Derivation of the Double Porosity Model of Single Phase Flow via Homogenization Theory

TLDR

The microscopic model describes Darcy flow in a reservoir with highly discontinuous porosity and permeability coefficients. The study derives a general double‑porosity model for single‑phase flow in naturally fractured reservoirs using homogenization theory. The authors scale the matrix coefficients by a small parameter ε, use weak‑limit extraction and a dilation operator to derive a macroscopic double‑porosity model that couples Darcy flow in matrix blocks with a fracture equation containing a matrix‑to‑fracture source term.

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 $\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.