Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

WATER RESOURCES RESEARCH, VOL. 49, 7911 7928, doi:10.1002/2013wr014055, 2013 Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location Qinzhuo Liao 1 and Dongxiao Zhang 2 Received 2 May 2013; revised 16 October 2013; accepted 23 October 2013; published 4 December 2013. [1] In this work, we propose a new collocation method for uncertainty quantification in strongly nonlinear problems. Based on polynomial construction, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific locations. In some cases, especially when the advection dominates, the model response has a strongly nonlinear profile with a discontinuous shock or large gradient. This nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain, which causes nonphysical oscillation and inaccurate estimation using the traditional PCM. To address this issue, a new location-based transformed probabilistic collocation method (xtpcm) is developed in this study, inspired by the Lagrangian point of view, in which model response is represented by an alternative variable, i.e., the location of a particular response value, which is relatively linear to the random parameter with a smooth profile. The location is then approximated by polynomial construction, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties. The advantage of the xtpcm is demonstrated through applications to multiphase flow and solute transport in porous media, which shows that the xtpcm achieves higher statistical accuracy than does the PCM, and produces more reasonable realizations without oscillation, while computational effort is greatly reduced compared to the direct sampling Monte Carlo method. Citation: Q. Liao and D. Zhang (2013), Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location, Water Resour. Res., 49, 7911 7928, doi:10.1002/2013wr014055. 1. Introduction [2] There has been increasing interest in uncertainty quantification of geophysical analysis in recent years, which estimates the statistics of system outputs, given the probabilistic description of the model characteristics and inputs [Zhang, 2002;Xiu, 2010;Le Ma^itre and Knio, 2010].The Monte Carlo (MC) method is one of the most common approaches, in which model inputs are randomly sampled and simulated to obtain corresponding outputs that can be further analyzed statistically. However, it often requires a large number of realizations to reduce sampling errors; thus, it may become prohibitively expensive, especially for largescale problems [Ballio and Guadagnini,2004]. 1.1. Polynomial Chaos Expansion (PCE) Techniques [3] As an attractive alternative, polynomial chaos expansion (PCE) techniques have found increased use in uncertainty quantification over the past decade. First introduced by Wiener [1938], the PCE constructs a model response 1 Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California, USA. 2 College of Engineering, Peking University, Beijing, China. Corresponding author: D. Zhang, College of Engineering, Peking University, 5 Yiheyuan Rd., Beijing 100871, China. (dxz@pku.edu.cn) 2013. American Geophysical Union. All Rights Reserved. 0043-1397/13/10.1002/2013WR014055 surface by polynomial of uncertain parameters and offers an efficient way of including nonlinear effects in stochastic analysis [Ghanem and Spanos, 1991]. For parametric distributions other than Gaussian distribution (e.g., Beta, Gamma, Uniform, etc.), Xiu and Karniadakis [2002] developed generalized polynomial chaos (gpc), where stochastic solutions are expressed as different types of orthogonal polynomials to achieve better convergence. [4] In general, the PCE techniques can be categorized as either intrusive or nonintrusive approaches [Le Ma^itre and Knio, 2010]. The former approaches have been termed intrusive, as they reformulate the governing equations that require new solvers. The best-known method in this group is the stochastic finite element method, which relies on Galerkin projection reformulations [Ghanem and Spanos, 1991], and has been applied to many areas, including fluid flow in porous media [Ghanem, 1998; Deb et al., 2001]. However, in this method, the PCE coefficients are governed by a set of coupled equations, which are difficult to solve when the number of coefficients is large. [5] Nonintrusive approaches employ deterministic sampling of the forward model run that can be treated as a black box. Tatang et al. [1997] developed the probabilistic collocation method (PCM), also referred to as the stochastic collocation method (SCM), which is successfully applied to uncertainty analysis [Webster et al., 1996; Pan et al., 1997]. In the PCM, the output random field is approximated either by Lagrange interpolation [Mathelin 7911

and Hussaini, 2003; Babuska et al., 2007], or by orthogonal basis function, whose coefficients are solved through pseudospectral projection [Le Ma^itre et al., 2002; Xiu, 2007] or matrix inversion/least-squares regression [Isukapalli et al., 1998; Li and Zhang, 2007]. Initially, the PCM is found to be quite promising for low-dimensional systems based on tensor product. Later, the choices of collocation points are extended to the Smolyak sparse grid [Xiu and Hesthaven, 2005] to gain efficiency for high-dimensional problems. In general, the PCM derives an uncoupled system; thus, it is highly parallelizable and can be easily implemented with existing simulators [Shi et al., 2009]. 1.2. Motivation: Effect of Strong Nonlinearity [6] It is well recognized that global PCE-based techniques have an inherent assumption of smoothness of the output response with respect to input parameters [Ghanem and Spanos, 1991; Le Ma^itre and Knio, 2010]. In some cases, where the relation is strongly nonlinear with a large gradient, or even discontinuity, the PCE approximation could be inaccurate and suffer from oscillation (i.e., the Gibbs phenomenon) [Gottlieb and Shu, 1997]. A straightforward approach to alleviate this problem of strong nonlinearity is increasing the order of the PCE, which requires more collocation points and model runs. Lin and Tartakovsky [2009] observed inaccurate concentration moments with small dispersivity, which they attributed to insufficient collocation points. Although they utilized the sparse grid algorithm and set a relatively large initial plume to reduce discontinuity, hundreds of collocation points were still required in a very high-order polynomial approach for only four random dimensions. Consequently, the traditional PCM may become too complicated and computationally unaffordable. Additionally, there is no guarantee that a high-order polynomial will provide a satisfactory result, since the oscillation problem does not disappear as the polynomial order increases. To address the limited utility of collocation methods when dealing with strong nonlinearity, Wan and Karniadakis [2005] formulated a multielement method to discretize the space of random inputs into multiple nonoverlapping subspaces, which was later combined with the collocation method by Foo et al. [2008]. Le Ma^itre et al. [2004] presented a wavelet-based method, which naturally leads to localized decompositions. Abgrall [2007] proposed a piecewise reconstruction approach via essentially nonoscillatory techniques. These methods suggest the possibility of a robust behavior, albeit at the expense of a slower rate of convergence, owing to a larger number of model runs in the subdomains. [7] In this work, we propose a new location-based transformed probabilistic collocation method (xtpcm) to quantify the uncertainty accurately and efficiently under strongly nonlinear conditions. The main difference between the traditional PCM and the new xtpcm is that the PCM approximates the model response directly, while the xtpcm approximates an alternative variable (i.e., the location of a particular response value) that is connected to the model response through transform. Since the relation of parameter and location is much more linear than that of parameter and response, this transform process greatly improves the performance of the collocation method. Case studies for multiphase flow and solute transport models are conducted to demonstrate the advantage of the xtpcm over the PCM, e.g., higher accuracy at the same level of computational effort. The results from the MC method are used as benchmarks to compare the effectiveness of the collocation methods. [8] The problem of strong nonlinearity, such as oscillation and inaccuracy, are observed in some fields and jeopardize the collocation methods. For example, in a pistonlike displacement of two-phase immiscible flow, the saturation at a given location and time can take on only two possible values (0 and 1), with a discontinuous shock at the flooding front. The probability density function (PDF) of the saturation near the front is bimodal, which is strongly non-gaussian, even when the input log-porosity is Gaussian distributed [Gu and Oliver, 2006; Chen et al., 2009]. Under this condition, when the traditional PCM tries to approximate the saturations by polynomials of Gaussian random variables (since the input is Gaussian), it may produce inaccurate estimations of the PCE coefficients. Therefore, the realizations generated from the PCE may provide nonphysical values outside of the boundaries of 0 and 1, and the statistical moments and the PDFs may deviate from the true values. Another similar instance in solute transport is that the point concentration has a bimodal PDF, with one of the modes being zero and the other being close to the concentration at the source, with small dispersion in early times [Dagan, 1982; Bellin et al., 1994]. This non- Gaussianity, therefore, creates difficulty in predicting the concentration statistics by traditional collocation methods. It was observed by Zhang et al. [2010] that the PCM may produce nonphysical realizations with negative concentration values under certain circumstances. Actually, such problems occur not only in the nonintrusive collocation method, but also in the intrusive Galerkin method [Tryoen et al., 2010] and moment equation method [Liu et al., 2007; Jarman and Tartakovsky, 2008]. We will present the new xtpcm to address the nonlinear problem in the collocation method in the following sections. [9] This paper is organized as follows: in section 2, we review some basic concepts. In section 3, we present the new xtpcm and compare it to the traditional PCM. In section 4, we perform case studies in multiphase flow and solute transport models. In section 5, we provide some discussion, including possible extensions and limitations. Finally, we present brief conclusions in section 6. 2. Review of Basic Concepts 2.1. Karhunen-Loeve Expansion (KLE) [10] Let the input m be a random function of space coordinate x and time coordinate t, and denoted by mðx; t; hþ5mðx; tþ1m 0 ðx; t; hþ, where mðx; tþ is the mean and m 0 ðx; t; hþ is the fluctuation. The covariance is described by C m ðx 1 ; x 2 ; tþ5hm 0 ðx 1 ; t; hþm 0 ðx 2 ; t; hþi, which can be decomposed as [Ghanem and Spanos, 1991]: C m ðx 1 ; x 2 ; tþ5 X1 n51 k n f n ðx 1 ; tþf n ðx 2 ; tþ (1) where k n and f n (x, t) are the eigenvalues and eigenfunctions, respectively, and can be solved from the Fredholm 7912

equation of the second type. Thus, the random field m(x, h) can be expressed by the Karhunen-Loeve expansion (KLE) in the truncated form as: mðx; t; hþ5mðx; tþ1 XN n51 pffiffiffiffi k nfn ðx; tþn n ðhþ (2) where n n are the independent random variables. 2.2. Polynomial Chaos Expansion (PCE) [11] Assume that the model output s is treated as a random process s(x, t, h). Since the process is usually non- Gaussian, and the covariance structure of the output field is not known in advance, the output cannot be expanded by using the KLE. Instead, the polynomial chaos expansion (PCE), as a more general representation of the random field, may be used to express the output as [Ghanem and Spanos, 1991]: ^sðx; t; hþ5a 0 ðx; tþ1 X1 1 X1 X i1 i 151 i 251 i 151 a i1 ðx; tþc 1 ðn i1 ðhþþ a i1i 2 ðx; tþc 2 ðn i1 ðhþ; n i2 ðhþþ1::: (3) where a 0 (x, t) and a i1 i 2 :::i j ðx; tþ are deterministic coefficients, and the basis C j ðn i1 ; n i2 ; :::; n ij Þ are a set of polynomial chaos with respect to the independent random variables n i1 ; n i2 ; :::; n ij [Wiener, 1938]. For independent Gaussian random variables, C j ðn i1 ; n i2 ; :::; n ij Þ are the multidimensional Hermite polynomials of degree j [Ghanem and Spanos, 1991]. In the case of other random distributions, generalized polynomial chaos (gpc) [Xiu and Karniadakis, 2002] can be used to represent the random field (e.g., Legendre-Chaos for uniform distribution, and Jacobi- Chaos for Beta distribution). Equation (3) is usually truncated by a finite random dimension N and degree M as: ^sðx; t; nðhþþ5 XQ a i ðx; tþw i ðnðhþþ; Q5ðN1MÞ!=ðN!M!Þ (4) where a i (x, t) are the PCE coefficients, and n5ðn i1 ; n i1 ; :::; n im Þ T and W i ðnðhþþ are the orthogonal basis functions such that hw i ; W j i5d ij. In general, the convergence rate of the PCE depends on the smoothness of the function s in terms of random parameter n. 2.3. Lagrange Interpolation [12] In the univariate case, let fn i g P 2 X be a set of prescribed nodes in the random space X, where P is the number of nodes. A Lagrange interpolation of the model response can be written as: ^sðx; t; nþ5 XP sðx; t; n i ÞL i ðnþ (5) where fl i ðnþg P are the corresponding Lagrange interpolation basis function. In the univariate case, the number of interpolation nodes is the same as the number of interpolation bases (also the number of orthogonal bases). Therefore, the interpolation approach is equivalent to a matrix inversion process, because of the uniqueness of the univariate interpolation structure [Xiu, 2010]. In the multivariate case, the model response can be interpolated from tensor product construction [Babuska et al., 2007] or sparse grid construction [Xiu and Hesthaven, 2005]. 3. Location-based Transformed Probabilistic Collocation Method (xtpcm) [13] The collocation method can be derived from the weighted residual method. Consider a stochastic partial differential equation Lsðx; t; hþ5f ðx; t; hþ (6) where L is a differential operator, s is the model response, f is a known function, and the random parameter h belongs to the event space X. Define a residual R as: Rðx; t; hþ5l^sðx; t; hþ2f ðx; t; hþ (7) [14] The notion in the weighted residual method is to solve equation (6) in a weak form by forcing the residual to zero in some average sense over the domain. That is: ð X Rðx; t; hþxðhþqðhþdh50 (8) where x is the weight function, and q is the probability density function. In the collocation method, the weight function is chosen as the Dirac delta function x j ðnðhþþ5d nðhþ2n j ðhþ, which leads to an uncoupled system as: Rðx; t; n j Þ50 (9) 3.1. Preprocessing [15] In the KLE based PCM or xtpcm, the input random field m is expanded by equation (2). To construct polynomial approximation for the output response, we choose P sets of collocation points n j ; j51; 2; :::; P for the random process of input, where n5ðn 1 ; n 2 ; :::; n N Þ T. By running the forward model P times correspondingly, we get P sets of output sðx; t; n j Þ; j51; 2; :::; P. The collocation points in the xtpcm could be exactly the same as those in the PCM, which means that we can improve the existing PCM results by the xtpcm without additional model runs. For better comparison, we will use the same collocation points in the PCM and the xtpcm for the case studies in this work. [16] The selection of collocation points is an important component of the collocation method, especially in highdimensional problems. Although tensor product construction makes mathematical analysis accessible [Babuska et al., 2007], the total number of nodes for an M-degree N- dimension polynomial construction is (M 1 1) N, which grows exponentially as N increases (i.e., the curse of dimensionality ). To reduce the computational burden, the sparse grid strategy based on the Smolyak algorithm, is employed by a linear combination of product formulas as [Xiu and Hesthaven, 2005]: 7913

Aðq; NÞ5 X q2n11jijq LIAO AND ZHANG: TRANSFORMED PROBABILISTIC COLLOCATION METHOD N21 ð21þ q2jij q2jij! U i1 U in (10) where U j is the one-dimensional interpolation formula for dimension j, N is the number of dimensions, the sparseness parameter q determines the order of the formula, and jij 5 i 1 1 i 2 1...1 i N. Set k 5 q N as the level of the Smolyak construction. To compute A(q, N), we only need to evaluate the function on the sparse grid: Hðq; NÞ5[ q2n11jijq H i1 1 3 3HiN 1 (11) where H i j 1 is the one-dimensional collocation point set for dimension index j, and Hðq; NÞ denotes the N-dimensionaln collocation point set. For functions ino space FN M5 f : ½21; 1ŠN! Rj@ jij f continuous; i j M; 8j, the interpolation error follows: ki N 2Aðq; NÞk 1 C N;l P 2M ðlog PÞ ðm12þðn21þ11 (12) where I N is the identity operator in an N-dimensional space, and P is the number of interpolation points. We can see that the convergence rate depends weakly on the dimension N, but strongly on the smoothness index M. Barthelmann et al. [2000] illustrate that the approximation result is poor for the unsmooth function and completely useless for the discontinuous function. It is noted that the sparse grid approximation may lose accuracy as the number of dimensions increases [Formaggia et al., 2013], and the extension of this approximation to high-dimensional problems is still an open research topic. In sum, the sparse grid strategy can somewhat reduce the number of collocation points, but the problems of high dimensionality and strong nonlinearity still exist and call for proper solutions. Other algorithms based on ad hoc selection of collocation points [Isukapalli et al., 1998; Li and Zhang, 2007] can also be considered. Such algorithms are motivated by using as few collocation points as possible while keeping a low approximation error, and are usually carried out through retaining the nodes with the highest probability for the input random variables. 3.2. Forward Transform From Response to Location [17] As we can see, the direct approximation of model response by the polynomial is not suggested in strongly nonlinear problems. A possible solution is to find an alternative variable to represent the model response, which is relatively linear to the random parameter. Intuitively inspired by the Lagrangian point of view, the location of a particular response value is a good candidate for transform from the Eulerian coordinates, especially for advectiondominated problems. For 1-D problems, it is actually a process of finding iso-points. If the output value is a monotonic function of location in the space domain, the approach is relatively simple (e.g., linear interpolation). Otherwise, there could be nonunique locations for a given response value, and these multiple locations can be analyzed individually. As for two or three-dimensional (2-D or 3-D) problems, the transform approach leads to searching for isolines (contours) or iso-surfaces, respectively. In some cases (e.g., homogeneous media), it is possible to parameterize the response interfaces (iso-lines or iso-surfaces) by certain simple geometrical shapes, which will be discussed in detail in section 4.5. Unfortunately, under general conditions, the interfaces are usually complicated and difficult to interpret. Possible solutions including level set method and motion analysis via digital image processing techniques, are ongoing and will be presented soon. [18] One issue that needs to be considered is that, if the minimum or maximum values of the output in the space domain are changed, we may not be able to determine the location of a certain value in some realizations. Therefore, we should first normalize the outputs into the same scale, and then perform interpolation. Define a relative value after normalization as: s r ðx; t; hþ sðx; t; hþ2s min ðt; hþ s max ðt; hþ2s min ðt; hþ s min ðt; hþ5min sðx; t; hþ x2d s max ðt; hþ5max sðx; t; hþ x2d (13) where the scaling factors s min and s max are the minimum and maximum value over the space domain D, respectively, which will be analyzed together with the location and used for generating samples in the following steps. Essentially, the location x of any given response value ~s is now represented by a level set, where the normalized model response s r (as a function of x) takes on the constant ~s at a certain time t as: xð~s; t; hþ fxjs r ðx; t; hþ5~s g (14) 3.3. Polynomial Construction [19] In the traditional PCM, the model response is approximated by the polynomial through two major approaches: the orthogonal basis approach [Le Ma^itre et al., 2002; Xiu, 2007], and the Lagrange interpolation approach [Mathelin and Hussaini, 2003; Babuska et al., 2007]. In the former orthogonal basis approach, the model response s is approximated by orthogonal polynomials as equation (4). According to equation (9), we set ^sðx; t; n j Þ5sðx; t; n j Þ and obtain: X Q a i ðx; tþw i ðn j Þ5sðx; t; n j Þ; j51; 2; :::; P (15) [20] To determine the PCE coefficients a i ðx; tþ; ; 2; :::; Q, a pseudospectral projection process can be applied as [Xiu, 2007]: a i ðx; tþ5 hsðx; t; nþ; W Ð iðnþi hw i ðnþ; W i ðnþi 5 X sðx; t; nþw iðnþqðnþdn hw i2 i P Q j51 sðx; t; n j ÞW i ðn j Þw j Q j51 hw i2 i (16) where n j ; w j are a set of nodes and associated weights, such that the summation approximates the integral. An 7914

alternative process is to treat equation (15) as a linear system in the matrix form [Li and Zhang, 2007]: Za5s (17) where a5ða 1 ; a 2 ; :::; a Q Þ T ; s5ðs 1 ; s 2 ;:::;s P Þ T and Z ji 5W i ðn j Þ is a P3Q Vandermonde-like matrix. For a well-posed problem, P Q is required. Thus, the PCE coefficients could be attained by weighted least-squares regression as: a5 Z T 21WZ WZ T s (18) where W is a diagonal weight matrix with W jj 5 w j. It can be shown that the solutions by equation (16) and (18) are equivalent, with the same set of collocation points and orthogonal bases. While in the Lagrange interpolation approach, the model response is approximated by Lagrange polynomials. [21] As for the xtpcm, the location in equation (14) as an alternative variable (instead of the model response in the PCM) is approximated by the polynomial. Similarly, there are two major approaches: the orthogonal basis approach and the Lagrange interpolation approach. In the former orthogonal basis approach, assume that the location is represented by the PCE similar to equation (3) as a random process: ^xð~s; t; hþ5b 0 ð~s; tþ1 X1 1 X1 X i1 i 151 i 251 i 151 b i1 ð~s; tþc 1 ðn i1 ðhþþ b i1i 2 ð~s; tþc 2 ðn i1 ðhþ; n i2 ðhþþ1::: (19) and truncated by a finite number similar to equation (4) as: ^xð~s; t; nðhþþ5 XQ b i ð~s; tþw i ðnðhþþ (20) [22] The PCE coefficients b i ð~s; tþ; ; 2; :::; Q are vectors, since the location x is a vector in space domain (e.g., x 5 (x, y, z) in 3-D space). According to equation (9), we set ^xð~s; t; n j Þ5xð~s; t; n j Þ and obtain: X Q b i ð~s; tþw i ðn j Þ5xð~s; t; n j Þ; j51; 2; :::; P (21) [23] The coefficients b i ð~s; tþ can also be solved through pseudospectral projection as: b i ðx; tþ5 hxð~s; t; nþ; W Ð iðnþi hw i ðnþ; W i ðnþi 5 X xð~s; t; nþw iðnþqðnþdn hw i2 i P Q j51 xð~s; t; nþw i ðn j Þw j hw i2 i or weighted least-squares regression: (22) B5 Z T 21WZ WZ T X (23) where Z ji 5W i ðn j Þ; B5ðb 1 ; b 2 ;:::;b Q Þ T ; X5ðx 1 ; x 2 ;:::;x P Þ T. [24] Similar to the location, the scaling factors s min and s max in equation (13) are approximated by the PCE, as well: ^s min ðt; nðhþþ5 XQ ^s max ðt; nðhþþ5 XQ b min ;i ðtþw i ðnðhþþ; b max ;i ðtþw i ðnðhþþ (24) [25] By setting ^s min ðt; n j Þ5s min ðt; n j Þ and ^s max ðt; n j Þ5 s max ðt; n j Þ, the PCE coefficients for scaling factors b min ;i ðtþ and b max ;i ðtþ can be solved through pseudospectral projection as: b min ;i ðtþ b max ;i ðtþ P Q j51 P Q j51 s min ðt; n j ÞW i ðn j Þw j ; hw i2 i s max ðt; n j ÞW i ðn j Þw j hw i2 i or weighted least-squares regression: (25) b min 5 Z T 21WZ WZ T s min ; b max 5 Z T 21WZ WZ T s max (26) [26] While in the Lagrange interpolation approach, the location and scaling factors are approximated by Lagrange polynomials similar to equation (5) in the univariate case as: ^s min ðt; nþ5 XP ^xð~s; t; nþ5 XP s min ðt; n i ÞL i ðnþ; xð~s; t; n i ÞL i ðnþ ^s max ðt; nþ5 XP s max ðt; n i ÞL i ðnþ (27) [27] As for the multivariate case, the location and scaling factors can be interpolated from tensor product construction [Babuska et al., 2007] or sparse grid construction [Xiu and Hesthaven, 2005]. [28] In the orthogonal basis approach, after substituting equation (22) or (23) into equation (20) and generating a sufficient number of realizations, we may obtain the location samples ^xð~s; t; hþ. Similarly, we can get the scaling factor samples ^s min ðt; hþ and ^s max ðt; hþ from combining equation (24) and equation (25) or (26). In the Lagrange interpolation approach, the samples of location and scaling factors can be generated through tensor product or sparse grid construction based on equation (27). 3.4. Backward Transform From Location to Response [29] The location samples ^xð~s; t; hþ may be transformed back to obtain the response samples over the domain ~sðx; t; hþ, by finding the output values in certain places (e.g., interpolation) as: ~sðx; t; hþ f~sj^xð~s; t; hþ5xg (28) [30] Recall that the outputs are normalized; thus, they have to be restored to the original state according to equation (13) as: 7915

Figure 1. Illustration of the difference between the traditional PCM and the xtpcm. ^sðx; t; hþ5^s min ðt; hþ1~sðx; t; hþ½^s max ðt; hþ2^s min ðt; hþš (29) [31] These responses then serve as the samples for statistical analysis of moments and PDFs. 3.5. Postprocessing [32] In the traditional PCM, the moments of model response, such as mean and variance, can be directly calculated from the PCE coefficients obtained from equation (16) or (18) based on the orthogonality of basis function W i as: hsi5a 1 ; r s 2 5 XM i52 a i 2 hw 2 i i (30) [33] An alternative way is to first generate a sufficient number of response samples from equation (4) or (5), then estimate the moments from the ensemble as: hsi5 1 N s X Ns s i ; r s 2 5 1 N s X Ns ðs i 2hsi Þ 2 ; E½ðs2hsi Þ n Š5 1 N s X Ns ðs i 2hsiÞ n (31) where N s is the ensemble size. The advantage of this sampling approach is that we can produce output realizations and obtain PDFs. [34] While in the xtpcm, we may estimate the statistical moments similar to equation (31), but from the transformed samples in equation (29). Furthermore, calculating response moments without the sampling process in the xtpcm may be impractical, since we would obtain the location moments instead of the model response moments. 3.6. Framework of the xtpcm [35] For better comparison, we first present the three major steps involved in the PCM: [36] 1. Preprocessing, i.e., representing the model input in terms of independent random parameters, selecting the collocation points and performing deterministic model runs at these points (section 3.1); [37] 2. Constructing polynomial of the model response, i.e., approximating the response by orthogonal basis function or Lagrange interpolation (section 3.3); [38] 3. Postprocessing, i.e., estimating the statistical properties of the model response either from the PCE coefficients directly or from sampling (section 3.5). [39] The five major steps involved in the xtpcm are: [40] 1. Preprocessing, i.e., representing the model input in terms of independent random parameters, selecting the collocation points and performing the deterministic model runs at these points (section 3.1); [41] 2. Transforming the model response to the location of a particular response value (along with scaling factor) (section 3.2); [42] 3. Constructing the polynomial of the location, i.e., approximating the location by orthogonal basis function or Lagrange interpolation, and generating location samples (section 3.3); [43] 4. Transforming the location samples back to the response samples (section 3.4); [44] 5. Postprocessing, i.e., estimating the statistical properties of the model response from the samples (section 3.5). [45] It can be seen that the first and last steps are exactly alike in the PCM and xtpcm. The differences, however, are: the xtpcm has two more steps for transforms between the response and the location, and the target of polynomial construction is changed from the response to the location in the middle step (Figure 1). 4. Illustrative Examples 4.1. Governing Equations [46] In this study, the properties of porous media, such as permeability and porosity, are treated either as a homogeneous random constant or as a heterogeneous random field. The saturation in multiphase flow or concentration in solute transport is considered as a model response. Thus, the governing equation becomes a stochastic partial differential equation whose solution is no longer deterministic, but probabilistic and characterized by statistical properties. 4.1.1. Multiphase Flow Model [47] The oil/water two-phase flow model for saturation tests is expressed by the following continuity equation as [Bear, 1972]: jðxþk ri r ðrp i 2q l i grzþ 1q i 5/ðxÞ @S i ; i5w; o (32) i @t where x and t denote location and time, respectively; w and o denote the two phases (i.e., water and oil); / is the porosity of the media; l i ; p i ; q i ; S i are the viscosity, pressure, density, and saturation of phase i, respectively; j is the absolute permeability; k ri is the relative permeability of phase i, which is a function of S i ; q i is the source or sink term; g is the gravitational acceleration; and z is the depth. Equation (32) is coupled with: S w 1S o 51 (33) p c ðs w Þ5p o 2p w where p c is the capillary pressure, which is a function of S w. 7916

Figure 2. Saturation profiles: directly generated from deterministic model runs by (a) MC method, (b) second-order PCM, and (c) fourth-order PCM; and randomly generated from polynomial approximation by (d) second-order xtpcm, (e) second-order PCM, and (f) fourth-order PCM. [48] The above equations can be simplified into a nonlinear advection-diffusion equation. When the capillary term is small compared to the advection term, the equation is hyperbolic and advection-dominated [Zhang, 2002]. With large capillarity, the shape of the saturation curve will be relatively smooth. With small capillarity, however, the curve would indicate the movement of a persistent sharp front between the two phases. It has been shown that capillarity may be neglected for studying oil/water displacement in oil reservoirs [Marle, 1981], which leads to a pureadvection problem and is usually called Buckley-Leverett displacement [Buckley and Leverett, 1942]. In the homogeneous case, analytical solutions to this problem are wellknown by the method of characteristics with a discontinuous shock wave [Marle, 1981]. This discontinuity in the space domain will be translated to discontinuity in the random domain, which makes it extremely difficult to quantify the saturation uncertainty by traditional collocation methods. Similar problems are observed in heterogeneous cases with spatially correlated random field, where the model is implemented with the commercial Eclipse black-oil simulator. 4.1.2. Solute Transport Model [49] The solute transport model for concentration tests under advection and dispersion is given as [Bear, 1972]: @cðx; tþ 5r D ij ðx; tþrcðx; tþ 2vðx; tþrcðx; tþ (34) @t where x is the location, t is the time, c is the solute concentration, D ij is the hydrodynamic dispersion tensor, v is the pore water velocity, computed by vðx; tþ5uðx; tþ=/. And / is the porosity, u is the Darcy velocity, computed by uðxþ52kðxþrhðxþ, where K(x) is the hydraulic conductivity, and h(x) is the hydraulic head. The hydrodynamic dispersion tensor is given by: D ij 5ða L 2a T v i v j Þ jvðx; tþj 1a T jvðx; tþjd ij 1D d d ij (35) where a L is the longitudinal dispersivity, a T is the transverse dispersivity, jvðx; tþj is the magnitude of the pore velocity, D d is the molecular diffusion coefficient, and d ij is the Kronecker operator (d ij 51; i5j; and d ij 50; i 6¼ j). [50] With large dispersivity, the concentration profile is relatively smooth. With small dispersivity, however, the profile has a large gradient. In the absence of dispersion, it would be similar to a Dirac delta function with strong discontinuity. This advection-dominated phenomenon in a space domain will lead to a strong nonlinearity in a random domain, hence strong non-gaussianity for model response. As pointed out by Dagan [1982] and Bellin et al. [1994], the point concentration at a specific location and time has a bimodal distribution with small dispersion at early times. Zhang et al. [2010] reported that the traditional PCM may produce nonphysical oscillatory realizations with negative concentration values under certain circumstances, due to nonlinearity and non-gaussianity. Such problems can be addressed by the new xtpcm, which will be discussed in detail in the following sections. 4.2. One-Dimensional Homogeneous Cases 4.2.1. Multiphase Flow Tests [51] Consider an oil/water two-phase flow problem involving water flooding from one end of a 1-D 7917

Figure 3. Saturation as a function of random parameter at locations: (a) x 5 0.2, (b) x 5 0.3, and (c) x 5 0.4; and corresponding PDF of saturation at these locations: (d) x 5 0.2, (e) x 5 0.3, and (f) x 5 0.4. The log-porosity ln(u) 5 0.2 1 0.316 h, where h N(0,1). homogeneous aquifer. The absolute permeability is constant, and the relative permeabilities are Corey-type functions as k rw 5S 2 w ; k ro5ð12s w Þ 2. The viscosity ratio is assumed to be 1. The injected volume is set to be 0.05 of the total volume. Gravity, capillary pressure and compressibility are neglected. In this standard Buckley-Leverett problem, let the log-porosity be Gaussian distributed with a mean of ln(0.2) and variance of 0.1, as ln(u) 5 ln(0.2) 1 0.316 h, where h N(0,1). The saturation as a model response is solved analytically by the method of characteristics. To obtain a converged MC reference, 10,000 samples of the parameter h are randomly generated to obtain the saturation realizations (Figure 2a). In the collocation methods, however, the KLE is not needed since there is only one random parameter in this homogeneous case. The collocation points are selected as the roots of Hermite polynomial according to Gaussian quadrature. Figures 2b and 2c show the corresponding realizations, where the saturation profile consists of a discontinuous shock and a continuous rarefaction. [52] In the MC approach, statistical properties are calculated directly from the 10,000 realizations. While in the PCM, we construct polynomials from the realizations on collocation points (Figures 2b and 2c) to obtain the relation between the parameter and the response (Figure 3). Figure 3a shows that the true relation (the MC reference) between the output saturation and the input parameter is strongly nonlinear at x 5 0.2. While the second-order PCM approximates this discontinuous/nonlinear function with a secondorder polynomial by fitting or interpolating the data at three collocation points, hence results in Gibbs oscillation. The fourth-order PCM approximates this function with a fourthorder polynomial through the data at five collocation points, and the result is not much improved as the number of collocation points increases. Figure 3d shows that the true saturation PDF from the MC method is bimodal, but the second or fourth-order PCM fails to capture this non- Gaussian phenomenon. Similar results can be observed in other locations (x 5 0.3 in Figures 3b and 3e, and x 5 0.4 in Figures 3c and 3f). [53] Since the function in Figure 3a is discontinuous at only one place in the random domain, it is natural to think that changing the collocation points (e.g., adding more points near the discontinuity, or moving the points next to the discontinuity) would help resolve the problem. However, it may not be practical in three aspects: (1) the position of discontinuity is not known beforehand and may be difficult to detect; (2) the position of discontinuity varies in different spatial locations, thus it is almost impossible to capture all discontinuities from one set of collocation points; (3) the oscillation occurs not only in discontinuous problems, but also in problems with steep profiles or large gradients, in which it is not clear how to change the collocation points. Similarly, the idea that decomposing the entire random space into a few smooth subdomains (e.g., on either side of the discontinuity) also suffers from these factors. [54] In the xtpcm, we analyze the location of particular saturation values, which can be obtained via forward transform from the realizations in Figure 2. In this test, the normalization process is not required, as the saturation boundaries are fixed. Figure 4 conveys the relation between the random parameter and the location, as well as the corresponding location PDF. It is seen that the location of particular response S w 5 0.2 in Figure 4a is much more linear to the random parameter than the response itself in Figure 3a. 7918

Figure 4. Location as a function of random parameter at particular saturation values: (a) S w 5 0.2, (b) S w 5 0.5, and (c) S w 5 0.8; and corresponding location PDF at these saturations: (d) S w 5 0.2, (e) S w 5 0.5, and (f) S w 5 0.8. The log-porosity ln(u) 5 0.2 1 0.316 h, where h N(0,1). Therefore, the polynomial approximation from the secondorder xtpcm with three collocation points almost overlaps the MC reference. Consequently, the corresponding location PDF (Figure 4d) is closer to Gaussian distribution and much easier to match than the response PDF (Figure 3d). Similar results can be observed in Figure 4 for S w 5 0.5 and 0.8, as well. Since the second-order xtpcm needs only three realizations from the model runs, which are the same as the second-order PCM (i.e., Figure 2b), computational effort is greatly reduced compared to the MC method. The fourth-order xtpcm is not shown here as the second-order xtpcm already performs adequately. [55] In the PCM, a sufficient number (we use 1000 here) of saturation realizations can be randomly generated from the constructed polynomial approximation. This sampling process (with only addition, subtraction and multiplication, but without division or matrix inversion) is actually very fast without deterministic model runs. As for the xtpcm, the same number of location samples is generated and transformed back to obtain the saturation samples through linear interpolation. Figure 2e shows 50 realizations generated from the second-order PCM by the same parameters as those used in Figure 2a. It can be seen that the results are nonphysical in three aspects: (1) oscillatory saturation value outside the boundary of 0 and 1, caused by polynomial fitting/interpolation near discontinuity; (2) lower value at the upstream and higher value at the downstream, because the saturation is analyzed individually at each location in the PCM; and (3) zero output variability at about x > 0.55, since the saturations from three deterministic model runs are all 0 at these places. While in the fourthorder PCM, although a higher-order polynomial and a larger number of collocation points are used, similar nonphysical problems are still observed in Figure 2f. On the contrary, 50 saturation realizations from the second-order xtpcm are much more reasonable, as shown in Figure 2d. To further illustrate the validity of the xtpcm, we also investigate the relation between the parameter and the saturation, as well as the saturation PDFs, as shown in Figure 3, which are obtained from the transformed saturation samples. Compared to the result from the PCM, the result from the xtpcm almost overlaps with the MC reference, even for the complex discontinuous relation and the bimodal PDF. [56] Since model uncertainties are usually quantified by statistical moments, we compare up to fourth-order moments of model response due to non-gaussianity. As illustrated in Figure 5, the results of the PCM may deviate from those of the MC method, with abnormal discontinuous curves. Such problems are also observed for the fourthorder PCM. On the other hand, the second-order xtpcm clearly outperforms both the second and fourth-order PCM. Note that the xtpcm, with only three realizations, generates almost the same results as the MC reference with 10,000 realizations. For better comparison, we also investigated the effect of ensemble size in the MC method, and observed that the moments are not accurate enough until the ensemble size reaches about 1000 (not shown here due to limited space). Two additional cases with different logporosity variability are implemented, where the means and variances of the saturation are compared (Figure 6). The mismatch between the PCM results and the MC results becomes more obvious as the variability grows, while the xtpcm results keep consistent with the MC results. 7919

Figure 5. Statistical moments of saturation by the MC method and collocation methods: (a) mean; (b) variance; (c) third-order moment; and (d) fourth-order moment. 4.2.2. Solute Transport Tests [57] In this section, we will apply the collocation methods to a solute transport model and analyze the concentration uncertainties. In the above examples, the responses are monotonic functions of locations; thus, there is only one possible location for a particular response value. To verify the xtpcm in examples with nonmonotonic functions, we design a 1-D solute transport test with instantaneous injection at the origin initially, whose analytical solution is given as [Bear, 1972]: c 0 cðx; tþ5 p 2/ ffiffiffiffiffiffiffiffiffiffi e 2ðx2vtÞ2 4D L t (36) pd L t [58] The domain is 30 m long, and the two boundaries are imposed with constant heads at 8 m and 5 m, respectively. The porosity is assumed to be constant at 0.1, the dispersivity a L is 0.1 m, and the input log-conductivity (in m/day) follows a Gaussian distribution with a mean of 0 and variance of 0.1, as ln(k) 5 0.316 h, where h N(0,1). The concentration at time t 5 10 days is investigated as a Figure 6. Statistical moments of saturation with different input variability, r 2 ln / 50:01: (a) mean and (d) variance; r 2 ln / 50:1: (b) mean and (e) variance; r 2 ln / 51: (c) mean and (f) variance. 7920

Figure 7. Concentration profiles: directly generated from deterministic model runs by (a) MC method, (b) second-order PCM, and (c) fourth-order PCM; and randomly generated from polynomial approximation by: (d) second-order xtpcm, (e) second-order PCM, and (f) fourth-order PCM. model response. Similar to the saturation tests, 10,000 realizations are carried out in the MC method as benchmarks. [59] Figure 7 shows the concentration profile from direct model simulation. Compared to Figure 2, there is no discontinuous shock wave, but a continuous steep front. Note that the maximum concentration in the domain varies among different realizations. To ensure that there are two locations for any given concentration value, each concentration profile is normalized by its maximum value c max (c min is zero as a constant).therefore, the profile is rescaled in the range [0, 1] as a relative concentration c r in equation (13). For a certain concentration level, there are two locations, which can be approximated by the polynomial individually as x 1 and x 2. Figure 8 shows the approximations of c max, x 1 and x 2 by the xtpcm in the forms of c max ðt; hþ, x 1 ð~c; t; hþ, andx 2 ð~c; t; hþ as equations (20) and (24). Here the illustrated x 1 and x 2 are related to the relative concentration level ~c50:5, while other levels can be treated similarly. It is seen that these transformed variables are relatively linear to the random parameter, and that the corresponding PDFs are close to Gaussian distribution. Therefore, the exact results from the MC approach are easily approximated by the xtpcm after transform, as in Figure 4. Figure 9 shows the relations between concentration and random parameter at various locations and corresponding PDFs, which are similar to Figure 3. The randomly generated realizations from the polynomial approximation by the PCM and the xtpcm are illustrated in Figures 7d 7f, with similar observations from Figure 2. Nonphysical problems are also observed for the PCM, while the xtpcm conveys reasonable results that resemble the MC method. [60] Two additional cases with different dispersivities are implemented to test the effect of dispersion from the statistical moments. When the dispersivity is small (a L 5 0.01 m), the problem is advection-dominated and the transform approach is necessary, as shown in Figures 10a and 10d. When the dispersivity is large (a L 5 1 m), the concentration profile becomes relatively smooth, and the results from the PCM are somewhat acceptable (still not as accurate as those from the xtpcm), as illustrated in Figures 10c and 10f. In general, the xtpcm clearly demonstrates an advantage over the PCM. 4.3. One-Dimensional Heterogeneous Case [61] Let the model input be a spatially correlated random field in this heterogeneous case. Consider a water-flooding problem with a 1-D domain of 5000 3 100 3 100 ft (1 ft 0.305 m), which is divided into 100 3 1 3 1 grid blocks. The initial reservoir pressure is 4000 psi (1 psi 6895 Pa). There is one water injection well at one end with a constant injection rate of 100 barrels per day, and one production well at the other end with a constant pressure of 3000 psi. The absolute permeability is assumed to be constant at 20 md (1 md 9.87e216 m 2 ), since saturation is not sensitive to permeability in this boundary condition. The irreducible water saturation (also the initial condition) is S wi 5 0.2, and the residual oil saturation is S or 5 0.3. The relative permeability and capillary pressure follows the Brooks-Corey type model [Brooks and Corey, 1964]: 31 k rw 5S 2 e k ; kro 5ð12S e Þ 2 11 ð12s 2 e k Þ; 2 pc 5p b S 1 e k ; Se 5 S w2s wi 12S or 2S wi (37) where p b is bubbling pressure and set as 0 here for simplicity, and k is a pore-size distribution index and set as 2. Assume that the log-porosity to be a second-order 7921

Figure 8. Transformed variables as a function of random parameter: (a) scaling factor c max, (b) first location x 1, and (c) second location x 2 ; and corresponding PDFs: (d) scaling factor c max, (e) first location x 1, and (f) second location x 2. The log-conductivity ln(k) 5 0.316 h, where h N(0,1). stationary Gaussian random field with a mean of 0.1, and covariance C ln / 5r 2 ln / exp ð2jx 12x 2 j=gþ, where r 2 ln / 50:5 and g=l50:2, where L is the domain length. The deterministic forward model run is implemented by the Eclipse simulator numerically. In the MC method, 1000 direct simulations are carried out. In the collocation method, the Figure 9. Concentration as a function of random parameter at locations: (a) x 5 7.5, (b) x 5 10, and (c) x 5 12.5; and corresponding PDFs of concentration at locations: (d) x 5 7.5, (e) x 5 10, and (f) x 5 12.5. The log-conductivity ln(k) 5 0.316 h, where h N(0,1). 7922

Figure 10. Moments of concentration with different dispersivities, a L 5 0.01 m: (a) mean, (d) variance; a L 5 0.1 m: (b) mean, (e) variance; and a L 5 1 m: (c) mean, (f) variance. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X Nb RMSE5t ðy estimation i 2y reference i Þ 2 (39) N b log-porosity random field is expressed by the KLE as equation (2) with truncated dimension N. The collocation points are selected based on a level-one Smolyak sparse grid as equation (11); other algorithms, such as tensor product or regular grid, can be used as well. The one-dimensional nodal set H i 1 is defined as the roots of Hermite polynomial with order 2i 2 1 (e.g., H 1 1 5f0g, H2 1 5f2 p ffiffiffi pffiffiffi 3 ; 0; 3 g) according to Gaussian quadrature. Therefore, the number of collocation points is P 5 2N 1 1as: n 1 5½0; 0; :::; 0Š T pffiffi n 2 5½0; 0; :::; 3 Š T p ffiffi ; n 3 5½0; :::; 3 ; 0Š T p ffiffi ; :::; n N11 5½ 3 ; 0; :::; 0Š T p n N12 5½0; 0; :::; 2 ffiffi 3 Š T p ; n N13 5½0; :::; 2 ffiffi 3 ; 0Š T ; :::; p n 2N11 5½2 ffiffi 3 ; 0; :::; 0Š T (38) [62] The polynomial approximating the model response is constructed as equation (10), with Lagrange interpolation approach (the orthogonal polynomial can be considered, as well). [63] Figure 11 shows that as N increases, the xtpcm will be more accurate. If N 5 10 dimensions of KLE are retained, then the xtpcm with P 5 21 times of model runs provide almost exactly the same results as the MC method from 1000 times of simulation. However, the traditional PCM even diverges with larger number of retained random dimension N and collocation points P. This is possibly because the polynomial fitting/interpolation process is more challenging for high-dimensional problems with strong nonlinearity/discontinuity. To better compare the performance of these two collocation methods with respect to the random dimension, we define a root mean square error (RMSE) corresponding to an L2 error norm as: where y denotes the target variable (i.e., mean saturation or saturation variance, in this case), i is the grid block index, and N b is the total number of blocks. The error measures the deviation of the solutions by collocation methods from the MC solutions. Following common sense, the result of the collocation methods is expected to converge to the MC reference, as the number of retained random dimensions in KLE increases and the target covariance function becomes more accurate [Rupert and Miller, 2007; Chang and Zhang, 2009]. However, in this strongly nonlinear case, the error from the PCM does not decline, but rather inclines (Figure 12). On the other hand, the error from the xtpcm keeps decreasing. 4.4. Two-Dimensional Case [64] Consider a homogeneous domain of 30 3 3 m. The left and right sides are assigned as constant heads of 8 m and 5 m, respectively; whereas, the top and bottom boundaries are impervious. If the instantaneous point source is released at the origin initially, the analytical solution is given as [Bear, 1972]: c 0 cðx; tþ5 p 4p/ ffiffiffiffiffiffiffiffiffiffiffiffi D L D T t e2 h i ðx2vtþ 2 4D L t 1 y2 4D T t (40) [65] The conductivity is assumed to be constant at 1 m/ day, and the random log-porosity is Gaussian distributed with a mean of 0.1 and variance of 0.1. The longitudinal dispersivity a L is 0.1 m, and the transverse dispersivity a T 7923