Electron. J. Diff. Equ., Vol. 2010(2010), No. 94, pp. 1-19.

A model for single-phase flow in layered porous media

Daniel J. Coffield Jr., Anna Maria Spagnuolo

Homogenization techniques are used to derive a double porosity model for single phase flow in a reservoir with a preferred direction of fracture. The equations in the microscopic model are the usual ones derived from Darcy's law in the fractures and matrix (rock). The permeability coefficients over the matrix domain are scaled, using a parameter $\epsilon$, based on the fracture direction in the reservoir. The parameter $\epsilon$ represents the size of the parts of the matrix blocks that are being homogenized and the scaling preserves the physics of the flow between matrix and fracture as the blocks shrink. Convergence to the macroscopic model is shown by extracting the weak limits of the microscopic model solutions. The limit (macroscopic) model consists of Darcy flow equations in the matrix blocks and fracture sheet, with additional terms in the fracture sheet equation. Together, these terms represent the fluid exchange between the matrix blocks and the fracture sheet.

Submitted February 23, 2010. Published July 13, 2010.
Math Subject Classifications: 76S05, 35B27.
Key Words: Layered media; naturally-fractured media; double-porosity model; homogenization.

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Daniel J. Coffield Jr.
University of Michigan-Flint, Mathematics Department
Flint, MI 48502-1950, USA
email: dcoffiel@umflint.edu, tel: (810) 762-3005, fax: (810) 766-6880
Anna Maria Spagnuolo
Oakland University, Department of Mathematics and Statistics
Rochester, MI 48309-4485, USA
email: spagnuol@oakland.edu

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