Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions

Abstract

The reduction from three- to two-dimensional analysis of the permeability of a fractured rock mass introduces errors in both the magnitude and direction of principal permeabilities. This error is numerically quantified for porous rock by comparing the equivalent permeability of three-dimensional fracture networks with the values computed on arbitrarily extracted planar trace maps. A method to compute the full permeability tensor of three-dimensional discrete fracture and matrix models is described. The method is based on the element-wise averaging of pressure and flux, obtained from a finite element solution to the Laplace problem, and is validated against analytical expressions for periodic anisotropic porous media. For isotropic networks of power law size-distributed fractures with length-correlated aperture, two-dimensional cut planes are shown to underestimate the magnitude of permeability by up to 3 orders of magnitude near the percolation threshold, approaching an average factor of deviation of 3 with increasing fracture density. At low-fracture densities, percolation may occur in three dimensions but not in any of the two-dimensional cut planes. Anisotropy of the equivalent permeability tensor varies accordingly and is more pronounced in two-dimensional extractions. These results confirm that two-dimensional analysis cannot be directly used as an approximation of three-dimensional equivalent permeability. However, an alternative expression of the excluded area relates trace map fracture density to an equivalent three-dimensional fracture density, yielding comparable minimum and maximum permeability. This formulation can be used to approximate three-dimensional flow properties in cases where only two-dimensional analysis is available.