water influx 1

8/16/2019 Water Influx 1

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Water InfuxWater Infux

Dr.Mostafa MahmoudDr.Mostafa Mahmoud

KinawyKinawy


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• Nearly all hydrocarbon reservoirs are surrounded

by water-bearing rocks called aquifers. Theseaquifers may be substantially larger than the oil orgas reservoirs they adjoin as to appear innite insize or they may be so small in size as to be

negligible in their e!ect on reservoir performance"• #s reservoir $uids are produced and reservoir

pressure declines a pressure di!erential developsfrom the surrounding aquifer into the reservoir"%ollowing the basic law of $uid $ow in porousmedia the aquifer reacts by encroaching acrossthe original hydrocarbon-water contact" &n somecases water encroachment occurs due tohydrodynamic conditions and recharge of the

formation by surface waters at an outcrop"


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• &n many cases the pore volume of the aquifer isnot signicantly larger than the pore volume of thereservoir itself" Thus the e'pansion of the water inthe aquifer is negligible relative to the overallenergy system and the reservoir behaves

volumetrically"

• &n this case the e!ects of water in$u' can beignored" &n other cases the aquifer permeabilitymay be su(ciently low such that a very large

pressure di!erential is required before anappreciable amount of water can encroach into thereservoir" &n this instance the e!ects of waterin$u' can be ignored as well"


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AQUIFERS

• )any gas and oil reservoirs produced by a mechanism

termed water drive. *ften this is called natural waterdrive to distinguish it from articial water drive thatinvolves the injection of water into the formation"+ydrocarbon production from the reservoir and thesubsequent pressure drop prompt a response from the

aquifer to o!set the pressure decline" This responsecomes in a form of water inux, commonly calledwater encroachment, which is attributed to,

• 'pansion of the water in the aquifer

• .ompressibility of the aquifer rock

• #rtesian $ow where the water-bearing formationoutcrop is located structurally higher than the payzone


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/eservoir-aquifer systems are commonlyclassied on the basis of,

0egree of pressure maintenance

• %low regimes

*uter boundary conditions

• %low geometries


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Maintenane

• 1ased on the degree of the reservoir pressure

maintenance provided by the aquifer thenatural water drive is often qualitativelydescribed as,

• #ctive water drive

• 2artial water drive

• 3imited water drive

The term active water drive refers to the water

encroachment mechanism in which the rateof water in$u' equals the reservoir totalproduction rate" #ctive water-drive reservoirsare typically characterized by a gradual and

slow reservoir pressure decline"


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Outer !oun"ar# Con"itions•The aquifer can be classied as innite or nite

4bounded5" 6eologically all formations are nite butmay act as innite if the changes in the pressure atthe oil-water contact are not 7felt8 at the aquiferboundary" &n general the outer boundary governsthe behavior of the aquifer and therefore,

• a" &nnite system indicates that the e!ect of thepressure changes at the oil9aquifer boundary cannever be felt at the outer boundary" This boundaryis for all intents and purposes at a constant pressure

equal to initial reservoir pressure"

• b" %inite system indicates that the aquifer outerlimit is a!ected by the in$u' into the oil zone andthat the pressure at this outer limit changes with

time"


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F$o% Regi&es

• There are basically three $ow regimes that

in$uence the rate of water in$u' into thereservoir" Those $ow regimes are,

• a" :teady-state

• b" :emi-steady 4pseudo-steady5-state

• c" ;nsteady-state


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F$o% 'eo&etries

Reser(oir)a*uier s#ste&s an +e$assi,e" on t-e +asis o fo% geo&etr#as.

a" dge-water drive

b" 1ottom-water drive

c" 3inear-water drive


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Infux

• Normally very little information is obtained during

the e'ploration-development period of a reservoirconcerning the presence or characteristics of anaquifer that could provide a source of water in$u'during the depletion period"

• Natural water drive may be assumed by analogywith nearby producing reservoirs but earlyreservoir performance trends can provide clues" #comparatively low and decreasing rate ofreservoir pressure decline with increasingcumulative withdrawals is indicative of $uid in$u'"


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Figure /0 %low geometries

Linear)%ater Dri(e

di i f $ id i $


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&ndications of $uid in$u'"

• arly water production from edge wells is

indicative of water encroachment" :uchobservations must be tempered by the possibilitythat the early water production is due to formationfractures< thin high permeability streaks< or to

coning in connection with a limited aquifer" Thewater production may be due to casing leaks"

• &f the reservoir pressure is below the oil saturationpressure a low rate of increase in produced gas-

oil ratio is also indicative of $uid in$u'"• .alculation of increasing original oil-in-place from

successive reservoir pressure surveys by using thematerial balance assuming no water in$u' is also

indicative of $uid in$u'"


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WATER INFLU1 MODELS

• :everal models have been developed for estimatingwater in$u' that are based on assumptions that

describe the characteristics of the aquifer"• The mathematical water in$u' models that are

commonly used in the petroleum industry include,

• 2ot aquifer

• :chilthuis= steady-state• +urst=s modied steady-state

• The >an verdingen-+urst unsteady-state

- dge-water drive

- 1ottom-water drive• The .arter-Tracy unsteady-state

• %etkovich=s method

- /adial aquifer

- 3inear aquifer


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T-e Pot A*uier Mo"e$

The simplest model that can be used to estimate thewater in$u' into a gas or oil reservoir is based onthe basic denition of compressibility" # drop in thereservoir pressure due to the production of $uidscauses the aquifer water to e'pand and $ow into thereservoir" The compressibility is dened

mathematically as,

> ? c > p @@"" 4A5

#pplying the above basic compressibility denitionto the aquifer gives,

Bater in$u' ? 4aquifer compressibility5 4initialvolume of water5 4pressure drop5


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orBe ? 4cw C cf 5 Bi 4pi - p5 @@@@@"4D5

where Be ? cumulative water in$u' bbl

cw ? aquifer water compressibility psi-A

cf ? aquifer rock compressibility psi-A

Bi ? initial volume of water in the aquifer bbl

pi

? initial reservoir pressure psip ? current reservoir pressure 4pressure at oil-

water contact5 psi


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• .alculating the initial volume of water in the

aquifer requires the knowledge of aquiferdimension and properties" These however areseldom measured since wells are not deliberatelydrilled into the aquifer to obtain such information"

%or instance if the aquifer shape is radial then,

22222343


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• quation 4D5 suggests that water is

encroaching in a radial form from alldirections" Euite often water does notencroach on all sides of the reservoir or thereservoir is not circular in nature"

• To account for these cases a modication toquation 4A5 must be made in order toproperly describe the $ow mechanism" *ne ofthe simplest modications is to include thefractional encroachment angle f in theequation to give,

• Be ? 4cw C cf 5 Bi f 4pi - p5 @@"4F5

@@""4G5


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• The above model is only applicable to a small

aquifer i"e" pot aquifer whose dimensions are ofthe same order of magnitude as the reservoiritself" 0ake 4AHIJ5 points out that because theaquifer is considered relatively small a pressure

drop in the reservoir is instantaneouslytransmitted throughout the entire reservoir-aquifer system" 0ake suggests that for largeaquifers a mathematical model is required whichincludes time dependence to account for the factthat it takes a nite time for the aquifer torespond to a pressure change in the reservoir"


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S-i$t-uis5 Stea"#)StateMo"e$

• :chilthuis 4AHKL5 proposed that for an aquiferthat is $owing under the steady-state $owregime the $ow behavior could be described

by 0arcy=s equation" The rate of water in$u'ew can then be determined by applying

0arcy=s equation,

@@@@@4L5


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• The last relationship can be moreconveniently e'pressed as,

@@@"4I5


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• The parameter . is called the water in$u'

constant and is e'pressed in bbl9day9psi" Thiswater in$u' constant . may be calculated fromthe reservoir historical production data over anumber of selected time intervals provided that

the rate of water in$u' ew has been determinedindependently from a di!erent e'pression"

• &f the steady-state appro'imation adequatelydescribes the aquifer $ow regime the calculated

water in$u' constant . values will be constantover the historical period"

• Note that the pressure drops contributing to in$u'are the cumulative pressure drops from the initial

pressure"


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• &n terms of the cumulative water in$u' Be thecommon :chilthuis e'pression for water in$u' is,

• quation 4J5 may be written in the followingform,

@@@@4J5


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Mo"e$

• *ne of the problems associated with the:chilthuis= steady-state model is that as the

water is drained from the aquifer the aquiferdrainage radius ra will increase as the timeincreases" +urst 4AHFK5 proposed that the7apparent8 aquifer radius ra would increasewith time and therefore the dimensionless

radius ra9re may be replaced with a timedependent function as,

ra9re ? at @@@@@@@4AM5

@@@@@@"4H5


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• :ubstituting quation 4AM5into quation 4L5gives,

@@@""

4AA5

@@@""4AD5

@@@""4AK5


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• The +urst modied steady-state equationcontains two unknown constants a and . thatmust be determined from the reservoir

aquifer pressure and water in$u' historicaldata" The procedure of determining theconstants a and . is based on e'pressingquation 4AA5 as a linear relationship"

@@@""4AF5

%igure D, )odied :teady :tate


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%igure D, )odied :teady :tateBater &n$u' )odel

• quation 4AF5 indicates that a plot of 4pi -p59ew versus ln4t5 will be a straight line with a

slope of A9. and intercept of 4A9.5ln4a5 asshown schematically in %igure 4D5"

0etermination of . and n


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) )Mo"e$

• The mathematical formulations that describe the$ow of crude oil system into a wellbore are identicalin form to those equations that describe the $ow ofwater from an aquifer into a cylindrical reservoir asshown in %igure 4K5

• Bhen an oil well is brought on production at a

constant $ow rate after a shut-in period the pressurebehavior is essentially controlled by the transient4unsteady-state5 $owing condition" This $owingcondition is dened as the time period during whichthe boundary has no e!ect on the pressure behavior"

• The dimensionless form of the di!usivity equation isbasically the general mathematical equation that isdesigned to model the transient $ow behavior inreservoirs or aquifers"


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Figure 40 Water infux into a #$in"ria$

reser(oir0


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• &n a dimensionless form the di!usivity equationtakes the form,

>an verdingen and +urst 4AHFH5 proposed solutionsto the dimensionless di!usivity equation for thefollowing two reservoir aquifer boundary

conditions,• .onstant terminal rate

• .onstant terminal pressure

@@@@@4AG5


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• %or the constant terminal rate boundary

condition the rate of water in$u' is assumedconstant for a given period< and the pressuredrop at the reservoir-aquifer boundary iscalculated" %or the constant terminal pressure

boundary condition a boundary pressure drop isassumed constant over some nite time periodand the water in$u' rate is determined"

• &n the description of water in$u' from an aquifer

into a reservoir there is greater interest incalculating the in$u' rate rather than thepressure"


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• This leads to the determination of the water in$u'as a function of a given pressure drop at the innerboundary of the reservoir-aquifer system"

• >an verdingen and +urst solved the di!usivityequation for the aquifer-reservoir system byapplying the 3aplace transformation to theequation" The authors= solution can be used todetermine the water in$u' in the following

systems,• dge-water-drive system 4radial system5

• 1ottom-water-drive system

• 3inear-water-drive system

E" W t D i


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E"ge)Water Dri(e

• %igure F" shows an idealized radial $ow systemthat represents an edge-water-drive reservoir"

The inner boundary is dened as the interface

between the reservoir and the aquifer"• The $ow across this inner boundary is

considered horizontal and encroachment occursacross a cylindrical plane encircling the

reservoir" Bith the interface as the innerboundary it is possible to impose a constantterminal pressure at the inner boundary anddetermine the rate of water in$u' across theinterface"

d l


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"model


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>an verdingen and +urst assumed that the aquifer ischaracterized by,

• ;niform thickness

• .onstant permeability

• ;niform porosity

• .onstant rock compressibility

• .onstant water compressibility• The authors e'pressed their mathematical relationship

for calculating the water in$u' in a form of adimensionless parameter that is called dimensionlesswater inux Be0" They also e'pressed the

dimensionless water in$u' as a function of thedimensionless time t0 and dimensionless radius r0

thus they made the solution to the di!usivity equationgeneralized and applicable to any aquifer where the

$ow of water into the reservoir is essentially radial"

Figure 80 Di&ension$ess %ater infux WeD


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Figure 80 Di&ension$ess %ater infux WeD or se(era$ (a$ues o ra9re0 (VanEverdingen and Hurst WeD. Permission to publish by the SPE.)

• The authors presented their solution in tabulatedand graphical forms "


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• The two dimensionless parameters t0 and r0

are given by,

3/:3

3/;<

3/=3


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• The water in$u' is then given by,

3/><

3?@<

Ta+$e /@)/


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Ta+$e /@)/

+$


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Ta+$e /@)?


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• quation 4DM 5assumes that the water is encroaching ina radial form" Euite often water does not encroach onall sides of the reservoir or the reservoir is not circularin nature" &n these cases some modications must bemade in quation 4DM5 to properly describe the $owmechanism" *ne of the simplest modications is tointroduce the encroachment angle to the water in$u'

constant 1 as,

• is the angle subtended by the reservoircircumference i"e" for a full circle ? KLM and forsemicircle reservoir against a fault ?AJM as shown

in %igure AM-AD"

3?/<

3??<

%igure L, Bater 0rive /eservoir


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%igure L, Bater 0rive /eservoir

water influx 1

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