Numerical Studies of Three-dimensional Stochastic Darcy’s ...

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J Sci Comput (2010) 43: 92–117 DOI 10.1007/s10915-010-9346-5 Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation G. Lin · A.M. Tartakovsky Received: 3 November 2008 / Revised: 7 July 2009 / Accepted: 8 January 2010 / Published online: 22 January 2010 © US Government 2010 Abstract Solute transport in randomly heterogeneous porous media is commonly de- scribed by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated. Keywords Probabilistic collocation method · Sparse grids · Porous media · Polynomial chaos 1 Introduction Natural subsurface environments cannot be deterministically characterized in all details due to high spatial variability or scarcity of data and should be modeled as spatially correlated random fields. This renders equations describing the flow and transport of contaminants in the subsurface stochastic [14]. G. Lin ( ) · A.M. Tartakovsky Computational Mathematics, Pacific Northwest National Laboratory, 902 Battelle Blvd., Box 999, Richland, WA 99352, USA e-mail: [email protected]

Transcript of Numerical Studies of Three-dimensional Stochastic Darcy’s ...

J Sci Comput (2010) 43: 92–117DOI 10.1007/s10915-010-9346-5

Numerical Studies of Three-dimensional StochasticDarcy’s Equation and StochasticAdvection-Diffusion-Dispersion Equation

G. Lin · A.M. Tartakovsky

Received: 3 November 2008 / Revised: 7 July 2009 / Accepted: 8 January 2010 /Published online: 22 January 2010© US Government 2010

Abstract Solute transport in randomly heterogeneous porous media is commonly de-scribed by stochastic flow and advection-dispersion equations with a random hydraulicconductivity field. The statistical distribution of conductivity of engineered and naturallyoccurring porous material can vary, depending on its origin. We describe solutions of athree-dimensional stochastic advection-dispersion equation using a probabilistic collocationmethod (PCM) on sparse grids for several distributions of hydraulic conductivity. Threerandom distributions of log hydraulic conductivity are considered: uniform, Gaussian, andtruncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève(K-L) decomposition as a second-order random process with an exponential covariancefunction. The convergence of PCM has been demonstrated. It appears that the accuracyin both the mean and the standard deviation of PCM solutions can be improved by using theJacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaosfor the Gaussian distribution. The effect of type of distribution and parameters such as thevariance and correlation length of log hydraulic conductivity and dispersion coefficient onleading moments of the advection velocity and solute concentration was investigated.

Keywords Probabilistic collocation method · Sparse grids · Porous media · Polynomialchaos

1 Introduction

Natural subsurface environments cannot be deterministically characterized in all details dueto high spatial variability or scarcity of data and should be modeled as spatially correlatedrandom fields. This renders equations describing the flow and transport of contaminants inthe subsurface stochastic [1–4].

G. Lin (�) · A.M. TartakovskyComputational Mathematics, Pacific Northwest National Laboratory, 902 Battelle Blvd., Box 999,Richland, WA 99352, USAe-mail: [email protected]

J Sci Comput (2010) 43: 92–117 93

The present paper describes a high-order probabilistic collocation method (PCM) ap-proach [5] that was used to solve stochastic flow and transport equations. PCM is an exten-sion of the polynomial chaos approach with a collocation projection. The classical polyno-mial chaos represents a second-order stationary stochastic process by a spectral expansionbased on Hermite orthogonal polynomials in terms of Gaussian random variables. Its use forsolving stochastic differential equations was pioneered by Ghanem and Spanos [6] who em-ployed a Galerkin projection to derive an equivalent system of deterministic equations thatcan be solved with standard numerical techniques. Xiu and Karniadakis [7] developed thegeneralized polynomial chaos (gPC) method, which employs a broader family of trial basesbased on orthogonal polynomials from the Askey scheme. In the gPC method, the Jacobi-chaos for truncated Gaussian distribution was constructed to represent Gaussian-like inputswith no tails to achieve fast convergence, which was first introduced in [8, 9]. The PCMapproach combines the simplicity of Monte Carlo (MC) methods and the fast convergenceof stochastic Galerkin methods. By taking advantage of the existing theory on multivariatepolynomial interpolations (see [10, 11]), fast convergence was achieved using the PCM ap-proach when the solutions possess sufficient smoothness in the random space. Additionally,implementing the PCM approach is straightforward because it only requires solutions of thecorresponding deterministic problems at pre-selected sampling points.

The tensor product of one-dimensional collocation point sets is a simple, straightforwardchoice, which is efficient for low random dimensional cases. However, the number of collo-cation points increases very rapidly as the number of random dimensions increases. In suchhigh-dimensional cases, using grid sets based on tensor products of one-dimensional collo-cation points leads to a prohibitively large number of collocation points. In the present work,to be able to deal with the problems with high random dimensions, the choice of collocationpoints was based on sparse grids obtained from the Smolyak algorithm [12], which wereconstructed from tensor products of one-dimensional quadrature formulas. Sparse grids usethe smoothness of the integrand to weaken the curse of dimensionality for functions withbounded mixed derivatives in low-to-moderate dimensions.

The PCM was first introduced by Tatang and McRae [13], and recently Xiu and Hes-thaven [5] have used Lagrange polynomial interpolation to construct high-order stochas-tic collocation methods. The properties of PCM were extensively studied in the past 10years. In [14–16], the errors of integrating or interpolating functions with Sobolev regularitywere analyzed for Smolyak constructions based on one-dimensional nested Clenshaw-Curtisrules. In [15], the degree of exactness of the Smolyak quadrature using Clenshaw-Curtisand Gaussian one-dimensional rules was investigated. In [5], the efficiency of Clenshaw-Curtis-based sparse grid stochastic collocation was demonstrated in comparison to otherstochastic methods on an elliptic problem. In 2003, Gerstner and Griebel [17] introducedthe dimension-adaptive tensor product quadrature method. In [18], sparse grid collocationschemes were applied to solving stochastic natural convection problems. In [19, 20], anadaptive hierarchical sparse grid collocation algorithm has been developed. Another par-ticularly interesting variation is the adaptive multi-element probabilistic collocation methoddeveloped by Foo [21]. The efficiency of the PCM approach highly depends on the choice ofthe collocation points. Recently, the PCM approach [22] on regular grids has been employedfor flow in porous media. However, PCM with sparse grids have not been widely applied tothe three-dimensional flow and transport in porous media.

In this paper we present a detailed analysis of the PCM solutions of stochastic flow andadvection dispersion equations for log conductivity having Gaussian, truncated Gaussianand uniform distributions. We recently demonstrated the computational efficiency of PCMfor solving transport equations with Gaussian distribution of log conductivity [23]. The

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Hermite-chaos was used to represent the Gaussian distribution in that work. The assumptionof Gaussian distribution of log conductivity is based on considerable field evidence and iscommonly used in stochastic analysis of flow and transport in geological porous media ap-plications. In the present work we show that the accuracy in both the mean and the standarddeviation of PCM solutions can be improved by using a truncated Gaussian distribution toapproximate the Gaussian distribution of log conductivity. The effect of the magnitude ofdispersion coefficient on the mean and standard deviation of the hydraulic head and soluteconcentration was also investigated. Additionally, we study how the statistical properties ofthe hydraulic head and solute concentration vary while using different correlation lengthsand variances of random hydraulic conductivity. We also study the effect of different formsof random distributions on the mean and variance of solute concentration.

This work provides a road map to add uncertainty quantification functionality into anypre-existing deterministic legacy code in a non-intrusive manner through the PCM approachon sparse grids. The sparse grids probabilistic collocation strategy also provides a scalablecomputational framework for flow and transport problems with multiple sources of uncer-tainty in an embarrassingly parallel way, which requires neither a particular effort to segmentthe problem into a very large number of parallel tasks nor any essential dependency betweenthose parallel tasks. The spectral convergence rate of the PCM approach was shown to becomputationally more efficient than the slow convergence rate of MC simulations for mod-erate correlation length cases. Furthermore, the magnitude of random perturbations in thePCM approaches can be much larger than in the conventional moment-equation approaches[24, 25].

In the next section, we present stochastic differential equations governing fluid flow andsolute transport in heterogeneous porous media. In Sect. 3, we describe a stochastic repre-sentation of randomly heterogeneous porous media using the Karhunen-Loève (K-L) expan-sion. In Sect. 4, we introduce a stochastic collocation method on sparse grids. In Sect. 5, wepresent numerical results.

2 Governing Equations

Consider a steady-state flow in saturated porous media satisfying

∇ · [K(x;ω)∇h(x;ω)] = 0, (1)

and subject to the boundary conditions

h(x;ω) = H(x), x ∈ �⊥, ω ∈ �

n(x) · ∇h(x;ω) = 0, x ∈ �‖, ω ∈ �(2)

where h(x;ω) is the hydraulic head, and H(x) is the prescribed hydraulic head on the bound-ary that is perpendicular to the flow �⊥. �‖ represents the boundary parallel to the flow, andn is a unit vector normal to �‖. K(x;ω) is the random hydraulic conductivity, and ω repre-sents a random event in the sample space �.

Solute transport in saturated porous media is governed by the advection-dispersion equa-tion [26]

∂C

∂t+ ∇ · (vC) = ∇ ·

[(Dw

τ+ α|v|

)∇C

], (3)

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which is subject to the initial and boundary conditions:

C(x,0;ω) = CI (x), x ∈ D,

n(x) · ∇C(x, t;ω) = 0, x ∈ �D,(4)

where C is the solute concentration, v = −K∇h/φ is the average pore velocity, φ isthe porosity, Dw is the diffusion coefficient, τ is the tortuosity, α is the dispersivity,

|v| =√

v2x + v2

y + v2z , CI (x) is the initial condition, and D is the flow domain. The dis-

persion coefficient is assumed to be a scalar for simplicity, (see for example Emmanuel andBerkowitz [26]). Extending the method for the dispersivity tensor is straightforward andwill not affect the conclusions with regard to the efficiency and accuracy of the method. Wedenote ∇∗ = Lo∇ , t∗ = t

to, v∗ = v

〈vo〉 , α∗ = αCo

, D∗w = Dw

τ 〈vo〉Coand K∗ = K

〈vo〉 . Here Lo, Co,and 〈vo〉 are characteristic length, characteristic solute concentration and characteristic fluidvelocity. Characteristic time, to = Lo

〈vo〉 , is the time for a solute plume to travel one charac-teristic length distance in the mean flow direction. After normalization, (1) and (3) take thedimensionless form:

∇∗ · [K∗(x∗;ω)∇∗h∗(x;ω)] = 0, (5)

∂C∗

∂t∗+ ∇∗ · (v∗C∗) = ∇∗ · (D∗

f ∇∗C∗), (6)

subject to the initial and boundary conditions:

C∗(x∗,0;ω) = C∗I (x

∗), x∗ ∈ D∗, ω ∈ �,

h∗(x∗;ω) = H(x∗), x∗ ∈ �∗⊥, ω ∈ �,

n(x∗) · ∇h∗(x∗;ω) = 0, x∗ ∈ �‖, ω ∈ �,

n(x∗) · ∇C∗(x∗, t∗;ω) = 0, x∗ ∈ �∗D ω ∈ �,

(7)

where D∗f = (D∗

w +α∗|v∗|). In the present work, the transport parameters are set to φ = 0.26,τ = 3, and Dw = 7.50 × 10−10 m2 s−1 [26]; the dispersivity was set to α = 0.08 m orα = 0.2 m; and the characteristic length, concentration and average pore-velocity are setto Lo = 1 m, Co = 1 mol/m3 and 〈vo〉 = 〈qo〉

φ= 0.003846 m s−1, respectively.

3 Stochastic Representation of Randomly Heterogeneous Porous Media

Let (�, F , P) be a probability space, where � is the sample space, F is the σ -algebraof subsets of �, and P is a probability measure. For ω ∈ �, let Y ∗(x∗;ω) = lnK∗(x∗;ω)

and assume that Y ∗(x∗;ω) is a second-order continuous stationary random process withmultivariate Gaussian distribution. The K-L expansion is easy to couple with the PCMapproach [27] and provides the best linear approximation in the mean square sense for asecond-order stationary random process [28]. Assume that Y ∗(x∗;ω) is continuous in themean square on the closure of D. That is, for x∗,y∗ ∈ D,

E[|Y ∗(y∗,ω)|2] < +∞ and

limy∗→x∗ E[|Y ∗(y∗,ω) − Y ∗(x∗,ω)|2] = 0.

(8)

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Then Y ∗(x∗;ω) can be expanded using the K-L expansion [28]:

Y ∗(x∗;ω) = Y∗(x∗) +

∞∑i=1

√λiψi(x∗)ξi(ω), (9)

where Y∗(x∗) is the mean of Y ∗(x∗;ω), and ξi(ω) are independent random variables with

zero mean and unit variance. In this work, we assume ξi to be random variables withGaussian, uniform, and Beta distributions. Here λi and ψi(x∗) are eigenvalues and corre-sponding eigenfunctions, which can be obtained from the Fredholm equation:

∫D

C∗Y ∗(t∗, s∗)ψ(t∗)dt∗ = λψ(s∗), (10)

where C∗Y ∗(t∗, s∗) is the covariance kernel [28]. In this work, an exponential form of

C∗Y ∗(t∗, s∗) was assumed.

In general, for a multi-dimensional problem with arbitrary geometry, the eigenvalue prob-lem for the K-L expansion has to be solved numerically. In the present work, we study flowand transport in a porous domain with cubic geometry, D = {(x∗, y∗, z∗) : 0 ≤ x∗ ≤ L1;0 ≤y∗ ≤ L2;0 ≤ z∗ ≤ L3}. We assume a separable exponential covariance kernel C∗

Y ∗(s∗, t∗)in the form C∗

Y ∗(s∗, t∗) = σ 2Y ∗ exp(−|s∗

1 − t∗1 |/η∗1 − |s∗

2 − t∗2 |/η∗2 − |s∗

3 − t∗3 |/η∗3), where

η∗1 , η∗

2 , and η∗3 are the correlation lengths in x∗, y∗, and z∗ coordinates and η∗

i = ηi/Li ,i = 1,2,3. With this assumption, the eigenvalues and corresponding eigenfunctions in athree-dimensional cubic box can be derived by combining one-dimensional eigenvalues andeigenfunctions that solve the one-dimensional Fredholm equation (10).

The one-dimensional covariance kernel can be expressed as

C∗Y ∗(t∗, s∗) = σ 2

Y ∗exp(−|t∗ − s∗|/η∗), (11)

where σ 2Y ∗ and η∗ are the variance and the correlation length of the stationary random

process, respectively. The eigenvalues and corresponding eigenfunctions in K-L expansionfor the one-dimensional covariance function (11) can be obtained analytically.

In numerical simulations, the series in (9) must be truncated at a finite index number M

as

Y ∗M(x∗;ω) = Y

∗(x∗) +

M∑i=1

√λiψi(x∗)ξi(ω). (12)

The corresponding optimal mean square error is:

∫D

E[(Y ∗(x∗;ω) − Y ∗M(x∗;ω))2]dx∗ =

∞∑i=M+1

λi. (13)

Thus, the truncation error is completely determined by the decay rate of the eigenvalues forfixed M (or, conversely, the choice of M to achieve a fixed truncation error is determined bythis rate). The decay rate of λi , which was studied for the covariance of stationary randomprocesses (C∗

Y ∗(t∗, s∗) = C∗Y ∗(t∗ − s∗)) in [29], depends on two factors: the regularity [30]

of C∗Y ∗(t∗, s∗) and the correlation length η∗. With the choice of exponential covariance, the

eigenvalue decay rate primarily depends on correlation length. We sort the combined eigen-

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values for the three-dimensional problem in monotonically decreasing order before truncat-ing the K-L expansion. The truncation error criterion is

M∑i=1

λi ≤ 90%∞∑i=1

λi, (14)

where 90% represents that 90% of the eigen-spectrum remains in the truncated K-L expan-sion. The criterion (14) allows for determining the appropriate number of random dimen-sions M from the eigenvalue decay rate before solving the stochastic flow and transportequations. For large correlation length, the eigenvalues decay very fast [31], so that only asmall number of terms must be kept in the K-L expansion. For a small correlation length,the eigenvalues decay slower, and more terms are needed in the K-L expansion. Moreover,the spatial resolution must be increased to capture the small-scale features associated withsmall correlation length and the time step in the transport equation must be decreased cor-respondingly for numerical stability. Thus, the computational cost of PCM grows very fastwith a decrease in the correlation length.

4 Stochastic Numerical Methods

The aim of this paper is to investigate the accuracy of PCM. We used an MC method toobtain high-accuracy numerical solutions of stochastic flow and transport equations that, weassume, represent the “true” moments, which were then used to evaluate the accuracy ofthe proposed PCM approach. The Monte Carlo and PCM approaches follow similar proce-dures for solving stochastic differential equations: (1) select sampling points in the prob-ability space of random parameters (e.g., hydraulic conductivity), (2) solve correspondingdeterministic equations at these points, and (3) estimate statistical moments of the solu-tions. The main difference between MC and PCM is in the choice of sampling points andthe corresponding weights associated with those points. We used a high-order numericalmethod (WENO) to solve the resulting deterministic equations in both MC and PCM meth-ods. Hence, the only source of discrepancy between PCM and MC (inaccuracy of PCM) isthe numerical errors in the random space. A brief description of PCM approaches and thealgorithms on how to choose sampling points follows.

4.1 Probabilistic Collocation Methods

Consider the stochastic equation:

L(x, t, ξ(ω);u) = f (x, t; ξ(ω)), (15)

subject to appropriate initial and boundary conditions. L is a general (nonlinear) differentialoperator acting on dependent variable u, x ∈ R

ds , ds = 1,2,3 is the physical location, ξ is arandom independent variable, and t is time.

Applying the weighted residual method to (15), we have∫

w(ξ)[L(x, t, ξ(ω);v) − f (x, t; ξ(ω))]dξ = 0, (16)

where v is the numerical approximation of u, w(ξ) are the weight functions and � representsthe random space. The Galerkin projection method, as one of the most important weighted

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residual methods, selects the weight functions to be the basis functions. In particular, wechose the weight functions to be the generalized polynomial chaos basis functions. Due tothe orthogonal properties of the generalized polynomial chaos basis functions, the stochasticintegral (16) can be reduced to a set of deterministic equations. For most nonlinear operatorsL, the form of the deterministic equations is different from the original (15). In contrast tothe Galerkin projection, in the collocation formulation we employ Delta functions δ(ξ − ξk)

as the weight functions, k = 0, . . . ,M − 1, where {ξk} is a proper set of points in samplespace on the support of ξ(ω) and M is the number of random dimensions. By applying thecollocation projection on both sides of (15), we obtain:

∫�

δ(ξ − ξk)[L(x, t, ξ(ω);v) − f (x, t; ξ(ω))]dξ

= L(x, t, ξk;v) − f (x, t; ξk) = 0. (17)

The main advantage of the collocation projection method over the Galerkin projectionmethod of Ghanem and Spanos [6] is that the resulting set of deterministic equations (17)is uncoupled in random space for any form of the nonlinear operator L, and these equationshave the same form as flow-and-transport equations with deterministic coefficients. By con-trast, the Galerkin projection formulation for the nonlinear operator L results in a systemof coupled ordinary or partial differential equations. Hence, for complex hydrogeochemi-cal processes, the collocation projection method can be much easier to implement than thestandard Galerkin projection method.

The general procedure for the PCM approach is similar to the procedure for MC simu-lations, with a difference in selecting the sampling points and corresponding weights. Theprocedure consists of three main steps:

(1) Generate Nc collocation points in probability space of random parameters as indepen-dent random inputs based on a quadrature formula;

(2) Solve a deterministic problem at each collocation point;(3) Estimate the solution statistics using the corresponding quadrature rule,

〈u(x, t)〉 =∫

u(x, t, ξ)ρ(ξ)dξ ≈Nc∑k=1

v(x, t, ξk)wk, (18)

σ(u)(x, t) =√∫

(u(x, t, ξ) − 〈u〉)2ρ(ξ)dξ ≈√√√√ Nc∑

k=1

v2(x, t, ξk)wk − 〈v〉2; (19)

where ρ(ξ) is the probabilistic distribution function (PDF) of random variable ξ , Nc is thenumber of quadrature points, {ξk} is the set of quadrature points and {wk} is the correspond-ing set of weights, which are the combination of quadrature weights in each random dimen-sion. In the second step of the PCM approach, as for MC, any existing code can be used tosolve deterministic flow-and-transport equations. Extensive reviews on the construction ofquadrature formulas may be found in [32] and [33]. Below, we provide a brief review of twodifferent methods for selecting collocation points.

4.2 Choices of Collocation Points

Without loss of generality, we assume that the random vector variables, ξ k , k = 1, . . . ,NM ,have a bounded support and can be represented by an M-hypercube, where M is the number

J Sci Comput (2010) 43: 92–117 99

of random dimensions (components) of vector ξ k = (ξ1,k, ξ2,k, . . . , ξM,k)T . The computa-

tional cost of the PCM approach is the number of collocation points times the cost of thecorresponding deterministic problem. Thus, for a given required accuracy, our goal is tochoose the collocation point set with the minimal number of collocation points. Two differ-ent methods for the selecting collocation-point sets are discussed here: (1) tensor products ofone-dimensional collocation point sets and (2) a sparse grid strategy for high dimensional-ity. In this work, sparse grid strategy was employed, and tensor products of one-dimensionalcollocation point sets are discussed only for the purpose of completeness.

4.2.1 Tensor Products of One-dimensional Collocation Point Sets

The tensor product of a one-dimensional point set is

U (f ) =∫

f (ξ1, . . . , ξM)ρ1(ξ1) · ρ2(ξ2) · · ·ρM(ξM)dξ1dξ2 · · ·dξM

≈(

U i11

⊗· · ·

⊗U iM

M

)(f ) =

ni1∑k1=1

· · ·niM∑

kM=1

f (ξi1k1

, . . . , ξiMkM

)

×(ω

i1k1

⊗· · ·

⊗ω

iMkM

), (20)

where f is any smooth function, ij ∈ N specifies the degree of the integration in dimension

j , nij is the number of quadrature points used in dimension j , ξijkj

is the kj -th quadrature

point in the j -th random dimension (kj ∈ [1, nij ]), and ωijkj

≡ ω(ξijkj

) is the corresponding

weight of quadrature point ξijkj

. Here U ijj is the one-dimensional quadrature formula for

dimension j ,

U ijj (f ) =

∫�j

f (ξj )ρj (ξj )dξj ≈nij∑k=1

f (ξijkj

) · ωijkj

. (21)

Evaluating (20) requires ZM collocation (quadrature) points, where ZM = ni1 · · ·niM . If wechoose the same k + 1 number of collocation points in each random dimension and denotethe total number of collocation points as Zk

M , we have ZkM = (k + 1)M .

For a small number of random dimensions, e.g., M ≤ 4, the tensor product of one-dimensional collocation point sets is a good choice for collocation point sets. However,Zk

M grows exponentially as M increases. Thus, quadrature using the tensor product of one-dimensional collocation point sets rapidly becomes less efficient as the number of randomdimensions increases.

4.2.2 High Dimensionality and Sparse Grids

The tensor product of one-dimensional collocation point sets can be efficient for a low ran-dom dimensional case. However, the number of random dimensions needed to control nu-merical error for both mean and variance estimates increases very rapidly as the correlationlength decreases. In such high-dimensional cases, using point sets based on tensor productsof one-dimensional collocation points leads to a prohibitively large number of collocationpoints. In this work, we use the Smolyak formula [12], which is a linear combination oftensor product formulas, and the resulting point set has a significantly smaller number ofpoints than the full tensor product set. Recently, Xiu and Hesthaven [5] have used Lagrange

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Fig. 1 Two random-dimensional (M = 2) sparse grid points v.s. full tensor products: (a) sparse grid levelk = 5, q = 7 (cross nodes) and k = 6, q = 8 (circle nodes); (b) corresponding full tensor products

polynomial interpolation to construct high order stochastic collocation methods based onsparse grids using the Smolyak formula [12]. Such sparse grids do not depend as stronglyon the dimensionality of the random space and as such are more suitable for applicationswith high-dimensional random inputs.

The one-dimensional quadrature formula (21) serves as a building block for the Smolyakformula. Smolyak’s formula for approximating the M-dimensional integral in (20) is givenby the linear combination of tensor product formulas as

I(f ) ≡ A(q,M) =∑

q−M+1≤|i|≤q

(−1)q−|i| ·(

M − 1q − |i|

)(U i1

1 ⊗ · ⊗ U iMM ), (22)

where U ijj is the one-dimensional quadrature formula obtained from (21) for dimension j ,

the sparseness parameter q ≥ M determines the order of the formula, and |i| = i1 + i2 +· · · + iM . Here nij (j = 1, . . . ,M) represents the number of collocation points in the j -thrandom dimension. We set k = q − M as the “level” of the Smolyak formula and choosen1 = 1 and nk = 2k−1 + 1 for k > 1. Then, to compute A(q,M), we only need to evaluate f

in the U ijj on the Smolyak “sparse grid”

�M ≡⋃

q−M+1≤|i|≤q

(�i11 × · · · × �

iM1 ), (23)

where �ij

1 represents a one-dimensional collocation point set at the j -th random dimension,and �M is denoted as the M-dimension collocation point set.

The criterion for convergence in terms of sparse grid level requires the absolute value ofthe difference of A(q,M) obtained with two successive sparse grid levels to be smaller thana specified threshold. We refer interested readers to [19] for extensive reviews on adaptivityof sparse grids.

Below we will introduce two main categories of sparse grids—nested and non-nested—corresponding to different random distributions. In particular, for a uniform distribution,sparse collocation point sets were generated based on the nested Chebyshev quadraturepoints using Clenshaw-Curtis formulas. For beta and Gaussian distributions, sparse colloca-tion point sets were generated based on Gaussian quadrature points. By choosing 2i−1 + 1quadrature points, a 2i degree of exactness can be achieved.

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Nested Sparse Grids One choice for the one-dimensional formula (21) is to use Clenshaw-Curtis points, which are the extreme points of the Chebyshev polynomials by Clenshaw-

Curtis formulas. The set of sparse grid points in the j -th random dimension, �ij

1 , is nested,

i.e.,�ij

1 ∈ �ij +11 . The number of points is given by nij = 2ij −1 + 1, ij ≥ 2. In practice, we

will use the same formulas in every dimension. Hence, we will drop the subscript j forsimplicity. These nodes are given as

Xik = − cos

(π(k − 1)

ni − 1

), X1

1 = 0, k = 1, . . . , ni . (24)

and the corresponding weights are

ωik = 2

ni − 1

(1 + 2

(ni−1)/2∑′

k=1

1

1 − 4k2cos

2πk(i − 1)

ni − 1

),

ωi1 = ωi

ni= 1

ni(ni − 2), 2 ≤ i ≤ ni − 1,

(25)

where∑′ represents that the last term of the sum is halved. With the choice of Clenshaw-

Curtis points, the one-dimensional quadrature formula (21) can integrate all polynomialsof degree less than ni exactly. Additionally, the nested Clenshaw-Curtis points can greatlyreduce the number of collocation points used in total.

For Smolyak Clenshaw-Curtis grid, the accuracy of the Smolyak formula can be obtainedas follows. Let σ = floor(q/M) and let τ = q mod M . Then A(q,M) has the degree ofexactness [15]

m(q,M) ={

2(q − M) + 1, if q ≤ 4M,

2σ−1 · (M + 1 + τ) + 2M − 1, otherwise.(26)

Non-Nested Sparse Grids An alternate way to generate a sparse nodal set is to useGaussian quadrature points by Gaussian formulas. By choosing 2i−1 + 1 Gaussian quadra-ture points, a 2i degree of exactness can be achieved. The nested property may be lost,which makes the nodal set larger than the nested nodal set. However, for random inputswith arbitrary PDFs, the sparse quadrature rule can be constructed in such a way that theweight function coincides with the PDF. Gaussian quadrature nodal sets for arbitrary weightfunctions can be obtained in an efficient way [34].

For a Smolyak Gaussian grid, the accuracy of the Smolyak formula can be obtained asfollows. Let σ = floor(q/M), and let τ = q mod M . Then A(q,M) has degree of exactness[15]

m(q,M) ={

2(q − M) + 1, if q ≤ 3M,

2σ−1 · (M + 1 + τ) − 1, otherwise.(27)

5 Results and Discussion

Stochastic numerical simulations of flow and transport were performed in the three-dimensional randomly heterogeneous porous cubic domain (6×3×3) shown in Fig. 2. Loghydraulic conductivity Y ∗ = ln(K∗) was treated as a random variable (infinite correlation

102 J Sci Comput (2010) 43: 92–117

Fig. 2 Sketch of the computational domain and initial shape and location of the plume

lengths in all directions) in one case and as a spatially dependent second-order stationaryrandom process (finite correlation lengths) in a second case. The effect of three differentrandom distributions (Gaussian, truncated Gaussian, which is a special case of beta distrib-ution, and uniform distributions) of the log hydraulic conductivity on the mean and varianceof the hydraulic head and solute concentration were studied. The dependence of leading mo-ments of the hydraulic head and concentration on the variance and correlation length of Y ∗

was also investigated. Simulations were conducted for three different variances σ 2Y ∗ = 0.1,

0.5, and 2.5. Additionally, three different correlation lengths were investigated in this work.First, the case of infinite correlation length is considered, which renders K∗ a scalar randomvariable, where the PCM solution is validated against the analytical solution. Second, twofinite correlation lengths η∗

j = 1 and 0.1, (j = 1,2,3) are employed in the simulations. Inall cases, the flow and advection-dispersion (5) and (6) were solved subject to the boundaryconditions

∂h∗

∂n∗ = 0 at boundary: y∗ = 0, y∗ = 3, z∗ = 0, and z∗ = 3,

h∗ = 6.0 at boundary: x∗ = 0, and h∗ = 0 at x∗ = 6, (28)

∂C∗

∂n∗ = 0 at all boundaries,

with initial solute concentration

C∗(x∗,0) = exp

[− (x∗ − x∗

o )2 + (y∗ − y∗

o )2 + (z∗ − z∗

o)2

2σ 2o

], (29)

J Sci Comput (2010) 43: 92–117 103

and with the center of the plume (x∗o , y

∗o , z

∗o) at time t = 0 located at (1.21,1.5,1.5) and

σo = 0.2. In the simulations, we employ a fifth-order WENO scheme [35] for spatial dis-cretization with 80 × 40 × 40 grid points. The stochastic simulations are based on the PCMapproach and multi-dimensional integration using sparse grids.

The numerical results section is organized as follows: to provide a large picture of howthe plume is transported, dispersed, and diffused in different randomly heterogeneous porousmedia, we first present some three-dimensional contour plots of flow and transport in theporous medium with uniform distribution of a hydraulic conductivity field obtained fromPCM simulations. 4 and 32 random dimensions were used for the correlation length η∗

j = 1and 0.1 cases, respectively. Next, we discuss several topics and quantitatively comparePCM results with MC simulations: (1) using the truncated Gaussian inputs to approximateGaussian inputs to improve the accuracy of PCM solutions for the same number of colloca-tion points, (2) the effect of the dispersion coefficient α, (3) the effect of correlation lengthand variance of log hydraulic conductivity, and (4) the effect of random distributions of loghydraulic conductivity on the leading moments of the solute concentration.

Transport in the porous medium with uniform random distribution of log conductivity

Figures 3 through 5 show numerical results of flow and transport simulations for a logconductivity field described as a uniform random process with correlation length η∗

j = 1,

(j = 1,2,3), and variances σ 2Y ∗ = 0.5 and 2.5, respectively. From comparison with MC

simulations, it was determined that 87 collocation points are sufficient to obtain an accuratesolution for correlation length η∗

j = 1, (j = 1,2,3). Figure 3(a, b) shows contours of meanand standard deviation of the non-dimensional hydraulic head for σ 2

Y ∗ = 0.5. Due to thechoice of the deterministic boundary conditions (28) the mean head linearly decreases in thedirection of flow, and the largest standard deviation of the hydraulic head was observed in themiddle of the computational domain. Figures 4 and 5 show the effect of the variance of Y ∗on the mean and variance of the solute concentration. It can be seen that the velocity of thecenter of mass of the plume decreases as the variance of Y ∗ increases from σ 2

Y ∗ = 0.5 to 2.5.As a result, the “average” plume gets less dispersed with σ 2

Y ∗ = 2.5 than with σ 2Y ∗ = 0.5,

i.e. the increase in the variance of conductivity reduces the mean and variance of the effec-

Fig. 3 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3)

and σ 2Y ∗ = 0.5 case: Three-dimensional contours of (a) mean 〈h∗〉, and (b) standard deviation, σh∗ , of

non-dimensional hydraulic head obtained from the PCM approach on a sparse grid (level k = 2, 87 collo-cation points)

104 J Sci Comput (2010) 43: 92–117

Fig. 4 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 0.5 case: Three-dimensional contours of (a) mean 〈C∗〉(x∗, y∗, z∗, t∗ = 2.5) and (b) standard devia-

tion σc∗ (x∗, y∗, z∗, t∗ = 2.5) of non-dimensional solute concentration obtained from the PCM approach onsparse grids (level k = 2, 87 collocation points)

Fig. 5 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 2.5 case: Three-dimensional contours of (a) mean 〈C∗〉 (x∗, y∗, z∗, t∗ = 2.5) and (b) standard devia-

tion σc∗ (x∗, y∗, z∗, t∗ = 2.5) of non-dimensional solute concentration obtained from the PCM approach onsparse grids (level k = 2, 87 collocation points)

tive dispersion coefficient. Additionally, the standard deviation of concentration is seen toincrease with an increasing variance of Y ∗.

Next, we investigate the PCM solution for a smaller correlation length, η∗j = 0.1,

(j = 1,2,3), as shown in Figs. 6 and 7, which is a more challenging problem due to thesignificantly increasing number of random dimensions, and more collocation points have tobe employed. Figures 6 and 7 show the effect of σ 2

Y ∗ on the mean and variance of concentra-tion. These simulations required 4721 collocation points to obtain accurate agreement withMC solution. Similar to the simulations with η∗

j = 1, (j = 1,2,3), in the simulations withη∗

j = 0.1, (j = 1,2,3) increase in σ 2Y ∗ leads to a decrease of average advective velocity, a

smaller dispersion of a plume, and a larger variance of the concentration. A spherical shapeof an “average” plume was observed for the mean solute concentration and a dumbbell shapeof the variance of concentration iso-contours was observed in the simulation with η∗

j = 0.1,

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Fig. 6 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 0.1, (j = 1,2,3) and

σ 2Y ∗ = 0.1 case: Three-dimensional contours of (a) mean 〈C∗〉(x∗, y∗, z∗, t∗ = 2.5) and (b) standard devia-

tion σc∗ (x∗, y∗, z∗, t∗ = 2.5) of non-dimensional solute concentration obtained from the PCM approach onsparse grids (level k = 2, 4721 collocation points)

Fig. 7 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 0.1, (j = 1,2,3) and

σ 2Y ∗ = 0.5: Three-dimensional contours of (a) mean 〈C∗〉(x∗, y∗, z∗, t∗ = 2.5) and (b) standard deviation

σc∗ (x∗, y∗, z∗, t∗ = 2.5) of non-dimensional solute concentration obtained from the PCM approach onsparse grids (level k = 2, 4721 collocation points)

(j = 1,2,3) and σ 2Y ∗ = 0.1. An increase in the variance of Y ∗ leads to the “egg shape” of the

variance of concentration iso-contours. As the correlation length decreases, the eigenvaluesobtained from the K-L expansion decrease more slowly. Thus, more eigen-modes have to beincluded in the K-L expansion. The number of eigen-modes is equal to the number of ran-dom dimensions. Therefore, the number of random dimensions increases as the correlationlength decreases.

In the next section, transport in porous media with Gaussian and truncated Gaussiandistributions of log hydraulic conductivity is described.

Log hydraulic conductivity: Truncated Gaussian Distribution—a Jacobi-chaos approxima-tion to Gaussian distribution

In a stochastic analysis of flow and transport in porous media it is common to represent loghydraulic conductivity as a random field with Gaussian distribution. It is also known that the

106 J Sci Comput (2010) 43: 92–117

long tails of Gaussian distribution can result in slow convergence [23] and ill-poseness ofHermite-chaos expansion for some applications where the finite boundness of the random in-puts is crucial [36, 37]. An alternative to the Gaussian distribution is the truncated Gaussiandistribution introduced in [8, 9], which is used to represent Gaussian-like inputs with notails using Jacobi-chaos. Denote ζ(ω) ∼ N(0,1) is a Gaussian random variable, which canbe represented by the Jacobi-chaos {φ(ξ)}. Here ξ ∼ B(α,β)(−1,1) is a beta random variablein (−1,1) with parameters α,β > −1 and the corresponding probability density function is

f (x;α,β) = �(α + β)

�(α)�(β)xα−1(1 − x)β−1, (30)

where � is the gamma function, �(z) = ∫ ∞0 t z−1e−t dt . If n is a positive integer, then

�(n) = (n − 1)!. ζ(ω) can be expanded using Jacobi-chaos. Due to the symmetry ofGaussian distribution, we set α = β in the Jacobi-chaos. By employing the orthogonalityproperties of Jacobi-chaos, we have,

ζ̃ (ω) =N∑

i=0

ζiφi(ξ), ζi = 〈ζ(ω),φi(ξ)〉〈φ2

i (ξ )〉 , (31)

where ζ̃ (ω) is the truncated Gaussian random variable, which is an approximation to theGaussian random variable, ζ(ω). Here ζ and ξ belong to two different probability spaces.The inner product 〈·, ·〉 can be evaluated by mapping both ζ and ξ to the space defined bythe uniform random variable, i.e.,

ζ(ω) = S−1(ϑ(ω)), ξ(ω) = T −1(ϑ(ω)), (32)

where S−1 and T −1 are the inverse cumulative distribution functions of ζ and ξ , respectively.In general, the analytical expression of the inversion (32) is not available, and we have toresort to numerical inversion. Here ϑ(ω) ∈ U(0,1) is the uniform random variable in (0,1).Thus, we can obtain the evaluation of the expansion coefficients as

ζi = 〈ζ,φi(ξ)〉〈φ2

i (ξ )〉 = 1

〈φ2i 〉

∫ 1

0S−1(ϑ)φi(T

−1(ϑ))dϑ. (33)

The integral in (33) can be evaluated by the Gaussian quadrature rule. The resulting trun-cated Gaussian random variable, ζ̃ (ω), is an approximation to the Gaussian random vari-able, ζ(ω).

Here we compare the performance of log truncated Gaussian inputs and log normal in-puts for hydraulic conductivity. Figure 8 demonstrates the first- and fifth-order truncatedGaussian distribution approximation to Gaussian distribution N(0,1). Good agreementswere observed between the truncated Gaussian and the Gaussian distributions, except aroundthe tails, because of the strictly bounded support of the truncated Gaussian distribution. Wesolve the stochastic transport equations for both Gaussian and truncated Gaussian distrib-utions of log conductivity. The fifth-order Jacobi-chaos with α = β = 10 was employed torepresent the truncated Gaussian distribution and Hermite-chaos was used for the Gaussiandistribution. The accuracy of approximation of the Gaussian distribution with a truncatedGaussian distribution was verified by comparing PCM solutions for the two distributionswith analytical solutions and MC solutions of the transport equations with Gaussian distrib-ution of log hydraulic conductivity.

J Sci Comput (2010) 43: 92–117 107

Fig. 8 Truncated Gaussiandistribution approximation toGaussian distribution N(0,1)

Fig. 9 Log hydraulic conductivity modeled as a truncated Gaussian vs. Gaussian random variablewith σ 2

Y ∗ = 0.5: The performance of log truncated Gaussian and log normal hydraulic conductiv-ity for (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration as a function of x∗ obtainedanalytically and from the PCM approach on a sparse grid with 32 collocation points

The infinite correlation length case was first investigated, which allows us to obtain ana-lytical solutions for the Gaussian distributions of Y ∗, as shown in Figs. 9 and 10. In Figs. 9and 10, we compare the numerical results and computational cost for truncated Gaussianand Gaussian distributions of Y ∗, and σ 2

Y ∗ = 0.5 and 2.5, respectively.The PCM results of both Gaussian and truncated Gaussian random inputs were in a close

agreement with the analytical solutions, even for σ 2Y ∗ larger than 1. The results show that the

truncated Gaussian distribution is a good approximation for the Gaussian log conductivitydistribution as far as its effect on the mean and variance of concentration is concerned.

Next, we investigated the finite correlation length case as shown in Figs. 11 and 12, wherewe compare the mean and the variance of the concentration obtained with PCM for thetruncated Gaussian and Gaussian distributions of Y ∗ with the MC solution. The correlationlengths η∗

j = 1, (j = 1,2,3) and variances σ 2Y ∗ = 0.5 and 2.5 were used in these simulations.

The PCM solutions with 411 Gaussian collocation points (collocation points for Gaussian

108 J Sci Comput (2010) 43: 92–117

Fig. 10 Log hydraulic conductivity modeled as a truncated Gaussian vs. Gaussian random variablewith σ 2

Y ∗ = 2.5: The performance of log truncated Gaussian and log normal hydraulic conductiv-ity for (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration as a function of x∗ obtainedanalytically and from the PCM approach on a sparse grid with 80 collocation points

Fig. 11 Log hydraulic conductivity modeled as a truncated Gaussian vs. Gaussian random process withcorrelation length η∗

j= 1, (j = 1,2,3) and σ 2

Y ∗ = 0.5: The performance of log truncated Gaussian and log

normal hydraulic conductivity for (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standarddeviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration as a functionof x∗ obtained analytically and from the PCM approach on a sparse grid with 87 collocation points for logtruncated Gaussian hydraulic conductivity and 411 collocation points for log normal hydraulic conductivity

random process of Y ∗) oscillate around the MC solutions for σ 2Y ∗ = 0.5 and 2.5, indicating

that this number of Gaussian collocation points is insufficient to obtain an accurate solu-tion, even for variances of Y ∗ smaller than 1. On the other hand, the PCM solution withonly 87 truncated Gaussian collocation points agrees very well with the MC solution forGaussian Y ∗ for σ 2

Y ∗ = 0.5. These results show that the approximation of Gaussian distribu-tion of Y ∗ with truncated Gaussian distribution significantly improves the efficiency of thePCM approach. Oscillations in the PCM solution with 1573 Gaussian collocation points forσ 2

Y ∗ = 2.5 indicate that the number of collocation points should increase with an increasingvariance of Y ∗.

Figure 13 shows the mean and the standard deviation of the breakthrough concentrationat x = 2.11 for two different σ 2

Y ∗ obtained with the PCM approach for truncated Gaussian

J Sci Comput (2010) 43: 92–117 109

Fig. 12 Log hydraulic conductivity modeled as a truncated Gaussian vs. Gaussian random process withcorrelation length η∗

j= 1, (j = 1,2,3) and σ 2

Y ∗ = 2.5: The performance of log truncated Gaussian and log

normal hydraulic conductivity for (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standarddeviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration as a functionof x∗ obtained analytically and from the PCM approach on a sparse grid with 411 collocation points for logtruncated Gaussian hydraulic conductivity and 1573 collocation points for log normal hydraulic conductivity

Fig. 13 Log hydraulic conductivity modeled as a truncated Gaussian vs. Gaussian random process withcorrelation length η∗

j= 1, (j = 1,2,3) and σ 2

Y ∗ = 0.5 and 2.5: The performance of log truncated Gaussian

and log normal hydraulic conductivity for (a) mean 〈C∗b〉 (x∗ = 2.11, t∗) and (b) standard deviation σC∗

b

(x∗ = 2.11, t∗) of the breakthrough concentration curves obtained from the PCM solutions

and Gaussian distributions and with the MC simulations for the Gaussian distribution of Y ∗.The correlation length η∗

j = 1, (j = 1,2,3) was used in these simulations. For σ 2Y ∗ = 0.5,

411 truncated Gaussian collocation points were needed to obtain a good agreement betweenPCM and the MC solutions while 1573 Gaussian collocation points were needed to obtainequally good agreement between PCM and the MC solutions. From a convergence study, wefound that 6000 realizations were required to obtain an accurate MC solution. For σ 2

Y ∗ = 2.5the agreement between the PCM results and the MC simulations is very good at early times,but deteriorates at later times, indicating that the number of collocation points becomesinsufficient at the later times.

In the next section, the effect of the dispersivity α on the mean and standard deviation ofsolute concentration was studied.

110 J Sci Comput (2010) 43: 92–117

Fig. 14 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 0.5: The effect of dispersion coefficient α on (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and

(b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration asa function of x∗ obtained from the PCM approach on a sparse grid with 411 collocation points

Fig. 15 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 2.5: The effect of dispersion coefficient α on (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and

(b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration asa function of x∗ obtained from the PCM approach on a sparse grid with 1573 collocation points

The effect of the dispersivity α on the mean and standard deviation of non-dimensionalsolute concentration

Figures 14 and 15 show the effect of dispersion coefficient α on the mean and standarddeviation of non-dimensional solute concentrations. In these simulations, the log hydraulicconductivity Y ∗ is assumed to be a uniform random process with η∗

j = 1, (j = 1,2,3) andσ 2

Y ∗ = 0.5 and 2.5. First, Figs. 14 and 15 reveal that an increase in the value of the dispersiv-ity coefficient α leads to a decrease in the mean and standard deviation of concentration. Sec-ond, for the same number of collocation points, oscillations were observed in PCM resultsfor a smaller dispersion coefficient α = 0.08 case, while good agreements were observedbetween PCM results and MC simulations for the larger dispersion coefficient α = 0.2 case,which illustrates that the large dispersion coefficient α can suppress the oscillations of PCMsimulations, and thus fast convergence can be achieved with fewer collocation points. In

J Sci Comput (2010) 43: 92–117 111

Fig. 16 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 0.5: The effect of dispersion coefficient α on (a) mean 〈C∗

b〉 (x∗ = 2.11, t∗) and (b) standard deviation

σC∗b

(x∗ = 2.11, t∗) of the breakthrough concentration curves as a function of t∗ obtained from the PCM

approach on a sparse grid with 411 collocation points

Fig. 17 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3) and

σ 2Y ∗ = 2.5: The effect of dispersion coefficient α on (a) mean 〈C∗

b〉 (x∗ = 2.11, t∗) and (b) standard deviation

σC∗b

(x∗ = 2.11, t∗) of the breakthrough concentration curves as a function of t∗ obtained from the PCM

approach on a sparse grid with 1573 collocation points

Figs. 16 and 17, we demonstrate the corresponding mean and standard deviation of thebreakthrough concentration curves for σ 2

Y ∗ = 0.5 and 2.5, respectively. It can be seen thatfor σ 2

Y ∗ = 0.5, a good agreement between PCM and MC solutions is achieved at all timeswith only 411 points in both α = 0.08 and α = 0.2 cases. However, for σ 2

Y ∗ = 2.5, the PCMsolution deteriorates with time even for 1573 collocation points in the α = 0.08 case, butnot the α = 0.2 case. Thus, smaller values of α can suppress oscillations when σ 2

Y ∗ is larger.Results of this section show that the number of collocation points required to achieve

convergence decreases with increasing dispersivity, α. In the following subsection, we in-vestigate how sensitive the mean and standard deviation of solute concentrations are to thecorrelation length and variance of the log hydraulic conductivity.

112 J Sci Comput (2010) 43: 92–117

Fig. 18 Log hydraulic conductivity modeled as a uniform random process with η∗j

= 1, (j = 1,2,3)

and σ 2Y ∗ = 0.5 and 2.5: The effect of σY ∗ of log hydraulic conductivity on (a) mean 〈C∗〉 at

(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5),of non-dimensional solute concentration as a function of x∗ obtained from the PCM approach on a sparsegrid with 411 and 1573 collocation points respectively

The effect of correlation length η∗j (j = 1,2,3) and variance σ 2

Y ∗ of log hydraulic conduc-tivity on the mean and standard deviation of non-dimensional solute concentration

In Fig. 18, we investigated the effect of σ 2Y ∗ on the mean and the standard deviation of non-

dimensional solute concentration for σ 2Y ∗ = 0.5 and 2.5, respectively. A uniform distribution

of log hydraulic conductivity was assumed. The number of realizations required to obtainthe MC solution was determined from a convergence study and was found to increase withthe variance of Y ∗. The figures also indicate that accurate PCM results can be achieved withfar fewer collocation points than MC sampling points. It is evident that the PCM approachis computationally more efficient than the MC simulations in moderate correlation-lengthcases. Figure 18 also shows that the velocity of the center of the mass of the plume decreaseswith increasing variance of Y ∗, and the “average” plume gets less dispersed with σ 2

Y ∗ = 2.5than with σ 2

Y ∗ = 0.5. The standard deviation of concentration increases with increasing vari-ance of Y ∗. In Fig. 19, we study the effect of the correlation length η∗

j (j = 1,2,3) of loghydraulic conductivity on the mean and standard deviation of non-dimensional solute con-centration. From Fig. 19, we observe that the mean and standard deviation of concentrationdecreases with decreasing correlation length η∗

j (j = 1,2,3).The results of this section show that the mean and standard deviation of solute concentra-

tion is more sensitive to variance than correlation length of the log hydraulic conductivity.In the following section, we study the effect of different forms of distributions on the meanand variance of concentration.

The effect of random distributions of log hydraulic conductivity on the mean and standarddeviation of non-dimensional solute concentration

Figures 20 and 21 show the effect of log hydraulic conductivity with uniform, beta andGaussian distributions on the mean and standard deviation of non-dimensional solute con-centration for the infinite correlation length (random variable) case. The log hydraulic con-ductivity is described as a random variable with equal means, but σ 2

Y ∗ = 0.5 and 0.25, re-spectively. A larger mean was observed for the uniform distribution. However, the threedistributions have close peak values of standard deviation.

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Fig. 19 Log hydraulic conductivity modeled as a uniform random process with η∗j

= ∞ (random variable),

1 and 0.1, (j = 1,2,3) and σ 2Y ∗ = 0.5: The effect of correlation length η∗

j(j = 1,2,3) of log hydraulic

conductivity on (a) mean 〈C∗〉 at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), of non-dimensional solute concentration as a function of x∗ obtainedfrom the PCM approach on a sparse grid with 32, 411 and 2113 collocation points respectively

Fig. 20 Log hydraulic conductivity modeled as a random variable with σ 2Y ∗ = 0.5: The effect

of log hydraulic conductivity with uniform, beta and Gaussian distributions on (a) mean 〈C∗〉 at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5),of non-dimensional solute concentration as a function of x∗ obtained from the PCM approach on a sparsegrid with 32 collocation points

In Figs. 22 and 23, we show the finite correlation length case. For all distributions, thecorrelation length is set to η∗

j = 1, (j = 1,2,3) and the variance σ 2Y ∗ = 0.5 and 2.5. To

estimate the accuracy of the PCM approach for beta, Gaussian and uniform distributions,PCM solutions were compared with MC solutions for corresponding distributions. A goodagreement is observed between the PCM solutions and MC simulations for beta and uniformdistributions, which indicates the fast convergence for uniform and beta distributions. ThePCM solutions with Gaussian random inputs for the standard deviation of C∗ are also in verygood agreement with the MC simulations everywhere except for large values of x∗ wheresmall numerical oscillations in the PCM solutions were observed. The PCM solutions re-

114 J Sci Comput (2010) 43: 92–117

Fig. 21 Log hydraulic conductivity modeled as a random variable with σ 2Y ∗ = 2.5: The ef-

fect of hydraulic conductivity with uniform, beta and Gaussian distributions on (a) mean 〈C∗〉 at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5),of non-dimensional solute concentration as a function of x∗ obtained from the PCM approach on a sparsegrid with 80 collocation points

Fig. 22 Log hydraulic conductivity modeled as a random process with η∗j

= 1, (j = 1,2,3) and σ 2Y ∗ = 0.5:

The effect of hydraulic conductivity with uniform, beta and Gaussian distributions on (a) mean 〈C∗〉 at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), ofnon-dimensional solute concentration as a function of x∗ obtained analytically and from the PCM approachon a sparse grid with 87 collocation points

quire larger numbers of Gaussian collocation points than the truncated Gaussian collocationpoints.

5.1 Summary

In this work, we have systematically studied the numerical solutions of the three-dimensional stochastic Darcy’s equations and stochastic advection-diffusion-dispersionequations using a PCM approach on sparse grids. Different distributions of random hy-draulic conductivity were considered. The accuracy in both the mean and the standarddeviation of PCM solutions can be improved by using the Jacobi-chaos representing thetruncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution.

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Fig. 23 Log hydraulic conductivity modeled as a random process with η∗j

= 1, (j = 1,2,3) and σ 2Y ∗ = 2.5:

The effect of hydraulic conductivity with uniform, beta and Gaussian distributions on (a) mean 〈C∗〉 at(x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), and (b) standard deviation, σC∗ at (x∗, y∗ = 1.5, z∗ = 1.5, t∗ = 2.5), ofnon-dimensional solute concentration as a function of x∗ obtained analytically and from the PCM approachon a sparse grid with 411 collocation points

The solutions of the stochastic advection-dispersion equation were found to be very similarfor the Gaussian and the truncated Gaussian distributions. As a result, we show that thetruncated Gaussian distribution is a good approximation of the Gaussian distribution thatpreserves the accuracy of solutions of stochastic equations and significantly increases theefficiency of PCM.

Regardless of the type of distributions we observed a faster convergence of the concen-tration mean and standard deviation solutions for larger dispersivity occurs. The numberof collocation points required to achieve a convergent solution increases with decreasingcorrelation lengths and/or increases the standard deviation of log hydraulic conductivity.Comparing the PCM results with analytical solutions and MC simulations indicates that thePCM approach rapidly converges and is highly accurate for variance of log conductivity aslarge as 2.5. We also observe that the numerical stochastic solutions converge much fasterfor uniform and beta distributions than the Gaussian distribution because of the boundednessof the random inputs for uniform and beta distributions. The fast convergence of the PCMapproach on sparse grids also relies on the smoothness of the solution in the random spacewith the assumption of equal importance in each direction. Ultimately, adaptivity in randomspace is necessary for regions with less regularity or directions with more importance. Anadaptive sparse-grids PCM approach such as in [17, 19–21] can improve efficiency. Theauthors thank the anonymous reviewers for their valuable comments and suggestions.

Acknowledgements This work was supported by the Advanced Scientific Computing Research Programof the U.S. Department of Energy Office of Science. Computations were performed using the computationalresources of the Environmental Molecular Sciences Laboratory. Pacific Northwest National Laboratory isoperated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830.

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