Exact integral solutions for two‐phase flow

Abstract

Exact integral solutions for the horizontal, unsteady flow of two viscous, incompressible fluids are derived. Both one‐dimensional and radial displacements are calculated with full consideration of capillary drive and for arbitrary capillary‐hydraulic properties. One‐dimensional, unidirectional displacement of a nonwetting phase is shown to occur increasingly like a shock front as the pore‐size distribution becomes wider. This is in contrast to the situation when an inviscid nonwetting phase is displaced. The penetration of a nonwetting phase into porous media otherwise saturated by a wetting phase occurs in narrow, elongate distributions. Such distributions result in rapid and extensive penetration by the nonwetting phase. The process is remarkably sensitive to the capillary‐hydraulic properties that determine the value of knw/kw at large wetting phase saturations, a region in which laboratory measurements provide the least resolution. The penetration of a nonwetting phase can be expected to be dramatically affected by the presence of fissures, worm holes, or other macropores. Calculations for radial displacement of a nonwetting phase resident at a small initial saturation show the displacement to be inefficient. The fractional flow of the nonwetting phase falls rapidly and, for a specific example, becomes 1% by the time one pore volume of water has been injected.