AN EXACT SOLUTION TO THE NONLINEAR DIFFUSION CONVECTION EQUATION FOR 2-PHASE FLOW - Sander, G; Norbury, J; Weeks, S (1993)
A new exact solution to the nonlinear diffusion-convection equation ∂θ/∂t = ∂/∂x(D(θ)∂θ/∂x)-V(t) ∂f(θ)/∂x, 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(θ) = D0/(1-νθ)2. The functional form of the fractional flow function f(θ) is then found from a condition which determines the integrability of the resulting nonlinear ordinary differential equation. This results in an f(θ)-function of the form f(θ) = (F1+F2θ+F3θ2)/ (1-νθ). Asymptotic expansions are derived for the fluid flux across the surface boundary x = 0 as a function of the parameters D0 and ν, and the surface boundary condition. These expansions show that a Buckley-Leverett-type profile can be obtained in the limit of ν→1 for a saturated surface boundary condition.