Dimensional model reduction for flow through fractures in poroelastic media

¹ Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Indiana 46556, USA.
² Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
³ MOX, Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy.
⁴ Department of Mechanical Engineer.ing and Materials Science, University of Pittsburgh, PA 15260, USA.
paolo.zunino@polimi.it

Received: 21 October 2015
Accepted: 7 November 2016

Abstract

We study the interaction between a poroelastic medium and a fracture filled with fluid. The flow in the fracture is described by the Brinkman equations for an incompressible fluid and the poroelastic medium by the quasi-static Biot model. The two models are fully coupled via the kinematic and dynamic conditions. The Brinkman equations are then averaged over the cross-sections, giving rise to a reduced flow model on the fracture midline. We derive suitable interface and closure conditions between the Biot system and the dimensionally reduced Brinkman model that guarantee solvability of the resulting coupled problem. We design and analyze a numerical discretization scheme based on finite elements in space and the Backward Euler in time, and perform numerical experiments to compare the behavior of the reduced model to the full-dimensional formulation and study the response of the model with respect to its parameters.