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, Volume 48 (2014) no. 5, p. 1451-1472

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.

DOI : https://doi.org/10.1051/m2an/2014003
Classification:  35J60,  76S05
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {5},
year = {2014},
pages = {1451-1472},
doi = {10.1051/m2an/2014003},
mrnumber = {3264361},
language = {en},
url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, pp. 1451-1472. doi : 10.1051/m2an/2014003. http://www.numdam.org/item/M2AN_2014__48_5_1451_0/

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