Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on soil foundation

Engineering Geology 133–134 (2012) 30–39

Contents lists available at SciVerse ScienceDirect

Engineering Geology journal homepage: www.elsevier.com/locate/enggeo

Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on soil foundation Sung Eun Cho ⁎ Dept. of Civil Engineering, Hankyong National University, 327 Chungang-Ro, Anseong-Si, Gyeonggi-Do 456-749, South Korea

a r t i c l e

i n f o

Article history: Received 3 June 2011 Received in revised form 8 February 2012 Accepted 26 February 2012 Available online 3 March 2012 Keywords: Seepage Monte Carlo simulation Probability Random field Embankment Hydraulic conductivity

a b s t r a c t In this study, probabilistic analysis of seepage through an embankment on soil foundation was performed. The traditional seepage analysis method was extended to develop a probabilistic approach that accounts for the uncertainties and spatial variation of the hydraulic conductivity in a layered soil profile. The hydraulic conductivity of soil shows significant spatial variations in different layers because of stratification; further, it varies on a smaller scale within each individual layer. It was assumed that the statistics of the hydraulic conductivity is different for different layers and that the hydraulic conductivity in a layer is uncorrelated with that in the other layers. Two-dimensional random fields were generated on the basis of the Karhunen– Loève expansion in a manner consistent with a specified marginal distribution function and an autocorrelation function. A series of seepage analyses of embankment–foundation systems were performed using the generated random fields to study the effects of uncertainty due to the spatial heterogeneity of the hydraulic conductivity on the seepage flow. The results showed that the probabilistic framework can be used to efficiently consider the various flow patterns caused by the spatial variability of the hydraulic conductivity in seepage assessment for an embankment on soil foundation. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Natural soils are highly variable and heterogeneous. Soils are formed in discrete layers related to particular field conditions and to a specific regional geology. Local and regional geological formations are mapped through site characterization, which creates cross sections, and infer the continuity and homogeneity of important deposits. This mapping attempts to divide the three dimensional site into formations or strata; identify, interpolate, and characterize structural features that form the boundaries among formations strata; and relate these formations or strata to regional geological processes and trends (Baecher and Christian, 2003). Soil regions in both natural and compacted soils are often characterized by a degree of spatial heterogeneity that takes the form of a random variation of material properties because of geomorphological processes, such as sedimentation or weathering in natural soils or poor construction control in earth structures. Soils naturally present heterogeneity at various scales of description. Generally, methods used for the seepage analysis are deterministic, with soil properties characterized as constants for a given soil layer and each specified layer assumed to be homogeneous (Gui et al., 2000). However, predicting seepage flow through embankments inevitably involves uncertainties because our knowledge on soil

⁎ Tel.: + 82 31 670 5149; fax: + 82 31 678 4674. E-mail address: [email protected]. 0013-7952/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2012.02.013

hydraulic properties is incomplete. Furthermore, the soil properties vary spatially, even within “homogeneous” layers, as a result of the depositional and post-depositional processes (e.g. Lacasse and Nadim, 1996). Since the first use of RFEM (Random Finite Element Method) in steady seepage by Fenton and Griffiths (1993) and Griffiths and Fenton (1993), several studies on seepage flow have considered the spatial fluctuations of a parameter by using random field theory (Fenton and Griffiths, 1996, 1997; Griffiths and Fenton, 1997, 1998; Gui et al., 2000; Ahmed, 2009; Srivastava et al., 2010). Several researchers have investigated the influence of spatial variability of hydraulic conductivity on seepage flow through earth dams. Fenton and Griffiths (1996) estimated the mean and variance of the total flow rate through the earth dam and free surface drawdown using a combination of Monte Carlo simulations, random field theory, and the finite element technique. The soil permeability of the earth dam was considered a spatially random field that followed a lognormal distribution, with a prescribed mean, variance, and spatial correlation structure. Fenton and Griffiths (1997) also investigated the stochastic nature of hydraulic gradients through an earth dam with simple drains by considering spatial variability of soil permeability. In particular, the magnitude and location of maximum hydraulic gradients were studied via Monte Carlo simulation to derive their probability distribution and assess their variability. Gui et al. (2000) performed probabilistic seepage analysis to investigate the effects of stochastic hydraulic conductivity on the slope stability of an embankment dam using a combination of random field simulation, seepage analysis, and slope stability analysis.

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

The hydraulic conductivity distribution was treated as a spatially stationary random field following a lognormal distribution. They used the turning band method to generate the spatial variability of the saturated hydraulic conductivity. The resulting pore-water pressures obtained from seepage analysis were then used to evaluate the factor of safety for the downstream slope of the earth dam. Srivastava et al. (2010) used commercially available finite difference numerical code for modeling the permeability parameter as a spatially correlated lognormally distributed random variable and studied its influence on the steady state seepage flow and on the slope stability analysis. Ahmed (2009) studied the free surface flow through earth dams using a probabilistic method. The spatial correlation of the field was considered by adopting the Local Average Subdivision (LAS) method (Fenton and Vanmarcke, 1990) to generate a random field of the hydraulic conductivity following a lognormal distribution, and the anisotropic heterogeneity, which arises when the scale of fluctuation is larger in the horizontal direction than the vertical direction, was considered. In all the studies, it was assumed that the soils are stationary, which means that the mean hydraulic properties are constant in the domain and that the covariance between any two points in the simulation domain depends on their distance rather than the actual locations of the two points. However, in an actual soil profile, the hydraulic properties exhibit spatial variations on various scales, for example, on a laboratory scale (because of variations in pore geometry), on a field scale (because of soil stratifications), and on a regional scale (because of large-scale geological variability) (Lu and Zhang, 2007). Therefore, it is important to extend the probabilistic analysis method for simulating seepage flow in random soil media with multiscale variability. In this study, the effect of spatial variability of the saturated hydraulic conductivity on the seepage flow in a two-layered embankment was studied by using the RFEM. Following the methodology used by Lu and Zhang (2007), random fields of the saturated hydraulic conductivity for a two-scale soil media were generated by the Karhunen–Loève expansion method. Statistical flow responses were then studied by carrying out a series of finite element steady state seepage analyses for the generated random fields.

31

equation governing the flow can (Papagiannkis and Fredlund, 1984):

 ∂ ∂h ∂ ∂h kx þ ky ¼0 ∂x ∂x ∂y ∂y

be

written

as

follows

ð1Þ

where h is the total head (the sum of pressure head and elevation head), and kx and ky are the hydraulic conductivity, in the x and y directions, respectively, depending upon the pore-water pressure. The steady state nonlinear differential equation (Eq. (1)) was solved using an iterative finite element scheme until the computed solution did not change by more than a specific value in successive iterations. Phreatic surfaces were determined by locating points of zero pressure head. A detailed description of the solution algorithm can be found in the literature (Fredlund and Rahardjo, 1995; Cho and Lee, 2001). Once the solution was obtained, the hydraulic gradients defined as the first derivative of the total head were calculated at each numerical integration point on the basis of the nodal total heads. 2.2. Hydraulic characteristics The generalized Darcy's Law was used to describe the velocities of pore-water. However, the hydraulic conductivity that is constant in the saturated soils depends on the degree of saturation or the matric suction in unsaturated soils. A number of empirical and semiempirical functions have been proposed to represent hydraulic conductivity functions. In this analysis, the soil hydraulic functions are described by a set of closed form equations (van Genuchten, 1980) Se ¼

θ−θr 1 ¼ θs −θr ½1 þ ðα ΨÞn  m

ð2–1Þ

where m = 1 − 1/n, n > 1 and 1=2

k ¼ ks Se

h   i 1=m m 1− 1−Se

2

ð2–2Þ

2. Seepage analysis

in which Ψ is the matric suction, Se is the effective water saturation, θr and θs denote the residual and saturated volumetric water contents, respectively, m, n, and α are retention parameters, k is the hydraulic conductivity function, and ks is the saturated hydraulic conductivity.

2.1. Governing equation

3. Monte Carlo simulation

For the seepage analysis in a two-dimensional domain, a finite element analysis routine was developed on the basis of the theory of saturated–unsaturated flow. The differential equation governing flow was derived assuming that flow follows Darcy's law regardless of the degree of saturation of the soil (Richards, 1931). Assuming steady state flow conditions, the net flow quantity from an element of soil must be equal to zero. For the case where the direction of the maximum or minimum hydraulic conductivity is parallel to either x or y axis, the differential

Although various stochastic methods have been proposed in the literature, currently, the only universal method for accurate solution of geotechnical problems is the Monte Carlo technique, mainly due to the large variability and strong non-linearity of soil properties. In a Monte Carlo simulation, a series of random fields are generated in a manner consistent with their probability distribution and correlation structure, and the response is calculated for each generated set. The process is repeated many times to evaluate various statistical properties such as mean, variance, coefficient of skewness, probability density

Fig. 1. Embankment cross-section and finite element discretization.

32

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

a)

Parameter

Mean μ

COV

Autocorrelation distance

ks (m/s) θs θr α (kPa− 1) n

1 × 10− 5 0.045 0.430 1.478 2.68

0.3, 0.5, 0.7, 1.0 – – – –

5 ≤ lh ≤ 30, 1 ≤ lv ≤ 10 – – – –

Mean flow rate Q(m3/sec/m)

Table 1 Statistical properties of soil parameters used for seepage analysis.

functions, and cumulative probability distribution functions, that can provide a broader perspective and a more comprehensive description of a given system. Despite the fact that this approach may require a large amount of computational effort, the mathematical formulation for the Monte Carlo simulation is relatively simple and the method has the capability of handling practically every possible case, regardless of its complexity.

4.1. Spatial variability of soil One of the main sources of heterogeneity is inherent spatial soil variability due to different depositional conditions and different loading histories (e.g. Elkateb et al., 2002). Spatial variations of soil properties are effectively described by their correlation structure within the framework of random fields (Vanmarcke, 1983). Autocorrelation distance, defined as the distance at which the autocorrelation function decays to 1/e (e is the base of natural logarithms), are used to describe the spatial extent within which soil properties show a strong correlation (DeGroot and Baecher, 1993). A large autocorrelation distance value implies that the soil property is highly correlated over a large spatial extent, resulting in a smooth variation within the soil profile. On the other hand, a small value indicates that the fluctuation of the soil property is large. Although an isotropic correlation structure was often assumed in previous studies, correlations in the vertical direction tend to have much shorter distances than those in the horizontal direction (e.g. Baecher and Christian, 2003). A Gaussian random field is completely defined by its mean μ(x),   variance σ 2(x), and autocorrelation function ρ x; x′ . In this study,

-5

1.80x10

-5

1.75x10

-5

1.70x10

-5

1.65x10

-5

1.60x10

-5

1

10

100

1000

10000

100000

10000

100000

Number of simulation

Stdv flow rate Q(m3/sec/m)

b)

4. Random field model Actual soil profiles may consist of two or more layers of soil with different properties and the properties within each single layer may vary spatially. In addition, even different soil layers that have the same mean value of a soil property may have very different spatial variability (Chok, 2009). Therefore, an algorithm for decomposing nonstationary fields is necessary to efficiently and accurately quantify uncertainty for flow in nonstationary heterogeneous soil media that included a number of zones with different statistics of hydraulic conductivity. In the framework for nonstationary soil media, the covariance between any two points within a single layer depends on their distance only, but the covariance between any two points that lie in different layers is zero (i.e., they are uncorrelated).

1.85x10

3.20x10

-6

2.80x10

-6

2.40x10

-6

2.00x10

-6

1.60x10

-6

1

10

100

1000

Number of simulation Fig. 3. Convergence of the estimated mean and standard deviation of the flow rate (COVks = 0.3, lh = 20 m and lv = 2 m for both layers). a. Mean vs. number of trials. b. Standard deviation vs. number of trials.

an exponential autocorrelation function was used and different autocorrelation distances in the vertical and horizontal directions were used as follows:   1 0   ′ ′   x−x  y−y  ′ A for two  dimensional fields ð3Þ ρ x; x ¼ exp@− − lh lv

where lh and lv are autocorrelation distances in the horizontal and vertical directions, respectively. 4.2. Simulation of random fields The spatial fluctuations of a parameter cannot be accounted for if the parameter is modeled by only a single random variable. Therefore, it is reasonable to use random fields for a more accurate representation of the variations when spatial uncertainty effects are directly included in the analysis. Because of the discrete nature of numerical methods such as finite element or finite difference formulation, a continuous-parameter

0.3 5 0.2

0.2

0.15

0.1

0.05

Fig. 2. Results of deterministic seepage analysis.

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

random field must also be discretized into random variables. This process is commonly known as discretization of a random field. Different approaches can be used to generate a random field with defined mean, standard deviation, and autocorrelation distance. Several methods have been developed to carry out this task, such as the spatial average method, the midpoint method, and the shape function method. These early methods are relatively inefficient, in the sense that a large number of random variables are required to achieve a good approximation of the field. More efficient approaches for discretization of random fields using series expansion methods such as the Karhunen–Loève expansion, the orthogonal series expansion, and the expansion optimal linear estimation method have been introduced. A comprehensive review and comparison of these discretization methods have been presented by Matthies et al. (1997) and Sudret and Der Kiureghian (2000). For geotechnical engineering applications LAS is widely used since it produces a piece-wise constant field of local average cells. Therefore, it is well suited to problems where the system is represented by a set of elements. Detailed explanations on the LAS can be found in Fenton and Vanmarcke (1990). In this study, the Karhunen–Loève expansion was adopted to simulate anisotropic random fields in two-dimensional space. The method was chosen since it requires the minimum number of terms for a specified level of accuracy compared to other mathematical

a

representations (Ghiocel and Ghanem, 2002) and is independent of the spatial discretization of the problem domain by finite elements. In addition, the method can be extended readily to non-stationary fields in a unified way as used in this study. Following Lu and Zhang (2007), the simulation domain is partitioned into two non-overlapping subdomains. Strictly within each subdomain, the soil is statistically stationary, implying that the covariance between any two points depends only on their distance and not on their location. The covariance between any two points from different regions is zero. Therefore the medium is globally nonstationary. The field of the saturated hydraulic conductivity can be exactly represented as a series involving random variables and deterministic spatial functions depending on the correlation structure of the field by the Karhunen–Loève expansion. The accuracy of the representation depends on the number of terms used in the series expansion. Detailed procedures can be found in Ghanem and Spanos (1991). Although a Gaussian random field is often used to model uncertainties with spatial variability for reasons of convenience and a lack of available data, the Gaussian model is not applicable in many situations where the random variable is always positive. For convenience, we find an underlying Gaussian random field that can be easily transformed into the target field. If the random variables are considered to be lognormally distributed, then appropriate lognormal random fields can be obtained by exponentiating the approximate Gaussian field.

b

1.00

33

0.80

l h= , l v= l h=20m, l v=4m l h=20m, l v=2m

0.90

Stdv Q/Qdet

Mean Q/Qdet

0.60

l h= , l v= l h=20m, l v=4m

0.80

l h=20m, l v=2m

l h=20m, l v=1m l h=10m, l v=2m l h=5m, l v=2m

0.40

0.20

l h=20m, l v=1m l h=10m, l v=2m l h=5m, l v=2m

0.00

0.70 0.2

0.4

0.6

0.8

0.2

1.0

0.4

COVks

c

d

0.96

1.0

0.8

1.0

l h= , l v= l h=20m, l v=4m

l h=20m, l v=4m

0.20

l h=20m, l v=2m

l h=20m, l v=2m l h=20m, l v=1m

l h=20m, l v=1m

Stdv max ie

Mean max ie

0.8

0.25

l h= , l v=

0.92

0.6

COVks

l h=10m, l v=2m l h=5m, l v=2m

l h=10m, l v=2m

0.15

l h=5m, l v=2m

0.10

0.88 0.05

Deterministic value 0.00

0.84 0.2

0.4

0.6

COVks

0.8

1.0

0.2

0.4

0.6

COVks

Fig. 4. Case 1: results of the Monte Carlo simulation (same COVks and autocorrelation distance for both layers, while the covariance between any two points that lie in different layers is zero). a. Variation in the estimated mean of the normalized flow rate with COVks. b. Variation in the standard deviation of the normalized flow rate with COVks. c. Variation in the mean of the maximum gradient with COVks. d. Variation in the standard deviation of the maximum gradient with COVks.

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

5. Analysis and results Hydraulic conductivity of embankment dams is used to calculate all the hydraulic parameters associated with embankment performance. Therefore, the uncertainty associated with the spatial variability of hydraulic conductivity results in uncertainties in the estimated parameters such as pore-water pressure, seepage rate, and hydraulic gradients (e.g. Gui et al., 2000). In this study, numerical modeling of water flow was conducted within a hypothetical earth embankment, in which the saturated hydraulic conductivity was treated as a random field. The hydraulic conductivity was assumed to be isotropic at any specific location. 5.1. Deterministic analysis Prior to the probabilistic analyses, a deterministic analysis was conducted to assess the ability of the numerical analysis to predict the seepage behavior. The analysis was performed using the mean value of saturated hydraulic conductivity for an embankment as shown in Fig. 1. Soil properties that were used in the analysis are displayed in Table 1. The embankment constructed on a 5 m thick soil foundation was 5 m high, with upstream and downstream slopes of 2 h:1v, and a water level in the reservoir was 4.5 m above the foundation. The studied domain was discretized to 3720 4-node quadrilateral elements with 3857 nodes. The results of the deterministic analysis are shown in Fig. 2. Vectors of seepage velocity and contours of hydraulic gradient at steady state flow are shown in the figure with the phreatic surface. In this study, only the magnitude of gradients was considered, and the direction was ignored. It means that the presented contours of hydraulic gradient are in the direction of flow. The flow rate calculated from the vertical seepage section that passes through the embankment and foundation was 1.9926 × 10 − 05 m 3/s/m and the maximum hydraulic gradient was 0.856 at the exit face located near the toe of the embankment. These values show relatively good agreement with the value of the flow rate (1.9947 × 10 − 05 m 3/s/m) and maximum gradient (0.863) obtained from the commercial seepage analysis package SEEP/W.

expressed in terms of the dimensionless coefficient of variation, defined as COVks = σks/μks, the mean and standard deviation of the underlying normal distribution of ln ks were given by σ lnks ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o ln 1 þ COV 2k

ð4Þ

s

2

μ lnks ¼ lnμ ks −0:5σ lnks :

ð5Þ

The Karhunen–Loève expansion method offers a random field to be represented in terms of a continuous function allowing for random property calculation at infinite points. Here, hydraulic conductivities were calculated at the numerical integration points of the element for the finite element analysis. The case with COVks = 0.3, lh = 20 m and lv = 2 m for both soil layers was considered as a base set, and the effect of incorporating different values of COVks (the coefficient of variation of saturated hydraulic conductivity) and autocorrelation distance in each layer was investigated, while the mean values of the saturated hydraulic conductivity were fixed. Fig. 3a and b shows the convergence of the estimated mean and standard deviation of the flow rate Q for the base set.

a 1.00

Mean Q/Qdet

34

0.90

0.80

l h=20m, l v=2m(1layer) l h=20m, l v=2m(2layer)

5.2. Stochastic simulations 0.70 0. 2

0.4

0. 6

0.8

1. 0

0.8

1.0

COV ks

b

0.80

l h=20m, l v=2m(1layer) l h=20m, l v=2m(2layer)

0.60

Stdv Q/Q det

In this section, stochastic responses for steady state of embankments on soil foundation are illustrated. As the saturated hydraulic conductivity of soil is spatially distributed, the saturated hydraulic conductivity of each layer was considered as a random field. Table 1 summarizes the statistical properties of the soil parameters for the embankment. In the process of simulating the two-layered soil profile, two independent random fields were simulated for the saturated hydraulic conductivity, using the Karhunen–Loève expansion method. The two distinct layers may have different values of mean, standard deviation and autocorrelation distance. A series of analyses were performed on the basis of the generated random fields. Analyses were carried out with the same mesh used for the deterministic analysis. Following the simulations that considered the spatial variability of the saturated hydraulic conductivity, the mean and standard deviation of the output quantities were computed. The main output quantities of interest from each realization were the total flow rate through the system Q and the maximum exit gradient ie. Accurate statistical responses were obtained from 15,000 sets of random fields that were generated for each case on the basis of the statistical information. Simulations of random field were performed using Matlab. The simulated random field was used through direct coupling with the seepage analysis routine written in FORTRAN. Saturated hydraulic conductivity was assumed to be characterized statistically by a lognormal distribution defined by a mean μks and a standard deviation σks. Once the mean and standard deviation were

0.40

0.20

0.00 0.2

0.4

0.6

COVks Fig. 5. Comparison of analyses with two uncorrelated random fields (2 layer random fields) and a single stationary random field (1 layer random field) for the case with lh = 20 m and lv = 2 m. a. Variation in the estimated mean of the normalized flow rate with COVks. b. Variation in the standard deviation of the normalized flow rate with COVks.

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

5.2.1. Uncorrelated random fields with the same statistical information and autocorrelation distance In the first part of the analyses (Case 1), both the upper and lower layers were assumed to have the same COVks and autocorrelation distance, while the covariance between any two points that lie in different layers was zero, i.e. they are uncorrelated. On the basis of the base set, COVks and autocorrelation distance for both layers were varied. Fig. 4a shows the estimated mean of normalized flow rate (by the deterministic flow rate) with the variation in saturated hydraulic conductivity. As the coefficient of variation of the input saturated hydraulic conductivity was increased, the estimated mean flow rate consistently decreased. Griffiths and Fenton (1993) observed a similar pattern in their study on seepage beneath water retaining structures founded on spatially random soil. The figure shows that as the horizontal autocorrelation distance increases (i.e., hydraulic conductivity is more strongly correlated in the horizontal direction) the mean flow rate increases, but no significant change in mean flow rate was observed with the increase of vertical autocorrelation distance. The reason is that the dominant flow in this problem is horizontal, so the formation of continuous horizontal flow channel is important. According to Fenton and Griffiths (1996), as the variance of the saturated hydraulic conductivity increases, and in the case of small autocorrelation lengths, the chances of getting a small hydraulic conductivity also increases, resulting in a decreased mean flow rate.

a

If the autocorrelation distances of both layers become infinite (lh = ∞,lv = ∞) the hydraulic conductivity of the two-layered soil profile was described by two random variables that are statistically independent to each other (i.e., by only a single random variable for each layer). Even in the case of infinite autocorrelation distances (lh = ∞, lv = ∞), the mean value decreased with the increase of COVks, which was different from the result of the single layered earth dam analysis, which becomes independent of variance in hydraulic conductivity and approaches the deterministic value predicted by mean hydraulic conductivity μk (Fenton and Griffiths, 1996; Srivastava et al., 2010). Fig. 4b shows the estimated standard deviation of Q/Qdet that increases with an increase of variation on input hydraulic conductivity. The figure shows the decrease in standard deviation of the normalized flow rate with a decrease in autocorrelation distance for a fixed COVks. The maximum standard deviation was obtained when the field of each layer was completely correlated (lh = ∞,lv = ∞), in which case the hydraulic conductivity within each layer was uniform at each realization. Fenton and Griffiths (1996) showed that flow rate variance exactly follows that of the hydraulic conductivity when the field becomes completely correlated (l = ∞) in a single layered embankment since the flow rate is proportional to the hydraulic conductivity. However, this statement does not apply to two-layered embankments. The variation of flow rate in a two-layered embankment becomes smaller than that of the hydraulic conductivity since

b

0.910

0.150

0.140

Stdv Q/Q det

Mean Q/Q det

0.908

0.906

0.904

0.902

0.130

0.120

0.110

lower layer upper layer

lower layer upper layer

0.900

0.100 0

2

4

6

8

10

0

l v (m)

c

35

2

4

6

8

10

l v (m)

d

0.854

0.066

lower layer upper layer

lower layer upper layer 0.064

Stdv max ie

Mean max i e

0.852

0.850

0.062

0.060

0.058

0.848 0

2

4

6

l v (m)

8

10

0

2

4

6

8

10

l v (m)

Fig. 6. Case 2: influence of the vertical autocorrelation distance in each layer on the estimated statistical response obtained from simulation (COVks = 0.3 and fixed lh = 20 m). a. Variation in the mean of the normalized flow rate. b. Variation in the standard deviation of the normalized flow rate. c. Variation in the mean of the maximum gradient. d. Variation in the standard deviation of the maximum gradient.

36

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

the variation of hydraulic conductivity through the whole flow domain was reduced as mentioned previously. Fig. 4c shows the relationship between the estimated mean of maximum exit gradient and COVks. The mean value increases with the increase of COVks. The figure shows a significant increase in the mean of the maximum gradient with the decrease of the vertical autocorrelation distance and with the increase of the horizontal autocorrelation distance. The mean values drop below the deterministic value only for small values of COVks, which suggests that the deterministic approach is not always conservative when considering the mean value of the maximum exit gradient. One can expect that the mean values return to deterministic values as the autocorrelation distance approaches infinity in the case of a single layered embankment. However, in the current case of a two-layered embankment, the mean values were greater than the deterministic value although the autocorrelation distances were infinite (lh = ∞,lv = ∞). The estimated standard deviation of the maximum exit gradient also increased as COVks increased (Figure 4d). It is also noted that as the autocorrelation distance increases the standard deviation reduces, but the sensitivity was not great. Extra analyses were performed to compare the result with two uncorrelated but otherwise similar random fields (i.e., the same statistical information and autocorrelation structure) with the result given by a single stationary random field over the entire domain. Compared to two uncorrelated random fields, a single stationary random field results in a stronger correlation in the vertical direction,

a

but similar correlation in the horizontal direction. No significant difference in mean flow rate between the two conditions is expected, but the standard deviation of flow rate obtained from a single stationary random field should be greater than that obtained from two uncorrelated random field. These can be inferred from Fig. 4a and b that show no significant difference in mean but increased standard deviation with an increase in vertical autocorrelation distance. Fig. 5a and b show the comparison of analyses with two uncorrelated random fields and a single stationary random field for the case with lh = 20 m and lv = 2 m, which agrees with the predictions. 5.2.2. Influence of lv in each layer The next analysis (Case 2) concerns the influence of vertical autocorrelation distance lv of each layer on the seepage behavior. For simplicity, the mean and variation of saturated hydraulic conductivity were assumed to be the same for both layers (COVks = 0.3). The estimated mean flow rate seems to slightly increase with the increase of lv in the upper layer (Figure 6a). This was because the flows in the upper layer include considerable vertical components. A more vertically continuous soil (in the case of larger lv) increases the vertical flow rate. However, the estimated mean flow rate slightly decreases with the increase of lv in the foundation since the horizontal flow was more likely to be blocked by vertically continuous less pervious regions. Fig. 6b shows increases in the estimated standard deviation of flow rate with the increase of vertical autocorrelation

b

0.910

0.140

Stdv Q/Q det

0.908

Mean Q/Q det

0.150

0.906

0.904

0.902

0.130

0.120

0.110

lower layer

lower layer

upper layer

upper layer

0.900

0.100 5

10

15

20

25

5

30

10

15

c

20

25

30

l v (m)

l v (m)

d

0.854

0.066

lower layer

lower layer

upper layer

upper layer

0.064

Stdv max ie

Mean max i e

0.852

0.062

0.850 0.060

0.058

0.848 5

10

15

20

l v (m)

25

30

5

10

15

20

25

30

l v (m)

Fig. 7. Case 3: influence of the horizontal autocorrelation distance in each layer on the estimated statistical response obtained from simulation (COVks = 0.3 and fixed lv = 2 m). a. Variation in the mean of the normalized flow rate. b. Variation in the standard deviation of the normalized flow rate. c. Variation in the mean of the maximum gradient. d. Variation in the standard deviation of the maximum gradient.

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

distances. As can be seen in the figures, the autocorrelation distance of the lower layer shows greater influences on the mean and standard deviation of flow rate than that of the upper one, but the sensitivities were not significant. Fig. 6c shows the change of the estimated mean for the maximum gradient due to the change of lv. The estimated mean of maximum exit gradient decreases as lv in the lower layer increases. Fig. 6d shows that the estimated standard deviation of the maximum exit gradient decreases with the increase of lv in the lower layer. Although the change in the mean and standard deviation was not significant, the result shows that a more vertically continuous foundation may decrease the possibility of the instability for seepage by reducing the mean and variation in maximum gradient. The vertical autocorrelation distance in the upper layer shows little influence on the maximum exit gradient. This was because the seepage gradients were usually greatest at the toe of the embankment where the seepage from the foundation was flowing to the ground surface. 5.2.3. Influence of lh in each layer Case 3 concerns the influence of horizontal autocorrelation distance lh of each layer on the seepage behavior. The mean and standard deviation of the flow rate increase with the increase of lh in both layers (Figure 7a and b). Larger lh values mean strongly correlated

a

hydraulic conductivity in the horizontal direction, which increases the chances of a continuous flow channel connecting more permeable regions that allows flow of water and results in an increased mean flow rate. Fig. 7c shows that the mean maximum exit gradient increases with the increase of lh in the lower layer, but the standard deviation decreases with the increase of lh in the lower layer (Figure 7d). The horizontal autocorrelation distance in the upper layer showed little influence on the maximum exit gradient. 5.2.4. Influence of the COVks in each layer In Case 4, the effect of varying COVks in each layer on the seepage behavior was investigated. The mean values of the hydraulic conductivity and autocorrelation distances in both layers were assumed to be the same (i.e., the condition of the base set). The results showed similar trends with those in Case 1. According to Fig. 8a and b, an embankment with a less variable soil mass overlying a more variable foundation (dotted line) gives a smaller mean and larger standard deviation in the flow rate compared to an embankment with a more variable soil mass overlying a less variable foundation (solid line). Fig. 8c and d show that an embankment with a more variable soil mass overlying a less variable foundation (solid line) gives larger values in the mean, but similar values in the standard deviation of

b

0.92

37

0.28 lower layer upper layer 0.24

Stdv Q /Qdet

Mean Q/Q det

0.90

0.88

0.86

0.20

0.16

lower layer upper layer 0.84

0.12 0. 2

0 .4

0. 6

0 .8

0. 2

1. 0

0.4

COV k s

c

d

0.92

0.8

1. 0

0 .8

1.0

0.20 lower layer upper layer

lower layer upper layer 0.90

0.16

Stdv max ie

Mean max ie

0. 6

COV k s

0.88

0.86

0.12

0.08

Deterministic value 0.84

0.04 0. 2

0.4

0. 6

COV k s

0 .8

1. 0

0.2

0 .4

0.6

COV k s

Fig. 8. Case 4: influence of the COVks in each layer on the estimated statistical response obtained from simulation (fixed lh = 20 m, lv = 2 m). a. Variation in the mean of the normalized flow rate. b. Variation in the standard deviation of the normalized flow rate. c. Variation in the mean of the maximum gradient. d. Variation in the standard deviation of the maximum gradient.

38

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

the maximum gradient compared to an embankment with a less variable soil mass overlying a more variable foundation (dotted line). 5.2.5. Typical realizations of the random field Fig. 9 shows typical realizations of the hydraulic conductivity fields, and their corresponding results of seepage analysis. In the figures, the lighter regions denote a smaller hydraulic conductivity value and darker regions indicate a larger hydraulic conductivity value. In Fig. 9a, the upper hydraulic conductivity field was assumed to have a smaller vertical autocorrelation distance than the lower layer. Therefore, the simulated profile represents an embankment with a more vertically erratic soil mass overlying a foundation layer with a more vertically continuous soil mass. Fig. 9b shows the profile with a more vertically continuous soil mass overlying a foundation layer with a more vertically erratic soil mass. Fig. 9c shows another profile with a more horizontally continuous soil mass overlying a foundation layer with a more horizontally variable soil mass. It can be observed that the different hydraulic conductivity fields result in various flow patterns. Fig. 9a shows that the flow through upper layer was more dominant because the vertically continuous less permeable regions blocked the horizontal flows in the lower layer. Fig. 9b shows another flow pattern where the flow in the upper layer was reduced by less permeable regions, resulting in a decreased elevation of downstream free surface exit.

6. Conclusions In this study, probabilistic analysis of seepage through an embankment on soil foundation was performed to study the effects of uncertainty due to the spatial heterogeneity of the hydraulic conductivity on the seepage flow. The results showed that the probabilistic framework can be used to efficiently consider the various flow patterns caused by the spatial variability of the hydraulic conductivity in the seepage assessment for an embankment on soil foundation. Although two layers share the same statistical information and autocorrelation structure, the predicted mean and standard deviation in the flow rate were always smaller than those obtained for a single layered profile. The reason is that the statistical independence of the two random fields results in reduced possibilities of realization of regions with larger hydraulic conductivity through the whole domain compared to the samples with a single soil layer. The effect of anisotropic heterogeneity was also considered in the probabilistic analysis of seepage through soils. The results showed that the influence of the autocorrelation distances on the seepage behavior is dependent on the dominant component of the flow vector. In this study, the effect of the horizontal autocorrelation distance on the mean flow rate that shows the global seepage response of the embankment was significant, while the influence of the vertical autocorrelation distance on the mean of the maximum exit gradient that represents the local seepage response at the exit near the ground

a

b

c Fig. 9. Typical realizations of the random field and the corresponding analysis results. a. Case 2 (lh, upper = 20 m, lv, upper = 2 m, and lh, lower = 20 m; lv, lower = 10 m). b. Case 2 (lh, upper = 20 m, lv, upper = 5 m, and lh, lower = 20 m; lv, lower = 2m). c. Case 3 (lh, upper = 20 m, lv, upper = 2 m, and lh, lower = 5 m; lv, lower = 2m).

S.E. Cho / Engineering Geology 133–134 (2012) 30–39

surface was greater. This can be inferred from the fact that the dominant flow was horizontal in the embankment–foundation system and vertical near the exit area. References Ahmed, A.A., 2009. Stochastic analysis of free surface flow through earth dams. Computers and Geotechnics 36 (7), 1186–1190. Baecher, G.B., Christian, J.T., 2003. Reliability and Statistics in Geotechnical Engineering. Wiley, New York. Cho, S.E., Lee, S.R., 2001. Instability of unsaturated soil slopes due to infiltration. Computers and Geotechnics 28 (3), 185–208. Chok, Y.H., 2009. Modeling the effects of soil variability and vegetation on the stability of natural slopes. Ph.D. thesis, The University of Adelaide, Australia. DeGroot, D.J., Baecher, G.B., 1993. Estimating autocovariance of in-situ soil properties. Journal of the Geotechnical Engineering 119 (1), 147–166. Elkateb, T., Chalaturnyk, R., Robertson, P.K., 2002. An overview of soil heterogeneity: quantification and implications on geotechnical field problems. Canadian Geotechnical Journal 40 (1), 1–15. Fenton, G.A., Griffiths, D.V., 1993. Statistics of block conductivity through a simple bounded stochastic medium. Water Resources Research 29 (6), 1825–1830. Fenton, G.A., Griffiths, D.V., 1996. Statistics of free surface flow through stochastic earth dam. Journal of Geotechnical and Geoenvironmental Engineering 122 (6), 427–436. Fenton, G.A., Griffiths, D.V., 1997. Extreme hydraulic gradient statistics in stochastic earth dam. Journal of Geotechnical and Geoenvironmental Engineering 123 (11), 995–1000. Fenton, G.A., Vanmarcke, E.H., 1990. Simulation of random fields via local average subdivision. Journal of Engineering Mechanics 116 (8), 1733–1749. Fredlund, D.G., Rahardjo, H., 1995. Soil Mechanics for Unsaturated Soils. John Wiley & Sons, New York. Ghanem, R.G., Spanos, P.D., 1991. Stochastic Finite Element—A Spectral Approach. Springer Verlag, New York. Ghiocel, D.M., Ghanem, R.G., 2002. Stochastic finite-element analysis of seismic soil– structure interaction. Journal of Engineering Mechanics 128 (1), 66–77.

39

Griffiths, D.V., Fenton, G.A., 1993. Seepage beneath water retaining structures founded on spatially random soil. Geotechnique 43 (4), 577–587. Griffiths, D.V., Fenton, G.A., 1997. Three-dimensional seepage through spatially random soil. Journal of Geotechnical and Geoenvironmental Engineering 123 (2), 153–160. Griffiths, D.V., Fenton, G.A., 1998. Probabilistic analysis of exit gradients due to steady seepage. Journal of Geotechnical and Geoenvironmental Engineering 124 (9), 789–797. Gui, S., Zhang, R., Turner, J.P., Xue, X., 2000. Probabilistic slope stability analysis with stochastic hydraulic conductivity. Journal of Geotechnical and Geoenvironmental Engineering 126 (1), 1–9. Lacasse, S., Nadim, F., 1996. Uncertainties in characterizing soil properties. Uncertainty in the geologic environment: from theory to practice. In: Shackleford, C.D., Nelson, P.P., Roth, M.J.S. (Eds.), Geotechnical Special Publication No. 58. ASCE, pp. 49–75. Lu, Z., Zhang, D., 2007. Stochastic simulations for flow in nonstationary randomly heterogeneous porous media using a KL-based moment-equation approach. Multiscale Modeling and Simulation SIAM 6 (1), 228–245. Matthies, H.G., Brenner, C.E., Bucher, C.G., Soares, C.G., 1997. Uncertainties in probabilistic numerical analysis of structures and solids-stochastic finite elements. Structural Safety 19 (3), 283–336. Papagiannkis, A.T., Fredlund, D.G., 1984. A steady state model for flow in saturated– unsaturated soils. Canadian Geotechnical Journal 21 (3), 419–430. Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics 1 (5), 318–333. Srivastava, A., Sivakumar Babu, G.L., Haldar, S., 2010. Influence of spatial variability of permeability property on steady state seepage flow and slope stability analysis. Engineering Geology 110 (3–4), 93–101. Sudret, B., Der Kiureghian, A., 2000. Stochastic finite element methods and reliability: a state-of-the-art report. (Technical Report No. UCB/SEMM-2000/08) Department of Civil and Environmental Engineering, Institute of Structural Engineering, Mechanics and Materials, University of California, Berkeley. van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44 (5), 892–898. Vanmarcke, E.H., 1983. Random Fields: Analysis and Synthesis. The MIT Press, Cambridge, MA.