We study mathematical models and their finite element approximations for solving the coupled problem arising in the interaction between fluid in a poroelastic material and fluid in a fracture. The fluid flow in the fracture is governed by the Stokes/Brinkman equations, while the poroelastic material is modeled using the Biot system. We present several approaches to impose the continuity of normal flux, including an interior penalty method and a Lagrange multiplier method. A dimensionally reduced fracture model based on averaging the equations over the cross-sections will also be presented.
Stability, accuracy, and robustness of the methods will be discussed.