A Generalized Finite Element Method for hydro-mechanically coupled analysis of hydraulic fracturing problems using space-time variant enrichment functions

In computational simulations of hydraulic fracturing problems, consideration of interactions between the propagating fracture zone and the fluid flow through the porous material requires an appropriate up-scaling procedure from the spatial scale of the local crack, which usually is much smaller compared to the scale of typical finite elements in poromechanics problems. This scale transition refers to both the displacement field (discontinuity across cracks) as well as to the fluid flow (accelerated flow within cracks and the interaction with the flow in the bulk material). To resolve the small and the large scale portion of the solution, the Generalized Finite Element Method (GFEM) exploiting the partition of unity property of shape functions is used. Accordingly, the displacements u and the liquid pressure pl are locally enriched to better resolve their distribution in the vicinity of cracks by means of an additive decomposition into a large and a small scale part. As far as the representation of cracks is concerned, the Extended Finite Element Method (XFEM) is used by enriching the displacement field by means of a jump function as well as crack tip functions. In the framework of the GFEM physically motivated enrichment functions for the local enrichment of the C1 discontinuity of the liquid pressure field across cracks are proposed in the paper. The space and time variant analytical solutions obtained from the 1D transient response of saturated porous materials subjected to the liquid pressure within the crack are applied as enrichment functions to locally improve the approximation of the liquid pressure field at discontinuities. Applying these space and time variant functions, which are the exact solutions of the pressure field in the vicinity of cracks, as local enrichment functions lead to a significant improvement in the local approximation of the pressure field at discontinuities. The new GFEM model is formulated in a poromechanics framework for fully saturated porous materials. Representative analyses demonstrate the improvement of the solution quality compared to existing FEM and XFEM models.