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

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Effective permeability is an essential parameter for describing fluid flow through fractured rock masses. This study investigates the ability of classical inclusion-based effective medium models (following the work of Sævik et al. in Transp Porous Media 100(1):115–142, 2013. doi: 10.1007/s11242-013-0208-0 ) to predict this permeability, which depends on several geometric properties of the fractures/networks. This is achieved by comparison of various effective medium models, such as the symmetric and asymmetric self-consistent schemes, the differential scheme, and Maxwell’s method, with the results of explicit numerical simulations of mono- and poly-disperse isotropic fracture networks embedded in a permeable rock matrix. Comparisons are also made with the Hashin–Shtrikman bounds, Snow’s model, and Mourzenko’s heuristic model (Mourzenko et al. in Phys Rev E 84:036–307, 2011. doi: 10.1103/PhysRevE.84.036307 ). This problem is characterised by two small parameters, the aspect ratio of the spheroidal fractures, $$\alpha $$ , and the ratio between matrix and fracture permeability, $$\kappa $$ . Two different regimes can be identified, corresponding to $$\alpha /\kappa <1$$ and $$\alpha /\kappa >1$$ . The lower the value of $$\alpha /\kappa $$ , the more significant is flow through the matrix. Due to differing flow patterns, the dependence of effective permeability on fracture density differs in the two regimes. When $$\alpha /\kappa \gg 1$$ , a distinct percolation threshold is observed, whereas for $$\alpha /\kappa \ll 1$$ , the matrix is sufficiently transmissive that such a transition is not observed. The self-consistent effective medium methods show good accuracy for both mono- and polydisperse isotropic fracture networks. Mourzenko’s equation is very accurate, particularly for monodisperse networks. Finally, it is shown that Snow’s model essentially coincides with the Hashin–Shtrikman upper bound.