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Homogenization theory and applications to filtration through porous media

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1734)

Keywords

  • Porous Medium
  • Weak Solution
  • Stokes System
  • Filtration Velocity
  • Seepage Velocity

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References

References for §1

  1. E. Acerbi, V. Chiadò Piat, G. Dal Maso, D. Percivale: An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., TMA, 18 (1992), pp. 481–496.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. G. Allaire: Homogenization of the Stokes Flow in a Connected Porous Medium, Asymptotic Analysis 2 (1989), 203–222.

    MathSciNet  MATH  Google Scholar 

  3. G. Allaire: Continuity of the Darcy's law in the low-volume fraction limit, Ann. Scuola Norm. Sup. Pisa, Vol. 18 (1991), 475–499.

    MathSciNet  MATH  Google Scholar 

  4. G. Allaire: Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482–1518.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. G. Allaire: One-Phase Newtonian Flow, in Homogenization and Porous Media, ed. U. Hornung, Springer, New-York, 1997, p. 45–68.

    CrossRef  Google Scholar 

  6. A.Yu. Beliaev, S.M. Kozlov: Darcy equation for random porous media, Comm. Pure Appl. Math., Vol. 49 (1995), 1–34.

    CrossRef  MathSciNet  Google Scholar 

  7. W. Borchers, H. Sohr: On the equations rot v=g and div u=f with zero boundary conditions, Hokkaido Mathematical Journal, Vol. 19 (1990), p. 67–87.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. A. Bourgeat, A. Mikelić: Note on the Homogenization of Bingham Flow through Porous Medium, J. Math. Pures Appl. 72 (1993), 405–414.

    MathSciNet  MATH  Google Scholar 

  9. A. Bourgeat, A. Mikelić, S. Wright: On the Stochastic Two-Scale Convergence in the Mean and Applications, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 456 (1994), pp. 19–51.

    MathSciNet  MATH  Google Scholar 

  10. Th.Clopeau, J.L.Ferrín, R.P.Gilbert, A.Mikelić: Homogenizing the Acoustic Properties of the Seabed, II, to appear in Math. Comput. Modelling, 2000.

    Google Scholar 

  11. H.I. Ene, E. Sanchez-Palencia: Equations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux, J. Mécan., 14 (1975), pp. 73–108.

    MathSciNet  MATH  Google Scholar 

  12. R.P. Gilbert, A. Mikelić: Homogenizing the Acoustic Properties of the Seabed, Part I, Nonlinear Anal., 40 (2000), 185–212.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. U. Hornung, ed., Homogenization and Porous Media, Interdisciplinary Applied Mathematics Series, Vol. 6, Springer, New York, 1997.

    Google Scholar 

  14. W. Jäger, A. Mikelić: On the Flow Conditions at the Boundary Between a Porous Medium and an Impervious Solid, in “Progress in Partial Differential Equations: the Metz Surveys 3”, eds. M. Chipot, J. Saint Jean Paulin et I. Shafrir, πPitman Research Notes in Mathematics no. 314, p. 145–161, Longman Scientific and Technical, London, 1994.

    Google Scholar 

  15. W. Jäger, A. Mikelić: On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid, Ann. Sc. Norm. Super. Pisa, Cl. Sci. — Ser. IV, Vol. XXIII (1996), Fasc. 3, p. 403–465.

    MathSciNet  MATH  Google Scholar 

  16. W. Jäger, A. Mikelić: On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness, Communications on Pure and Applied Mathematics, Vol. LI (1998), p. 1073–1121.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. W.Jäger, A.Mikelić: On the boundary conditions at the contact interface between two porous media, in “Partial differential equations, Theory and numerical solution”, eds. W. Jäger, J. Nečas, O. John, K. Najzar and J. Stará, π Chapman and Hall/CRC Research Notes in Mathematics no. 406, p. 175–186, CRC Press, London, 1999.

    Google Scholar 

  18. V.V. Jikov, S. Kozlov, O. Oleinik: Homogenization of Differential Operators and Integral Functionals, Springer Verlag, New York, 1994.

    CrossRef  Google Scholar 

  19. J. L. Lions: Some Methods in the Mathematical Analysis of Systems and their Control, Gordon and Breach, Science Publishers, Inc, New York, 1981.

    MATH  Google Scholar 

  20. R. Lipton, M. Avellaneda: A Darcy Law for Slow Viscous Flow Past a Stationary Array of Bubbles, Proc. Royal Soc. Edinburgh 114A, 1990, 71–79.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. E. Marušić-Paloka, A. Mikelić: An Error Estimate for Correctors in the Homogenization of the Stokes and Navier-Stokes Equations in a Porous Medium, Boll. Unione Mat. Ital., A(7) 10 (1996), no. 3, p. 661–671.

    MathSciNet  MATH  Google Scholar 

  22. A. Mikelić, L. Paoli: Homogenization of the inviscid incompressible fluid flow through a 2D porous medium, Proc. Amer. Math. Soc., Vol. 127 (1999), pp. 2019–2028.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. G. Nguetseng: A General Convergence Result for a Functional Related to the Theory of Homogenization, SIAM J. Math. Analysis, vol. 20(3), (1989), 608–623.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. E. Sanchez-Palencia: Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer Verlag, 1980.

    Google Scholar 

  25. L. Tartar: Convergence of the Homogenization Process, Appendix of [SP80].

    Google Scholar 

  26. R. Temam: Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B. V., Amsterdam, 1984.

    Google Scholar 

  27. V.V. Zhikov: On the homogenization of the system of Stokes equations in a porous medium, Russian Acad. Sci. Dokl. Math., Vol. 49 (1994), 52–57.

    MathSciNet  MATH  Google Scholar 

References for §2

  1. Bellout, H.; Bloom, F.; Nečas, J.: Young measure-valued solutions for non-Newtonian incompressible fluids. Comm. Partial Differential Equations, 19 (1994), no. 11–12, 1763–1803.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. R. B. Bird, W. E. Stewart, E. N. Lightfoot: Transport Phenomena, Wiley and Sons, New York, 1960.

    Google Scholar 

  3. R. B. Bird, R. C. Armstrong, O. Hassager: Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics, Wiley and Sons, New York, 1987.

    Google Scholar 

  4. A. Bourgeat, A. Mikelić: Note on the Homogenization of Bingham Flow through Porous Medium. Journal de Mathématiques Pures et Appliquées, Vol. 72 (1993), pp. 405–414.

    MathSciNet  MATH  Google Scholar 

  5. A. Bourgeat, A. Mikelić: Homogenization of the Non-Newtonian Flow through Porous Medium, Nonlinear Anal., TMA, Vol. 26 (1996), p. 1221–1253.

    CrossRef  MATH  Google Scholar 

  6. A. Bourgeat, A. Mikelić, R. Tapiéro: Dérivation des équations moyennées décrivant un écoulement non-newtonien dans un domaine de faible épaisseur, Comptes Rendus de l'Académie des Sciences, Série I, t. 316 (1993), p. 965–970.

    Google Scholar 

  7. R. H. Christopher, S. Middleman: Power-law through a packed tube, I & EC Fundamentals, Vol. 4 (1965), p. 422–426.

    CrossRef  Google Scholar 

  8. D. Cioranescu: Quelques Exemples de Fluides Newtoniens Generalisés, in “Mathematical topics in fluid mechanics”, eds. J.F. Rodrigues and A. Sequeira, π Pitman Research Notes in Mathematics Series no. 274, Longman, Harlow, 1992, p. 1–31.

    Google Scholar 

  9. Th. Clopeau, A. Mikelić: On the non-stationary quasi-Newtonian flow through a thin slab, “Navier-Stokes Equations, Theory and Numerical Methods”, ed. R. Salvi, πPitman Research Notes in Mathematics no. 388, Addison Wesley Longman, Harlow, 1998, p. 1–15.

    Google Scholar 

  10. Frehse, J.; Málek, J.; Steinhauer, M.: An existence result for fluids with shear dependent viscosity-steady flows, to appear in Proceedings of the second world congress of nonlinear analysis, preprint no 482 of SFB 256 Bonn, 1996

    Google Scholar 

  11. J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969.

    Google Scholar 

  12. J. L. Lions, E. Sanchez-Palencia: Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux, J. Math. Pures Appl. 60 (1981), 341–360.

    MathSciNet  MATH  Google Scholar 

  13. R. Lipton, M. Avellaneda: A Darcy Law for Slow Viscous Flow Past a Stationary Array of Bubbles, Proc. Royal Soc. Edinburgh 114A, 1990, 71–79.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. E. Marušić-Paloka, A. Mikelić: The Derivation of a Non-Linear Filtration Law Including the Inertia Effects Via Homogenization, Nonlinear Anal., 42 (2000), 97–137.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. A. Mikelić: Homogenization of Nonstationary Navier-Stokes Equations in a Domain with a Grained Boundary, Annali di Matematica pura ed applicata (IV), Vol. CLVIII (1991), 167–179.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Mikelić, A.; Tapiéro R.: Mathematical derivation of the power law describing polymer flow through a thin slab. Modél. Math. Anal. Numér. 29 (1995), no. 1, 3–21.

    MathSciNet  MATH  Google Scholar 

  17. A. Mikelić: Non-Newtonian Flow, in “Homogenization and Porous Media” ed. U. Hornung, Interdisciplinary Applied Mathematics Series, Vol. 6, Springer, New York, 1997, p. 69–95.

    Google Scholar 

  18. Raugel, G.; Sell, G. R.: Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions. J. Amer. Math. Soc. 6 (1993), no. 3, 503–568.

    MathSciNet  MATH  Google Scholar 

  19. Ch. Shah, Y. C. Yortsos: Aspects of Non-Newtonian Flow and Displacement in Porous Media, Topical Report for U.S. DOE, University of Southern California, Los Angeles, 1993.

    Google Scholar 

  20. R. Temam: Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B. V., Amsterdam, 1984.

    Google Scholar 

  21. Y. S. Wu, K. Pruess, A. Witherspoon: Displacement of a Newtonian Fluid by a Non-Newtonian Fluid in a Porous Medium, Transport in Porous Media, Vol. 6, (1991), pp. 115–142.

    CrossRef  Google Scholar 

References for § 3

  1. G.I. Barenblatt, V.M. Entov, V.M. Ryzhik: Theory of Fluid Flows Through Natural Rocks, Kluwer, Dordrecht, 1990.

    CrossRef  MATH  Google Scholar 

  2. J. Bear: Hydraulics of Groundwater, McGraw-Hill, Jerusalem, 1979.

    Google Scholar 

  3. A. Bourgeat, E. Marušić-Paloka, A. Mikelić: The Weak Non-Linear Corrections for Darcy's Law, Math. Models Methods Appl.Sci. (M3AS), 8 (6) (1996), 1143–1155.

    CrossRef  MATH  Google Scholar 

  4. M. Firdaouss, J.L. Guermond: Sur l'homogénéisation des équations de Navier-Stokes à faible nombre de Reynolds, C.R. Acad. Sci. Paris, t.320, Série I (1995), 245–251.

    MathSciNet  MATH  Google Scholar 

  5. M. Firdaouss, J.L. Guermond, P.Le Quéré: Nonlinear Corrections to Darcy's Law at Low Reynolds Numbers, J.Fluid Mech., Vol. 343 (1997), 331–350.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  6. D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983.

    CrossRef  MATH  Google Scholar 

  7. S.M. Hassanizadeh, W.G. Gray: High Velocity Flow in Porous Media, Transport in Porous Media, Vol. 2 (1987), 521–531.

    CrossRef  Google Scholar 

  8. J.L. Lions: Some Methods in the Mathematical Analysis of Systems and Their Control, Gordon and Breach, New York, 1981.

    MATH  Google Scholar 

  9. E. Marušić-Paloka, A. Mikelić: The derivation of a non-linear filtration law including the inertia effects via homogenization, Nonlinear Anal., 42 (2000), 97–137.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. C.C. Mei, J.-L. Auriault: The Effect of Weak Inertia on Flow Through a Porous Medium, J.Fluid Mech., Vol. 222 (1991), 647–663.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  11. A. Mikelić: Homogenization of Nonstationary Navier-Stokes Equations in a Domain with a Grained Boundary, Annali di Mat. pura ed appl. (VI), Vol.158 (1991), 167–179.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. A. Mikelić: Effets inertiels pour un écoulement stationnaire visqueux incompressible dans un milieu poreux, C.R. Acad. Sci. Paris, t.320, Série I (1995), 1289–1294.

    Google Scholar 

  13. M. Rasolarijaona, J.-L. Auriault: Non-linear Seepage Flow Through a Rigid Porous Medium, Eur.J.Mech., B/Fluids, 13, No 2, (1994), 177–195.

    Google Scholar 

  14. D.W. Ruth, H. Ma: On the Derivation of the Forchheimer Equation by Means of the Averaging Theorem, Transport in Porous Media, Vol. 7 (1992), 255–264.

    CrossRef  Google Scholar 

  15. E. Sanchez-Palencia: Non-Homogeneous Media and Vibration Theory, Springer Lecture Notes in Physics 127, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  16. A.E. Scheidegger: Hydrodynamics in Porous Media, in Encyclopedia of Physics, Vol. VIII/2 (Fluid Dynamics II), ed. S. Flügge, Springer-Verlag, Berlin, 1963, 625–663.

    Google Scholar 

  17. J.-C. Wodié, T. Levy: Correction non linéaire de la loi de Darcy, C.R. Acad. Sci. Paris, t.312, Série II (1991), 157–161.

    MathSciNet  MATH  Google Scholar 

References for § 4

  1. G.S. Beavers, D.D. Joseph: Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), pp. 197–207.

    CrossRef  ADS  Google Scholar 

  2. C. Conca: Étude d'un fluide traversant une paroi perforée I. Comportement limite près de la paroi, J. Math. pures et appl., 66 (1987), pp. 1–44. II. Comportement limite loin de la paroi, J. Math. pures et appl., 66 (1987), pp. 45–69.

    MathSciNet  MATH  Google Scholar 

  3. H.I. Ene, E. Sanchez-Palencia: Equations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux, J. Mécan., 14 (1975), pp. 73–108.

    MathSciNet  MATH  Google Scholar 

  4. G. Dagan: The Generalization of Darcy's Law for Nonuniform Flows, Water Resources Research, Vol. 15 (1981), p. 1–7.

    CrossRef  ADS  Google Scholar 

  5. W. Jäger, A. Mikelić: Homogenization of the Laplace equation in a partially perforated domain, prépublication no. 157, Equipe d'Analyse Numérique Lyon-St-Etienne, September 1993, published in “Homogenization, In Memory of Serguei Kozlov”, eds. V. Berdichevsky, V. Jikov and G. Papanicolaou, p. 257–284, World Scientific, Singapore, 1999.

    Google Scholar 

  6. W. Jäger, A. Mikelić: On the Flow Conditions at the Boundary Between a Porous Medium and an Impervious Solid, in “Progress in Partial Differential Equations: the Metz Surveys 3”, eds. M. Chipot, J. Saint Jean Paulin et I. Shafrir, πPitman Research Notes in Mathematics no. 314, p. 145–161, Longman Scientific and Technical, London, 1994.

    Google Scholar 

  7. W. Jäger, A. Mikelić: On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid, Ann. Sc. Norm. Super. Pisa, Cl. Sci. — Ser. IV, Vol. XXIII (1996), Fasc. 3, p. 403–465.

    MathSciNet  MATH  Google Scholar 

  8. W. Jäger, A. Mikelić: On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness, Communications on Pure and Applied Mathematics, Vol. LI (1998), p. 1073–1121.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. W. Jäger, A. Mikelić: On the boundary conditions at the contact interface between two porous media, in “Partial differential equations, Theory and numerical solution”, eds. W. Jäger, J. Nečas, O. John, K. Najzar and J. Stará, π Chapman and Hall/CRC Research Notes in Mathematics no. 406, p. 175–186, CRC Press, London, 1999.

    Google Scholar 

  10. W. Jäger, A. Mikelić: On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. Appl. Math.,60 (2000), 1111–1127.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. W. Jäger, A. Mikelić: On the roughness-induced effective boundary conditions for a viscous flow, accepted for publication in Journal of Differential Equations, 1999.

    Google Scholar 

  12. W. Jäger, A. Mikelić, N. Neuß: Asymptotic analysis of the laminar viscous flow over a porous bed, preprint, Universität Heidelberg, Germany, January 1999.

    MATH  Google Scholar 

  13. Th. Levy, E. Sanchez-Palencia: On Boundary Conditions for Fluid Flow in Porous Media, Int. J. Engng. Sci., Vol. 13 (1975), p. 923–940.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J.L. Lions: Some Methods in the Mathematical Analysis of Systems and Their Control, Gordon and Breach, New York, 1981.

    MATH  Google Scholar 

  15. E. Marušić-Paloka, A. Mikelić: An Error Estimate for Correctors in the Homogenization of the Stokes and Navier-Stokes Equations in a Porous Medium, Boll. Unione Mat. Ital., (7) 10-A (1996), no. 3, p. 661–671.

    MathSciNet  MATH  Google Scholar 

  16. P.G. Saffman: On the boundary condition at the interface of a porous medium, Studies in Applied Mathematics, 1 (1971), pp. 93–101.

    CrossRef  MATH  Google Scholar 

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Mikelić, A. (2000). Homogenization theory and applications to filtration through porous media. In: Fasano, A. (eds) Filtration in Porous Media and Industrial Application. Lecture Notes in Mathematics, vol 1734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103977

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