Modelling well leakage in multilayer aquifer systems using the extended finite element method

The extended finite element method (XFEM) is applied to the problem of predicting the steady-state leakage from layered sedimentary aquifer systems perforated by abandoned wells. Multi-aquifer systems are modelled using a quasi-three-dimensional model where the head distribution in each aquifer is assumed to be two-dimensional. The standard finite element method is locally enriched in the vicinity of both injection and abandoned wells so that the logarithmic singularity in the solution at the wells is more accurately approximated. Two versions of the discrete equations are developed. In the first, no assumptions are made about the relative alignment of the meshes in adjacent aquifers and a relatively complex, but general, system of equations results. In the second, the meshes in adjacent aquifers are assumed to be aligned and system of equations is less complex. In the simplified case, the model can be broken down into a set of three building blocks—aquifer elements, aquitard elements and well elements. Numerical examples show that the standard Finite Element Method (FEM) poorly approximates the head and the flux rate at the abandoned wells and that the convergence rate, in terms of leakage through the abandoned wells, of the FEM is on the order of O(h0.4). In contrast, the XFEM is shown to be more than two orders of magnitude more accurate for coarse meshes. Furthermore, the XFEM solution converges four times faster, at a rate of about O(h1.8). The excellent computational efficiency and flexibility of the XFEM model makes it an attractive alternative to both the standard finite element analysis and to semi-analytical methods for predicting well leakage in multi-aquifer systems.