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

Published online by Cambridge University Press:  13 August 2014

Peter Knabner  and
Jean E. Roberts
Peter Knabner
Affiliation:
University of Erlangen-Nuremberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany.. knabner@am.uni-erlangen.de
Jean E. Roberts
Affiliation:
Inria Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France.; jean.roberts@inria.fr
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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.

Keywords

Flow in porous mediafracturesDarcy−Forchheimerflowsolvabilityregularizationmonotone operators
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1451 - 1472
Copyright
© EDP Sciences, SMAI 2014

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