The research is a study on the microscopic scale of the immiscible displacement of oil by water in a porous medium such as sandstone. Of particular interest (with application to the oil industry) are the residual saturation of oil, the permeability to water at residual oil saturation and the maximum trapped blob size. Initially the effects of gravity, surface tension and distribution of pore sizes were studied in a computer simulation of a buoyancy driven, quasi-static invasion. The rock was modelled as a three-dimensional lattice of spherical pores connected by narrow cylindrical throats. With the rock water-wet, the tendency of the surface tension to favour the invasion of smaller pores led to a larger residual oil saturation by pore volume than by pore numbers. Also bourne out were some scaling arguments based on percolation theory for the maximum trapped blob size as a function of the relative strength of buoyancy and surface tension forces. The second part of the research investigated the interaction of viscous and surface tension forces. As this is a much more complicated problem, involving the solution of flow equations, the invasion process was first simulated with exact equations of motion on small networks (up to 10x10), where surface tension effects dominate. From these simulations a simplified set of rules was developed to determine which pore in a locality on the oil-water interface is invaded and how long the invasion takes. These rules include a viscous correction to the dominant surface tension forces. Finally, some theory has been developed for the inclusion of the small-scale analysis into a larger model, allowing a full simulation of the viscous dominated invasion to be performed.