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Mathematical analysis of a discrete fracture model couplingDarcy 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.. [email protected]
Jean E. Roberts
Affiliation:
Inria Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France.; [email protected]
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Abstract

We consider a model for flow in a porous medium with a fracture in which the flow in thefracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed byDarcy’s law. We give an appropriate mixed, variational formulation and show existence anduniqueness of the solution. To show existence we give an analogous formulation for themodel in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for the model withDarcy’s law in the matrix is the weak limit of solutions of the model with theDarcy−Forchheimerlaw in theentire 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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