Three-dimensional flow in fractured porous media: A potential solution based on singular integral equations

Governing equations for flow in three-dimensional heterogeneous and anisotropic porous media containing fractures or cracks with infinite transverse permeability are described. Fractures are modeled as zero thickness curve surfaces with the possibility of multiple intersections. It is assumed that flow obeys to an anisotropic Darcy’s law in the porous matrix and to a Poiseuille type law in fractures. The mass exchange relations at fractures intersections are carefully investigated as to establish a complete mathematical formulation for the flow problem in a fractured porous body. A general potential solution, based on singular integral equations, is established for steady state flow in an infinite fractured body with uniform and isotropic matrix permeability. The main unknown variable in the equations is the pressure field on the crack surfaces, reducing thus from three to two the dimension of the numerical problem. A general transformation lemma is then given that allows extending the solution to matrices with anisotropic permeability. The results lead to a simple and efficient numerical method for modeling flow in three-dimensional fractured porous bodies.