A space–time multiscale method for computing statistical moments in strongly heterogeneous poroelastic media of evolving scales

Abstract

SUMMARYA new multiscale procedure is proposed to compute flow in compressible heterogeneous porous media with geology characterized by power-law covariance structure. At the fine scale, the deformable medium is modeled by the partially coupled formulation of poroelasticity with Young's modulus and permeability treated as stationary random fields represented by their Karhunen-Loève decompositions. The framework underlying the multiscale procedure is based on mapping these random parameters to an auxiliary domain and constructing a family of equivalent stochastic processes at different length scales characterized by the same ensemble mean and covariance function. The poromechanical variables inherit a space-time version of the scaling relations of the random input parameters which allows for constructing a set of multiscale solutions of the same governing equations posed at different space and time scales. A notable feature of the multiscale method proposed herein is the feasibility of solving both the poroelastic model and the Fredholm integral equation for the eigenpairs of the Karhunen-Loève expansion in an auxiliary domain with much lower computational effort and then derive the long term behavior at a coarser scale from a straightforward rescaling of the auxiliary solution. Within the framework of the finite element approximation, in conjunction with the Monte Carlo algorithm, numerical simulations of fluid withdrawal and injection problems in a heterogeneous poroelastic reservoir are performed to illustrate the potential of the method in drastically reducing the computational burden in the computation of the statistical moments of the poromechanical unknowns in large-scale simulations.

Spectral Element Approximation of Fredholm Integral Eigenvalue Problems

“…The spectral element method was employed to numerically compute the eigenvalues and eigenfunctions of the Fredholm integral equation (1). The use of Gauss-Lobatto-Legendre (GLL) collocation points and reduced integration with GLL quadrature rendered the implementation similar to piecewise constant finite elements in the sense that the resulting discrete equations are a standard eigenvalue problem rather than a generalized one.…”

“…If the covariance kernel C is symmetric and positive definite, there is a monotonically decreasing set of eigenvalues and an orthonormal set of eigenfunctions satisfying (1). The solution of the eigenvalue problem (1) is relevant to the study of differential equations with random input data.…”

“…The next section reviews the piecewise-constant finite element method for (1) according to [1], and points out some computational issues. Section 4 introduces the spectral element method with Gauss-Lobatto-Legendre collocation points.…”

A wavelet Galerkin approximation of Fredholm integral eigenvalue problems with bidimensional Haar functions

“…is, up to index permutations, the matrix of the piecewise-constant finite element method for (4) [1]. In the following we describe a pyramid algorithm [2] to recover K…”

“…This representation is obtained from the eigenvalues and eigenfunctions of a homogeneous Fredholm integral equation of the second kind whose kernel is given by the covariance function of the random process. This approach has been widely used in the parametrization of parameters for elasticity problems, heat and mass transfer, fluid mechanic and acoustic [1,6,9].…”