Consistent MPFA Discretization for Flow in the Presence of Gravity

M Starnoni,E Keilegavlen,J M Nordbotten,I Berre

doi:10.1029/2019wr025384

Abstract

AbstractA 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.

Highlights

There exist several methods for solving numerically the single-phase and multiphase flow equations in porous media
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 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, obtaining an expression of the fluxes in terms of jumps in cell-center gravities

Summary

Introduction

There exist several methods for solving numerically the single-phase and multiphase flow equations in porous media. MPFA is a control volume method introduced independently by two different research groups in 1994 (Aavatsmark et al, 1994; Edwards & Rogers, 1994). We only consider the so-called O-method developed by Aavatsmark and coworkers They first introduced MPFA for general quadrilateral grids in Aavatsmark et al (1994) and Aavatsmark et al (1996) and extended the method to triangular and polygonal grids in Aavatsmark et al (1998a, 1998b). The reader can refer to Aavatsmark (2002) for an excellent review on MPFA methods for quadrilateral grids and to Aavatsmark et al (2007) for a numerical investigation on its convergence properties. Local criteria which ensure monotonicity for general control volume methods on heterogeneous media are given in Nordbotten et al (2007)

Methods

Results

Conclusion