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

P S Lang,A Paluszny,R W Zimmerman

doi:10.1002/2014jb011027

Abstract

AbstractThe 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.

Highlights

The hydraulic response of fractured rock to pressure gradients is of vital interest to various Earth science and engineering disciplines, as the majority of host rocks to hydrocarbon reservoirs and nuclear waste disposal sites are heavily fractured [e.g., van Golf-Racht, 1982; Wu et al, 1999]
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
Percolation Threshold Densities Three-dimensional fracture networks exhibit percolation in the range of 2.2 < ρ′3-D,c < 4.2, which becomes apparent by the increase in the minimum and maximum equivalent permeability by an order of magnitude

Summary

Introduction

The hydraulic response of fractured rock to pressure gradients is of vital interest to various Earth science and engineering disciplines, as the majority of host rocks to hydrocarbon reservoirs and nuclear waste disposal sites are heavily fractured [e.g., van Golf-Racht, 1982; Wu et al, 1999]. With respect to fractured rock, equivalent or block permeability [e.g., Renard and de Marsily, 1997; Amaziane and Hontans, 2002] quantifies the ensemble flow response of fractures and rock matrix. Hydraulic analysis of fractured rock is often reduced to a two-dimensional problem, owing to computational limitations, especially if multiphysics effects are incorporated in the model [Nick et al, 2011; Rutqvist et al, 2013]. This paper will focus on quantifying the difference between two- and three-dimensional permeability calculations, by means of direct computation of the full equivalent permeability tensor of three-dimensional networks, and extracted planar trace maps

Methods

Results

Discussion

Conclusion