A refined two-scale model for Newtonian and non-Newtonian fluids in fractured poroelastic media

A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance and the velocity profile in the integration points at the discontinuity is coupled to surrounding poroelastic material model in a two-scale approach. The resulting monolithic scheme allows for complex fluid behaviour to be included, which is demonstrated via the inclusion of the fluid inertia and the use of a Carreau fluid. The governing equations are discretised using T-splines, while the fracture is represented by spline-based interface elements, and an implicit time discretisation scheme is used. Mesh refinement studies are carried out for a typical case, which contains a pressurised and propagating fracture. A coarse macro-scale mesh is sufficient to obtain the correct propagation velocities, but a finer mesh is needed to prevent pressure oscillations. Time step refinement studies show the capabilities of the model to capture pressure waves within the fracture. Finally, it is shown that if inertial effects are limited and a Newtonian fluid is used, a fracture scale discretisation using a single element is sufficient. However, for a Carreau fluid or when the problem is inertia-dominated, smaller elements are needed to correctly represent the velocity profile.