Reliability analysis of seepage using an applicable procedure based on stochastic scaled boundary finite element method

Engineering Analysis with Boundary Elements 94 (2018) 44–59

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Reliability analysis of seepage using an applicable procedure based on stochastic scaled boundary finite element method A. Johari∗, A. Heydari Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran

a r t i c l e

i n f o

Keywords: Seepage Scaled boundary finite-element method Groundwater flow quantities Random field Reliability index

a b s t r a c t This paper presents a practical approach for reliability analysis of steady-state seepage by modeling spatial variability of the soil permeability. The traditional semi-analytical method; named Scaled Boundary Finite-Element Method (SBFEM) is extended by a coded program to develop a stochastic SBFEM coupled with random field theory. The domain is discretized into several non-uniform SBFEM sub-domains. The flow quantities such as exit gradient, flow rate, and the reliability index of piping safety factor are estimated. The precision of the outputs and the accuracy of the method are verified with the Finite-Element Method (FEM). A set of stochastic analysis is performed in three illustrative examples to illuminate the applicability of the proposed method. In these examples, the effect of the variations in the position of the sub-domain discretization center, the cutoff location, and the cutoff length are investigated stochastically. Further, the influence of the permeability’s Coefficient of Variation (COVk ) and the correlation length is evaluated. The results are shown acceptable agreement with those obtained by the conventional Stochastic Finite-Element Method (SFEM). The proposed approach has potential to model the complex geometries and cutoffs in different locations without additional efforts to deal with the spatial variability of the permeability.

1. Introduction Seepage analysis has a crucial importance in geotechnical engineering. The flow results play a key role in assessment of flow rate, exit gradient, and safety factor against piping. The mentioned quantities may be required for the safe design of various engineering structures such as dams, bridges and retaining walls. Over the last decades, several methods such as classical, analytical, and numerical methods have been utilized to analyze seepage problems. A succinct explanation of each method with relevant flaws and recent contributions is presented in the following. The electrical analogue is of classical methods. Ohm’s law of an electric current is the counterpart of Darcy’s law in a flow regime. Limited size of electro-conducting materials seriously restricts the method performance in large-scale problems [1]. The flow net is a drawing classical solution. In fact, the flow net is a plot of two perpendicular sets of streamlines and equipotential lines which can be obtained by tedious sketching techniques through trial-and-error [1]. Although analytical methods are required to attain a profound concept of the basic physics [2], a few cases with simple geometry can be solved analytically. He [3] has proposed a model for seepage flow in porous media with fractional derivatives.

∗

Numerical approaches are capable of analyzing more complicated seepage problems; however, these methods still have some pertinent deficiencies. The Finite-Difference Method (FDM) [4,5], the FiniteVolume Method (FVM) [7,8], the Finite-Element Method (FEM) [9–13], the Boundary Element Method (BEM) [14–16] and meshless methods [17–19] are used as the most conventional numerical procedures in seepage problems. The FDM is the earliest numerical procedure to solve partial differential equations. The studies of Bardet and Tobita [4] and Jie et al. [5] can be named among the important contributions using the FDM. Bardet and Tobita [4] presented a practical FDM for unconfined seepage. Jie et al. [5] applied FDM to deal with steady seepage analysis in the homogeneous isotropic medium. The FVM is mainly employed for the numerical solution of problems in fluid mechanics that was introduced by McDonald in 1971 [6]. Darbandi et al. [7] developed a moving-mesh FVM capable of solving the seepage problem in domains with arbitrary geometries. Bresciani et al. [8] applied a method based on a finite volume scheme to analyze steady-state flow in porous media. Like the FD and FV methods, the FEM is a mesh generation based approach, which has substantial restrictions to model singularities related to adjacency of sharp edges and angles. Bathe and Khoshgoftaar [9] used the FEM to study the seepage through porous media. Ataie-Ashtiani et al. [10] employed the FEM to simulate the groundwater flow in unconfined aquifers

Corresponding author. E-mail address: [email protected] (A. Johari).

https://doi.org/10.1016/j.enganabound.2018.05.015 Received 17 December 2017; Received in revised form 6 May 2018; Accepted 29 May 2018 0955-7997/© 2018 Elsevier Ltd. All rights reserved.

A. Johari, A. Heydari

Engineering Analysis with Boundary Elements 94 (2018) 44–59

with a periodic boundary condition. Ouria and et al. [11] studied a nonlinear analysis of transient seepage beneath a dam considering the effect of the change in permeability of soil by the coupled FEM. Chen et al. [12] proposed a numerical solution based on the FEM to solve complex drainage systems. Kazemzadeh-Parsi and Daneshmand [13] numerically analyzed three-dimensional unconfined seepage problems in homogeneous and anisotropic domains with arbitrary geometry using the fixed grid finite-element method. As a feature of the BEM, complicated geometries are simply discretized just by boundary discretization. However, a fundamental solution is needed to satisfy the governing equations in the domain. This is regarded as the main defect of this approach. The investigations of Brebbia and Chang [14], Chen et al. [15], and Rafiezadeh and Ataie-Ashtiani [16] can be named as the outstanding contributions. Brebbia and Chang [14] applied boundary elements to seepage problems in inhomogeneous and anisotropic soil mediums. Chen et al. [15] employed the BEM based on the formulation of the dual integral equations to analyze the seepage flow under a dam with sheet piles. Rafiezadeh and Ataie-Ashtiani [16] used the BEM to analyze transient free-surface seepage problems in anisotropic materials. Meshless methods are used to solve fluid mechanic problems by utilizing unstructured nodes. The accuracy of these approaches is acceptable, howbeit they are well known as time-consuming methods that is a remarkable imperfection. Hashemi and Hatam [17] utilized radial basis function-based differential quadrature method as a mesh-free approach to solve two-dimensional transient seepage. Jie et al. [18] investigated the application of the natural element method in seepage analysis with a free surface. Zhang et al. [19] combined the moving Kriging interpolation and the Monte Carlo integration to analyze unconfined transient seepage through homogeneous and inhomogeneous media. Recently, Song and Wolf [20] have developed the SBFEM to overcome the limitations of the previous methods. It is a novel semianalytical approach for the dynamic analyzing of unbounded domains. This method combines significant advantages of the FEM and the BEM. Over the last two decades, the SBFEM has been successfully applied to various problems in bounded and unbounded domains [21-24]. Nevertheless, just a few studies focused on seepage problems using the SBFEM. Bazyar and Graili [25] extended SBFEM to analyze confined and unconfined seepage problems. Bazyar and Talebi [26,27] applied the SBFEM to analyze transient seepage and heat conductivity in anisotropic soils. In the mentioned studies, the sources of uncertainty of input parameters were not considered and seepage problems were analyzed deterministically. However, due to the inherent uncertainties of the soil characteristics, indicate that seepage problem has a probabilistic nature rather than deterministic. Thus, a stochastic assessment of seepage analysis would be useful as an alternative or a supplement of the deterministic analysis to provide better engineering decisions. Griffiths and Fenton [28–30] studied the effect of stochastic soil permeability on confined seepage beneath water retaining structures based on spatially random soil for two-dimensional and three-dimensional seepage problems. Ahmed [31] extended stochastic analysis of free surface flow through earth dams by the FEM. Ahmed et al. [32] investigated the problem of confined flow under dams and water-retaining structures using stochastic finite-element modeling in anisotropic heterogeneous soils. Srivastava et al. [33] studied the influence of spatial variability of the permeability property on steady state seepage flow and slope stability analysis using finite difference numerical code. Luo et al. [34] expressed a simplified procedure for reliability analysis considering the spatially varying of soil parameters. Rohaninejad and Zarghami [35] combined the Monte Carlo and the finite difference methods to evaluate the physical behavior of embankment dams stochastically. Cho [36] probabilistically analyzed the seepage to consider the spatial variability of permeability for an embankment on soil foundation in a layered soil profile. Long et al. [37] explored a sensitivity analysis of the SBFEM for the elastostatics. Long et al. [38] analyzed the fracture of cracked structures stochastically considering the random field properties by the SBFEM. Ahmed

et al. [39] scrutinized the flow beneath water-retaining structures under heterogeneous conditions. Hekmatzadeh et al. [40] investigated the effect of uncertainty in the soil properties, earthquake coefficients, and sediment characteristics in the stability of a diversion dam. The common feature of the mentioned studies was the consideration of spatial variability for simple problems, while they had their corresponding constraints especially for complex geometry, sharp corners, and singular points. The main goal of this paper is to overcome the mentioned limitations and develop a method for reliability analysis of the steady-state seepage using stochastic SBFEM to consider the spatial variability of the permeability. On one hand, it is worth mentioning that the typical SBFEM is not capable of discretizing the whole domain except the boundary. On the other hand, the entire domain needs to be discretized to take into account the spatially varying of soil properties due to random field theory. For this purpose, a coded computer program is required to couple the SBFEM with random field theory, which is provided in this research. The outputs of the proposed method are verified by FEM solution in another coded program. In further part of this study, to clarify the efficiency and accuracy of the proposed method in reliability analysis of seepage flow, three stochastic examples are conducted. The influence of variations in the location of sub-domain discretization center, the cutoff location, and the cutoff length are elucidated in the examples. Furthermore, the effect of the COVk and the correlation length on the groundwater flow quantities are investigated as well.

2. Steady-state two-dimensional seepage The flow of water within the pores of materials is an intricate phenomenon. Henry Darcy discovered an efficient law to properly explain that how water flows through the porous medias. Darcy’s law attributes the velocity and the rate of water flow in a porous medium to the hydraulic gradient and the permeability coefficient as [41]: 𝑣𝑋 = −𝑘𝑋

𝜕ℎ 𝜕𝑋

(1)

where vX stands for velocity in the X direction, kX represents the permeability in the corresponding direction, and h holds the values of the hydraulic heads. The seepage problems can be solved based on boundary conditions. In one-dimensional problems, the water flows through direct columns of soil. In this case, Darcy’s law can be applied to analyze the seepage problems directly. However, in two-dimensional conditions like the seepage beneath dams and water retaining structures, nonlinear streamlines are constructed. Thus, Darcy’s law cannot be exerted straightly. Therefore, driving a governing differential equation is essential to solve the seepage problems. The detailed elucidation of pertinent formulations is expressed in Mariño’s book [41]. The two-dimensional fundamental differential equation of seepage flow in porous media can be expressed as [41]: 𝑘ℎ

𝜕2 ℎ 𝜕2 ℎ 𝜕ℎ + 𝑘𝑣 = 𝑆𝑠 2 𝜕𝑡 𝜕𝑋 𝜕𝑌 2

(2)

where X and Y indicate the horizontal and vertical directions. kh and kv are the permeability in the horizontal and vertical directions, respectively. Ss is specific storage coefficient, and t stands for the time. For steady-state groundwater flow, the specific storage coefficient and time conditions are vanished due to the conservation of water mass. Therefore, Eq. (2) transforms into the Laplace’s equation as [42,40]: ( ) ( ) 𝜕 𝜕ℎ 𝜕 𝜕ℎ 𝑘 + 𝑘 =0 𝜕𝑋 ℎ 𝜕𝑋 𝜕𝑌 𝑣 𝜕𝑌

(3)

A solution of Laplace’s equation is required to analyze the steadystate groundwater flow problems. The output of Eq. (3) is the values of hydraulic head. Some important seepage quantities such as flow rate, exit gradient, and safety factor against piping can be assessed using determined values of hydraulic head. 45

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Engineering Analysis with Boundary Elements 94 (2018) 44–59

|J| is the determinant of the Jacobian matrix on the boundary and can be expressed as: |𝐽 | = 𝑥(𝜂)

𝜕 𝑦(𝜂) 𝜕 𝑥(𝜂) − 𝑦(𝜂) 𝜕𝜂 𝜕𝜂

(10)

The Cauchy–Euler equation in Eq. (4) leads to a linear eigenvalue problem as [47]: [𝑍 ][𝜙] = [𝜙][𝜆] where [Z] is a Hamilton matrix and is defined as: [ [ 0 ]−1 [ 1 ]𝑇 [ ]−1 ] − 𝐸0 𝐸 𝐸 [𝑍 ] = [ 2 ] [ 1 ][ 0 ]−1 [ 1 ]𝑇 [ 1 ][ 0 ]−1 −𝐸 + 𝐸 𝐸 −𝐸 𝐸 𝐸 Fig. 1. The SBFEM local coordinates.

Eq. (3) is properly solved by the SBFEM [43]. The complete details of the SBFEM theory are represented in the literature [e.g. 44–46]. Here a concise summary of fundamental concepts and substantial equations of the method is stated. Unlike the FEM, only the boundary of the domain requires mesh discretization and the dimension reduces by one. As well as no fundamental solution is needed in comparison with the BEM. In simple geometries, the whole boundary is scaled due to a single point named scaling center that must be exactly visible from the entire domain, except the boundary passing through the scaling center. For complicated geometry, which chosen a single scaling center is not possible; the entire domain can be separated into sub-domains with their own scaling centers in a similar manner. With utilizing this method, the points that located in adjacency of sharp corners or singular nodes can be modeled directly with high accuracy. By applying the principle of virtual work [47], the method of weighted residuals [48] to the governing equations, or with an analytical solution [49] the SBFEM equations were obtained. As demonstrated in Fig. 1, the SBFEM presents a local coordinate system (𝜉,𝜂). This is described by scaling the boundary respect to the scaling center "O" with the dimensionless radial coordinate 𝜉 that increases from 0 at the scaling center "O" to 1 on the boundary. The circumferential coordinate 𝜂 forms the boundary. For one element in the boundary, 𝜂 varies from −1 to 1 in the isoparametric element. The head field within the domain can be computed analytically by determining the head function on the boundary. According to the virtual work principle, the SBFE equation of seepage can be derived as [47]: [ 0 ] 2 𝜕 2 {ℎ(𝜉)} ([ 0 ] [ 1 ] [ 1 ]𝑇 ) 𝜕 {ℎ(𝜉)} [ 2 ] 𝐸 𝜉 + 𝐸 − 𝐸 + 𝐸 − 𝐸 {ℎ(𝜉)} = 0 (4) 𝜉 𝜕𝜉 𝜕𝜉2

where diagonal matrices [𝜆n ] and [𝜆p ] indicate that all real components are negative and positive, respectively. The stiffness matrix of the bounded domain is obtained as: [ ][ ]−1 (14) [𝐾 ] = 𝜙21 𝜙11 On the boundary (𝜉 = 1) the relation between head functions and flow rates is expressed as: {𝑄} = [𝐾 ]{ℎ}

[ 2] 𝐸 =

∫𝜂

[ 2 ]𝑇 [ 1 ] 𝐵 (𝜂) [𝑘] 𝐵 (𝜂) |𝐽 |𝑑𝜂

(6)

[ 2 ]𝑇 [ 2 ] 𝐵 (𝜂) [𝑘] 𝐵 (𝜂) |𝐽 |𝑑𝜂

(7)

𝑄𝑁𝑜𝑟. =

(16)

4. Stochastic analysis A distinctive feature of geotechnical engineering is the spatial variability of soil properties that vary randomly within soil layers whether heterogeneous or even homogeneous. Spatial distribution of soil properties can be modeled using the theory of random field [50]. The random field theory has the capability to expand the deterministic SBFEM to a stochastic phase for investigating geotechnical problems such as seepage analysis.

𝜕𝜂

} { −𝑦(𝜂) 𝜕 [𝑁 (𝜂)] 𝑥(𝜂) 𝜕𝜂

𝑄 𝜇𝑘 Δ𝐻

where QNor . is non-dimensional flow rate, 𝜇 k is the mean of conductivity, ΔH is the difference between the values of the head at upstream and downstream of the dam.

where [E0 ] is a positive definite, [E2 ] is a semi-positive definite, and [E1 ] is a non-symmetric matrix. [k] is the permeability tensor. [B1 (𝜂)] and [B2 (𝜂)] represent the head gradient-head relationship as: { 𝜕 𝑦(𝜂) } [ 1 ] 1 𝜕𝜂 (8) 𝐵 (𝜂) = [𝑁 (𝜂)] |𝐽 | −𝜕 𝑥(𝜂) [ 2 ] 1 𝐵 (𝜂) = |𝐽 |

(15)

Stiffness matrix obtained, from Eq. (14), can be employed to determine the nodal heads on the boundary using Eq. (15). The flow rate is directly calculated by substituting the output of seepage analysis into Eq. (15). A summation of flow rate values, related to downstream boundary nodes, is required to achieve the total flow rate. The potential dropped between two adjacent equipotential lines divided by distance between them is known as a hydraulic gradient. The maximum exit gradient ie is the hydraulic gradient at the downstream end of the flow line where percolating water leaves the soil mass and emerges into the free water at the downstream. The critical hydraulic gradient ic is the hydraulic gradient at which flotation of particles begins. It can be defined as a ratio of the submerged unit weight of soil 𝛾’ and the unit weight of water 𝛾 w . The factor of safety against piping is directly calculated as a ratio of ie and ic . A safety factor of 3 to 5 is considered adequate for the safe performance of the structure for preventing piping failure [41]. In order to achieve better evaluation, all the outputs are calculated in dimensionless form. The exit gradient and safety factor are nondimensional. While flow rate has to change into dimensionless value as follows [42]:

where {h(𝜉)} is the head function in the radial direction, [E0 ], [E1 ], and [E2 ] are coefficient matrices and all are formulated only in the circumferential coordinate. The coefficient matrices are introduced as: [ 0] [ 1 ]𝑇 [ 1 ] (5) 𝐸 = 𝐵 (𝜂) [𝑘] 𝐵 (𝜂) |𝐽 |𝑑𝜂 ∫𝜂 ∫𝜂

(12)

By using Eq. (11), the eigenvalues and eigenvectors are sorted properly as: ] [[ ] [ ]] [[ ] [ ]][[ ] 𝜙11 𝜙12 𝜙11 𝜙12 𝜆𝑛 (13) [𝑍 ] [ ] [ ] = [ ] [ ] [ ] 𝜙21 𝜙21 𝜆𝑝 𝜙22 𝜙22

3. Scaled boundary finite-element method

[ 1] 𝐸 =

(11)

(9) 46

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Engineering Analysis with Boundary Elements 94 (2018) 44–59

4.1. Random field theory

over the past years to obtain the reliability index. On one hand, the Cornell method only provides a precise solution for linear limit state functions. On the other hand, the Cornell method, the FORM, and the FOSM are known as analytical methods, while the requirement of a mathematical form of the limit state function is considered as the greatest disadvantage of these methods. In addition, the spatial variability of the random parameters and corresponding correlation between the parameters are not considered in these methods. Moreover, the artificial neural network and other genetic algorithm based approaches require a training and testing datasets and hence need a relatively long computational time for numerical iterations to meet the minimum reliability index. Further, the design point (which is used in topics such as designing based on the reliability index) cannot be obtained in simulation methods. Furthermore, Griffiths et al. [54] studied the probability of slope failure once using the FORM without spatial distribution of the random variables and another time with MCS with random field properties. They discovered that considering the spatial distribution will lead to conservative predictions of the probability of slope failure. Moreover, Tun et al. [55] compared the results of reliability analyses of 4 study cases for slope stability based on the genetic algorithm with the results of the MCS and Hasofer Lind reliability analyses. The outputs indicated that the genetic algorithm produced similar results to those of the MCS and Hasofer Lind. Therefore, through using spatial variability of the parameters by the random field theory, the reliability index can be estimated for a normal space as follows [40]:

Mean, standard deviation (Std.), and correlation function are used to describe the random field. In this study, the soil is assumed isotropic so that the permeability k will be similar in the horizontal and vertical directions. The permeability is supposed to have lognormal distribution (due to the geological nature of the permeability) with mean μk , Std. 𝜎 k , and spatial correlation length in vertical ϴv and horizontal ϴh directions. For convenience in calculations, the dimensionless parameter COVk is introduced as a ratio of the Std. and mean of permeability as COVk = 𝜎 k /μk . The mean value 𝜇 lnk and Std. 𝜎 lnk of the underlying normal field are explained as follows [40]: ( ( )) 1 𝜎ln 𝑘 = 𝑙𝑛 1 + 𝐶𝑂𝑉𝑘 2 2

(17)

( ) 1 2 𝜇ln 𝑘 = 𝑙𝑛 𝜇𝑘 − 𝜎ln 2 𝑘

(18)

The correlation function is used for considering the spatial correlation of soil properties at two different locations. In this study, a two dimensional Markov correlation function (𝜌) with correlation length in horizontal and vertical directions is adopted due to its simplicity and conservatively approach as follows [51]: ( | ) 𝑥1 − 𝑥2 || ||𝑦1 − 𝑦2 || 𝜌 = exp − | − (19) 𝜃ℎ 𝜃𝑣 where |x1 −x2 | and |y1 −y2 | are the absolute distance between the two points in a horizontal and vertical direction, respectively. The spatial correlation length ϴ is defined as the segregation distance, which within the length values of a random parameter at two vicinal locations is considerably correlated, while the values beyond ϴ are negligibly correlated. The Covariance matrix Cholesky decomposition [52] is adopted in this study for discretization the random field. The final lognormal random field can be formed as [40]: ( ) 𝑘𝑖 = exp 𝜇ln 𝑘 + 𝜎ln 𝑘 𝐺𝑖 (20)

𝛽=

(23)

where E and 𝜎 represent the values of the mean and standard deviation of the performance function, respectively. When the performance function has a lognormal distribution the Eq. (23) is converted to the following equation [56]: 𝜎2

𝑙𝑛(𝐸 [𝐺(𝑋 )]) − 𝑙𝑛𝐺2(𝑋 ) 𝛽ln = √ ( ) 𝑙𝑛 1 + 𝐶𝑂𝑉𝐺(𝑋 ) 2

where ki is the allocated conductivity in the ith element and G is a standard normal random field.

(24)

5. Implementation of random field in the SBFEM

4.2. Reliability index calculation

Based on random field theory, it is necessary to allocate the uncertainties of the soil properties across the domain in order to insert the soil inhomogeneity into the problem. This means that the parameters have a spatial distribution and vary from one location to another one. Therefore, the whole domain must be discretized to consider the sources of uncertainty inside the domain. As mentioned in the introduction section, the FEM is a universal method for the stochastic analysis of the seepage problems. However, the abundance of required elements, along with the inability to analyze problems with complex geometry, have been considered as some of the numerous limitations of this method. Hence, the SBFEM seems to be a powerful method for solving this kind of problems. On one hand, the typical SBFEM only discretizes the boundary using two-node linear elements. On the other hand, complex geometry can be easily discretized into the sub-domains. The main difficulties of the current problem are how to discretize a complex domain into subdomains, as well as a large number of sub-domains which are required to consider the spatial distribution of the permeability. To overcome this problem, the authors have used a non-uniform mesh generation based on the SBFEM for adjustment the random field theory with the SBFEM. In this way, a complicated domain is discretized into a finite number of non-uniform meshes. Each mesh is supposed as a sub-domain and then is separately modeled according to the corresponding scaling center which is located somewhere within the sub-domain. Through using this procedure, the properties of stochastic variables are independently allocated in each sub-domain. More details about the stochastic scaled boundary finite-element method can be seen in Ref. [22].

The performance function or limit state function (G(X)) can be used to separate the safe and failure states of the dam stability arising from the safety factor against piping. In fact, the limit state function is defined as the subtraction of the demanded factor of safety (DF) from the available factor of safety (AF) as follows [53]: 𝐺(𝑋 ) = 𝐴𝐹 (𝑋 ) − 𝐷𝐹 (𝑋 )

(21)

where X stands for the vector of random variables. The demanded factor of safety is mostly intended to be equal to 1 for analyzing the dam stability. Therefore, Eq. (21) is changed to the following form with different possible states of the dam stability: ⎧𝐺(𝑋) > 0, ⎪ 𝐺(𝑋) = 𝐴𝐹 (𝑋) − 1⎨𝐺(𝑋) = 0, ⎪𝐺(𝑋) < 0, ⎩

𝐸 [𝐺(𝑋 )] 𝜎[𝐺(𝑋 )]

𝑆𝑡𝑎𝑏𝑙𝑒 𝐿𝑖𝑚𝑖𝑡 𝐹 𝑎𝑙𝑖𝑢𝑟𝑒

𝑠𝑡𝑎𝑡𝑒 𝑠𝑡𝑎𝑡𝑒 𝑠𝑡𝑎𝑡𝑒

(22)

The reliability index is defined as the distance of the design mode from the mean of limit state function by the number of standard deviations. Therefore, it is essential to obtain the probability distribution of the performance function for determining the reliability index. In this way, various methods such as Cornell method, First Order Reliability Method (FORM or Hasofer Lind method), First Order Second Moment (FOSM or Taylor series method), simulation approaches (random field considering spatial variability method and Monte Carlo Simulation (MCS)), and more advanced techniques including artificial neural network and genetic algorithm have been utilized by different researchers 47

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Fig. 2. The geometry of a domain: (a) Discretizing the domain into sub-domains and (b) a single sub-domain with linear elements. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 1 The determinstic soil properties of the verification example. Parameter

k (m/s)

𝛾 w (kN/m3 )

𝛾 (kN/m3 )

ΔH (m)

Value

1.7E−4

9.81

20

0.50

Fig. 5. The non-uniform SBFEM sub-domains.

Fig. 6. Hydraulic heads at the bottom of the domain. Fig. 3. The geometry of the verification example [17].

Fig. 7. Contours of hydraulic heads. Fig. 4. Domain discretization in the FEM solution.

can be arbitrary moved within the domain, which is considered as an important advantage of this sub-domain generation. This key feature can be used to model dams with sheet piles or trenched core conveniently by considering the center of sub-domain generation beneath the water retaining structures. For a single realization of the random field (after using Eq. (20) to assign the permeability into each sub-domain independently) the stiffness matrix for each sub-domain is obtained utilizing the equilibrium

Fig. 2(a) shows a domain that is divided into non-uniform subdomains. The red nodes represent the scaling centers related to corresponding sub-domains, which are located exactly at the geometry center. Fig. 2(b) illustrates the geometry of a single sub-domain. To increase the accuracy of the solution, each side of the sub-domain is divided into two linear elements with equal length, which is demonstrated via blue nodes. The location of the center of the mesh generation is not fixed and 48

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Engineering Analysis with Boundary Elements 94 (2018) 44–59

Fig. 8. The selected geometry to assess the influence of the location of the sub-domain discretization center.

equation from Eq. (14) exactly the same as the deterministic solution. The global stiffness matrix of the domain for each realization will be acquired through assembling the stiffness matrices of the whole subdomains just exactly similar to the FEM. This process will be sufficiently repeated until the desired probability distribution of the performance function is achieved. It is worth mentioning that although the SBFEM equation of seepage problems, Eq. (4), has been completely weakened in the circumferential direction; this equation still remained in strong form in the radial direction. In the process of the classical SBFEM, in order to compute the head values in the circumferential direction, first the values of the head are calculated numerically on the boundary and then computed analytically using Eq. (12) in the radial direction. For this reason, the traditional SBFE method was considered as a semi-analytical method. However, through utilizing the proposed procedure, not only the number of elements that are needed in a FEM solution is reduced, but also only, the two-node linear elements are used. Moreover, using the proposed method leads to reduce the spatial dimensions of the problem by one unit, just as the way of the traditional SBFEM. In addition, the proposed process is a complete numerical approach, like the FEM, and the analytical solution in the radial direction of the usual SBFEM has been completely eliminated.

3) Developing seepage important quantities such as flow rate, exit gradient, and safety factor against piping. 4) Evaluation the influence of variations in the location of the subdomain discretization center, cutoff location, and cutoff length. 5) Assessment the effect of the different COV of stochastic parameters and correlation length on the mean and Std. of the outputs as well as reliability index of safety factor against piping. 6.1. Verification of the program To verify the precision and performance of the developed programs, the outputs including potential heads, normalized flow rate, exit gradient, and safety factor against piping are compared with the results of the FEM method. For this purpose, a small and simple dam from literature [17] is chosen. The geometry of the dam is shown in Fig. 3. The deterministic properties of a saturated silty soil layer, which are selected for this verification, are presented in Table 1. The upstream and downstream water levels are 0.50 m and 0.00 m, respectively. A 1.2 m by 3.6 m horizontal layer, which is shown in Fig. 4, is considered for the FEM solution. The domain is discretized into 432 (12 by 36) uniform squares with a size of 0.1 m. Each square forms an element in FEM solution. Furthermore, the problem is solved by 960 non-uniform SBFEM based sub-domains, which is demonstrated in Fig. 5, by the RFSBFEM. The center of sub-domain generation is assumed at the center of the domain. The nodal potential heads are computed by seepage analysis for steady-state. These head values are then used to determine the normalized flow rate, exit gradient, and safety factor against piping. The head results of the RFSBFEM and the FEM on the floor of the domain are compared in Fig 6. The results show a perfect conformity between the RFSBFEM and the FEM. The potential contour of these models is indicated in Fig. 7. This figure shows a good agreement between the results. The exit gradient, normalized flow rate, and safety factor against piping in downstream are computed. These quantities are represented in Table 2. The outputs of the models show a great compatibility between the FEM and the RFSBFEM in square mesh and non-uniform SBFEM based sub-domains.

6. Computer program To perform this study, two computer programs are developed based on the Random Field Scaled Boundary Finite-Element Method (RFSBFEM) and the FEM. The equations, which are used for the FEM solution, are represented in the literature [44]. These programs are coded by MATLAB to take full advantages of its matrix operations. The RFSBFEM program is consisting of a sub-domain generation code, which is capable to discretize complicated domains. This program is able to consider the sources of the uncertainties of the soil parameters, while others programs e.g. SEEP W have limitations in stochastic analysis. The FEM program is used to verify the efficiency and accuracy of the RFSBFEM in deterministic solution by considering the Std. of permeability equal zero. The deterministic outputs including heads, flow rate, exit gradient, and safety factor against piping are compared between two programs. The major capability of the RFSBFEM program can be mentioned as follows:

7. Stochastic illustrative examples

1) Discretizing complex geometries into sub-domains by selecting the center of employed sub-domain discretization in an arbitrary and appropriate location within the domain. 2) Implementation of uncertainties in soil parameters and spatial variability based on random field theory.

In this section, three stochastic numerical examples are presented to illuminate the efficiency and accuracy of the proposed method. All examples are conducted by merging the random field with scaled boundary finite-element method for considering uncertainties of input variables. In the first example, the effect of location variations of sub-domain 49

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Table 2 Comparison of the model’s outputs. Method

Exit gradient (ie )

Normalized flow rate (QNor. )

Safety factor against piping (FSpiping )

FEM RFSBFEM

0.6525 0.6522

0.5836 0.5837

1.5919 1.5927

Table 3 Deterministic inputs for all the examples. Parameter

𝛾 w (kN/m3 )

𝛾 (kN/m3 )

Upstream water level (m)

Downstream water level (m)

Value

9.81

20

5

0

Table 4 Stochastic lognormal parameters of all the examples. Parameter

Mean

COVk

Correlation length (m)

Value

9.81

0.125, 0.250, 0.500, 1.000, 2.000, 4.000, 8.000, 16.000

2.00, 4.00, 8.00, 16.00

discretization center is assessed. In addition, the influences of variations in the COVk value and correlation length on the mean and Std. of the outputs are evaluated. The second example is allocated to investigate the influence of cutoff location on the output quantities. The effect of cutoff length is studied in the third example. 7.1. Assessment of the location variation of the sub-domain discretization center In the first numerical example, as shown in Fig. 8, a simple dam without cutoff is considered. The domain is discretized into 972 nonuniform SBFEM based sub-domains for all cases with respect to a single node that can be chosen anywhere in the domain. The locations of 5 different center nodes are demonstrated in Fig. 8. The deterministic inputs and the relevant values are prepared in Table 3. As reported in this table, the saturated unit weight of soil and unit weight of water are selected equal to 20 kN/m3 and 9.81 kN/m3 , respectively, and the upstream and downstream water are at the levels of 5.00 m and 0.00 m, respectively. Moreover, as mentioned by Griffiths and Fenton [28–30], Ahmed [31], Cho [36], and Ahmed et al. [39] the soil permeability follows a lognormal distribution due to the variability nature. As listed in Table 4, the mean value of permeability is taken 1E-5 m/s, which is constant in all cases. The value of COVk is selected as 0.125, 0.250, 0.500, 1.000, 2.000, 4.000, 8.000, and 16.000. The correlation length in both horizontal and vertical directions are considered as 2.00 m, 4.00 m, 8.00 m, 16.00 m to conduct the stochastic analyzes. It should be mentioned that the deterministic and stochastic inputs of the soil which given in Tables 3 and 4 are considered to be the same for two other numerical examples. To reduce the computational time and in order to reach a sufficient number of random field realizations, the problem is firstly solved for a different number of realizations. For this purpose, the sub-domain discretization center is selected at the point A. The COVk = 1 and ϴh = ϴv = 2 are considered. The results are shown in Figs. 9 and 10 for the mean and Std. of the exit gradient, respectively. The figures demonstrate that after 1000 realizations the results are become converge to a constant value. However, the results still have fluctuation up to 10,000 realizations, which can affect the results. Therefore, 10,000 realizations are selected in this study for random field model in all cases. To demonstrate the influence of location of the sub-domain discretization center on the outputs in a stochastic analysis, the first example is solved for different cases where the sub-domain discretization center is located at the points B, C, D, and E with considering COVk = 1 and ϴh = ϴv = 2. Fig. 11 shows the spatial variability of the random permeability for a sample realization related to the different positions of the sub-domains discretization center B, C, D, and E. The blue sub-domains indicate the

Fig. 9. The mean variations of exit gradient with respect to the number of realizations.

fewer values of permeability and conversely. The contours of potential headlines of 10 different realizations and the equipotential lines of the mean values of the potential head related to different cases are illustrated in Fig. 12. In these figures, the contours of the mean values of the potential head are demonstrated by thicker contour lines. The results show that the mean contours of the potential lines are approximately similar for all locations of the sub-domain discretization center and the location of this point has a negligible effect on the outputs. The Probability Density Functions (PDFs) of exit gradient related to different positions of the sub-domain discretization center are illustrated in Fig. 13. It can be seen in the figure that the PDFs of this output have a lognormal distribution and are very close to each other. The PDFs of flow rate and safety factor against piping show similar trend. In addition, for a better comparison, the mean and Std. values of the seepage outputs are given in Table 5. The results have insignificant differences in the values. It was shown that location of the sub-domain discretization center had no very significant influences on the results. Therefore, the location of the sub-domain manufacturing center is chosen at the center of the domain (point A) to examine the effects of the stochastic input parameter on the outputs. Before doing the stochastic analysis, a deterministic 50

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Table 5 Comparison of the results for different sub-domain discretization center. Location of sub-domain center

𝜇𝑖𝑒

𝜎𝑖𝑒

𝜇𝑄𝑁𝑜𝑟.

𝜎𝑄𝑁𝑜𝑟.

𝜇𝐹 𝑆𝑝𝑖𝑝𝑖𝑛𝑔

𝜎𝐹 𝑆𝑝𝑖𝑝𝑖𝑛𝑔

Point B Point C Point D Point E

0.5353 0.5416 0.5381 0.5328

0.2347 0.2375 0.2354 0.2330

0.3303 0.3312 0.3285 0.3341

0.0694 0.0700 0.0652 0.0689

2.3833 2.3931 2.3912 2.3738

1.2437 1.2577 1.2511 1.2346

Table 6 Deterministic results of illustrative example 1. Exit gradient (ie )

Normalized flow rate (QNor. )

Safety factor against piping (FSpiping )

0.5181

0.4364

2.0049

A stochastic steady-state analysis is conducted with the same soil properties and sub-domain numbers. The COVk = 1 and ϴh = ϴv = 2 are considered. The Std. contour of the potential head is shown in Fig. 14. Based on this figure, it can be concluded that the Std. is increased steadily from its lowest value near the boundaries of the upstream and downstream sides to its largest value in the middle of the seepage regime. This behavior is originated from the constant values of the potential head of the upstream and downstream boundaries. The results are compatible with the results of previous studies in the literature [28–30], which shows the accuracy of the proposed method. This part is dedicated to investigate the influence of different COVk values and correlation lengths (ϴh and ϴv ) on the exit gradient, normalized flow rate, and reliability index of the safety factor against piping. For this purpose, a set of stochastic analysis is performed using the nonuniform SBFEM based sub-domains by considering the location of the sub-domain discretization center at point A. In this set, the values of COVk is selected as 0.125, 0.250, 0.500, 1.000, 2.000, 4.000, 8.000, and 16.000 as well as ϴh = ϴv are considered as 2.00 m, 4.00 m, 8.00 m, 16.00 m, and also deterministic values. 7.1.1. Effect of COVk and correlation length on the mean and Std. of the exit gradient The effect of different COVk and ϴ on the mean of the exit gradient is portrayed in Fig. 15. The mean values of the exit gradient remain approximately close to the deterministic value for typical COVk (COVk < 1) and then are continuously increased with respect to the growth of the COVk . The variations are completely rational because as the COVk values are increased, the uncertainties are intensified and therefore the

Fig. 10. The Std. variations of exit gradient with respect to the number of realizations.

solution is performed by taking the Std. of the permeability equal to zero. The results are given in Table 6. These values will be compared with stochastic outputs.

Fig. 11. Sample realizations representing the spatial variation of random permeability for different positions of the sub-domain discretization center: (a) Point B, (b) Point C, (c) Point D and (d) Point E. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) 51

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Fig. 12. Contours of potential headlines for locations of the sub-domain discretization center at: (a) Point B, (b) Point C, (c) Point D and (d) Point E.

Fig. 14. The Std. contour of the potential head.

Fig. 13. The PDF of exit gradient for different locations of the sub-domain discretization center.

mean values are moved farther away from the deterministic value. Furthermore, by increasing the correlation length to a specific value (ϴ ≃12), the mean values of the exit gradient are first growth and after that are begun to decrease for bigger correlation lengths. When correlation length is tended to the infinity, the mean values related to all the COVk values come close to the deterministic value. As a reason for this behavior, when the correlation length is increased to the infinity, the problem nature is met the deterministic solution. The variations of the exit gradient Std. are demonstrated in Fig. 16. The Std. values are shown an increasing trend for all the COVk values. While the Std. values of the exit gradient are nearly insensitive to the variations of the correlation length. However, for the infinite amount of the ϴ, the problem is inclined to the deterministic essence and therefore the Std. values are reached to zeros.

Fig. 15. Mean variations of the exit gradient.

7.1.2. Influence of the COVk and 𝜃 on the mean and Std. of the normalized flow rate Fig. 17 indicates the influence of the different COVk values and correlation lengths on the mean of the normalized flow rate. The mean values of the normalized flow rate are evidently fallen down with respect 52

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Fig. 16. Std. variations of the exit gradient.

Fig. 18. Variations of the normalized flow rate Std.

Fig. 17. Variations of the normalized flow rate mean.

Fig. 19. Variation of reliability index with respect to COVk and correlation length.

to the deterministic value for all variations. For the typical COVk values, the insignificant divergence between deterministic and stochastic analysis is observed. With increasing the COVk , the divergence is grown incessantly. Furthermore, the mean variations of normalized flow rate are decreased slightly for ϴ = 2–4. The mean variations are unceasingly increased for greater correlation lengths. As correlation length is gone to the infinity, the mean values of this quantity are approached to deterministic value. The explanation of this is similar to the exit gradient behavior. The variations of the Std. values of the normalized flow rate related to the various COVk values and correlation lengths are depicted in Fig. 18. The trend of this parameter is started from zero at the COVk = 0.0 and is increased with the COVk to meet its maximum value and after that is begun to decrease and return to zero at the infinite COVk . This parameter is experienced a continuous increasing trend with growing in correlation length. The maximum Std. occurs in the infinite correlation length somewhere between COVk equal to zero and the infinity. When

the COVk and correlation length both are equaled to zero, the Std. values are reached to zero due to the similar conditions to the deterministic analysis. 7.1.3. Variations of the reliability index with respect to the COVk and correlation length The results of variations of the reliability index with respect to the random field parameters are shown in Fig. 19 when the COVk and correlation length are varied. For the typical COVk , large values of the reliability index are obtained. This is caused by the negligible Std. of the safety factor in typical COVk values. As expected, with increasing the COVk (the Std.), the mean and Std. of FSpiping are increased, which is caused to reduce the reliability index values. The reliability index values are started to converge to a constant value after typical COVk values for all cases. The reason for this behavior can be interpreted that after COVk = 2, the ratio of the mean and Std. of the safety factor against 53

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Fig. 20. The geometry of the illustrative example 2.

Fig. 21. Sample realization representing the spatial variation of permeability and employed sub-domains for (a) cutoff location A and (b) cutoff location E. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 22. Effect of cutoff location on the mean of exit gradient.

Fig. 23. Effect of cutoff location on the Std. of exit gradient.

It should be noted that as it was mentioned by Griffiths and Fenton [28–30], Ahmed [31], and Ahmed et al. [39], in order to numerically make probabilistic interpretations from a random field, it is possible either to consider the high value of the coefficient of variation in a numerical example that results in the high exit gradient and unallowable safety factor. A comparison between the results of the current study with

piping (COVFS ) remains constant. The variations of the reliability index related to ϴ≲6 are reduced. After this approximate correlation length, the reliability index is subjected to incremental variations, which is originated from the behaviors of the exit gradient and safety factor against piping. 54

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Fig. 24. Influence of cutoff location on the mean of normalized flow rate.

Fig. 26. Effect of cutoff location on reliability index.

As one result of the previous example, the dams are modeled merely by selecting the sub-domain discretization center exactly under the cutoff location. As another feature of the applied method, the cutoff thickness is supposed to be equal to zero precisely, while the most conventional methods have considered the cutoff thickness, albeit negligible, to form the elements. This domain is discretized into 972 sub-domains for all cases. The correlation length is presumed equal to two in all cases. The stochastic analyzes are conducted for the different values of COVk . For the sake of conciseness, a sample realization representing the spatial distribution of the random permeability and employed subdomains for the cases that the cutoff is placed at locations A and E are portrayed in Fig. 21. Pinker sub-domains indicate more permeability. 7.2.1. Influence of the variation in cutoff location on the mean and Std. of exit gradient Fig. 22 shows the variations of the exit gradient’s mean relevant to different cutoff locations. As mentioned earlier in the previous example, the mean values for all cases are increased gradually with increasing COVk . When the cutoff is situated at location C, the maximum mean values of the exit gradient are obtained, which are nearly close to the results of the cutoff location B. Afterwards, lower mean values of the exit gradient are achieved for the cutoff locations A and D, with a slight difference. However, the lowest mean values are attained for the cutoff location E with a considerable discrepancy with respect to the other locations. The variations of the Std. of this parameter with regard to different cutoff locations are demonstrated in Fig. 23. Insignificant divergence is observed in Std. trend for four cutoff locations A, B, C, and D. In this figure, the cutoff location B with an inconsequential difference has the biggest Std. Like the mean, the minimum Std. values are related to cutoff location E.

Fig. 25. Influence of cutoff location on the Std. of normalized flow rate.

those of the previous researches demonstrated a complete satisfactory agreement for all high values of the COVk and correlation length. However, the results of this study indicated that COVk > 2 had negligible influence on the reliability index. 7.2. Evaluation the influence of variation in the cutoff location One advantage of the RFSBFEM is modeling the water retaining structures simply (e.g. cutoffs and sheet piles). The cutoff location has a significant effect on the dam designing. In this section, an illustrative example is brought to the reader to clarify the influence of variations of the cutoff location including the uncertainties of the input parameters. For this purpose, five identical cutoffs A, B, C, D, and E in different locations with a similar length of 5 m are assumed as demonstrated in Fig. 20. The deterministic and stochastic parameters of the soil are similar to those of the first example that is listed in Tables 3 and 4. The proposed method is employed to model the dam for each cutoff, separately.

7.2.2. Influence of variation in cutoff location on the mean and Std. of normalized flow rate The influence of cutoff location on the mean of normalized flow rate for different COVk values is shown in Fig. 24. Whatever the COVk value is increased further, the mean values of the normalized flow rate for all cutoff locations are converged continuously. For the COVk equal to 16 the mean values are completely converged to a constant value for all cutoff locations. The influence of cutoff location on the Std. of normalized flow rate for different COVk is displayed in Fig. 25. 55

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Fig. 27. Considered geometry for illustrative example 3.

Fig. 28. Sample realization representing the spatial variation of permeability and domain discretization for: (a) Cutoff length A and (b) Cutoff length E.

Fig. 29. Mean variations of exit gradient for different cutoff’s length.

Fig. 30. Std. variations of exit gradient for cutoff’s length.

At first, the Std. values of all cutoff locations are approximately similar. By increasing the COVk value to 2 the Std. values related to cutoff locations A and E diverges from the rest of cutoffs. For greater COVk value (COVk ≥ 2), the Std. values of the different cutoff locations are approached each other again. As shown in these figures, the cutoff locations A and E have the lowest values of the mean and Std. of the normalized flow rate, while others have approximately the same values.

7.2.3. Variations of reliability index with respect to the location of the cutoff The effect of variation in the cutoff location on the reliability index values is depicted in Fig. 26. The utmost values of this parameter are related to the cutoff location E, which is located exactly at the dam toe. The other locations of the cutoffs are shown negligible differences in the values of reliability index. It can be seen in this figure that for COVk ≥ 2 the reliability index values of all cutoffs are begun to converge to a 56

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Fig. 31. Mean variations of normalized flow rate for different cutoff length.

Fig. 33. Influence of cutoff length on reliability index.

Fig. 32. Std. variations of normalized flow rate for different cutoff length. Fig. 34. The PDF of exit gradient for different cutoff length.

constant value. It can be construed that for huge COVk , the reliability index values are not impressed by the values of the COVk .

For the sake of brevity, sample realization including spatial distribution of the random permeability and domain discretization for different cutoff’s length A and E with the length of 3 and 7 m, respectively, are shown in Fig. 28.

7.3. Assessment of the effect of variation in the cutoff length The RFSBFEM is provided an applicable procedure to model the complicated geometries directly without additional efforts and expenses. In this example, an earth core dam with complex geometry is considered to demonstrate this potential. The selected geometry of the problem is illustrated in Fig. 27. In this case, the problem is only solved for the seepage beneath dam stochastically and seepage through the dam body is ignored due to concrete surface of the dam body. A set of stochastic analysis is performed for the different length of the cutoff to elucidate the influence of cutoff length on the outputs. The input properties of the soil are similar to the previous examples (Tables 3 and 4) with a difference that the correlation length is considered equal to two for all cases. The domain is discretized into 632 sub-domains.

7.3.1. Influence of variations in the cutoff length on the mean and Std. of exit gradient Fig. 29 represents the variations of the mean values of exit gradient for different cutoff lengths. The cutoff A has 3.0 m length and others (B–E) are lengthened 1.0 m than the previous cutoff. As expected, by increasing the length of cutoff, the exit gradient’s mean is decreased remarkably. This reduction is almost the same for all COVk values. The differences between the mean variations related to the different cutoff lengths remain constant. The variations of the exit gradient Std. related to different cutoffs A–E are portrayed in Fig. 30. For typical COVk , the length of the cutoff has no important effect on the Std. This figure 57

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Fig. 36. The PDF of FSpiping for different cutoff length.

Fig. 35. The PDF of normalized flow rate for different cutoff length.

8. Conclusions

represents that the smallest values of the Std. are related to the longest cutoff. As the length of the cutoff is decreased, the Std. values are slightly increased. As COVk values are increased further, the variations of Std. related to different cutoff lengths are diverged.

The deterministic analysis of seepage, due to model and soil uncertainties, cannot provide a reliable value for the flow quantities. Therefore, calibration of the deterministic method to consider the different sources of the uncertainties is essential. This paper proposed an efficient method for reliability analysis of seepage by focusing on the exit gradient, flow rate, and reliability index. For this purpose, a program is coded in MATLAB for coupling the random field theory with the SBFEM. The domain is discretized into several sub-domains to consider the spatial variability of the permeability. In this paper, to demonstrate the performance of the proposed method, three stochastic examples were presented. For reliability analysis, 10,000 realizations were determined as the sufficient number.

7.3.2. Influence of variations in cutoff length on the mean and Std. of normalized flow rate The mean variations of normalized flow rate for different cutoff lengths are demonstrated in Fig. 31. As shown in this figure, the mean of this parameter is experienced a considerable reduction by increasing the length of the cutoff. For COVk greater than 16, the mean values associated with the different lengths of the cutoff, are gradually converged to a constant value. Fig. 32 displays the Std. variations of the normalized flow rate for different cutoff lengths. No significant variations are observed on the Std. values for typical COVk . With increasing COVk , the Std. related to different length is started to diverge. The maximum divergence happened somewhere between COVk 1 and 2. After that the Std. variations are begun to converge again and for the infinite COVk the Std. values are converged completely. Furthermore, the highest cutoff has the least values of mean and Std.

1. The first example was designed to evaluate the effect of position variations of the sub-domain discretization’s center. For this purpose, five different positions for discretization’s center were selected. The results were shown that the center position had insignificant influence on the outputs. Therefore, the subdomain-manufacturing center was chosen at the domain center for a set of stochastic analysis. It was observed that increasing the COVk increased the mean and Std. of exit gradient. Likewise, correlation length firstly increased the mean and Std. values, and after a specific correlation length decreased these values. The mean value of flow rate experienced a reduction with COVk . As the correlation length increased, the mean value firstly decreased and then increased and converged to deterministic value. The reliability index of FSpiping decreased with increase COVk and converged to a constant value for COVk ≥ 2. The minimum reliability index is related to ϴ = 6. 2. In the second example, the RFSBFEM was employed to clarify the influence of the variations in cutoff location. Five identical dams, with zero thickness cutoff in different locations, was modeled only by selecting sub-domain discretization center beneath the cutoff. The maximum mean and Std. of exit gradient were related to a cutoff, which was located in the middle of dam foundation. As cutoff was shifted to the dam toe, these values were reduced incessantly. The minimum mean and Std. values of flow rate were pertinent to cutoffs, which was placed at the heel and toe of the dam. Further, for infinite COVk the mean value of flow rate for all cutoff locations was converged to a constant value. It is worth noting that the dam

7.3.3. Investigation of variations of the reliability index with respect to the cutoff length Fig. 33 illustrates the influence of cutoff length on the reliability index values. As the cutoff length is increased, the values of reliability index are increased for all cases. This figure shows that differences between the variations of reliability index for all COVk are approximately remained fixed. In order to obtain a better understanding, the effect of variations in the cutoff length on the PDFs of outputs consisting of exit gradient, normalized flow rate, and safety factor against piping are represented in Figs. 34–36, respectively. It can be seen in Fig. 34, as the cutoff length is increased, the mean value of exit gradient is decreased and the PDF peak is moved to the left. According to Fig. 35, the normalized flow rate has a similar behavior to exit gradient and the PDF apex is slightly shifted to left by increasing the cutoff length. Fig. 36 shows unlike other outputs, the mean value and the peak of FSpiping PDF are transferred to the right. 58

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toe cutoff shown maximum reliability index for typical COVk . For COVk ≥ 2 all cutoffs were reached to a fixed value. 3. The third example showed the capability of the RFSBFEM to analyze the seepage problem of complex geometry considering the uncertainties. As the cutoff length increased, the mean and Std. values of exit gradient and flow rate, related to a fixed COVk increased. For infinite COVk the mean of flow rate approached a constant value. Furthermore, the reliability index of FSpiping increased continuously for a constant COVk for longer cutoff.

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