Analysis of nonequilibrium effects and flow instability in immiscible two-phase flow in porous media

Abstract

Two-dimensional, high-resolution, numerical solutions for the classical formulation and two widely accepted nonequilibrium models of multiphase flow through porous media are generated and compared with experimental observations from literature. Flow equations for simultaneous flow of two immiscible phases through porous media are written in a vorticity stream-function form. In the resulting system of equations, the vorticity stream-function equation is solved using a spectral method and the transport equation is discretized in space using a central-upwind scheme. The system of equations is solved for a two-dimensional domain using a semi-implicit time-stepper. The solutions reveal behavior that is not apparent in one-dimensional solutions, namely that sharpening of the saturation front caused by the inclusion of a dynamic capillary pressure results in propagation of viscous fingers compared to the classical formulation. The inclusion of nonequilibrium effects in the constitutive relations, in the form of effective saturation, introduces dispersion and smears the otherwise highly resolved viscous fingers in the saturation front. Once developed, the length of the mixing zone in the numerical solutions remains constant with time regardless of the degree of instability. This is contrary to the evolution of the mixing zone observed in unstable flow experiments where, unlike the numerical solutions, the propagation speed of the leading edge of the front appears to increase with time.