Single-phase and multi-phase flow in porous media has important applications for petroleum reservoir engineering and groundwater processes. Both applications, in particular the latter, may involve multiple time and spatial scales, long simulation time periods, and many coupled nonlinear components. In particular, the advection-dominated component and the nonlinear coupling of compressibility, capillary pressure and relative permeabilities often result in sharp saturation fronts, which demands steep gradients to be preserved with minimal oscillation and numerical diffusion.

In this talk, we consider Discontinuous Galerkin (DG) methods for simulating single-phase and multi-phase flow in porous media. DG methods, as specialized finite element methods that utilize discontinuous spaces to approximate solutions, are locally mass conservative by construction. The advantages of DG methods include small numerical diffusion and little oscillation as well as their abilities to capture the discontinuities and sharp fronts in the solution very well. To reduce computational cost without losing the fine scale resolution, we propose a two-scale formulation of the DG methods. Using a certain closure assumption, the two-scale DG algorithm is able to capture fine scale flow phenomena in strongly heterogeneous porous media and yet only requires a computational cost slightly larger than that for the coarse scale. A number of numerical examples are presented to illustrate computational advantages of this multiscale DG method for porous media flow.