Generalized multiscale finite element method for highly heterogeneous compressible flow

TL;DR

This paper develops a generalized multiscale finite element method for simulating highly heterogeneous compressible flow in porous media, offering efficient and accurate coarse-grid solutions with online basis enhancements.

Contribution

The paper introduces a new GMsFEM approach with offline and online basis construction, including rigorous convergence analysis and application to 3D heterogeneous media.

Findings

Convergence rates depend on coarse mesh size and spectral decay.

Online basis significantly improves accuracy for singular sources.

Method demonstrates computational efficiency in 3D simulations.

Abstract

In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize and the eigenvalue decay of the local spectral problem. To further increase the accuracy of multiscale method, residual driven online multiscale basis is added to the offline space. The construction of online multiscale basis is based on a carefully design error indicator motivated by the analysis. We find that online basis is particularly important for the singular source. Rich numerical tests on typical 3D highly heterogeneous medias are presented to demonstrate the impressive computational advantages of the proposed multiscale method.