Fluid flow and effective permeability of an infinite matrix containing disc-shaped cracks

A basic equation governing the steady-state flow around a single crack in an infinite porous body is given. The flow through the crack obeys to the Poiseuille’s law and the matrix has an anisotropic permeability. A semi-analytical solution is established for this equation in the case of elliptical disc-shaped crack. This solution takes a closed-form expression for the case of superconducting circular cracks. The results are compared to those obtained for flattened ellipsoidal inclusions obeying to the Darcy’s flow law, which are in some works supposed to represent the cracks. It is shown that the flow solution for an elliptical disc-shaped crack obeying the Poiseuille’s law is different from that obtained as the limiting case of flattened ellipsoidal inclusions. The results are then used to establish dilute Mori–Tanaka and self-consistent estimates of the effective permeability of porous media containing Poiseuille’s type elliptical cracks.