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.
Mots clés : flow in porous media, fractures, Darcy−Forchheimerflow, solvability, regularization, monotone operators
@article{M2AN_2014__48_5_1451_0,
author = {Knabner, Peter and Roberts, Jean E.},
title = {Mathematical analysis of a discrete fracture model coupling {Darcy} flow in the matrix with {Darcy-Forchheimer} flow in the fracture},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1451--1472},
publisher = {EDP-Sciences},
volume = {48},
number = {5},
year = {2014},
doi = {10.1051/m2an/2014003},
mrnumber = {3264361},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2014003/}
}
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 PY - 2014 SP - 1451 EP - 1472 VL - 48 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2014003/ DO - 10.1051/m2an/2014003 LA - en ID - M2AN_2014__48_5_1451_0 ER -
%0 Journal Article %A Knabner, Peter %A Roberts, Jean E. %T Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy-Forchheimer flow in the fracture %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1451-1472 %V 48 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2014003/ %R 10.1051/m2an/2014003 %G en %F M2AN_2014__48_5_1451_0
Knabner, Peter; 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 , Tome 48 (2014) no. 5, pp. 1451-1472. doi : 10.1051/m2an/2014003. http://archive.numdam.org/articles/10.1051/m2an/2014003/
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