Probabilistic collocation method for flow in porous

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WATER RESOURCES RESEARCH, VOL. 43, W09409, doi:10.1029/2006WR005673, 2007

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Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods Heng Li1,2 and Dongxiao Zhang1,3 Received 29 October 2006; revised 7 June 2007; accepted 15 June 2007; published 15 September 2007.

[1] An efficient method for uncertainty analysis of flow in random porous media is

explored in this study, on the basis of combination of Karhunen-Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen-Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment-equation approach based on Karhunen-Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head. Citation: Li, H., and D. Zhang (2007), Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods, Water Resour. Res., 43, W09409, doi:10.1029/2006WR005673.

1. Introduction [2] The prediction of subsurface flow often involves some degree of uncertainty, which can result from the heterogeneity of the medium and our incomplete knowledge about the medium properties. Such uncertainty necessitates a stochastic description of the medium properties, which leads to stochastic partial differential equations governing flow and transport. A large number of studies on flow and transport in random porous media have been performed in the last two decades and many stochastic approaches have been developed [e.g., Dagan, 1989; Gelhar, 1993; Cushman, 1997; Neuman, 1997; Zhang, 2002; Rubin, 2003]. [3] Monte Carlo method is the most common approach to solve the stochastic differential equations numerically. In the Monte Carlo method, one needs to generate a large number of realizations of the random input field. Monte Carlo method is a statistical sampling approach, thus its accuracy depends on the number of realizations that one chooses [e.g., Ballio and Guadagnini, 2004]. The direct sampling Monte Carlo method is conceptually straightfor1

Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing, China. 2 Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, Norman, Oklahoma, USA. 3 Department of Civil and Environmental Engineering, and Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California, USA. Copyright 2007 by the American Geophysical Union. 0043-1397/07/2006WR005673$09.00

ward and easy to implement. However, its main disadvantage is the requirement of large computational efforts, especially for large-scale problems. [4] An alternative is the method based on moment equations [e.g., Graham and McLaughlin, 1989; Neuman, 1993; Zhang and Neuman, 1995; Zhang, 1998, 1999; Guadagnini and Neuman, 1999]. It derives a system of deterministic differential equations governing the statistical moments (usually the first two moments) of the random variables, with the method of perturbation or some type of closure approximation. However, the computational cost for the (conventional) moment equation method is still high for large-scale problems. In computing the hydraulic head covariance to first-order in s2Y, the variability of the log hydraulic conductivity, Y = ln [Ks(x)], one needs to solve a set of deterministic equations on a grid of N nodes for about 2N times: N times for solving the cross-covariance between the hydraulic head and the log hydraulic conductivity, and about N times for the hydraulic head covariance [Zhang, 2002; Lu and Zhang, 2006]. [5] Another alternative approach, the polynomial chaos expansion (PCE) method, is pioneered by Ghanem and Spanos [1991] and applied to various problems of mechanics [Ghanem, 1998; Xiu and Karniadakis, 2002]. This technique includes representing the random variables using polynomial chaos basis and deriving appropriate discretized equations for the expansion coefficients using the Galerkin technique. The polynomial chaos expansion allows high order approximation of random variables and possesses the property of fast convergence under certain conditions. However, the deterministic coefficients of the polynomial

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LI AND ZHANG: PROBABILISTIC COLLOCATION METHOD FOR FLOW IN POROUS MEDIA

chaos expansion are governed by a set of coupled equations, which are difficult to solve when the number of coefficients is large (owing to a large random dimensionality or a high polynomial order). [6] The moment equation approach based on KarhunenLoeve expansion (KLME) is recently developed by Zhang and Lu [2004] for single phase flow in porous media and extended to unsaturated and multiphase flows [Yang et al., 2004; Chen et al., 2005, 2006], flow in nonstationary random hydraulic conductivity fields [Lu and Zhang, 2007], and solute transport [Liu et al., 2007]. The log hydraulic conductivity is represented with the KarhunenLoeve expansion and the flow and transport quantities such as the hydraulic head, velocity, and concentration are decomposed by the perturbation expansions. Unlike the PCE, the KLME leads to uncoupled equations governing the coefficients, which can be used to construct the required statistical moments. With the KLME approach, high order terms of the mean and variance of hydraulic head (or other flow and transport quantities) can be obtained with relatively small computational efforts [Zhang and Lu, 2004] and can be easily implemented with existing simulators [Lu and Zhang, 2006; Liu et al., 2006, 2007]. The KLME approach is generally more efficient than the traditional moment equation approach. Although it is derived with polynomial perturbation expansions, the KLME is found to be applicable to large variability in log hydraulic conductivity for single phase flow. However, its validity range is not yet clear for multiphase flow. [7] In this study, we present a new efficient approach, probabilistic collocation method (PCM), with the combination of Karhunen-Loeve and polynomial chaos expansions. In this approach, the perturbation expansion is not formally invoked. We explore the validity of this approach for various degrees of spatial variability of the media and different correlation lengths. We also compare this approach with existing stochastic approaches for efficiency and accuracy.

2. Mathematical Formulation 2.1. Governing Equations [8] We consider steady state flow in saturated media satisfying thefollowing equation: r ½Ks ðxÞrhðxÞ þ g ðxÞ ¼ 0;

ð1Þ

differential equation, whose solutions are no longer deterministic values but probability distributions or related moments. We aim to efficiently and accurately estimate the statistical properties of the hydraulic head in terms of the statistical moments of log transformed Ks(x). 2.2. Karhunen-Loeve Expansion [10] Let Y(x, q) = ln Ks(x, q) be a random space function, where x 2 D and q 2 Q (a probability space). One may write Y(x, q) = Y (x) + Y0(x, q), where Y (x) is the mean and Y0(x, q) is the fluctuation. The spatial structure of the random field may be described by the covariance CY (x, y) = hY0(x, q)Y0(y, q)i. Since the covariance is bounded, symmetric and positive-definite, it may be decomposed as [e.g., Ghanem and Spanos, 1991] CY ðx; yÞ ¼

Ks ðxÞrhðxÞ nðxÞ ¼ QðxÞ;

ð2Þ

x 2 GN ;

ð4Þ

Z CY ðx; yÞf ðxÞdx ¼ lf ðyÞ:

ð5Þ

D

Then the random process Y(x, q) can be expressed as Y ðx; qÞ ¼ Y ðxÞ þ

1 pffiffiffiffiffi X ln fn ðxÞx n ðqÞ;

ð6Þ

n¼1

where x n(q) are orthogonal Gaussian random variables with zero mean and unit variance. The expansion in equation (6) is called the Karhunen-Loeve (KL) expansion. The KL expansion, which is a spectral expansion, is optimal with mean square convergence when the underlying process is Gaussian [e.g., Ghanem and Spanos, 1991]. [11] Although, in general, the eigenvalue problem (5) has to be solved numerically, there exist analytical or semianalytical solutions under certain conditions. For a onedimensional stochastic process with a covariance function CY(x1, y1) = s2Yexp( jx1 y1j/h), where s2Y and h are the variance and the correlation length of the process, respectively, the eigenvalues and their corresponding eigenfunctions can be expressed as [e.g., Zhang and Lu, 2004], 2hs2Y ; þ1

h2 w2n

ð7Þ

and 1 ffi ½hwn cosð wn xÞ þ sinð wn xÞ; fn ð xÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 w2n þ 1 L=2 þ h

ð3Þ

where h(x) is hydraulic head, Ks(x) is the hydraulic conductivity, and g(x) is the source (or sink) term. H(x) is the prescribed head-on Dirichlet boundary segment GD, Q(x) is the prescribed flux across Neumann boundary segments GN, and n(x) = (n1,. . .,nd)T is an outward unit vector normal to the boundary G = GD [ GN. [9] In this study, Ks(x) is treated as a random process (space function), thus equation (1) becomes a stochastic

ln fn ðxÞfn ðyÞ;

where ln and fn(x) are eigenvalues and deterministic eigenfunctions, respectively, and can be solved from the following Fredholm equation:

ln ¼ x 2 GD ;

1 X n¼1

subject to boundary conditions hðxÞ ¼ H ðxÞ;

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ð8Þ

where wn are positive roots of the characteristic equation

h2 w2 1 sinð wLÞ ¼ 2hw cosð wLÞ:

ð9Þ

For problems in multidimension, if we assume that the covariance function CY (x, y) is separable, for example CY (x, y) = sY2 exp( jx1 y1j/h1 jx2 y2j/h2) in a

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rectangular domain D = {(x1, x2): 0 x1 L1; 0 x2 L2}, the eigenvalues and eigenfunctions can be obtained by combining those in each dimension. For a non-separate covariance in a domain of arbitrary shape, the eigenvalue problem of (5) has to be solved numerically. Furthermore, the KL expansion is not limited to stationary random fields [Lu and Zhang, 2007]. P1 [12] From equation (6), one can get Ds2Y = n¼1 ln, where D is the domain size, which indicates the total variance s2Y is decomposed by an infinite series of eigenvalues ln. Equation (9) has infinite number of positive roots. If the roots wn are sorted in an increasing order, the related eigenvalues ln are monotonically decreasing, which allows us to truncate the KL expansion to finite terms. The rate of decay of ln determines the number of terms that are retained in the Karhunen-Loeve expansion, which determines the random dimensionality of the problem. 2.3. Polynomial Chaos Expansion [13] Since the covariances of the dependent random processes such as head and flux are yet to be found, the Karhunen-Loeve expansion cannot be used to represent their random structures. Instead, the polynomial chaos expansion, introduced by Wiener [1938], may be used to effectively express the dependent (output) random fields. This approach is now widely used in structural mechanics and other fields [e.g., Ghanem and Spanos, 1991; Ghanem, 1998]. [14] With the polynomial chaos expansion, each random field of interest can be expressed as yðx; qÞ ¼ a0 ðxÞ þ

1 X

ai1 ðxÞG1 x i1 ðqÞ

i1 1 X X

ai1 i2 ðxÞG2 x i1 ðqÞ; x i2 ðqÞ

i1 ¼1 i2 ¼1

þ

i1 X i2 1 X X

where x is a vector of dimension N. There is a one-to-one correspondence between the terms in equations (10) and (12). The total number of terms P is determined by the random dimensionality N and the degree of the polynomial chaos expansion d, P¼

Lyðx; qÞ ¼ f ðxÞ;

ai1 i2 i3 ð xÞG3 x i1 ðqÞ; x i2 ðqÞ; x i3 ðqÞ þ . . . ;

^yðx; qÞ ¼ ð10Þ

where the coefficients a0(x) and ai1i2,. . .,id(x) are deterministic functions of x, and Gd(xi1,. . .,xid) are orthogonal polynomial chaos of order d with respect to the random variables (x i1,. . .,xid). For independent standard Gaussian random variables (x i1,. . .,x id), Gd(xi1,. . .,xid) are the multidimensional Hermite Polynomials of degree d expressed as

d

1 T 2x x

Gd x i1 ; . . . ; x id ¼ ð 1Þ e

h 1T i @d e 2x x ; @x i1 . . . @x id

P X

cj ðxÞYj ðxÞ;

ð14Þ

P X

ci ðxÞYi ðxðqÞÞ:

ð15Þ

i¼1

Define the residual R as Rðfci g; xÞ ¼ L^y f :

ð16Þ

The weighted-residual method in the random space is expressed as Z Rðfci g; xÞwj ðxÞpðxÞdx ¼ 0;

ð11Þ

where x is a vector denoting (x i1,. . .,xid )T. Hermite polynomials form the best orthogonal basis for Gaussian random variables [e.g., Ghanem and Spanos, 1991]. In case of other random distributions, generalized polynomial chaos expansions [Xiu and Karniadakis, 2002] can be used to represent the random field as in (10). [15] In practice, equation (10) is usually truncated by finite terms, and it can be written simply as yðx; qÞ ¼

ð13Þ

2.4. Weighted-Residual Method [16] The Galerkin scheme is usually used to evaluate the deterministic coefficients in the polynomial chaos expansion approach [e.g., Ghanem, 1998; Xiu and Karniadakis, 2002]. A new approach is the stochastic collocation method (SCM) developed by Mathelin et al. [2005], which has been applied to problems with small number of random variables. If a stochastic process (random field) has to be described with a large number of random variables, the SCM with a tensor product implementation may suffer from the problem of high dimensionality. The probabilistic collocation method (PCM), also called the deterministic equivalent modeling method (DEMM), is introduced by Tatang et al. [1997] and successfully applied to the uncertainty analysis in some fields [Webster et al., 1996; Kumar et al., 2005]. [17] Both the Galerkin scheme and the PCM used to determine the coefficients of the polynomial chaos expansions can be derived with the weighted-residual method. For a stochastic differential equation

i1 ¼1 i2 ¼1 i3 ¼1

ð N þ d Þ! : N!d!

where y(x, q) is the unknown random space function and f(x) is the source term. The operator L involves differentiations in space and can be nonlinear. If y(x,q) is approximated by the polynomial chaos expansion and the approximation is denoted as ^y(x, q),

i1 ¼1

þ

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ð17Þ

x

where wj(x) is the weighting function, j = 1,. . ., P, and p(x) is the joint probability density function of x. [18] With the Galerkin scheme, the weighting function is chosen as the basis function of the approximation, wj ðxÞ ¼ Yj ðxÞ:

ð18Þ

Substituting equation (18) into equation (17) and expressing it in the form of ensemble mean yields,

ð12Þ

j¼1

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ð19Þ


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which leads to a coupled system of deterministic equations. A particular example of the Galerkin PCE scheme is given in section 4.1. [19] In the probabilistic collocation method, the weighting function is chosen as the Dirac delta function, wj ðxÞ ¼ d x xj ;

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a second-order polynomial chaos expansion and express it as ^ h(x), ^ hðxÞ ¼ a0 ðxÞ þ

N X

ai ðxÞx i þ

i¼1

þ

ð20Þ

N 1 X N X

N X

aii ðxÞ x 2i 1

i¼1

aij ðxÞ x i x j ;

i¼1 j>i

ð23Þ

where xj is a particular set selected with certain algorithm out of the random vector x. The elements in xj are called the collocation points. Then equation (17) becomes,

R fci g; xj ¼ 0;

where N is the dimensionality of the input random field. This can be written as ^ hðxÞ ¼

ð21Þ

which results in a set of independent equations, evaluated at the given sets of collocation points, xj, where j = 1, 2, , P. It is seen that P sets of collocation points are needed and thus equation (21) has to be solved for P times in order to obtain the P coefficients {ci}, where i = 1, 2, ,P. As indicated in the next section, these collocation points may be selected with algorithm similar to that for selecting integration points in Gaussian quadrature.

P X

ci ðxÞYi ðxÞ;

where x is a vector of dimension N. [24] Using the probabilistic collocation method, we let the residual be zero at some selected sets of collocation points, specified as xj = (x 1,j,x 2,j,. . .,x N,j)T, where j = 1,2,. . ., P. It follows from equation (21) that at the jth set of collocation point, ^ hj(x) satisfies the following equation, ( r

" exp Y ðxÞ þ

# ) N pffiffiffiffiffi X hj ðxÞ þ g ðxÞ ¼ 0; ln fn ðxÞx n;j ðqÞ r^ n¼1

ð25Þ

3. Kl Based Probabilistic Collocation Method 3.1. Implementation [20] In previous studies [e.g., Webster et al., 1996; Tatang et al., 1997], PCM is used for parametric sensitivity and uncertainty analysis and is limited to independent scalar random variables. When uncertain parameters are correlated and exist as a random field, the PCM may be combined with the KL expansion. Sarma et al. [2005] made use of the PCM and the KL expansion for optimizing reservoir production in the presence of an uncertain permeability field. In this work we combine the PCM with the KL expansion for analyzing uncertainty propagation for flow in random porous media, where random process and stochastic differential equation have to be considered. [21] Four major steps are involved in implementation of this approach: (1) Representation of the input random field using the KL expansion in terms of a set of independent standard random variables (srv’s); (2) approximation of the outputs of interest using the polynomial chaos expansion in terms of the set of srv’s; (3) determination of the unknown coefficients in the polynomial chaos expansion using the PCM technique; (4) evaluation of the statistical properties of the outputs. [22] Substituting the KL expansion of Y(x) in equation (6) into the governing equation (1) yields ( r

" exp Y ðxÞ þ

#

N pffiffiffiffiffi X ln fn ðxÞx n ðqÞ rhðxÞ

ð24Þ

i¼1

) þ gðxÞ ¼ 0:

n¼1

ð22Þ

[23] Now we take the second-order PCM for example to perform uncertainty analysis in random porous flow. We consider the approximation of the random field h(x) with

where j = 1, 2,. . ., P. It means that we only need to choose P sets of collocation points for the random variables x = (x 1,x 2,. . .,xN)T, and to solve equation (25) P times (one at each set of collocation points xj). It should be noted that these values of hydraulic head ^ hj(x) are solved independently through equation (25), which is identical to the original equation (11). That is, the codes for solving deterministic differential equations could be employed directly, like in the Monte Carlo method. [25] With these P sets of collocation points, we can solve equation (25) with the finite difference method, or any other methods, to obtain the P sets of the hydraulic head field ^ hj(x). Here we denote the coefficients in equation (24) by c1(x), c2(x),. . ., cP(x) in sequence, and C(x) = [c1, c2,. . ., cP]T. The corresponding hydraulic head field at each set of collocation points (like each realization in the Monte Carlo h2(x),. . ., ^ hP(x), and h(x) = [^ h1, ^ h2,. . ., method) are ^ h1(x), ^ ^ hP]T. In PCM, each of these hydraulic head fields is called a ‘‘representation’’ because they do not carry the same weight unlike the realizations in the direct sampling Monte Carlo method. Then we can rewrite (24) as, Z CðxÞ ¼ hðxÞ;

ð26Þ

where Z, of elements Zji = Yi(xj), is a space-independent matrix of dimension P P, consisting of Hermite polynomials evaluated at the selected sets of collocation points. By solving the linear system of equations (26), the deterministic coefficients C(x) could be obtained readily. Note that the sets of collocation points should be so selected that the matrix Z satisfies the condition of rank(Z) = P. 3.2. Selection of Collocation Points [26] The performance of PCM strongly depends on the choice of the collocation points. One particular scheme is, in

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Figure 1. Series of eigenvalues and their finite sums, for h = 4 and h = 2. analogy to Gaussian quadrature, to select the collocation points at a given order of polynomials from the roots of the next higher order orthogonal polynomial for each uncertain parameter [e.g., Webster et al., 1996; Tatang et al., 1997; Villadsen and Michelsen, 1978]. If the polynomial order is d, then the number of collocation points available is (d + 1)N, which is always larger than the number of collocation points needed. For example, the third-order =ffiffiffix 3p

ffiffiffi3x, and the three roots Hermite polynomial is H3(x) p of this polynomial are 0, 3, 3, respectively. For the second-order PCM, the P sets of collocation points (x 1,1, x 2,1,. . .,xN,1), (x 1,2, x2,2,. . .,x N,2),. . ., (x 1,P, x2,P,. . .,x N,P) are chosen from all the combinations of the three roots. Here we demonstrate how to choose the P sets of collocation points. We rank pffiffiffi these pffiffiffi three roots in order of decreasing probability, (0, 3, 3), since 0 has the highest probability for the standard Gaussian random variable with zero mean and unit variance. The first set of collocation point, i.e., (0, 0,. . .,0) will always contain the roots with the highest probability for each random variable. The other sets are selected by keeping as many of the variables of high probability as possible. In order to satisfy the condition of rank(Z) = P in (algebraeq), each new set xj + 1 of proposed collocation points must possess the property that the ( j + 1)th row of Z is linearly independent with the previous j rows in Z. Otherwise, the proposed set is rejected and the next set with equal or lower probability is tested. The last step is to make sure that all of its columns are also linearly independent for the purpose of satisfying the condition of full rank for the matrix Z. When N or d is large, the selection process can be computationally demanding. However, for a given type of polynomial chaos of known N and d, the P sets of collocation points are independent of physical problems. As for integration points in Gaussian quadrature, the selection of collocation points only has to be done once and can hence be tabulated beforehand. An alternative, which circumvents the requirement of full rank for the matrix, is to estimate the set of coefficients in the polynomial chaos expansion via a regression based method called stochastic response surface method [Isukapalli et al., 1998]. In the regression method, a set of sample points is chosen and the model outputs are equated to the estimates of the approximation at these points. However, the number of sample points must be more than the number of unknown

coefficients to be estimated, resulting in a high computational effort. 3.3. Post-Processing [27] Once we obtain the coefficients of equation (23), we could easily evaluate the statistical quantities, such as various moments and the probability density function, of the outputs based on expansion (23) or (24) by certain sampling methods such as Monte Carlo. Since the expansion is reduced to a polynomial form and it does not involve solving equations, the evaluation is computationally efficient. Alternatively, one may directly derive the statistical moments of h(x) from equation (24), such as the mean and variance, hhðxÞi ¼ c1 ðxÞ;

s2h ¼

P X

D E cj ðxÞ2 Y2j :

ð27Þ

ð28Þ

j¼2

Higher order moments can be obtained similarly. 3.4. Results Compared With Monte Carlo Simulations [28] This and the following sections show some examples to illustrate the KL based PCM and examine its validity and applicability to flow in porous media. We first consider steady state flow in a one-dimensional domain of length L = 10 [L] (where [L] denotes any consistent length unit) and assume the forcing term to be zero. The boundary conditions are prescribed heads at the two ends, H0 = 7 [L] and HL = 5 [L]. The correlation length is given as h = 4 [L]. The mean of the log hydraulic conductivity is given as hYi = 0.0. Two cases with different spatial variability (i.e., s2Y = 1.0 or 2.0) are performed. [29] The eigenvalue and eigenfunction ln and fn(x),n = 1,2,. . ., can be determined analytically by solving equations (7) and (8). The eigenvalues are monotonically decreasing as illustrated in Figure 1a, for cases with different correlation lengths (h = 2 and 4). Figure 1b shows the sum of eigenvalues as a function of number of terms included. Owing to the rapid decay, only the first 6 terms are retained in the KL expansion for h = 4. That is, the random dimensionality of {x n} is N = 6. One can see from

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Figure 2. Comparisons of the variance of hydraulic head derived from PCM up to the third-order and MC, with different spatial variabilities.

Figure 1b that about 90% energy of the process is preserved in this case. For the second-order PCM, the total number of collocation points is P = 28. More precisely, the PCM involves 28 sets of collocation points. [30] We next test the sensitivity of the order of polynomials in PCM. If we consider the third-order polynomial chaos expansion and the number of retained terms in the KL expansion is also 6, the total number of collocation point sets grows to 84. In this paper, the finite difference method is used to solve equation (25) discretized with 151 physical nodes. Owing to the particular boundary conditions in our examples, the mean head obtained from different approaches are very close to each other. We thus focus our discussion only on the head variance in the following sections. Figures 2a and 2b show the variance of hydraulic head for s2Y = 1.0 and 2.0, respectively, obtained from the second- and third-order PCM as well as Monte Carlo method. Note that in our Monte Carlo simulation, we implement the KL expansion to generate the random fields of the log hydraulic conductivity Y(x), based on equation (6) with 60 terms, which is found to be adequate in reproducing the ensemble statistics of Y(x). We choose 10,000 realizations in the Monte Carlo simulation to ensure statistical convergence and thus accurate results, which are used as the benchmarks in our study. It is seen that both the second- and third-order PCM solutions are in agreement with those from Monte Carlo simulations in the case of s2Y = 1.0. As the spatial variability becomes large, i.e., s2Y = 2, the secondand third-order PCM agree with the Monte Carlo result fairly well in spite of some deviations among the three. This observation is encouraging as the variance of s2Y = 2, being equivalent to the coefficient of variation of hydraulic conductivity CvKs = sKs/hKsi = 253%, represents a large variability in hydraulic conductivity. We will discuss the case of larger spatial variability in section 5. [31] A comparison on computational effort and solution accuracy can be made between PCM and Monte Carlo method. The computational effort in solving (25) for each set of collocation points in the PCM is almost the same as solving (1) for each realization in the Monte Carlo method. We only need to solve (25) for 28 times in the second-order PCM and 84 times in the third-order PCM, much less than the number of realizations needed in Monte Carlo method.

Figure 3. (a) 1000 realizations of hydraulic head from MC simulations; (b) 28 representations of hydraulic head from second-order PCM for the case of h = 4.0 and sY2 = 1.0.

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Figure 4. Comparisons of head variance derived from PCE and MC, with different spatial variabilities. Figure 3a shows the 1000 realizations of hydraulic head from Monte Carlo simulations for the case of s2Y = 1.0, which are solved from randomly generated Y(x) and are equally probable. Figure 3b depicts the 28 representations of hydraulic head from the second-order PCM for the same case, which are among the range of the 1000 realizations shown in Figure 3a. Each representation in PCM could be a realization from the Monte Carlo simulation, but contrary to the realizations in the latter, the representations carry different weights. It is seen from Figure 2 that the 28 sets of hydraulic head from PCM represent very well the random field in terms of the statistical moments (see also section 5.3 for the probability density functions). Both the Monte Carlo method and PCM involve sampling. The difference is that in the direct sampling Monte Carlo method the realizations are equally probable whereas in the PCM a structural expression (i.e., the polynomial chaos expansion) of the output random field is generated at first and then the collocation technique is adopted according to the Gaussian quadrature rule [e.g., Villadsen and Michelsen, 1978]. Owing to the fast convergence of the polynomial chaos expansion and the high accuracy of Gaussian quadrature, PCM can yield quite good results with much less samplings than the direct sampling Monte Carlo method.

4. Comparisons With Other Stochastic Methods 4.1. Polynomial Chaos Expansion Approach [32] In this section, we compare PCM with the traditional polynomial chaos expansion (PCE) approach. The tradi-

tional PCE approach utilizes the Galerkin scheme to determine the coefficients of the PCE [Ghanem and Spanos, 1991; Ghanem, 1998]. For the same problem as in the previous section, we also use the KL expansion to represent the random field of the log hydraulic conductivity in the PCE approach. The governing equation is (22) and the PCE of hydraulic head is expressed as equation (24). [33] For the particular one-dimensional example on hand, assuming Y (x) = 0, substituting equation (24) into equation (22) and simplifying yields P P N pffiffiffiffi X X X d 2 cj ð xÞ li fi0 ð xÞx i Yj þ 2 dx j¼1 j¼1 i¼1

!

dcj ð xÞ Yj ¼ 0: dx

ð29Þ

With the Galerkin technique, multiplying equation (29) by each of the Yk, k = 1, 2, 3,. . ., P, and taking the inner product, one has the following equation: P X N pffiffiffiffi d 2 ck ð xÞ 2 X dcj ð xÞ li fi0 ð xÞ Yk þ x i Yj Yk ¼ 0; 2 dx dx j¼1 i¼1

ð30Þ

where the orthogonality of Yj, i.e., hYjYki = hY2k id jk, is utilized. The finite difference method is also used to discretize this equation in physical space. Equation (30) results in a coupled system of P equations. Note that hxiYjYki should be calculated ahead of time, according to each term of x i, Yj and Yk. After solving the coupled system of equations for the deterministic coefficients ci(x), the

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Figure 5. Comparisons of head variance derived from KLME and MC, with different spatial variabilities. statistical properties, such as the mean and variance, of the hydraulic head can be evaluated with equations (27) – (28). The post-processing is the same as in PCM. [34] The head variances derived from the second- and third-order polynomial chaos expansion approach, denoted as PCE, are presented in Figure 4-effectvar, where the results from Monte Carlo simulations are added as the benchmarks. Three cases with different spatial variabilities are shown. It is seen that when the spatial variability is small, i.e., s2Y = 0.3, the results from both the second- and third-order PCE are in good agreement with those from Monte Carlo method. As the spatial variability grows, the results from the second-order PCE have some deviations from those from Monte Carlo method. Also, the results from the third-order PCE are closer to those from Monte Carlo method. [35] Since both PCM and PCE use the same polynomial chaos expansion, their orders have the same meaning and the difference between them is how the coefficients of the expansion are evaluated, as discussed in section 2.4. PCE utilizes the Galerkin scheme involved with the integration on each of the basis functions, which is accurate up to the given order of polynomials (subject to discretization errors). However, the PCM utilizes the optimal collocation technique according to the Gaussian quadrature, with which the accuracy of the solutions can be equal to or even higher than the given order of polynomials [e.g., Villadsen and Michelsen, 1978]. The superior performance of the second-

order PCM relative to the second-order PCE may stem from the optimal collocation scheme utilized in the Gaussian quadrature. With the same number of terms, P, in the same polynomial chaos expansion, PCM only needs to run the deterministic differential equation independently for P times; in PCE the P terms of coefficients are governed by P coupled equations, which lead to a large computational effort, especially for large-scale problems. Also, computing all the terms of hxiYjYki in PCE takes of order N P2 operations, where N and P are the retained number of terms in the KL expansion and the number of terms in polynomial chaos expansion, respectively. In addition, since the PCM approach is non-intrusive in that the resulting equations are the same as the original equations, it can be easily implemented with existing codes and can be naturally parallelized. This advantage of PCM is even more pronounced for nonlinear problems. 4.2. KLME Approach [36] Another efficient approach, the moment-equation approach based on the Karhunen-Loeve expansion (KLME) is recently developed by Zhang and Lu [2004]. Here we compare the KLME approach with the PCM. In the KLME approach, the log hydraulic conductivity Y(x) is expressed by the KL expansion and the hydraulic head h(x) is represented by the perturbation expansion, h(x) = h(0)(x) + h(1)(x) + h(2)(x) + . In this series, the order of each term is with respect to sY, the standard deviation of Y(x). Also, each

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Figure 6. Comparisons of head variance derived from PCM, KLME and MC, at h = 2 with different spatial variabilities. term h(m)(x) (m 1) is expanded in terms of the product of m orthogonal Gaussian random variables, i.e., ðm Þ

h

ðxÞ ¼

1 X

m Y

i1 ;i2 ;...;im ¼1

j¼1

ðm Þ

hi1 ;i2 ;:::;im ðxÞ ¼ 0;

x 2 GD ;

ð36Þ

! ðmÞ

x ij hi1 ;i2 ;...;im ðxÞ;

ð31Þ ðmÞ

rhi1 ;i2 ;:::;im ðxÞ nðxÞ ¼

(m)

where hi1,i2,. . .,im are deterministic coefficients to be determined. Deriving from equations (1) – (3), one can finally obtain the following equations with boundary conditions that govern the coefficients [Zhang and Lu, 2004]: r2 hð0Þ ðxÞ þ rhY ðxÞi rhð0Þ ðxÞ ¼

g ðxÞ ; KG ðxÞ

ð32Þ

m ð 1Þmþ1 QðxÞ Y fi ðxÞ; m!KG ðxÞ j¼1 j

x 2 GN ; ð37Þ

where KpGffiffiffiffiffi (x) = exp(hY(x)i) and fn(x) is used to represent ~f n(x) = ln fn(x) for simplicity. It is seen that the equations for different order terms of the hydraulic head are recursive and the equations at the same order are independent. To (m)

obtain the coefficients hi1,i2,. . .,im(x), where ij = 1,. . .,n, the x 2 GD ; ð33Þ number of times required to solve independent differential hð0Þ ðxÞ ¼ H ðxÞ; equations is Sm = n(n + 1). . .(n + m 1)/m!. Once these coefficients are solved, one can directly calculate the mean, x 2 GN ; ð34Þ variance, and the covariance of hydraulic head [Zhang and rhð0Þ ðxÞ nðxÞ ¼ QðxÞ=KG ðxÞ; Lu, 2004]. [37] We solve the same problem as in the previous and for m 1, sections using the KLME approach and present the results m ð 1Þmþ1 g ðxÞ Y ðmÞ ðmÞ in Figure 5. We also examine three cases with different r2 hi1 ;i2 ;:::;im ðxÞ þ rhY ðxÞi rhi1 ;i2 ;...;im ðxÞ ¼ fi ðxÞ m!KG ðxÞ j¼1 j spatial variabilities s2Y and add the Monte Carlo results as benchmarks. Up to second-order (in s2Y) correction for the 1 X ðm 1Þ

rfi1 ðxÞ rhi2 ;:::im ðxÞ; head variance, one needs to solve for the coefficients h(1) i , m Pi ;:::;i (3) m h(2) 1 ij , and hijk . The numbers of terms included are chosen as (1) ð35Þ 6, 4 and 4, respectively, i.e., index i in hi running up to 6 (2) and each index in hij running up to 4, and so on. Owing to 9 of 13

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can find that the KLME has the same construction of expansion as the polynomial chaos expansion with the PCM or PCE. In the KLME, the number of terms included in approximating h(i)(x) can be distinctly selected for each order i [Zhang and Lu, 2004]. Generally, the number decreases as the increase of order i in h(i)(x). While in the PCM or PCE, for different order terms in a polynomial chaos expansion the dimension of the random vector remains to be the same as the number of terms retained in the KL expansion.

5. Discussions

Figure 7. Head variance derived from PCM (up to the fourth-order) and MC, at sY2 = 4.0 for different h. the symmetry of the coefficients such as h2ij, the computational times are reduced. The total number of times to (2) (3) solve the deterministic equations for h(1) i , hij , and hijk is 6 + 10 + 20 = 36. [38] From Figure 5, one can see that the first-order (in sY2) approximations of head variance have some deviations from the Monte Carlo results. When s2Y is small or moderate, i.e., s2Y = 0.3 or 1.0, the second-order head variances from the KLME approach are in good agreement with the Monte Carlo results. However, when the spatial variability becomes large, i.e., s2Y = 2.0, the second-order head variance from the KLME approach deviates from the Monte Carlo result. We try to increase the number of terms included in h(i)(x), i = 1, 2,. . ., consequently bringing about a larger computational effort, but do not find much improvement. It has been found by Zhang and Lu [2004] that for similar cases the approximation can be improved with third-order corrections for s2Y = 2. [39] It should be noted that the order in the KLME is different from that in the PCM (or PCE). The order in KLME refers to that in the perturbation expansion and is with respect to sY, whereas that in PCM or PCE is with respect to the degree of the polynomial chaos expansion. If all terms in the perturbation expansion are collected, one

5.1. Effect of Correlation Length h [40] As shown in Figure 1, the rate of decay in the eigenvalues is dependent on the correlation length h relative to the domain length L. To further test the effect of correlation length on the recently developed approaches, i.e., the PCM and the KLME, we perform 3 cases for h = 2, with different spatial variabilities, i.e., s2Y = 0.3, 1.0, and 2.0. The head variance obtained with PCM and KLME are presented in Figure 6. We also include the Monte Carlo simulation results for comparison. Note that the number of physical nodes is also chosen as 151 and the number of realizations in Monte Carlo simulations is 10,000. A smaller number of physical nodes or realizations in Monte Carlo simulations could lead to unstable results. The retained number in the KL expansion for PCM is also chosen as 6, the same as that for h = 4. Also, for the second-order (2) (3) KLME, the numbers of terms included in h(1) i , hij , and hijk are also chosen as 6, 4, and 4, respectively. From Figure 6, we observe that for small or moderate spatial variability, i.e., s2Y = 0.3 or 1, the results obtained from both the secondorder PCM and the second-order KLME agree well with those from Monte Carlo method. When s2Y is large, i.e., sY2 = 2, the second-order PCM result is still close to the Monte Carlo result while the second-order KLME result exhibits some deviation. 5.2. Effect of Large Spatial Variability sY2 [41] As illustrated in the previous sections, for moderate spatial variability, the second- or third-order PCM (or PCE) and the second-order KLME can obtain quite accurate results. However, for large variability there are some deviations. In this section, we examine some cases with an even larger spatial variability, i.e., s2Y = 4.0, corresponding to the coefficient of variation of hydraulic conductivity CvKs = 732% [Zhang, 2002]. We explore the higher order PCM solutions for the purpose of improving the accuracy. For the cases of s2Y = 4.0 and with different correlation lengths h, we perform the PCM from the second- up to the fourth-order. The retained number in the KL expansion is 6, thus the total number of collocation point sets for the fourth-order PCM reaches 210. Figures 7a and 7b show the head variances for h = 4 and h = 2, respectively, derived from the secondto fourth-order PCM and Monte Carlo method with 10,000 realizations. From Figures 7a and 7b, it is seen that the head variances from both the second- and fourth-order PCM are quite close to those from Monte Carlo simulations, for both h = 4 and h = 2. However, the third-order solutions deviate significantly from their Monte Carlo counterparts. The same general observation on the relative performance of the second- and third-order PCM can be made from

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Figure 8. Comparisons of the probability density function (PDF) of hydraulic head at three positions (i.e., x = 2, 4, and 6) obtained with PCM, KLME, and MC. Figure 2b for s2Y = 2.0, although less pronounced. The inferior performance of the third-order PCM is related to the fact that its collocation points are selected from the roots of fourth-order Hermite polynomial, which do not contain the origin point that corresponds to the region of highest probability for a standard Gaussian random variable. This indicates that an even order PCM may perform better than the next higher odd order PCM in the presence of large variability in hydraulic conductivity when the particular scheme is invoked for selecting collocation points. 5.3. Probability Density Function [42] Once the dependent random field such as the hydraulic head is approximated by either the polynomial chaos expansion (24) in the PCM (or the PCE) or the perturbation polynomial expansion (31) in the KLME, the probability density function (PDF) of the random field can be simulated with certain sampling methods. Taking the first case of h = 4 and s2Y = 1.0 for example, we calculate the PDF of h(x) at three positions (i.e., x = 2, 4, and 6), with the PCM, KLME, and Monte Carlo method. The comparisons are presented in Figure 8. One can see that the results obtained from both the second-order PCM and the second-order KLME are close to those from Monte Carlo simulations. With the direct sampling Monte Carlo simulations, a large number of realizations are needed to achieve the statistically accurate

results (even 10,000 realizations yield a not entirely smooth PDF). However, with the PCM or the KLME approach, one only needs to solve the differential equations for 28 or 36 times in order to construct the polynomial expansions, based on which 10,000 samplings are then drawn to approximate the PDF. Since the dependent random field is explicitly expressed in a polynomial form in PCM and KLME, the computational time is significantly reduced compared to solving original differential equations with the Monte Carlo method. The PCM and KLME approaches provide an efficient and accurate way to evaluate the PDF of the dependent random fields. 5.4. Illustrative Examples in 2D [43] In this section, we consider two cases in a twodimensional domain of saturated heterogeneous medium, which is a square of size L1 = L2 = 10 [L], uniformly discretized into 40 40 square elements. The non-flow conditions are prescribed at two lateral boundaries. The hydraulic head is prescribed at the left and right boundaries as 10.5 [L] and 10.0 [L], respectively. The mean of the log hydraulic conductivity is given as hYi = 0.0. Assume the covariance function of the log hydraulic conductivity is CY(x, y) = s2Y exp( jx1 y1j/h1 jx2 y2j/h2), where h1 = h2 = h = 4.0 and s2Y = 1.0. For the first case, there is no source term, i.e., g(x) = 0. For the second case, there is a

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Figure 9. Comparison of head variance at x2 = 5 in 2D obtained with PCM and MC. pumping well extracting water at the flow rate of 1 [L3/T] at the center of the domain. For the first case without well, Figure 9 shows the head variance along the section of x2 = 5. For the second case with well, Figures 10a and 10b show the mean and variance of hydraulic head along the diagonal line, respectively. In both cases, we found it suffice to retain 7 modes (N = 7) in KL expansion for the twodimensional log transformed hydraulic field, thus the total number of terms in the second-order polynomial chaos expansion is 36. Results from the second-order PCM with 36 simulations agree quite well with those from the Monte Carlo method with 10,000 simulation runs. It is seen that the computational efficiency and accuracy observed in one dimension also carries to two dimensions.

6. Conclusions [44] We explored a method for uncertainty analysis of flow in random porous media by combining the KarhunenLoeve (KL) expansion and the probabilistic collocation method (PCM). The underlying (input) random field is

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represented by the KL expansion and the dependent random field such as the hydraulic head is expressed by the polynomial chaos expansion. The PCM is used to determine the coefficients in the polynomial chaos expansion by evaluating the hydraulic head at different sets of collocation points. This approach is appealing because it results in independent deterministic differential equations, which, similar to the Monte Carlo method, can be implemented with existing codes. We applied the PCM to several cases of flow in random porous media in one and two dimensions, with different spatial variabilities and correlation lengths. We also compared it with the Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment-equation approach based on Karhunen-Loeve expansion. This study leads to the following conclusions: [45] 1. With a relatively small computational effort, the probabilistic collocation method (PCM) based on Karhunen-Loeve expansion is feasible for accurately quantifying uncertainty associated with flow in random porous media, where the random process and stochastic differential equation have to be considered. [46] 2. While both the PCM and the Galerkin PCE approach rely on polynomial chaos expansion to approximate the hydraulic head, the difference depends on how the coefficients of the expansion are evaluated. The traditional PCE approach uses a Galerkin scheme and leads to a set of coupled equations whereas the PCM uses a collocation approach and results in independent deterministic differential equations. The PCM is thus generally more efficient than the Galerkin PCE approach. [47] 3. Similar to the Monte Carlo method, the PCM can be easily implemented with existing codes and naturally parallelized. Both the direct sampling Monte Carlo method and the PCM involve sampling. The Monte Carlo method resorts to the law of large numbers by generating a large number of equally probable realizations and is thus computationally demanding; the PCM builds upon an optimal approximation evaluated at selected sets of collocation points and hence reduces the computational efforts significantly. [48] 4. The moment-equation approach based on KarhunenLoeve expansion (KLME) first decomposes the hydraulic

Figure 10. Comparisons of mean and variance of hydraulic head along the diagonal line in 2D with a pumping well obtained with PCM and MC. 12 of 13

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head into terms of different order of sY with the perturbation expansion and then expresses each term with a polynomial expansion. If all terms in the perturbation expansion are collected, the KLME has the same construction as the polynomial chaos expansion, which is used by the PCM and the PCE approach. The difference is how the approximations are made in all these methods. In the KLME, the equations for different order terms of the hydraulic head are recursive, i.e., the equations for the higher order terms depend on the lower order terms; the equations at the same order are independent. The number of terms retained in the KL expansion of the log hydraulic conductivity can be different for different order terms of hydraulic head. [49] 5. The PCM, the Galerkin PCE approach, and the KLME are all fairly accurate at least for the spatial variability s2Y as large as 2.0 (corresponding to the coefficient of variation of hydraulic conductivity as 253%). Also, the PCM and the KLME are more efficient than the Galerkin PCE of coupled equations. With small computational efforts, the probability density functions (PDF) of the dependent random fields can be obtained with the PCM and the KLME. In general, for moderate spatial variability the second-order PCM (or KLME) is enough to obtain quite accurate results while for larger spatial variability, a higher order PCM (or KLME) may be needed. It should be noted that PCM is not restricted to the flow type and configuration considered in this study. Our ongoing study indicates that the approach is applicable to complex flow configurations and multiphase flow in porous media. [50] Acknowledgments. This work is partially supported by the National Natural Science Foundation of China through grant 50688901, by the Outstanding Oversea Scholar Foundation of the Chinese Academy of Sciences through grant 2005-1-1, and by the U.S. National Science Foundation through grant 0620631. The authors would like to thank the two anonymous reviewers and the associate editor for their constructive comments on an earlier version, which led to a much improved paper.

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H. Li, Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, Norman, OK, USA. D. Zhang, Department of Civil and Environmental Engineering, and Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089, USA. ([email protected])

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