A New Finite-Volume Approach to Efficient Discretization on Challenging Grids

Abstract

Abstract MPFA methods were introduced to solve control-volume formulations on general simulation grids for porous media flow. While these methods are general in the sense that they may be applied to any grid, their convergence properties vary. An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper [1], conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations which lead to smaller flux stencils are desirable for grids with high aspect ratio or severe skewness and for media with strong anisotropy or strong heterogeneity. The ideas were recently pursued in [2] where the L-method was introduced for general media in 2D. For homogeneous media and uniform grid, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencils appear as a consequence of adapting the method to the closest neighboring cells. Here, we extend the ideas for discretization on 3D grids, and ideas and results are shown for both conforming and nonconforming grids. The ideas are particularly desirable for simulation grids which contain faults and local grid refinement. We present numerical results herein which include convergence results for single phase flow on challenging grids in 2D and 3D, and some simple two-phase results. Also, we compare the L-method with the O-method. Introduction In reservoir simulation, control-volume formulations with two-point flux approximations (TPFA) are generally used to approximate the elliptic operator Equation 1 which we here consider in both 2D and 3D. The tensor K is the permeability, and it is assumed to be symmetric and positive definite. The potential u is the pressure. Two-point flux approximations are physically intuitive, and the discretization of the operator (1) results in an M-matrix. However, for non-K-orthogonal grids, TPFA gives an error in the solution which does not vanish as the grids are refined, cf. [3,4].