Mixed Multiscale Methods for Compressible Flow

Multiscale methods are a robust and accurate alternative to traditional upscaling methods. Multiscale methods solve local problems to numerically construct a set of basis functions that later can be used to compute global solutions that describe the flow on both the coarse computational scale and the underlying fine parameter scale. This way, one is able to account for both effective coarse-scale properties and sub-scale variations. The methods are particularly efficient when the flow field must be updated repeatedly. Because temporal changes in the flow equations are moderate compared to the spatial variability, it is seldom necessary to recompute basis functions each time the global flow field is recomputed. Herein, we discuss and compare two ways of extending a multiscale mixed method that was originally developed for incompressible flow to compressible flow. The first approach is based upon a mixed residual formulation with a fine-scale domain-decomposition corrector. The second approach is to associate more than one basis function for each coarse face and coarse cell and use bootstrapping to dynamically build a basis function dictionary that spans the evolving flow patterns. We present and discuss several numerical examples, from simplified 1D cases to 3D cases with realistic reservoir geometries.