Some analytical solutions for fluid flow in and around a single fracture in a porous formation based on singular integral equation

This study considers the fluid flow through a porous formation containing discontinuities (fault, fracture, crack, microcrack), which is usually much more conductive than the surrounding matrix. The discontinuity is mathematically represented by a 1D smooth function of curvilinear abscise and physically characterized by its aperture. Fluid flow is assumed to obey Poiseuille's law in the discontinuity and Darcy's law in the parent porous rock. The solution for the fluid potential within a finite fractured porous medium is established under a singular integral equation form. Explicit solutions of flow and pressure field around a superconductive discontinuity within an infinite matrix with an anisotropic permeability, are derived by a development of a singular integral in a conventional Cartesian coordinate system. The solution shows that the fluid flow transported by a single crack only depends on the determinant of permeability tensor of the host rock but not on its components. A numerical simulation is performed to show that this result is also true for the discontinuity with a finite conductivity. The case of pressurized crack is also considered, discussed and compared to available numerical solutions in the literature.