We consider the thermal flow problem occuring in a fractured porous medium. The incompressible filtration flow in the porous matrix and the viscous flow in the fractures obey the Boussinesq approximation of Darcy-Forchheimer law and respectively, the Stokes system. They are coupled by the Saffman’s variant of the Beavers–Joseph condition. Existence and uniqueness properties are presented. The use of the energy norm in describing the Darcy-Forchheimer law proves to be appropriate. In the ε-periodic framework, we find the two-scale homogenized system which governs their asymptotic behaviours when ε → 0 and the Forchheimer effect vanishes. The limit problem is mainly a model of two coupled thermal flows, neither of them being incompressible.
Keywords: Fractured porous media, $$-domes, two-scale homogenized system, Darcy-Forchheimer law, Boussinesq approximation, Beavers–Joseph condition
@article{M2AN_2021__55_3_789_0,
author = {Gruais, Isabelle and Poli\v{s}evski, Dan},
title = {Thermal flows in fractured porous media},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {789--805},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2020087},
mrnumber = {4253166},
zbl = {1487.35317},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020087/}
}
TY - JOUR AU - Gruais, Isabelle AU - Poliševski, Dan TI - Thermal flows in fractured porous media JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 789 EP - 805 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020087/ DO - 10.1051/m2an/2020087 LA - en ID - M2AN_2021__55_3_789_0 ER -
%0 Journal Article %A Gruais, Isabelle %A Poliševski, Dan %T Thermal flows in fractured porous media %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 789-805 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020087/ %R 10.1051/m2an/2020087 %G en %F M2AN_2021__55_3_789_0
Gruais, Isabelle; Poliševski, Dan. Thermal flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 789-805. doi: 10.1051/m2an/2020087
[1] , Homogenization of the Stokes flow in a connected porous medium. Asymptotic Anal. 2 (1989) 203–222. | MR | Zbl | DOI
[2] , Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. | MR | Zbl | DOI
[3] , and , On basic conceptions of the theory of homogeneous fluids seepage in fractured rocks (in Russian). Prikl. Mat. i Mekh. 24 (1960) 852–864. | Zbl
[4] , and , Theory of Fluid Flows Through Natural Rocks. Kluwer Acad. Pub., Dordrecht (1990). | Zbl | DOI
[5] and , Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967) 197–207. | DOI
[6] and , Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590–607. | MR | Zbl | DOI
[7] and , An introduction to homogenization. In: Vol. 17 of Oxford Lecture Series in Mathematics and Its Applications. Oxford Univ. Press, New York (1999). | MR | Zbl
[8] and , Thermal Flow in Porous Media. D. Reidel Pub. Co., Dordrecht (1987). | DOI
[9] and , Model of diffusion in partially fissured media. ZAMP 53 (2002) 1052–1059. | MR | Zbl
[10] , Wasserbewegung durch boden. Z. Ver. Dtsch. Ing. 45 (1901) 1782–1788.
[11] and , Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. Rev. Française Automat. Informat. Recherche Opérationnelle, Sér. Rouge Anal. Numér. 9 (1975) 41–76. | MR | Zbl | Numdam
[12] and , Fluid flows through fractured porous media along Beavers-Joseph interfaces. J. Math. Pures Appl. 102 (2014) 482–497. | MR | Zbl | DOI
[13] and , Heat transfer models for two-component media with interfacial jump. Appl. Anal. 96 (2017) 247–260. | MR | Zbl | DOI
[14] and , Model of two-temperature convective transfer in porous media. Z. Angew. Math. Phys. 68 (2017) 11. | MR | Zbl | DOI
[15] and , Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization. Transp. Porous Med. 78 (2009) 489–508. | MR | DOI
[16] , Low Reynolds number flow past a porous spherical shell. Proc Camb. Phil. Soc. 73 (1973) 231–238. | Zbl | DOI
[17] and , An introduction to Variational Inequalities and Their Applications. Academic Press, New-York (1980). | MR | Zbl
[18] and , Darcy’s law for slow viscous flow past a stationary array of bubbles. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 71–79. | MR | Zbl | DOI
[19] , and , Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35–86. | MR | Zbl
[20] , Partial Differential Equations. Mir Publishers, Moscow (1978). | MR
[21] , The Flow of Homogeneous Fluids through Porous Media. Edwards, MI (1946).
[22] , A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. | MR | Zbl | DOI
[23] , On the homogenization of fluid flows through periodic media. Rend. Sem. Mat. Univers. Politecn. Torino 45 (1987) 129–139. | MR | Zbl
[24] , Basic homogenization results for a biconnected -periodic structure. Appl. Anal. 82 (2003) 301–309. | MR | Zbl | DOI
[25] , The regularized diffusion in partially fractured porous media. In: Vol. 2 of Current Topics in Continuum Mechanics. Ed. Academiei, Bucharest (2003). | MR
[26] , On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 50 (1971) 93–101. | Zbl | DOI
[27] and , Micro-structure models of diffusion in fissured media. J. Math. Anal. Appl. 155 (1991) 1–20. | MR | Zbl | DOI
[28] , Régularité de la solution d’une équation non linéaire dans . Journées d’Analyse non linéaire. Proceedings, Besançon, France, 1977. In: Vol. 665 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1978) 205–227. | MR | Zbl | DOI
[29] , Flow in porous media I: A theoretical derivation of Darcy’s law. Transp. Por. Med. 1 (1986) 3–25. | DOI
[30] , Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer-Verlag, New York (1986). | MR | Zbl
Cité par Sources :
