Asymptotic and numerical modelling of flows in fractured porous media
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) no. 2, p. 239-275
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 fault interfaces and the model finally consists in a coupling between a 2D 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 interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps 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 flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.
DOI : https://doi.org/10.1051/m2an/2008052
Classification:  76S05,  74S10,  35J25,  35J20,  65N15
@article{M2AN_2009__43_2_239_0,
     author = {Angot, Philippe and Boyer, Franck and Hubert, Florence},
     title = {Asymptotic and numerical modelling of flows in fractured porous media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     pages = {239-275},
     doi = {10.1051/m2an/2008052},
     zbl = {1171.76055},
     mrnumber = {2512496},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_2_239_0}
}
Angot, Philippe; Boyer, Franck; Hubert, Florence. Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) no. 2, pp. 239-275. doi : 10.1051/m2an/2008052. http://www.numdam.org/item/M2AN_2009__43_2_239_0/

[1] P.M. Adler and J.-F. Thovert, Fractures and Fracture Networks. Kluwer Acad., Amsterdam (1999).

[2] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D-meshes. Numer. Methods Partial Differential Equations 23 (2007) 145-195. | MR 2275464 | Zbl 1111.65101

[3] 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. | MR 1786657 | Zbl 0980.65129

[4] P. Angot, A model of fracture for elliptic problems with flux and solution jumps. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003) 425-430. | MR 2015088 | Zbl 1113.76455

[5] 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. | MR 2062141 | Zbl 1052.65557

[6] J. Bear, C.-F. Tsang and G. De Marsily, Flow and Contaminant Transport in Fractured Rock. Academic Press, San Diego (1993).

[7] B. Berkowitz, Characterizing flow and transport in fractured geological media: A review. Adv. Water Resour. 25 (2002) 861-884.

[8] C. Bernardi, M. Dauge and Y. Maday, Compatibilité de traces aux arêtes et coins d'un polyhèdre. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 679-684. | MR 1797751 | Zbl 0976.52008

[9] C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. (2007) http://hal.archives-ouvertes.fr/hal-00153795.

[10] 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).

[11] F. Boyer and F. Hubert, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032-3070. | MR 2439501 | Zbl pre05653755

[12] Y. Caillabet, P. Fabrie, P. Landereau, B. Noetinger and M. Quintard, 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. | MR 1740139 | Zbl 0969.76051

[13] Y. Caillabet, P. Fabrie, D. Lasseux and M. Quintard, Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method. Comput. Geosci. 5 (2001) 121-150. | MR 1878560 | Zbl 0988.86002

[14] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203-1249. | Numdam | MR 2195910 | Zbl 1086.65108

[15] R. Eymard and T. Gallouët, H-convergence and numerical schemes for elliptic equations. SIAM J. Numer. Anal. 41 (2003) 539-562. | MR 2004187 | Zbl 1049.35015

[16] 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. | MR 1804748 | Zbl 0981.65095

[17] 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. | MR 2008978 | Zbl 1055.86001

[18] 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).

[19] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, Boston (1985). | MR 775683 | Zbl 0695.35060

[20] F. Hermeline, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Engrg. 192 (2003) 1939-1959. | MR 1980752 | Zbl 1037.65118

[21] 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. | MR 2007435 | Zbl 1177.76231

[22] V. Martin, J. Jaffré and J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667-1691. | MR 2142590 | Zbl 1083.76058

[23] V. Mityushev and P.M. Adler, Darcy flow arround a two dimensional lense. Journal Phys. A: Math. Gen. 39 (2006) 3545-3560. | MR 2219997 | Zbl 1086.76068

[24] V. Reichenberger, H. Jakobs, P. Bastian and R. Helmig, A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29 (2006) 1020-1036.