A comparison of multiscale methods for elliptic problems in porous media flow

Abstract

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method -the adaptive local-global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic twophase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation.

A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms

“…The method is an extension of the numerical subgrid upscaling technique proposed in [4] and, as we show here, has a clear connection with the multiscale mixed finite element method [28]. The key ingredients of our method are:…”

“…This observation was made in [6] for the case when subgrid communication was disallowed (they also treated a method with oversampling that leads to a nonconforming fine-scale velocity field In the absence of a source term with multiscale character, this method is precisely the multiscale mixed finite element method proposed in [28]. Our method differs in its derivation (the variational multiscale framework rather than the multiscale finite element method) and, more importantly, in the treatment of fine-scale sources.…”

“…Different definitions of the multiscale basis functions exist [16,1,2], and in this work we have adopted the recent one proposed by [28], where the multiscale basis function is the solution to a flow problem restricted to a pair of adjacent coarse elements with source terms specified in such a way that the flow through the interface is identically one (see Figure 2-1). More precisely, the multiscale basis functions (N Hh, f'H,h) for interface Fa (common to coarse elements Ei and Ej) are the solution to the following local problem:…”

“…It has been reported recently [28,30] that some multiscale methods have difficulty in producing high quality solutions in the presence of permeability anisotropy or large aspect ratios of the grid (both cases are common in reservoir models). Here, we demonstrate the ability of our proposed method to cope with these scenarios.…”

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

“…Adaptive grids also allow important flow features with small spatial support, such as frontal regions, to be resolved accurately without sacrificing computational efficiency, see for example [29,28,14]. We think of adaptive grid strategies as part of the class of multi-scale methods, as discussed in, for example, [21] and references therein. Multiscale methods, designed to capture multiple scales involved in the reservoir fluid flow processes, include approaches that upscale to a coarse simulation grid and reconstruct the solution on a finer scale within each coarse grid cell using a functional representation, but also approaches based on grid refinement.…”