Relationship between Isotropic Fracture Network Permeabilities Calculated in 2D and 3D
Philipp S. Lang, Adriana Paluszny, Robert W. Zimmerman
Department of Earth Science and Engineering
Imperial College, London, UK
Permeability calculations of fracture networks based on two-dimensional fracture trace maps will typically underestimate the permeability of the three-dimensional network by a substantial amount. A novel implementation based on the finite element method allows us to directly compute the full permeability tensor of a discrete fracture network in a permeable matrix, requiring no a priori assumptions regarding the orientation of the principal directions. For isotropic networks having fractures whose lengths are distributed according to a power-law distribution, we have conducted a parametric study of the macroscopic permeability using discrete fracture and matrix models. The hydraulic aperture of fractures is assumed to scale sublinearly with length. Permeabilities computed in two dimensions based on randomly extracted cut planes show consistently lower values as compared to their three-dimensional sources, throughout a range of fracture densities. We find a correction formula for networks having fracture densities above the percolation threshold, to allow us to obtain three-dimensional permeabilities from two-dimensional analysis. The proposed model uses as input the 2D permeability, an estimate of the power-law exponent with fracture trace length bounds, the dimensionless fracture density, and the matrix permeability (if it is non-zero). Moreover; we present bandwidths to estimate errors in permeability results over a range of porous fractured media properties.
Figure 1. (a) three-dimensional fracture network model; (b) randomly extracted cut plane; (c) hydraulic aperture of fractures scales sublinearly with size, where r m
and r
M
are the minimum and maximum fracture radii used to populate networks.
Figure 2. (a) evolution of effective permeability with increasing density of three-(line) and two-(dots) dimensional permeability. (b) close-up at higher densities, with a curve fit to the two-dimensional results.