The Multiscale Finite Element Method (MsFEM) and Its Applications

Abstract

Multiscale problems occur in many scientific and engineering disciplines, in petroleum engineering, material science, etc. These problems are characterized by the great deal of spatial and time scales which make it difficult to analyze theoretically or solve numerically. On the other hand, the large scale features of the solutions are often of main interest. Thus, it is desirable to have a numerical method that can capture the effect of small scales on large scales without resolving the small scale details. In the first part of this work we analyze the multiscale finite element method (MsFEM) introduced in [28] for elliptic problems with oscillatory coefficients. The idea behind MsFEM is to capture the small scale information through the base functions constructed in elements that are larger than the small scale of the problem. This is achieved by solving for the finite element base functions from the leading order of homogeneous elliptic equation. We analyze MsFEM for different situations both analytically and numerically. We also investigate the origin of the resonance errors associated with the method and discuss the ways to improve them. In the second part we discuss flow based upscaling of absolute permeability which is an important step in the practical simulations of flow through heterogeneous formations. The central idea is to compute the upscaled, grid-block permeability from fine scale solutions of the flow equation. It is well known that the grid block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of grid blocks. We analyze the effects of the boundary conditions and grid block sizes on the computed grid block absolute permeabilities. Moreover, we employ the ideas developed in the analysis of MsFEM to improve the computed values of absolute permeability. The last part of the work is the application of MsFEM as well as upscaling of absolute permeability on upscaling of two-phase flow. In this part we consider coarse models using MsFEM. We demonstrate the efficiency of these models for practical problems. Moreover, we show that these models improve the existing approaches.