Finite element discretization of Darcy's equations with pressure dependent porosity

Vivette Girault; François Murat; Abner Salgado

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 6, page 1155-1191
  • ISSN: 0764-583X

Abstract

top
We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.


How to cite

top

Girault, Vivette, Murat, François, and Salgado, Abner. "Finite element discretization of Darcy's equations with pressure dependent porosity." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1155-1191. <http://eudml.org/doc/250842>.

@article{Girault2010,
abstract = { We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.
},
author = {Girault, Vivette, Murat, François, Salgado, Abner},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Porous media flows; Darcy equations; finite elements; exponential porosity; bounded porosity; convergence; splitting scheme},
language = {eng},
month = {10},
number = {6},
pages = {1155-1191},
publisher = {EDP Sciences},
title = {Finite element discretization of Darcy's equations with pressure dependent porosity},
url = {http://eudml.org/doc/250842},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Girault, Vivette
AU - Murat, François
AU - Salgado, Abner
TI - Finite element discretization of Darcy's equations with pressure dependent porosity
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/10//
PB - EDP Sciences
VL - 44
IS - 6
SP - 1155
EP - 1191
AB - We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.

LA - eng
KW - Porous media flows; Darcy equations; finite elements; exponential porosity; bounded porosity; convergence; splitting scheme
UR - http://eudml.org/doc/250842
ER -

References

top
  1. R.A. Adams, Sobolev spaces. Academic Press (1975).  Zbl0314.46030
  2. G. Allaire, Homogeneization of the Navier-Stokes equations with slip boundary conditions. Comm. Pure Appl. Math.44 (1991) 605–641.  Zbl0738.35059
  3. M. Azaïez, F. Ben Belgacem, C. Bernardi and N. Chorfi, Spectral discretization of Darcy's equations with pressure dependent porosity. Report 2009-10, Laboratoire Jacques-Louis Lions, France (2009).  Zbl05817268
  4. I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math.20 (1973) 179–192.  Zbl0258.65108
  5. W. Bangerth, R. Hartman and G. Kanschat, deal.II – a general-purpose object-oriented finite element library. ACM Trans. Math. Softw.33 (2007) 24.  
  6. J. Berg and J. Löfström, Interpolation spaces: An introduction, Comprehensive Studies in Mathematics223. Springer-Verlag (1976).  Zbl0344.46071
  7. D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk and M. Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics939. Springer-Verlag, Berlin, Germany (2008).  
  8. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics15. Third edition, Springer-Verlag (2008).  
  9. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér.R2 (1974) 129–151.  Zbl0338.90047
  10. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics. Springer-Verlag, New York (1991).  Zbl0788.73002
  11. F. Brezzi, J. Rappaz and P.-A. Raviart, Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math.36 (1980) 1–25.  Zbl0488.65021
  12. P.-G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical AnalysisII, Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 17–351.  
  13. D. Cioranescu, P. Donato and H.I. Ene, Homogeneization of the Stokes problem with non-homogeneous boundary conditions. Math. Appl. Sci.19 (1996) 857–881.  Zbl0869.35012
  14. H. Darcy, Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris, France (1856).  
  15. J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp.29 (1975) 689–696.  Zbl0306.65072
  16. H.I. Ene and E. Sanchez-Palencia, Équations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux. J. Mécanique14 (1975) 73–108.  Zbl0304.76037
  17. A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences159. Springer-Verlag, New York, USA (2004).  Zbl1059.65103
  18. G.B. Folland, Real analysis, modern techniques and their applications. Second edition, Wiley Interscience (1999).  Zbl0924.28001
  19. P. Forchheimer, Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing.45 (1901) 1782–1788.  
  20. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations – Theory and algorithms, Springer Series in Computational Mathematics5. Springer-Verlag, Berlin, Germany (1986).  Zbl0585.65077
  21. V. Girault and M.F. Wheeler, Numerical discretization of a Darcy-Forchheimer model. Numer. Math.110 (2008) 161–198.  Zbl1143.76035
  22. V. Girault, R. Nochetto and L.R. Scott, Maximum-norm stability of the finite-element Stokes projection. J. Math. Pure. Appl.84 (2005) 279–330.  Zbl1210.76051
  23. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics24. Pitman, Boston, USA (1985).  Zbl0695.35060
  24. F. Hecht, A. Le Hyaric, O. Pironneau and K. Ohtsuka, Freefem++. Second Edition, Version 2.24-2-2. Laboratoire J.-L. Lions, UPMC, Paris, France (2008).  
  25. A.Ya. Helemskii, Lectures and exercises on functional analysis, Translations of Mathematical Monographs233. American Mathematical Society, USA (2006).  
  26. L.V. Kantorovich and G.P. Akilov, Functional analysis. Third edition, Nauka (1984) [in Russian].  Zbl0555.46001
  27. D. Kim and E.J. Park, Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities. Comput. Math. Appl.51 (2006) 793–804.  Zbl1134.65367
  28. J.L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris, France (1968).  Zbl0165.10801
  29. E.J. Park, Mixed finite element methods for nonlinear second order elliptic problems. SIAM J. Numer. Anal.32 (1995) 865–885.  Zbl0834.65108
  30. S.E. Pastukhova, Substantiation of the Darcy Law for a porous medium with condition of partial adhesion. Sbornik Math.189 (1998) 1871–1888.  Zbl0932.35162
  31. K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. M3AS17 (2007) 215–252.  Zbl1123.76066
  32. J.E. Roberts and J.-M. Thomas, Mixed and Hybrid methods in Handbook of Numerical AnalysisII: Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 523–639.  Zbl0875.65090
  33. J. Schöberl and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems. SIAM J. Matrix Anal. Appl.29 (2007) 752–773.  Zbl1154.65029
  34. E. Skjetne and J.L. Auriault, Homogeneization of wall-slip gas flow through porous media. Transp. Porous Media36 (1999) 293–306.  
  35. L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana3. Springer-Verlag, Berlin-Heidelberg (2007).  Zbl1126.46001
  36. W. Zulehner, Analysis of iterative methods for saddle point problems: a unified approach. Math. Comp.71 (2001) 479–505.  Zbl0996.65038

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.