Weakly singular integral equations for Darcy's flow in anisotropic porous media

The weakly singular, weak-form fluid pressure and fluid flux integral equations are developed for steady-state Darcy's flow in a porous media which possesses generally anisotropic permeability. A systematic technique is established first to regularize the conventional fluid pressure and fluid flux integral equations in which the former contains a singular kernel (of order 1/r2) and the latter contains a strongly singular kernel (of order 1/r3). The key step in the regularization procedure is to construct decompositions for the fluid velocity fundamental solution and the strongly singular kernel in a form well-suited for performing an integration by parts via Stokes' theorem. Weakly singular kernels appearing in these decompositions are obtained in an explicit form by the Radon transform method. The final integral equations possess following important features: (1) they contain only weakly singular kernels of order 1/r; (2) their validity requires only the continuity of pressure boundary data; and (3) they are applicable for modeling steady-state flow in porous media possessing generally anisotropic permeability and containing surfaces of discontinuity. A proper use of this set of weak-from integral equations forms a symmetric formulation for a weakly singular, symmetric Galerkin boundary element method (SGBEM). The numerical implementation is then carried out and certain examples are solved to demonstrate the accuracy of the method.