The epsilon method: analysis of seepage beneath an impervious dam with sheet pile on a layered soil

Transcription

1 The epsilon method: analysis of seepage beneath an impervious dam with sheet pile on a layered soil Zheng-yi Feng and Jonathan T.H. Wu 59 Abstract: An approximate solution method, referred to as the epsilon method, allows flow characteristics such as flow rate and exit gradient to be determined for seepage through a two-layer soil system. The finite element program SEEP was employed to analyze flow characteristics of an impervious dam with sheet pile on a layered soil. Extensive analyses were performed for different conditions, including soil layer thickness, soil hydraulic conductivity, dam width, and sheet pile depth. The flow rate and exit gradient were determined for each condition analyzed. The results were reduced to simple charts, called the epsilon curves. The epsilon curves allow a designer to obtain solutions to the seepage problem without a computer code and to verify solutions obtained from a computer code. They are especially useful when searching for an optimum design of a masonry dam. The epsilon curves can be extended to a soil system comprising more than two layers. An example of a single-row sheet pile structure in a three-layer system is given to illustrate how to use the method for multiple-layer systems. The method was verified by comparing the results with those 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 caractéristiques d écoulement telles que le débit et le gradient de sortie pour l écoulement à travers un système bicouche de sol. Un programme d éléments finis SEEP a été utilisé pour analyser les caractéristiques d un barrage imperméable avec palplanches d acier reposant sur un sol multicouche. On a réalisé des analyses élaborées pour différentes conditions, incluant l épaisseur de la couche de sol, la conductivité hydrauliques du sol, la largeur du barrage, et la profondeur des palplanches d acier. Le débit et le gradient de sortie ont été déterminés pour chaque condition analysée. Les résultats ont été réduits dans de simples graphiques, appelés «courbes epsilon.» Les courbes epsilon permettent au concepteur d obtenir des solutions aux problèmes d écoulement sans avoir recours à un programme d ordinateur, et permettent également au concepteur de vérifier les solutions obtenues avec un programme d ordinateur. Elles sont particulièrement utiles lorsqu on cherche à définir une conception optimale d un barrage en maçonnerie. L application des courbes epsilon peut être élargie à une système de sol comprenant plus de deux couches. On donne un exemple d une structure comprenant une seule rangée de palplanches d acier dans un système de trois couches de sol pour illustrer comment utiliser la méthode pour des systèmes multicouche. La méthode a été vérifiée en comparant les résultats avec ceux 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 approximate solution method for seepage through layered soil based on her closed-form solutions. The approximate solution method is referred to as the epsilon method, in which a dimensionless parameter ε is defined as k2 [1] tan πε = k 1 where k 1 and k 2 are the hydraulic conductivity of the upper and lower layers, respectively, of a two-layer soil system. As the ratio k 2 /k 1 varies from 0 to, the value of ε will vary from 0 to There are three distinctive values of ε, namely 0, 0.25, and 0.50, with each corresponding to a single-layer system. Consider a two-layer system in which the thickness of the upper and lower layers is d 1 and d 2, respectively, overlying an impervious base. The flow conditions for the three distinctive ε values will be as follows: (i) ε =0, which leads to k 2 = 0 and corresponds to a single-layer sys- Received 20 October Accepted 19 October Published on the NRC Research Press Web site at on 3 January 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 , USA. 1 Corresponding author ( tonyfeng@dragon.nchu.edu.tw). Can. Geotech. J. 43: (2006) doi: /T05-092

2 60 Can. Geotech. J. Vol. 43, 2006 Fig. 1. Cross section of the masonry dam over a soil system with two layers of equal thickness. tem of thickness d 1 with hydraulic conductivity k 1 over an impervious base; (ii) ε = 0.25, which leads to k 1 = k 2 and corresponds to a single-layer system of thickness d 1 + d 2 with hydraulic conductivity k 1 over an impervious base; and (iii) ε = 0.50, which leads to k 2 = and corresponds to a single-layer system of thickness d 1 with hydraulic conductivity k 1 over an infinitely pervious stratum. The basic principle of the epsilon method is that if the flow characteristics (e.g., flow rate, water pressure, or exit gradient) for the three distinctive single-layer systems corresponding to ε = 0, 0.25, and 0.50 can be obtained, the solutions of the two-layer systems with any hydraulic conductivities can be readily determined. This is accomplished by first calculating the operative ε value for the given flow problem using eq. [1]; the flow characteristics are then determined by interpolation from the flow characteristics of the three distinctive ε values. A study was undertaken to investigate the flow characteristics of masonry dams with a sheet pile cutoff wall at the downstream toe and situated over a layered soil system. Masonry dams of different widths and sheet piles of different lengths were investigated. The results were reduced to a series of charts, referred to as the epsilon curves, based on the concept of the epsilon method. With the epsilon curves, a designer can determine the flow characteristics of the seepage problem without using a computer code, verify the validity of solutions obtained from a computer code, and determine an optimum design without a cumbersome trial-anderror analysis. To obtain the epsilon curves for the dams investigated in this study, a finite element program SEEP developed by Wong and Duncan (1983) was employed. The dams were assumed to rest on the surface of a soil system with two layers of equal thickness underlain by an impervious base. Extensive analyses were performed to examine the flow characteristics of dams of different dimensions and foundation soils of different hydraulic conductivities. Specifically, the parameters varied included the thickness of the soil layer, the hydraulic conductivities of the soil layers, the width of the dam, and the depth of the sheet pile at the downstream toe. The application of the epsilon curves can be extended beyond a two-layer soil system. The method can be used to determine flow characteristics of a multiple-layer soil system with different layer thickness by simple hand calculations. An example problem with a single row of sheet piles in a three-layer soil system is presented to illustrate the procedure. The results were compared with those obtained from Fig. 2. A typical mesh for the numerical analyses. The discretization comprised a total of 1720 elements (43 columns 40 rows). All dimensions in metres.

3 Feng and Wu 61 Fig. 3. Epsilon curves of I e T/h versus ε for a single row of sheet pile (B/T = 0). Fig. 4. Epsilon curves of k 1 h/q versus ε for a single row of sheet pile (B/T = 0). the SEEP program. Excellent agreement between the two was noted. Problem under investigation and the numerical model The typical layout of a masonry dam with a single row of curtain grouting sheet pile investigated in this study is depicted in Fig. 1. The foundation beneath the dam consists of two soil layers of equal thickness d, with the total thickness of the stratum T =2d, the depth of the sheet pile S, the width of the dam B, and the total head loss of the system h. The hydraulic conductivity of the upper and lower layers is k 1 and k 2, respectively. The exit gradient was evaluated as the average hydraulic gradient at the Gauss points in the exit element. The flow rate, or seepage quantity per unit time, of the system is denoted by q. The flow characteristics under various conditions were investigated 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), and B/T (0, 1, 2, 3, 4, 5), giving a total of 270 different scenarios. It should to be noted that S/T values greater than indicate that the sheet pile will penetrate into the lower layer, and there will be no flow in the system when ε =0 and S/T > The finite element program SEEP was employed to obtain the flow rate and exit gradient for all cases investigated in this study. The program is capable of analyzing steady-state confined or unconfined flow problems in a two-dimensional configuration. The basic finite element discretization in this study comprised a total of 1720 elements. A typical mesh is shown in Fig. 2. Quadrangular elements, formed by static condensation of four constant-gradient triangular elements, were employed for simulation of the soil. To minimize the effort of node numbering for the different meshes of the 270 analyses, dummy nodes were employed for different depths of sheet piles. In the analyses, the sheet pile is assumed to be of zero thickness. Along the vertical sheet pile line, two node numbers are assigned to each node along the sheet pile location. For a given length of sheet pile, the two nodes were utilized to connect to the left and right sides of soil elements adjacent to the sheet pile. For soil elements below the sheet pile, however, only one node number was used. The unused node numbers in the mesh are referred to as dummy nodes. Note that the assemblage of the global matrix will not be affected when dummy nodes are present. There will not be any dummy nodes for S/T =1.

4 62 Can. Geotech. J. Vol. 43, 2006 Fig. 5. Epsilon curves of I e T/h versus ε for B/T =5. Fig. 6. Epsilon curves of k 1 h/q versus ε for B/T =5. The single-row sheet pile structure is a special case of the dam configuration depicted in Fig. 1, where the width of the dam is regarded as zero (B = 0). For single-row sheet pile structures, closed-form solutions of exit gradient and flow rate for ε = 0, 0.25, and 0.50 are readily available and have been presented by Harr (1962). To verify the accuracy of the results obtained from the SEEP program, the single-row sheet pile structure problems presented in Figs. 3 and 4 were first analyzed. Figure 3 shows the resulting nondimensional epsilon curves in terms of I e T/h versus ε, where I e is the exit gradient; and Fig. 4 shows the nondimensional epsilon curves in terms of k 1 h/q versus ε. These results were found to be in very good agreement with the analytical solutions of Polubarinova-Kochina (1941). For comparison, the values from the solutions of Polubarinova-Kochina have been plotted in Figs. 3 and 4. The vertical axis k 1 h/q is presented as the inverse of that provided by Polubarinova-Kochina (as cited by Harr 1962). It is important to note that as the value of ε becomes larger than about 0.45, the numerical solutions tend to become somewhat unstable due to the very large deviation between k 1 and k 2. Nevertheless, when ε = 0.50, the lower layer becomes infinitely pervious; therefore, the Dirichlet constant-head condition (i.e., head loss = h/2) can be imposed on the interface of the two layers. This means that only the upper layer needs to be analyzed for ε = The solution for ε = 0.50 can be determined reliably without any numerical instability problems. The complete epsilon curves were obtained by drawing smooth curves with broken lines in Fig. 3 between ε = 0.45 and ε = Analysis of results and discussions A series of epsilon curves were generated for the 270 scenarios and plotted using five dimensionless parameters, namely ε, I e T/h, k 1 h/q, S/T, and B/T. There are numerous ways to present the correlations among the five parameters. Only selected curves are presented in this paper. The closedform solutions by Polubarinova-Kochina (1941) were developed for a single row of sheet pile in a soil system of two equal-thickness layers. The same condition was adopted by Polubarinova-Kochina in the description of the epsilon method. To be consistent with these solutions, the analyses carried out in this study for masonry dams also assumed the soil to be of two equal-thickness layers. Although the epsilon curves developed in this paper are limited to layers of equal thickness, the curves can be extended to masonry dams on soil systems of more than two soil layers and with different layer thicknesses using a procedure similar to that illustrated in the example in the next section. The epsilon curves of I e T/h versus ε for B/T = 5 and S/T varying from to are shown in Fig. 5. The corre-

5 Feng and Wu 63 Fig. 7. Epsilon curves of I e T/h versus S/T for B/T =5. Fig. 8. Epsilon curves of k 1 h/q versus S/T for B/T =5. sponding curves for the flow rate, i.e., k 1 h/q versus ε curves, are shown in Fig. 6. Note that when ε approaches 0, some of the k 1 h/q versus ε curves become unbounded. If the vertical axis had been plotted with q/k 1 h, the inverse of k 1 h/q, some of the q/k 1 h versus ε curves will become unbounded as ε approaches Depending on the range ε of interest, one way of plotting the flow rate would be more appropriate. Plotting k 1 h/q will be more desirable if the operative ε value is close to 0.50; on the other hand, if the operative ε value is close to 0, plotting q/k 1 h will be preferred. The I e T/h versus S/T curves for B/T = 5 and ε varying from 0 to 0.50 are shown in Fig. 7. For sheet pile penetrating into the lower layer, the curves are to the right of the broken line. For S/T > and ε = 0 (i.e., with an impervious lower layer), there will be no flow, and I e will be equal to zero. If S/T > and ε = 0.50 (i.e., with an infinitely pervious lower layer), the flow rate will be infinite, and I e T/h will be equal to 1, which agrees with the solution of Polubarinova-Kochina (1941) for a single-row sheet pile structure. The k 1 h/q versus S/T curves corresponding to Fig. 7 are depicted in Fig. 8, which shows that k 1 h/q approaches infinity for ε = 0 and S/T near In Fig. 9, the ratio B/T is plotted against I e T/h for different values of ε for S/T = The influence of dam width can be evaluated with such a figure. As can be expected, the value of I e decreases as B/T increases, since the flow path increases with an increase in S/T. The k 1 h/q versus B/T curves corresponding to Fig. 9 are shown in Fig. 10. The effect of increasing B/T on k 1 h/q is seen to increase as ε decreases from 0.50 to The epsilon curves for B/T = 1 and B/T = 3 for ε varying from 0 to 0.50 are presented in Figs. A1 A12 in the Appendix. These curves can be used for interpolations in design computations. The applications of the epsilon curves can be used to (i) determine the flow rate and exit gradient for a dam with geometry shown in Fig. 1, (ii) verify solutions obtained from seepage computer codes, and (iii) obtain an optimum design without performing cumbersome trial-and-error analyses. The epsilon method is not limited to a two-layer soil system. It can also be applied to multiple-layer soil systems with layers of unequal thickness. An example is presented in the following section to illustrate the computation procedure. Multiple-layer systems The epsilon method and the epsilon curves presented in this paper can be extended to include multiple-layer soil systems with layers of variable thickness. The procedure was first introduced by M.E. Harr (class notes from Groundwater

6 64 Can. Geotech. J. Vol. 43, 2006 Fig. 9. Epsilon curves of I e T/h versus B/T for S/T = Fig. 10. Epsilon curves of k 1 h/q versus B/T for S/T = and seepage, 1977). The procedure is based on the assumption that the exit gradient and flow rate of a multiple-layer soil system can be determined by interpolation using multiple values of ε defined between the layers. For a soil system that consists of n layers, n 1 values of ε will need to be defined to obtain the flow characteristics. An example is given here to illustrate the procedure. The example problem involves a single-row sheet pile structure in a three-layer soil system (i.e., n = 3), with each layer having a different thickness and a different hydraulic conductivity. The configuration and the associated hydraulic conductivities of the system are shown in Fig. 11. Two (i.e., n 1)εvalues are defined as follows: [2] ε 1 = tan k π k and 1 [3] ε 2 = tan k π k where k 3 is the hydraulic conductivity of the bottom layer of the three-layer soil system. The solution procedure can be described by the following steps: (1) Determine the operative ε values from the hydraulic conductivities. For the example problem, the operative ε values are ε 1 = 0.30 and ε 2 = (2) Compute the values of I e and k 1 h/q for the three distinctive ε values, namely ε 1 = 0, 0.25, 0.50 and ε 2 = 0, 0.25, There are a total of nine combinations for the three values of ε 1 and ε 2 as listed in Table 1. Each of the nine 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 I e and k 1 h/q as shown in Figs. 3 and 4 can be used directly. The T value of the corresponding system in Table 1 should be determined first to obtain S/T, I e, and k 1 h/q of the nine combinations. For sets 1 3 (see Table 1), since the bottom of the top 34 m layer is impervious (ε 1 = 0), T = 34 m. Therefore, S/T = 42/34 > 1. This implies that there will be no flow in the system. For sets 7 9, since the bottom of the top 34 m layer is infinitely pervious (ε 1 = 0.50), T = 34 2=68m.From Fig. 3, I e T/h = 1, and thus I e = h/t = 100/68 = 1.47 and k 1 h/q = 0 (from Fig. 4). For set 4(ε 1 = 0.25 and ε 2 = 0), the corresponding system is a uniform layer with T =54m.Forset5(ε 1 = 0.25 and ε 2 = 0.25), the corresponding system is a uniform layer with T =80m.Forset6(ε 1 = 0.25 and ε 2 = 0.50), the top 54 m layer lies on a very pervious stratum, with T =

7 Feng and Wu 65 Fig. 11. Example problem for a three-layer system. Fig. 12. Curves of I e versus ε 1 for different values of ε 2 for B/T =0. Table 1. Values of I e and k 1 h/q for the example problem for the three values of ε 1 and ε 2. Corresponding system Set ε 1 ε 2 I e k 1 h/q T (m) S/T > > > =108m.Once the values of T are determined for sets 4 6, the values of S/T, I e, and k 1 h/q can be obtained from Figs. 3 and 4. Table 1 lists the values of I e and k 1 h/q for the nine cases along with the respective single-layer properties. (3) Plot I e and k 1 h/q versus ε 1 curves for the three distinctive values of ε 2. The epsilon curves for the example are shown in Figs. 12 and 13. (4) Plot I e and k 1 h/q versus ε 2 interpolation curves for the operative ε 1 value. The interpolation curves for the example problem are shown in Figs. 14 and 15. (5) Determine I e and q from the I e and k 1 h/q versus ε 2 interpolation curves. For the example problem, I e = 0.82 and k 1 h/q = 1.9, thus q = m 3 /(s m). For the purpose of verification, an independent analysis of the example problem was conducted using the SEEP program. The results of the SEEP analysis indicated that I e = and q = 1.04 m 3 /(s m). These results are in excellent agreement 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 determining the flow characteristics (such as flow rate and exit gradient) of multiple-layer systems with different layer thicknesses and hydraulic conductivities without the use of a computer code. (2) The epsilon curves developed in this study allow a design engineer to obtain solutions to seepage problems at a masonry dam situated over a two-layer soil system with equal layer thickness without using a computer code. The epsilon curves also allow a design engineer to verify solutions obtained from a computer code and are especially useful when searching for an optimum design. (3) The epsilon curve and the solution method can be extended to soil systems comprising more than two layers.

8 66 Can. Geotech. J. Vol. 43, 2006 Fig. 13. Curves of k 1 h/q versus ε 1 for different values of ε 2 for B/T =0. Fig. 15. Interpolation curve of k 1 h/q versus ε 2 for the example problem. (4) The concept of the epsilon method may be applied to other seepage problems involving layered soils, including both confined and unconfined seepage problems. References Harr, M.E Groundwater and seepage. McGraw-Hill Book Company, New York. Polubarinova-Kochina, P.Y Concerning seepage in heterogeneous (two-layered) media. Inzhenernii Sbornik, Vol. 1, No. 2. Wong, K.S., and Duncan, J.M SEEP: a computer program for seepage analysis of saturated free surface or confined steady flow. Virginia Polytechnic Institute and State University, Blacksburg, Va. Fig. 14. Interpolation curve of I e versus ε 2 for the example problem. Appendix A (See following pages.)

9 Feng and Wu 67 Fig. A1. Epsilon curves of I e T/h versus ε for B/T =3. Fig. A3. Epsilon curves of k 1 h/q versus ε for B/T =3. Fig. A2. Epsilon curves of I e T/h versus ε for B/T =1. Fig. A4. Epsilon curves of k 1 h/q versus ε for B/T =1.

10 68 Can. Geotech. J. Vol. 43, 2006 Fig. A5. Epsilon curves of I e T/h versus S/T for B/T =3. Fig. A7. Epsilon curves of k 1 h/q versus S/T for B/T =3. Fig. A6. Epsilon curves of I e T/h versus S/T for B/T =1. Fig. A8. Epsilon curves of k 1 h/q versus S/T for B/T =1.

11 Feng and Wu 69 Fig. A9. Epsilon curves of I e T/h versus B/T for S/T = Fig. A11. Epsilon curves of k 1 h/q versus B/T for S/T = Fig. A10. Epsilon curves of I e T/h versus B/T for S/T = Fig. A12. Epsilon curves of k 1 h/q versus B/T for S/T =