Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture

Peter Knabner; Jean E. Roberts

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 5, page 1451-1472
  • ISSN: 0764-583X

Abstract

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We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed by Darcy’s law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy’s law in the matrix is the weak limit of solutions of the model with the Darcy−Forchheimerlaw in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.

How to cite

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Knabner, Peter, and Roberts, Jean E.. "Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1451-1472. <http://eudml.org/doc/273308>.

@article{Knabner2014,
abstract = {We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed by Darcy’s law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy’s law in the matrix is the weak limit of solutions of the model with the Darcy−Forchheimerlaw in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.},
author = {Knabner, Peter, Roberts, Jean E.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {flow in porous media; fractures; Darcy−Forchheimerflow; solvability; regularization; monotone operators; Darcy-Forchheimer flow},
language = {eng},
number = {5},
pages = {1451-1472},
publisher = {EDP-Sciences},
title = {Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture},
url = {http://eudml.org/doc/273308},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Knabner, Peter
AU - Roberts, Jean E.
TI - Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1451
EP - 1472
AB - We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed by Darcy’s law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy’s law in the matrix is the weak limit of solutions of the model with the Darcy−Forchheimerlaw in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.
LA - eng
KW - flow in porous media; fractures; Darcy−Forchheimerflow; solvability; regularization; monotone operators; Darcy-Forchheimer flow
UR - http://eudml.org/doc/273308
ER -

References

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