Mixed Multiscale Finite Element Methods on Adaptive Unstructured Grids Using Limited Global Information

We study mixed multiscale finite element methods (MsFEM) on unstructured coarse grids. Unstructured grids are often used when highly heterogeneous reservoirs are discretized via irregular anisotropic fine grids. Our study is motivated by the development of coarse-scale models for coupled flow and transport equations in a multi-phase system. An unstructured coarse grid is often used to upscale the transport equation with hyperbolic nature in a highly heterogeneous reservoir. Solving the flow equation on the same coarse grid provides a general robust coarse-scale model for the multiphase flow and transport at a low CPU cost. We present numerical results when both the flow and transport equations are solved on the coarse grid. Numerical examples involve highly channelized permeability as well as a 3- D reservoir model using an unstructured fine grid. In all examples, we show that our approach can provide an accurate approximation of the resolved solution at a much lower cost. We also study the convergence of the mixed multiscale finite element method on unstructured coarse grids.