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

摘要

We consider a model for flow in a porous medium with a fracture in which theflow in the fracture is governed by the Darcy-Forchheimer law while that in thesurrounding matrix is governed by Darcy's law. We give an appropriate mixed,variational formulation and show existence and uniqueness of the solution. Toshow existence we give an analogous formulation for the model in which theDarcy-Forchheimer law is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for themodel with Darcy's law in the matrix is the weak limit of solutions of themodel with the Darcy-Forchheimer law in the entire domain when the Forchheimercoefficient in the matrix tends toward zero.