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Modeling fractures as interfaces: a model for Forchheimer fractures

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Abstract

In this paper we are concerned with modeling single phase flow in a medium with known fractures. In particular we are interested in the case in which the flow rate in the fractures is large enough to make it appropriate to use Forchheimer’s law for modeling the flow in the fractures even though the flow in the surrounding domain is such that Darcy’s law is adequate. We describe a model in which the fractures are treated as interfaces. We also consider the case of intersecting fractures and the case of nonconforming meshes.

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References

  1. Alboin, C., Jaffré, J., Roberts, J.E.: Domain decomposition for flow in fractured porous media. In: Domain Decomposition Methods in Sciences and Engineering, pp. 365–373. Domain Decomposition Press, Bergen (1999)

    Google Scholar 

  2. Amir, L., Kern, M., Martin, V., Roberts, J.E.: Décomposition de domaine pour un milieu poreux fracturé: un modèle en 3D avec fractures qui s’intersectent. ARIMA 5, 11–25 (2006)

    Google Scholar 

  3. Angot, Ph., Gallouet, T., Herbin, R.: Convergence of finite volume method on general meshes for non smooth solution of elliptic problems with cracks. In: Vilsmeier, R., Benkhaldoun, F., Hanel, D. (eds.) In: Proceedings of the 2nd International Symposium Finite Volumes for Complex Applications, pp. 215–222. Hermès Duisberg (1999)

  4. Arbogast, T., Cowsar, L.C., Wheeler, M.F., Yotov, I.: Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37, 1295–1315 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Brezis, H., Lions, J.-L. (eds.) In: Nonlinear Partial Differential Equations and Their Applications, pp. 13–51. Pitman, NewYork (1994)

    Google Scholar 

  6. Douglas, J., Paes Leme, P.J.S., Giorgi, T.: Generalized Forchheimer flow in porous media. In: Lions, J.-L., Baiocchi, C. (eds.) Boundary Value Problems for Partial Differential Equations and Applications. Research Notes in Applied Mathematics, vol. 29, pp. 99–113. Masson, Paris (1993)

    Google Scholar 

  7. Douglas, Jr. J., Arbogast, T.: Dual porosity models for flow in naturally fractured reservoirs. In: Cushman, J.H. (ed.) Dynamics of Fluids in Hierarchial Porous Formations, pp. 177–221. Academic Press (1990)

  8. Douglas, J., Arbogast, T., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Erhel, J., Dreuzy, J.-R.: Efficient algorithms for the determination of the connected fracture network and the solution of the steady-state flow equation in fracture networks. Comput. Geosci. 29(1), 107–111 (2003)

    Article  Google Scholar 

  10. Faille, I., Flauraud, E., Nataf, F., Pegaz-Fiornet, S., Schneider, F., Willien, F.: A new fault model in geological basin modelling, application to nite volume scheme and domain decomposition methods. In: Herbin, R., Kroner, D. (eds.) In: Finite Volumes for Complex Applications III, pp. 543–550. Hermès Penton Sci. (2002)

  11. Forchheimer, P.: Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45, 1782–1788 (1901)

    Google Scholar 

  12. Frih, N., Roberts, J.E., Saâda, A.: Un modèle Darcy–Forchheimer pour un écoulement dans un milieu poreux fracturé. ARIMA 5, 129–143 (2006)

    Google Scholar 

  13. Giorgi, T.: Derivation of the Forchheimer law via matched asymptotic expansions. Trans. Porous Media 29, 191–206 (1997)

    Article  Google Scholar 

  14. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006)

    Article  Google Scholar 

  16. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Ciarlet et, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis 2. Finite Element Methods – part 1, pp. 523–639. Elsevier, North Holland (1991)

    Google Scholar 

  17. Panfilov, M., Fourar, M.: Physical splitting of nonlinear effects in high-velocity stable flow through porous media. Adv. Water Resour. 29, 30–41 (2006)

    Article  Google Scholar 

  18. Quateroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Claderon Press, Oxford (1999)

    Google Scholar 

  19. Wheeler, M.F., Yotov, I.: Multigrid on the interface for mortar mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 184, 287–302 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Witaker, S.: The Forchheimer equation: theoretical development. Trans. Porous Media 25, 27–61 (1996)

    Article  Google Scholar 

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Correspondence to Najla Frih.

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Frih, N., Roberts, J.E. & Saada, A. Modeling fractures as interfaces: a model for Forchheimer fractures. Comput Geosci 12, 91–104 (2008). https://doi.org/10.1007/s10596-007-9062-x

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  • DOI: https://doi.org/10.1007/s10596-007-9062-x

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