Unequal order T-spline meshes for fracture in poroelastic media

Spline-based meshes allow for a higher inter-element continuity. For coupled problems, e.g. poroelasticity, different meshes with different orders of interpolation are normally used for the various fields in order to avoid spurious oscillations. When including discontinuities in these meshes, there exist several options for the discretisation. Herein we will discuss two options which use T-splines, one aiming at a minimum number of degrees of freedom around the crack tip, the other trying to maximise this number. Both meshes retain a higher-order continuity along the fracture, but the mesh which maximises the number of degrees of freedom mesh introduces two additional degrees of freedom around the crack tip to allow for a sharper crack. The two discretisations are used to simulate a pressurised fracture inside a poroelastic material and the results are compared to results obtained using a Non-Uniform Rational B-Spline (NURBS) mesh. A comparison between the two discretisations shows the effect of including additional degrees of freedom close to the crack tip. However, both meshes yield similar results further away from the crack tip. It is shown that both T-spline meshes capture a fully closed discontinuity at the fracture tip, whereas the NURBS mesh retains a small opening due to the discontinuity which exists for the cracked as well as the intact elements. A fully closed fracture aperture results in T-splines with a lower discontinuity pressure compared to NURBS, making T-splines more suitable for simulations in which the fracture propagation is limited by the fluid transport within the fracture.