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.