A multiscale finite-volume method for three-phase flow influenced by gravity

Abstract

A multi-scale finite-volume (MSFV) method for solving multiphase flow problem in highly heterogeneous media was recently developed. In contrast with classical upscaling techniques, the goal of multiscale methods is not simply to capture the large-scale effects of the fine-scale heterogeneity, but to provide an efficient tool for solving large flow problems with fine-scale resolution. The MSFV is based on a fractional flow formulation of the problem: first an equation for the total velocity is solved, then a fine-scale velocity field is reconstructed, finally the phase-saturation distribution is obtained by solving the nonlinear transport equations. In addition to the original fine grid the MSFV method employs an imposed coarse grid and a dual coarse grid. The first step is to compute the effective parameters that have to be used for solving the global flow problem on a coarse grid. This is done by means of a set of basis functions, which are numerical solutions computed on the cells of the dual grid. From these basis functions, the fluxes across the coarse-block boundaries are computed and the transmissibilities are extracted. Then the conservative fine-scale total-velocity field is reconstructed by solving a local flow problem in each coarse cell. In this paper we extend the MSFV method to account for gravity effects, which enables our method to work for fairly realistic problems, e.g. dead-oil problems or black-oil problems. To correctly account for gravity effects an additional local problem has to be solved on each block of the dual grid. This problem provides an additional basis function (gravity basis function) that represents a local correction to the fluxes computed in the absence of gravity.