FLOW OF WATER THROUGH SOILCHAPTER 3

Learning outcomeAt the end of this lecture, student should be able to:

Define the concept of pressure head and hydraulic gradientUnderstand the definition of permeability and 1-D flow of water through soil using Darcy LawDetermine the coefficient of permeability using constant head and falling head test

Content Definition of pressure head and hydraulic gradientPermeability and 1-D flow of water through soil Darcy LawDetermination of the coefficient of permeability constant head, falling headField permeability test confined aquifer, unconfined aquiferSeepage and 2D-flow in isotropic and homogeneous soil. Seepage calculation using flow net under concrete Dam/sheet piles and through earth dam

INTRODUCTIONThe study of the flow of water through permeable soil media is important in soil mechanicsIt is necessary for estimating the quantity of underground seepage under various hydraulic conditionsE.g. for investigating problems involving the pumping of water for underground construction, and for making stability analyses of earth dams and earth retaining structures that are subject to seepage forces

INTRODUCTIONWater is free to flow within a soil massIn porous media, water will flow from the zones of higher to lower pore pressureWhen considering problems of water flow, it is usual to express a pressure as a pressure head or head, measured in meter of water

PRESSURE HEAD AND HYDRAULIC GRADIENTBernoullis equation states 3 heads components, total head (h) causing a water flow

Where h = total headu = pressurev = velocityg = acceleration due to gravityw = unit weight of waterZ = vertical distance of a given point above or below datum

PRESSURE HEAD AND HYDRAULIC GRADIENTIf Bernoullis eq is applied to the flow of water through a porous medium, the velocity head can be neglected because the seepage velocity is very smallAnd the total head at any point can be as follows

PRESSURE HEAD AND HYDRAULIC GRADIENTFigure 1: Pressure, elevation and total heads for flow of water through soil

PRESSURE HEAD AND HYDRAULIC GRADIENTThe loss of head between two points, A and B, can be given byThe head loss, h can be expressed in a nondimensional form as

Where i = hydraulic gradientL = distance between points A and B (the length of flow over which the loss of head occurred)h = hA hB

i = h L

PRESSURE HEAD AND HYDRAULIC GRADIENTFigure 2: Nature of variation of v with hydraulic gradient, i

PRESSURE HEAD AND HYDRAULIC GRADIENTIn general, the variation of the velocity v with the hydraulic gradient i is divided into 3 zonesi/ laminar flow zone (Zone I)ii/ transition zone (Zone II)iii/ turbulent flow zone (Zone III)In most soils, the flow of water through the void spaces can be considered laminar

In fractured rock, stones, gravels and very coarse sands, turbulent flow conditions may existV i

COEFFICIENT OF PERMEABILITYPermeability (hydraulic conductivity) the capacity of a soil to allow water to pass through itHydraulic conductivity is generally expressed in cm/sec or m/sec

COEFFICIENT OF PERMEABILITYThe value k is used as a measure of the resistance to flow offered by soil, and affected by several factor:

i/ the porosity of the soilii/ the particle size distributioniii/ the shape and orientation of soil particlesiv/ the degree of saturation/presence of airv/ the type of cation and thickness of adsorbed layers associated with clay minerals (if present)vi/ the viscosity of the soil water, which varies with temperature

COEFFICIENT OF PERMEABILITYHydraulic conductivity (k) varies widely for different soilsThe hydraulic conductivity of unsaturated soils is lower and increases rapidly with the degree of saturation

Soil typek (cm/sec)k (ft/min)Clean gravel100-1.0200-2.0Coarse sand1.0-0.012.0-0.02Fine sand0.01-0.0010.02-0.002Silty clay0.001-0.000010.002-0.00002Clay

Darcys LawIn 1856, Darcy published a simple equation for the discharge velocity of water through saturated soils

Where v = discharge velocity, which the quantity of water flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction flowk = hydraulic conductivity

v = ki

Darcys LawThis equation is on observation about the flow of water through clean sandsValid for laminar flow conditions and applicable for a wide range of soils

EFFECT OF TEMPERATUREThe value of the coefficient of permeability will be affected by changes in temperatureIt may be shown theoretically that for a laminar flow condition in a saturated soil mass:

Where w = unit weight of water = viscosity of waterk w

EFFECT OF TEMPERATUREA correction for the effect of temperature , may be obtained as follows:

Where kt = value k corresponding to a temperature of tk20 = value of k corresponding to a temperature of 20Ckt = temperature correction coefficient

kt = kt k20

EFFECT OF TEMPERATURE

LABORATORY MEASUREMENTSCoefficient of permeability (k) can be measured using field tests or tests conducted in the laboratoryThe aim to produce similar results as using field testsIn laboratory, errors may occur due to:i/ the presence of air bubbles in the waterii/ Variations in sample density and porosityiii/ variations in temperature and viscosity of water2 test constant head test and falling head test

The constant head test

The constant head testTo determine the coefficients of permeability (k) of coarse-grained soils such as gravels and sands having value of k above 10-4m/s

The constant head testIn this type of laboratory setup, the water supply at the inlet is adjusted in such a way that the difference of head between the inlet and the outlet remains constant during the test periodAfter a constant flow rate is established, water is collected in a graduated flask for a known duration

The constant head test

Where Q = volume of water collectedA = area of cross section of the soil speciment =duration of water collection

Where q = quantity flowing in unit time

Q = Avt = A(ki)tq = Av = Aki

The constant head testAnd because

The equation can be substituted as belowi = h Lk = QL Aht

Example 1During a test using a constant-head permeameter, the following data were collected. Determine the average value k

Diameter of sample = 100mmTemperature of water = 17Distance between manometer tapping points =150mm

Quantity collected in 2 min. (ml)541503509474Difference in manometer levels (mm)76726865

Solution

Solution

Example 2With the aid of sketch, derive the formula for the permeability of a soil using a constant head apparatus

SolutionBy applying Darcys equation

q = kAik = q/Aibut q = Q/t,i = h/Lk = 1/A x Q/t x L/Hk = QL/Ath (mm/s)

Where,q = flow rate (mm3/s)Q = quantity collected in time (s) = Q (ml) x 103 (mm3)A = cross sectional area (mm2)H = different in manometer levels (mm)L = distance between manometer tapping points (mm)

The falling head test

The falling head testTo determine the coefficient of permeability of fine soilsFor these soils, the rate of flow of water through them is too small to enable accurate measurements using the constant head permeameter

ProcedureThe test is conducted by filling the standpipe with de-aired water and allowing seepage to take place through the sampleThe height of water in the standpipe is recorded at several time intervalsTest repeated using standpipes of different diameter

The falling head test

Wherea = cross sectional area of the standpipeA = cross sectional area of the samplet = time intervalh1 = initial standpipe readingh2 = final standpipe readingL = length of samplek = 2.303 aL log10 h1 At h2

Procedure Specimen 100mm diameter undisturbed sampleSpecimens can also be prepared by compaction in a standard mouldA wire mesh and gravel filter is provided at the top and bottom of the sampleThe base of the cylinder is stood in a water reservoir fitted with a constant-level overflow and the top connected to a glass standpipe of known diameter

Example 3During a test using falling-head permeameter, the following data were recorded. Determine the average value of k.

Diameter of sample = 100mmLength of sample = 150mm

Recorded dataStandpipe diameter (mm)Level in standpipe (mm)Time interval (s)Initial, h1Final, h25.001200800828004001499.00120090017790070016970040036812.501200800485800400908

SolutionCross sectional area of sample, A = 1002 x 4

Cross sectional area of standpipe a = d2 x 4k = 2.303 aL log10 h1 At h2

Solution

Recorded dataStandpipe diameter (mm)Level in standpipe (mm)Time interval (s)Initial, h1Final, h25.001200800828004001499.00120090017790070016970040036812.501200800485800400908

Computed log10 h1 h2k (mm/s)x 10-30.17611.8540.30101.7440.12491.9750.10911.8070.24301.8470.17611.9590.30101.789Average k = 1.85 x 10-3 mm/s = 1.85 x 10-6 m/s

Exercise Question 1The following data were recorded during a constant-head permeability test:

Internal diameter = 75 mmHead lost over a sample length of 180 mm = 247 mmQuantity of water collected in 60 s = 626 mlCalculate the coefficient of permeability for the soilQuestion 2In a falling-head permeability test the following data were recorded:

Internal diameter of permeameter = 75.2 mmLength of sample = 122.0 mmInternal diameter of standpipe = 6.25 mmInitial level in standpipe = 750.0 mmLevel in standpipe after 15 min = 247.0 mmCalculate the permeability of the soil

Learning outcomeAt the end of this lecture, student should be able to:

Determine the coefficient of permeability for field permeability test confined aquifer and unconfined aquifer

FIELD PERMEABILITY TESTComprehensive multiple-well pumping tests can be expensive to be carry out, but offer a high level of reliabilityThe use of site investigation boreholes can be economically advantageous

FIELD PERMEABILITY TESTSteady state pumping testsPumping tests involve the measurement of a pumped quantity from a well, together with observations in other wells of the resulting drawdown of the ground levelSteady state is achieved when a constant pumping rate, the levels in observation wells are then notedThe analysis of the results depends on whether the aquifer is confined or unconfined

Pumping test in a confined aquiferThe average hydraulic conductivity for a confined aquifer can be determined by conducting a pumping test from a well with a perforated casing that penetrates the full depth of the aquiferThe pumping rate must not be high enough to reduce the level in the pumping well below the top of the aquiferPumping is continued at a uniform rate q until a steady state is reachedThe arrangement of a pumping well and two observation wells is shown here

Pumping test in a confined aquifer

Pumping test in a confined aquiferWater can enter the test well only from the aquifer of thickness HThe hydraulic conductivity is given as follows k = q log10 (r1/r2) 2.727H (h1-h2)

Pumping test in a confined aquiferApproximation may be derived from a consideration of the radius of influence (r0) of the pumpingIt may be assumed that no drawdown of the piezometric head takes place outside the radius of influencer = ro and h =ho

Example 4 A permeability pumping test was carried out from a well sunk into a confined stratum of dense sand. The arrangement of pumping well and observation wells are shown below. When a steady state was achieved at a pumping rate of 37.4m3/hr, the following drawdown were observed:

pumping well: d = 4.46mobservation well 1: d = 0.42mobservation well 2: d = 1.15m

a)Calculate a value for the coefficient of permeability of the sand using the observation well data

b) Estimate the radius of influence at this pumping rate

Solution a) Observation well data: r1 = 50m r2 = 15m

ho = 11.7 + 7.4 2.5 = 16.6mh1 = 16.6 0.42 = 16.18mh2 = 16.6 1.15 = 15.45m

q = 37.4 / 3600 = 10.39 x 10-3 m3/sH = 11.7m

k = q log10 (r1/r2) 2.727 H (h1-h2)

= (10.39 x 10-3 ) log 10 (50/15) 2.727 x 11.7 x (16.18 15.45) = 2.33 x 10-4 m/s

(b) No drawdownThen, putting r1 =50m and h1 = 16.18m

k = q log10 (ro/r1) 2.727 H (ho-h1)

Log10 (ro/50) = 2.33 x 10-4 x 2.727 x 11.716.6 16.18 10.39 x 10-3ro = 100m

Pumping test in an unconfined aquiferAn unconfined aquifer is a free-draining surface layer underlain by an impervious baseDuring the test, water is pumped out at a constant rate from a test well that has a perforated casingSeveral observations wells at various radial distance are made around the test wellContinuous observation of water level in the test well are made after the start of pumping until a steady state is reached

Pumping test in an unconfined aquifer

Pumping test in an unconfined aquiferThe hydraulic conductivity is given as follows

k = 2.303q log10 (r1/r2)(h12-h22)

Example 5A permeability test was carried out from well sunk through a surface layer of medium dense sand. Initially, the water table was located at a depth of 2.5m. When a steady state was achieved at a pumping rate of 23.4m3/hr, the following draw-downs were observed

Pumping well:d = 3.64mObservation well 1:d = 0.48mObservation well 2:d = 0.96m

(a) Calculate value for the coefficient of permeability of the sand using the observation well data(b) Estimate the radius of influence at this pumping rate

Solution (a) Observation data :r1 = 62mr2 = 18mho = 12 2.5 = 9.5mh1 = 9.5 0.48 = 9.02mh2 = 9.5 0.96 = 8.54mq = 23.4/3600 = 6.5 x 10-3 m3/sk = 2.303q log10 (r1/r2)(h12-h22)

= 3.04 x 10-4 m/s

(b) Putting r1 =62m and h1 = 9.02m

k = 2.303q log10 (ro/r1) (ho2-h12)

ro = 229 m

ExerciseQuestion 1For a field pumping test a well was sunk through a horizontal layer of a sand which proved to be 14.4 m thick and to be underlain by a stratum of clay. Two observation wells were sunk, respectively 18 m and 64 m from the pumping well. The water table was initially 2.2 m below the ground level. At a steady state pumping-rate of 328 litres/min, the drawdowns in the observation wells were found to be 1.92 and 1.16 m respectively. Calculate the coefficient of permeability of the sand.Question 2A horizontal layer of sand of 6.0 m thickness is overlain by a layer of clay with a horizontal surface thickness of 4.8 m. An impermeable layer underlies the sand. In order to carry out a pumping test, a well was sunk to the bottom of the sand and two observation wells were sunk through clay just into the sand at distances 12 m and 40 m from the pumping well. At a steady pump rate of 600 litres/min., the water levels in the observation wells were reduced by 2.28 m and 1.79 m respectively. Calculate the coefficient of permeability of the sand if the initial piezometric surface level lies 1.0 m below the ground surface.

Seepage and 2-D Flow in Isotropic and Homogeneous Soil

In preceding lesson, we considered some simple cases for which direct application of Darcys law was required to calculate the flow of water through soilIn many instances, the flow of water through soil is not in one direction only, nor is it uniform over the entire area perpendicular to the flowThe seepage taking place around sheet-piling, dams, under other water-retaining structures and through embankments and earth dams is two dimensionalVertical and horizontal velocity components vary from point to point within the cross-section of the soil massGraphical representation known as a flow net will be introduced

The flow of water through soils is described by Laplaces equation.

Where H =total head kx and kz = hydraulic conductivities in X and Y directionsLaplaces eq expresses the condition that the changes of hydraulic gradient in one direction are balanced by changes in the other directions

Laplaces equation is also called the potential flow equation because the velocity head is neglectedIf the soil is isotropic with respect to the hydraulic conductivity that is kx = kz, the preceding continuity equation for 2-D flow simplifies to

There are 2 techniques for Laplaces equation. One of it is an approximate method called flownet sketching

Flow Net

Flow line - a line along which a water particle will travel from upstream to the downstream side in the permeable soil medium

Equipotential line - a line along which the potential head at all points is equalA flownet is a graphical representation of a flow field that satisfies Laplaces equation and comprises a family of flow lines and equipotential linesA combination of a number of flow lines and equipotential lines is called a flow net

Completed flow net

Construction of Flow NetDraw the structure and soil mass to suitable scaleIdentify impermeable and permeable boundariesSketch a series of flow lines (4 or 5) and then sketch an appropriate number of equipotential lines such that the area between a pair of flow lines and a pair of equipotential lines is approximately a curvilinear squareTheoritically, any no. of flow lines may be drawn and the greater the no., the more accurate should be the calculations that follow.However, from a practical point of view the task is simplified by drawing only few flow lines; it is not often that more than 5 or 6 will be necessary

Seepage CalculationIn any flow net, the strip between any two adjacent flow lines is called a flow channel.Let h1, h2, h3, h4,hn be the piezometric levels corresponding to the equipotential linesThe rate of seepage through the flow channel per unit length (perpendicular to the vertical section through the permeable layer) can be calculated as followsq1=q2=q3=..=qn

From Darcys law, the flow rate is equal to kiA. Thus

If the number of flow channels in a flow net is equal to Nf, the total rate of flow through all the channels per unit length can be given by

Or

Where H =head difference between the upstream and downstream sidesNd = number of potential dropNf = number of flow channels in flow netn = b1/l1=b2/l2=b3/l3==n (i.e. the elements are not square)

EXAMPLE

ANSWER

SUMMARYIn this chapter, weve discussed Darcys Law, definition of hydraulic conductivity, laboratory and field determination of hydraulic conductivity The accuracy of the values of k determined in the laboratory depends on several factorTemperature of the fluidViscosity of the fluidTrapped air bubbles present in the soil specimenDegree of saturation of the soil specimenMigration of fines during testingDuplication of field conditions in the laboratoryThe actual value of the hydraulic conductivity in the field may also be somewhat different than that obtained in the laboratory because of the nonhomogeneity of the soilHence, proper care should be taken in assessing the order of the magnitude of k for all design consideration.