Effective Permeability of Fractured Rocks by Analytical Methods: A 3D Computational Study

Analytical upscaling methods have been proposed in the literature to predict the effective hydraulic permeability of a fractured rock from its micro-scale parameters (fracture aperture, fracture orientation, fracture content, etc.). In this presentation, we put special emphasis on three effective medium methods (the symmetric and asymmetric self-consistent methods, and the differential method), and evaluate their accuracy for a wide range of parameter values. The analytical predictions are computed using our recently developed effective medium formulations, which are specifically adapted for fractured media. Compared to previous formulations, the new expressions have improved numerical stability properties, and require fewer input parameters. To assess their accuracy, the analytical predictions have been compared with 3D finite element simulations. Specifically, we generated realizations of several different fracture geometries, each consisting of 102 fractures within a unit cube. We applied unit potential difference on two opposing sides, and no-flux conditions on the remaining sides. A commercial finite-element solver was used to calculate the mean flux, from which the effective conductivity was found. This process was repeated for fracture densities up to ɛ = 1.0. Also, a wide range of fracture permeabilities was considered, from completely blocking to infinitely permeable fractures. The results were used to determine the range of applicability for each analytical method, which excels in different regions of the parameter space. For blocking fractures, the differential method is very accurate throughout the investigated parameter range. The symmetric self-consistent method also agrees well with the numerical results on sealed fractures, while the asymmetric self-consistent method is more unreliable. For permeable fractures, the performance of the methods depends on the dimensionless quantity λ = (K_frac a)/(r K_mat ), describing the contrast between fracture and matrix permeability. For λ ≈ 1, all the analytical methods agree well with the numerical results, even for large fracture densities. When the contrast between fracture and matrix permeability is increased (λ ≥ 10), the differential method is only accurate for fracture densities well below the percolation threshold, i.e., when the fracture network is disconnected. The symmetric and asymmetric self-consistent methods have an acceptable accuracy for both small and high densities, even for very high values of λ. The symmetric method may be somewhat more accurate for moderate densities, but only the asymmetric method has the correct limiting behavior for very high densities. The asymmetric method is also surprisingly accurate at predicting the percolation thresholds of the fracture geometries we have studied.Nomenclature