Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 94, pp. 1-19.
Title: A model for single-phase flow in layered porous media
Authors: Daniel J. Coffield Jr. (Univ. of Michigan, Flint, MI, USA)
Anna Maria Spagnuolo (Oakland Univ., Rochester, MI, USA)
Abstract:
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.