Geodesic
Equations of liquid filtration in double porosity media as a~reiterated homogenization of Stokes equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 161-169.

Voir la notice de l'article provenant de la source Math-Net.Ru

An exact double porosity model for the liquid filtration in an absolutely rigid body is derived from homogenization theory. The governing equations of fluid dynamics on the microscopic level consist of the stationary Stokes system for a slightly compressible viscous fluid filling voids in a solid skeleton. In turn, this domain (voids) is a union of two independent periodic systems of cracks (fissures) and pores. We suppose that the dimensionless size $\delta$ of pores depends on the dimensionless size $\varepsilon$ of cracks: $\delta=\varepsilon^r$ with $r>1$. As a result we derive the usual Darcy equations of filtration for the liquid in cracks, while the liquid in pores is blocked and unmoved. 
@article{TM_2012_278_a14,
     author = {Anvarbek Meirmanov},
     title = {Equations of liquid filtration in double porosity media as a~reiterated homogenization of {Stokes} equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {161--169},
     publisher = {mathdoc},
     volume = {278},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_278_a14/}
}
TY  - JOUR
AU  - Anvarbek Meirmanov
TI  - Equations of liquid filtration in double porosity media as a~reiterated homogenization of Stokes equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2012
SP  - 161
EP  - 169
VL  - 278
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2012_278_a14/
LA  - ru
ID  - TM_2012_278_a14
ER  - 
%0 Journal Article
%A Anvarbek Meirmanov
%T Equations of liquid filtration in double porosity media as a~reiterated homogenization of Stokes equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2012
%P 161-169
%V 278
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2012_278_a14/
%G ru
%F TM_2012_278_a14
Anvarbek Meirmanov. Equations of liquid filtration in double porosity media as a~reiterated homogenization of Stokes equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 161-169. http://geodesic.mathdoc.fr/item/TM_2012_278_a14/

[1] Allaire G., Briane M., “Multiscale convergence and reiterated homogenisation”, Proc. R. Soc. Edinb. A, 126:2 (1996), 297–342 | DOI | MR | Zbl

[2] Barenblatt G.I., Zheltov Yu.P., Kochina I.N., “Ob osnovnykh predstavleniyakh teorii filtratsii odnorodnykh zhidkostei v treschinovatykh porodakh”, PMM, 24:5 (1960), 852–864 | MR | Zbl

[3] Barenblatt G.I., Entov V.M., Ryzhik V.M., Teoriya nestatsionarnoi filtratsii zhidkosti i gaza, Nedra, M., 1972

[4] Biot M.A., “Generalized theory of acoustic propagation in porous dissipative media”, J. Acoust. Soc. Amer., 34:9A (1962), 1254–1264 | DOI | MR

[5] Burridge R., Keller J.B., “Poroelasticity equations derived from microstructure”, J. Acoust. Soc. Amer., 70:4 (1981), 1140–1146 | DOI | Zbl

[6] Kazemi H., “Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution”, Soc. Pet. Eng. J., 9:4 (1969), 451–462 | DOI

[7] Meirmanov A.M., “Metod dvukhmasshtabnoi skhodimosti Nguetsenga v zadachakh filtratsii i seismoakustiki v uprugikh poristykh sredakh”, Sib. mat. zhurn., 48:3 (2007), 645–667 | MR | Zbl

[8] Meirmanov A., “Double porosity models for liquid filtration in incompressible poroelastic media”, Math. Models Methods Appl. Sci., 20:4 (2010), 635–659 | DOI | MR | Zbl

[9] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[10] de Swaan A., “Analytic solutions for determining naturally fractured reservoir properties by well testing”, Soc. Pet. Eng. J., 16:3 (1976), 117–122 | DOI

[11] Warren J.E., Root P.J., “The behavior of naturally fractured reservoirs”, Soc. Pet. Eng. J., 3:3 (1963), 245–255 | DOI