This work is concerned with the application of finite difference simulation to modelling the pressure response in partially penetrating oil wells. This has relevance to the oil and hydrology industries where pressure behaviour is used to infer the nature of aquifer or reservoir properties, particularly permeability. In the case of partially penetrating wells, the pressure response carries information regarding the magnitude of permeability in the vertical direction, a parameter that can be difficult to measure by other means and one that has a direct influence on both the total volumes of oil that can be recovered and on the rate of recovery. The derivation of the non-linear differential equations that form the basis for multiphase fluid flow in porous media is reviewed and it is shown how they can be converted into a set of finite difference equations. Techniques used to solve these equations are explained, with particular emphasis on the approach followed by the commercial simulation package used in this study. This involves use of Newton's method to linearize the equations followed by application of a pre-conditioned successive minimization technique to solve the resulting linear equations. Finite difference simulation is applied to a hypothetical problem of solving pressure response in a partially penetrating well in an homogenous but anisotropic medium and the results compared with those from analytical solutions. Differences between the results are resolved, demonstrating that the required level of accuracy can be achieved through selective use of sufficiently small grid blocks and time-steps. Residual discrepancies with some of the analytical methods can be traced to differences in the boundary conditions used in their derivation. The simulation method is applied to matching a complex real-life well test with vertical and lateral variation in properties (including fluid saturation). An accurate match can be achieved through judicious adjustment of the problem parameters with the proviso that the vertical permeability needs to be high. This suggests that the recovery mechanism in the oil field concerned can be expected to be highly efficient, something that has recently been confirmed by production results. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 2002.