The Use of Variable Weighting to Eliminate Numerical Diffusion in Two-Dimensional Two-Phase Flow in Porous Media

Abstract

A simple numerical method has been developed that largely eliminates numerical diffusion errors associated with saturation discontinuities or shocks for two-phase flow in one and two dimensions. The important aspect of the approach is the computation of a variable weighting factor for the interface fractional flow between grid blocks. In order to eliminate numerical diffusion at shocks, the appropriate weighting has been found to be downstream weighting just ahead of the shock and upstream weighting otherwise. In one dimension, the Buckley-Leverett problem as well as the motion of an oil bank and a more general initial saturation have been analyzed. In two dimensions, five-spot pattern floods with both homogeneous and heterogeneous permeability distributions have been modelled. In all cases examined, shocks are confined to a single grid block in the direction of flow. Numerical diffusion is present only where there are steep saturation gradients that are not shocks. The approach appears to be generalizable to the multicomponent, multidimensional case including gravity and capillarity.

The Extension of the Variable-Flux-Weighting Method to Multidimensional Problems Including Gravity and Capillarity

“…Fo~ a two-phase (oil-water) system, the time evolutIOn of the saturation equation is determined by a single equation: ~~ __ qdF ~ at dS ax (1) where S is the water saturation F is the volume fractional flow of water (water fl~x divided by total flux), and q is the total fluid velocity. A saturation distribution at any time consists of propagating compression and rarefaction waves.…”