Derivation of the double porosity model of single phase flow via homogenization theory. (English) Zbl 0698.76106

Arbogast, Todd; Douglas, Jim jun.; Hornung, Ulrich

SIAM J. Math. Anal. 21, No. 4, 823-836 (1990).

Summary: A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter \(\epsilon\) representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as \(\epsilon\) tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as \(\epsilon\) tends to zero.

MSC:

76S05	Flows in porous media; filtration; seepage
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

double porosity model; single phase flow; naturally fractured reservoir; homogenization theory; Darcy flow; matrix blocks; effective macroscopic limit; Hilbert spaces