Treatment of time derivative and calculation of flow when solving groundwater flow problems by Galerkin finite element methods

Two techniques connected with the use of the finite element Galerkin method for solving the linear parabolic differential equation describing unsteady groundwater flow in an anisotropic non-homogeneous aquifer are introduced. The first is a mode superposition technique for dealing with the time derivative which involves computing the smallest eigenvalues and associated eigenvectors of the matrices arising from the Galerkin method. It is shown how such a technique allows us to interpret the response of the groundwater level to input in terms of parallel linear reservoirs. It is further argued that if properly implemented, the technique will have computational advantages over standard finite difference methods, e.g. in the case when the input function is constant over relatively large time subintervals. The second is a technique based on so-called generalized flow formulae for calculating flow values across external or internal boundaries, posterior to obtaining the groundwater level values. The implementation of the technique in the case of linear triangular elements on an irregular grid is discussed. It is finally argued from simplified cases that, apart from guaranteeing a match with prescribed input, the technique may often be expected to give more accurate flow values than those obtained directly from the groundwater gradients.