Weakly-Singular Integral Equations for Steady State Flow in Anisotropic Porous Media Containing Discontinuity Surfaces

In this paper, we present the development of the weakly-singular, weak-form fluid pressure and fluid flux integral equations for steady state Darcy’s flow in porous media. The integral equation for fluid flux is required for the treatment of flow in a domain which contains surfaces of discontinuities (e.g. cracks and impermeable surfaces), since the pressure integral equation contains insufficient information about the fluid flux on the surface of discontinuity. In this work, a systematic technique has been established to regularize the conventional fluid pressure and fluid flux integral equations in which the pressure equation contains a Cauchy singular kernel and the fluid flux equation contains both Cauchy and strongly-singular kernels. The key step in the regularization procedure is to construct a special decomposition for the fluid velocity fundamental solution and the strongly-singular kernel such that it is well-suited for performing an integration by parts via Stokes’ theorem. These decompositions involve weakly-singular kernels where their explicit form can be constructed, for general anisotropic permeability tensors, by the integral transform method. The resulting integral equations possess several features: they contain only weakly-singular kernels of order 1/r; their validity requires only that the pressure boundary data is continuous; and they are applicable for modeling fluid flow in porous media with a general anisotropic permeability tensor. A suitable combination of these weakly-singular, weak-form integral equations gives rise to a symmetric weak-form integral equation governing the boundary valued problem, thereby forming a basis for the weakly-singular, symmetric Galerkin boundary element method (SGBEM). As a consequence of that the integral equations are weakly-singular, the SGBEM allows standard C° elements to be employed everywhere in the discretization.