  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.
