Abstract
Accurate and robust discretization of the fluid flow equations is required to account for the extreme heterogeneity of oil and gas reservoirs, and the combined effect of anisotropy and grid distortion, necessary to adapt the grid to the geology. There is a growing need for handling distorted unstructured grids and full permeability tensors that appear after upscaling of the fine-scale permeability field. The classical cell-centered finite difference method results in a 7-point stencil and is insufficient to account for these effects. A number of closely related methods, coined as multipoint flux approximations (MPFA), have been proposed recently and are currently under active development. The basic principle of MPFA is that the flux across an interface between two gridblocks depends on the state variables (pressure and saturations) of more than two gridblocks. MPFA leads naturally to an enhanced finite volume method with a 27-point stencil.
In this paper, we implement a variant of the MPFA method for corner-point geometry hexahedral grids. Motivated by the very high aspect ratio of gridblocks in typical reservoir models (ratios as high as 100:1 are not uncommon), we propose a hybrid method that employs a multipoint flux approximation in the areal direction and a two-point flux approximation in the vertical direction. This restricted MPFA method leads to an 11-point stencil, therefore reducing the computational effort significantly. We discuss the implementation of the method in detail. We evaluate its performance on a number of test cases and show that, for typical applications, this simplification does not greatly compromise the accuracy of the solution.
