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Geotechnical News, June 1998 52

Unsaturated Seepage Modeling Made EasyMurray D. Fredlund

Department of Civil Engineering University of Saskatchewan, Saskatoon, Sask., Canada

IntroductionFresh out of university, one of the firstchallenges I was presented with was aseepage modeling problem. My employerhad, of course, purchased

thousands ofdollars of seepage modeling software and itwas my responsibility to recover the cost ofthis software by demonstrating its worth. Ihad experience in both seepage modeling aswell as the programming of finite elementseepage software so I eagerly pursued theproblem at hand. The problem involvedtransient flow of water in both the saturatedand unsaturated zones of a soil profile and, assuch, required two soil functions for eachsoil. A soil-water characteristic curve,(SWCC), and a hydraulic conductivity curvewere required as input into the finite elementseepage program. Two soil types wereinvolved in the problem and I approached myemployer for the four required soil propertyfunctions. I naturally expected these functionswould be provided as they consistently werein university.

I then learned what most graduates learnwhen attempting to do seepage modeling. At$3000 to $5000 per test, there was little or nomoney in the budget to experimentallydetermine these soil functions. A search fortypical ways to estimate these soil functionssimilarly came up dry. In the end, a rough,hand-drawn, estimate of these soilfunctions was used and the analysisproceeded. Several thousand dollars of state-of-the-art seepage analysis software waslimited by the lack of information.

Role of the Soil-WaterCharacteristic CurveThe soil-water characteristic curve (i.e.,relationship between water content andsuction) has become of great value inestimating unsaturated soil propertyfunctions. The characterization of seepage,

for example, in terms of a hydraulic headgradient and a coefficient of permeabilityfunction appears to be generally accepted(Fredlund, 1995). Figure 1 illustrates therelationship between the soil-watercharacteristic curve and the coefficient ofpermeability function for the unsaturatedportion of the soil profile. The use ofnonlinear soil property functions foranalyzing unsaturated soils problems appearsto be gaining general acceptance. This articleprimarily addresses indirect procedures thatcan be used to estimate unsaturated soilproperty functions for use in the numericalmodeling of saturated/ unsaturated soilsystems in engineering practice.

Figure 1. Visualization of therelationship between the coefficient ofpermeability function and the soil-watercharacteristic curve.

Properties of the Soil-WaterCharacteristic CurveThe behavior of unsaturated soils (i.e.,unsaturated soil property functions) arestrongly related to the pore size geometry andthe pore size distribution. The soil-watercharacteristic curve becomes a dominant

relationship for understanding unsaturatedsoil behavior. The soil-water characteristiccurve defines the degree of saturationcorresponding to a particular suction in thesoil and becomes a measure of the pore sizedistribution of the soil. Figure 2 shows thegeneral features of the desorption andadsorption branches of a soil-watercharacteristic curve. An equation proposed byFredlund and Xing (1994) to empiricallybest-fit the soil-water characteristic curve isas follows:

( )( )( )

w a w

s

a w fn

mC u u

e u u af f

=

+

ln /

[1]

where: w = volumetric water content, s =volumetric water content at saturation, e =2.718...., ( ua - uw ) = soil suction, af = soilparameter related to the air entry of the soiland equal to the inflection point on the curve,nf = soil parameter related to the rate ofdesaturation, mf = soil parameter related toresidual water content conditions, C(ua- uw) =correction factor to ensure that the functiongoes through 1,000,000 kPa of suction at zerowater content.

The soil-water characteristic curve can beused to compute approximate soil propertyfunctions for unsaturated soils. Examples arethe coefficient of permeability function, thecoefficient of water volume change functionand the shear strength function (Fredlund,1995). While it is relatively easy to measurethe soil-water characteristic curve in thelaboratory, it is still quite costly and the testhas not found its way into most conventionalsoils laboratories. For this reason, estimationof the soil-water characteristic using grainsize distribution and volume-mass propertiesis beneficial.

Soil-watercharacteristic curve

Soil suction (kPa)

Wat

er co

nte

nt (%

)

Soil suction (kPa)101 102

103

104 105 106

101 102

103

104 105 106

Coe

ffici

en

t of p

erm

eab

ility,

k w

(m

/s)

0

1020304050

10-5

10-6

10-7

10-8

10-9

10-10


Approaches to Obtain UnsaturatedSoil Property FunctionsAt the start of my graduate studies I set outwith the intent of solving the problem ofcoupled seepage and volume change forunsaturated soils. I soon discovered that themost difficult part of solving coupledproblems in unsaturated soils was not solvingthe coupled equations but rather obtaininginput that is reasonable and physicallyrealistic. I noticed that most of the problemswith non-convergence and instability in finiteelement software were due to unreasonableinput.

This realization led to the development of atheoretical framework which outlinesproposed methods for working withunsaturated soil properties and functions. Theframework was later developed into asoftware knowledge-based database systemcalled SoilVision. The primary purpose ofthe software is to provide the scientificcommunity with a large database ofunsaturated soils information as well as

theoretical methods of estimating thebehavior of unsaturated soils. The system willthen allow unsaturated soil analysis wheredata is limited. The software allows thefollowing methods to be implemented.

Several approaches can be taken towards thedetermination of unsaturated soil propertyfunctions (Fig. 3). The term, unsaturatedsoil property functions, refers to suchrelationships as: 1.) coefficient ofpermeability versus soil suction, 2.) waterstorage variable versus soil suction, and 3.)shear strength versus soil suction. Laboratorytests can be used as a direct measure of therequired unsaturated soil property. Forexample, a (modified) direct shear test can beused to measure the relationship betweenmatric suction and shear strength. These testscan be costly and the necessary equipmentmay not be available. Therefore, it may besufficient to revert to an indirect laboratorytest involving the measurement of the soil-water characteristic curve for the soil. Thesoil-water characteristic curve can then beused in conjunction with the saturated shearstrength properties of the soil, to predict the

relationship between shear strength andmatric suction. Some accuracy will likely belost in reverting to this approach; however,the trade-off between accuracy and cost maybe acceptable for many engineering projects.

Figure 3 also shows the possibility of using aclassification test for the prediction of thedesired unsaturated soil property function. Aclassification test such as a grain size analysisis used to estimate the soil-watercharacteristic curve which in turn is used todetermine the unsaturated soil propertyfunction. A theoretical curve could be fittedthrough the data from a grain size analysis(Fig. 4). The theoretical grain size curve isthen used for predicting the soil-watercharacteristic curve. A comparison of theestimated soil-water characteristic curve withexperimental data is shown in Fig. 5. Whilethere may be a further reduction in theaccuracy of the estimated unsaturated soilproperty function, the engineer must assesswhether or not the approximated soil functionis satisfactory for the analyses which must beperformed.

Figure 2 Definition of variables associated with the soil-water characteristic curve

Figure 3 Approaches that can be used in the laboratory to determine the unsaturated soilproperties.

P resen t th eU n sa tu ra ted

S o il F un c tion s

D irec t M easu rem entE xperim en ts

C om p u te theU n sa tu ra ted

S o il F u n c tion s

S o il-W ate rC h arac teris t ic C urve

M easu rem en ts

C om p u te th eU n sa tu rated

S o il F u nc tion s

E s tim ation o fS o il-W ate r

C h arac teris t icC u rve

C lassifica tion Tes ts(G ra in -S ize D is trib u tion )

L A B O R A TO R Y TE S TS

0.50Air-entry value

Residual water content

0.45

0.40

0.35

0.30

0.250.20

0.15

0.10

0.05

0.00

Volu

me

tric

wa

ter c

on

tent

Soil suction (kPa)

0.1 1 10 100 1000 10000 100000 1000000


0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1 10

Particle diameter (mm)

% P

assin

g

Sieve analysis

Fit with FredlundEquation

Figure 4 Grain-size distribution curve fit for a Silty Clay (#10838).

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.1 1 10 100 1000 10000

Soil suction (kPa)

Volu

met

ric w

ater

co

nte

nt Estimated byFredlund methodExperimental

Figure 5 Comparison between experimental and estimated soil-water characteristiccurves for Silty Clay (#10838

Figure 6 illustrates how one of several approaches can be used to determine the unsaturatedsoil property functions when using the classification and/or soil-water characteristic curve inconjunction with a knowledge-based system, to compute the unsaturated soil property

functions. Plausible procedures can bestbe viewed within the context of adatabase of soil-water characteristic curveinformation and a knowledge-basedsystem. Ongoing use is made of dataaccumulated from other laboratorystudies. The first suggested procedureinvolves matching measured soil-watercharacteristic curves with soil-watercharacteristic curves already in thedatabase. The measured soil-watercharacteristic curves can be either used tocompute unsaturated soil propertyfunctions or can be used to selectunsaturated soil property functionsalready in the database.

The second suggested procedure involvesmatching measured classificationproperties (i.e., grain size curves) withclassification properties already in thedatabase. Once one or more similar soilshave been found, corresponding soil-water characteristic curves can beretrieved from the database. These soil-water characteristic curves data can beused to compute suitable unsaturated soilproperty functions or existing unsaturatedsoil property functions can be retrievedfrom the database.

The third suggested procedure involvesworking directly with the measuredgrain-size distribution curve. There mayalso be some value in comparing thegrain size curve to grain size curves in thedatabase. Soil-water characteristic curvescan then be computed and compared tosoil-water characteristic curves in thedatabase. A decision must be maderegarding a reasonable soil-watercharacteristic curve and then theunsaturated soil property functions can becomputed. Each of the above suggestedprocedures becomes increasingly lessprecise.

The advantages to this approach arenumerous.


Com puteUnsa tura te d

S oil Func tions

S e le c tUnsa tura te d

S oil Func tions

M atc h S o il-W a te rC hara c teristic C urve s

C om pute theU nsa turate d

S oil Functions

S ele ctU nsa turate d

S oil Functions

S e le c t S o il-W a terCha ra c te ristic C urve s

M a tc h S o il C la ssific a tionor O the r Fa c tors

C om pute theU nsa tura te d

S oil Func tions

C om pa re C om pute d Curve sw ith S im ila r S o il-W a ter

C ha rac te ristic Curves inD a ta ba se ; the n S e lec t

C om pute S o il-W a te rC ha rac te ristic Curve

M a tc h G ra in -S izeD istribution

U S E O F A KN OW LE D G E -BAS E D S YS T E M

Figure 6 Approaches that can be used to determine the unsaturated soil property functionswhen using classification tests and a data base.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170-10

0

10

Figure 7 Problem definition for site in Papua New Guinea.

Firstly, an estimate of the unsaturated behavior of a certain soil is quickly available.Unsaturated soil mechanics has often been avoided due to complexity. The SoilVision systemalleviates this complexity. Secondly, the cost of estimation of soil behavior is greatly reduced.Testing of unsaturated soil property functions can cost thousands of dollars. SoilVisionprovides estimates without the high cost of experimental testing. Thirdly, SoilVision makesthe estimation of behavior of unsaturated soils easy so that inexperienced professionals canwork in this difficult area.

An Example of an Environmental ApplicationAn example application of this technology is the modeling of water seepage through minetailings. A mine site in Papua, New Guinea is presented in this example. A eroded drainageditch through mining tailings over a clay layer forms the problem (Figure 7). Two types ofanalysis are required; steady state and transient state. A simulated rainfall of 5.3 meters peryear is simulated in the steady state analysis. A high rainfall is chosen to simulate the wetclimate found in Papua New Guinea. The purpose of the steady state analysis is to determinethe location of the water table. The water content of the shoulders of the drainage ditch understeady state is unknown. Finite element seepage analysis will be performed to determine thewater content throughout the drainage ditch under steady state conditions.

A drought is simulated in the transient analysis to analyze how long it would take to fullydesaturate the tailings. The results from the steady state analysis will be used as a startingpoint for the transient analysis. An evaporation rate of 1.0 meter per year is placed as a fluxon top of the tailings. The information given is the volume-mass properties and grain-sizedistributions for both the mining tailings and the underlying clay layer. From the giveninformation it is necessary to estimate a soil-water characteristic curve and hydraulic

conductivity curve for both the clay andthe mining tailings in order to perform anadequate seepage analysis.

Soil PropertiesVolume-mass properties of void ratio =0.80, saturation = 98%, and specificgravity = 2.66 were given for the minetailings. A grain-size distribution asshown in Figure 8 was also given for themine tailings. The clay underlying themine tailings had given volume-massproperties dry density = 1430 kg/m3saturation = 100%, and specific gravity =2.65. A grain-size distribution was alsogiven for the underlying clay and can beseen in Figure 9. The mentionedinformation formed the basis for therequired analysis.

Input into SoilVisionThe first task at hand is to input the giveninformation into SoilVision. Volume-mass, geography, and a description of thesoil are entered into the database system.A typical page from the database can beseen inFigure 10. Secondly, the grain-sizeinformation must be input and fit with anequation. The final fit can then be seen inFigure 8.

Once the grain-size distribution isentered, the soil must be classified.Classification is necessary forSoilVision's Rule Base to properly extractsimilar soils from the database.Classification by the USDA methodclassifies the mining tailings as a SAND.


0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 1010

Particle size (mm)

Figure 8 Given grain-size distribution for the mine tailings #11505.

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 101010

Particle size (mm)

Figure 9 Given grain-size distribution for underlying clay #11638..


Figure 10 Sample input form for soils information.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

1.0E-2 1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+61.0E+6

Soil suction (kPa)

Figure 11 Soil-water characteristic curve estimated by the Fredlund and Wilson method (1997) for mine tailings #11505.

The soil-water characteristic curve must now be estimated. An accurate way of estimating thesoil-water characteristic curve is by the Fredlund et al. (1997) method provided withinSoilVision. The algorithm estimates the soil-water characteristic curve from volume-massproperties and the grain-size distribution of a soil. A packing porosity was chosen for themine tailings and this produced a soil-water characteristic curve as shown in Figure 11.

A graph of the estimated soil-water characteristic curve can be seen in Figure 11. If someuncertainty exists regarding the prediction, the estimated results can be compared toexperimental results in the database by querying the database and graphing groups ofexperimentally measured soil-water characteristic curves. An example of this is shown inFigure 12. The database was queried for soils with similar grain-size distributions. Thecorresponding experimentally measured soil-water characteristic curves were plotted alongwith the estimation in Figure 13. The database contains over 600 soils with matchingexperimentally measured grain-size distributions and soil-water characteristic curves. The

knowledge-based system not onlyprovides the user with an estimate of thesoil-water characteristic curve but alsoprovides an indication for the possiblevariance in a certain situation.

Estimating the HydraulicProperties of SoilIt is now necessary to estimate thehydraulic conductivity of the tailings andthe clay. The most variable parameter ofa soil is its saturated hydraulicconductivity. SoilVision provides severalways of estimating this parameterbecause of this variation. Hazensequation, the Kozeny-Carmen equation,and experimental values from thedatabase are three ways that have beenimplemented in SoilVision to determinethe saturated hydraulic conductivity of asoil. Saturated values of hydraulicconductivity for the mine tailings and theunderlying clay were experimentallytested. A saturated hydraulic conductivityof 1.1 x 10-5 m/s was used for the minetailings and a value of 8 x 10-9 m/s wasused for the underlying clay. Once thesaturated hydraulic conductivity isestimated,. the entire hydraulicconductivity curve can be estimatedbased on the soil-water characteristiccurve and the saturated hydraulicconductivity. A graph of the finalequation can then be viewed in Figure 14.


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.001 0.01 0.1 1 10

Particle size (mm)

Perc

ent p

assin

g, %

Original soil

Figure 12 Grain-size distributions selected from the SoilVision database similar to thecurrent soil

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100 1000 10000 100000 1000000

Soil suction (kPa)

Satu

ratio

n

Fredlund (1996) estimation

Figure 13 Comparison between the estimated and experimentally measured soil-watercharacteristic curves selected in the query

Analysis of the problem can now begin with the functions now provided by SoilVision. Thesoil property functions were input into the program SEEP/W and both the steady state, andthe transient state problem were solved. The steady state analysis showed the location of thewater table under the heavy rainfall experienced in Papau, New Guinea and the transientanalysis showed the saturation levels in the tailings in the event of a long drought. Thesolution for the steady state analysis can be seen in Figure 15 while the solution for thetransient state analysis can be seen in Figure 16.

Potential for Improvement in Engineering ImplementationThe soil-water characteristic curve (relationship between water content and suction) hasbecome of great value in estimating unsaturated soil property functions. The characterizationof seepage, for example, in terms of a hydraulic head gradient and a coefficient ofpermeability function appears to be generally accepted (Fredlund, 1995). The use of nonlinearsoil property functions for analyzing unsaturated soils problems appears to be gaininggeneral acceptance. The soil-water characteristic curve then becomes a dominant relationship

for understanding unsaturated soilbehavior. This concept allows the soil-water characteristic curve can be used tocompute approximate soil propertyfunctions for unsaturated soils.

The advantages to this approach arenumerous. Firstly, an estimate of theunsaturated behavior of a certain soil isquickly available. The SoilVisionknowledge-based database systemalleviates this complexity. Secondly, thecost of estimation of soil behavior isgreatly reduced. Testing of unsaturatedsoil property functions can cost thousandsof dollars. A knowledge-based systemprovides estimates without the high costof experimental testing. Thirdly, aknowledge-based system makes theestimation of behavior of unsaturatedsoils easy so that inexperiencedprofessionals can work in this difficultarea.

AcknowledgementsThe author would like to acknowledgethat SoilVision is a product ofSoilVision Systems Ltd., Saskatoon,Saskatchewan, Canada and SEEP/W is aproduct of Geo-Slope International Ltd.


0.0000001

0.000001

0.00001

0.0001

0.01 0.1 1 10 100 1000

Soil suction (kPa)

Coef

ficie

nt o

f per

mea

bility

, k (m

/s)

Campbell's method (1973)

Figure 14 Estimated hydraulic conductivity curve for the mine tailings #11505.

4

10

10

12

14 14

18

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170-10

0

10

Figure 15 Results from SEEP/W of steady state analysis

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170-10

0

10

20

Figure 16 Results from SEEP/W of transient state analysis showing the location of the watertable over time.

ReferencesCampbell, J.D. (1973), Pore Pressures

And Volume Changes In UnsaturatedSoils. Ph.D. Thesis, University ofIllinois at Urbana-Champaign,Urbana-Champaign, Ill.

Fredlund, D. G. (1995), The Scope ofUnsaturated Soils Problems, Proc.First Int. Conf. on Unsaturated Soils,Paris, September 6 - 8, Vol. 3.

Fredlund, D. G. (1996), Microcomputersand Saturated/UnsaturatedContinuum Modelling inGeotechnical Engineering,Symposium on Computers inGeotechnical Engineering,INFOGEO 96, Sao Paulo, Brazil,August 28 - 30, Vol. 2, pp. 29 - 50.

Fredlund, D. G. and Rahardjo, H. (1993),Soil Mechanics for Unsaturated Soils,John Wiley & Sons, New York, 560p.

Fredlund, D. G. and Xing, A. (1994),Equations for the Soil-WaterCharacteristic Curve, CanadianGeotechnical Jour., Vol. 31, pp. 521 -532.

Fredlund, M. D., (1996), The Design OfA Knowledge-Based System ForUnsaturated Soils, Master's Thesis,University of Saskatchewan,Saskatoon, Saskatchewan, Canada

Fredlund, M. D., G.W. Wilson and D.G.Fredlund (1995), A Knowledge-basedSystem for Unsaturated Soils,Proceedings of the Canadian Societyof Civil Engineering Conference,August Montreal, Quebec

Fredlund, M.D., G.W. Wilson, and D.G.Fredlund, (1997), Prediction of theSoil-Water Characteristic Curve fromthe Grain-Size Distribution Curve,Proceedings of the 3rd Symposiumon Unsaturated Soil, Rio de Janeiro,Brazil, April 20 - 22, pp. 13-23.

For further information you may contactSoilVision Systems Ltd. at:http://www.quadrant.net/soilvision

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