Effective Permeability of Media with a Dense Network of Long and Micro Fractures

Abstract

This article presents a new methodology to estimate the effective permeability of random fractured media of any anisotropy containing both microfractures and a large number of long fractures crosscutting the representative volume element. The fractures are replaced by fictitious permeable materials for which the tangential permeability is deduced from a Poiseuille flow. A self-consistent scheme is proposed to derive the macroscopic permeability. On the one hand, the contribution of long fractures to the effective permeability writes by simple superposition of the fracture tangential permeabilities. On the other hand, the contribution of microfractures needs to resort to auxiliary problems requiring the computation of second-order Hill (or Eshelby) tensors related to ellipsoids embedded in an anisotropic matrix, for which a complete procedure is detailed. The effect of the microfracture normal permeability is put in evidence in the upscaling scheme and analyzed. In particular, it is shown that it must be chosen large enough to allow the connections between families. Examples are finally developed and compared to numerical simulations in the 2D case.Keywords Anisotropic effective permeability · Self-consistent scheme · Fracture permeability · Double scale fracture network Nomenclature XVector notation 7 Second-order tensor notation p Pressure field ∇ P Macroscopic pressure gradient qMicroscopic filtration vector Q Macroscopic filtration vector µ Fluid viscosity

A 3D Computational Study of Effective Medium Methods Applied to Fractured Media

“…That is, given the input data (inclusion shapes, orientations, permeabilities and volume fractions), it may not exist a consistent microstructural configuration such that the permeability estimate is attained. We can obtain a realizable self-consistent scheme by treating both the matrix and the fractures as ellipsoidal inclusions (Torquato, 2002;Barthélémy, 2008). In other words, we set…”

“…To calculate R 0 , we must construct an ellipsoidal shape (with shape tensor H 0 ) that somehow resemble the geometry of the space between the inclusions. For fractured media, Barthélémy (2008) suggested using a weighted average of the inclusion shape tensors {H 1 , . .…”

“…The equivalent permeability in the normal direction is therefore larger than the tangential component, but in this paper we set them to be equal. Earlier work by Barthélémy (2008) shows that this simplification has no effect on the upscaled rock permeability.…”

“…A 3D computational study of effective medium methods applied to fractured media Some numerical tests of effective medium theory have been conducted, with encouraging results (Zimmerman and Yeo, 2000;Pozdniakov and Tsang, 2004;Barthélémy, 2008;Tawerghi and Yi, 2009). These works are limited to spherical/near-spherical inclusions, or isotropic/transversely isotropic media, or two-dimensional geometries.…”

“…Thus, we require the solution of the single-inclusion problem with an anisotropic background permeability, which is slightly more complicated. Here, we only describe the resulting formula, and refer to Barthélémy (2008) for a complete derivation. To avoid cluttering the formulas, we ignore the inclusion family subscript i for the remainder of this section.…”