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

Abstract

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 Saevik 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). 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.

Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

“… We did not discuss the application of percolation theory, CPA, and the EMA to characterization of fractures and fracture networks, and modeling of their flow and transport properties, as the problems need a review of their own. We do, however, point out that much progress has been made (see, for example, Darcel et al, ; de Dreuzy et al, ; Davy et al, , ; Ebigbo et al, ; Maillot et al, ; Pyrak‐Nolte & Nolte, ). Much of the progress has been reviewed by Meakin and Tartakovsky (), Adler and Thovert (), Sahimi (), Rutqvist and Tsang (), and Adler et al ().Meakin and Tartakovsky address complications to modeling flow and transport in porous media due to complex geometries, density contrasts of the fluids, and dynamic changes in contact angles.…”

Relationship Between the Orientation of Maximum Permeability and Intermediate Principal Stress in Fractured Rocks

“…However, these models have assumed isotropic transmissivity of individual fractures and do not account for the effects of intermediate principal stress. Most numerical studies of the mechanical deformation of fractured rocks are two‐dimensional (Baghbanan & Jing, ; Bisdom et al, ; Jing et al, ; Min, Rutqvist, et al, ; Rutqvist et al, ; Ucar et al, ; Zhang & Sanderson, ; Zhang et al, ), while most three‐dimensional studies of permeability have been restricted to fluid flow and have not included mechanical effects (Bogdanov et al, ; ; Ebigbo et al, ; Lang et al, ; Mourzenko et al, ; Sævik et al, ). Some recent models have been extended to three dimensions (Garipov & Tchelepi, ); Lei, Latham, Xiang, et al, , but ignore shear‐induced dilation (Garipov & Tchelepi, ), or implement constitutive models that account for shear‐induced dilation in an isotropic manner (Lei et al, ).…”

“…The permeability of fractured rock masses is often modeled as a function of fracture network geometry (Bogdanov et al, ; Davy et al, , De Dreuzy et al, ; Ebigbo et al, ; Min, Jing, et al, ; Mourzenko et al, ; Olson et al, ; Sævik et al, , ). The transmissivity of individual fractures is often defined as a constant, is correlated with fracture length, or follows a direction‐dependent distribution.…”

Estimating fluid flow rates through fracture networks using combinatorial optimization

“…The fracture transmissivity is often represented using the cubic law -or some variation thereof [Zimmerman & Bodvarsson, 1996, Liu et al, 2016 -which states that the fracture transmissivity is proportional to the cube of the average aperture of the fracture. At the reservoir scale, more connected fractures yield more potential flow paths, and hence a higher effective permeability of the rock [Ebigbo et al, 2016]. This connectivity of fractures, or planar features in 3D space, is dependent on the dimensions (e.g.…”