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Adaptive Heterogeneous Multiscale Methods for immiscible two-phase flow in porous media

In this contribution we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible two-phase flow system with degenerate permeabilities. The method is in a general formulation which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an a-posteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved in [C. Canc{è}s, I. S. Pop, and M. Vohral\'ık. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp., 2014].