Analytical expressions for the permeability of random two‐dimensional Poisson fracture networks based on regular lattice percolation and equivalent media theories

Abstract

The permeability of random two‐dimensional Poisson fracture networks can be studied using a model based on percolation theory and equivalent media theory. Such theories are usually applied on regular lattices where the lattice elements are present with probability p. In order to apply these theories to random systems, we (1) define the equivalent to the case where p = 1, (2) define p in terms of the statistical parameters of the random network, and (3) define the equivalent of the coordination number z. An upper bound for permeability equivalent to the case of p = 1 is found by calculating the permeability of the fracture network with the same linear fracture frequency and infinitely long fractures. The permeability of networks with the same linear fracture frequency and finite fractures can be normalized by this maximum. An equivalent for p is found as a function of the connectivity ζ, which is defined as the average number of intersections per fracture. This number can be calculated from the fracture density and distributions of fracture length and orientation. Then the equivalent p is defined by equating the average run length for a random network as a function of ζ to the average run length for a lattice as a function of p. The average run length in a random system is the average number of segments that a fracture is divided into by intersections with other fractures. In a lattice, it is the average number of bonds contiguous to a given bond. Also, an average coordination number can be calculated for the random systems as a function of ζ. Given these definitions of p and z, expressions for permeability are found based on percolation theory and equivalent media theory on regular lattices. When the expression for p is used to calculate the correlation length from percolation theory, an empirical formula for the size of the REV can be developed. To apply the models to random length systems, the expression for ζ must be modified to remove short fractures which do not contribute to flow. This leads to a quantitative prediction of how permeability decreases as one removes shorter fractures from a network. Numerical studies provide strong support for these models. These results also apply to the analogous electrical conduction problem.