A nonlinear test case for the finite element method in two-phase flow

The ability of the finite element method to compute the motion of sharp interfaces in two-phase flow is examined by applying it to a test problem for which an analytical solution can be found. The problem is one of imbibition, the nonlinear diffusion of a fixed amount of water into an oil filled porous medium and can be solved exactly by similarity and the separation of variables method used by Boyer. The finite element program used was of the Galerkin type and employed a self-adaptive time stepping algorithm with both linear and quadratic isoparametric triangular elements. Results are presented for both elements and show that there is little difficulty in this type of diffusion problem in following the oil-water interface to accuracies of 2 or 3 percent.