WATER RESOURCES RESEARCH, VOL. 46, W11562, doi:10.1029/2010WR009066, 2010
A multiscale probabilistic collocation method for subsurface flow in heterogeneous media Liangsheng Shi,1,2 Dongxiao Zhang,2,3 Lin Lin,1 and Jinzhong Yang1 Received 1 January 2010; revised 5 August 2010; accepted 15 September 2010; published 30 November 2010.
[1] Owing to the spatial variability of the media properties, uncertainty quantification for subsurface flow and solute transport usually requires high‐resolution simulations. In this work, a multiscale probabilistic collocation method (MSPCM) is developed for solving such problems in a computationally efficient manner. The subsurface flow problem is cast in a stochastic framework, and probabilistic collocation strategy is used to represent the original stochastic differential equation. The resulting equations are a set of decoupled deterministic equations with respect to collocation points. A multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh. Coarse‐scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points. The coarse‐scale solution is then obtained by solving a modified coarse formulation that takes into account the fine‐scale heterogeneity. The fine‐scale solution is reconstructed after the coarse‐scale solution is available. Since the PCM and multiscale finite element method are implemented at different levels, the MSPCM inherits their respective advantages, in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid. The performance of the proposed method is demonstrated with numerical examples. The capability of MSPCM in reproducing the probability density functions (PDFs) of head and velocity is investigated. The numerical results show that the MSPCM with proper coarsening level is able to capture small‐scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity. Citation: Shi, L., D. Zhang, L. Lin, and J. Yang (2010), A multiscale probabilistic collocation method for subsurface flow in heterogeneous media, Water Resour. Res., 46, W11562, doi:10.1029/2010WR009066.
1. Introduction [2] Subsurface flow and solute transport in porous media may be treated as a random process because of uncertainty in the formation parameters, the source/sink terms, and the boundary/initial conditions [Dagan, 1989; Gelhar, 1993; Zhang, 2002]. With increasing interests in predicting uncertainty of flow, many numerical approaches have been developed in the past decades. Existing numerical methods for the solution of stochastic partial differential equation (PDE) fall under the categories of Monte Carlo simulation (MC), moment equation method (MEM), spectral stochastic finite element method (SSFEM), and stochastic collocation method (SCM). Recently, the stochastic collocation method has been paid much attention to because of its fast convergence and decoupled nature. In previous studies [Webster et al., 1996; Tatang et al., 1997; Xiu and Hesthaven, 1
National Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China. 2 Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA. 3 Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing, China. Copyright 2010 by the American Geophysical Union. 0043‐1397/10/2010WR009066
2005; Xiu, 2007; Foo et al., 2007, Huang et al., 2007, Ganapathysubramanian and Zabaras, 2007a; Li and Zhang, 2007, 2009], the stochastic collocation method has been applied in different disciplines. The principal steps of the stochastic collocation method are described as follows. First, the polynomial expansion and Karhunen‐Loeve expansion are employed to decompose the random outputs and inputs, respectively. Then stochastic collocation points are generated by using the existing theory of multivariate polynomial interpolations. The stochastic partial differential equations are reduced into a series of deterministic equations defined at each set of collocation points. These equations have the same form as the original equations and can be solved by existing deterministic solvers. Thus the numerical implementation of stochastic collocation method is straightforward as it only requires repetitive solutions of the deterministic problems similar to Monte Carlo simulation [Xiu and Hesthaven, 2005; Li and Zhang, 2009]. There are several distinct ways to select the collocation points. The collocation technique proposed by Tatang et al. [1997] is used in this study, and other techniques such as those discussed by Xiu and Hesthaven [2005] can also be considered. [3] Monte Carlo simulation and other sampling methods require spatial discretization that can resolve the small‐ scale heterogeneity in the medium properties. However, the high‐resolution simulation requires a tremendous computational effort. It is thus prohibitively expensive to capture
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the small‐scale variations without resorting to parallelization and domain decomposition. Multiscale algorithm aims at the derivation of a coarse‐scale equation to avoid the fully resolved computation [Hou and Wu, 1997; Ganapathysubramanian and Zabaras, 2007b]. The multiscale finite element method (MSFEM) was developed for this purpose [Hou and Wu, 1997]. This method considers the fine‐scale features by introducing local PDEs subject to special boundary condition. The large‐scale behavior from small‐scale variation is characterized via multiscale basis functions. The multiscale finite element methodology has been considerably advanced in the past 10 years. He and Ren [2005, 2006a] employed the MSFEM to simulate both saturated and unsaturated flow in heterogeneous media. The MSFEM was extended to two‐phase flow by Jenny et al. [2005] and Efendiev et al. [2006]. Chu et al. [2010] recently developed a modified MSFEM to characterize the effect of small‐scale inclusions of low or high conductivity with irregular boundary. We refer to the book by Efendiev and Hou [2009] for more detailed discussion on the theory and applications of multiscale finite element method. [4] The major limitation of MSFEM is that the coarse‐ scale basis functions are constructed from local information (for example, hydraulic conductivity). This process is no longer effective in the problems with strong nonlocal effects (such as channel flow). To take into account some global information, Efendiev and Hou [2007] used the global fine‐ scale solution at initial time to determine the basis functions. He and Ren [2006b] introduced a similar technique to capture the head variation in the well vicinity. Another shortcoming of MSFEM is the overhead for computing the coarse‐scale basis functions [Hou and Wu, 1997]. This overhead is comparable to the cost for solving the fully resolved problem. Since the coarse‐scale basis functions are independent of each other, they can be constructed independently. The computational efficiency can thus be improved with parallel algorithms. Moreover, coarse‐scale basis functions are computed at initial time and used repeatedly in the subsequent time. This leads to significant computational savings for unsteady state flow where the governing equation needs to be solved at different times. It is emphasized that usually constructing basis function takes most of computational time of MSFEM [Hou and Wu, 1997]. In steady flow, it has similar computational cost as fine‐scale problem if without resorting to parallel computation. [5] Most studies of multiscale finite element method are restricted in a deterministic context. There is no published work to extend the MSFEM to the uncertainty quantification in subsurface flow. In the present study, we combine the multiscale finite element method (MSFEM) with probabilistic collocation method to develop a new framework to stochastic groundwater flow problems. We would like to note that although each of the two techniques has been developed, this paper introduces a framework to combine them to derive a multiscale method for stochastic analysis. We choose the probabilistic collocation method to decouple the stochastic system because of possible fewer realizations resulted from the PCM. Owing to the potential non‐ Gaussian property of longitudinal velocity, this paper focuses on the probability distribution functions (PDFs), rather than just the first two statistical moments, of head and velocity. Unlike in most of the other stochastic approaches,
the PDFs can be constructed from the multiscale algorithm. For simplicity, this combined approach is referred to as the multiscale probabilistic collocation method (MSPCM). [6] This paper is organized as follows: A mathematical description of the groundwater flow is given in section 2. A review to probabilistic collocation method is presented in section 3, including a simple introduction to the Karhunen‐ Loeve expansion and polynomial chaos expansion. Section 4 describes the MSFEM technique for the resulting collocation equations. Several illustrative examples are given in section 5 and some conclusions in section 6.
2. Problem Description [7] Let W and Y denote the flow domain and a suitable probability space, respectively. The two‐dimensional confined flow in heterogeneous medium with a spatially varying conductivity can be described with the following stochastic partial differential equation: r½ K ðx; !Þrhðx; !Þ þ g ðxÞ ¼ 0 ; x 2 W; ! 2 Y
ð1Þ
subject to the boundary conditions hðx; !Þ ¼ H ðxÞ;
x 2 GD
K ðx; !Þrhðx; !Þ nðxÞ ¼ QðxÞ;
ð2Þ x 2 GN
ð3Þ
where K(x) is hydraulic conductivity; h(x,w) is the hydraulic head; g(x) is the source/sink term; H(x) and Q(x) are the prescribed head on the Dirichlet boundary GD, and the flux across the Neumann boundary GN, respectively; n(x) is the outward unit vector normal to the boundary G = GN[GD. In this study, Y = lnK is treated as multi‐Gaussian field with known mean and covariance function. [8] Monte Carlo simulation is a straightforward but computationally demanding method for solving equation (1) by dealing with a large number of conductivity realizations. An alternative is to reformulate the original stochastic equation by representing the variable h and Y with different kinds of expansions. In previous studies, the perturbative expansion [Dagan, 1989; Gelhar, 1993; Zhang, 1998, 2002; Zhang and Winter, 1998], the Taylor expansion [Dettinger and Wilson, 1981; Protopapas and Bras,1990], and the Karhunen‐Loeve and polynomial expansions [Ghanem and Spanos, 1991; Ghanem and Dham, 1998; Zhang and Lu, 2004] have been extensively explored. In the present work, the random input (the log conductivity field) is decomposed by a truncated Karhunen‐Loeve expansion in terms of linear summation of uncorrelated random variables [Ghanem and Spanos, 1991]. And the random outputs (here, hydraulic head and velocity) are represented with the polynomial chaos expansion [Wiener, 1938] or the generalized polynomial chaos expansions [Xiu and Karniadakis, 2003].
3. Probabilistic Collocation Method [9] In this part, we present a brief discussion on the Karhunen‐Loeve expansion and the polynomial chaos expansion. A detailed description about these theories can be found in work by Ghanem and Spanos [1991], Wiener [1938], and Xiu and Karniadakis [2003].
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3.1. Karhunen‐Loeve Expansion [10] Assume a random field Y(x, w) to have mean hY(x)i and variance s2Y. Its Karhunen‐Loeve expansion can be expressed as Y ðx; !Þ ¼ hY ðx; !Þi þ
1 pffiffiffiffi X i fi ðxÞi ð!Þ
ð4Þ
i¼1
where x i(w) are orthogonal random variables i.e., hxi(w)i = 0 and hxi(w) xi(w)i = d ij, li and fi(x) are the eigenvalues and the eigenfunctions, respectively, and can be solved from the following Fredholm equation:
where Gp(xi1,…,xip) denotes polynomial of order p, and a0, ai1,… are the deterministic coefficients. The polynomials should be chosen from the hypergeometric polynomials of the Askey scheme to maintain the best convergence [Xiu and Karniadakis, 2003]. For example, for Gaussian, Gamma, and uniform variables the Gp(x i1,…,xip) should be chosen as Hermite, Laguerre, and Legendre polynomials, respectively. For example, the multidimensional Hermite Polynomials corresponding to Gaussian random variables is given as
Z
W
CY ðx1 ; x2 Þf ðx2 Þdx2 ¼ f ðx1 Þ
where CY(x1. x2) is the covariance function of random input Y(x). Equation (5) may be solved analytically for special cases such as a separable covariance function defined on a rectangular domain. However, in multidimensions with general covariance functions, (5) must be solved numerically. Ghanem and Spanos [1991] presented a Galerkin type procedure for solving equation (5). This procedure is computationally demanding because it Rrequires high‐ R dimensional numerical integrations such as W WCY(x1, x2) f1(x1)f2(x1)dx1dx2. Recently developed multipole method provides a potentially efficient way to implement KL expansion numerically [Schwab and Todor, 2006]. If Y(x) is further restricted to a Gaussian field, {x i} are a set of independent Gaussian random variables. Representation of non‐Gaussian field using the KL expansion requires uncorrelated standardized random variables xi(w) such that equation (4) produces the desired non‐Gaussian marginal distribution [Li et al., 2007]. In practice, a truncated KL expansion is used. The number of termsPretained, M, 2 depends on the spectrum energy measured by M i¼1 li/(s YD), where D is the area of two‐dimensional flow domain. In a previous work, we provided some guidance on M under various correlation lengths [Chang and Zhang, 2009]. It is noted that for the case with high ratio between domain length and correlation length (L/g), M needs to be maintained at a big value. 3.2. Polynomial Chaos Expansion [11] The polynomial chaos expansion (PCE) of stochastic process was first introduced by Wiener [1938]. In the original study, the polynomial chaos comprised Hermite polynomials in Gausssian random variables. Though the convergence rate of the Hermite chaos is exponential for Gaussian variables and related variables (e.g., lognormal), the convergence is considerably slower for other kinds of variables (e.g., Gamma and uniform). Xiu and Karniadakis [2003] proposed some generalized polynomial chaos expansions for different types of input variables. The general polynomial chaos expansions can be represented as hðx; t; !Þ ¼ a0 ðx; tÞ þ
1 X i1 ¼1
þ
i1 1 X X i1 ¼1 i2 ¼1
þ
ai1 ðx; tÞG1 ði1 ð!ÞÞ
ai1 i2 ðx; tÞG2 ði1 ð!Þi2 ð!ÞÞ
i1 X i2 1 X X i1 ¼1 i2 ¼1 i3 ¼1
1 T Hp i1 ; . . . ; ip ¼ ð1Þn e2x x
ð5Þ
ai1 i2 i3 ðx; tÞG3 ði1 ð!Þi2 ð!Þi3 ð!ÞÞ þ . . . ð6Þ
@n 1 T e2x x @i1 @ip
ð7Þ
[12] In this study, the random log conductivity is assumed to be represented by a truncated KL expansion via a finite number of uncorrelated Gaussian random variables [x 1,,…, x M,], where M is the number of terms retained in the KL truncation. [13] The solution to the stochastic partial differential equation (1) can be written as hðx; !Þ ¼ hðx; xÞ; ¼ ½1 ; . . . ; M ;
ð8Þ
[14] Since the log conductivity is assumed to be second‐ order Gaussian process, we approximate the hydraulic head h(x,t) by a Hermite chaos in this study. For notational simplicity, equation (6) can be rewritten as hðx; xÞ ¼
N X
ci ðxÞy i ðxÞ
ð9Þ
i¼1
where there is a one‐to‐one correspondence between the functions Gp(xi1,…,x ip) and yi(x) as well as their corresponding coefficients, x is a vector of dimension M, and N can be expressed as N¼
ð p þ M Þ! M !p!
ð10Þ
[15] As can be observed from equation (10), a large M leads to a tremendously large N. This property limits the application of PCE in a high‐degree‐of‐freedom system where hundreds of uncorrelated random variables are included to characterize the stochasticity. In order to alleviate rapid increase of N, Li and Ghanem [1998] proposed adaptive polynomial chaos expansion. This technique removes the insignificant terms in equation (9) by detecting the contribution of each coefficient ci(x) to the whole system. Blatman and Sudret [2008] also developed sparse polynomial chaos expansion to obtain a rather low number N. Recently, Li et al. [2009] proposed the leading term approximation to polynomial chaos expansion by neglecting all the cross terms such as xixj (i ≠ j). All existing study about the adaptive (or sparse) PCE is based on the numerical experiments, while no rigorous convergence analysis is provided. In section 5, we will make use of the leading term approximation to handle the case with a large truncation term M.
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3.3. Implementation of Probabilistic Collocation Method [16] For the ease of description, we substitute the following equation for stochastic differential equation (1), Lhðx; Þ þ g ðxÞ ¼ 0
ð11Þ
where L is an operator involving differentiations in space. We seek the approximation b h (x, x)using polynomial chaos expansion. Minimizing the residual of equation (11) in the random space yields Z h i Lb hðx; Þ þ g ðxÞ !j ðxÞðxÞd ¼ 0
ð12Þ
where wj(x) is the weighting function, and r(x) is the joint probability density function of x. By choosing Dirac delta function d(x − xj) as the weighting function, equation (12) becomes Lb h x; j þ g ðxÞ ¼ 0
ð13Þ
which leads to a series of independent equations at given collocation points xj. Substituting the KL expansion of Y(x) into the flow equation results in "
# " # M pffiffiffiffi X ^ i fi ðxÞi;j rh x; xj r exp Y x; xj þ i¼1
þ g ðxÞ ¼ 0
;x 2 W
ð14Þ
where xj = (x 1,j,x 2,j…,x M,j)T (j = 1, 2, …, N). [17] There are many algorithms to generate the sets of collocation points. In this work, we employ the probabilistic collocation technique [Tatang et al., 1997; Isukapalli et al., 1998]. Other algorithms include tensor product method, Smolyak algorithm and Stroud’s cubature method [Xiu and Hesthaven, 2005; Ding et al., 2008; Chang and Zhang, 2009]. The total number of collocation points depends on the number of coefficients ci(x) to be determined, or specifically dependent on the M and p in equation (10). Similar to Monte Carlo simulation, the PCM works with independent equations at a discrete set of points in the random space. Monte Carlo simulation obtains realizations by random sampling while the PCM generates realizations on the basis of the Karhunen‐Loeve expansion and polynomial chaos expansion. Also, the realizations in PCM carry different weights [Li and Zhang, 2007] while the realizations in Monte Carlo simulation are equally probable. We also note that in this study, upscaling in PCM and Monte Carlo simulation is performed for each individual realization. The computational efficiency may be enhanced by constructing basis functions from a small family of realizations that are sufficiently scattered in stochastic space [Aarnes and Efendiev, 2008].
4. Derivation of Multiscale Finite Element Method [18] The collocation method reduces the stochastic problem into a series of deterministic collocation equations with spatially varying hydraulic conductivity, which can be solved by any existing deterministic solvers. However, care must be taken to ensure the quality of numerical solutions by studying the sensitivities of grid resolution. In Monte
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Carlo simulation, the convergence of statistical moments within respect to mesh size (D) has been investigated extensively. The characteristic mesh size (D) should be inversely related to the level of spatial variability, e.g., the standard deviation of log hydraulic conductivity sY. Ababou et al. [1989] suggested the following relationship between g Y/D (the number of cells within each correlation length g Y) and sY:g Y /D ≥ 1 + sY. Several studies indicated that at least five cells with each correlation length should be used in order to minimize the filtering (local averaging) effect of discretization scale [Ababou et al., 1989; Bellin et al., 1992; Van Lent and Kitanidis, 1996; Salandin and Fiorotto, 1998]. Since a larger number of cells may be required to resolve the small‐scale heterogeneity as sY increases, performing a fully resolved calculation on fine girds becomes computationally demanding. Although these studies about the discretization level are based on Monte Carlo realization, numerical examples in section 5 will show similar conclusions for probabilistic collocation method. [19] We consider a computational scheme that captures the large‐scale structures of the solutions on a coarse‐scale mesh without resolving all the small‐scale features. The multiscale finite element method [Hou and Wu, 1997; Hou et al., 1999] provides an effective way to account for the small‐scale features on a coarse‐scale mesh. The basic idea of this method is to construct the coarse‐scale basis function, which implies the fine‐scale heterogeneity, while the standard finite element method constructs the coarse‐scale basis function according to the spatial coordinates of the nodal points. By using the coarse‐scale basis function and reformulating equation (14) on the coarse mesh, the new matrix form can be developed. Below we discuss how the deterministic collocation equation (14) can be solved with the multiscale finite element method. 4.1. Basis Functions in Multiscale Finite Element Methods [20] Assume the flow region W to be divided by a coarse mesh with Nc triangular elements. Further, let each coarse element E be discretized using Nf small triangular elements (see Figure 1, wherein the dash lines correspond to fine mesh and the solid line to the coarse mesh. Only one refinement over a coarse element is given in Figure 1). We consider a coarse element E with three vertexes I, J and K, and define a set of basis functions {i, i = 1,…,3} at each vertex on the coarse element. The boundary of E is denoted by ∂E. [21] The multiscale finite element method (MSFEM) incorporates the fine‐scale information into coarse‐scale basis function, which can be solved from [Hou and Wu, 1997] r K x; xj r i x; xj ¼ 0; in E
ð15Þ
i ¼ gi on @E
ð16Þ
M pffiffiffiffi P i fi(x)xi,j], gi is the where K(x,xj) = exp[hY(x,x j)i + i¼1
boundary function of coarse‐scale basis function i along ∂E. The function gi can be chosen to vary linearly along ∂E or to be the solution of the reduced elliptic problems on each side of ∂E with boundary conditions 1 and 0 at the two end points [Hou and Wu, 1997; Hou et al., 1999]. To be consistent with
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dition may deteriorate for problems in which global effects are crucial. Chu et al. [2010] developed a new oversampling technique to reflect the actual flow boundary conditions and other global information (such as sink/source term, channel flow). Despite the superiority of oversampling technique, this work only employs oscillatory boundary condition due to the relatively simple flow considered. [24] One important feature of MSFEM is that the construction of the basis function is a local operation within the coarse element. Thus, the construction in one coarse element is independent of that in other coarse elements. As a result, a fully resolved simulation is broken into many smaller and independent pieces. 4.2. Formulation of Multiscale Finite Element Methods [25] The variational form of equation (14) at collocation points xj can be given as Z W
Figure 1. Schematic showing fine‐ and coarse‐scale elements.
K x; xj rh x; xj n dx þ
@g1 x; xj @ K x; xj ¼0 @x @x
Z g1 ¼ 1 jG1 ¼
x2
x
d K ; xj
,Z
x2
x1
d K ; xj
ð18Þ
[22] Other boundary conditions along border IK and KJ are obtained in the same way. In general, gi is oscillatory due to the oscillation in K(x,x j). When gi on ∂E is available, equation (15) can be solved numerically. As usual, it is required that 3 X
i x; xj ¼ 1
ð19Þ
i¼1
For the stochastic problem discussed here, the boundary conditions and coarse‐scale basis functions must be updated at each set of collocation points x j. [23] It has been observed that the boundary condition gi on the coarse‐scale basis function is an important issue in the multiscale finite element method [He and Ren, 2006b; Efendiev and Hou, 2007]. The oscillatory boundary condition seems to be useful for many problems [Hou et al., 1999]. However, the validity of oscillatory boundary con-
g ðxÞn dx ¼ 0
; x 2 W ð20Þ
Nc X
i ðx; xu Þ^ hi ðx; xu Þ
ð21Þ
i¼1
where Nc is the total number of coarse nodes on the flow region W. [27] Since elliptic problem (15) on coarse element can be solved by the standard finite element method, each coarse basis function can be expressed as Nf X n x; xj ¼ ’k x; xj ckn
ð17Þ
The equation has the following analytical solution
W
where n is the multiscale basis function. h (x,xu) on the coarse [26] The approximated solution b mesh is written as ^ hðx; xu Þ ¼
existing literatures, we call the former boundary condition as linear boundary condition and the latter one as oscillatory boundary condition. As an example of oscillatory boundary condition, we focus on the specific coarse element E in Figure 1. The coordinates of vertexes I, J and K are (x1, y1), (x2, y2), and (x3, y3), respectively. The reduced elliptic equation along G1 (border IJ) can be written as [Hou and Wu, 1997]
Z
ð22Þ
k¼1
where ’k (x, xu) is the standard bilinear basis function at collocation points xj, Nf is the number of fine nodes on a coarse element. For brevity, the arguments (x, xj) will be omitted in the following derivation. [28] For a specific coarse mesh with Nc coarse elements, equation (20) is given in the matrix form Ahj ¼ B
ð23Þ
where hj = [b h1,…,b hNc]T, subscript j denotes the solution at collocation points x j, B is the vectors with the nth bn (n = 1, …, Nc), and A is the matrix with elements amn (m, n = 1,…, Nc). By estimating the multiscale coarse basis function with standard bilinear basis functions, bn and amn can be rewritten as bn ¼ 1 ð x; y; t Þ
3 XX e
amn ¼
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e
p¼1
Z cnp
Ge
’pE;e dG
n Tr cm p cq
p¼1 q¼1 r¼1
p
@’E;e @’qE;e @’pE;e @’qE;e þ dx ’r @x @x @y @y We Z
ð24Þ
ð25Þ
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where w could be h, vx and vy. As in Li and Zhang [2007, 2009], the probability distribution functions of h, vx and vy can be obtained by sampling their respective polynomial chaos expansions. [32] To extend current simulation to the stochastic analysis of solute transport, one can expand the concentration with polynomial chaos expansion, and each velocity realization corresponding to each collocation point is used to run the advection‐dispersion equation. The concentration statistics are then obtained in the same way as head and velocity [Zhang et al., 2010]. Lin and Tartakovsky [2009] recently also presented the detailed steps for solute transport based on probabilistic collocation method. Moreover, a multiscale simulation of solute transport may need additional multiscale approach for advection‐dispersion equation. Two different approaches have been reported by Efendiev and Hou [2007] to implement the multiscale simulation of solute transport.
5. Numerical Experiments Figure 2. Schematic diagram of the simulated region (cases 1–3).
where the summation with respect to E is an finite element assembly. Equation (25) or its extension to 3D problem is a fast computation since it only involves basic arithmetical calculation. 4.3. Reconstruction of the Fine‐Scale Solution [29] Given the coarse solution hj and coarse‐scale basis function i, the fine‐scale solution rj is reconstructed by rj ¼
Nc X
hj i
ð26Þ
i¼1
[30] To compute the statistical moments on the fine grids, the fine‐scale head should be reconstructed at each set of collocation points x j. After obtaining the fine‐scale head, we then compute the fine‐scale velocity by using the reconstructed fine‐scale head realizations. 4.4. Postprocessing [31] After obtaining the head h, longitudinal velocity vx and transverse velocity vy, the coefficients in equation (9) can be determined by ZCðxÞ ¼ WðxÞ
ð27Þ
where Z, of elements zij = yj(xi), is a space‐independent matrix of dimension N*N, C(x) is a space‐dependent vector of coefficients ci(x), and W(x) is a space‐dependent vector of h(x), vx(x) or vy(x). The mean and the variance of the head, longitudinal velocity and transverse velocity can be written as hwðxÞi ¼ c1 ðxÞ
2w ¼
N X i¼2
c2i ðxÞ y 2i
ð28Þ
ð29Þ
[33] In this section, we present some numerical experiments to compare the performance of multiscale probabilistic collocation method (MSPCM), standard probabilistic collocation method (PCM), and Monte Carlo simulation (MCS). In all cases the log conductivity is assumed to be a second‐ order Gaussian field with known mean and covariance function. 5.1. Cases 1 and 2 [34] In case 1, the flow domain is a square of 200 m by 200 m. As shown in Figure 2, the western and eastern borders are specified as Dirichlet boundaries with heads of 6 m and 4m, respectively; the northern and southern borders are specified as impervious boundaries. In all the following examples, the log conductivity is assumed to follow a separable exponential covariance function, " CY ðx; yÞ ¼
2Y
exp
jx1 x2 j ðY Þ
1
jy1 y2 j ðY Þ
2
# ð30Þ
where g (Y) and g (Y) are correlation lengths at x and y 1 2 directions, respectively. We set hYi = 2,s2Y = 1, and g (Y) 1 = g (Y) 2 = 20 m. The Karhunen‐Loeve expansion is truncated at M = 50, and second‐order polynomial chaos expansion is used to expand head and velocities, thus there are totally 1326 collocation points for the PCM. A convergence analysis of Monte Carlo simulation reveals that around 4000 realizations are needed to obtain stable PDF for this problem. To ensure statistical convergence, the reference solution is obtained by Monte Carlo simulation with 10,000 realizations. [35] Since there is no previous work showing the role of ratio h of correlation length (g Y) to mesh size (D) in the PCM, we experimentally test the convergence property of head and velocity PDFs with respect to h. As an example, Figure 3 shows the PDFs of head, longitudinal velocity and transverse velocity at node (80, 80) for four ratios h = 1, 2, 5, 10. No obvious difference is seen for the head PDFs with different mesh sizes (Figure 3a). However, the estimated PDF of longitudinal velocity shows a mild deviation when each correlation length is covered by one mesh (Figure 3b). And the PDF of transverse velocity seems to be more sen-
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Figure 4. One conductivity realization from PCM (case 1). fine triangular elements in the x and y direction, respectively, on which each coarse basis function is constructed. The results from the “accurate PCM,” MSPCM with CL = 25 and CL = 100 are named as PCM‐100, MSPCM‐100‐20 and MSPCM‐100‐10, respectively. [37] To examine the implementation of the multiscale finite element method, one PCM realization, shown in Figure 4, is solved by PCM‐100, MSPCM ‐100‐10 and MSPCM ‐100‐20. Figure 5 presents the head realizations from PCM‐100, MSPCM‐100‐10 and MSPCM‐100‐20. It is seen that MSPCM‐100‐20 gives satisfactory head contours, while the result from MSPCM‐100‐10 shows a slight deviation from the “accurate PCM solution.” Longitudinal velocity realizations from the accurate PCM (PCM‐100) and
Figure 3. The PDF of (a) head, (b) longitudinal velocity, and (c) transverse velocity at point (80, 80) for different h (case 1).
sitive to the discretization (Figure 3c). All PDFs become stable after h reaches 5. [36] We regard the solution from the PCM with Nx = Ny = 100 as the “accurate PCM solution,” where Nx and Ny are the numbers of elements in x and y direction. For the MSPCM, we test two different coarsening levels CL = 25 and 100. That is to say, each coarse mesh contains 20 and 10
Figure 5. Head realizations from PCM‐100 (solid lines), MSPCM‐100‐10 (dotted lines), and MSPCM‐100‐20 (long‐dashed lines) (case 1).
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Table 1. Relative Error of Water Balance From Different Methods
Region I Region II Global
PCM‐100
MSPCM‐100‐10
MSPCM‐100‐20
0.28% 1.41% 0.06%
0.48% 1.42% 0.10%
0.24% 1.43% 0.05%
multiscale PCM (MSPCM‐100‐10 and MSPCM‐100‐20) are compared in Figure 6. Velocity realization from MSPCM‐100‐10 (Figure 6b) only reveals some basic features of the real longitudinal velocity realization, while MSPCM‐100‐20 (Figure 6c) is able to capture all the main features and parts of the velocity details. The local mass balance in domain I and domain II as well as the global mass balance is investigated for PCM‐100, MSPCM‐100‐10 and
Figure 6. Longitudinal velocity realizations from (a) PCM‐100, (b) MSPCM‐100‐10, and (c) MSPCM‐100‐20 (case 1).
Figure 7. The PDF of (a) head, (b) longitudinal velocity, and (c) transverse velocity at node (80, 80) from different methods (case 1).
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Table 2. Error R1 and R2 in Case 1 R1 Quantiles
P30
MSPCM‐100‐10 MSPCM‐100‐20
0.04% 0.04%
MSPCM‐100‐10 MSPCM‐100‐20
5.47% 3.31%
MSPCM‐100‐10 MSPCM‐100‐20
23.87% 8.42%
R2
P50
P80
Error for Head PDF 0.02% 0.03% 0.03% 0.05% Error for Longitudinal Velocity PDF 0.88% 9.34% 0.90% 3.13% Error for Transverse Velocity PDF ‐ 1.61% ‐ 1.90%
P30
P50
P80
0.07% 0.07%
0.15% 0.06%
0.05% 0.03%
8.31% 6.09%
0.29% 2.09%
10.38% 4.23%
20.23% 4.05%
‐ ‐
2.83% 0.63%
MSPCM‐100‐20. Relative errors of mass balance from these schemes are shown in Table 1. It is seen that MSPCM‐ 100‐20 achieves similar relative errors of mass balance as accurate PCM (PCM‐100), while MSPCM‐100‐10 brings about relatively large errors in domain I and in the global domain. On the basis of the observations from head and velocity realizations and the mass balance, we may conclude that the multiscale finite element method with proper coarsening level is able to obtain the accurate flow field and retain a satisfactory mass balance. [38] Simulated PDFs of head, longitudinal velocity and transverse velocity at node (80, 80) are presented in Figure 7. The head PDF from MSPCM‐100‐10 shows agreeable match with the reference solution obtained with 10,000 realizations, while the PDFs of longitudinal and transverse velocity from MSPCM‐100‐10 show some discrepancy. It indicates a multiscale simulation with coarsening level as high as 100 may lead to large errors for the PDFs of velocity. Three quantiles P30, P50 and P80 are read from the PDFs. The relative error (R1) between the quantiles from the MSPCM and those from the PCM are calculated to estimate the capability of MSPCM in recovering the accurate PCM. The relative error (R2) between the quantiles from the MSPCM and those from Monte Carlo simulation are calculated to estimate the capability of MSPCM in approximating the reference solution. R1 and R2 are listed in Table 2. It can be noticed that R1 and R2 of P30, P50 and P80 from the multiscale PCM grow bigger for the velocity PDFs. Also, large relative errors in the transverse velocity PDF are observed for MSPCM‐100‐10. [39] The mean and variance of head and velocity along the intersection A‐B (y = 80, 0 ≤ x ≤ 200) are given in Figures 8 and 9. Dashed columns denote the locations of the coarse nodes for MSPCM‐100‐10. The points intersecting these columns and the mean (variance) curve are the results solved from the coarse‐scale equation of MSPCM‐100‐10, and the other points in the curve are reconstructed by equation (26). The relative error between the moments from the multiscale PCM and the moments from the accurate PCM is defined as REr ¼
Nn 1 X jvms vf j vf Nn i¼1
ð31Þ
where Nn is the total node number, vms is the solution from the MSPCM, and vf is the solution from the fully resolved (accurate) PCM. As shown in Figure 8, the MSPCM gives quite good mean head. MSPCM‐100‐10 and MSPCM‐100‐ 20 lead to an error of 0.05% and 0.00%, respectively, while
Figure 8. The simulated (a) mean head, (b) mean longitudinal velocity, and (c) mean transverse velocity along intersection A‐B from different methods (case 1).
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Figure 10. Comparison of reconstructed longitudinal velocity (along cross section A‐B) derived from different methods (case 1). respectively, 21.31% and 16.71% error for the longitudinal velocity variance, and 35.96% and 22.71% for the transverse velocity variance. With a high coarsening level of 100, MSPCM‐100‐10 results in a large deviation from the accurate variance. Accuracy is improved by decreasing CL
Figure 9. The simulated variance of (a) head, (b) mean longitudinal velocity, and (c) mean transverse velocity along intersection A‐B from different methods (case 1). the relative errors increase to the respective 3.00% and 1.39% for the mean longitudinal velocity. Although the multiscale probabilistic collocation method is able to duplicate the mean properties at a high accuracy, the relative errors for head and velocity variances grow bigger. As illustrated in Figure 9, MSPCM‐100‐10 and MSPCM‐100‐ 20 produce 12.00% and 0.53% error for the head variance,
Figure 11. The PDF of (a) longitudinal velocity and (b) transverse velocity at node (100, 80) from different methods (case 1).
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Figure 12. The PDF of (a) head, (b) longitudinal velocity, and (c) transverse velocity at node (88, 80) from different methods (case 2).
to 25 (MSPCM‐100‐20). An interesting phenomenon in Figures 8 and 9 is that the simulated velocity moments from the multiscale algorithm show an abrupt oscillation around the coarse nodes while the fully resolved PCM exhibits continuous profiles. As shown in Figures 8 and 9, the higher the coarsening level, the stronger is the oscillation. Figure 10 is given to further illustrate this phenomenon. Figure 10 compares the reconstructed longitudinal velocity realization by different methods but with the same conductivity realization as input. The velocity realizations derived from PCM‐100, MSPCM‐100‐10 and MSPCM‐ 100‐20 are extracted from Figures 6a, 6b, and 6c, respectively. We also add the longitudinal velocity realizations computed from PCM‐10 and PCM‐20 which resolve the flow field only by Nx = Ny = 10 and Nx = Ny = 20 while without employing the multiscale algorithm. It is seen that compared to PCM‐10 and PCM‐20, MSPCM‐100‐10 and MSPCM‐100‐20 (especially MSPCM‐100‐20) lead to an improved agreement with the reference velocities at the coarse nodes. The fine‐scale velocities reconstructed from the multiscale simulations also exhibit discontinuity. Since the fine‐scale velocities are computed by using the fine‐ scale heads, a highly accurate reconstruction to fine‐scale heads seems necessary to obtain an accurate reconstruction of the fine‐scale velocity and velocity moments. Although in Figure 5, MSPCM‐100‐10 only results in very small errors for reconstructing the head realization, the errors are amplified for reconstructing the velocity and its moments (Figures 8–10), indicating that a lower coarsening level (lower than the requirement for head and head moments) is helpful to alleviate the abrupt oscillation in the velocity and its moments. [40] We also examine the accuracy of estimated PDFs at fine‐scale nodes, which are approximated with the reconstructed fine‐scale solutions. The fine‐scale node (100, 80) in MSPCM‐100‐10 is discussed as an example. It is noted that in MSPCM‐100‐20, node (100, 80) is a coarse‐ scale node. The estimated PDFs of longitudinal and transverse velocity at node (100, 80) are shown in Figure 11. In comparison to the estimation at coarse‐scale node (80, 80) in Figure 7, it seems that MSPCM‐100‐10 has the similar performance when predicting the PDFs for fine‐scale nodes. [41] We also investigate a case (case 2) with s2Y = 2 and other parameters being the same as in case 1. Comparisons of PDFs at node (88, 80) from the Monte Carlo simulation, PCM‐100, MSPCM‐100‐10, and MSPCM‐100‐20 are shown in Figure 12. Compared to MSPCM‐100‐20, all PDFs from MSPCM‐100‐20 have an improved match with those from PCM‐100. The relative errors R1 and R2 of P30,
Table 3. Error R1 and R2 in Case 2 R1 Quantiles
P30
MSPCM‐100‐10 MSPCM‐100‐20
0.21% 0.06%
MSPCM‐100‐10 MSPCM‐100‐20
5.98% 1.52%
MSPCM‐100‐10 MSPCM‐100‐20
16.89% 8.65%
P50
R2 P80
P30
Error for Head PDF 0.10% 0.15% 0.25% 0.02% 0.08% 0.10% Error for Longitudinal Velocity PDF 12.78% 3.43% 9.78% 0.15% 0.58% 6.39% Error for Transverse Velocity PDF ‐ 37.05% 3.64% ‐ 18.79% 13.92%
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P50
P80
0.29% 0.17%
0.13% 0.06%
6.86% 5.79%
6.78% 2.07%
‐ ‐
47.13% 27.53%
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summarized as follows: (1) The multiscale PCM with a proper coarsening level is able to reproduce the results of the fine‐scale PCM; (2) It is easier to obtain agreeable head PDF than the velocity PDF. With or without the multiscale technique, the velocity PDFs from the probabilistic collocation method show difference to the reference solution when the variance is higher (than 2). This difference is caused by the properties of PCM, because the problem of higher variance usually requires a higher‐order polynomial chaos expansion [Chang and Zhang, 2009] while the current simulation only employs a second‐order expansion. However, a complete fourth‐order PCM simulation with M = 50 requires 316251 sets of collocation points. Obviously, the number of collocation sets is even larger than the required number of realizations in Monte Carlo simulation. It is expected that the collocation sets under high‐order expansion can also be reduced in a similar fashion to the work of Li and Ghanem [1998], Blatman and Sudret [2008], and Li et al. [2009]. However, this dimensionality reduction technique has not been tested under the condition of huge variances. This task is beyond the scope of the current study but shall be investigated in future work.
Figure 13. The PDF of (a) head, (b) longitudinal velocity, and (c) transverse velocity at node (62, 100) from different methods (case 3). P50 and P80 from the MSPCM are given in Table 3. The error reductions from the multiscale simulation with a lower coarsening level are observed. In addition, we find that although the velocity PDFs from the MSPCM agree well with the fully resolved PCM, all the velocity PDFs from the accurate PCM and the multiscale PCM show a mild difference from the reference solutions. The example with s2Y = 3 (not shown) also confirm these findings, which can be
5.2. Case 3: High L/g Ratio [42] The stochastic collocation method suffers from the “curse of dimensionality” for the cases of high L/g ratio. As the ratio L/g increases, the required random dimensionality M can reach hundreds, which leads to the rapid increase of the number of collocation points. As observed by Li and Ghanem [1998] and Blatman and Sudret [2008], some coefficients in the polynomial chaos expansion play a minor role due to their insignificant contributions to uncertainty. We recently proposed the leading term approximation to polynomial chaos expansion so that the computational burden can be reduced for high random dimensionality M [Li et al., 2009]. The leading term approximation ignores the cross terms in the original polynomial chaos expansion. We will use this method to simulate case 3. (Y) [43] In the next case (case 3), we set g (Y) 1 = g 2 = 10m and retain the same remaining parameters as in case 1. The KL expansion is truncated at M = 250, which leads to 1281 sets of collocation points by adopting the leading term approximation, compared to around 5000 realizations needed in the Monte Carlo simulation. We regard the solution from the PCM with Nx = Ny = 200 as the “accurate PCM solution.” Monte Carlo simulation with 10,000 realizations is also run with this mesh to obtain the reference solution. Multiscale simulations with CL = 25 and 100 are tested. There are 40 and 10 coarse triangular elements in x and y direction for CL = 25 and 100, respectively. The results from the “accurate PCM,” MSPCM with CL = 25 and CL = 100 are named as PCM‐ 200, MSPCM‐200‐40 and MSPCM‐200‐20, respectively. [44] The estimated head and velocity PDFs at node (60, 100) are presented in Figure 13. It is seen that PCM‐200 and MSPCM‐200‐40 produce very similar PDFs of head, and longitudinal and transverse velocity. However, MSPCM‐ 200‐20 leads to a larger deviation. Table 4 presents the relative errors R1 and R2 of P30, P50 and P80 from the MSPCM. We see that the multiscale PCM with a lower coarsening level generally brings about a reduction in R1 errors while the reduction to R2 is not obvious. We notice
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Table 4. Error R1 and R2 in Case 3 R1 Quantiles
P30
MSPCM‐100‐10 MSPCM‐100‐20
0.09% 0.15%
MSPCM‐100‐10 MSPCM‐100‐20
1.20% 1.42%
MSPCM‐100‐10 MSPCM‐100‐20
30.65% 12.77%
R2
P50
P80
Error for Head PDF 0.09% 0.11% 0.16% 0.16% Error for Longitudinal Velocity PDF 7.31% 11.73% 4.22% 5.08% Error for Transverse Velocity PDF ‐ 5.07% ‐ 1.69%
that in case 1 both R1 and R2 errors can be reduced by decreasing the coarsening level and the only difference between the setups of case 1 and case 3 is the correlation length. Hence, the performance of R2 error in case 3 may stem from the difference in correlation scale and the leading term approximation to polynomial chaos expansion. 5.3. Case 4: Composite Porous Media [45] In case 4, the mean log conductivity has five zones: Z1 with hY1i = 0 (20 ≤ x ≤ 40, 0 ≤ y ≤ 20), Z2 with hY2i = 4 (40 ≤ x ≤ 60, 100 ≤ y ≤ 140), Z3 with hY3i = 4 (100 ≤ x ≤ 120, 40 ≤ y ≤ 80), Z4 with hY4i = 0 (140 ≤ x ≤ 160, 160 ≤ y ≤ 80), and Z5 with hY5i = 2 (the remaining domain). We assume the perturbation of log conductivity in the five zones honors the same covariance function as in case 1 with variance s2Y = 1. The mean flow is nonuniform because of the spatially varying mean log conductivity. The KL expansion is truncated at M = 50 as in case 1, which leads to 1326 sets of collocation points, compared to around 8000 realizations needed in Monte Carlo simulation. This case is simulated by the PCM‐100, MSPCM‐100‐20, MSPCM‐100‐10, and Monte Carlo simulation. [46] It is of great interest to examine the capability of multiscale algorithm in simulating the nonuniform flow field. We take a careful examination of the area around Z3 where the velocity shows a rapid change in the y direction. The mean flow fields from the different methods are plotted in Figure 14. Flow fields in Figures 14b and 14c are reconstructed from their respective coarse‐scale solutions. It is seen that MSPCM‐100‐20 obtains a decent flow field and the reconstructed velocity in the middle part of mesh (100 ≤ x ≤ 120, 80 ≤ y ≤ 100) shows a slightly abrupt orientation change compared to Figure 14a. We notice that the middle part is close to the interface of inclusion Z3, which causes the obvious mismatch when the coarsening level is high. [47] The estimated head and velocity PDFs at node (112, 84) are presented in Figure 15. As shown in Figures 15a and 15b, MSPCM‐100‐20 obtains quite accurate head and longitudinal velocity PDFs. But the transverse velocity PDF (Figure 15c) from MSPCM‐100‐20 shows mild deviation from the reference solution. It is noticed that in Figure 15c the fine‐scale PCM (PCM‐100) provides an agreeable match to the reference solution, so the mismatch of multiscale PCM comes from the multiscale equation. The poor accuracy of transverse velocity PDF is attributed to the fast change of velocity in the y direction (see Figure 14). The orientation of transverse velocity changes rapidly from
P30
P50
P80
0.07% 0.13%
0.09% 0.15%
0.10% 0.15%
4.79% 4.62%
0.59% 3.94%
8.65% 1.76%
16.68% 4.81%
‐ ‐
6.28% 10.05%
downward to upward, so it is more challenging to reconstruct the fine‐scale property of transverse velocity by the multiscale algorithm. Table 5 lists the relative errors R1 and R2 of P30, P50 and P80 from the MSPCM. 5.4. Application to Solute Transport [48] In this subsection, the velocity fields obtained in case 4 are used to solve the solute transport. The boundary conditions for solute transport are specified third type on the left and second type on the right. The total mass flux and dispersive mass flux are zero on boundaries. The local dispersivity in the longitudinal and transverse direction is set to be 20 and 2 m, respectively. Three initial plumes are located at (38 ≤ x ≤ 46, 58 ≤ y ≤ 62), (18 ≤ x ≤ 26, 98 ≤ y ≤ 102), and (98 ≤ x ≤ 106, 118 ≤ y ≤ 122) with initial concentration c0 = 1 g/m3. The simulation employs a constant molecular diffusion coefficients of 5 × 10−4 m2/d and constant porosity of 0.3. The simulation period is 100 days. [49] Two concentration realizations solved by fine‐scale PCM and MSPCM‐100‐20 are compared in Figure 16. The black contours (dashed lines) represent the MSPCM simulation, while the color scale depicts the PCM results. It is seen that velocity fields obtained from standard finite element method and multiscale finite element method lead to almost identical concentration contours. The comparison reconfirms the validity of the flow field reconstructed by MSPCM. Figure 17 shows the mean and standard deviation of concentrations obtained from the fine‐scale PCM and MSPCM simulations. The MSPCM results are in an excellent agreement with those obtained from the fine‐scale PCM. Hence, the multiscale PCM gives very similar results as the traditional fine‐scale PCM. [50] The reference mean and standard deviation of concentrations obtained from MC simulation are also shown in Figure 18. Comparison between Figures 17 and 18 indicates that the mean concentrations obtained from the PCM and MSPCM show a satisfactory match with the MC simulation, while the higher moments start to deteriorate slightly. The high‐order collocation method can be used to improve the accuracy of the higher moments, which has been discussed in detail by Lin and Tartakovsky [2009]. 5.5. Computational Advantages [51] For the fine‐scale PCM, the computer memory needed for solving the flow equation is O(N2fn), where Nfn is the number of nodes in the fine‐scale simulation. The MSPCM requires O(N2c ) memory for coarse‐scale equation
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triangular elements) and the MSPCM (solved by multiscale finite element method with triangular elements) is written as ¼
Nfn2 Nc2
þ Ncg Nf2
ð32Þ
Since N2fn (N2c + NcgNf2), it is clear that the multiscale algorithm offers great savings in the computer memory. In the cases discussed above, the computer memory for fine‐ scale PCM is about 66 times of that for MSPCM‐100‐20 in case 1 (where we have Nfn = 2601, Nc = 121, Ncg = 200, Nf = 21), and around 198 times of that for MSPCM‐200‐40
Figure 14. Velocity field from (a) PCM‐100, (b) MSPCM‐100‐10, and (c) MSPCM‐100‐20. (Nc is the number of nodes in coarse‐scale equation) and * O(N2f ) memory for the construction of coarse‐scale Ncg basis functions, where Ncg is the number of coarse grids, Nf is the number of fine nodes on each coarse element. The ratio between the computer memory for the regular fine‐ scale PCM (solved by standard finite element method with
Figure 15. The PDFs of (a) head, (b) longitudinal velocity, and (c) transverse velocity at node (112, 88) from different methods (case 4).
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Table 5. Error R1 and R2 in Case 4 R1 Quantiles
P30
MSPCM‐100‐10 MSPCM‐100‐20
0.77% 0.63%
MSPCM‐100‐10 MSPCM‐100‐20
84.66% 0.03%
MSPCM‐100‐10 MSPCM‐100‐20
225.02% 41.14%
R2
P50
P80
Error for Head PDF 0.87% 0.94% 0.65% 0.77% Error for Longitudinal Velocity PDF 84.60% 81.24% 1.65% 5.52% Error for Transverse Velocity PDF ‐ 138.09% ‐ 13.87%
P30
P50
P80
0.69% 0.70%
0.74% 0.79%
0.89% 0.82%
98.18% 7.36%
86.13% 0.84%
67.13% 12.87%
251.29% 36.38%
‐ ‐
140.04% 9.46%
in case 3 (where we have Nfn = 10101, Nc = 441, Ncg = 800, Nf = 21). [52] The MSPCM saves a considerable CPU time due to the smaller matrices during the simulation. For each realization, the fine‐scale PCM handles the fine‐scale flow equation with the matrix of size N2fn, while the MSPCM solves the coarse‐scale flow equation with the matrix of size N2c and Ncg local equations with matrices of size N2f . The local equations are independent and can thus be solved in parallel. In this study, all the coarse‐scale basis functions (determined by local equations) are preconditioned by parallelization. The CPU time for the fine‐scale PCM in case 1 and case 3 is 196.9 s and 1375154.1 s, respectively, while the CPU time for MSPCM‐100‐20 (case 1) and MSPCM‐ 200‐40 (case 3) is 6.4 s and 19.5 s, respectively.
6. Conclusions [53] The stochastic collocation method has recently been studied extensively for different problems. Its kernel idea is to represent the random inputs by the Karhunen‐Loeve expansion and decompose the random outputs by polynomial chaos expansions. Then the original stochastic partial differential equation is reduced into a set of deterministic equations. The resulting equations are uncoupled and can be
Figure 16. Contours of concentration realizations at t = 100 d obtained from PCM and MSPCM.
Figure 17. Contours of (a) mean concentration hCi and (b) standard deviation sC at t = 100 d obtained from PCM and MSPCM.
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Figure 18. Contours of (a) mean concentration hCi and (b) standard deviation sC at t = 100 d obtained from MC simulation. solved by an existing simulator. However, in the stochastic collocation method high‐resolution simulations are still needed to resolve the fine‐scale heterogeneity. For large‐ scale problems, it would be computationally expensive to carry out the numerical simulation with highly fine spatial discretizations. In this work, we developed the multiscale probabilistic collocation method by combining the probabilistic collocation method and multiscale finite element technique. Without a fine‐scale simulation, the effect of the small‐scale effect to the flow is included by the coarse‐scale basis functions. At each set of collocation point, a new formulation is derived at the coarse mesh. Several numerical experiments are studied to examine the proposed methodology and to investigate the capability of MSPCM in estimating the PDFs of head and velocity. The reconstructed velocity fields are also used to implement the stochastic
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analysis of solute transport. This study leads to the following major conclusions: [54] 1. The MSPCM combines the advantages of probabilistic collocation method and multiscale finite element technique. The resulting decoupled equations can be computed on the coarse mesh by incorporating the small‐scale heterogeneity into the coarse‐scale basis functions. [55] 2. Multiscale finite element with a proper coarsening level is able to generate satisfactory head fields, while it is more challenging to produce accurate velocity fields. Velocity PDFs from the multiscale probabilistic collocation method show larger errors than the head PDF. The reconstructed fine‐scale velocity and its moments exhibit abrupt oscillation and discontinuity. The reduced accuracy in the velocity field stems from the multiscale algorithm because the fine‐scale velocity filed is reconstructed by interpolation algorithm. However, the numerical examples show that by choosing an appropriate coarsening level, the local oscillation of the reconstructed velocity only results in a slight mass imbalance locally and the reconstructed velocity is able to produce a satisfactory concentration field. [56] 3. Two coarsening levels (25 and 100) were tested in this work. The illustrative examples 1–3 reveal that multiscale simulation with both levels can obtain very accurate PDFs for head and decent PDFs for velocity. That is to say, covering each correlation length with one or two coarse grids in the numerical computation is acceptable for these cases if the multiscale finite element method is employed. A lower coarsening level generally reduces the error (see Tables 1–3). Especially, when there are high‐ or low‐ conductivity inclusions in the porous media, the multiscale simulations with two different levels differ significantly. As shown in the composite porous media case, the performance of MSPCM also depends on the specific flow configuration. If the flow detail is strongly dependent on the local heterogeneity (such as high‐ or low‐permeability inclusion), it is expected that the multiscale simulation with a high coarsening level will result in obvious loss of accuracy due to the rapid change of velocity at a very small scale. The oversampling technique developed by Chu et al. [2010] may be used to circumvent this obstacle. [57] 4. Unfortunately, the MSPCM also suffers from the “curse of dimensionality” as in probabilistic collocation method. The computational cost grows fast with the increase of the random dimensionality retained in the KL expansion. In case 3, we employed the leading‐term approximation for the PCM to perform high‐L/g simulations. However, there is not yet a rigorous convergence theory to support this algorithm. In this study, we try to weaken the “curse” by decreasing the number of terms in polynomial chaos expansion. Recently, several new methods focus on the dimension reduction techniques after realizing that different dimensions (typically KL dimensions) actually have different importance. Nobile et al. [2008] proposed the anisotropic collocation method on the basis of relative importance of each dimension. The analysis‐of‐variance (ANOVA) decomposition was used by Winter et al. [2006] to effectively break the “curse of dimensionality.” Bieri and Schwab [2009] developed a stochastic collocation method based on high Dimensional Model Reduction (HDMR) techniques. Most recently, Foo and Karniadakis [2010] combined multielement probabilistic collocation method with analysis of variance (ANOVA) functional decomposi-
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tion. They proved the new probabilistic collocation method is efficient to handle the cases with up to 600 dimensions. The emergence of these new techniques for probabilistic collocation method enables possible extension of the multiscale probabilistic collocation method developed in this study to large‐scale problems. [58] Acknowledgments. This work is partially supported by Natural Science Foundation of China through grants 40672164, 40701071, 0620631, and 50688901.
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[email protected])
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