Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T15:03:23.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic and numerical modelling of flows in fractured porous media

Published online by Cambridge University Press:  07 February 2009

Philippe Angot ,
Franck Boyer  and
Florence Hubert
Philippe Angot
Affiliation:
Université de Provence and CNRS, Laboratoire d'Analyse Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected]; [email protected]; [email protected]
Franck Boyer
Affiliation:
Université de Provence and CNRS, Laboratoire d'Analyse Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected]; [email protected]; [email protected]
Florence Hubert
Affiliation:
Université de Provence and CNRS, Laboratoire d'Analyse Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected]; [email protected]; [email protected]
Get access

Abstract

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale,so that we can asymptotically reduce them to immersed polygonal faultinterfaces and the model finally consists in a coupling between a2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.A cell-centered finite volume scheme on general polygonal meshes fitting the interfacesis derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocityjumps through the interfaces.We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures.Some numerical results are reported showing different kinds of flowsin the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numericalsolutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

Keywords

Fractured porous mediaDarcy flowfinite volume methodasymptotic models of flow.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 239 - 275
Copyright
© EDP Sciences, SMAI, 2009

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

P.M. Adler and J.-F. Thovert, Fractures and Fracture Networks. Kluwer Acad., Amsterdam (1999).
Andreianov, B., Boyer, F. and Hubert, F., Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D-meshes. Numer. Methods Partial Differential Equations 23 (2007) 145–195. CrossRef
P. Angot, Finite volume methods for non smooth solution of diffusion models; application to imperfect contact problems, in Recent Advances in Numerical Methods and Applications, O.P. Iliev, M.S. Kaschiev, S.D. Margenov, B.H. Sendov and P.S. Vassilevski Eds., Proc. 4th Int. Conf. NMA'98, Sofia (Bulgarie), World Sci. Pub. (1999) 621–629.
Angot, P., A model of fracture for elliptic problems with flux and solution jumps. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003) 425–430. CrossRef
P. Angot, T. Gallouët and R. Herbin, Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks, in Finite Volumes for Complex Applications II, R. Vilsmeier, F. Benkhaldoun and D. Hänel Eds., Hermès (1999) 215–222.
J. Bear, C.-F. Tsang and G. de Marsily, Flow and Contaminant Transport in Fractured Rock. Academic Press, San Diego (1993).
Berkowitz, B., Characterizing flow and transport in fractured geological media: A review. Adv. Water Resour. 25 (2002) 861–884. CrossRef
Bernardi, C., Dauge, M. and Maday, Y., Compatibilité de traces aux arêtes et coins d'un polyhèdre. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 679–684. CrossRef
C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. (2007) http://hal.archives-ouvertes.fr/hal-00153795.
I.I. Bogdanov, V.V. Mourzenko, J.-F. Thovert and P.M. Adler, Effective permeability of fractured porous media in steady-state flow. Water Resour. Res. 107 (2002).
Boyer, F. and Hubert, F., Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032–3070. CrossRef
Caillabet, Y., Fabrie, P., Landereau, P., Noetinger, B. and Quintard, M., Implementation of a finite-volume method for the determination of effective parameters in fissured porous media. Numer. Methods Partial Differential Equations 6 (2000) 237–263. 3.0.CO;2-W>CrossRef
Caillabet, Y., Fabrie, P., Lasseux, D. and Quintard, M., Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method. Comput. Geosci. 5 (2001) 121–150. CrossRef
Domelevo, K. and Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. CrossRef
Eymard, R. and Gallouët, T., H-convergence and numerical schemes for elliptic equations. SIAM J. Numer. Anal. 41 (2003) 539–562. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in Handbook of Numerical Analysis VII, P.G. Ciarlet and J.L. Lions Eds., North-Holland (2000) 713–1020.
I. Faille, E. Flauraud, F. Nataf, S. Pégaz-Fiornet, F. Schneider and F. Willien, A new fault model in geological basin modelling. Application of finite volume scheme and domain decomposition methods, in Finite Volumes for Complex Applications III, R. Herbin and D. Kröner Eds., Hermes Penton Sci. (HPS) (2002) 543–550.
B. Faybishenko, P.A. Witherspoon and S.M. Benson, Dynamics of Fluids in Fractured Rock, Geophysical Monograph Series 122. American Geophysical Union, Washington D.C. (2000).
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, Boston (1985).
Hermeline, F., Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Engrg. 192 (2003) 1939–1959. CrossRef
J. Jaffré, V. Martin and J.E. Roberts, Generalized cell-centered finite volume methods for flow in porous media with faults, in Finite Volumes for Complex Applications III, R. Herbin and D. Kröner Eds., Hermes Penton Sci. (HPS) (2002) 357–364.
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. CrossRef
Mityushev, V. and Adler, P.M., Darcy flow arround a two dimensional lense. Journal Phys. A: Math. Gen. 39 (2006) 3545–3560. CrossRef
Reichenberger, V., Jakobs, H., Bastian, P. and Helmig, R., A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29 (2006) 1020–1036. CrossRef