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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. angot@cmi.univ-mrs.fr; fboyer@cmi.univ-mrs.fr; fhubert@cmi.univ-mrs.fr
Franck Boyer
Affiliation:
Université de Provence and CNRS, Laboratoire d'Analyse Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. angot@cmi.univ-mrs.fr; fboyer@cmi.univ-mrs.fr; fhubert@cmi.univ-mrs.fr
Florence Hubert
Affiliation:
Université de Provence and CNRS, Laboratoire d'Analyse Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. angot@cmi.univ-mrs.fr; fboyer@cmi.univ-mrs.fr; fhubert@cmi.univ-mrs.fr
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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 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.

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

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