyshows theconcept.Thestoragewatervolumevaries from0.08 to2.21m3,butabout half of it is usually taken by sediment. Due to the small volume, theireffectivenessinwaterqualityislimited.Thereisaneedforregularmaintenanceandcleaningatleasttwiceayear,inwhichcasetheremovalofsolidsandheavymetalscanreach25%.

Figure7.18 Aschematicdiagramofoil/greaseseparator.(AdaptedfromSchueler,T.R.,ControllingUrbanRunoff: A Practical Manual for Planning and Designing Urban BMPs, WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

Figure 7.19 A schematic diagram of a water quality inlet. (Adapted from Schueler, T.R.,ControllingUrban Runoff: A Practical Manual for Planning and Designing Urban BMPs,WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

BMP17–Grids:Gridsareusuallyplacedattheoutletofmainstormsewerstothereceivingwaters.AtypicaldesignisdepictedinFigure7.20.Theirfunctionis tocollectmostly floatingsolidwaste (e.g. foamcups)anddebris (branches,wood) from spreading at the surfaceof the aquatic system.Theyneed regularmaintenancewhichconsistsofcleaningandremovingthecollectedmaterial toavoidcloggingand,possibly,upstreamflooding.BMP18–Channelstabilizationstructures:Suchstructuresareused toprotect

open channels from erosion.They are presented in detail in various textbooks(e.g.Goldman et al., 1986; Simons andSenturk, 1992;Vanoni, 2006) and arealsobrieflydiscussedinChapter8.

Figure7.20 Typicalgridattheendofastormsewer.

BMP19–Vegetatedcanals:Amethodofprotectionofchannelsandcanalsfromerosionistousegrassandvegetation.Thespecifictheoryanddesignispresentedinvarioustextbooks(e.g.Chow,1959;French,1985).

BMP 20 – Riprap, gabions and other methods of channel protection: Suchmethods to protect channels from erosion are presented in various textbooks(e.g.Goldmanetal.,1986).BMP 21 – Storage, retention/detention in the storm sewer: BMPs under thiscategoryofferindirectimprovementtorunoffqualitybyreducingvolumeand/orflowpeak.Fourmethodsareused:

1. Temporary or permanent storage of runoff in underground storage tanks(Figure 7.21). Most often, reinforced concrete or metal pipes of largediameter are used, which are placed in parallel to the storm sewer andcommunicatewithitthroughorificesorweirs.Theentireorportionoftherunoffisdivertedandstored.Thedrainageoftheundergroundstoragetanksis(a)throughinfiltrationtothesubsoil, inwhichcasethetanksshouldbeperforated, and (b) by drainage back to the storm sewer after the floodpasses.

2. Temporary storage in a detention tank within the storm sewer, which iscreatedbyenlargingthesewer(Figure7.22).Sizingofthedetentiontankisbasedonfloodroutingtheory(Pulsmethod),presentedlaterinthischapter

(seealsoBMPs26or27).3. Storage ina tunnelwhichcanbeconstructedunder theseweratagreater

depth(Figure7.23).This solution isusefulwhen there is space limitationfor a regular surface or subsurface detention pond. Since the tunnel isusuallyexcavatedinrock,drainageispossibleonlythroughpumpingofthestoredwaterbacktothestormsewerafterthefloodhaspassed.ThisisalsoasolutionthatcanbeusedinsolvingCSOproblems(seefollowingBMPs30to32).

4. Storagewithinthestormsewerorcanal.Thisisimportantwhenthesewerislarge.ThecomputationsfollowtheMuskingummethodtheory,presentedinthefollowing.

Figure7.21 Storageinundergroundpipes.

Figure7.22 Undergrounddetentionbasin.

Ifthehydrographsattheupstreamanddownstreamsidesofastormsewerreachare measured simultaneously, one will observe that these differ in flow peak,time base and generally shape, even if there is no inflow or outflow ofwaterfromthestormsewerreach.Thehydrographpeakatthedownstreamendofthereachislowerthanthatattheupstreamendandthetimebaseislonger.Thisisaresult ofwater storage in the storm sewer.Thehydrograph at thedownstreamsectioncanbecomputediftheoneattheupstreamsectionandthegeometryofthe conduit are known, using flow routing. A common method used is theMuskingummethod.Inacanalorstormsewer reach flowingopen, thewatersurfacehasaslope

andthewaterstoredinthereachisafunctionofwaterstageatboththeupstreamanddownstreamsectionsof thereach(Figure7.24).The figure shows that thetotalstoredvolumecomprisestwoparts:prismstorageSp(definedasthevolumeof water between conduit invert and the line parallel to the invert from thedownstreamwater surface) andwedge storageSw (defined as thewater storedabovetheparallellinetothebedandthefreewatersurface).Prismstorageisafunction of the downstream water stage and, therefore, outflow O from thedownstream section of the reach. Wedge storage depends on the differencebetweeninflowIandoutflowOfromthereach.Therefore,thetotalvolumecanbewrittenasfollows:

(7.49)

Figure7.23 Useofatunnelforstorage.

Figure7.24 Definitionofwedgeandprismstorageinacanalandacircularstormsewer.

whereIandOareinflowandoutflowfromthereach(m3/s)f1,f2implyfunctionsof

TheMuskingummethodassumesthatfunctionsf1andf2arelinear:

(7.50)

Inthiscase,Equation7.49gives

wherex=β/Kisadimensionlesscoefficient,whichshowstherelativeweightofthe inflow and outflow in determining the storage in the conduit reach. Thiscoefficienttakesvaluesbetween0.0and0.5.The0.0valueimpliesthatinflowisnot important and the 0.5 value implies that inflow and outflow are equallyimportant. In the Muskingum method, to solve the flow routing problem,Equation7.51iscombinedwiththewaterbudgetequationforthereach.LetusconsideratimeintervalΔt.IfIm(t)andOm(t)are themean inflowandoutflowratesintheconduitreachduringΔt,thewaterbudgetcanbewrittenasfollows:

(7.52)

where

(7.53)

(7.54)

(7.55)

where

I(t),O(t),andS(t)areinflowrate,outflowrateandstoragewithinthereachattimet

I(t−Δt),O(t−Δt)andS(t−Δt)aretherespectivequantitiesattimet−Δt

ThecombinationofEquations7.51through7.55leadstothefollowing:

(7.56)

wherecoefficientsC0,C1andC2aredefinedasfollows:

(7.57)

(7.58)

(7.59)

Itisobviousthat

(7.60)

Equation7.56canbeusedtocomputestepbystep,inaniterativeprocedure,theoutflowhydrographO(t)atthedownstreamendofthereachforagiveninflowhydrographI(t),ifKandx(andthus,C0,C1,C2)areknown,andthevalueintheprevioustimestepO(t−Δt)hasbeencomputed(orisknownfort=0).First,thevaluesofcoefficientsC0,C1andC2havetobecomputedusingEquations7.57through7.59.TodeterminethevaluesofparametersKandx, the followingprocedurecan

beused:

1. Hydrographs are measured, simultaneously, at the upstream anddownstreamsectionsoftheconduitreach.

2. For each time interval Δt, the change in storage ΔS is computed usingEquation7.52.

3. Avalueisassumedforparameterx(intheinterval[0,0.5]),andthevalueof

thequantity[xI+(1−x)O]iscomputed(inRHSofEquation7.51).4. ThestorageS(t)iscomputedbyaccumulatingΔS,S=∑ΔS.5. AplotofS(fromstep4)versus[xI+(1−x)O](fromstep3)isprepared.6. According to Equation 7.51, the plot should represent a straight line of

slopeK,iftheselectedvalueforparameterx(step3)iscorrect.7. If thedatadeviate from thestraight line, theassumptionof thexvalue is

incorrect,andanotherguessisneeded.Then,steps3–7arerepeatedtogettheplotwherethepointsfallmostcloselytothestraightline.

8. Thestraightregressionlineisfitted,anditsslopeiscomputedwhichgivestheKvalue.

BMP22–Storm sewerwashing:Several times solidsdeposit in storm sewersandalsoabiologicalslimedevelops(particularlyincombinedsystems).Sewerwashingmaybe important,particularlyofcombinedsystems, toremovesolidsand/orslime.Thisshouldbetakingplaceduringdryweather,forcingthewashwater to end into theWWTP (NovotnyandOlem,1994).High-pressurewaterjetscanbeusedforeffectivewashing.BMP 23 – Control of illicit sanitary sewer connections: Illicit connections ofhouseplumping into separate storm sewers are a fact inmanycities.This hasadverseimpactsonreceivingwaterssinceuntreatedsewageendsthere.Itisthenimportantforpublicagenciestofindandminimize/eliminateillicitconnections(either through discharge and/or water qualitymonitoring at various locationsalong the storm sewer, or by using specialized cameras). Furthermore, publicawarenessonillicitconnectionadverseimpactsandimposingofstrictfinesmayalsobenecessaryinsomeinstances.

7.4.4Pollutioncontrolwithsurfacedetention/retentionandstorage

BMP 24 – Flood detention basins in series (or online) with the storm sewer:Theyofferwaterqualityadvantagesonlyindirectlybyreducingthehydrographpeak,which is themainreasonof theirconstruction,while the totalvolumeofthehydrographisunaffected.Asaresult,theyreducethedownstreamfloodrisk,thevelocityintheconduitandtheerosionrisk.Severaltimes,theirconstructionoffers opportunities for aesthetics improvements through the existence ofponding water, development of parks and recreation areas and support ofwetlandvegetation.Figure 7.25 presents a schematic plan view, indicating how such a basin is

locatedinrelationtothestormsewer,‘inseries’, i.e. thestormsewerendsandemptiesintothedetentionbasinandtheoutletofthedetentionbasinconnectsto

thedownstreamsewer.Theoutletofthedetentionbasinshouldbeofrestrictedcapacitycomparedtotheinflowingsewer,inorderforincomingwatertopondandbestoredinthebasin,i.e.astheinflowinghydrographcomesintothebasin,the water-level in the basin rises and reaches a maximum level, and as theinflowinghydrograph recedes, thewater-level in thebasin falls.Asa resultofwaterstorage, theoutflowhydrographfromthebasin ismodifiedcompared tothe inflowinghydrograph,andmorespecifically, theoutflowhydrographhasalowerpeakanda larger timebase.Since there isno lossorgainofwater, theinflowingandoutflowinghydrographshavethesamevolume.Thecomputationoftheoutflowinghydrographisbasedonthetheoryorflood

routingthroughareservoir(Pulsmethod).Tosolvethisproblem,theinflowinghydrographand thegeometryof thedetentionbasinand theoutflowstructure,i.e. thestoragevolumeand theoutflowrateasa functionofwater-level in thebasin, have tobeknown.Similar to theMuskingummethod, let us define I(t)andO(t) as the inflowandoutflow rates to and from thebasinandS(t) as thevolumeofwaterstoredattimet.Thenitisgenerallyvalidthatbothoutflowrateandstorageinthebasindependonwater-levelinthebasin:

(7.61)

wheref3andf4implyfunctionsofHisthewater-levelinthedetentionbasin

Similar to the Muskingum method, the water budget Equation 7.52 holds. IfEquations7.53through7.55areusedagaintoexpressmeaninflowandoutflow,andchangeofstoredwaterinthebasin,thenEquation7.52reducesto

(7.62)

Figure7.25 Detentionbasininseries(planview).

IfwedefinethefollowingparameterΓ(t)as

(7.63)

Then,Equation7.62reads

(7.64)

AccordingtoEquations7.61and7.63

(7.65)

and

(7.66)

where f5 is some functionwhichdependsondetentionbasingeometryand theproperties of the outflow structure. Equations 7.64 and 7.65 or 7.66 form asystemoftwoequationswithtwounknowns.TheunknownsareΓ(t)andO(t)ifthe values of Γ(t − Δt),O(t − Δt) and Im(t) are known. This implies that thesystemcanbe solved ina step-by-step iterativeprocedurewhere thevaluesofthe previous time step are used to compute those of the next time stepΔt. Inaddition,thegeometryexpressedbyfunctionf5andthefirstvalueofO(0)att=0shouldbeknown.Theprocedureisappliedveryquicklyinatableandcanbeveryfastifaspreadsheetisused.BMP27–Flooddetentionbasinsinparallel(oroffline)withthestormsewer:They have themain advantages of the previously described basins, consistingmainlyofreducingthehydrographpeak.Thelocationofthebasin(inplanview)inrelationtothestormsewerisshownschematicallyinFigure7.26.Inthiscase,thestormsewerdoesnotenterthebasinasinthepreviouscase(Figure7.25).Iftheaimistoreducethepeakofthehydrograph(i.e.floodcontrol),thereisa

sideweirplacedalongthestormsewer(Figure7.27);inotherwords,thesideofthe storm sewer has an opening at a certain distance above the invert, whichallowspassageofwater,whenitexceedsacertaindepth(i.e.discharge)tospilloutintothebasin.Figure7.28showsthatlowflows(Q<Q*)passthebasinandcontinuedownstreaminthemainstormsewer,whilepartofhighflows(Q>Q*,t1 < t < t2) is transferred to the detention basin. The detention basin is either

designedtoinfiltratethestoredwaterorthiscanreturntothemainstormsewer,afterthefloodpasses,throughapipethatconnectstothestormseweratalowerpoint(Figure7.27).AdetailofthisstructureissimilartothatpresentedinFigure7.21.

Figure7.26 Detentionbasininparallel(planview).

Figure7.27 Reductionofthepeakofthehydrographinadetentionbasinplacedinparallel.

Figure7.28 Determinationofhydrographroutedtoaninfiltrationbasininparallel.

BMP26–Basin for infiltrationof first flush inparallelwith the stormsewer:The basin is again placed in parallel with the storm sewer (Figure 7.26).However, the hydraulic structure that transfers water to the basin is differentfromtheonedescribedinthepreviousBMP.Theaimhereisnottoreducethehydrographpeak(Figure7.27)buttotransfertothebasinthefirstflush,i.e.theinitialpartof thehydrograph,orabout13–25mmofrunoff(Figure7.28).ThedifferenceisseeneasilybycomparingFigures7.27and7.28.Atypicalhydraulicstructuredesign to route thepartof thehydrograph to thebasin isdepicted inFigure 7.29. Instead of the side weir of the previous BMP, which is used totransferflowshigherofacertainflow,aweir isplacedacross thestormsewerwidth to block flows below a certain value (Q <Q*). In addition, a side orbottom orifice has to be placed in the storm sewer upstream of the weir toreceivethisflowandtransferitintothebasin.TheconceptispresentedinFigure7.29.Thebedof thebasin is constructedat the same level as the stormsewerinvertat thepointofconnection; inthiscase, themaximumwaterdepthinthebasinwill be thedifferenceof the levelof the crestof theweir and the stormsewerinvert.Thereshouldalsobeareturnflowcontrolvalve(flapgatetowardsthebasin)whichpermitswatertoenterthebasinfromthestormsewerbutdoesnotallowflowintheoppositedirection;thus,wateristrappedinthebasin.

Figure7.29 Basinplanviewandhydraulicstructureforaninfiltrationbasinforfirstflush.

BMP27–Extendeddrybasin:Asmentioned,waterqualitybenefitsfromusingdetention basins are indirect and only come from reducing the peak of thehydrograph.Themainreasonisthatsuchstructuresaremostlydesignedhavinginmindfloodcontrol;therefore,thedesignisbasedoninfrequentstormevents.Asa result, runoff from frequentevents carryingpollutantspasses through thebasinrelativelyfastandpollutantsarenotretained.Extendeddrybasinsaredetentionbasinsproperlyconfiguredanddesignedto

offerwaterqualityimprovementofrunoff.Wateroffrequentstormsisretainedforabout24h,somethingthatleadstoabout90%removalofsolids.However,nitrogenandphosphorus arenot removed.Aconfigurationof anextendeddrybasinisshowninFigure7.30.Thehydraulicstructureoftheoutflowisdesignedto provide the required hydraulic retention time (HRT) for the runoff of afrequentfloodtoremaininthebasin,butalsoitprovidesthestorageforalargerevent.Thisisdoneasfollows:(1)Thebasinhastwooperationalparts,theupperandthelowerones.Theloweroneisintermittentlyfloodedandretainswaterforatleast24htoimprovewaterquality.Theupperpartisusedforflooddetentionof less frequent events. (2) The outflow hydraulic structure drains slowly thelowerpart (in24hormore).For this,acommondesign is touseaperforated

vertical or horizontal pipe (Figure 7.31) placed in gravel, or an orifice, or acombinationofboth.

Figure7.30 Extendeddrypond.

Figure7.31 Alternativeoutflowhydraulicstructuredetailsfromthelowerpartoftheextendeddrypond:(a)Controlbyorificeand(b)controlbyperforatedstandpipe.

Thestoragecapacityof the lowerpartcanbedeterminedbasedoneitherofthefollowingcriteria(Schueler,1987):(1)runofffromastormeventofa1-yearreturnperiodanddurationof24hwithanHRTof24h,(2)runofffromastormeventofa2-yearreturnperiodanddurationof24hwithanHRTof24hand(3)firstflush(i.e.13–25mmorrunoff)withanHRTof24h.TheupperpartofthebasinisdimensionedusingthemethodspresentedinBMPs24or25.The main mechanism of the extended dry pond in pollutant removal is

deposition.Therefore,theeffectivenessisonsolidremoval,whiletheremovalofdissolved pollutants is limited. According to Schueler (1987), the removal of

solidsdependsonHRTandvariesfrom60%–70%forHRTof6hto80%–90%forHRTof48h.GreaterHRTscanresultinalmostcompleteremoval.Observedlong-termremovalwas65%forHRTbetween6and12h.Otherpollutantsareremoved in lower percentages. For example, a maximum of 40%–50% wasobservedforphosphorus,nitrogen,BODandCODforHRTof32–48h.Metalsareremovedathighpercentageswheneasilyadsorbedtosolids(likePb)andtoalesserdegreewhentheyareindissolvedform(likeZn).Finally,extendeddrypondsremovebacteriabyanorderofmagnitudeandhydrocarbonsby60%–70%atHRTof32h.Figure7.32presentsremovalsforvariousconstituentsinrelationtotheHRT.

Figure7.32 PercentremovalofvariouspollutantsasafunctionofHRT.(FromSchueler,T.R.,ControllingUrban Runoff: A Practical Manual for Planning and Designing Urban BMPs, WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

The design of the extended dry pond is based on the following criteria(Schueler,1987):

1. The lower level shouldbedesigned for at least the first fush (R = 13–25mm). Better performance can be achieved when designing to detain therunoff from the1-yearor2-year stormofa24hduration. Ingeneral, thelowerlevelshoulddetain50%–90%oftheannualrainfallevents.

2. BasedonFigure7.32,theresidencetime,i.e.thetimeittakesforthelowerpart toempty,shouldbeat least24hupto48h.Asmentioned,aspecialdesignoftheoutletstructureisneeded,sothatdrainageisslowandtakesthe required time. One option is the one schematically shown in Figure7.31a; in this case,oneorificecontrols theoutflow.Therefore, thedesignparameteristhediameterofthisorifice.Theotheroptionwouldbetohavea structure similar to the one shown in Figure 7.31b, where designparametersarethediameteroftheholesoftheperforatedpipe,thelevelof

eachholeandthenumberofholes.Thefirstcaseiseasiertodesignbutmaysuffer fromcloggingproblems, since it isembeddedunderneath thebasinbed,andsedimentsanddebrismaybeaproblem.Theotheroptionismoredifficulttodesign,butitmayworkbetterwithlowercloggingrisk.

3. Plantingofthelowerpartofthebasinandcreationofawetlandmayleadtoincreasednitrogenandphosphorusremovals.Commondepths,inthiscase,shouldnotexceed15–30cm.

4. Riprap shouldbeplacedon thebanksandwherever there is a chance forerosion.

5. Thebasinbanksshouldbebetween3:1and20:1.6. Aclearzone8mwideshouldsurround thebasin.Treesaround thebasin

areaincreaseaestheticsandecologicalvalues.7. An access road should be provided (3 m wide) for maintenance, which

shouldtakeplaceatleasttwiceayear,andshouldincluderepairs,cleaningofoutletstructureandsedimentremoval,amongothers.

8. Basincatchmentareashouldbeoftheorderof4–8ha.

Figure7.33 ASchematicdiagramofalongitudinalsectionofawetbasin.

BMP28–Wetbasin:Wetbasins(Figure7.33)aregoodalternativestoextendeddrybasins,becausetheyaremoreeffectiveinremovingsediments,BODandtracemetals.Theyalsoremove,toalargedegree,dissolvedpollutants,suchasnitrateandorthophosphate.Finally,duetothepermanentwater,theyoffermoreecologicalandaestheticsopportunities,e.g.theycanbeusedinparks.Dependingonthedesign,theycanreducethehydrographpeakandprobably,toalesserdegree,reducethevolumeofrunoff.Theminimumsizeofadrainagebasintoincludeawetpondis8ha.

Themechanismsactinginwetpondsinclude(1)deposition:thepondactsasa

settling basin for solids‒solid deposition is themain process enhanced by theexistence of permanent water and, possibly, by growing hydrophytes; (2)biological processes: these occur mainly through growing plants and/or algaeand contribute to dissolved constituent removal (i.e. nutrients); and (3)decomposition by bacteria: these occur in the bottom sediments and concerndecomposition of phosphorus, nitrogen and organic compounds. Typicalpercentagesofpollutantremovalsinwetpondsareasfollows(Schueler,1987):54% for solids, 20% for COD, 51% for Zn, 65% for Pb, 20% for organicnitrogen,20%fororganicphosphorus,60%fordissolvednitrogenand80%fordissolvedphosphorus.Theperformanceofthewetbasininremovingpollutantsdependsonitssize

comparedtothecatchmentarea.Figures7.34and7.35showthisdependence.InFigure7.34,pollutantremovalisrelatedtotheratioofbasinwetvolume(Vb)torunoff volume (VR) from the mean precipitation over the entire watershed(Schueler, 1987). Alternatively, Driscoll (1988) presented the relation ofpollutant removal as a functionofthepercentageof thebasinwet surface area(Ab) to the watershed areaA (Figure 7.35). In both graphs, one can see thatcurves reach equilibrium, i.e. pollutant removals do not increase significantlyafteracertainbasinsize,andspecifically,whenthebasinvolumeexceedsaboutfourtimestherunoffofthemeanprecipitation,ortheareaofthebasinexceedsabout1–2%ofthewatershedsurfacearea.The effectiveness of the wet basin in removing solids can be evaluated by

treatingitasasettlingbasin,i.e.onehastocalculatethesettlingvelocityoftheexpectedsolidparticlesbasedonStokes law;aparticleofacertainsizesettleswhentheverticalcomponentoftheflowvelocityislessthantheparticlesettlingvelocity.

Figure7.34 Wetbasinperformanceasafunctionoftheratioofwetbasinvolumetothevolumeofrunofffrom the mean precipitation over the entire watershed. (Schueler, T.R., Controlling Urban

Runoff: A Practical Manual for Planning and Designing Urban BMPs, WashingtonMetropolitanWaterResourcesPlanningBoard,Washington,DC,July1987.)

Figure7.35 Removalofvariouspollutantsasafunctionoftheratioofthesurfaceareaofthewetbasintothe drainage area. (From Driscoll, E.D., Long-term performance of water quality ponds, inDesign of Urban Runoff Quality Controls, Proceedings of the Engineering FoundationConference,ASCE,NewYork,1988,pp.145–162.)

Thedesignofthewetbasinsshouldfollowcertaincriteria(Schueler,1987):

1. Longandnarrowbasinsarepreferredovershortandwidetoavoid‘short-circuiting’,i.e.flowpassingquicklythroughthebasinwithoutreplacingtheexistingwater.Minimumratiooflengthtowidthis3to1.Whenthisisnotpossibletokeep,thenbaffescanbeplacedinthebasintoguidetheflowtofollowasinuouspathinsteadofastraightone.

2. Shallowbasinsarepreferredoverdeeponestoenhancesolidsdeposition.Aminimum depth of 0.6 m should be considered to minimize bottomsediment resuspension, and amaximum depth of 2.5m to avoid thermalstratification.Thedepthshouldbeshallower(around0.3m)attheedgesofthebasin towards thebanks toenhancehydrophytegrowth; this increasesefficiency in treatment, contributes to aesthetics improvement andrecreation opportunities, offers ecological values and increases safety. Asurroundingzonewithvegetation,8–10mwide,isalsorecommended.

3. Thebanksshouldhaveamaximumslope3:1toavoiderosion,andshouldbepreferablyplantedwithgrass.Riprapshouldbeplacedaterosion-proneareas.Areasdownstreamofpipeoutletsshouldalsobeprotected.

4. Wetbasinsarerecommendedatwatershedswithaminimumsizeof8ha.5. Soil typesbelow thewetbasin shouldbeof categoriesCorD.Basins in

soils of categories A or B are not recommended unless they are isolatedusingageomembraneoraclaylayer.

6. Frequentmaintenance ismandatory, consistingofgrass cutting, removingofsolidandotherwaste,andcheckingforerosion,settlementsandstructurestability, clogging of hydraulic structures, good operation, etc. An accessroadisrequired.

7. Thedesignofthewetbasin,particularlywhenitisofrelativelylargesize,should also consider the effectsof thermal impacts andanoxic conditionsonthedownstreamstreamwherethebasindischarges.

BMP 29 – Wetlands: Wetlands and mostly constructed wetlands have beensuccessfully used in treating stormwater runoff from urban and agriculturalareas. They are effective systems in removing organic matter, nitrogen andphosphorus,andheavymetals.Thedepthofwaterinthesesystemsvariesfrom15 to 30 cm. Common vegetation used includes Typha sp., Scirpus sp. andPhragmitessp.Therearenowpublishedseveralspecializedbooksonwetlandsand constructed wetlands (e.g. Reed et al., 1995; Kadlec andWallace, 2009;Steanakisetal.,2014).BMP30–Detentionbasinsforcombinedsewers(inseries):Detentionbasinsforcombined sewers are designed either in series or in parallel to the sewer line,similartothoseforstormsewers.Theyareusuallysizedforthefirstflush.TheyareusedtocontrolCSO,butalsotominimizehumanhealthandodourproblems;thus, since combined sewers are old systems, in most cases, basins areconstructed underground, mostly because usually above-ground space is notavailablefortheirconstruction;nevertheless,openbasinscanbeconstructedatlocationsdistantfromsettlements.Figure 7.36 presents the operation of a combined sewer and includes the

sewer, theWWTPand theaquatic system.Ahydraulic structure (flowdivider,hydraulicswitch)isalsoshowninFigure7.36,whichisusedtodirectthefloweither to theWWTPor to theaquatic system. In the secondcase,CSOoccursand untreated wastewater ends in the aquatic system. Combined sewers areusuallydesigned fora5-to10-year returnperiod.Thediameterof the influentpipe to the WWTP usually has much lesser capacity than the sewer and isusually designed to bring the peak of the sewage flow and probably somenuisance water draining on the streets, i.e. five to six times, the daily meansewageflow.Thisflowiscalledcriticalflow(Qc).Duringdryweatherorlightprecipitation,theflowQinthesewerislessthanQc,andtheflowdividerdirectstheflowtowardstheWWTP(Figure7.36a).Aftertreatment,theflowreturnstothe combined sewer and, finally, outflows to the aquatic system. During wetweather with Q exceeding Qc, the flowdivider allows only Qc to go to the

WWTPandtheremainingflowendsdirectlyanduntreatedtotheaquaticsystem(Figure7.36b),whichcanhaveadverseimpactsonbothhumansandaquaticlife.This is done to avoid flooding of theWWTPandwashing out of the bacteriawithadverseimpactonitsoperation.Detentionbasinsforcombinedsewersinseriesoperatesimilarlytothosefor

separatesewers(BMP26).However,incasetheyareplacedunderground,theymayneedpumpingofthestoredCSOinsteadofdrainagebygravity.AtypicallayoutofabasinplacedinseriesisshowninFigure7.37.Asseen,thebasinisplaceddownstreamoftheflowdivider;therefore,itoperatesonlywhenQ>Qc.WhenQ≤Qc, thesystemoperatesaspresented inFigure7.36a,and thebasinstays dry since the entire flow is diverted by the flow divider to theWWTP.WhenQ>Qc,theflowcontinuespassingthroughtheflowdividerandreachesthebasin. Initially, thebasinfillsupasshowninFigure7.37a, and there isnoflow in the downstream sewer ending into the aquatic system. Therefore, thebasincontrolsCSOoutflowtotheaquaticsystem.FlowQcisdivertedfromthebasin to the WWTP, which, after treatment, is discharged into the aquaticsystem. CSO to the aquatic system starts after the capacity of the basin isexceeded (Figure 7.37b). Nevertheless, there is still water stored above thespillwaycrest,and,eveninthiscase,thereisimprovementsincetheflowpeaksareloweredandthetimebaseofthehydrographislengthened.ComputationcanbedoneusingthePulsmethodforfoodrouting.

Figure7.36 Typicaloperationofacombinedsewer.(a)DryweatherQQcand(b)wetweatherQQc.

BMP31–Detentionbasinsforcombinedsewers(inparallel):Theseareplacedinparalleltothecombinedsewer;therefore,ahydraulicstructureshouldbeusedtodiverttheflowtothebasin.AtypicallayoutisshowninFigure7.38whichisself-explanatory.BMP32–Detentionbasinsforcombinedsewer(insidetheaquaticsystem):ThismethodwasdevelopedandisusedinSweden(Field,1990;Fieldetal.,1990).Theadvantageisthatthestructureisinsidetheaquaticsystem,solvingproblemsof space availability; furthermore, these are light structures (in contrast toundergroundbasins)andtheconstructioncostislower.A typical layout is shown in Figure 7.39. The basin comprises nine

compartments communicating with each other. Depending on the area of thewatershedandtherunoffvolume,thenumberandsizeofcompartmentscanbedesigned accordingly. The last compartment communicates directly with theaquaticsystem.WhenQ≤Qc, thecombinedsewer flowends into theWWTPandthebasinisfullofcleanwaterfromtheaquaticsystem.WhenQ>Qc,CSOreaches the first compartment of the basin, pushing clean water out into the

secondcompartment.ThiscontinuesasCSOcontinuesuntilsomeorallof thecompartmentsarefilled.IftheCSOvolumeislessthanthecapacityofthebasin,CSO remains entrapped in the basin; otherwise, a portion of it remains in thebasin and some escapes into the aquatic system. After the end of runoff,entrappedCSO ispumpedback to theWWTPfor treatment (Figure7.39) andcleanwaterfillsthebasinagain(NovotnyandOlem,1994).

Figure7.37 DetentionbasinforCSOcontrolplacedinseries.(a)Thebasinisfilling(Q>Qc)and(b)thebasinisfullandspills(Q>Qc).

7.4.5Runofftreatment

Conventional runoff treatment methods, similar to those used in wastewatertreatment,areusuallynotanoption.Thisisduetotherelativelyhighvolumeofrunoff,whichwouldmakeapplicationofsuchmethodsexpensive.Toreducethecost, such methods could be employed only for treatment of the first flush.

Alternatively, treatment can follow collection of runoff in basins, and thenpumpingatslowratesinthetreatmentfacility.Mostofthetime,suchmethodsare used to treat collected combined sewer flow, particularly those based onbiochemicalprocesses.Mostmethodscanbefoundintextbooksonwastewatertreatment and will not be described here. They include those based onphysicochemical processes (e.g. settling, separation) and biochemicalmethods(e.g. biodiscs, anaerobic ponds, aerated ponds, facultative ponds, constructedwetlands).Particularly,forCSO,disinfectionisalsoused.

Figure7.38 DetentionbasinforCSOcontrolplacedinparallel.(a)Thebasinisfilling(Q>Qc)and(b)thebasinisfull(Q>Qc).

Figure7.39 Detentionbasinforcombinedsewerinsidetheaquaticsystem.

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Chapter8Sedimenttransportanderosion

8.1INTRODUCTION

Apart of hydraulic engineering is devoted to the so-called sediment transportscienceorsedimenttransportengineeringorsedimenttransporttechnology.Allthesearenearlysynonymous terms,describing theengineeringscience that (1)dealswiththephenomenaandprocessestakingplacenotonlyinfluvialsystems(on which we concentrate in this chapter) but also on bare soil subjected toprecipitation where sediments are generated, lakes or reservoirs and estuariesreceiving river sediment-laden water and the coastal environment wheresediments are finally deposited and transported alongshore, and (2) providesengineering solutions to alleviate adverse problems of erosion, sedimenttransportanddeposition.Therefore,sedimenttransportphenomenaorprocessesdealwiththeinterrelationshipbetweenthesedimentparticlesandtheflowingornearly stagnantwater.Typical processes include (1) the impact of rainfall thatdrops on soil, soil grain detachment and erosion, (2) sediment entrainment byrunoffandtransportation,(3)channelaggradation/degradationand(4)sedimentdeposition.Theseprocessesareoftendescribedinbrieforinasimplifiedway,aserosion,sedimentmovementortransport,depositionorsedimentation.We will deal with some of these in this chapter. Our emphasis will be on

presentingthebasictheory,addressingsediment-relatedproblemsandprovidingengineeringsolutions.Ofcourse,thechapterisintroductory,doesnotaddressallthese exhaustively, and presents only representative and commonly usedmethods;thereasonforthisisthatthereareseveralbookswrittenspecificallytothe subject of erosion and/or sediment transport. Such books are listed in thereferencelistfortheinterestedreader.So, what may be these sediment-related problems that need engineering

solutions? The following list is not exhaustive but presents the most typical

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problemsrequiringtheknowledgeoferosionandsedimenttransporttheoryandpractice:

Uphillsoilerosionmaycauseslopeinstability.Soilerosionofagriculturallandmayresultinlossoffertilesoils.Urbanization and road or highway construction increase erosion andsedimentyields.River general degradationnear bridges and/or local scourmay endangerbridgestability.Riverbankerosionmayresultinbankfailureandlossofland.Riveraggradationmaycauselocalorgeneralflooding.Sedimentdepositionmayaffectrivernavigabledepth.Sediment-ladenriverscausereservoirsiltingandlossofstoragecapacity,affectingreservoirusefuloroperationallife.Navigableriversiltingmayrequiredredging.Sediment trapping in a reservoir may result in downstream riverdegradation.Sediment trapping in a reservoir may cause coastal sediment starvationandlossofvaluablelandfromcoastlineerosionandretreat.Sediment and turbiditymayhaveadverse ecological impactson riverorlakewildlife(e.g.affectfish).Sedimentsmaybethecarriersandstorageagentsofotherpollutants(e.g.phosphorus,metals,pesticides),whichcanbereleased in timeinaquaticsystemswheresedimentsaredeposited,causingwaterqualityproblems.Sedimentsmayinfluencewaterqualityforwateruses,suchasirrigation,potablesuppliesandrecreation.The presence of sediments may affect natural water aesthetics or theperceptionaboutitsgoodquality.

Typical examples of engineering designs requiring the knowledge of sedimenttransportincludesoilerosionprotectionmeasures,designofdebrisbasins,riverchannelization,hydraulicdesignofbridges,designofriverdegradationcontrolandstabilitymeasures, channelbankprotectionmeasures, foodcontroldesign,reservoir design to minimize silting, and in-stream sand and gravel miningplanning,amongothers.

8.2PROPERTIESOFSEDIMENT

8.2.1Sizeandshape

Size is the basic property of a single particle. However, when dealing withsediments, because of the great variation of particles in a sediment sample,average size values over groups of sizes are commonly used. Themostwell-acceptedandcommonlyusedsedimentsizescaleistheonedevelopedbyLaneetal.(1947),whichispresentedinTable8.1.Todeterminesize,eithersieveanalysis(forsandandlargersizes;Table8.1)

orvisual-accumulationtubeanalysis(forsmallerthansandsizes)isused.Suchanalysesresultinthedeterminationofthesievediameterforasingleparticleinthe sample or the group, and the development of the so-called particle sizedistributioncurve(orsedimentgradationcurve),whichpresentsthevariationofthepercentagesof theweightof thevarioussizes ina sediment sample that issmallerthanagivensievesizeasfunctionofthesize.Threetypicalcurvesarepresented inFigure8.1.Theverticalaxisof this figurepresents thepercentofthe total sampleweight passing, i.e. being finer than a certain sieve size. Thehorizontalaxisofthegraphshowsthissievesize.CurveAinFigure8.1showsasedimentsamplewithfinermaterialcomparedtocurvesBandC.CurvesBandChave,ontheaverage,comparablesedimentsizes;however,thesizesofcurveBarespreadoverawiderrangecomparedtothoseofcurveC,i.e.thesampleofcurveCismoreuniformintermsofparticlesizes.ThesedimentgradationcurveallowsforthedefinitionofdiameterDx,where

x implies a certain percentage on the vertical axis of the graph (Figure 8.1).DiameterDximpliesthesievediameterthatallowsxpercent(%)ofthesamplebyweight topass, i.e. the sievediameter forwhichxpercentof the sample isfiner.Acommonlyuseddiameter isD50,whichimplies thesievediameter thatallows 50% of the sample to pass. Other diameters used in various sedimenttransport formulasandotherequationsareD10,D35,D40,D65,D85,D90,D15.9,D84.1 and others. As an example, in Figure 8.1, for curve A, we haveapproximatelyD50=0.8mm,D80=1.1mmandD10=0.5mm.Similarly, forcurveC,D50=2.3mm,D80=2.65mmandD10=1.95mm.

Table8.1Lane’ssedimentgradesizesClassname Range(mm)

Boulders Verylarge 4000–2000

Large 2000–1000

Medium 1000–500

Small 500–250

Cobbles Large 250–125

Small 125–64

Gravel Verycoarse 64–32

Coarse 32–16

Medium 16–8

Fine 8–4

Veryfine 4–2

Sand Verycoarse 2–1

Coarse 1–1/2

Medium 1/2–1/4

Fine 1/4–1/8

Veryfine 1/8–1/16

Silt Coarse 1/16–1/32

Medium 1/32–1/64

Fine 1/64–1/128

Veryfine 1/128–1/256

Clay Coarse 1/256–1/512

Medium 1/512–1/1024

Fine 1/1024–1/2048

Veryfine 1/2048–1/4096

Source:Lane,E.W.,Trans.Am.Geophys.Union,28(6),936,1947

Figure8.1Typicalsedimentgradationcurves.

Othercommondefinitionsofthesizeofasinglesedimentparticleincludethemean diameter (i.e. the arithmetic average of the longest, intermediate andshortest dimensions of the particle in three perpendicular axes), the nominaldiameter(i.e.thediameterofaspherewiththesamevolumeastheparticle)andthefalldiameter(i.e.thediameterofasphereofspecificgravity2.65whichfallsatthesameterminalvelocitywiththeparticleindistilledwaterat24°C).Inaddition tosize,an importantparameterdescribingparticlecharacteristics

istheshape,whichdescribeshowtheparticlelooks,i.e.itsformregardlessofitssize.Toquantifyshape,therehavebeenseveralparametersproposed.Themostcommonparameter is theshape factor,most frequentlygivenby the followingexpression:

(8.1)

wherea, b and c are the longest, intermediate and shortest dimensions of theparticles,respectively,inthreeperpendicularaxes.

8.2.2Density,specificweightandspecificgravity

Thedensityρs(i.e.particlemassperunitvolume[kg/m3]),thespecificweightγs(i.e.particleweightperunitvolume[N/m3])andthespecificgravitySg(i.e.theratioofparticlespecificweightγstothatofwaterγat4°C)ofsedimentparticlesarealsoimportantparametersenteringsedimenttransportcomputations.Ausualvalueforthespecificgravity,commonlyusedincomputations,is2.65;thisisthespecificgravityofquartzwhichisthemostabundantmineralinriverbedsandsediments.

8.2.3Fallvelocity

The fall velocity is the terminal velocity of a single particle settling in stilldistilledwaterextendinginfinitely.Itcanbecomputedbybalancingtheparticlebuoyantweight that tends tomove the particle downwards and the drag forceresistingthemotion.Themainparametersthenaffectingthefallvelocityaretheparticle size, shape, density and surface roughness, and the fluid density andviscosity. Expressions for fall velocity have been originally developed forsphericalparticles.Then,solutionshavebeenpresentedconsideringthevariousfactors affecting fall velocity of non-spherical particles, including effect ofparticleshape,boundaries, turbulence,sedimentconcentrationandparticlesizedistribution,amongothers.Theearlierdiscussionsareoutsidethescopeofthischapterandcanbefound

in various specific textbooks (i.e. Simons and Sentürk, 1992; Yang, 1996;Vanoni,2006).Here,wewillpresenttwoformulastocomputethefallvelocity,which,however,arevalidwhencertainconditionshold.Thefirstformula,usedinvariousapplicationsinvariousfields,istheclassic

Stokes law (Stokes, 1851). It predicts the fall velocity of a spherical particleaccordingtothefollowingformula:

(8.2)

whereVfisthefallvelocity(m/s)gisthegravitationalacceleration(9.81m/s2)Distheparticlediameter(m)visthekinematicviscosity(m2/s)γsandγarethespecificweightsofparticleandwater(N/m3),respectively

TheformulaisvalidforvaluesoftheparticleReynoldsnumberRep<0.1(whereRepisdefinedbyRep=VfD/v).Thisrangeincludesquartzparticles(Sg=2.65)intherangeofsiltandclay(D<0.062mm;Table8.1).ThesecondformulaisRubey’sformula(Rubey,1933),whichcanbeusedto

computefallvelocitiesofquartzsilt,sandandgravelparticles(therefore,ithasawiderapplication,particularlyforfluvialengineeringapplications),andreads

(8.3)

wherealltheparametershavebeendefinedalreadyandλisgivenby

(8.4)

8.3FLOWRESISTANCE

Comparedtolined(i.e.rigidwall,non-erodible)openchannels,flowresistanceinalluvialriversandstreamsisdifficulttodescribeanddetermine.Onereasonisthatthecross-sectionalshapeandthebedconfigurationcontinuouslychangeastheflowtransportssediments.Various typesofbedformsdevelop;which type

prevails depends on the interrelationship of velocity, Froude number andresistance to flow, among other parameters.While in lined channels the flowresistanceisonlyduetoskinroughness,inalluvialchannelstheso-calledformdragprovidesadditionalresistancetoflow,andtheresultingtotalresistance,alsodependingontheflowdepth,isinmostcasesgreater.

8.3.1Channelflowresistance

Resistance to flow in open channels is presented in various publications (e.g.Chow, 1959; French, 1985; Simons and Sentürk, 1992; Yen, 1992). Threetraditionalequationsareusedforflowresistancecomputationinopenchannels:theDarcy-Weisbach,theChézyandtheManningequations.These,respectively,read

(8.5)

(8.6)

(8.7)

whereVisthemeanflowvelocity(m/s)Risthehydraulicradiusofthecrosssection(m),definedastheratioofflowareatothewettedperimeter

S is the slopeof the energygrade line (m/m),which in the caseofuniformflowcoincideswiththebedandwatersurfaceslope

fistheDarcy-WeisbachfrictionalfactorCistheChézyroughnesscoefficient(m1/2/s)

nistheManningroughnesscoefficient(s/m1/3)

Thethreeresistancecoefficientsf,CandndependontheflowReynoldsnumberRe,theboundaryrelativeroughnessandtheshapeofthechannel(Chow,1959;TsihrintzisandMadiedo,2000).fisexpressedintermsoftheReynoldsnumberin a Moody-type diagram (e.g. Chow, 1959) for the three flow states (i.e.laminar,transitionalandturbulent).SimilardiagramshavealsobeendevelopedforC andn (e.g.Henderson, 1966).However, usually values forC and n aretakenfromtablesasconstants(seethefollowingdiscussion),dependingmostlyonthechannelwallmaterial,conditionandgeometry(e.g.Cowan,1956;Chow,1959;Barnes,1967;ArcementandSchneider,1989;Yen,1992;Tsihrintzisand

Madiedo,2000).Thereasonfor this is that fullydevelopedturbulentflow(i.e.turbulentflowthatdoesnotdependontheReynoldsnumberbutonlyonrelativeroughness) is assumed, which is the case in almost all practical applications.Finally, a channel does not have a constant resistance coefficient under alloccasionsandconditions,butsatisfactorycharacterizationintermsoffrictionalfactorshasbeenachievedforthecaseofsteady,uniform,sediment-freeflowsinchannels of impervious rigid boundaries, with densely distributed, nearlyhomogeneous roughness on the wetted perimeter (Chow, 1959; Coyle, 1975;SimonsandSentürk,1992;Guillen,1996;TsihrintzisandMadiedo,2000).IfwedefnetheshearvelocityV*as

(8.8)

thenEquations8.5through8.7become

(8.9)

(8.10)

(8.11)

From these, one can easily determine the relation between the three flowresistancecoeffcients:

(8.12)

Theshearvelocityisrelatedtotheshearstressт0(orunittractiveforce)exertedbytheflowonthechannelboundarybythefollowingequation:

(8.13)

CombiningEquations8.8and8.13,

(8.14)

From the three presented resistance equations, the most commonly used inpractical applications is the Manning equation. However, its application islimited to channels, smooth or rough, under fully turbulent conditions. In

applying this equation, the proper determination of the Manning roughnesscoefficient n is very important. Chow (1959) presents the following factorsaffecting then value: (1) the surface roughness of thematerial of the channelsection,whichisduetothesizeandshapeofgrains;(2)thevegetationpresent,whichincreaseschannelroughness,withdeterminingfactorstheheight,density,distributionandtypeofthevegetation(TsihrintzisandMadiedo,2000);(3)thechannel irregularity, accounting for factors such as changes in cross-sectionalshapeandsize,and irregularitiesof thebedalong thechannel; (4) thechannelalignment,where sharpchangesofcurvature, turnsandmeandering increasenvalues; (5) silting and scouring,with both changing then values of lined andunlinedchannels;(6)obstructions,suchasbridgepiers,whichincreasenvalues;(7)waterdepth,withngenerallydecreasingasdepthincreases;and(8)sizeandshape,whicharefactorswithonlyaminorornoinfluenceonnvalues.Cowan (1956) developed a procedure to estimate then value, based on the

followingformula,whichconsidersmostoftheaforementionedfactors:(8.15)

wheren0 is thebasicnvalueforastraightuniformchannelofacertaincross-sectionalmaterialandn1,n2,n3andn4areincreasesofn0accountingforsurfaceirregularities,variationinshapeandsizeofthechannelsection,obstructionsandvegetation, respectively. Factorm accounts for channelmeandering. Table 8.2(Chow, 1959) provides values of factors n1– n4 and m based on channelconditions. Cowan’s (1956) method applies according to the followinglimitations (Chow, 1959): (1) Water is sediment free. (2) The method is notapplicableforchannelswithhydraulicradiuslargerthan5m.(3)Channelsareunlinedwithaminimumn0of0.020.Acomprehensivesetofnvaluesforvariouschannelmaterialsandconditions

arepresentedinTable8.3(afterChow,1959).Photographsofchannelsectionsassociatedwithn values areprovidedbyChow (1959),Barnes (1967),French(1985), Arcement and Schneider (1989) and Chaudhry (1993), among others,which canhelp to determine, by comparison, the propern for a given stream,riverorfloodplain.

Table 8.2 Cowan (1956) procedure for the computation of the Manningroughnesscoefficient

ParameterinEquation8.15 Condition Description Values

n0–materialofcrosssection Earth Basicnvalueforstraight,uniform,smoothchannelsinnaturalmaterials

0.020

Rockcut 0.025

Finegravel 0.024

Coarsegravel 0.028

n1–degreeofirregularity Smooth Bestattainableforthematerialsinvolved 0.000

Minor Gooddredgedchannels,slightlyerodedorscouredsideslopesofcanals

0.005

Moderate Fair-to-poordredgedchannels,moderatelysloughedorerodedsideslopesofcanals

0.010

Severe Badlysloughedbanksofnaturalstreams,badlysloughedorerodedsideslopesofcanals,unshaped,jaggedandirregularsurfaceofchannelsinrock

0.020

n2–sectionshapeandsizevariation

Gradual Gradualchangeinsizeorshapeofcrosssection

0.000

Alternatingoccasionally

Occasionallyalternatinglargeandsmallsectionsoroccasionalshapechangesandshiftingormainflowfromsidetoside

0.005

Alternatingfrequently

Frequentlyalternatinglargeandsmallsectionsorfrequentshapechangesandshiftingormainflowfromsidetoside

0.010–0.015

n3–obstructions Negligible Presenceandcharacteristicsofobstructions,i.e.debris,treeroots,bouldersandlogs,withfactorsconsideredsuchasthedegreeofreductionofwaterarea,characterofobstructionsininducingturbulence(sharpedge,smoothsurfaced)andposition/spacingofobstructionsinlongitudinalandtransversedirections

0.000

Minor 0.010–0.015

Appreciable 0.020–0.030

Severe 0.040–0.060

n4–vegetation Low Densegrowthsoffexibleturfgrasses,flowdepth2–3timesthevegetationheight;orfexibletreeseedlings,flowdepth3–4timesthevegetationheight

0.005–0.010

Medium Turfgrasses,flowdepth1–2timesthevegetationheight;orstemmygrasses,weedsortreeseedlings,moderatecover,flowdepth2–3timesthevegetationheight;orbrushygrowths,moderatelydenseonchannelsidesslopes,novegetationonthechannelbottom,hydraulicradiusgreaterthan0.6m

0.010–0.025

High Turfgrasses,flowdepthaboutvegetationheight;ordormantseason(similartowillowsorcottonwood8–10yearsold,someweedsandbrushes,nofoliage,hydraulicradiusgreaterthan0.6m);orgrowingseason(similartobushywillows1-yearold,someweedsinfullfoliage,denseonchannelsidesslopes,nosignificantvegetationonthechannelbottom,hydraulicradiusgreaterthan0.6m)

0.025–0.050

Veryhigh Turfgrasses,flowdepthlessthanhalfofvegetationheight;orgrowingseason(similartobushywillows1-yearold,weedsinfullfoliageonchannelsideslopesordensegrowthofcattailsonthechannelbottom,hydraulicradiusupto3–5m);orgrowingseason(similartotreeswithweedsandbrush,allinfullfoliage,hydraulicradiusupto3–5m)

0.050–0.100

m–meandering Minor Ratio(meanderlengthtothestraightlengthofthechannelreach)1.0–1.2

1.000

Appreciable Ratio1.2–1.5 1.150

Severe Ratio>1.5 1.300

Source:AdaptedfromChow,V.T.,Open-ChannelHydraulics,McGraw-HillInc.,NewYork,1959.

TheManningnforalluvialchannelshasbeenrelatedbyvariousresearcherstothesedimentparticlesizeofthebed.Obviously,thisnvaluereferstotheskinorsurface roughness of the bed. Typical suggested formulas have the followinggeneralform:

(8.16)

whereDx, as mentioned earlier, is a certain representative diameter (m), with xrepresenting the percentage of finer material according to the gradationcurveofthebedmaterial

bisanempiricalcoefficient

Table8.4summarizesthecoefficientsandunitsofvariousformulasofEquation8.16 (Simons and Sentürk, 1992). Table8.5 presents a comparison ofn valuepredictionsbyalltheseformulas,whichisalsodepictedinFigure8.2.Onecansee that although a different representative diameter Dx is used in various

formulas, predictions are relatively close to each other, with the exceptionprobably of Keulegan’s Equations 8.16d and e. The Meyer-Peter and Müller(1948)formula(Equation8.16c)compareswellwithStrickler’s(1923)formulas(Equations8.16a andb) andKeulegan’s formula (1949) (Equation8.16f). TheMeyer-Peter and Müller (1948) formula (Equation 8.16c) is a well-testedequationandisrecommendedforsandandgravelbedsofalluvialstreams.TheLaneandCarlson(1953)formula(Equation8.17e)predictsslightlyincreasednvaluesandisapplicableforcobbles.

8.3.2Flowresistanceinalluvialstreamsandrivers

As mentioned earlier, in addition to skin resistance discussed in the previoussection, excess flow resistance occurs in alluvial channels with sedimenttransport due to the creation of bed forms in the movable bed. The resultingoverallresistancemaybeconsiderablyhighercomparedtoskinfrictionwithoutbedforms.

Table 8.3 Chow’s (1959) values for the Manning roughness coefficient n forexcavatedearthenandnaturalstreams

Manningnvalues

Description Min Normal Max

A.Excavatedordredgedchannels

a.Earth,straightanduniform

1.Clean,recentlycompleted 0.016 0.018 0.020

2.Clean,afterweathering 0.018 0.022 0.025

3.Gravel,uniformsection,clean 0.022 0.025 0.030

4.Withshortgrass,fewweeds 0.022 0.027 0.033

b.Earth,windingandsluggish

1.Novegetation 0.023 0.025 0.030

2.Grass,someweeds 0.025 0.030 0.033

3.Denseweedsoraquaticplantsindeepchannels 0.030 0.035 0.040

4.Earthbottomandrubblesides 0.028 0.030 0.035

5.Stonybottomandweedybanks 0.025 0.035 0.040

6.Cobblebottomandcleansides 0.030 0.040 0.050

c.Dragline—excavatedordredged

1.Novegetation 0.025 0.028 0.033

2.Lightbrushorbanks 0.035 0.050 0.060

d.Rockcuts

1.Smoothanduniform 0.025 0.035 0.040

2.Jaggedandirregular 0.035 0.040 0.050

e.Channelsnotmaintained,weedsandbrushuncut

1.Denseweeds,highasflowdepth 0.050 0.080 0.120

2.Cleanbottom,brushonsides 0.040 0.050 0.080

3.Same,higheststageoffow 0.045 0.070 0.110

4.Densebrush,highstage 0.080 0.100 0.140

B.Naturalstreams

B.1Minorstreams(topwidthatfoodstage<30m)

a.Streamsonplain

1.Clean,straight,fullstage,noriftsordeeppools 0.025 0.030 0.033

2.Sameas1above,butmorestonesandweeds 0.030 0.035 0.040

3.Clean,winding,somepoolsandshoals 0.033 0.040 0.045

4.Sameas3above,butsomeweedsandstones 0.035 0.045 0.050

5.Sameas4above,lowerstages,moreineffectiveslopesandsections 0.040 0.048 0.055

6.Sameas4above,butmorestones 0.045 0.050 0.060

7.Sluggishreaches,deeppools 0.050 0.070 0.080

8.Veryweedyreaches,deeppoolsorfloodwayswithheavystandoftimberandunderbrush

0.075 0.100 0.150

b.Mountainstreams,novegetationinchannel,banksusuallysteep,treesandbrushalongbankssubmergedathighstages

1.Bottom:gravels,cobblesandfewboulders 0.030 0.040 0.050

2.Bottom:cobbleswithlargeboulders 0.040 0.050 0.070

B.2Floodplains

a.Pasture,nobrush

1.Shortgrass 0.025 0.030 0.035

2.Highgrass 0.030 0.035 0.050

b.Cultivatedareas

1.Nocrop 0.020 0.030 0.040

2.Maturerowcrops 0.025 0.035 0.045

3.Maturefieldcrops 0.030 0.040 0.050

c.Brush

1.Scatteredbrush,heavyweeds 0.035 0.050 0.070

2.Lightbrushandtrees,inwinter 0.035 0.050 0.060

3.Lightbrushandtrees,insummer 0.040 0.060 0.080

4.Medium-to-densebrush,inwinter 0.045 0.070 0.110

5.Medium-to-densebrush,insummer 0.070 0.100 0.160

d.Trees

1.Densewillows,summerstraight 0.110 0.150 0.200

2.Clearedlandwithtreestumps,nospouts 0.030 0.040 0.050

3.Sameas2above,butwithheavygrowthofspouts 0.050 0.060 0.080

4.Heavystandoftimber,afewdowntrees,littleundergrowth,foodstagebelowbranches

0.080 0.100 0.120

5.Sameas4above,butwithfoodstagereachingbranches 0.100 0.120 0.160

B.3Majorstreams(topwidthatfoodstage>30m)withthenvaluelessthanthatforminorstreamsofsimilardescription,becausebanksofferlesseffectiveresistance

a.Regularsectionwithnobouldersorbrush 0.025 — 0.060

b.Irregularandroughsection 0.035 — 0.100

Table8.4Coefficients ofEquation8.16 for surface roughnessn determinationbasedonthesedimentparticlesizeaccordingtovariousstudies

Note:Ift=0.3048m;Iin.=0.0254m.

Table8.5ComparisonofpredictionsofManningn by thevarious formulasofEquation8.16

Figure8.2ComparisonofpredictionsofManningnbythevariousformulasofEquation8.16.

Thevariousformsthatcanbedevelopedinanalluvialchannelwithsedimenttransportarethefollowing(SimonsandSentürk,1992;Yang,1996):(1)ripples(i.e. smallbed formsoccurringas sandwaveson thebedwithwavelengthsofabout30cmandheightof5cm);(2)dunes(i.e.largersandwavesthanripples);(3) plane bed, where there are no elevations or depressions on the bed; (4)antidunesorstandingwaveswith their flowmovingdownstreamandbedsand

wavesandsurfacewavesinphasemovingupstream;(5)chutesandpoolswhicharelargeandlongaccumulationsofsediment;and(6)barswhicharesedimentaccumulations having dimensions comparable to the channel width andmeandepth.When thematerial ismedium sand and finer (D50 < 0.6mm), the bedforms are created by the flow in the following order as the stream power(definedasτ0Vwithτ0 takenfromEquation8.14)increases:planebedwithoutsediment transport, ripples, ripples superimposed on dunes, dunes,washed-outdunes, plane bed with sediment transport, antidune standing wave, antidunebreaking wave, chutes and pools. Figure 8.3 depicts this. For coarser bedmaterial(D50>0.6mm),thefirstbedformatsmallstreampowerisdunes(noripplesareformed).AsFigure8.3shows,similartothestreampowerincrease,aswemovefromripplestoantidunes,thevelocityandtheFroudenumber(Fr)increase while the resistance to flow generally increases from a small valueequivalenttotheskinfrictionvalue(e.g.estimatedbyEquations8.16)providedbythesedimentgrains(plainbed—nosedimenttransport)tothelargestvaluewhendunesarecreated.Then,flowresistancefallstovaluescomparabletotheskinfrictionroughnessforplainbedwithsediment transport,andthenagain itincreaseswhenantidunesaretheprevailingbedforms.AqualitativeschematicdiagramforthischangeisshowninFigure8.4.

Figure8.3Bedformsinalluvialstreams.

Figure8.4Bedformsandqualitativeflowresistance.

Generally, ripples and dunes are associatedwith the lower flow regime forsubcriticalflows(smallstreampower),whileplanebedwithsedimenttransport,antidunes,andchutesandpoolsoccurintheupperflowregimeforsupercriticalflows (larger streampower).There is also the transition regimewhere thebedconfigurationchangesfromdunestoplanebedandtheflowisclosetocritical.Asmentioned, the lower flow regime is characterized by increased resistanceandlowsedimenttransport.Theopposite(smallresistance,significantsedimenttransport)istruefortheupperflowregime.Which bed form develops depends on various parameters, including the

(Yang, 1996) (1) flow depth and mostly the relative roughness (i.e. ratio ofroughness element dimension to flowdepth), (2) channel slope, (3)sediment–water mixture density, (4) bed material gradation, (5) fall velocity, (6)temperatureand(7)shapeofthechannelcrosssection.Topredictthebedform,severalmethods,mostlygraphical, havebeendeveloped, relatingvarious flowandsedimentparameters.AnevaluationofthesemethodsispresentedbyYang(1996). Commonmethods include that by Engelund andHansen (1966), whoused the mean velocity, the shear velocity (Equation 8.13) and the Froudenumberinagraphtopredictbedforms.Similarly,SimonandRichardson(1966)used a graphical method based on the stream power (Equation 8.14) and themedianfalldiametertomapareaswhereeachbedformdevelops(Figure8.5).Asmentioned,bedformssignificantlyaffectresistancetoflow(Figure8.4)by

exertingbedformdraginadditiontograinskinfriction.Onewaytoaddressthisproblemisbycomputingseparately theresistancedueto thegrainsanddueto

the bed forms, and adding the two together. Therefore, the total shear stress(Equation8.14)canbewrittenas

(8.17)

where and aretheshearstressesduetoskinfrictionandbedformroughness,respectively,andR′andR″arethepartsofthehydraulicradiusR(R=R′+R″)which are related to skin and form resistance, respectively. Using the sameapproach, the roughness coefficient (either Darcy–Weisbach f or Chézy C orManningn)canbeseparatedintotworesistancecoefficientparts.Forexample,forManningn

(8.18)

Asmentioned,valuesforn′canbeestimatedusingeitherEquation8.16or thevarious tablespresented (e.g.Table8.3).Figure8.6 presents typical values fortotaln(Equation8.18)forvariousbedformsinafine-to-mediumsandchannel.Onecanrelate thesevalues to thequalitativeManningnvariationpresentedinFigure8.4.Thegraphpresentsarange(minimumtomaximum)ofnvalues.Forfood control design in alluvial streams, the maximum curve is recommendedbecause it gives conservative depths, while for sediment transport and scourdesign, the minimum or intermediate value curves can be used to computeconservativevelocities.

Figure 8.5 Bed form as function of stream power and median fall diameter. (From Simons, D.B. andRichardson, E.V., Resistance to Flow in Alluvial Channels, U.S. Geological SurveyProfessionalPaper422-J,1966.)

Figure8.6BedformManningn″valuesforfine-to-mediumsandchannels.

Several methods have been developed, based on the approach to separategrainandbedformroughness, tocompute total resistance inalluvialchannels.The well-known methods include those by Einstein (1950), Einstein andBarbarossa (1952), Engelund and Hansen (1966), Alam and Kennedy (1969),RichardsonandSimons(1967),andShen(1962),amongothers.WepresenthereEinstein’s method, while details on the others can be found in specializedsedimenttransporttextbooks(e.g.SimonsandSentürk,1992).AccordingtoEinstein(1950),themeanflowvelocityViscomputedbasedon

thehydraulicradiusR′duetograinroughnessasfollows:

(8.19)

where is the shear velocity due to grain roughness (calculated by Equation 8.8usingR′)(m/s)

ksistheequivalentgrainroughnessdimension(m)(Einsteinsetks=D65)x(−)isaparameterdependingontheratioks /δ=D65/δ,whereδ(m)is thelaminarboundarysub-layerthicknessgivenby

(8.20)

wherevisthekinematicviscosity(m2/s).TheparameterxisgiveninFigure8.7as function of ks/δ =D65/δ. Rao and Kumar (2009) developed the followingequation to predict Einstein’s parameter x, whichwas testedwithNikuradse’s

dataforpipesandapproximatesthegraphofFigure8.7;soinsteadofthegraphofFigure8.7,thefollowingequationcanbeused:

(8.21)

Figure 8.7 Values of parameter x in Equation 8.19. (From Einstein, H.A., The bed load function forsedimenttransportationinopenchannelflows,TechnicalBulletinNo.1026,U.S.DepartmentofAgriculture,Washington,DC,1950.)

Similartothegrainroughness,Einstein(1950)proposedthefollowingfunctiontocomputethemeanflowvelocityVbasedonthehydraulicradiusR″duetobedformroughness:

(8.22)

1.

2.3.4.5.6.

7.

wherefimplies‘functionof’Sgisthespecificgravity(−)Sistheslope(−)

ThefunctionfwasdevelopedbyEinsteinandBarbarossa(1952)basedonfielddata,andispresentedinFigure8.8.

Figure8.8Einstein’sfunctionf(Equation8.22).(FromEinstein,H.A.andBarbarossa,N.L.,Trans.ASCE,117,1121,1952.)

TheEinsteinmethodcanbeusedbasedon thefollowingstepssuggestedbyEinsteinandBarbarossa(1952)torelatedischargeandtotalhydraulicradiusofagivenalluvialstreamandflowcondition.IfthedischargeQisgiven,performthefollowing:

Collect data on slope S, sediment sizes (D65, D35) and cross-sectionaldimensions. Cross-sectional dimension data can be used to produce arelationship (i.e. a rating curve) between flow areaA and total hydraulicradiusR,i.e.A=f1(R)(wheref1implies‘functionof’).AssumeavalueforthegrainroughnesspartofthehydraulicradiusR′.Computetheshearvelocity basedonEquations8.13and8.14: .UseeitherFigure8.7orEquation8.21todetermineparameterx.DeterminethemeanflowvelocityVusingEquation8.19settingks=D65.

ComputefromEquation8.22Einstein’sparameter: .UsethecomputedΨ′valuetodeterminetheratio fromFigure8.8.

8.9.10.11.12.13.

UsethemeanflowvelocityVvaluecomputedinStep5todetermine .UseEquations8.13and8.14 todetermineR″.ComputeR=R′+R″.UsetherelationshipA=f1(R)fromStep1tocomputetheflowareaA.Computethedischarge:Q=AV.IfthecomputedQinStep12isequaltothegivenone,thentheprocedureisfinished.Otherwise,gotoStep2andassumeanewvalueforR′andrepeattheprocedureuntilitconverges.

If the totalhydraulic radiusR isnowgiven insteadof thedischargeQ, thenfollowthesameSteps1–10andcheckifthecomputedvalueR(Step10)agreeswiththegivenone.Ifitdoes,theprocedureisfinished;otherwise,assumeanewvalue for R′ and repeat the procedure. Then, follow Steps 11 through 12 tocomputeAandQ.

8.4INCIPIENTMOTION

Incipientmotion is thecommonlyused term(other terms includebeginningofmotion,initiationofmotion,initialmotionandcriticalmovement)fordescribingtheflowconditionsleadingtothemovementofasedimentparticlelyingonthestream bed. The theory presented here is important, because, in addition tosedimenttransportandchanneldegradationprocesses,itcanbeusedtodesignastablechannelortosizeriprapforbankprotection.Severalmethodshavebeenproposedtoaccessincipientmotion.Herewewill

presentsomecommonmethodswhileseveralotherscanbefoundinspecializedtextbooks.Onewell-knownmethodhasbeendevelopedbyShields(1936)basedon similarity analysis. Although it has received criticism, it still findsapplication. The method uses two parameters in a graph (Figure 8.9). TheverticalaxiscontainsthedimensionlessShieldsparameter,τ*,definedby

(8.23)

whereτcisthecriticalshearstressformotion(N/m2)γsandγarethesedimentandwater-specificweightsDsisthemeansedimentsize

ThehorizontalaxiscontainsthecriticalboundaryReynoldsnumber,R*,defined

by

(8.24)

wherev is the kinematic viscosity (m2/s) and the other parameters are alreadydefined.TheoriginalShieldsgraphcontainedonlyexperimentaldata.The linethroughthedatawasaddedlaterbyRouse(1939).ThegraphofFigure8.9alsocontains a family of parallel lines, added later by Vanoni (1964) and notcontained in the original graph (Shields, 1936), representing the followingdimensionlessparameter (Equation8.25).Theuseof thisparametermakes theuseofthegrapheasieravoidingnecessaryiterationsbecausetheparameterDsiscontainedinbothaxes:

(8.25)

It is noticed that to the right (fully developed turbulent flow), τ* tends to theconstant0.060,whileMeyer–PeterandMüller(1948)suggestedthevalue0.047instead, a value also suggested by several other researchers. ForR*= 10, thelowestvalueofτ*occurs,whichisabout0.033.Thus,ithasbeensuggestedtouse this value, irrespectiveof theR value, as a conservative criteriondefiningmotion or no motion (Shen and Julien, 1993). Alternatively, a safety factor(usually2–4)canbeusedwhenflowisturbulent(ShenandJulien,1993).

Figure 8.9 Shields graph of incipientmotion. (Adapted fromVanoni,V.,Measurements of critical shearstress for entraining fine sediments in a boundary layer, Report KH-R-7, W.M. KeckLaboratoryofHydraulicsandWaterResources,California InstituteofTechnology,Pasadena,CA, 1964;Vanoni, V.A., ed., Sedimentation Engineering, ASCE Manuals and Reports on

EngineeringPracticeNo.54,AmericanSocietyofCivilEngineers,Reston,VA,2006.)

TheShieldsmethodappliesfornearlyuniformbedmaterialsofsizeDs.Whenthe material is non-uniform, the phenomenon becomes more complex. Thefactors affecting it include, among others, armouring of the bed (i.e. largerparticlesprotect finer particles frommovingby the flow), and considering thefactthatlargerparticlesprotrudemorethanfinerparticles,thus,largerparticlesmay move first. The following criterion has been proposed in these cases todefinestability(ShenandJulien,1993):

(8.26)

Another method for incipient motion determination was developed by Yang(1973) based on the balance of forces acting on the sediment particle. ThemethodcomputesacriticalvelocityVcatincipientmotion:

(8.27)

whereR*isthecriticalboundaryReynoldsnumber(Equation8.24)Vfisthefallvelocity(m/s)

Finally,Chow (1959) presents permissible velocity, i.e.maximumnon-erosivemean velocity (originally developed by Fortier and Scobey, 1926) and criticalunit tractive force values (Table 8.6), while the U.S. Bureau of Reclamationpresents the graph of Figure 8.10 to compute critical unit tractive force asfunctionofthesedimentsize.The aforementioned methods (values of permissible velocities or shear

stresses computed from the Shields graph or values of critical unit tractiveforces) can be used to design stable channels. Regarding the shear stress at achannelsection,thisisnotconstantthroughoutthecrosssection.ThemaximumshearstressiscomputedbyEquation8.14whenthechannel iswide(width-to-depth ratio greater than 6.0). In other cases, the unit tractive force is notuniformly distributed in the channel section but presents different maximumvaluesonthesidesandthebed.Then,Equation8.14reads

(8.28)

where αmax is a coefficient less than one. Values for αmax were computed

theoretically for various channel sections (i.e. trapezoidal and rectangular) andare given in the graph of Figure 8.11 adopted from the U.S. Bureau ofReclamation(Chow,1959)asafunctionoftheratiobedwidthbtoflowdepthd.Asanexample,maximumvaluesforαmaxare0.97and0.75forthebedandthebank,respectively,ofatrapezoidalcanalforb/d=4andsideslopeofthebank1.5–1(horizontal tovertical).Theaforementionedmethodscanalsobeusedtodesignappropriateriprapsizeforchannelprotection.

Table 8.6 Permissible velocities and critical tractive forces recommended byFortierandScobey(1926)

Source:Chow,V.T.,Open-ChannelHydraulics,McGraw-HillInc.,NewYork,1959.

8.5SEDIMENTTRANSPORTFORMULAS

The sediment transport formulas are used to compute the sediment transportcapacity of a given stream cross section or reach under specific hydraulicconditions and bed characteristics. Several formulas have been developed andare presented in various textbooks.Here, only typical classic andwell-knownformulaswillbepresentedand theirusewillbeexplained.Wewillutilize thefollowingcommonlyusedterms:

•••

•

•

•

•

•

•

•

Bedlayer:Theflowlayeradjacentandimmediatelyabovethebed,whichisusuallyconsideredabouttwograindiametersthick.Bedmaterial:Themixtureofallsedimentsizesfoundinthestreambed.Sedimentload:Thesedimenttransportrateexpressedaseitherweightperunittimeorvolumeperunittime.Bedload(BL):Thesedimentquantitythatmoveswithinthebedlayer.Themechanisms of movement include saltation (i.e. picking from anddropping on the bed of a particle by turbulence), rolling and sliding ofsedimentparticles.Suspendedload(SL):Thesedimentquantitythatstaysduetoturbulenceinsuspensioninthewatercolumn(i.e.betweenthetopofthebedlayerandthewatersurface)andistransportedbythestreamflow.Total sediment load (TSL): The total sediment quantity in the stream,whichis thesumofBLandSL.Usually, theBLvariesbetween5%and25%oftheSL.Washload(WL):Thepartofthetotalloadcomprisingparticlesfinerthanthoseexistinginthebedmaterialmixture.Mostpartofthisloadoriginatesfromerosion in theupperwatershed and thebanks.TheWLmakesonepart of the SL, the other part being the suspended bed material load(SBML; i.e. coarser particles in suspension also existing in the bedmaterial).Bed material load (BML): The part of the total load comprising sizesfoundinthebedmaterial.Mosttimes,theBMLisconsideredtohavesizesgreater than 0.0625mm (i.e. the limit between sand and silt; see Table8.1).TheSBMLandtheBLmaketheBML.Sedimentconcentration:Thequantityof sediment in agivenquantityoffluvial water transporting the sediment. The sediment and/or waterquantity can be measured by weight or by volume. Usually, sedimentconcentrationisexpressedastheratioofsedimenttransportrate(masspertime) to discharge (volume per time), in units mg/L or kg/m3.Alternatively,concentrationhasalsobeenexpressedastheratioofweightofsediment to total (water+sediment)weightorvolumeofsediment tototalvolume.Thesedimentconcentrationhasaverticaldistributioninthewater column,which generally differs from that of the velocity. Typicalqualitative velocity and sediment concentration profiles are shown inFigure8.12.Sediment discharge: The quantity of sediment passing a given streamcross section in unit time. We have bed material, suspended and totalsedimentdischarges,respectively.

Figure8.10Criticalunittractiveforceasfunctionofsedimentsize.(FromLane,E.W.,ProceedingsoftheASCE,79,1953;U.S.BureauofReclamation,DesignofSmallDams,U.S.DepartmentoftheInterior,BureauofReclamation,Washington,DC,816pp.,1977.)

Figure8.11Ratioof amaximumunit tractive forcedividedby shear stress (Equation8.28) in a channelsection as functionof ratiob/d. (FromChow,V.T.,Open-ChannelHydraulics,McGraw-HillInc.,NewYork,1959.)

Figure8.12Typicalqualitativevelocityandsedimentconcentrationprofilesinariver.

The relation between the various types of sediment load is presented in thefollowingequations:

(8.29)

where(8.30)

(8.31)

8.5.1Bedload

BL is the quantity of transported bed material; transport occurs when criticalconditions for motion of the bed material (see previous section on incipientmotion)prevail.Severalformulashavebeenproposedbyvariousresearchers,assummarizedinvariousspecializedtextbooks.Here,weindicativelypresenttworepresentative formulas: the Meyer–Peter and Müller (1948) formula and theEinstein(1950)formula.

8.5.1.1Meyer–PeterandMüllermethod

TheMeyer–PeterandMüller (1948) formula,which isoneof themostwidelyanddiachronicallyusedandacceptedformulas,reads

(8.32)

whereqBm is theBL transport ratebydryweight (mass)perunitwidthofchannel(kg/s/m) (to express the transportbyBLvolumeqBv (m3/s/m),onehas todividetheright-handsideoftheEquation8.32bythespecificweightγs)

τo is the boundary shear stress (unit tractive force) given by Equation 8.14(N/m2)

τc(N/m2)isthecriticalshearstress,which,forfullydevelopedturbulentflow,isgivenby(seealsotheShieldsdiagram;Figure8.9)

(8.33)

AcommonlyusedformofEquation8.32(transportbyvolume),whichassumesSg=2.65(i.e.quartz),is

(8.34)

In computing τo, the hydraulic radius due to grain friction R′ and thecorrespondingManningn′areused.Wenotethat,unlikeEinstein’sapproachforgrainandbedformroughness (i.e.Equations8.17and8.18),Meyer–PeterandMüller(1948)assumedthat

(8.35)

whereR′andRare therespectivehydraulic radiidue tograinfrictionand the total(m)

S′andSaretherespectiveenergyslopesduetograinfrictionandtotaln′andnaretherespectiveManningnvaluesduetograinfrictionandtotal

TocomputeManningn′,Equation8.16c(Table8.4) isusedwithDx=D90 (m)andb=26.Equation8.32canalsobeusedtocomputethetransportofmaterialcontaining

certain different sizes. In this case, amean sediment sample diameter is usedinsteadofDs,asfollows:

(8.36)

wherepiisthepercentagebyweightofacertainDsiinthesedimentsample.

8.5.1.2Einsteinmethod

Einstein’s(1950)formulaisbasedonthemostcomprehensivetheoryamongallformulas.Furthermore,ithasbeendemonstratedthatseveralotherproposedBLformulas, followingmathematicalmanipulation, canbeconverted toEinstein’s(ShenandJulien,1992).Einstein’s(1950)BLformularelatestwodimensionlessparameters:oneisthedimensionlesstransportrate,Φ,definedby

(8.37)

andtheotheristhedimensionlessflowintensity,Ψ′,alreadydefinedinEquation8.22

(8.38)

whereqBwistheweightoftransportedBLperunittimeandchannelunitwidth(N/s/m)andtheotherparametershavealreadybeendefined.ThetwoparametersΦ andΨ′ of Equations 8.37 and 8.38 can be used to predict the transport ofnearly uniform bed material, in which caseD35 is used as the representative

sedimentsize.Einstein(1950)proposedforthisthefunctionrelatingΦandΨ′,whichisbasedonfielddataandispresentedinFigure8.13.AccordingtoShenandJulien(1992),severalotherBLformulas(e.g. thosebyEngelund–Fredsoe,Yalin,Meyer–Peter, Ackers andWhite andBagnold) can be converted to fallaroundoronthiscurve,showingthegeneralvalidityofEinstein’sprocedure.In the case of non-uniform bedmaterial, Einstein (1950) proposed a separatecomputationofBLtransportofeachsizeclass.Inthiscase,Equation8.37isalsousedtocomputethedimensionlesstransportrate forclass i, i.e. inEquation8.37qBwisreplacedbyqBwi,whichreferstothetransportofthemeangrainsizeDSiofclass iwhosepercentagebyweightinthebedsampleispBi(thus,Ds inEquation 8.37 is replaced by Dsi). The dimensionless flow intensity ofEquation8.38alsoreferstoDsi(therefore,D35isreplacedbyDsi).Furthermore,thecorrecteddimensionlessflowintensity iscomputedinthiscase,whichisgivenby

(8.39)

whereξi is thehiding correction factor, accounting forhidingof small grainswithsizes less than sizeX by larger grains (Einstein’s Figure 8.14 is used toestimateξiforagivensizeDsiasfunctionoftheratioDsi/X)

Y is the pressure correction factor, accounting for the different resistance toflowprovidedbygrainsofdifferentsizes(Einstein’sFigure8.15isusedtoestimateYasfunctionof theratioD65/δ,whereδ is the laminarboundarysub-layerthicknessgivenbyEquation8.20)

x is the parameter already presented in Equation 8.19, which is given asfunction of the laminar boundary sub-layer thickness δ in Figure 8.7 (or,alternatively,itcanbecomputedbyEquation8.21)

Figure8.13FunctionrelatingΨ′andΦinEinstein’s(1950)bedloadformula.

Finally,thegrainsizeXiscomputedbythefollowingformulas:

(8.40)

Accordingtothepreviousdiscussion,theproceduretoestimateBLtransportisas follows: Compute δ from Equation 8.20. Compute x from Figure 8.7 orEquation8.21.ComputeX fromEquation8.40.Compute ξi for each sedimentclassfromFigure8.14.ComputeYfromFigure8.15.Compute fromEquation8.38foreachsedimentclass.Compute fromEquation8.39.Compute usingFigure8.13foreachsedimentclass.Then,theBLtransportrateqBwiscomputedfollowingEquation8.37asfollows:

(8.41)

Figure 8.14 Estimate of hiding factor ξ. (From Einstein, H.A., The bed load function for sedimenttransportation in open channel flows, Technical Bulletin No. 1026, U.S. Department ofAgriculture,Washington,DC,1950.)

Figure 8.15 Estimate of pressure correction factor Y. (From Einstein, H.A., The bed load function forsedimenttransportationinopenchannelflows,TechnicalBulletinNo.1026,U.S.DepartmentofAgriculture,Washington,DC,1950.)

8.5.1.3Einstein–Brownmethod

ThismethodwasbasedonEinstein(1942)formulaandwaspresentedbyBrown(1950). It isbasedon thefunctional relationofΦandΨwhich ispresented inFigure8.16,where

(8.42)

is a dimensionless transport rate by mass (see Equation 8.37) and Ψ is theinverseoftheShieldsparameterτ*(Equation8.23):

(8.43)

Finally, λ is a term we have also seen in Rubey’s formula for fall velocity(Equation8.4):

(8.44)

Figure 8.16 Einstein–Brown function . (From Brown, C.B.: Sediment transportation, inEngineering Hydraulics, H. Rouse, ed. 1950. Copyright Wiley-VCH Verlag GmbH & Co.KGaA.)

AsseeninFigure8.16,when ,Equation8.42reads

(8.45)

8.5.2Suspendedload

AspreviouslymentionedandpresentedinFigure8.12, theverticaldistributionof the suspended sediment concentration in the water column has a specialprofile,showinglowvaluesatthesurfaceandincreasedvaluesatthebed.Thisisaresultoftransportedsedimentparticlesfallingduetogravitytowardsthebedandotherparticlesbrought into suspensionby turbulence from thebed.Underequilibrium conditions, Rouse (1937) derived the following equation for thesedimentconcentrationdistributionwithdistanceabovethebed:

(8.46)

whereC(y)isthesuspendedsedimentconcentrationatdistanceyabovethebed(m)Caisthesedimentconcentrationatareferencelevelaabovethebed

ThelevelaisthatbelowwhichwehaveBLtransportandisusuallytakenasa=2Ds;d is the water depth (m) (note that for a wide channelR′ ≈ d); and theexponentzisgivenby

(8.47)

whereVf is the fallvelocity (m/s) (Rubey’s formula–Equations8.3and8.4, canbeusedtocalculateit)

V*istheshearvelocity(m/s)κ is the von Karman constant (≈0.4 for clear water, but it decreases withincreasing sediment concentration, as presented by Einstein and Chien,1953,1954)

Einstein(1950)replacedVfwith inEquation8.47.Tocomputethesuspendedsediment loaddischarge fora specific sediment sizeclass,he integrated in thedirectionperpendiculartothebedabovethereferencelevela:

(8.48)

whereqswiistheSLdischargeperunitchannelwidthbyweight(N/S/m)

V(y)isthevelocityatdistanceyfromthebed(m/s)C(y)thesedimentconcentrationbyvolumeatdistancey(m3/m3)

ForthevelocityV(y),thefollowingformulawasused:

(8.49)

wherexistheparameterofEquation8.19andFigure8.7.Notethatthisequationwas used after integration to derive Equation 8.19 for the mean velocity.IntroducingEquations8.47and8.49intoEquation8.48,Einstein(1950)got

(8.50)

where I1i and I2i are Einstein’s integral functions (Yang, 1982; Dimons andSentürk,1992;Vanoni,2006),definedas

(8.51)

and

(8.52)

whereEistheratioE=ai/ddisthedepthofflow(m)

ThenumericalvalueofthetwointegralscanbeevaluatedusingFigures8.17and8.18 as function of parameters z and E. As mentioned, ai = 2Dsi. Theconcentration Cai is computed by the following equation which was derivedbasedonEinstein’s(1950)experimentaldata:

(8.53)

ThisleadstothefollowingequationforSLtransport,relatingittoBLtransportqBwi:

(8.54)

AndtheTSLiscomputedbyadditionofthatofeachsizeclass:

(8.55)

One should notice that in Equations 8.54 and 8.55, Einstein’s BL equation(Equation8.41)oranyotherBLequation (e.g.Meyer–PeterandMüller’s)canbeused.

Figure 8.17 Evaluation of Einstein’s (1950) integral function I1. (From Einstein, H.A., The bed loadfunctionforsedimenttransportationinopenchannelflows,TechnicalBulletinNo.1026,U.S.DepartmentofAgriculture,Washington,DC,1950.)

8.5.3Totalsedimentload

8.5.3.1Einsteinmethod

Taking intoaccountEquation8.29, theTSL can be calculated fromEquations8.41and8.54as

(8.56)

whereqTistheTSLrateperunitwidthofthestream(kg/s/m).

Figure 8.18 Evaluation of Einstein’s (1950) integral function I2. (From Einstein, H.A., The bed loadfunctionforsedimenttransportationinopenchannelflows,TechnicalBulletinNo.1026,U.S.DepartmentofAgriculture,Washington,DC,1950.)

8.5.3.2AckersandWhitemethod

The method is based on computing the mean sediment concentration Cm byweight (kg/kg) based on the following empirical equation determined from alargenumberofexperimentaldata,whichcontainsfourempiricalcoefficientsa1,a2,a3,a4:

(8.57)

whereVisthemeanflowvelocity(m/s)Fisgivenbythefollowingequation

(8.58)

Theempiricalcoefficientsa1,a2,a3anda4dependon thedimensionlessgraindiameterD*:

(8.59)

ForD* ≥ 60, they are constants assuming the following values: a1 = 0,a2 =0.025, a3 = 0.17 and a4 = 1.5, and for 1.0 <D* ≤ 60, they are given by thefollowingequations:

(8.60)

(8.61)

(8.62)

(8.63)

Asmentioned, the sedimentconcentrationCm isbyweightdefinedasmassofsedimentdividedbytotalmassofwaterandsedimentor

(8.64)

whereqisthewaterdischargeperunitchannelwidth(m3/s/m)qTvisthesedimentdischargeperunitchannelwidth(m3/s/m)

ThisequationcanbeusedtoexpressqTvasfunctionofaknowndischargeqasfollows:

(8.65)

The totalBL discharge per unit channelwidth byweightqTm (kg/s/m) can bedetermined from this equationbymultiplying the sedimentdensityρs (kg/m3).The steps to apply themethod are the following: ComputeD* fromEquation8.59 for a given grain size (usuallyD50). Determine the coefficients ai fromEquations8.60through8.63.ComputeparameterFfromEquation8.58and theflowparameters.Compute themeansedimentconcentrationCm fromEquation8.57.UseEquation8.65 to compute theunitTSLdischarge (byvolumeor byweight).

8.5.3.3Yang’smethod

Yang (1972) derived an empirical equation to predict the mean sedimentconcentrationCmbasedonregressionanalysisoflaboratoryfumedata,relatingTSLtransportofsandtotheunitstreampower(i.e.productofmeanvelocityVandenergyslopeS):

(8.66)

whereCmisthemeansedimentconcentrationbyweight(ppm).Toconvertittokg/kg, the value predicted by Equation 8.66 should be multiplied by the unitconversion factor 10−6;Vcr is the critical velocity for incipient motion (m/s),

•

•

•

whichhas alsobeenpresented (Equation8.27), and the other parameters havealready been defined. After computingCm based on Equation 8.66, Equation8.65canbeusedtocomputethetotalBLeitherbyvolumeorbyweight.

8.6LANDEROSIONANDWATERSHEDSEDIMENTYIELD

8.6.1Introduction

The soil erosion from natural, agricultural or other lands is a very importantprocessaffectingboththemorphologyoftheerodinglanditselfandtheerosionandsedimentationprocessesinthereceivingstreamswherethelanddrainsandsuppliessediments.Extensivelanderosioncanresult,amongothers,inthelossof topsoil in agricultural areas, leading to lower productivity, and rilling andgullyingofthelandsurface,creatinglanduseproblems.Transportedbystreams,eroded sediments can cause reservoir silting, while sediments deposited instreamscanleadtoincreasedfoodstages.Ontheotherhand,measurestocontrollanderosionmayleadtosedimentstarvationinthereceivingstreamandexcessbed degradation and/or bank instability. These examples demonstrate the needforaccurateestimationofsoilerosionfromlandandwatershedsedimentyieldsandtheimportanceofquantifyingtheseprocessesinsedimentationstudies.In this section, we present the basics of erosion, while comprehensive

treatmentsofthesubjectcanbefoundinspecializedtextbooks(e.g.KirkbyandMorgan,1980;Goldmanetal.,1986;SimonsandSentürk,1992).The erosion process comprises two parts: (1) loosening of soil grains by

rainfallimpact,somethingaidedbythecyclesoffreezing–thawingandwetting–drying,and(2)transportationofloosesoilgrainsbyflowingrainwater(runoff).Thereareseveraltypesofsoilerosionfromland(Goldmanetal.,1986):

Splasherosion: This type occurs on unvegetated bare soil and is due toraindrop impact on soil and splashing away of soil grains. It is moresignificantinheavyintenserainfallsoflargeraindropsizeandonslopingground.Sheeterosion:Thistypeoccursonrelativelyfatslopinggroundunderthinsheetflow(overlandflow)whichtransportssoilgrainswhichwereearlierdetachedduetorainfallimpact.Rill flow:Due toground irregularities, sheet flow starts concentrating indeeperparts.Asflowconcentratesthere,velocityandturbulenceincrease,soil grains are detached and transported more easily, and small well-defined channels (rills) are formed. Rill dimensions are of a few

•centimetres.Gullyerosion:Gulliesarelargerchannelsformedeitherwhenrillsarecutdeeperduetoheavyrainfallrunofforwhenrillsconfluencetoformdeeperandwiderchannels.

Thereareseveralfactorsaffectingsoilerosion.Themostimportantonesincludeclimatic and soil characteristics, topography and ground cover (Kirkby andMorgan, 1980; Goldman et al., 1986). Climate is the main factor that affectserosionsinceerosion isa rain-drivenprocess.Themain rainfall characteristicsaffectingerosionisrainfallintensity(withshortdurationhighlyintenserainfallbeing more erosive) and rainfall droplet size (the larger the size, the moreerosivetherainfall).Climateindirectlyaffectserosionbyaffectingthegrowthofvegetation, which is themost effective shield to erosion. As a result, erosionratesarelowerinmildtemperatureandrainyclimates,wherevegetationgrowsfast andprovides adensegroundcover; on theotherhand, in cold and indryclimates,vegetationissparseand,thus,erosionratesarehigher.Soil characteristics influencing erosion include texture, organic matter

content,structureandpermeability.Withrespecttotexture(seeFigure7.6), themost erodible soils are those with high content of fine sand and silt and loworganicmatterandclaycontent.Sandysoilsandclaysoilsarelesserodible,theformerbecauseofthehighpermeabilityandthelatterbecauseclayparticlestendto stick together and coagulate.Organicmatter tends to improve soil structureand increase permeability, thus making the soil less erodible.With respect tostructure, i.e. the tendency of soil particles to aggregate, a loose granularstructureenhances infiltration leading to less runoffanderosion.Similarly, themorepermeablethesoil,thelesstherunoffandtheerosion.The topography of the area is also a key factor, with slope length and

steepnessbeingimportantparameters,becausetheyinfluencerunoffvelocity.Asboth length and slope increase, so does flow velocity and erosion potential.Finally, ground cover (i.e. vegetation or protection measures) is important inreducing runoff.Vegetation reduces raindrop impact,decreases runoffvelocity,reinforces the soil through the plant roots which hold particles together, andenhancesinfiltration.

8.6.2Universalsoillossequation(USLE)

TheUniversalSoilLossEquation(USLE)(SmithandWischmeier,1957;MeyerandMonke,1965;WischmeierandSmith,1965,1978;Wischmeieretal.,1971;Wischmeier,1972,1973)isthemostcommonlyusedempiricalmodeltopredict

land erosion. It has been developed by the Soil Conservation Service (nowNaturalResourcesConservationService)oftheU.S.DepartmentofAgriculturebasedonmonitoringofexperimentalplotsthroughouttheUnitedStates,whichstartedinthe1930sandlastedforseveraldecades.DetailedpresentationsoftheUSLE can be found in specialized textbooks (e.g. Kirkby andMorgan, 1980;Goldmanetal.,1986;SimonsandSentürk,1992).TheformoftheUSLEequationis

(8.67)

whereAistheannualorforaspecifictimeperiodsoilloss(U.S.tons/acreormetrictonnes/ha)

R is the rainfall erosivity factor (or rainfall erosion index) (−), whichexpressestheeffectivenessofrainfallincausingerosion

K is the soil erodibility factor (tons/acre),whichexpresses inoneparameterthesoilcharacteristicsrelevanttoerosion

LS is the slope length and steepness factor (−), which accounts for slopelengthandsteepnessinerosionpotential

C is the vegetative cover and cropping-management factor (−), whichexpressestheeffectofvegetationormulchonreducingerosionandcanbevariableduringtheyearrelatedtovegetationcanopygrowth

P is the erosion control practices factor, i.e. it describes how humaninterventions,suchascontouringandterracing,controlerosion(−)

Finally,βisaunitconversionfactorwhichisequalto1.0whenparameterAisintons/acreor2.241whenAisinmetrictonnes/ha.

8.6.2.1RainfallerosivityfactorR

The rainfall erosivity index R is the product of rainfall kinetic energy at thegroundandmaximumintensity for30minduration (I30) andcanbeevaluatedfor either one rainfall event or as a mean annual value for a series of years.AccordingtoKirkbyandMorgan(1980),Rforasinglestormcanbecomputedbythefollowingequation:

(8.68)

where

Ristherainfallerosivityfactor(−)Ijistherainfallintensityforaspecifictimeincrementofthestorm(mm/h)Δtjisthecorrespondingtimeincrement(h)I30isthemaximum30minintensity(mm/h)jisthetimeincrementnisthenumberoftimeincrementsinthestorm

R, although it has units from its definition, for simplicity, is considered non-dimensionalandthedimensionsaregiventothefollowingsoilerodibilityfactorK (sameas forA).Thisway,conversionbetweenEnglishandSIunits iseasyand only based on the value of β (Equation8.67). Specifically for theUnitedStates,Wischmeier and Smith (1978) have produced the map of Figure 8.19presentingaverageannualRvalues.

8.6.2.2SoilerodibilityfactorK

ThesoilerodibilityfactorKcanbepredictedbythenomographofFigure8.20given byWischmeier andSmith (1978) or by simply usingTable8.7. Percentorganicmattershouldbemeasured.PermeabilityclassdescriptionsaregivenbyWischmeier et al. (1971). Table 8.7 gives K values for the main soil textureclassesofFigure7.6andothers.

Figure8.19AverageannualvaluesforrainfallerosivityfactorRfortheUnitedStates.(FromWischmeier,W.H. and Smith, D.D., Predicting Rainfall Erosion Losses – A Guide to Conservation

Planning,AgriculturalHandbookNo.537,USDA,1978.)

Figure 8.20Nomograph to estimate the soil erodibility factorK. (FromWischmeier,W.H. et al., J. SoilWaterConserv.,26(5),189,1971.)

8.6.2.3SlopelengthandsteepnessfactorLS

The slope length and steepness factor LS is given by the following equation(SmithandWischmeier,1957;WischmeierandSmith,1965):

(8.69)

wherel is theslope lengthfromthepointwhereoverland(sheet) flowstarts to thepointwhereslopeiseithersignificantlymildertoallowdepositionortothepoint where a structure is set that interrupts runoff (e.g. a drainage pipeacrosstheslope)(m)

θistheangleoftheslopewhichisdefinedaccordingto

(8.70)

wheresistheslopegradient(%)m is an exponent depending on slope (Wischmeier and Smith, 1978), asshowninTable8.8

Table8.7SoilerodibilityfactorKvaluesOrganicmattercontent(%)

<0.5% 2% 4%

K

Sand 0.05 0.03 0.02

Finesand 0.16 0.14 0.10

Veryfinesand 0.42 0.36 0.28

Loamysand 0.12 0.10 0.08

Loamyfinesand 0.24 0.20 0.16

Veryloamyfinesand 0.44 0.38 0.30

Sandyloam 0.27 0.24 0.19

Finesandyloam 0.35 0.30 0.24

Veryfinesandyloam 0.47 0.41 0.33

Loam 0.38 0.34 0.29

Siltyloam 0.48 0.42 0.33

Silt 0.60 0.52 0.42

Sandyclayloam 0.27 0.25 0.,21

Clayloam 0.28 0.25 0.21

Siltyclayloam 0.37 0.32 0.26

Sandyclay 0.14 0.13 0.12

Siltyclay 0.25 0.23 0.19

Clay 0.13–0.29

Sources:Wischmeier,W.H. et al., J. Soil Water Conserv., 26(5), 189, 1971; Kirkby, M.J. andMorgan,R.P.C.,SoilErosion,JohnWiley&Sons,Chichester,U.K.

Table8.8ValuesoftheexponentmofEquation8.69s(%) M

<1 0.2

1–3 0.3

3–5 0.4

>5 0.5

8.6.2.4VegetativecoverandcroppingmanagementfactorC

Thecropping-management factorC shows theeffectofcropsand/ormulchonreducingerosioncomparedtocultivatedbaresoil.TheCfactorcanalsobeusedin construction sites to evaluate the effect of seeding of grass or mulch incontrolling erosion during the construction period. In other words, this factorshows the effect of measures to reduce the raindrop impact. Comprehensivetables withC values are given byWischmeier and Smith (1978) for specificcrops and mulch application. General values can be found in Table 8.9 forpasture,rangeandidleland,Table8.10forforestlandandTable8.11formulchapplication in construction sites. The last table also presents mulch rateapplication for various mulches, slope steepness range and maximum slopelengthforwhichtheCvaluesareapplicable.

Table8.9FactorCvaluesforpermanentpasture,rangeandidleland

Source:Wischmeier,W.H.andSmith,D.D.,PredictingRainfallErosionLosses–AGuidetoConservationPlanning,AgriculturalHandbookNo.537,USDA,1978.

Notes:G–Theslopecoverisgrassorgrass-likeplants,decayingcompacteddufforlitter50mmdeep.W–Theslopecoverisbroadleafherbaceousplantswithlittlelateralrootnetworknearthesurfaceorundecayedresiduesorboth.

Table8.10FactorCvaluesforundisturbedforestlandPercentofareacoveredbycanopyoftreesandundergrowth

Percentofareacoveredbydecayingforestduff(litter)atleast5cmthick

Cvalue

100–75 100–90 0.0001–0.001

70–45 85–75 0.002–0.004

40–20 70–40 0.003–0.009

Source:Wischmeier,W.H.andSmith,D.D.,PredictingRainfallErosionLosses–AGuidetoConservationPlanning,AgriculturalHandbookNo.537,USDA,1978.

8.6.2.5ErosioncontrolpracticesfactorP

ThevalueoftheerosioncontrolpracticesfactorPquantifiestheeffectoftillageand ploughing practices which reduce the velocity of runoff and affect itstendency to flow downhill. The no-practice (baseline) condition is ploughingdirectlyupanddowntheslopeforwhichP issetequalto1.Soilconservationpractices includecontouring, terracing,crop rotationsand retentionof residuesonthesurface.WischmeierandSmith(1978)presentdetaileddescriptionsofthevariouspracticesandTable8.12summarizesPvalues.

Table8.11FactorCvalues,lengthlimitsandslopeforconstructionsitesTypeofmulch Mulchrate

(tons/ha)Landslope

(%)FactorC Lengthlimit

(m)

None 0.0  All 1.00

Straworhaytieddownbyanchoringandtackingequipment

2.2  1–5 0.20 61

2.2   6–10 0.20 30

3.4  1–5 0.12 91

3.4   6–10 0.12 46

4.5  1–5 0.06 122

4.5   6–10 0.06 61

4.5   11–15 0.07 46

4.5   16–20 0.11 30

4.5   21–25 0.14 23

4.5   26–33 0.17 15

4.5   34–50 0.20 11

Crushedstone,1/4–11/2in. 303 <16 0.05 61

303   16–20 0.05 46

303   21–33 0.05 30

303   34–50 0.05 23

538 <21 0.02 91

538   21–33 0.02 61

538   34–50 0.02 46

Woodchips 16 <16 0.08 23

16   16–20 0.08 15

27 <16 0.05 46

27   16–20 0.05 30

27   21–33 0.05 23

56 <16 0.02 61

56   16–20 0.02 46

56   21–33 0.02 30

56   34–50 0.02 23

Source:Wischmeier,W.H.andSmith,D.D.,PredictingRainfallErosionLosses–AGuidetoConservationPlanning,AgriculturalHandbookNo.537,USDA,1978.

Table8.12ValuesoftheerosioncontrolpracticesfactorPLandslope(%) Contouring Stripcropping Terracing

1–2 0.60 0.30 0.12

3–5 0.50 0.25 0.10

6–8 0.50 0.25 0.10

9–12 0.60 0.30 0.12

13–16 0.70 0.35 0.14

17–20 0.80 0.40 0.16

21–25 0.90 0.45 0.18

8.6.3Modifieduniversalsoillossequation

ThemainlimitationoftheUSLEisthatitcomputesannualaveragesoillossdueto sheet and rill erosion in farmlandandconstruction sites.The applicationofUSLE has limitations in arid areas (e.g. southwestern United States) whichreceive a significant portion of annual rainfall in the form of short-durationintensestorms.Toaddress thisproblem,WilliamsandBerndt (1972)modifiedthe USLE and developed a procedure to compute sediment yields fromwatersheds produced from single-storm events. In the produced ModifiedUniversal Soil Loss Equation (MUSLE), the rainfall erosivity factor wasreplacedbyarunofftermconsideringvolumeandpeakofrunoff.TheformoftheMUSLEequationis

(8.71)

whereYsisthesedimentyieldforaspecificstorm(U.S.tonsortonnes)QVistherunoffvolume(acre-feetorm3)qpisthepeakflowrateofrunoff(cfsorm3/s)

Allotherparameters(i.e.K,LS,CandP)areasdefinedforUSLE.Finally,β′isaunitconversionfactorwhichisequalto1.0whenYsiscomputedinU.S.tons

or 124 when Ys is computed in metric tonnes. The coefficient 95 and theexponent0.56werebasedoncalibrationof themethod inwatersheds inTexasandNebraska.

8.6.4Erosioncontrolmeasures

Erosioncontrolisdesirableduetothementionedadverseimpactsofsediments.There are two general types of measures to adopt to control erosion: (1)measurestoreduceerosionatthelocationswhereitoccurs,i.e.wheresedimentsareproduced(controlat thesource),and(2)structuralmeasures to traperodedandtransportedsediments.

8.6.4.1Erosioncontrolatthesource

Suchmeasuresareusuallytakenattheupperslopingpartsofthewatershed.OnecangetagoodfeelingaboutwhatkindofmeasurestheyshouldbebyanalysingthetermsoftheUSLE.

8.6.4.1.1ChangingtheRandKfactorsFactors R (rainfall erosivity index) and K (soil erodibility index) cannot beaffectedmuchsince thefirstonedependsonrainfallcharacteristicsof theareaand the other on local soil texture, organic content and permeability (Figure8.20).Attempts to addorganicmatter (decomposedvegetation residues) couldresultinlowerKvaluesanderosionrates.

8.6.4.1.2ChangingtheLSfactorAs mentioned, the length of the slope is an important factor in erosionproduction. Reducing the length can reduce erosion rates. Possible ways toreducelengthisbyprovidingbetterdrainageoftheslopeandthiscanbedoneby the addition of open side drains (nearly parallel to the contours andperpendicular to the main slope direction) which confluence to downdrains(along the slope direction). Such a method is shown in Figure 8.21, wherepractically the slope length between side drains is reduced to one-third of theentireslopelength.Thevegetationdevelopmentbetweensidedrainscanfurther,andmoreeffectively,reduceerosionasdiscussedinthenextparagraph.

Figure8.21Sidedrainsanddowndrainsusedtoreduceslopelength.(PhotobyV.A.Tsihrintzis.)

8.6.4.1.3ChangingtheCfactorChangingtheCfactoristhebestwaytoreduceerosion,andthiscanbedone(asseen inTables8.9and8.11)byeitherenhancing thevegetationcoverorusingmulch.BothmeasuressignificantlyreducethevalueoftheCfactor,particularlyvegetation.Details on the typeofvegetation and sitepreparation aregivenbyGoldmanetal.(1986).

8.6.4.1.4ChangingthePfactorAs Table 8.12 shows, contouring (and minimum tillage), strip cropping andterracingareeffectiveerosioncontrolpractices,significantlyreducingthevalueofP factor.Terraces (Figure8.22) reduceboth slope steepness and length andalsoenhance infiltration, thus reducing runoff.Terracescanalsobeconsideredas a structural measure; however, it is presented here since it can controlsedimentsatthesource.

8.6.4.1.5MeasuresduringconstructionConstruction is a major activity producing sediments. Temporary measuresshouldbetakenduringconstructiontominimizesoillosses.MeasurestochangetheLS,CandP factorsareeffective.ExamplesareshowninFigures8.23and

8.24. In Figure 8.23, the slope, in the rainy season during construction, iscoveredwithplastic,andsandbagsareusedtokeepplasticinplace.Figure8.24demonstratespoorconstructionpracticesinasmallurbanproject.

Figure8.22Terracingtoreduceslopesteepness.(PhotobyV.A.Tsihrintzis.)

Figure8.23Plasticusedtocoverslopeduringconstruction.(PhotobyV.A.Tsihrintzis.)

Figure 8.24 (a) Poor construction practices leave trench unprotected from rainfall impact to erode; (b)erodedmaterialfromthetrenchistransporteddownslope.(PhotosbyV.A.Tsihrintzis.)

8.6.4.2Structuralmeasurestocontrolerosion

These can be of three types: (1) indirect,which involves installation ofwaterconveyance and energy dissipation structures—such structures provideconveyancemeansforwatertoflowin,safelyminimizingerosionpotential;(2)channel protection from degradation (channel stabilization); and (3) direct,which involves building special structures whose purpose is to trap erodedsediments(sedimentretentionstructuressuchasdebrisbasins[Figure8.25]).Wewill discuss them briefly in Table 8.13, since there are specialized textbooksexhaustingtheissue(e.g.Goldmanetal.1986;SimonsandSentürk,1992).

Figure8.25Debris basin: (a)Looking from thewatersheddownslope, theupstream slopeof the earthendam,whichiscoveredwithconcrete,isshownaswellastheriserandtheemergencyspillway.(b)Lookingupslope, thedownstreamslopeof thedamand the emergency spillwayare alsoshown.Duringconstruction,erosionmeasuresareinstalledbycoveringthefaceoftheearthendamwithplastic.Therightsideofthepictureshowssomeerosionintheuncoveredfaceofthedam.(PhotosbyV.A.Tsihrintzis.)

Table8.13StructuralmeasurestocontrolerosionTypeofmeasure Structure Shortdescription

Waterconveyancestructures–outletenergydissipation

Dikes Dikesaresmalltemporarynon-engineeredcompactedsoilembankmentsplacedontheheadoftheslopetopreventorminimizewaterflowingdowntheslope.

Swales Swalesaresmallnon-engineeredchannelscollectingandconveyingdrainage.Theyareearthenorgrassed.

Sidedrainsanddowndrains

Theyareplacedontheslopealongthecontoursandinthemainslopedirection,aspresentedinFigure8.21.Theycaneitherbecorrugatedmetalorpavedwithconcreteorgunite.

Pipeslopedrains Theyareplasticpipesusedtocollectandconveyrunoffdownaslopepreventingerosion.Theyarealsoeffectivewheninstalledinsidegulliestominimizerunoffand,thus,gullyerosion.

Culverts Theyareplacedunderneathroadsatstreamorgullycrossingstoprovidedrainageconveyanceandprotectroadfromerosionandfailure.

Outletprotection Alltheaforementionedstructuresrequireprotectionattheoutlet,toeitherresistorreducetheenergyoftheflowandpreventlocalscour.Eitherripraporsmallenergydissipatorsareused.

Channelstabilization

Channellinings Severalmaterialscanbeusedtolineachannelandmakeiterosionresistant.Theseincluderiprap(i.e.gravelandrock),groutedriprap,gabions,reinforcedconcreteandgrass,amongothers.

Bedstabilization Specializedstabilizationorgradecontrolstructures(concreteorriprap)canbeusedatpointwheremajorheadcuttingandbeddegradationisexpected.Suchstructurescanbedesignedasretainingwallsacrossthechanneltomakeupstreamslopemilderandretainsediments.

Localprotection Specializedstructurescanbeusedatplacesinthechannelwherelocalscouroccurs.Examplesincludebankstabilizationmethods(e.g.jetties,groynes)andbridgepierprotectionfromlocalscourandundermining.

Sedimentretention Sedimentordebrisbasins

Thesedonotstoperosionbuttraperodedsedimentsfrommovingdownstreamwithrunoff.Theyessentiallycompriseasmalldamwithariserandanemergencyspillway(Figure8.25).Theriserisavertical,usuallyperforatedcorrugatedmetalpipe,whichactsasa‘strain’allowingrunofftopassthroughbutretainingsediments.Thesedimentsthenaccumulatebehind.Obviously,coarsermaterialsaretrappedandfinerpassthrough.EquationsliketheUSLE(Equation8.67)orMUSLE(Equation8.71)canbeusedtocomputethevolumeoftheaccumulatedsedimentswithtimeandsizethedebrisbasin.Accessshouldbeprovidedforcleanup.

Sedimentbarriers Thesearetemporarynon-engineeredstructureswhichretainsediments.Theycanbeeitherpermeablefencesalongthecontoursmadeofpermeablefabricorstrawbalesanchoredatthebaseoftheslope.Theyarebothusuallyattachedtoawirefence.

Source:AdaptedfromGoldmanetal.,1986.

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Index

Above-groundrunoff,101AckersandWhitemethod,432–434Activereservoirvolume,269–270

cumulativeinflowcurve,270cumulativeinflows,280,282designrisk,278–279estimation,270–271methodofmaximumshortage,288–290monthlyinflows,280–281non-conventionalsizingmethod,advantagesof,279non-conventionalstochasticmethodsARIMAmodel,276ARMAmodel,276autoregressivemodels,274–275cumulativecurvemethod,274FFGNmodel,277movingaveragemodels,275–276naturalinflows,273

persistence,286–288Rippl’sconventionaldesign,disadvantagesof,271–272sensitivity,279–280sequentpeakanalysismethod,272–273syntheticdataofinflows,277–278synthetictimeseriesofinflows,283–286

Actualevapotranspirationaverageactualevapotranspiration,90Coutagnemethod,92–93Turcmethod,91–92

Alluvialstreamsandriversbedforms,413discharge,418Einstein’sfunction,417–418Einstein’sparameter,417fine-to-mediumsandchannels,415–416floodcontroldesign,416forms,412grainroughness,416grainskinfriction,414kinematicviscosity,416parameters,414qualitativeflowresistance,414roughnesscoefficient,415sedimentgrains,413skinresistance,409streampowerandmedianfalldiameter,414–415

Anisotropicalluvialdeposits,260–261APRconicalscan,47Aquiferrecharge,263–264Arealrainfallcalculation,surfaceintegration,68–70

directintegrationmethodaveragingmethod,63Thiessenmethod,63–64,67

surfaceadaptationmethodisohyetalcurvemethod,64–68optimuminterpolationmethod,70–74

timedistributionsofrainfalllimited-scalephenomena,72medium-scalephenomena,72synoptic-scalephenomena,73,75–76

Arithmeticmeanmethod,57,66Atmospheric precipitation, formation mechanisms of, 19–20; see also

PrecipitationAtmosphericscrubbing,366Averageriverflowrateestimation

averagewater-levelestimation,131ratingcurveextensionof,129

preparation,127–129inunsteadyflowconditions,130

fromwater-levelmeters/recorders,131–135Averagingmethod,forsurfacerainfallcalculation,63

Basinmethod,263Bedload(BL)

Einstein-Brownmethod,428–429Einsteinmethod,425–428Meyer-PeterandMüllermethod,424–425

Bedmaterialload(BML),422Bernoullitrialsanddistribution,190Binomialdistribution,190–191Blaney-Criddlemethod,84–86,91Boundedrecessionbaseflowmethod,161

Capillarypotential,93–94Channelflowresistance

excavatedearthenandnaturalstreams,407,410–411Manningequation,407Manningroughnesscoefficient,407–409openchannels,405particlesize,409,411predictionsofManningequation,409,412resistancecoefficients,406shearstress,407shearvelocity,406skin/surfaceroughness,409

Channelstabilizationstructures,378Chézyformula,129Chowmethod,251–252,255–256ClarkUHtransform,161CloudsandtheEarth’sradiantenergysystem(CERES),45,49Cokrigingmethod,72Compositehydrographseparation,115Conductivecooling,23–24Confinedaquifer

Chowmethod,251–252,255–256Cooper–Jacobmethod,248–254,256–257

description,227–228steadyflow,232–235Theismethod,247–248,254–255

Constantloss/gainmethod,162Constantmonthlybaseflowmethod,162Continuousdistributions

exponentialdistribution,193gammadistribution,193–195log-Pearsondistribution,195–196uniformdistribution,193

Coolingmechanismsandprecipitationtypes,23–24Cooper-Jacobmethod,248–254,256–257Coutagnemethod,92–93Cycloniccooling,23

Data(rainfall)completionmethodarithmeticmeanmethod,57correlationandregressiondescription,58simplelinearregression,58–59,61–62

inversedistancemethod,57–58,60–61normalratiosmethod,57rainfallgradient,62

Deadreservoirvolumesizingactivevolumelossestimation,296–301deadstorage,293–294estimation,294–296sediments,292

Deadstorage,293–294Deficitandconstantlossmethod,160–161Designstormandflood,306–307Detentionbasins

combinedsewer,aquaticsystem,393–395parallelcombinedsewer,393,395seriescombinedsewer,392–394

Deterministichydrology,4,173Discretedistributions

Bernoullitrialsanddistribution,190binomialdistribution,190–191

geometricdistribution,191Poissondistribution,191–192uniformdistribution,192

Distributionscontinuousexponentialdistribution,193gammadistribution,193–195log-Pearsondistribution,195–196uniformdistribution,193

discreteBernoullitrialsanddistribution,190binomialdistribution,190–191geometricdistribution,191Poissondistribution,191–192uniformdistribution,192

Gumbeldistribution,198–200,210–214–215Kolmogorov–Smirnovtest,202–203log-normaldistribution,188–190,210–212normaldistribution,184–188,214X2test,202,211–213

Doublecumulativecurves,52–56Doublenegativeexponentialdistribution,seeGumbeldistribution

Einstein–Brownmethod,428–429Einsteinmethod,425–428,431Energybalancemethod,78–79Engineeringhydrology,4Environmentaleducationandawareness,366Erosioncontrol

practicesfactor,440–441atsourceCfactor,443constructionpractices,443–444LSfactors,442–443Pfactor,443RandKfactors,442

structuralmeasures,445–446Evaporation,10

definition,77

energybalancemethod,78–79masstransfermethod,79–80Penmanmethod,80–81waterbalancemethod,77–78

Evapotranspiration,5actualevapotranspiration,90–93Blaney-Criddlemethod,84–86Hargreavesmethod,87HEC-HMSmodel,163Jensen-Haisemethod,86Makkink’smethod,86–87Penman–Monteithmethod,81–82Priestley–Taylormodel,87–89Thornthwaite’smethod,82–83waterbalancemethod,81

Exponentialdistribution,193Exponentiallossmethod,161Extendeddrybasins,387–390

Floatingtypegauge,26Flooddetentionbasins

parallel,384–386series,384–385

Floodhydrograph,307–308Floodsafetystructures

designflood,302,308–316designstormandflood,306–307floodhydrograph,307–308hydrologicrisk,302nonlinearbasins,303probablemaximumflood,306rainfall-runoffmodel,303spillwaydesigncriteria,305–306spillwaydesignflood,304spillwayhydrologicdesign,303–304

Floodvolume,268,302Floodingmethod,264Frequencystormmethod,163Frontalcycloniccooling,23

Gammadistribution,193–195Gaugeweightsmethod,163Geometricdistribution,191Grassfilters,368–370,372,374–375Grassswales,375–376Gravimetrictypegauge,26Green-Amptmodel,95–96GreenandAmptlossmethod,161Griddeddeficitconstantlossmethod,161GriddedGreenandAmptlossmethod,161Griddedprecipitation,163GriddedSCScurvenumberlossmethod,161Grids,378Groundwaterhydrology,8

aquiferrecharge,263–264aquifers,223–224classificationofaquifers,226–228expressionofgroundwaterflow,230–232facilitiesandoperationalcost,223fieldmeasurements,228–229filters,224mathematicalproblem,229–230non-uniformflowanisotropicalluvialdeposits,260–261Chowmethod,251–257Cooper–Jacobmethod,248–251drawdowntests,247Theismethod,247–248unconfinedaquifer,257–260wellhydraulics,243–246

pipelines,224reservoirtanks,223–224salinationinterfaceofsaltwaterandfreshwater,264–265saltwaterelevationcone,265–266

soilandaquiferparameters,224–226spatialdistribution,223steadyflowconfinedaquifer,232–235

semi-confinedaquifer,237–239unconfinedaquifer,235–237

surfacewaterflowregulators,224temporalvariability,223theoryofimages,239–240wellnearimpermeableboundaries,242–243wellnearriver,240–242

waterquality,223watersupplysources,223welllosses,261–263

Gullyerosion,435Gumbeldistribution,210–211

maximumdistribution,198–199minimumdistribution,199–200

Hail,22–23Hailstones,22–23Hargreavesmethod,87HEC-HMSmodel

componentsbaseflowmethod,161–162channelflowrouting,162controllingsearchtolerance,164controlspecificationmanager,163–164directsurfacerunoff,161hydrologicalloss,160–161loss/gainmethod,162meteorologicalmodel,162–163Nelder–Meadmethod,164objectivefunction,164optimizationtrial,164simulationrun,164univariategradientmethod,164

description,160inRafinaBasinbasinmodel,165calibratedvalues,168–170controlspecificationmanagerformat,166–167curvenumber,166

directrunoffestimation,166gaugeweightsmethod,166initialestimatedparameters,166–167Muskingummethod,166observedandsimulatedfloodhydrograph,168–169recessionbaseflowmethod,166SCScurvenumberloss,165simulationrunandresults,167–168studyarea,164–165

Holtanmodel,96Horton’smodel,94Horton’sstreamclassification,102Huggins-Monkemodel,96Hydrographs

components(parts)of,105confluencetime,107descendinglimb,107directrunoff,107–108floodduration/hydrographbasetime,107floodhydrographfloodrunoffcalculation,114andhyetograph,113–114ofobservedrunoff,113–114

lagtime,107peakflowdischarge,106peakof,106φindex,108–110risinglimb,106separation,110compositehydrographseparation,115indirectrunoffandbaseflow,111methodoflogarithms,112–113methodsofbaseflowseparation,111–112

shapeofbellshape,105factorsinfluencing,116–122ofsinglefloodevent,106timeofconcentration,107

Hydrologicalloss,9–10evaporationdefinition,77energybalancemethod,78–79masstransfermethod,79–80Penmanmethod,80–81waterbalancemethod,77–78

evapotranspirationactualevapotranspiration,90–93Blaney–Criddlemethod,84–86Hargreavesmethod,87Jensen–Haisemethod,86Makkink’smethod,86–87Penman–Monteithmethod,81–82Priestley–Taylormodel,87–89Thornthwaite’smethod,82–83waterbalancemethod,81

infiltration,74,77interception,74,77retention,74,77surfacerunoff,74,77

Hydrologicalmodelsblack-boxmodels,159conceptualmodels,159description,159deterministicmodels,159distributedmodels,159HEC-HMSmodelcomponents,160–164description,160inRafinaBasin,164–170

lumpedmodels,159physicalmodels,159semi-distributedmodels,159semi-lumbedmodels,159timeresolution,159

Hydrologicdesignfloodsafetystructuresdesignflood,302,308–316

designstormandflood,306–307floodhydrograph,307–308hydrologierisk,302nonlinearbasins,303probablemaximumflood,306rainfall–runoffmodel,303spillwaydesigncriteria,305–306spillwaydesignflood,304spillwayhydrologiedesign,303–304

riverdiversiondesigncriteria,317diversionchannel,315estimationofdesignflood,318–319openchannel/closedtunnel,315

sizingofreservoir(seeSizingofreservoirs)waterstructure-specificissuesdischargepredictions,323–327duration-dischargecurve,327–328irrigationsystem,319operationaldepth-duration-frequencycurves,320–322wastewatertreatmentplant,322–323waterresourcesprojects,319

Hydrologyannualrainfalldepth,11annualrunoffcoefficient,12classificationof,3–4descriptionof,2dischargevolume,13historicalevolutionof,3hydrologicalbalance,9hydrologicalcycle,4–5objectiveof,2parametersandunits,5–6relevancetootherscientificareas,2riverbasin,6–7runoffcoefficient,13–14scaleininsertionof,6spatialscale,7

timescales,7–8water,worldwidedistributionof,8

Hydrometricstationdischargemeasurementbymethodofvelocityfield,127crosssectionofriver,125currentmeterconfiguration,125–126velocitydistributionprofileinriversection,124verticalvelocityprofile,124–125

installationcriteriafor,122–123water-levelmeasurement,123–124

Hydrometryhydrometricstationdischargemeasurementbymethodofvelocityfield,124–127installationcriteriafor,122–123water-levelmeasurement,123–124

purposeof,122

idfcurves,seeIntensityduration-frequency(idf)curvesIncipientmotion

criticalboundaryReynoldsnumber,419dimensionlessShieldsparameter,419permissiblevelocitiesandcriticaltractiveforces,420–421permissiblevelocity,420shearstress,420Shieldsgraph,419–420uniformbedmaterials,420

IndicatorKrigingapproach,72Inducedwatersupply,264Infiltration

capillarypotential,93–94description,93distributionof,74,77inlets,371,374rateestimationGreen–Amptmodel,95–96Holtanmodel,96Horton’smodel,94Huggins–Monkemodel,96Kostiakovmodel,96–97

Philipmodel,97SoilConservationServicemethod,97–98

roleinstormevent,94trenches,370–373watersupply,seasonaldistributionof,94

Initialandconstantlossmethod,161InstantaneousUH(IUH)

Clark’smethod,156–157byconvolution/Duhamelintegral,155Nashmodel,156byS-curve,155time-areadiagrams,156

Intensityduration-frequency(idf)curves,215–217constructionof,217–218floodprotectionprojectexample,218–222

Interception,74,77Inversedistancemethod,57–58Isohyetalcurvemethod,forarealrainfallcalculation,64–68

Jensen-Haisemethod,86Joss-Waldvogeldisdrometers,28–38

Kinematicwaveroutingmethod,162Kinematicwavetransform,161Kolmogorov–Smirnovtest,202–203Kostiakovmodel,96–97Krigingmethod

meanannualrainfallforwesternGreece,72,74piezometricmapforCentralGreece,72–73principleof,70variationsof,72variogrammodelexponentialmodel,71Gaussianmodel,70nuggeteffect,70–71powermodel,71rangeofinfluence,70–71sphericalmodel,71

Lagroutingmethod,162Laggedhydrographmethod,145Landerosionandwatershedsedimentyield

climate,435erosioncontrolmeasuressource,442–444structuralmeasures,445–446

gullyerosion,435modifieduniversalsoillossequation,442rillflow,435sheeterosion,435splasherosion,434topography,435universalsoillossequation(seeUniversalsoillossequation(USLE))

Lightningimagingsensor(LIS),45,49Linearreservoirbaseflowmethod,162Log-normaldistribution,188–190,210–213Log-Pearsondistribution,195–196Low-efficiencyreservoir,269Loweratmosphere,20

Makkink’smethod,86–87Manningequation,129,344–347Masstransfermethod,79–80Mesosphere,20Meteorologicalhomogeneitytest,51–52Meteorologicalradar,26–28Meyer-PeterandMüllermethod,424–425Microhail,23ModClarktransform,161ModifiedPulsroutingmethod,162Modifieduniversalsoillossequation(MUSLE),442Muskingummethod,166Muskingumroutingmethod,162Muskingum-Cungeroutingmethod,162

NeighbourhoodKrigingmethod,72Nelder-Meadmethod,164Non-frontalcycloniccooling,23

NonlinearBoussinesqbaseflowmethod,162Normaldistribution,184–188,208–209,214Normalratiosmethod,57

Oil/greaseseparators,373,376–377Optimuminterpolationmethod,seeKrigingmethodOrdinarysimpleKrigingmethod,72Orographiccooling,23Overlandflow,341–343

Penmanmethod,80–81,90Penman-Monteithmethod,81–82,91Percolationloss/gainmethod,162φindex,108–110Philipmodel,97Phreaticaquifers,seeUnconfinedaquifersPhysicalhydrology,4Pointmeasurementdevices,networkinstallationof,49–51Poissondistribution,191–192Pollutants

chlorides,338generationprocesses,337heavymetals,338oils,grease,338polycyclicaromatichydrocarbons,338suspendedsolids,337

Pollutioncontrolseweranddrainageditcheschannelstabilizationstructures,378controlofillicitsanitarysewerconnections,383grids,378oil/greaseseparators,373,376–377riprap,gabions,379storage,retention/detention,stormsewer,379–383stormsewerwashing,383vegetatedcanals,379waterqualityinlets,377–378

sourceatmosphericscrubbing,366

controlofchemicals,367coveringwithgrass,mulchandspecialmeshes,366–367environmentaleducationandawareness,366erosioncontrolmeasures,366grassfilters,368–370,372,374–375grassswales,375–376infiltrationinlets,371,374infiltrationtrenches,370–373porouspavements,367–370rainwaterharvesting,372–374retentiononroof,372streetandimpermeablesurfaceflushing,366streetsweeping,366

surfacedetention/retentionandstoragebasinparallelwithstormsewer,385–387detentionbasins,combinedsewer,aquaticsystem,393–395detentionbasins,parallelcombinedsewer,393,395detentionbasins,seriescombinedsewer,392–394extendeddrybasins,387–390parallelflooddetentionbasins,384–386seriesflooddetentionbasins,384–385wetbasins,390–392wetlands,392

Population,173Porousmedia,226Porouspavements,367–370PR,seePrecipitationradar(PR)Precipitation,1

datahomogeneitytest,51–52doublecumulativecurve,52–56HEC-HMSmodel,163measurementofJoss-Waldvogeldisdrometers,28–38meteorologicalradar,26–28precipitationsensors,24–25raingaugeinstallation,38raingaugerecorders,25–26snowfallmeasurement,26

spacemeasurements,40,43–49pointmeasurementdevices,networkinstallationof,49–51rainfallmeasurements,completionof,54arithmeticmeanmethod,57correlationandregression,58–62inversedistancemethod,57–58normalratiosmethod,57rainfallgradient,62sourcesoferror,53

terminology,19typesofcoolingmechanismsand,23–24hail,22–23rain,21snow,22

Precipitationradar(PR),44Kaband,47–48Kuband,47–48parametersof,45–46

Precipitationsensors,24–25Pressurizedaquifer,seeConfinedaquiferPriestley–Taylormodel,87–89Primaryrainfallsensors,scangeometriesof,45–46Probabilistichydrology,4,173Probability,1,178

Bayes’theorem,177conditionalprobability,176–177definition,174event,174experimentsandsamplespace,173–175floodexample,205–207outcome/samplepoint,173probabilityfunction,176randomvariableasymmetry,181–182expectedvalueofafunction,180frequencydistribution,184functionforcreationofmoments,183kurtosis,181,183

meanandstandarddeviation,203mean(expected)value,179medianandmostprobablemode,180propertiesofmean,180standarddeviation,181standarderror,183statisticalmoments,functionsof,183variance,180–181variationcoefficient,181variationproperties,181

totalprobability,177Probablemaximumflood(PMF),306Probablemaximumprecipitation(PMP),306

Rain,21Rainfallerosivityfactor,436–437Rainfallgradient,62Rainfall-runoffanalysis

availabledata,135basinareaandothercharacteristics,135empiricalmethods,forrunoffpeakestimation,138–139purpose,135revitalizedfloodhydrographmodel,156–158runoffpeakestimation,rationalmethodforGiandottiequation,137Kirpichequation,136principle,135rainfallintensity,136runoffcoefficient,136U.S.SoilConservationServiceequation,137–138

UHtheoryassumptions,139–140ofcompositerainfall,142–145floodhydrograph,149–151instantaneousUH,155–156laggedhydrographmethod,145principleofproportionality,140–141principleofsuperposition,140–141restrictions,158

S-curvecalculation,145–149fromsinglerainfallevent,140–142syntheticUH(seeSyntheticUHs)

Rainfallsensor,24Raingauges

10-foldraingauges,24–25installationof,38networkinstallation,49–51recorders,25–26tippingbucketraingauge,25–26volumetricraingauges,24–25

Rainwaterharvesting,372–374RD-69disdrometers

description,28hydro-meteorologicalparameters,29–31raincharacteristics,derivationofcumulativerainfalldepth,33,36dropnumbersandlower/upperlimitsofdiameters,30,32finalraindropspeedvs.meandiameterofeachclass,30,33kineticenergyflux,33,39–40linearregressionofZ–Rrelationship,38,43liquidwatercontent,33,37–38meandiameterandclassrangecalculation,31,33–34radarreflectivity,36,41–42rainintensity,33,35–36raindropsizedistributions,29–30,33

RD-80disdrometersdescription,28–29hydro-meteorologicalparameters,29–31raindropsizedistributions,29–30specifications,29–30

Recessionbaseflowmethod,162Rechargewells,264Regionalhydrologicmodels,290–291Retention,74,77Revitalizedfloodhydrograph(ReFH)model,156–158Rillflow,435Rippl’sconventionaldesign,disadvantagesof,271–272

Riverbasin,6–7alignmentandlongitudinalprofileofstream,104averageslope,104basinboundary,101dischargeof,105drainagenetworkdensity,103drainagenetworkontopographicmap,102drainagesurface,102hypsometriccurve,103reservoirconstructionwaterbalancedata,14–15waterlevelestimation,15–17

ridge,101streamorder,102surfaceofstreams,104

Riverdiversiondesigncriteria,317diversionchannel,315estimationofdesignflood,318–319openchannel/closedtunnel,315

Runoffcomponents,1

Salinationinterfaceofsaltwaterandfreshwater,264–265saltwaterelevationcone,265–266

Saltwaterandfreshwater,interfaceof,264–265Saltwaterelevationcone,265–266Sample,173SCScurvenumberlossmethod,161SCSmethod,seeSoilConservationService(SCS)methodSCSUHtransform,161Sedimentation,293–294Sedimenttransportanderosion

flowresistancealluvialstreamsandrivers,409,412–418channel,405–412

incipientmotioncriticalboundaryReynoldsnumber,419

dimensionlessShieldsparameter,419permissiblevelocitiesandcriticaltractiveforces,420–421permissiblevelocity,420shearstress,420Shieldsgraph,419–420uniformbedmaterials,420

landerosionandwatershedsedimentyieldclimate,435erosioncontrolmeasures,442–446gullyerosion,435modifieduniversalsoillossequation,442rillflow,435sheeterosion,435splasherosion,434topography,435universalsoillossequation,435–441

propertiesdensity,specificweightandspecificgravity,404fallvelocity,404–405sizeandshape,402–404

sediment-relatedproblems,401–402sedimenttransportformulasbedlayer,421bedload,424–429bedmaterial,421sedimentconcentration,423sedimentdischarge,423–424suspendedload,429–431totalsedimentload,422,431–434washload,422

Semi-confinedaquifer,237–239Sequentpeakanalysismethod,272–273s-graph,161Shapeofhydrographs

bellshape,105climaticfactorsformsofprecipitation,118intensityanddurationofrainfall,116–117

raindirection,116,118raindistributionincatchment,116–117rainfalltypes,118–119timedistributionofrainfall,116–117

geologicalfactorsdeepergeologicalformations,122surfacesoillayers,121–122

topographicfactorsdistributionanddensityofstreams,119sizeandshapeofbasin,119–120slopeofbasin,119–120slopeofmainwatercourse,120–121terrainofcatchmentarea,121vegetationcover,percentageandspeciesof,121

Sheeterosion,435Shreve’sstreamclassification,102Sizingofreservoirs

activereservoirvolume,269–270cumulativeinflowcurve,270cumulativeinflows,280,282designrisk,278–279estimation,270–271methodofmaximumshortage,288–290monthlyinflows,280–281non-conventionalsizingmethod,advantagesof,279non-conventionalstochasticmethods,273–277persistence,286–288Rippl’sconventionaldesign,disadvantageof,271–272sensitivity,279–280sequentpeakanalysismethod,272–273syntheticdataofinflows,277–278synthetictimeseriesofinflows,283–286

deadvolumeactivevolumelossestimation,296–301deadstorage,293–294estimation,294–296sediments,292

floodvolume,302powergeneration,268

reliabledesign,268riverdam,267riversitewithoutmeasurementsderivationofmonthlyinflows,291–292regionalhydrologicmodels,290–291sectionsofvolumeandreservoirexploitation,268–269waterstorage,267

Slopelengthandsteepnessfactor,438–439Smith–Parlangelossmethod,161Snow,22Snowbanks,26Snowfallmeasurements,26Snowmaturation,22Snowmelt,HEC-HMSmodel,163SnyderUHtransformmethod,161Soilandaquiferparameters

porosity,224specificyield/effectiveporosity,225storagecapacity,225–226voidsratio,224–225

SoilConservationService(SCS)method,97–98agriculturalwatersheds,349,351–352antecedentmoisturecondition,349,352–353cumulativesurfacerunoffdepth,348hydrograph,354hydrologicsoilclassification,345,348permeablesurfacesinurbanareas,349,353soiltextureclassification,345,349urbanlanduses,349–350

Soilerodibilityfactor,436,439Soilmoistureaccountinglossmethod,161Spacemeasurements,ofprecipitation

themicrowaveimager,44,48precipitationradar,44Kaband,47–48Kuband,47–48parametersof,45–46

satellitescoverageandresolution,40,43

low-inclinationorbits,40mid-inclinationorbits,40sun-synchronousorbit,40TropicalRainfallMeasuringMission,43–45

Spillwaydesigncriteria,305–306flood,304hydrologiedesign,303–304

Splasherosion,434Statisticalanalysisofextremes

Gumbelmaximumdistribution,198–199Gumbelminimumdistribution,199–200limitdistributions,196pointfrequencyanalysisgraphicalmethod,197methodoffrequencyfactor,197–198

Weibulldistribution,200–201,214–215Statisticalhydrology,4,173;seealsoProbabilityStochastichydrology,4,173Storage/levelpoolroutingmethod,seeModifiedPulsroutingmethodStormsewerwashing,383Stoutcorrection,131–134Straddlestaggerroutingmethod,162Strahler’sstreamclassification,102Stratosphere,20Stream–canalmethod,263Streams;seealsoRiverbasin

alignmentandlongitudinalprofile,104componentsmodulatingstreamdischarge,105streamorder,102surfaceof,104

Streetandimpermeablesurfaceflushing,366Streetsweeping,366Supplypits,264Surfacehydrology,4Surface integration method, for areal rainfall calculation, see Areal rainfall

calculation,surfaceintegrationSurfacerunoff,5

hydrologicalloss,74,77

quantityandqualitymanagementcategoriesandtypesofBMPs,364–365pollutioncontrol(seePollutioncontrol)runoffattenuation,363–364runoffconveyance,363runoffpre-treatment,363–364runofftreatment,364,394,396secondaryimpactmitigation,364systemmaintenance,364

Surfacewaterrecharge,263–264Suspendedload(SL),429–432Synoptic-scalephenomena,73,75–76SyntheticUHs

characteristics,151SCSdimensionlesshydrographmethod,154Snyder’smethod,151–153triangularSCShydrographmethod,154

10-foldraingauges,24–25Terminalvelocity,21Theismethod,247–248,254–255Themicrowaveimager(TMI),44,48Thermosphere,20TheWorldMeteorologicalOrganization(WMO),50Thiessenmethod,forsurfacerainfallcalculation,63–64,67Thornthwaite’smethod,82–83Timedistributionsofrainfall

limited-scalephenomena,72medium-scalephenomena,72synoptic-scalephenomena,73,75–76

Tippingbucketraingauge,25–26Totalsedimentload(TSL),422

AckersandWhitemethod,432–434Einsteinmethod,431Yang’smethod,434

Transmissivity,230Trenchmethod,264TropicalRainfallMeasuringMission(TRMM)satellite

indayperiod,43–44

installedinstruments,44–45positionof,44–45

Troposphere,20Turcmethod,91–92

Unconfinedaquifersdrawdown,258–260flowdelay,257specificyield,258steadyflow,235–237verticalflow,257

Undergroundpercolation,5Uniformdistribution,192–193Univariategradientmethod,164UniversalKrigingmethod,72Universalsoillossequation(USLE)

erosioncontrolpracticesfactor,440–441landerosionprediction,435rainfallerosivityindex,436–437slopelengthandsteepnessfactor,438–439soilerodibilityfactor,436,439vegetativecoverandcroppingmanagementfactor,439–441

Urbanhydrologyandstormwatermanagementdefinitions,331–332description,333history,332–333impactsofurbanization,334–336pollutantgenerationprocesses,337receivingwaters,338–339surfacerunoffquantityandqualitymanagementcategoriesandtypesofBMPs,364–365pollutioncontrol(seePollutioncontrol)runoffattenuation,363–364runoffconveyance,363runoffpre-treatment,363–364runofftreatment,364,394,396secondaryimpactmitigation,364systemmaintenance,364

typesandsourcesofpollutants,337–338

urbanrunoffquality,336–337urbanrunoffqualitycomputationspollutantaccumulation,streetsurface,359–361simplemethodofWashington,DC,356–359USEPAmethod,354–356USGSmethod,356–357washoffofaccumulatedpollutants,361–363

urban runoff quantity computations (see Urban runoff quantitycomputations)

Urbanization,impactsof,334–336Urbanrunoffqualitycomputations

pollutantaccumulation,streetsurface,359–361simplemethodofWashington,DC,356–359USEPAmethod,354–356USGSmethod,356–357washoffofaccumulatedpollutants,361–363

UrbanrunoffquantitycomputationsrationalmethodBransbyWilliamsequation,342circularpipes,345–347FederalAviationAgencyequation,342geometriesforchannelsections,344idfcurves,340–341Izzardequation,342Kerbyequation,341Manningkinematicwaveequation,342Manningroughnesscoefficient,342–343overlandflow,341rainfallintensity,340rectangularandtrapezoidalchannelsections,344–345surfacerunoffcoefficient,339–340typicalirregularopenchannelsection,345,347urbandrainagebasin,340wettedperimeter,344

soilconservationservicemethod

agriculturalwatersheds,349,351–352antecedentmoisturecondition,349,352–353cumulativesurfacerunoffdepth,348hydrograph,354hydrologicsoilclassification,345,348permeablesurfacesinurbanareas,349,353soiltextureclassification,345,349urbanlanduses,349–350

USEPAmethod,354–356User-specifiedS-graphtransformmethod,161User-specifiedUHtransform,161USGSmethod,356–357USLE,seeUniversalsoillossequation(USLE)

Variables,173covariance,203–204randomvariable,179–184asymmetry,181–182expectedvalueofafunction,180frequencydistribution,184functionforcreationofmoments,183kurtosis,181,183meanandstandarddeviation,203mean(expected)value,179medianandmostprobablemode,180propertiesofmean,180standarddeviation,181standarderror,183statisticalmoments,functionsof,183variance,180–181variationcoefficient,181variationproperties,181

Variogram,70Vegetatedcanals,379Vegetativecoverandcroppingmanagementfactor,439–441Verticalstructureofatmosphere,20–21Visibleandinfraredradiationscanner(VIRS),44,48–49Volumetricraingauges,24–25

Washload(WL),422Washoffofaccumulatedpollutants,361–363Wastewaterdisposal,264Waterbalancemethod

evaporation,77–78evapotranspiration,81

Water-levelrecorders,123–124Waterquality,223Waterqualityinlets,377–378Waterstructure-specificissues

dischargepredictions,323–327duration-dischargecurve,327–328irrigationsystem,319operationaldepth–duration–frequencycurves,320–322wastewatertreatmentplant,322–323waterresourcesprojects,319

Water,worldwidedistributionof,8Weibulldistribution,200–201,214–215Wellhydraulics,243–246Welllosses,261–263Wellnearimpermeableboundaries,242–243Wellnearriver,240–242Wetbasins,390–392Wetlands,392

X2test,202,211–213

Yang’smethod,434