An exact solution to the nonlinear diffusion-convection equation for two-phase flow

TLDR: In this paper, a new exact solution to the nonlinear diffusion-convection equation dt dx V dxjdx for two-phase fluid flow in porous media is derived.

Abstract: SUMMARY A new exact solution to the nonlinear diffusion-convection equation dt dx V dxj dx for two-phase fluid flow in porous media is derived. The solution is sought for Dirichlet boundary conditions and a diffusivity of the form D(8) = D0/(l — v6) 2. The functional form of the fractional flow function f(6) is then found from a condition which determines the integrability of the resulting nonlinear ordinary differential equation. This results in an /(0)-function of the form f(9) = (Ft + F26 + F302)/(l — v6). Asymptotic expansions are derived for the fluid flux across the surface boundary x = 0 as a function of the parameters Do and v, and the surface boundary condition. These expansions show that a Buckley-Levere tt-type profile can be obtained in the limit of v -> 1 for a saturated surface boundary condition.