The development and testing of a computer based mathematical model simulating simultaneous transport of heat (or a dissolved solute) and water in a saturated single-phase groundwater flow system are described. The formulation of the underlying flow and transport equations is presented and the discretization of these equations is developed using the standard Galerkin finite element method, and the upstream weighted residual finite element technique. A computer code is developed to solve the discretized equations; instructions for data preparation and use of the code are included. The new upstream weighted residual technique is evaluated by comparing the simulated results with the analytical solution for a plume dispersing from a point source in a uniform flow. These comparisons demonstrate that the upstream weighted residual technique has distinct advantages over the standard Galerkin finite element method in terms of accuracy and computational efficiency.