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

ABSTRACT The variable-flux-weighting method has been shown to largely eliminate numerical diffusion at saturation discontinuities in two-phase flow in two-dimensional problems1. The method has been extended to multidimensional problems including gravity and capillarity. The method is derived from fractional flow theory and involves 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 saturation distribution for two-phase flow including gravity has been solved and compared with the analytical solution. The comparison is very good with no numerical diffusion apparent at the saturation discontinuity. Problems including capillarity nave also been solved. In this case there is no saturation discontinuity, but the method introduces downstream weighting at the upstream edge of the capillary fringe which effectively eliminates any significant numerical diffusion. Two- and three-dimensional problems have also been solved which demonstrate the accuracy of the method. For three-dimensional problems, the current implementation of the VFW method is ten times slower than an upstream-weighting IM PES method.