the epsilon method: analysis of seepage beneath an...

The epsilon method: analysis of seepage beneathan impervious dam with sheet pile on a layeredsoil

Zheng-yi Feng and Jonathan T.H. Wu

Abstract: An approximate solution method, referred to as the epsilon method, allows flow characteristics such as flowrate and exit gradient to be determined for seepage through a two-layer soil system. The finite element program SEEPwas employed to analyze flow characteristics of an impervious dam with sheet pile on a layered soil. Extensive analy-ses were performed for different conditions, including soil layer thickness, soil hydraulic conductivity, dam width, andsheet pile depth. The flow rate and exit gradient were determined for each condition analyzed. The results were re-duced to simple charts, called the epsilon curves. The epsilon curves allow a designer to obtain solutions to the seep-age problem without a computer code and to verify solutions obtained from a computer code. They are especiallyuseful when searching for an optimum design of a masonry dam. The epsilon curves can be extended to a soil systemcomprising more than two layers. An example of a single-row sheet pile structure in a three-layer system is given to il-lustrate how to use the method for multiple-layer systems. The method was verified by comparing the results withthose obtained from the SEEP program, and excellent agreement was noted.

Key words: seepage, dam, sheet pile, layered soil, hydraulic conductivity.

Résumé : Une méthode de solution approximative, appelée la « méthode epsilon », permet de déterminer les caracté-ristiques d’écoulement telles que le débit et le gradient de sortie pour l’écoulement à travers un système bicouche desol. Un programme d’éléments finis SEEP a été utilisé pour analyser les caractéristiques d’un barrage imperméableavec palplanches d’acier reposant sur un sol multicouche. On a réalisé des analyses élaborées pour différentes condi-tions, incluant l’épaisseur de la couche de sol, la conductivité hydrauliques du sol, la largeur du barrage, et la profon-deur des palplanches d’acier. Le débit et le gradient de sortie ont été déterminés pour chaque condition analysée. Lesrésultats ont été réduits dans de simples graphiques, appelés « courbes epsilon. » Les courbes epsilon permettent auconcepteur d’obtenir des solutions aux problèmes d’écoulement sans avoir recours à un programme d’ordinateur, et per-mettent également au concepteur de vérifier les solutions obtenues avec un programme d’ordinateur. Elles sont particu-lièrement utiles lorsqu’on cherche à définir une conception optimale d’un barrage en maçonnerie. L’application descourbes epsilon peut être élargie à une système de sol comprenant plus de deux couches. On donne un exemple d’unestructure comprenant une seule rangée de palplanches d’acier dans un système de trois couches de sol pour illustrercomment utiliser la méthode pour des systèmes multicouche. La méthode a été vérifiée en comparant les résultats avecceux obtenus avec le programme SEEP. On a noté une excellente concordance.

Mots clés : écoulement, barrage, palplanches d’acier, couches de sol, conductivité hydraulique.

[Traduit par la Rédaction] Feng and Wu 69

Introduction

Polubarinova-Kochina (1941) proposed an approximatesolution method for seepage through layered soil based onher closed-form solutions. The approximate solution methodis referred to as the epsilon method, in which a dimensionlessparameter ε is defined as

[1] tan πε = 2kk1

where k1 and k2 are the hydraulic conductivity of the upperand lower layers, respectively, of a two-layer soil system. Asthe ratio k2/k1 varies from 0 to ∞, the value of ε will varyfrom 0 to 0.50. There are three distinctive values of ε,namely 0, 0.25, and 0.50, with each corresponding to a sin-gle-layer system. Consider a two-layer system in which thethickness of the upper and lower layers is d1 and d2, respec-tively, overlying an impervious base. The flow conditions forthe three distinctive ε values will be as follows: (i) ε = 0,which leads to k2 = 0 and corresponds to a single-layer sys-

Can. Geotech. J. 43: 59–69 (2006) doi: 10.1139/T05-092 © 2006 NRC Canada

59

Received 20 October 2004. Accepted 19 October 2005. Published on the NRC Research Press Web site at http://cgj.nrc.ca on3 January 2006.

Z.-Y. Feng.1 Department of Soil and Water Conservation, National Chung-Hsing University, 250 Kuo-Kuang Road, Taichung,402 Taiwan.J.T.H. Wu. Department of Civil Engineering, University of Colorado at Denver, Denver, CO 80217-3364, USA.

1Corresponding author (e-mail: [email protected]).

tem of thickness d1 with hydraulic conductivity k1 over animpervious base; (ii) ε = 0.25, which leads to k1 = k2 andcorresponds to a single-layer system of thickness d1 + d2with hydraulic conductivity k1 over an impervious base; and(iii) ε = 0.50, which leads to k2 = ∞ and corresponds to asingle-layer system of thickness d1 with hydraulic conduc-tivity k1 over an infinitely pervious stratum.

The basic principle of the epsilon method is that if theflow characteristics (e.g., flow rate, water pressure, or exitgradient) for the three distinctive single-layer systems corre-sponding to ε = 0, 0.25, and 0.50 can be obtained, the solu-tions of the two-layer systems with any hydraulicconductivities can be readily determined. This is accom-plished by first calculating the operative ε value for the givenflow problem using eq. [1]; the flow characteristics are thendetermined by interpolation from the flow characteristics ofthe three distinctive ε values.

A study was undertaken to investigate the flow character-istics of masonry dams with a sheet pile cutoff wall at thedownstream toe and situated over a layered soil system. Ma-sonry dams of different widths and sheet piles of differentlengths were investigated. The results were reduced to a se-ries of charts, referred to as the epsilon curves, based on theconcept of the epsilon method. With the epsilon curves, adesigner can determine the flow characteristics of the seep-age problem without using a computer code, verify thevalidity of solutions obtained from a computer code, and de-termine an optimum design without a cumbersome trial-and-error analysis.

To obtain the epsilon curves for the dams investigated inthis study, a finite element program SEEP developed byWong and Duncan (1983) was employed. The dams were as-

sumed to rest on the surface of a soil system with two layersof equal thickness underlain by an impervious base. Exten-sive analyses were performed to examine the flow character-istics of dams of different dimensions and foundation soilsof different hydraulic conductivities. Specifically, the pa-rameters varied included the thickness of the soil layer, thehydraulic conductivities of the soil layers, the width of thedam, and the depth of the sheet pile at the downstream toe.

The application of the epsilon curves can be extended be-yond a two-layer soil system. The method can be used to de-termine flow characteristics of a multiple-layer soil systemwith different layer thickness by simple hand calculations.An example problem with a single row of sheet piles in athree-layer soil system is presented to illustrate the proce-dure. The results were compared with those obtained from

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60 Can. Geotech. J. Vol. 43, 2006

Fig. 1. Cross section of the masonry dam over a soil systemwith two layers of equal thickness.

Fig. 2. A typical mesh for the numerical analyses. The discretization comprised a total of 1720 elements (43 columns × 40 rows). Alldimensions in metres.

the SEEP program. Excellent agreement between the twowas noted.

Problem under investigation and thenumerical model

The typical layout of a masonry dam with a single row ofcurtain grouting “sheet pile” investigated in this study is de-picted in Fig. 1. The foundation beneath the dam consists oftwo soil layers of equal thickness d, with the total thicknessof the stratum T = 2d, the depth of the sheet pile S, the widthof the dam B, and the total head loss of the system h. Thehydraulic conductivity of the upper and lower layers is k1and k2, respectively. The exit gradient was evaluated as theaverage hydraulic gradient at the Gauss points in the “exit”element. The flow rate, or seepage quantity per unit time, ofthe system is denoted by q.

The flow characteristics under various conditions were in-vestigated by varying ε (0, 0.10, 0.20, 0.25, 0.30, 0.40, 0.45,0.50), S/T (0.250, 0.375, 0.525, 0.625, 0.750, 0.875), andB/T (0, 1, 2, 3, 4, 5), giving a total of 270 different scenar-ios. It should to be noted that S/T values greater than 0.500indicate that the sheet pile will penetrate into the lower

layer, and there will be no flow in the system when ε = 0and S/T > 0.500.

The finite element program SEEP was employed to obtainthe flow rate and exit gradient for all cases investigated inthis study. The program is capable of analyzing steady-stateconfined or unconfined flow problems in a two-dimensionalconfiguration. The basic finite element discretization in thisstudy comprised a total of 1720 elements. A typical mesh isshown in Fig. 2. Quadrangular elements, formed by staticcondensation of four constant-gradient triangular elements,were employed for simulation of the soil.

To minimize the effort of node numbering for the differentmeshes of the 270 analyses, “dummy nodes” were employedfor different depths of sheet piles. In the analyses, the sheetpile is assumed to be of zero thickness. Along the verticalsheet pile line, two node numbers are assigned to each nodealong the sheet pile location. For a given length of sheetpile, the two nodes were utilized to connect to the left andright sides of soil elements adjacent to the sheet pile. Forsoil elements below the sheet pile, however, only one nodenumber was used. The unused node numbers in the mesh arereferred to as dummy nodes. Note that the assemblage of theglobal matrix will not be affected when dummy nodes arepresent. There will not be any dummy nodes for S/T = 1.

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Feng and Wu 61

Fig. 3. Epsilon curves of IeT/h versus ε for a single row of sheetpile (B/T = 0).

Fig. 4. Epsilon curves of k1h/q versus ε for a single row of sheetpile (B/T = 0).

The single-row sheet pile structure is a special case of thedam configuration depicted in Fig. 1, where the width of thedam is regarded as zero (B = 0). For single-row sheet pilestructures, closed-form solutions of exit gradient and flowrate for ε = 0, 0.25, and 0.50 are readily available and havebeen presented by Harr (1962).

To verify the accuracy of the results obtained from theSEEP program, the single-row sheet pile structure problemspresented in Figs. 3 and 4 were first analyzed. Figure 3shows the resulting nondimensional epsilon curves in termsof IeT/h versus ε, where Ie is the exit gradient; and Fig. 4shows the nondimensional epsilon curves in terms of k1h/qversus ε. These results were found to be in very good agree-ment with the analytical solutions of Polubarinova-Kochina(1941). For comparison, the values from the solutions ofPolubarinova-Kochina have been plotted in Figs. 3 and 4.The vertical axis k1h/q is presented as the inverse of that pro-vided by Polubarinova-Kochina (as cited by Harr 1962).

It is important to note that as the value of ε becomeslarger than about 0.45, the numerical solutions tend to be-come somewhat unstable due to the very large deviation be-tween k1 and k2. Nevertheless, when ε = 0.50, the lower layerbecomes infinitely pervious; therefore, the Dirichlet con-stant-head condition (i.e., head loss = h/2) can be imposedon the interface of the two layers. This means that only theupper layer needs to be analyzed for ε = 0.50. The solution

for ε = 0.50 can be determined reliably without any numeri-cal instability problems. The complete epsilon curves wereobtained by drawing smooth curves with broken lines inFig. 3 between ε = 0.45 and ε = 0.50.

Analysis of results and discussions

A series of epsilon curves were generated for the 270 sce-narios and plotted using five dimensionless parameters,namely ε, IeT/h, k1h/q, S/T, and B/T. There are numerousways to present the correlations among the five parameters.Only selected curves are presented in this paper. The closed-form solutions by Polubarinova-Kochina (1941) were devel-oped for a single row of sheet pile in a soil system of twoequal-thickness layers. The same condition was adopted byPolubarinova-Kochina in the description of the epsilonmethod. To be consistent with these solutions, the analysescarried out in this study for masonry dams also assumed thesoil to be of two equal-thickness layers. Although the epsi-lon curves developed in this paper are limited to layers ofequal thickness, the curves can be extended to masonrydams on soil systems of more than two soil layers and withdifferent layer thicknesses using a procedure similar to thatillustrated in the example in the next section.

The epsilon curves of IeT/h versus ε for B/T = 5 and S/Tvarying from 0.250 to 0.875 are shown in Fig. 5. The corre-

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Fig. 5. Epsilon curves of IeT/h versus ε for B/T = 5. Fig. 6. Epsilon curves of k1h/q versus ε for B/T = 5.

sponding curves for the flow rate, i.e., k1h/q versus ε curves,are shown in Fig. 6. Note that when ε approaches 0, some ofthe k1h/q versus ε curves become unbounded. If the verticalaxis had been plotted with q/k1h, the inverse of k1h/q, someof the q/k1h versus ε curves will become unbounded as ε ap-proaches 0.50. Depending on the range ε of interest, one wayof plotting the flow rate would be more appropriate. Plottingk1h/q will be more desirable if the operative ε value is closeto 0.50; on the other hand, if the operative ε value is close to0, plotting q/k1h will be preferred.

The IeT/h versus S/T curves for B/T = 5 and ε varyingfrom 0 to 0.50 are shown in Fig. 7. For sheet pile penetratinginto the lower layer, the curves are to the right of the brokenline. For S/T > 0.500 and ε = 0 (i.e., with an imperviouslower layer), there will be no flow, and Ie will be equal tozero. If S/T > 0.500 and ε = 0.50 (i.e., with an infinitely per-vious lower layer), the flow rate will be infinite, and IeT/hwill be equal to 1, which agrees with the solution ofPolubarinova-Kochina (1941) for a single-row sheet pilestructure. The k1h/q versus S/T curves corresponding toFig. 7 are depicted in Fig. 8, which shows that k1h/q ap-proaches infinity for ε = 0 and S/T near 0.500.

In Fig. 9, the ratio B/T is plotted against IeT/h for differentvalues of ε for S/T = 0.750. The influence of dam width canbe evaluated with such a figure. As can be expected, the

value of Ie decreases as B/T increases, since the flow pathincreases with an increase in S/T. The k1h/q versus B/Tcurves corresponding to Fig. 9 are shown in Fig. 10. The ef-fect of increasing B/T on k1h/q is seen to increase as ε de-creases from 0.50 to 0.10.

The epsilon curves for B/T = 1 and B/T = 3 for ε varyingfrom 0 to 0.50 are presented in Figs. A1–A12 in the Appen-dix. These curves can be used for interpolations in designcomputations.

The applications of the epsilon curves can be used to(i) determine the flow rate and exit gradient for a dam withgeometry shown in Fig. 1, (ii) verify solutions obtained fromseepage computer codes, and (iii) obtain an optimum designwithout performing cumbersome trial-and-error analyses.

The epsilon method is not limited to a two-layer soil sys-tem. It can also be applied to multiple-layer soil systemswith layers of unequal thickness. An example is presented inthe following section to illustrate the computation procedure.

Multiple-layer systems

The epsilon method and the epsilon curves presented inthis paper can be extended to include multiple-layer soil sys-tems with layers of variable thickness. The procedure wasfirst introduced by M.E. Harr (class notes from Groundwater

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Feng and Wu 63

Fig. 7. Epsilon curves of IeT/h versus S/T for B/T = 5. Fig. 8. Epsilon curves of k1h/q versus S/T for B/T = 5.

and seepage, 1977). The procedure is based on the assump-tion that the exit gradient and flow rate of a multiple-layersoil system can be determined by interpolation using multi-ple values of ε defined between the layers. For a soil systemthat consists of n layers, n – 1 values of ε will need to be de-fined to obtain the flow characteristics.

An example is given here to illustrate the procedure. Theexample problem involves a single-row sheet pile structurein a three-layer soil system (i.e., n = 3), with each layer hav-ing a different thickness and a different hydraulic conductiv-ity. The configuration and the associated hydraulicconductivities of the system are shown in Fig. 11. Two (i.e.,n – 1) ε values are defined as follows:

[2] ε =π1

21 1

1

tan− kk

and

[3] ε =π2

1 3

2

1tan− k

k

where k3 is the hydraulic conductivity of the bottom layer ofthe three-layer soil system.

The solution procedure can be described by the followingsteps:

(1) Determine the operative ε values from the hydraulicconductivities. For the example problem, the operative εvalues are ε1 = 0.30 and ε2 = 0.15.

(2) Compute the values of Ie and k1h/q for the three distinc-tive ε values, namely ε1 = 0, 0.25, 0.50 and ε2 = 0, 0.25,0.50. There are a total of nine combinations for thethree values of ε1 and ε2 as listed in Table 1. Each of thenine combinations corresponds to a single-layer system.Since the example is for a single row of sheet pile (i.e.,B/T = 0), the epsilon curves for Ie and k1h/q as shown inFigs. 3 and 4 can be used directly. The T value of thecorresponding system in Table 1 should be determinedfirst to obtain S/T, Ie, and k1h/q of the nine combina-tions. For sets 1–3 (see Table 1), since the bottom of thetop 34 m layer is impervious (ε1 = 0), T = 34 m. There-fore, S/T = 42/34 > 1. This implies that there will be noflow in the system. For sets 7–9, since the bottom of thetop 34 m layer is infinitely pervious (ε1 = 0.50), T =34 × 2 = 68 m. From Fig. 3, IeT/h = 1, and thus Ie =h/T = 100/68 = 1.47 and k1h/q = 0 (from Fig. 4). For set4 (ε1 = 0.25 and ε2 = 0), the corresponding system is auniform layer with T = 54 m. For set 5 (ε1 = 0.25 andε2 = 0.25), the corresponding system is a uniform layerwith T = 80 m. For set 6 (ε1 = 0.25 and ε2 = 0.50), thetop 54 m layer lies on a very pervious stratum, with T =

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Fig. 9. Epsilon curves of IeT/h versus B/T for S/T = 0.750. Fig. 10. Epsilon curves of k1h/q versus B/T for S/T = 0.750.

54 × 2 = 108 m. Once the values of T are determinedfor sets 4–6, the values of S/T, Ie, and k1h/q can be ob-tained from Figs. 3 and 4. Table 1 lists the values of Ieand k1h/q for the nine cases along with the respectivesingle-layer properties.

(3) Plot Ie and k1h/q versus ε1 curves for the three distinc-tive values of ε2. The epsilon curves for the example areshown in Figs. 12 and 13.

(4) Plot Ie and k1h/q versus ε2 interpolation curves for theoperative ε1 value. The interpolation curves for the ex-ample problem are shown in Figs. 14 and 15.

(5) Determine Ie and q from the Ie and k1h/q versus ε2 inter-polation curves. For the example problem, Ie = 0.82 andk1h/q = 1.9, thus q = 1.053 m3/(s·m).

For the purpose of verification, an independent analysis ofthe example problem was conducted using the SEEP pro-gram. The results of the SEEP analysis indicated that Ie =0.813 and q = 1.04 m3/(s·m). These results are in excellentagreement with those obtained from the epsilon curves.

Concluding remarks

(1) The epsilon method proposed by Polubarinova-Kochina(1941) is a simple yet accurate method for determiningthe flow characteristics (such as flow rate and exit gra-dient) of multiple-layer systems with different layerthicknesses and hydraulic conductivities without the useof a computer code.

(2) The epsilon curves developed in this study allow a de-sign engineer to obtain solutions to seepage problems ata masonry dam situated over a two-layer soil systemwith equal layer thickness without using a computercode. The epsilon curves also allow a design engineer toverify solutions obtained from a computer code and areespecially useful when searching for an optimum de-sign.

(3) The epsilon curve and the solution method can be ex-tended to soil systems comprising more than two layers.

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Feng and Wu 65

Fig. 11. Example problem for a three-layer system.

Corresponding system

Set ε1 ε2 Ie k1h/q T (m) S/T

1 0 0 0 ∞ 34 >1.0002 0 0.25 0 ∞ 34 >1.0003 0 0.50 0 ∞ 34 >1.0004 0.25 0 0.63 3.2 54 0.7705 0.25 0.25 0.71 2.1 80 0.5306 0.25 0.50 0.99 0 108 0.3907 0.50 0 1.47 0 68 0.6208 0.50 0.25 1.47 0 68 0.6209 0.50 0.50 1.47 0 68 0.620

Table 1. Values of Ie and k1h/q for the example problem for thethree values of ε1 and ε2.

Fig. 12. Curves of Ie versus ε1 for different values of ε2 forB/T = 0.

(4) The concept of the epsilon method may be applied toother seepage problems involving layered soils, includ-ing both confined and unconfined seepage problems.

References

Harr, M.E. 1962. Groundwater and seepage. McGraw-Hill BookCompany, New York.

Polubarinova-Kochina, P.Y. 1941. Concerning seepage in heteroge-neous (two-layered) media. Inzhenernii Sbornik, Vol. 1, No. 2.

Wong, K.S., and Duncan, J.M. 1983. SEEP: a computer programfor seepage analysis of saturated free surface or confined steadyflow. Virginia Polytechnic Institute and State University,Blacksburg, Va.

Appendix A

(See following pages.)

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Fig. 13. Curves of k1h/q versus ε1 for different values of ε2 forB/T = 0.

Fig. 14. Interpolation curve of Ie versus ε2 for the example prob-lem.

Fig. 15. Interpolation curve of k1h/q versus ε2 for the exampleproblem.

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Feng and Wu 67

Fig. A1. Epsilon curves of IeT/h versus ε for B/T = 3. Fig. A3. Epsilon curves of k1h/q versus ε for B/T = 3.

Fig. A2. Epsilon curves of IeT/h versus ε for B/T = 1. Fig. A4. Epsilon curves of k1h/q versus ε for B/T = 1.

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Fig. A5. Epsilon curves of IeT/h versus S/T for B/T = 3.

Fig. A6. Epsilon curves of IeT/h versus S/T for B/T = 1.

Fig. A7. Epsilon curves of k1h/q versus S/T for B/T = 3.

Fig. A8. Epsilon curves of k1h/q versus S/T for B/T = 1.

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Feng and Wu 69

Fig. A9. Epsilon curves of IeT/h versus B/T for S/T = 0.525.

Fig. A10. Epsilon curves of IeT/h versus B/T for S/T = 0.250.

Fig. A11. Epsilon curves of k1h/q versus B/T for S/T = 0.525.

Fig. A12. Epsilon curves of k1h/q versus B/T for S/T = 0.250.

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