Inclusion-Based Effective Medium Models for the Permeability of a 3D Fractured Rock Mass

Abstract:

Following the work of Saevik et al. (Transp. Porous Media, 2013; Geophys. Prosp., 2014), we investigate the ability of classical inclusion-based effective medium theories to predict the macroscopic permeability of a fractured rock mass. The fractures are assumed to be thin, oblate spheroids, and are treated as porous media in their own right, with permeability k_f, and are embedded in a homogeneous matrix having permeability k_m. At very low fracture densities, the effective permeability is given exactly by a well-known expression that goes back at least as far as Fricke (Phys. Rev., 1924). For non-trivial fracture densities, an effective medium approximation must be employed. We have investigated several such approximations: Maxwell’s method, the differential method, and the symmetric and asymmetric versions of the self-consistent approximation. The predictions of the various approximate models are tested against the results of explicit numerical simulations, averaged over numerous statistical realizations for each set of parameters.

Each of the various effective medium approximations satisfies the Hashin-Shtrikman (H-S) bounds. Unfortunately, these bounds are much too far apart to provide quantitatively useful estimates of k_eff. For the case of zero matrix permeability, the well-known approximation of Snow, which is based on network considerations rather than a continuum approach, is shown to essentially coincide with the upper H-S bound, thereby proving that the commonly made assumption that Snow’s equation is an “upper bound” is indeed correct. This problem is actually characterized by two small parameters, the aspect ratio of the spheroidal fractures, α, and the permeability ratio, κ = k_m/k_f. Two different regimes can be identified, corresponding to α < κ and κ < α, and expressions for each of the effective medium approximations are developed in both regimes. In both regimes, the symmetric version of the self-consistent approximation is the most accurate.