Abstract
This article deals with the discretization of hybrid dimensional model of Darcy flow in fractured porous media. These models couple the flow in the fractures represented as the surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of Gradient schemes which accounts for a large family of conforming and nonconforming discretizations. The Vertex Approximate Gradient (VAG) scheme and the Hybrid Finite Volume (HFV) scheme are applied to such models and are shown to verify the Gradient scheme framework. Our theoretical results are confirmed by a few numerical experiments performed both on tetrahedral and hexahedral meshes in heterogeneous isotropic and anisotropic media.
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References
Alboin, C., Jaffré, J., Roberts, J., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. Contemp. Math. 295, 13–24 (2002)
Brenner, K., Groza, M., Guichard, C., Masson, R.: Vertex approximate gradient scheme for hybrid dimensional two-phase darcy flows in fractured porous media. In: Proceedings of FVCA 7 (2014)
Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive flows in porous media. ESAIM: M2AN 46, 265–290 (2010)
Eymard, R., Herbin, R., Gallouet, T.: Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30, 1009–1043 (2010). doi:10.1093/imanum/drn084
Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resources 29, 1020–1036 (2006)
Tunc, X., Faille, I., Gallouet, T., Cacas, M.C., Havé, P.: A model for conductive faults with non matching grids. Comput. Geosci. 16, 277–296 (2012)
Acknowledgments
The authors would like to thank GDFSuez EP and Storengy for partially supporting this work, and Robert Eymard for fruitful discussions during the elaboration of this work.
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© 2014 Springer International Publishing Switzerland
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Brenner, K., Groza, M., Guichard, C., Lebeau, G., Masson, R. (2014). Gradient Discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics & Statistics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-05591-6_52
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DOI: https://doi.org/10.1007/978-3-319-05591-6_52
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