Skip to main content
SpringerLink
Log in

An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media

  • Chapter
  • First Online:
Non-Newtonian Fluid Mechanics and Complex Flows

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2212))

Abstract

The flow of complex fluids through porous media is common to many engineering applications. The upscaling is a powerful tool for modeling nonhomogeneous media and we consider homogenization of quasi-Newtonian and electrokinetic flows through porous media. For the quasi-Newtonian polymeric fluids, the incompressible Navier-Stokes equations with the invariants dependent viscosity is supposed to hold the pore scale level. The two-scale asymptotic expansions and the two-scale convergence of the monotone operators are applied to derive the reservoir level filtration law, given as a monotone relation between the filtration velocity and the pressure gradient. The second problem, we consider, is the quasi-static transport of an electrolyte through an electrically charged medium. The physical chemistry modeling is presented and used to get a dimensionless form of the problem. Next the equilibrium solutions are constructed through solving the Poisson-Boltzmann equation. For the solutions being close to the equilibrium, the two-scale convergence is applied to obtain the Onsager relations linking gradients of the pressure and of the chemical potentials to the filtration velocity and the ionic fluxes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Minty’s lemma (see [33]) Let F be a convex lower semi-continuous and proper functional on a reflexive Banach space B. Then for u ∈ B the following three conditions are equivalent to each other:

    1. (a)

      u solves the problem infv ∈ B F(v).

    2. (b)

      < F′(u), v − u >≥ 0, ∀ v ∈ B.

    3. (c)

      < F′(v), v − u >≥ 0, ∀ v ∈ B.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler, P.M.: Macroscopic electroosmotic coupling coefficient in random porous media. Math. Geol. 33(1), 63–93 (2001)

    Google Scholar 

  3. Adler, P.M., Mityushev, V.: Effective medium approximation and exact formulae for electrokinetic phenomena in porous media. J. Phys. A: Math. Gen. 36, 391–404 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allaire, G. : Homogenization of the stokes flow in a connected porous medium. Asymptot. Anal. 2, 203–222 (1989)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Allaire, G.: One-phase newtonian flow. In: Hornung, U. (ed.) Homogenization and Porous Media, pp. 45–68. Springer, New-York (1997)

    Google Scholar 

  7. Allaire, G., A. Damlamian, A., Hornung, U.: Two-scale convergence on periodic surfaces and applications. In: Bourgeat, A., Carasso, C., Luckhaus, S., Mikelić, A. (eds.) Mathematical Modelling of Flow through Porous Media, pp. 15–25. World Scientific, Singapore (1995)

    Google Scholar 

  8. Allaire, G., Mikelić, A., Piatnitski, A.: Homogenization approach to the dispersion theory for reactive transport through porous media. SIAM J. Math. Anal. 42, 125–144 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Allaire, G., Mikelić, A., Piatnitski, A.: Homogenization of the linearized ionic transport equations in rigid periodic porous media. J. Math. Phys. 51, 123103 (2010). Erratum in the same journal, 52, 063701 (2011).

    Google Scholar 

  10. Allaire, G., Brizzi, R., Mikelić, A., Piatnitski, A.: Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media. Chem. Eng. Sci. 65, 2292–2300 (2010)

    Article  Google Scholar 

  11. Allaire, G., Dufrêche, J.-F., Mikelić, A., Piatnitski, A.: Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media. Nonlinearity 26, 881–910 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Allaire, G., Bernard, O., Dufrêche, J.-F., Mikelić, A.: Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling. Comp. Appl. Math. 36, 1431–1462 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Allaire, G., Brizzi, R., Dufrêche, J.-F., Mikelić, A. Piatnitski, A. : Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients. Comp. Geosci. 17, 479–495 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Allaire, G., Brizzi, R. , Dufrêche, J.-F., Mikelić, A., Piatnitski, A.: Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling. Phys. D. 282 , 39–60 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Auriault, J.L. , Strzelecki, T.: On the electro-osmotic flow in a saturated porous medium. Int. J. Engng Sci. 19, 915–928 (1981)

    Article  MATH  Google Scholar 

  16. Baranger, J. , Najib, K.: Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau. Numer. Math. 58, 35–49 (1990)

    Google Scholar 

  17. Bird, R.B., Stewart, W.E.N., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (1960)

    Google Scholar 

  18. Bird, R.B., Armstrong R.C., Hassager, O.: Dynamics of Polymeric Liquids, Vol.1, Fluid Mechanics. Wiley, New York, (1987)

    Google Scholar 

  19. Bourgeat, A., Mikelić, A.: Note on the homogenization of Bingham flow through porous medium. J. Math. Pures Appl. 72, 405–414 (1993)

    Google Scholar 

  20. Bourgeat, A., Mikelić, A.: Homogenization of the non-newtonian flow through porous medium. Nonlinear Anal. Theory Methods Appl. 26, 1221–1253 (1996)

    Google Scholar 

  21. Bourgeat, A., Mikelić, A., Tapiero, R.: Dérivation des équations moyennées décrivant un écoulement non newtonien dans un domaine de faible épaisseur. C. R. Acad. Sci. Paris, Sér. I 316, 965–970 (1993)

    Google Scholar 

  22. Bourgeat, A., Gipouloux, O., Marusic-Paloka, E.: Filtration law for polymer flow through porous media. Multiscale Model. Simul. 1, 432–457 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer Science and Business Media, New York (2010)

    Book  Google Scholar 

  24. Bulíček, M. , Gwiazda, P. , Málek, J., Świerczewska-Gwiazda, A.: On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2, 109–136 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rendiconti Seminario Matematico della Università di Padova, 31, 308–340 (1961)

    MathSciNet  MATH  Google Scholar 

  26. Chan, D.Y., Horn, R.G.: The drainage of thin liquid films between solid surfaces. J. Chem. Phys. 83, 5311–5325 (1985)

    Article  Google Scholar 

  27. Christopher, R.H., Middleman, S.: Power-law flow through porous media. Ind. Eng. Cheni. Fund. 4, 422 (1965)

    Article  Google Scholar 

  28. Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  29. Coelho, D., Shapiro, M., Thovert, J.-F. , Adler, P.M. : Electro-osmotic phenomena in porous media. J. Colloid Interface Sci. 181, 169–90 (1996)

    Article  Google Scholar 

  30. Dufrêche, J.-F. , Bernard, O. , Durand-Vidal, S. , Turq, P.: Analytical theories of transport in concentrated electrolyte solutions from the MSA. J. Phys. Chem. B 109, 9873 (2005)

    Article  Google Scholar 

  31. Dufrêche, J.-F., Marry, V. , Malikova, N. , Turq, P., : Molecular hydrodynamics for electro-osmosis in clays: from Kubo to Smoluchowski. J. Mol. Liq. 118, 145 (2005)

    Article  Google Scholar 

  32. Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer, Heidelberg (1976)

    Book  MATH  Google Scholar 

  33. Ekeland, I., Temam, R. : Analyse Convexe et Problèmes Variationnels. Gauthier-Villars, Paris (1973)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  35. Ern, A. , Joubaud, R. , Lelièvre, T.: Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes. Nonlinearity 25, 1635–1652 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gipouloux, O., Zine, A.M.: Computation of the filtration laws through porous media for a non-Newtonian fluid obeying the power law. Comput. Geosci. 1, 127–153 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gupta, A.K. , Coelho, D. , Adler, P.M.: Electroosmosis in porous solids for high zeta potentials. J. Colloid Interface Sci. 303, 593–603 (2006)

    Article  Google Scholar 

  38. Hornung, U. (ed.) : Homogenization and Porous Media. Interdisciplinary Applied Mathematics Series, vol. 6. Springer, New York (1997)

    MATH  Google Scholar 

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

    Book  Google Scholar 

  40. Kaloušek, M. : Homogenization of incompressible generalized Stokes flows through a porous medium. Nonlinear Anal. Theory Methods Appl. 136, 1–39 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Karniadakis, G., Beskok, A., Aluru, N.: Microflows and Nanoflows. Fundamentals and Simulation. Interdisciplinary Applied Mathematics, Vol. 29. Springer, New York (2005)

    Google Scholar 

  42. Khuzhayorov, B., Auriault, J.-L., Royer, P. : Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. Int. J. Eng. Sci. 38, 487–504 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  44. Lions, J.L., Sanchez-Palencia, E.: Écoulement d’un fluide viscoplastique de Bingham dans un milieu poreux. J. Math. pures et appl. 60, 341–360 (1981)

    MathSciNet  MATH  Google Scholar 

  45. Lipton, R., Avellaneda, M. : A darcy law for slow viscous flow past a stationary array of bubbles. Proc. Royal Soc. Edinburgh 114A, 71–79 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  46. Looker, J.R. : Semilinear elliptic Neumann problems and rapid growth in the nonlinearity. Bull. Aust. Math. Soc., 74, 161–175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Looker, J.R., Carnie, S.L.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Media 65, 107–131 (2006)

    Article  MathSciNet  Google Scholar 

  48. Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. Sci. 2, 35–86 (2002)

    MathSciNet  MATH  Google Scholar 

  49. Marciniak-Czochra, A., Ptashnyk, M.: Derivation of a macroscopic receptor-based model using homogenization techniques. SIAM J. Math. Anal. 40, 215–237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Marušić-Paloka, E., Piatnitski, A.: Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection. J. Lond. Math. Soc. 72, 391–409 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Marino, S., Shapiro, M., Adler, P.M. : Coupled transports in heterogeneous media. J. Colloid Interface Sci. 243, 391–419 (2001)

    Article  Google Scholar 

  52. Marry, V., Dufrêche, J.-F., Jardat, M., Turq, P.: Equilibrium and electrokinetic phenomena in charged porous media from microscopic and mesoscopic models: electro-osmosis in montmorillonite. Mol. Phys. 101, 3111 (2005)

    Article  Google Scholar 

  53. Mei, C.C. , Vernescu, B.: Homogenization Methods for Multiscale Mechanics. World Scientific Publishing Company, Singapore (2010)

    Book  MATH  Google Scholar 

  54. Mikelić, A.: Non-newtonian flow. In: Hornung, U. (ed.) Homogenization and Porous Media, pp. 69–95. Springer, New-York (1997)

    Google Scholar 

  55. Mikelić, A., Tapiero, R. : Mathematical derivation of the power law describing polymer flow through a thin slab. Math. Modell. Numer. Anal. 29, 3–22 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  56. Moyne, C., Murad, M. : Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure. Int. J. Solids Struct. 39, 6159–6190 (2002)

    Article  MATH  Google Scholar 

  57. Moyne, C., Murad, M.: Macroscopic behavior of swelling porous media derived from micromechanical analysis. Transp. Porous Media 50, 127–151 (2003)

    Article  MathSciNet  Google Scholar 

  58. Moyne, C., Murad, M. : A Two-scale model for coupled electro-chemomechanical phenomena and Onsager’s reciprocity relations in expansive clays: I homogenization analysis, Transp. Porous Media 62, 333–380 (2006)

    Article  MathSciNet  Google Scholar 

  59. Moyne, C., Murad, M. : A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager’s reciprocity relations in expansive clays: II. Computational validation. Transp. Porous Media 63(1), 13–56 (2006)

    Article  Google Scholar 

  60. Moyne, C., Murad, M. : A dual-porosity model for ionic solute transport in expansive clays. Comput. Geosci. 12, 47–82 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  61. Neuss-Radu, M. : Some extensions of two-scale convergence. C. R. Acad. Sci. Paris Sér. I Math. 322(9), 899–904 (1996)

    MathSciNet  MATH  Google Scholar 

  62. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  63. O’Brien, R.W., White, L.R.: Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 2 74(2) , 1607–1626 (1978)

    Article  Google Scholar 

  64. Oleinik, O.A.E., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization, vol. 2. Elsevier, Amsterdam (2009)

    MATH  Google Scholar 

  65. Orgéas, L., Idris, Z., Geindreau, C., Bloch, J.-F., Auriault, J.-L.: Modelling the flow of power-law fluids through anisotropic porous media at low-pore Reynolds number. Chem. Eng. Sci. 61, 4490–4502 (2006)

    Article  Google Scholar 

  66. Pavliotis, G.A., Stuart, A.: Multiscale Methods: Averaging and Homogenization. Springer, New York (2008)

    MATH  Google Scholar 

  67. Pearson, J., Tardy, P.: Models for flow of non-newtonian and complex fluids through porous media. J. Non-Newtonian Fluid Mech. 102, 447–473 (2002)

    Article  MATH  Google Scholar 

  68. Ray, N., Muntean, A., Knabner, P.: Rigorous homogenization of a Stokes-Nernst-Planck-Poisson system. J. Math. Anal. Appl. 390(1), 374–393 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  69. Ray, N. , Eck, Ch., Muntean, A., Knabner, P.: Variable Choices of Scaling in the Homogenization of a Nernst-Planck-Poisson Problem. Preprint no. 344, Institut für Angewandte Mathematik, Universitaet Erlangen-Nürnberg (2011)

    Google Scholar 

  70. Rosanne, M., Paszkuta, M., Adler, P.M.: Electrokinetic phenomena in saturated compact clays. J. Colloid Interface Sci. 297, 353–364 (2006)

    Article  Google Scholar 

  71. Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Heidelberg (1980)

    Google Scholar 

  72. Shah, C.B. , Yortsos, Y.C.: Aspects of flow of power-law fluids in porous media. AIChE J. 41, 1099–1112 (1995)

    Article  Google Scholar 

  73. Shahsavari, S., McKinley, G.H.: Mobility of power-law and Carreau fluids through fibrous media. Physical Review E 92, 063012 (2015)

    Article  Google Scholar 

  74. Shahsavari, S., McKinley, G.H.: Mobility and pore-scale fluid dynamics of rate-dependent yield-stress fluids flowing through fibrous porous media. J. Non-Newtonian Fluid Mech. 235, 76–82 (2016)

    Article  MathSciNet  Google Scholar 

  75. Schmuck, M.: Analysis of the Navier–Stokes–Nernst–Planck–Poisson system. Math. Models Methods Appl. Sci. 19, 993–1014 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  76. Schmuck, M.: Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multiple-scale approach. Commun. Math. Sci. 9(3), 685–710 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  77. Schmuck, M.: First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations. ZAMM (J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik). 92, 304–319 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  78. Schmuck, M.: New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials. J. Math. Phys. 54, 021504 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  79. Temam, R.: Navier–Stokes Equations. Elsevier, Amsterdam (1984)

    MATH  Google Scholar 

  80. Wu, Y.S., Pruess, K. , Witherspoon, P.A. : Displacement of a newtonian fluid by a non-newtonian fluid in a porous medium. Trans. Porous Media. 6, 115 (1991)

    Article  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the MOCOMIPOC project (Modélisation multiéchelles des écoulements complexes en présence de gaz dans les milieux chargés) from the NEEDS program (Projet fédérateur Milieux Poreux MIPOR), part of CNRS, France and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Author information

Authors and Affiliations

  1. Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne cedex, France

    Andro Mikelić

Authors
  1. Andro Mikelić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andro Mikelić .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze , Firenze, Italy

    Angiolo Farina

  2. Département de Mathématiques, Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France

    Andro Mikelić

  3. Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Firenze, Italy

    Fabio Rosso

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mikelić, A. (2018). An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media. In: Farina, A., Mikelić, A., Rosso, F. (eds) Non-Newtonian Fluid Mechanics and Complex Flows. Lecture Notes in Mathematics(), vol 2212. Springer, Cham. https://doi.org/10.1007/978-3-319-74796-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation