The mathematical description of groundwater flow through a deformable porous medium has been obtained by combining Darcy's law with the mass conservation equations of both the groundwater and the soil mass. The partial differential equation governing the transient fluid flow with its appropriate initial and boundary conditions is the result. The numerical implementation of these differential equations for a rigid porous medium under partially saturated conditions has been achieved by converting them into integral equations by applying Galerkin's "weighted residual formulation" and Green's theorem. The finite element programs have been implemented in the FORTRAN "90" numerical environment. The numerically simulated one-dimensional compressible groundwater flows under saturated conditions were validated by comparing their results with analytical solutions. Satisfactory agreements were found between their results and the corresponding analytical solutions. For the partially saturated condition steady state and transient groundwater flow severe limitations in the formulation were encountered, in particular for the higher suction range in coarse granular materials.