A model problem for two-phase fluid flow and heat transfer with phase change in a porous medium is described. The model is based on a steam-water mixture in sand. Under certain conditions, a two-phase zone, in which liquid and vapour coexist, is separated from a region of only vapour by an interface. A numerical method for locating the interface in the one-dimensional, steady-state problem is described. The results from the steady-state computations are used as benchmarks for the numerical results for the transient problem. It is shown that methods such as front tracking and the level-set method are not practical for the solution of the transient problem, due to the indeterminate nature of the interface velocity, in common with similar degenerate diffusion problems. An interface-capturing method, based on a two-phase mixture formulation, is presented. A finite volume method is developed, and numerical results show evolution to the correct steady state. Furthermore, similarity solutions are found, and the interface is shown to propagate at the correct velocity, by way of a numerical convergence study. Numerical results for the two-dimensional problem are also presented.