Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

TL;DR

This paper introduces a novel finite volume discretization method for fractured poroelastic media with contact mechanics, effectively handling nonlinear inequalities and coupling deformation, contact, and fluid pressure in 2D and 3D.

Contribution

It combines Lagrange multipliers with finite volume discretization to directly impose inequality constraints on fracture faces, improving modeling of contact and fluid flow.

Findings

Convergence rates are maintained across complex geometries.

The method effectively couples deformation, contact, and fluid pressure.

The approach is validated in 2D and 3D simulations.

Abstract

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law while slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complimentary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2d and 3d, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.