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A Stochastic Mortar Mixed Finite Element Method for Flow in Porous Media with Multiple Rock Types

Benjamin Ganis∗† Ivan Yotov∗ Ming Zhong∗

March 29, 2010

Abstract This paper presents an efficient multiscale stochastic framework for uncertainty

quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcy’s law with nonstationary stochastic permeabil- ity represented as a sum of local Karhunen-Loève expansions. The approximation uses stochastic collocation on either a tensor product or a sparse grid, coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method. The latter method requires solving a coarse scale mortar interface problem via an itera- tive procedure. The traditional implementation requires the solution of local fine scale linear systems on each iteration. We employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration. In the stochastic setting, the basis is further reused over multiple re- alizations, leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude. Error analysis and numerical experiments are presented.

Keywords. uncertainty quantification, stochastic collocation, multiscale basis, mortar finite element, mixed finite element, porous media flow, Smolyak sparse grid

1 Introduction

Accurately predicting physical phenomena often involves incorporating uncertainties into a model’s input, due to both natural randomness and incomplete knowledge of various physical properties, and then following those uncertainties into the model’s output. In this paper we simulate single-phase flow though porous media, by modeling the permeability as a spatially random function. As a result, the equations governing the flow are stochastic. The goal is uncertainty quantification (UQ) via the computation of the expectation and variance of the stochastic solution, with the latter giving a measure of confidence of the former. To compute these statistical moments, we employ the stochastic collocation method [8, 36, 27, 18] coupled with the multiscale mortar mixed finite element method (MMMFEM) [6] implemented with a multiscale flux basis [19].

∗ Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania

15260, USA; {bag8@pitt.edu, yotov@math.pitt.edu, miz17@pitt.edu}. † Corresponding Author.

1

Stochastic modeling methods can be classified into three groups: (1) sampling meth- ods [15], (2) moment/perturbation methods [38], and (3) non-perturbative methods based either on polynomial chaos expansions [37] or stochastic finite elements [13, 20]. A brief survey of these methods can be found in [32], where an extensive reference list is given. In this order, these methods range from being non-intrusive to very intrusive in terms of modifications to the deterministic model. The stochastic collocation method is a member of the first category along with the well known Monte Carlo (MC) method [15]. Whereas MC simulations require generating a large number of realizations at random points in the stochastic event space, the stochastic collocation method instead performs realizations at specifically chosen collocation points. This technique obtains better accuracy than MC with fewer realizations. While the non-sampling methods such as moment/perturbation and stochastic finite elements are known to be highly accurate, in practice they are only suitable for systems with relatively small dimensions of random inputs. Their intrusive character complicates implementation, and the resulting large coupled systems may be dif- ficult to parallelize. Conversely, sampling methods generate systems of the same size as their deterministic equivalents that are completely decoupled from each other and hence very easy to parallelize.

In our model, the mean removed log permeability function is parameterized using in- dependent identically distributed random variables in a truncated Karhunen-Loève (KL) expansion. The eigenvalues and eigenfunctions forming this series are computed from a given covariance relationship in which statistical properties such as variance and correla- tion lengths are assumed to be experimentally determined.

This work builds upon the framework for stochastic collocation and mixed finite ele- ments that was developed in [18]. There, the porous media was assumed to be stationary, meaning that the statistical properties of the permeability were assumed to be constant throughout the domain. In this work we follow [26], see also [35] for a related perturbation- based approach, in extending this framework to allow nonstationary porous media with different covariance functions for different parts of the domain. These statistically inde- pendent zones are used to represent multiple rock types, motivated by geologic features such as stratification. We shall refer to these zones as KL regions. In this framework for nonstationary porous media, the covariance between any two points within a single KL region depends on their distance only, but the covariance between any two points which lie in different KL regions is zero, i.e. they are uncorrelated.

In porous media problems, resolving fine scale accuracy is oftentimes computationally in- feasible, necessitating multiscale approximations, such as the variational multiscale method [24, 4] and multiscale finite elements [23, 10, 2]. Both have been applied to stochastic problems in [7, 17] and [14, 1] respectively.

This paper employs for each stochastic realization the MMMFEM [6], with the recently proposed multiscale flux basis implementation [19]. As a mixed method, it provides accurate approximation of both pressure and velocity and element-wise conservation of mass, which are advantageous properties for porous media flow. The MMMFEM uses non-overlapping domain decomposition to break up the physical domain into subdomains controlled by separate computer processors, giving a natural parallelization within a fixed realization, thereby enabling UQ for very large problems.1 Within each subdomain, there is a fine scale

1 It should be noted that one of the benefits of non-intrusive UQ techniques is the “embarassingly parallel”

2

discretization that may be spatially non-conforming to its neighboring subdomains. On subdomain interfaces, a coarse scale mortar discretization is used to impose weak continuity of the discrete normal velocities. Using these varying scales, the global fine scale problem is reduced to a coarse scale interface problem and solved in parallel using an iterative method. We present error analysis for the stochastic multiscale approximation of the pressure and the velocity. We refer the reader to [25] for work on overlapping Schwarz domain decomposition methods for stochastic partial differential equations.

Notice that the physical domain has two decompositions: KL regions for the statistical representation of the nonstationary random permeability, and subdomains for the domain decomposition of the MMMFEM. The former is a physical decomposition depending on geologic structure, and the latter is a computational decomposition depending on available computing resources. It is our choice in implementation that the subdomains conform to the KL regions, meaning that each subdomain belongs to a single KL region. Therefore the number of KL regions NΩ is less than or equal to the number of subdomains ND, and each KL region can be expressed as a union of one or more disjoint subdomains. This approach allows for utilizing more processors than the physically dependent number of KL regions.

In a deterministic setting, the traditional implementation of the interface iteration in the MMMFEM requires solving one Dirichelet-to-Neumann problem on each subdomain (a linear system) for each interface iteration. Solving these subdomain problems is the dominant computational cost of the MMMFEM, and therefore this cost worsens with the condition number of the problem. In [19], a new approach was proposed called multiscale flux basis implementation, in which one subdomain problem is solved for each mortar de- gree of freedom before the interface iteration begins. The solutions to these problems form a basis of flux responses containing all the necessary information to solve the subdomain problem. The computational cost in forming the basis is a fixed and controllable quantity, and therefore does not worsen with the condition number of the problem. Linear com- binations of multiscale basis functions are used during the interface iteration so that no additional subdomain problems are required, except for one or more additional solves to recover the local fine scale information at the completion of the iteration. Therefore the multiscale flux basis implementation is more efficient in cases where the number of interface iterations strictly exceeds the number of mortar degrees of freedom per subdomain. This gain in computational efficiency increases with the number of subdomains.

In this paper we propose possible ways that extend the concept of a multiscale flux basis to the stochastic flow problem, where the permeability is a nonstationary random field. To this end, we investigate three algorithms that combine stochastic collocation and the MMMFEM with varying degrees of the multiscale flux basis implementation. The first collocation algorithm uses the MMMFEM with its traditional implementation, requiring solving one subdomain problem per interface iteration, on every stochastic realization. The second collocation algorithm forms a deterministic multiscale basis to s