### Description

#### Introduction (dummy text)

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, α, and the ratio between matrix and fracture permeability, κ. Two different regimes can be identified, corresponding to α/κ<1 and α/κ>1. The lower the value of α/κ, 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 α/κ≫1, a distinct percolation threshold is observed, whereas for α/κ≪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.

#### Results

In predicting the effective permeability of a three-dimensional fractured rock mass, two distinct regimes can be distinguished, depending on the relative size of two small, dimensionless numbers: the aspect ratio of the spheroidal fractures, , and the permeability ratio, . When , effective permeability is linearly dependent on fracture density without a distinct percolation threshold. With increasing , this relationship becomes increasingly nonlinear at low and intermediate fracture densities, but remains linear at high fracture densities. The ratio can therefore be interpreted as a measure of the relative contributions of the fracture network and matrix to the overall flow, similar to the flux ratio discussed by Paluszny and Matthai (2010), and Nick et al. (2011). For polydisperse fracture networks, a characteristic value of can be determined by weighting the individual values of

of the various fracture sets with fracture volume while averaging.

Comparison to explicit numerical simulations of mono- and polydisperse isotropic fracture networks (with fracture-size dependent fracture permeabilities) shows that the self-consistent effective medium methods are generally capable of predicting effective permeability over a wide range of conditions. While the symmetric self-consistent method is particularly accurate at low fracture densities, the asymmetric self-consistent method predicts the correct asymptotic behaviour (cf. Sævik et al. 2013). The differential method is only useful when

. In that regime, even the Hashin–Shtrikman and Snow upper bounds appear to give good approximations. Maxwell’s approximation is only reliable at very low fracture densities.

These effective medium models have been shown to be powerful tools. They are valid at both low and high fracture densities. When considering polydisperse fracture networks, the type and characteristics of the fracture-size (as well as aperture and fracture-permeability) distribution do not affect the applicability or accuracy of these models. These results are perhaps surprising, since intuitively, fracture intersections would seem to be a crucial factor in controlling the effective permeability, and yet these models do not explicitly contain fracture intersections as a parameter. Moreover, they are all based on the problem of a single “inclusion”, for which the concept of intersection is not even meaningful.

It should be noted that the heuristic model proposed by Mourzenko et al. (2011) is good at predicting effective permeability. It is very accurate for monodisperse fracture networks, but suffers a slight loss of accuracy for polydisperse networks, that appears to depend on the attributes of fracture-size distribution. The present work has in effect shown a link between predictions of methods based on inclusions-in-a-matrix and methods based on fracture networks.

#### References

Mourzenko, V.V., Thovert, J.F., Adler, P.M.: Permeability of isotropic and anisotropic fracture networks, from the percolation threshold to very large densities. Phys. Rev. E **84**, 036–307 (2011). doi:10.1103/PhysRevE.84.036307

Nick, H.M., Paluszny, A., Blunt, M.J., Matthai, S.K.: Role of geomechanically grown fractures on dispersive transport in heterogeneous geological formations. Phys. Rev. E **84**, 056–301 (2011). doi:10.1103/PhysRevE.84.056301

Paluszny, A., Matthai, S.K.: Impact of fracture development on the effective permeability of porous rocks as determined by 2-D discrete fracture growth modeling. J. Geophys. Res. Solid Earth **115**(B02), 203 (2010). doi:10.1029/2008JB006236

Sævik, P.N., Berre, I., Jakobsen, M., Lien, M.: A 3D computational study of effective medium methods applied to fractured media. Transp. Porous Media **100**(1), 115–142 (2013). doi:10.1007/s11242-013-0208-0