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TECHNICAL PAPERS

# Upscaling Fractured Heterogeneous Media: Permeability and Mass Exchange Coefficient

[+] Author and Article Information
Moussa Kfoury

Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France and  Institut de Mécanique de Fluides de Toulouse, Allée Prof. C. Soula, 31400 Toulouse, Francekfoury_moussa@yahoo.fr

Rachid Ababou, Michel Quintard

Institut de Mécanique de Fluides de Toulouse, Allée Prof. C. Soula, 31400 Toulouse, France

Benoît Noetinger

Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France

J. Appl. Mech 73(1), 41-46 (May 08, 2005) (6 pages) doi:10.1115/1.1991864 History: Received June 15, 2004; Revised May 08, 2005

## Abstract

In order to optimize oil recuperation, to secure waste storage, $CO2$ sequestration and describe more precisely many environmental problems in the underground, we need to improve some homogenization methods that calculate petrophysical parameters. In this paper, we discuss the upscaling of fluid transport equations in fractured heterogeneous media consisting of the fractures themselves and a heterogeneous porous matrix. Our goal is to estimate precisely the fluid flow parameters like permeability and fracture/matrix exchange coefficient at large scale. Two approaches are possible. The first approach consists in calculating the large-scale equivalent properties in one upscaling step, starting with a single continuum flow model at the local scale. The second approach is to perform upscaling in two sequential steps: first, calculate the equivalent properties at an intermediate scale called the ”unit scale,” and, second, average the flow equations up to the large scale. We have implemented the two approaches and applied them to randomly distributed fractured systems. The results allowed us to obtain valuable information in terms of sizes of representative elementary volume associated to a given fracture distribution.

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## Figures

Figure 1

Two different up-scaling paths: (I) Direct upscaling from the local-scale (dx) to the block-scale (Lb), and (II) Up-scaling in two stages passing through the intermediate unit-scale(lu)

Figure 2

This figure shows the faces quotation used in this paper for a square porous medium

Figure 3

Lognormal permeability map (Darcy) generated using FFTMA for 200×200 cells (at left) and the pressure (bar) evolution at the impermeable edges for a horizontal confined flow for the real medium and its anisotropic homogeneous equivalent (at right). At bottom, pressure maps for the real medium (at left) and its equivalent homogeneous medium (at right).

Figure 4

Above: example of a fractured porous medium used in this work (200×200 cells). Below: histogram of matrix-fracture exchange coefficient α (left), and histogram of first component KEQXX of equivalent permeability (right).

Figure 5

First component of the equivalent permeability for a fracture network at the block-scale (kffxx,m2)

Figure 6

Sequential upscaling: First component of fracture network permeability in each cell at the unit-scale for all partitions

Figure 7

Sequential upscaling: Arithmetic and harmonic values of exchange coefficient for each partition at the unit scale

## Errata

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