Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T09:13:23.999Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical analysis of a discrete fracture model couplingDarcy flow in the matrix with Darcy–Forchheimer flow in the fracture∗

Published online by Cambridge University Press:  13 August 2014

Peter Knabner  and
Jean E. Roberts
Peter Knabner
Affiliation:
University of Erlangen-Nuremberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany.. knabner@am.uni-erlangen.de
Jean E. Roberts
Affiliation:
Inria Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France.; jean.roberts@inria.fr
Get access

Abstract

We consider a model for flow in a porous medium with a fracture in which the flow in thefracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed byDarcy’s law. We give an appropriate mixed, variational formulation and show existence anduniqueness of the solution. To show existence we give an analogous formulation for themodel in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for the model withDarcy’s law in the matrix is the weak limit of solutions of the model with theDarcy−Forchheimerlaw in theentire domain when the Forchheimer coefficient in the matrix tends toward zero.

Keywords

Flow in porous mediafracturesDarcy−Forchheimerflowsolvabilityregularizationmonotone operators
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1451 - 1472
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Adams, Sobolev Spaces, vol. 65 of Pure and Appl. Math. Academic Press, New York (1975).
C. Alboin, J. Jaffré, J. Roberts and C. Serres, Domain decomposition for flow in porous media with fractures, in Proc. of the 11th Int. Conf. on Domain Decomposition Methods in Greenwich, England (1999).
Allaire, G., Homogenization of the stokes flow in a connected porous medium. Asymptotic Anal. 2 (1989) 203–222. Google Scholar
G. Allaire, One-phase newtonian flow, in Homogenization and Porous Media, vol. 6 of Interdisciplinary Appl. Math., edited by U. Hornung. Springer-Verlag, New York (1997) 45–69.
Amirat, Y., Ecoulements en milieu poreux n’obeissant pas a la loi de darcy. RAIRO Modél. Math. Anal. Numér. 25 (1991) 273–306. Google Scholar
Angot, P., Boyer, F. and Hubert, F., Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 43 (2009) 239–275. Google Scholar
M. Balhoff, A. Mikelic and M. Wheeler, Polynomial filtration laws for low reynolds number flows through porous media. Transport in Porous Media (2009).
J. Bear, Dynamics of Fluids in Porous Media. American Elsevier Pub. Co., New York (1972).
Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. RAIRO: Modél. Math. Anal. Numér. 8 (1974) 129–151. Google Scholar
Fabrie, P., Regularity of the solution of Darcy−Forchheimer’s equation. Nonlinear Anal., Theory Methods Appl. 13 (1989) 1025–1049. Google Scholar
I. Faille, E. Flauraud, F. Nataf, S. Pegaz-Fiornet, F. Schneider and F. Willien, A new fault model in geological basin modelling, application to finite volume scheme and domain decomposition methods, in Finie Volumes for Complex Appl. III. Edited by R. Herbin and D. Kroner. Hermés Penton Sci. (2002) 543–550.
Forchheimer, P., Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45 (1901) 1782–1788. Google Scholar
Frih, N., Roberts, J. and Saada, A., Un modèle darcy-frochheimer pour un écoulement dans un milieu poreux fracturé. ARIMA 5 (2006) 129–143. Google Scholar
Frih, N., Roberts, J. and Saada, A., Modeling fractures as interfaces: a model for forchheimer fractures. Comput. Geosci. 12 (2008) 91–104. Google Scholar
P. Knabner and G. Summ, Solvability of the mixed formulation for Darcy−Forchheimer flow in porous media. Submitted.
Martin, V., Jaffré, J. and Roberts, J.E., Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691. Google Scholar
Showalter, R. and Morales, F., The narrow fracture approximation by channeled flow. J. Math. Anal. Appl. 365 (2010) 320–331. Google Scholar
G. Summ, Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien. Ph.D. thesis. Friedrich-Alexander-Universität Erlangen-Nürnberg (2001).
L. Tartar, Convergence of the homogenization process, in Non-homogeneous Media and Vibration Theory, vol. 127 of Lect. Notes Phys. Edited by E. Sancez-Palencia. Springer-Verlag (1980).
E. Zeidler, Nonlinear function anaysis and its applications – Nonlinear monotone operators. Springer-Verlag, Berlin, Heidelberg, New York (1990).