Integrating the 3D-GMTSN criterion with XFEM to simulate the spatial propagation trajectory of hydraulic fractures interacting with natural fractures

The accurate prediction of the spatial propagation trajectory of hydraulic fractures (HFs) that interact with natural fractures (NFs) remains a fundamental challenge in hydraulic fracturing engineering. A notable challenge arises from the formation of complex three-dimensional (3D) fracture networks at the intersections of HFs and NFs, and precise 3D fracture criteria and effective simulation techniques are lacking. In research conducted previously, the 3D-generalized maximum tangential strain (3D-GMTSN) criterion was developed to accurately predict both the direction and onset of 3D fracture initiation. In this study, the 3D-GMTSN criterion was first systematically integrated with the extended finite element method (XFEM) to simulate the spatial propagation trajectory of 3D fractures. To overcome numerical challenges associated with 3D fracture intersections, a redundant enrichment node removal technique to mitigate stiffness matrix singularity, and a coupled element-by-element approach with the conjugate gradient method that eliminates explicit global stiffness matrix assembly. This framework was further enhanced using OpenMP-based parallelization, achieving substantial computational efficiency gains for large-scale engineering simulations. Finally, the spatial propagation trajectory of the interaction between HFs and NFs was successfully simulated. This work establishes an integrated framework combining physics-driven criteria, robust numerical algorithms, and high-performance parallel computing. The numerical model used in this study can effectively simulate complex fracture interactions, thereby overcoming a notable challenge in hydraulic fracturing engineering.