Skip to main content
SpringerLink
Log in

On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

This paper is concerned with the global existence, uniqueness and homogenization of degenerate partial differential equations with integral conditions arising from coupled transport processes and chemical reactions in three-dimensional highly heterogeneous porous media. Existence of global weak solutions of the microscale problem is proved by means of semidiscretization in time deriving a priori estimates for discrete approximations needed for proofs of existence and convergence theorems. It is further shown that the solution of the microscale problem is two-scale convergent to that of the upscaled problem as the scale parameter goes to zero. In particular, we focus our efforts on the contribution of the so-called first order correctors in periodic homogenization. Finally, under additional assumptions, we consider the problem of the uniqueness of the solution to the homogenized problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, A., Fournier, J.F.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 140. Academic Press, San Diego (2003)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)

    Article  MathSciNet  Google Scholar 

  4. Amaziane, B., Antontsev, S., Pankratov, L., Piatnitski, A.: Homogenization of immiscible compressible two-phase flow in porous media: application to gas migration in a nuclear waste repository. Multiscale Model. Simul. 8, 2023–2047 (2010)

    Article  MathSciNet  Google Scholar 

  5. Amaziane, B., Jurak, M., Vrbaški, A.: Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure. Appl. Anal. 92, 1417–1433 (2013)

    Article  MathSciNet  Google Scholar 

  6. Beneš, M., Pažanin, I.: Homogenization of degenerate coupled fluid flows and heat transport through porous media. J. Math. Anal. Appl. 446, 165–192 (2017)

    Article  MathSciNet  Google Scholar 

  7. Beneš, M., Štefan, R.: Hygro-thermo-mechanical analysis of spalling in concrete walls at high temperatures as a moving boundary problem. Int. J. Heat Mass Transf. 85, 110–134 (2015)

    Article  Google Scholar 

  8. Cao, H., Yue, X.: Homogenization of Richard’s equation of van Genuchten-Mualem model. J. Math. Anal. Appl. 412, 391–400 (2014)

    Article  MathSciNet  Google Scholar 

  9. Di Luzio, G., Cusatis, G.: Hygro-thermo-chemical modeling of high performance concrete. I: theory. Cem. Concr. Compos. 31, 301–308 (2009)

    Article  Google Scholar 

  10. Filo, J., Kačur, J.: Local existence of general nonlinear parabolic systems. Nonlinear Anal. 24, 1597–1618 (1995)

    Article  MathSciNet  Google Scholar 

  11. Gawin, D., Pesavento, F., Schrefler, B.A.: Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: hydration and hygro-thermal phenomena. Int. J. Numer. Methods Eng. 67, 299–331 (2006)

    Article  Google Scholar 

  12. Heuser, P.: In: Homogenization of Quasilinear Elliptic-Parabolic Equations with Respect to Measures, Ruprecht-Karls-Universitat, Heidelberg (2012)

    MATH  Google Scholar 

  13. Kačur, J.: On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces. I. Math. Z. 203, 153–171 (1990)

    Article  MathSciNet  Google Scholar 

  14. Kačur, J., Mahmood, M.S.: Solution of solute transport in unsaturated porous media by the method of characteristics. Numer. Methods Partial Differ. Equ. 19, 732–761 (2003)

    Article  MathSciNet  Google Scholar 

  15. Kačur, J., Mahmood, M.S.: Galerkin characteristics method for convection-diffusion problems with memory terms. Int. J. Numer. Anal. Model. 6, 89–109 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague (1997)

    MATH  Google Scholar 

  17. Li, X., Wu, W., Cescotto, S.: Contaminant transport with non-equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme. Int. J. Numer. Anal. Methods Geomech. 24, 219–243 (2000)

    Article  Google Scholar 

  18. Li, X., Li, R., Schrefler, B.A.: A coupled chemo-thermo-hygro-mechanical model of concrete at high temperature and failure analysis. Int. J. Numer. Anal. Methods Geomech. 30, 635–681 (2006)

    Article  Google Scholar 

  19. Mazya, V.G., Rossmann, J.: Elliptic Equations in Polyhedral Domains. Mathematical Surveys and Monographs, vol. 162 (2010)

    Book  Google Scholar 

  20. Ožbolt, J., Balabanić, G., Periškić, G., Kušter, M.: Modelling the effect of damage on transport processes in concrete. Constr. Build. Mater. 24, 1638–1648 (2010)

    Article  Google Scholar 

  21. Pinder, G.F., Gray, W.G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, New York (2008)

    Book  Google Scholar 

  22. Rektorys, K.: The Method of Discretization in Time and Partial Differential Equations. Reidel Co, Dodrecht, Holland (1982)

    MATH  Google Scholar 

  23. Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  24. Wu, Z., Yin, J., Wang, Ch.: Elliptic and Parabolic Equations. World Scientific, Singapore (2006)

    Book  Google Scholar 

Download references

Acknowledgements

This research has been performed in the Center of Advanced Applied Sciences (CAAS), financially supported by the European Regional Development Fund (project No. CZ.02.1.01/0.0/0.0/16_019/0000778).

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, Department of Mathematics, Czech Technical University in Prague, Thákurova 7, 166 29, Prague 6, Czech Republic

    Michal Beneš

Authors
  1. Michal Beneš
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michal Beneš.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beneš, M. On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media. Acta Appl Math 171, 12 (2021). https://doi.org/10.1007/s10440-020-00378-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-020-00378-y

Keywords

Search

Navigation