Consistent MPFA Discretization for Flow in the Presence of Gravity

Abstract

A standard practice used in the industry to discretizing the gravity term in the two-phase Darcy flow equations is to apply an upwind strategy. In this paper, we show that this can give a persistent unphysical flux field and an incorrect pressure distribution. As a solution to this problem, we present a new consistent discretization of flow, termed Gravitationally Consistent Multipoint Flux Approximation (GCMPFA), which is valid for both single- and two-phase flows. The discretization is based on the idea that the gravitational term in the flow equations is treated as part of the discrete flux operator and not as a right-hand side. Here, the traditional formulation representing pressure as a potential is extended to the case including gravity by introducing an additional set of right-hand side to the local linear system solved in the MPFA construction, thus obtaining an expression of the fluxes in terms of jumps in cell-center gravities. Numerical examples showing the convergence of the method are provided for both single- and two-phase flows. For two-phase flow, we show how our new method is capable of eliminating the unphysical fluxes arising when using a standard upwind scheme, thus converging to the correct pressure distribution.