Reiterated Homogenization and the Double-Porosity Model

Authors: Peter Shi¹; Anna Spagnuolo¹; Steve Wright²

Source: Transport in Porous Media, Volume 59, Number 1, April 2005 , pp. 73-95(23)

Publisher: Springer

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Abstract:

We consider the effects of a hierarchical, multiple layered system of fractures on the flow of a single-phase, slightly compressible fluid through a porous medium. A microscopic flow model is first defined which describes precisely the physics of the flow and the geometry of the fracture system and porous matrix, all of which depends on a positive parameter egr

that determines the scale of the various fracture-level thicknesses. We then show by a rigorous mathematical argument that the unique solution of this microscopic problem converges as egr

0 to the solution of a double-porosity model of the global macroscopic flow. Our techniques make use of the concept of reiterated homogenization and essentially consist of an adaptation of the methods of extension and dilation operators to the reiterated-homogenization context. We show how the porosities and permeability tensor of the porous medium are determined in a precise way by certain physical and geometric features of the microscopic fracture domain, the microscopic matrix blocks, and the interface between them.

Keywords: double-porosity model; reiterated homogenization; fracture domain; matrix blocks; dilation operator; fluid flow

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s11242-004-1121-3

Affiliations: 1: Department of Mathematics and Statistics, Oakland University, 48309-4485, Rochester, Michigan, U.S.A., 2: Department of Mathematics and Statistics, Oakland University, 48309-4485, Rochester, Michigan, U.S.A., Email: wright@oakland.edu

Publication date: 2005-04-01