Effective permeability of three-dimensional porous media containing anisotropic distributions of oriented elliptical disc-shaped fractures with uniform aperture

Solution for the fluid potential within a fractured porous medium is described. This solution is derived numerically for the particular case of a Pouseuille's elliptical fracture plane with uniform aperture. A closed-form solution of total flow integrated over a single fracture is obtained as a function of matrix permeability, fracture permeability and fracture geometry parameters. This solution allows firstly the comparison between two approaches: Poiseuille's fracture and Darcy's flattened ellipsoidal inclusion. This shows the difference of pressure and flow fields in the fractures and the equivalence of total flow transported by fractures. Then, a semi-analytical solution is used to establish an effective permeability model based on the self-consistent scheme for a porous medium containing an anisotropic distribution of oriented elliptical plane fractures. The proposed model is in good agreement with numerical solutions reported in the literature for a random distribution of oriented fractures. The present model exhibits a percolation threshold for a three-dimensional fractured network. The percolation threshold is a critical value of fracture density, beyond which the effective permeability is greater than zero for the case of an impermeable parent material.