Asymptotic and numerical modelling of flows in fractured porous media

Philippe Angot; Franck Boyer; Florence Hubert

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 2, page 239-275
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Angot, Philippe, Boyer, Franck, and Hubert, Florence. "Asymptotic and numerical modelling of flows in fractured porous media." ESAIM: Mathematical Modelling and Numerical Analysis 43.2 (2009): 239-275. <http://eudml.org/doc/250659>.

@article{Angot2009,
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. },
author = {Angot, Philippe, Boyer, Franck, Hubert, Florence},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Fractured porous media; Darcy flow; finite volume method; asymptotic models of flow.; global solvability; cell-centered finite volume scheme; convergence; double-permeability model},
language = {eng},
month = {2},
number = {2},
pages = {239-275},
publisher = {EDP Sciences},
title = {Asymptotic and numerical modelling of flows in fractured porous media},
url = {http://eudml.org/doc/250659},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Angot, Philippe
AU - Boyer, Franck
AU - Hubert, Florence
TI - Asymptotic and numerical modelling of flows in fractured porous media
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/2//
PB - EDP Sciences
VL - 43
IS - 2
SP - 239
EP - 275
AB - 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.
LA - eng
KW - Fractured porous media; Darcy flow; finite volume method; asymptotic models of flow.; global solvability; cell-centered finite volume scheme; convergence; double-permeability model
UR - http://eudml.org/doc/250659
ER -

References

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Citations in EuDML Documents

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  1. Luca Formaggia, Alessio Fumagalli, Anna Scotti, Paolo Ruffo, A reduced model for Darcy’s problem in networks of fractures
  2. Carlo D’Angelo, Anna Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids
  3. Peter Knabner, Jean E. Roberts, Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture
  4. Carlo D’Angelo, Anna Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

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