 
Summary: SIAM J. MATH. ANAL. c 1990 Society for Industrial and Applied Mathematics
Vol. 21, No. 4, pp. 823836, July 1990 001
DERIVATION OF THE DOUBLE POROSITY MODEL
OF SINGLE PHASE FLOW VIA HOMOGENIZATION THEORY*
TODD ARBOGAST, JIM DOUGLAS, JR., and ULRICH HORNUNG
Abstract. 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
representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix
as 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 tends to zero.
Key words. porous medium, double porosity, fractured reservoir, homogenization
AMS(MOS) subject classifications. 76S05
1. Introduction. It has long been known that the porous rock that composes
a petroleum reservoir may contain many cracks or fractures. A naturally fractured
reservoir is one that has throughout its extent many interconnected fracture planes.
