From the macroscopic equations of flow through a porous medium, a model is developed for the subsidence or decay of a mound of fluid over a horizontal impervious barrier. Three finite difference approaches are used to investigate the resulting moving boundary problem. In the first method, a coordinate transformation is used that fixes the toe or leading edge of the fluid location. The resulting problem is solved on a regular grid with interpolation near the moving boundary. In the second method, a coordinate transformation is employed that fixes the location of the entire boundary while in the third method, a transformed polar coordinate formulation is employed that maps the domain onto a quarter circle. For the latter two methods the resulting equations are solved using a regular grid. It is demonstrated that for all three methods, a predictor-corrector scheme gives satisfactory results for the boundary position without iteration. It was found that the first two methods were more accurate than the third for a given spatial discretization. A new similarity solution to the Bousinesq approximation to the problem for an initially parabolic mound is presented and is found to compare favourably to the numerical solution to the full problem for mounds with large initial aspect ratios.;The model is refined so that the effect on the mobility of the fluid of heating along the interface is incorporated. The fluid viscosity and density are assumed to be dependent on temperature. The problem is investigated using the latter two fully transformed finite difference grids above. The backwards Euler method is used for the temporal derivatives of both the temperature and boundary position equation and a successive substitution scheme with relaxation is used to iterate between the various coupled equations. Again it is found that the scheme based on rectangular coordinates is more accurate for a given spacial discretization than the scheme based on polar coordinates. Depending upon the choice of parameters in the problem the mound takes on a \u27kinked\u27 appearance with the quickly heated toe area spreading out with a linear like profile and the central body of the fluid dropping slowly
