The aim of this Note is to quantify the change of characteristics of the media of an Excavated Damaged Zone (EDZ) affected by several fractures. For this, we consider Darcy flow through matrix blocks and fractures with permeability of order ε2δθ and 1 respectively. ε is the size of a typical porous block, δ representing the relative size of the fracture and θ is a parameter characterising the permeability ratio. We derive the global behavior from the limit as ε and δ tend to zero. The resulting homogenized equation is of dual-porosity type for θ=2, but it is a simple-porosity model with effective coefficients for θ>2, and there is no flow at the macroscopic level when 0<θ<2.
Le but de cette Note est de quantifier les changements dans les paramètres de l'écoulement au sein d'un milieu poreux lorsque celui-ci est endommagé par l'apparition de fissures en grand nombre. Pour cela, on considère l'écoulement d'un fluide régi par la loi de Darcy avec une perméabilité de l'ordre de ε2δθ dans les matrices et d'ordre un dans le réseau de fissures. Pour décrire les diverses situations nous avons caractérisé respectivement par et θ la taille des blocs, l'épaisseur relative des fractures et le rapport des perméabilités. On étudie alors le comportement asymptotique de ce problème lorsque ε et δ tendent vers zéro. On montre que le problème homogénéisé est un modèle à double porosité pour θ=2, un modèle à simple porosité avec des coefficients effectifs lorsque θ>2 mais qu'il n'y a pas d'écoulement pour le modèle globalement équivalent avec 0<θ<2.
Accepted:
Published online:
Mots-clés : Milieu poreux, EDZ, Milieu poreux fracturé, Modèle de double porosité
Brahim Amaziane 1; Alain Bourgeat 2; Mariya Goncharenko 3; Leonid Pankratov 3
@article{CRMECA_2004__332_1_79_0,
author = {Brahim Amaziane and Alain Bourgeat and Mariya Goncharenko and Leonid Pankratov},
title = {Characterization of the flow for a single fluid in an excavation damaged zone},
journal = {Comptes Rendus. M\'ecanique},
pages = {79--84},
publisher = {Elsevier},
volume = {332},
number = {1},
year = {2004},
doi = {10.1016/j.crme.2003.11.006},
language = {en},
}
TY - JOUR AU - Brahim Amaziane AU - Alain Bourgeat AU - Mariya Goncharenko AU - Leonid Pankratov TI - Characterization of the flow for a single fluid in an excavation damaged zone JO - Comptes Rendus. Mécanique PY - 2004 SP - 79 EP - 84 VL - 332 IS - 1 PB - Elsevier DO - 10.1016/j.crme.2003.11.006 LA - en ID - CRMECA_2004__332_1_79_0 ER -
%0 Journal Article %A Brahim Amaziane %A Alain Bourgeat %A Mariya Goncharenko %A Leonid Pankratov %T Characterization of the flow for a single fluid in an excavation damaged zone %J Comptes Rendus. Mécanique %D 2004 %P 79-84 %V 332 %N 1 %I Elsevier %R 10.1016/j.crme.2003.11.006 %G en %F CRMECA_2004__332_1_79_0
Brahim Amaziane; Alain Bourgeat; Mariya Goncharenko; Leonid Pankratov. Characterization of the flow for a single fluid in an excavation damaged zone. Comptes Rendus. Mécanique, Volume 332 (2004) no. 1, pp. 79-84. doi : 10.1016/j.crme.2003.11.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2003.11.006/
[1] Basic consepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., Volume 24 (1960), pp. 1286-1303
[2] Fundamentals of Fractured Reservoir Engineering, Development in Petroleum Science, vol. 12, Elsevier, Amsterdam, 1982
[3] Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Appl. Math., Volume 21 (1990), pp. 823-826
[4] A general double porosity model, C. R. Acad. Sci. Paris, Sér. IIb, Volume 327 (1999), pp. 1245-1250
[5] Overall behaviour of fractured porous media versus fractures' size and permeability ratio, Contemp. Math., Volume 295 (2002), pp. 75-92
[6] Singular double porosity model, Appl. Anal., Volume 82 (2003), pp. 103-116
[7] A. Bourgeat, L. Pankratov, M. Panfilov, Study of the double porosity model versus the fissures thickness, Asymptotic Anal. (2003) in press
[8] Homogenization and Porous Media (U. Hornung, ed.), Interdisciplinary Appl. Math., vol. 6, Springer-Verlag, New York, 1997
[9] Homogenization of parabolic equations with contrasting coefficients, Izv. Math., Volume 63 (1999) no. 5, pp. 1015-1061 (English translation from Izv. Ross. Akad. Nauk Ser. Mat., 63, 4, 1999, pp. 179-224)
[10] Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb., Volume 194 (2003) no. 1, pp. 121-146
[11] Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968
[12] Homogenization of thin structures by two-scale method with respect to measure, SIAM J. Math. Anal., Volume 32 (2001), pp. 1198-1226
[13] Homogenization of lattice-type domains: L-convergence (D. Cioranescu; J.-L. Lions, eds.), Nonlinear Partial Differential Equations, Vol. XIII, Res. Notes Math., vol. 220, Longman, 1997, pp. 259-280
[14] On one extension and application of the method of two-scale convergence, Russian Acad. Sci. Sb. Math., Volume 191 (2000) no. 7, pp. 31-72
[15] Homogenization of Reticulated Structures, Appl. Math. Sci., vol. 136, Springer-Verlag, New York, 1999
[16] An extension theorem from connected sets and homogenization in general periodic domains, Nonlinear Anal., Volume 18 (1992), pp. 481-496
[17] Homogenization of parabolic equations in domains of degenerating measure, Appl. Anal., Volume 81 (2002), pp. 753-769
Cited by Sources:
Comments - Policy