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Spatial variability in seepage flow through earth-fill dams
Conference Paper · October 2013
DOI: 10.13140/2.1.1245.3769
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CANADIAN DAM ASSOCIATION
ASSOCIATION CANADIENNE DES BARRAGES
CDA 2013 Annual Conference
Congrès annuel 2013 de l’ACB
Montréal, Québec
October 5-10, 2013
Du 5 au 10 octobre 2013
SPATIAL VARIABILITY IN SEEPAGE FLOW THROUGH EARTH-FILL DAMS
Melih Calamak, Res. Asst., Dept. of Civil Eng., Middle East Technical University, Ankara, Turkey
Elcin Kentel, Assoc. Prof. Dr., Dept. of Civil Eng., Middle East Technical University, Ankara, Turkey
A. Melih Yanmaz, Prof. Dr., Dept. of Civil Eng., Middle East Technical University, Ankara, Turkey
ABSTRACT:
Dams made of earth materials show highly variable hydraulic properties because of uncertain nature of the earthen
embankment. Uncertain nature of soil properties may result in variations in flow fields in embankment dams. Determination
of seepage characteristics of a dam is crucial since it directly affects the stability. Underestimation of the seepage may lead
to catastrophic failures of dams. In this paper, uncertainty in the hydraulic conductivity of the embankment is evaluated
through Monte Carlo simulations (MCS) to reveal its effects on unsaturated seepage flow. A software utilizing finite
element method for seepage analysis is modified to conduct MCS. Also a code is written for generating
saturated/unsaturated random hydraulic conductivity values from defined probability density functions (PDFs). The solution
of the flow problem yields the free surface through the dam body and the quantity of seepage for each simulation. The
proposed method is applied on a 20 m high simple zoned earth-fill dam. Steady state, two-dimensional and unsaturated
seepage flow characteristics are determined for different PDFs of the hydraulic conductivity. The focus of the study is on
the discussion of the seepage flux variance. Resulting probability density functions of the flow rate are determined and
probability distributions are fitted to characterize them statistically. It is found that most of the probability density functions
of flow rate follow beta or gamma distributions. Also, if the mean hydraulic conductivity of the core material of the simple
zoned embankment is relatively very small, the uncertainty of the flow rate may be considered insignificant when compared
to that of a homogeneous earth-fill dam.
RÉSUMÉ:
Les barrages en remblai présentent des propriétés hydrauliques très variables en raison de la nature incertaine des matériaux
qui les composent. L'incertitude sur les propriétés des matériaux de remblai peut entraîner des variations au niveau du
domaine d'écoulement des ouvrages en remblai. La détermination des caractéristiques du régime d'écoulement d'un barrage
est cruciale car elle affecte directement sa stabilité. Une sous-estimation des infiltrations peut conduire à des ruptures
catastrophiques de barrages. Dans cet article, l'incertitude sur la conductivité hydraulique des matériaux de remblai est
évaluée par des simulations de type Monte Carlo (MC) afin de révéler ses effets sur les le régime d'écoulement non saturé.
Un logiciel, utilisant la méthode des éléments finis pour la réalisation d'analyses d'écoulement a été modifié pour mener les
simulations MC. De plus, un code a été écrit pour générer des valeurs de conductivité hydraulique aléatoires saturées / nonsaturées selon des fonctions de densité de probabilité (FDP) définies. L'application de cette méthode permet de définir la
surface libre à travers le corps du barrage et le débit de fuite pour chaque simulation. La méthode proposée est appliquée à
un barrage zoné en remblai d'une hauteur de de 20 m. Les caractéristiques de l'écoulement non-saturé bidimensionnel en
régime permanent sont définies pour les différentes FDP de la conductivité hydraulique. Dans cette étude, on portera une
attention particulière à la variance des débits de fuite. Les fonctions de densité de probabilité des débits de fuite sont
déterminées et sont caractérisées au moyen de distributions de probabilités. On montre que la plupart des fonctions de
densité de probabilité de débit suivent des distributions béta ou gamma. En outre, si la conductivité hydraulique moyenne du
matériau du noyau d'un barrage zoné est relativement très faible, l'incertitude sur le débit peut être considérée comme
négligeable par rapport à celle d'un barrage homogène en terre.
1 INTRODUCTION
All soils are heterogeneous in some degree in nature and hydraulic properties of natural earth materials may
show extremely high spatial variability. From field test, it can be said that properties governing the seepage flow
through an earthen dam body are uncertain in nature and this uncertainty may have a strong effect on seepage
flow. Spatial heterogeneity with the combination of inadequate number of field observations may be the main
reason of uncertainty in seepage flow (Zhang, 1999). Since the seepage flow greatly affects the stability and the
performance of the dam body, it is important to conduct a stochastic study while determining the seepage.
Many researchers have conducted stochastic studies on groundwater flow through last fifty years by considering
a number of statistical techniques and numerical and analytical simulation tools (Warren and Price, 1961, Freeze,
1975, Bakr et al. 1978, Gutjahr et al. 1978, Smith and Freeze 1979, Gutjahr and Gelhar, 1981, Fenton and
Griffiths, 1996, Tartakovsky, 1999, Lin and Chen, 2004). More recent of these studies have focused on the effect
of spatial variability on unsaturated flow (Yeh et al. 1985a, 1985b, Mantoglou and Gelhar, 1987, Mantoglou,
1992, Zhang, 1999, Li et al. 2009, Le et al. 2012). Findings of some of these studies are summarized below.
First researchers have focused on stochastic solution of one dimensional, steady or unsteady groundwater flow
problems by Monte Carlo simulation (MCS) (Warren and Price, 1961, Freeze, 1975, Smith and Freeze 1979,
Gutjahr and Gelhar, 1981). Then studies considering stochastic model of unsaturated flow were presented in the
literature. Yeh et al. (1985b) analyzed steady state, unsaturated flow stochastically using perturbation
approximation. A simple approximation with the Mualem’s soil catalog data was used for the derivation of
unsaturated hydraulic conductivity. Mantoglou and Gelhar (1987) proposed a general methodology for
large-scale transient unsaturated flow in spatially varying soils. They used soil moisture retention curves and a
parametric approximation for modeling the flow in unsaturated zone, in addition to the hysteresis effect in local
variables of the soil. Li et al. (2009) presented the probabilistic collocation method (PCM) for stochastic analysis
of steady and unsteady flow in unsaturated zones. Models of van Genuchten and Mualem were adopted for the
estimation of unsaturated hydraulic conductivity. Also, a comparison was made between the results of PCM and
MCS. They found that PCM can accurately calculate the flow with less computational effort than MCS. Le et al.
(2012) investigated the unsaturated seepage through earth embankments by considering random field of porosity.
Random field was generated by using local average subdivision (LAS) technique. MCS approach was adopted
with the finite element method and van Genuchten model for unsaturated hydraulic conductivity.
Studies on stochastic seepage analysis through earth dams have started with Fenton and Griffiths (1996) and
continued with Ahmed (2009). Both studies considered the random field of hydraulic conductivity having lognormal distribution with known mean, variance and correlation structure. LAS method were used for the
simulation of the random field. In the later one, it was found that the seepage flow found from stochastic analysis
is smaller than that is found by deterministic analysis. It was also found that if the earthen material of the dam is
highly heterogeneous, the construction of a core structure may not be required.
The objective of the study is to investigate the effect of soil uncertainty on saturated and unsaturated seepage
flow through earthen embankments. In order to achieve this, a random field simulation method with Monte Carlo
simulation is proposed and applied on a simple zoned earth-fill dam. By the recent enhancements in computer
processors, MCS technique become the most widely used approach in solving stochastic problems. In this
approach, the input parameters of the problem are randomly generated from prescribed distributions repeatedly.
Each of the generated set of inputs are called realizations and for every realization, governing equations of the
problem are solved and the random field of the output is obtained. For statistical convergence, a large number of
realizations are needed. In the study, one thousand realizations are performed for each combination of variances
of conductivity. For each realization of spatially varying hydraulic conductivity, a set of values are generated
from prescribed mean and variance and analyzed separately to obtain the set of seepage flow rates and free
surface profiles. Box-Muller Transformation algorithm is used for random number generation. The van
Genuchten Method is adopted for calculating the unsaturated seepage flow. Seepage flow results are evaluated
statistically. The statistical distributions and descriptive statistics of the seepage flow are derived for various
statistical distributions of the hydraulic conductivity.
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
2
2 METHOD
The method used for modeling saturated and unsaturated seepage flow with the random field model of hydraulic
conductivity and the numerical method are briefly introduced herein.
2.1 Saturated/unsaturated Seepage Flow Model
The partial differential equation governing two-dimensional, steady-state free surface seepage flow can be
written as
div  K grad   0
(1)
where  is the summation of pressure head (h) and elevation head (z) and defined as the total head; K is the
conductivity tensor (K=K(x,z)). The hydraulic conductivities of the saturated and unsaturated zones greatly
differ from each other. The mechanism and modeling of unsaturated flow is more complex than that of saturated
flow. Unsaturated hydraulic conductivity is dependent on the volumetric water content, the volumetric water
content is dependent on the pressure, and the pressure is dependent on the hydraulic conductivity (Gelhar, 1993).
The functional relationship between the water content and the soil suction is given by soil water retention curve
(SWRC) and it can be used to understand the behavior of an unsaturated soil (Pham, et al. 2005). There are many
physical based and empirical models used to estimate the hydraulic conductivity of the unsaturated part of the
soil with the characteristic of SWRC. However, the most commonly used one is the van Genuchten Method.
Closed form analytical expressions for the relative hydraulic conductivity, K r are introduced by using water
content and pressure head curve, relationship in this method (van Genuchten, 1980). This empirical SWRC
model contains three shape parameters namely, α, n, and m. The SWRC function has the following form in van
Genuchten Method.
s  r 

 r 
n m
h   
1  h 

s

( h  0)

(2)
( h  0)
where,  is the volumetric water content, s is the saturated water content, r is the residual water content, h is the
pressure head, and α, n and m are shape parameters, in which
m  11 n
(3)
The hydraulic conductivity function of saturated and unsaturated soils is given by the following expression.
( h  0)
K s K r ( h )
K h   
K s
(4)
( h  0)
in which, K is the hydraulic conductivity, Ks is the saturated hydraulic conductivity and Kr(h) is the relative
hydraulic conductivity function. The function of Kr(h) can be written both in terms of volumetric water content
and pressure head. The following equation was established by Mualem (1976) and relates (h) with Kr(h).

K r    1 / 2 1 1 1/ m


m
2
(5)

In above equation  is the effective saturation where,
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
3

 h   r
(6)
s   r
The relative hydraulic conductivity can also be expressed by the following equation (van Genuchten, 1980) and
in this study the below equation is utilized for calculating unsaturated hydraulic conductivity.
1  h  1  h   
K h  
n m
n 1
r
2
(7)
1  h  
n m/2
2.2 Random Field Model for Hydraulic Conductivity
The use of random field models allows understanding the behavior of spatially variable soils and becoming ever
more important (Fenton and Griffiths, 2007). There are many approaches used to generate random fields to
model soils. In this study, for spatially varying hydraulic conductivity, a random number generation algorithm
utilizing Box-Muller Transformation is written in C# language. The random field generator of the study provides
random hydraulic conductivity values generated from a known probability density function which is defined
with a mean and a coefficient of variation (COV), which is the ratio between the standard deviation and mean.
Random field is generated without a correlation structure. The shape parameters of the van Genuchten Method
are assumed to be deterministic in the study. Random field generation procedure is explained herein.
Hydraulic conductivity (K) of soils follow log-normal distribution in nature (Law, 1944; Bulnes, 1946; Warren
et al. 1961; Willardson and Hurst, 1965; Bennoin and Griffiths, 1966) and it is statistically defined with a mean,
K, and variance, σK2. Then, it can be said that ln K follows a normal (Gaussian) distribution with mean μ lnK and
variance σ2lnK. These parameters can be obtained using the following transformations (Ang and Tang, 1975):

2 
 ln2 K  ln 1  2K 
K 

(8)
 ln K  ln  K  
(9)
1 2
ln K
2
Then, log-normally distributed random values of hydraulic conductivity can be produced using the following
transformation.
K i  exp(  ln K   ln K ri )
(10)
where, Ki is the hydraulic conductivity of the i th element of the finite element mesh and ri is the standard normally
distributed random number generated obtained from Box-Muller Transformation (Box and Muller, 1958):
ri   2 ln u 1i sin 2 u 2 i
(11)
in which u1i and u2i are independently and uniformly distributed random variables within the interval (0,1].
2.3 Finite Element Model
In this study, Monte Carlo simulation is adopted with the finite element method for stochastic modeling. A large
number of random field realizations for hydraulic conductivity is generated and seepage flow is solved in the
problem domain with finite element method. For each realization, the seepage flow rate through the body and the
location of the free surface is calculated using SEEP/W software which utilizes an iterative finite element
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
4
method. The finite element equations are obtained by applying Galerkin’s weighted residual approach to Eq. (1).
A repeated substitution technique is utilized to compute the total heads and flow rates. A detailed information
about the finite element method and the iterative process of this program can be found in Geo-Slope Intl. Ltd.
(2008). The generation of the random field is handled by the use of the C# code which is employed as an add-in
to the software. Also, for the solution of the large number of realizations, a code written in Windows command
prompt line is utilized for calling the software. For acquiring and interpreting the results, some macro codes are
written in Visual Basic language.
The finite element mesh used is composed of quadrilateral and triangular elements and formed automatically by
the software. A detailed presentation about meshing is supplied in Geo-Slope Intl. Ltd. (2008). Computed head
values are used to predict the hydraulic conductivity of the elements. In a realization, the random hydraulic
conductivity values of the elements are kept the same in every iteration. However, there is an exception for the
elements changing its zone from saturated to unsaturated through the iteration procedure. Their hydraulic
conductivity values are changed using Eqs. (4), (7) and (10). Iteration is stopped when the convergence is
satisfied in the relative change of the total head of the element.
3 APPLICATION PROBLEM AND SIMULATION RESULTS
In the study, a simple zoned, 20 m high earth-fill dam shown in Figure 1, is adopted to apply the proposed
method. The dimensions of the core and the side slopes are determined according to the small dam design
specifications given in USBR (1987). The upstream and downstream slopes are 1:3 and 1:2.5, respectively and
the crest thickness is 7 m. The dam is assumed to be constructed on an impervious foundation. There is a
constant 16 m of head at the upstream of the dam and there is no tailwater. The hydraulic and statistical
properties of the dam materials are selected based on several studies given in the literature. The mean hydraulic
conductivity of the shell material is determined using EPA (1986), checked with Rice (2007) and fixed at
μKsh=6.49E-4 m/s. Besides, the coefficient of variation values of the shell material are determined from Cho
(2012) and verified from Deb and Shukla (2012). The values vary between 0.3 and 1.0. For the core material, the
mean hydraulic conductivity is selected from Benson (1993) and checked with Rice (2007) and fixed at
μKc=1.0E-10 m/s. The coefficient of variation of hydraulic conductivity of clay is stated to vary between
0.68-2.40 in Duncan (2000). However, Benson (1993) showed that the COV values of compacted fine-grained
soils have a larger range varying between 0.3 and 7.7. In the study, to consider the big variance in the clay
material, the later stated range is selected as COV of clay. The van Genuchten’s soil water retention parameters
are assumed to be deterministic and determined from the previous studies. For the shell material, the study of
Wolf et al. (2007) and for the core material, the studies of Yates et al. (1989) and Ghanbarian-Alavijeh et al.
(2010) are used to specify van Genuchten’s α, n and m parameters. The input parameters of the application
problem are presented in Table 1.
2m 3m
2m
1V:3H
1V:2.5H
1V:0.425H
16 m
Coarse sand to
coarse gravel
Shell
Impervious foundation
1V:0.425H
Clay
Core
20 m
20 m
Coarse sand to
coarse gravel
Shell
Impervious foundation
Figure 1: The geometry and material properties of the application problem
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
5
Table 1: The input parameters of the application problem
References
Core Material
EPA (1986), Rice (2007)
μKc
1.0E-10 m/s
Cho (2012),
{0.3,0.5,0.7,1.0}
COV(Kc) {0.3,1.0,2.0,7.7}
Deb and Shukla (2012)
0.175
α
0.03
2.85
Wolf et al. (2007)
n
1.70
0.65
m
0.41
Shell Material
6.49E-4 m/s
μKsh
COV(Ksh)
α
n
m
References
Benson (1993), Rice (2007)
Benson (1993)
Yates et al. (1989),
Ghanbarian-Alavijeh et al.
(2010)
In the study, 1000 realizations (simulations) are conducted for each combination of COV values of both shell
and core material. Therefore, there exist 16000 realizations in total for the application problem. For each
realization, using random field generator, hydraulic conductivity field is mapped to the problem region and then
solved by finite element method. A sample mapped random hydraulic conductivity for a realization is presented
in Figure 2. Points having the same hydraulic conductivity are joined by contours. The higher variance in the
core results in smaller areas of similar hydraulic conductivity. Large part of the downstream side of the shell
remains unsaturated; therefore, relatively smaller conductivity values are observed there.
Figure 2: A sample hydraulic conductivity variation in the dam body when COV(Ksh;Kc)={1.0;7.7}
In Monte Carlo simulations, the number of the realizations affects the accuracy of the results. When the variance
of the resulting parameter stabilizes, the number of the realizations can be said to be adequate. In the study, the
adequacy of the number of realizations is checked by calculating the coefficient of variation of the flow rate.
Figure 3 shows the change of COV(Q) with respect to number of realizations. It is clear that for all combinations
of COV(Ksh;Kc), values of COV(Q) values stabilizes after around 500 iterations. The figure also proves that
when the variation in the hydraulic conductivity of core material increases the variation in flow rate increases.
0.14
0.14
0.12
0.12
COV(Ksh;Kc)={1.0;7.7}
COV(Ksh;Kc)={0.3;0.7}
0.1
COV(Ksh;Kc)={0.5;0.7}
0.08
COV (Q)
COV (Q)
0.1
COV(Ksh;Kc)={0.5;2.0}
0.06
COV(Ksh;Kc)={0.3;1.0}
0.04
COV(Ksh;Kc)={0.3;2.0}
COV(Ksh;Kc)={0.3;0.3}
COV(Ksh;Kc)={0.7;2.0}
0.06
COV(Ksh;Kc)={0.7;1.0}
0.04
COV(Ksh;Kc)={0.3;0.3}
COV(Ksh;Kc)={1.0;0.3}
0
100
200
300
400
500
600
No. of realizations
COV(Ksh;Kc)={1.0;1.0}
0.02
0
0
COV(Ksh;Kc)={1.0;2.0}
COV(Ksh;Kc)={0.7;0.3}
COV(Ksh;Kc)={0.5;1.0}
0.02
COV(Ksh;Kc)={0.7;7.7}
0.08
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
No. of realizations
Figure 3: Change of the coefficient of variation of the flow rate with respect to number of realizations for COV(Ksh;Kc)
combinations
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
6
The location of the free surface and the flow rate passing through the dam body are calculated for all
realizations. However, the study only dealt with the flow rate results statistically. Figure 4 demonstrates the free
surface of the seepage flow and the contours of total head for both the deterministic and a stochastic solution.
Flow rates passing through the body are shown at the toe of the dam. In the deterministic solution, mean
hydraulic conductivity values are assumed to be the deterministic values. The flow rate computed in the
deterministic solution is 10.14E-10 m3/s and the phreatic line through the core follows a regular pattern.
However, there is an irregular phreatic surface in the stochastic solution. Flow tends to avoid low permeability
regions and follow routes having higher hydraulic conductivity (see Figure 2 for hydraulic conductivity mapping
for the same realization). Also, the computed flow rate is relatively smaller than that of deterministic solution.
The high variation in core material resulted in decrease in the flow rate.
(a)
(b)
Figure 4: Total head contours and water flux through the body (a) in deterministic solution, (b) in a stochastic solution when
COV(Ksh;Kc)={1.0;7.7}
After holding Monte Carlo simulations with finite element analysis, sets of the response variable are obtained.
The response variable, namely the flow rate can be characterized by a statistical distribution which can be fitted
to a probability density function. Then, this probability density function can be used to predict the likelihood of
an event, such as piping and slope failures. In the study, the sets of flow rate values obtained from Monte Carlo
simulation are statistically analyzed at first. Descriptive statistics of the flow rate passing through the dam body
is provided in Table 2. For each combination of the coefficient of variation of the shell and core material, the
maximum and minimum, the range, the mean, COV, skewness and kurtosis of the flow rate data are given in this
table. Although the COV of K varies between 0.3 and 7.7, the coefficient of variation of the flow rate varies
between 0.02 and 0.10 and it becomes larger when COV of K increases. This may be attributed to the fact that
when the COV of K is small, it tends to be uniform through the body, resulting in smaller uncertainty in the flow
rate. The variation in the flow rate is not much affected by the variation of the conductivity and its variation
range is relatively small when compared to the range of COV of K.
The mean flow rate decreases with the increase in the variance of the hydraulic conductivity of the core material.
This may be explained by the fact that when the variance of K increases, the hydraulic conductivity rapidly
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
7
changes from one point to another and the chance of getting a small conductivity increases. This results in
preferential flow paths, avoiding from low permeability regions and smaller flow quantities. Almost all mean
flow rates computed with MCS are smaller than the deterministic flow rate result. This means the variability in
soil results in decrease in the seepage flux through it. However, the mean flow rate resulting from relatively
small variances of conductivity may not be smaller than that of the deterministic solution.
Table 2: Descriptive statistics of the flow rate passing through the dam body for the combinations of the coefficient of
variations of the shell and core material.
Max
Min
Range
μ
Fitted
COV values
(*E-10) (*E-10) (*E-10) (*E-10) COV Skewness Kurtosis
distribution
Shell
Core
(m3/s)
(m3/s)
(m3/s)
(m3/s)
type
0.3
0.3
10.85
9.75
1.10
10.23
0.02
0.17
-0.91
Beta
0.3
1.0
10.41
8.28
2.13
9.28
0.04
0.14
0.02
Gamma
0.3
2.0
9.33
6.54
2.80
7.73
0.06
0.21
0.16
Normal
0.3
7.7
5.78
2.91
2.86
4.06
0.10
0.38
0.22
Gamma
0.5
0.3
10.78
9.69
1.09
10.22
0.02
0.15
-0.90
Beta
0.5
0.5
1.0
2.0
10.50
9.20
8.35
6.36
2.15
2.84
9.26
7.70
0.04
0.06
0.18
0.14
-0.13
-0.05
Beta
Gamma
0.5
0.7
0.7
7.7
0.3
1.0
5.62
10.78
10.72
2.76
9.75
8.34
2.86
1.02
2.37
4.02
10.23
9.29
0.10
0.02
0.04
0.27
0.10
0.26
0.24
-0.94
0.23
Gamma
Beta
Gamma
0.7
0.7
2.0
7.7
9.94
6.11
6.43
2.72
3.51
3.39
7.71
4.02
0.06
0.10
0.28
0.48
0.22
0.93
Beta
Gen. Ex. Val.
1.0
1.0
1.0
1.0
0.3
1.0
2.0
7.7
10.77
10.41
9.18
5.70
9.63
7.44
5.97
2.91
1.14
2.97
3.20
2.80
10.22
9.27
7.71
4.05
0.02
0.04
0.06
0.10
0.15
-0.01
0.21
0.40
-0.73 Gen. Ex. Val.
0.69
Gamma
-0.07 Gen. Ex. Val.
0.13
Pearson V
Then, the frequency histograms of the flow rate data are produced for each combination of COV values and to
describe them statistically some distributions are fitted. For this purpose, common distribution functions, i.e.,
gamma, beta, generalized extreme value (Gen. Ex. Val.), normal, and Pearson V, are used. The distribution that
best fits to the data is selected and plotted on the histograms. Figure 5 shows the probability density functions of
the flow rate data and fitted statistical distributions for all combinations of COV(K sh;Kc). The types of the
distributions and related distribution parameters are shown on the figures. The distribution fitting process and the
determination of the distribution parameters are held with a software called EasyFit (Mathwave, 2013). Out of
sixteen distributions, six of them fitted to gamma, five of them fitted to beta, three of them fitted to generalized
extreme value and the remaining are fitted to normal and Pearson V distributions (see Table 2 for fitted
distribution types and related COV combinations). Almost all distributions are positively skewed; therefore,
normal distribution was insufficient to describe them. Skewness increases with the increase of the variance of the
core material. Some of the distributions are platykurtic (Kurtosis<0), whereas the others are leptokurtic
(Kurtosis>0). For small variance of the core material, the shape of the statistical distributions of the flow rate are
more flat.
The probability distributions derived for the flow rate can be used for dealing with uncertainty. In a design
process of an earth fill dam, they may provide a good approximation for assessing failure risks. However, any
risk computation is out of the scope of this study.
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
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Beta
2.87
3.87
9.74E-10
1.09E-9
Probability density
α1
α2
a
b
(b)
Probability density
(a)
Normal
4.39E-11
7.73E-10
σ
μ
Probability density
Probability density
(c)
(d)
Q (m3/s)
α
β
γ
Gamma
26.73
8.11E-12
1.89E-10
α1
α2
a
b
Beta
10.35
16.96
7.84E-10
1.16E-9
Q (m3/s)
Probability density
Beta
3.31
3.61
9.66E-10
1.08E-9
(f)
Probability density
α1
α2
a
b
(e)
Q (m3/s)
Q (m3/s)
Gamma
272.06
2.83E-12
0
α
β
γ
(h)
Probability density
α
β
γ
Probability density
Gamma
131.46
3.11E-12
5.19E-10
Q (m3/s)
Q (m3/s)
(g)
α
β
γ
Q (m3/s)
Gamma
101.68
3.96E-12
0
Q (m3/s)
Figure 5: Histograms of the flow rate and fitted distributions when COV(Ksh;Kc) is equal to;
(a) {0.3;0.3} (b) {0.3;1.0} (c) {0.3;2.0} (d) {0.3;7.7} (e) {0.5;0.3} (f) {0.5;1.0} (g) {0.5;2.0} (h) {0.5;7.7} (i) {0.7;0.3} (j)
{0.7,1.0} (k) {0.7;2.0} (l) {0.7;7.7} (m) {1.0;0.3} (n) {1.0;1.0} (o) {1.0;2.0} p) {1.0;7.7}
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
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Probability density
Beta
2.90
3.36
9.73E-10
1.08E-9
Q (m3/s)
Q (m3/s)
Probability density
Beta
17.99
60.90
5.49E-10
1.52E-9
k
σ
μ
(l)
Probability density
α1
α2
a
b
(k)
Gamma
65.43
4.34E-12
6.45E-10
α
β
γ
(j)
Probability density
α1
α2
a
b
(i)
Q (m3/s)
Gen. Ex. Val.
-0.17
3.84E-11
3.86E-10
Q (m3/s)
k
σ
μ
α
β
γ
(n)
Gamma
747.81
1.24E-12
0
Probability density
Probability density
(m)
Gen. Ex. Val.
-0.20
1.90E-11
1.01E-9
Q (m3/s)
Q (m3/s)
Probability density
(o)
Q (m3/s)
Gen. Ex. Val.
-0.20
5.36E-11
7.54E-10
α
β
γ
(p)
Probability density
k
σ
μ
Figure 5: cont’d
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CDA 2013 Annual Conference, Montréal, Qc, Canada - October 2013
Pearson V
93.03
3.72E-8
0
Q (m3/s)
10
4 CONCLUSIONS
The effect of uncertainty due to random hydraulic conductivity field on seepage flow of an earth-fill dam is
investigated using Monte Carlo simulations. Random fields of hydraulic conductivity are generated using an
algorithm and a software utilizing finite element method is used to conduct seepage analysis. The proposed
method is applied for two-dimensional, steady state, saturated/unsaturated seepage flow through a simple zoned
earth-fill dam considering various coefficient of variation values of shell and core material. Then, the statistics of
the flow rate through the body is evaluated for the combinations of the coefficient of variations of the shell and
core.
Results indicated that the variance of the flow rate increases when the variance of the core material increases.
Also, the mean flow rate decreases with the increase in the variance of the hydraulic conductivity of the core
material. It is seen that the flow rate passing through a simple zoned earth-fill dam body is only affected by the
hydraulic properties and the uncertainty of the core material. The uncertainty of the flow rate solely depends on
the statistical properties of the core material. The uncertainty of the shell material has almost no effects on the
flow rate statistics.
It is found that the flow rate may have different types of probability distributions. They cannot be described by
only the first and second central moments of the data. Most of the probability density functions of the flow rate
can be defined by beta or gamma distributions; however, they may not be the best fitting distribution in every
time. The skewness of the distribution increases when the variance of the core increases. The statistical
distributions become flattened when the variance of the core decreases. The derived distributions can be used to
estimate the probability of a failure event. Flow rate related risk analysis may be the main subject of a further
study. For a complete seepage related risk analysis, stochastic solution of transient seepage flow problems
occurring during rapid fill and drawdown of the reservoir should also be considered by taking the hysteresis
effect in the soil into account.
It may be concluded that, in simple zoned embankment dams, if the mean hydraulic conductivity of the core
material is relatively very small (~106 times smaller than the shell), the uncertainty of the flow rate may be
insignificant. Because, the water flux through the body is highly minimized with the core and the seepage face is
very small. Also, the effect of hydraulic conductivity uncertainty on water flux is minor. Intuitively, soil
uncertainty in a homogeneous embankment dam without a drainage system is much more important. Its effects
on water flux and especially on the seepage face is thought to be more crucial since the stability of the
downstream slope depend on the exit location of the free surface and its uncertainty surely comes from the soil
variability.
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