On a singular diffusion equation originating from porous media flow

Abstract

In this paper we consider the singular boundary value problem, −z^1/pz″=D(f)/(1+p) for 0<f<1 with z(0)=z(1)=0. Here p>0 and D: [0,1]→R is a given function. This problem arises in a model for two-phase capillary induced flow in porous media. Considering the special case D(f)=f^α_w(1−f)^α₀, with α₀,α_w>−2, we investigate the singular behaviour of the solution z(f) as α₀,α_w↓−2. We show that the solution then becomes unbounded. We investigate the behaviour of z and z′ in this limit process. The results are incorporated in an algorithm which we use to solve the problem numerically. The numerical results show significant improvement over standard discretisation techniques near the limit. Non-existence arises for α₀ or α_w≤−2.