A vertex scheme for two-phase flow in heterogeneous media

This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux up-winding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions. For both structured and unstructured meshes, we observe the high-accuracy wetting saturation profile that ensures minimal numerical diffusion at the front. Performing several examples of quarter-five spot problems in two and three dimensions, we show that the method can easily handle heterogeneities in the permeability field. Two distinct features that make the method appealing to reservoir simulators are: (i) maximum principle is satisfied, and (ii) mass balance is locally conserved.