The Unsteady Flow of a Weakly Compressible Fluid in a Thin Porous Layer III: Three-dimensional Computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary-value and eigenvalue problems. In this paper we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme. 1. Introduction A problem with significant applications in the oil and gas industry, and also to water resource management, is that of computing the pressure and velocity fields for a weakly compressible fluid flowing in a porous medium, with wells injecting or extracting fluid considered as line sources and sinks respectively. The (in general) heterogeneity of the porous medium, compressibility of the fluid, singularities induced by the source and sink terms, large size of the computational domain and long time interval over which simulations are often required make accurate and efficient modelling of such a scenario an extremely challenging task. Problems such as this have been very widely considered in the literature; we refer to e.g. (1, 2) and the many references therein for a detailed summary of the modelling and computational issues that must be resolved. Here, we consider the case of fluid flowing in a porous three-dimensional (3-d) layer of inhomogeneous and anisotropic permeability, with variable upper and lower boundaries and vertical side walls and wells. Numerical solution of the full equations of motion throughout the layer can be prohibitively expensive. The approach we present in this paper is based on the key observation that in geophysical applications the depth scale h of the layer is often small compared to the length scale l. For example, in (3) numerical results for single-phase