A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids

We analyse and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid. Earlier work has shown that the multiscale method performs excellently on highly heterogeneous cases using uniform coarse grids. In this paper, we extend the methodology to nonuniform and unstructured coarse grids and discuss various formulations for generating the coarse-grid basis functions. Moreover, we focus on the impact of large-scale features such as barriers or high-permeable channels and discuss potentially problematic flow cases. To improve the accuracy of the multiscale solution, we introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or on adapting the coarse grid more directly to large-scale permeability structures of arbitrary shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner. The suggested strategies are illustrated in several numerical experiments.