N-dimensional first integral and similarity solutions for two-phase flow

Abstract

This paper considers similarity solutions of the multi-dimensional transport equation for the unsteady flow of two viscous incompressible fluids. We show that in plane, cylindrical and spherical geometries, the flow equation can be reduced to a weakly-coupled system of two first-order nonlinear ordinary differential equations. This occurs when the two phase diffusivity D(θ) satisfies (D/D′)′ = I/α and the fractional flow function f(θ) satisfies df /dθ = κDn/2, where n is a geometry index (1, 2 or 3), α and κ are constants and primes denote differentiation with respect to the water content θ. Solutions are obtained for time dependent flux boundary conditions. Unlike single-phase flow, for two-phase flow with n = 2 or 3, a saturated zone around the injection point will only occur provided the two conditions ∫01D/(1 - f)dθ