“…Entering the 21st century, with the rapid development of computer technology and numerical calculation methods, numerical simulation methods have become an important tool in engineering practice and become an important method in the study of shale reservoir fracturing engineering. Currently, the numerical simulation methods used for the study of hydraulic fracture propagation mechanism in the shale fracturing process are finite element method, extended finite element method, discrete element method, boundary element method, phase field method, and other methods [2,13,14,32,63,[69][70][71][82][83][84]. Among them, the first four are commonly used methods, while the phase field method is an emerging method that needs to be further studied, and based on these numerical methods, some models were established, such as the finite element model, extended finite element model, discrete fracture model, displacement discontinuity model, unconventional fracture model, etc.…”