At present,finite element methods have been widely applied to solving a variety of engineering problems.In the fields of petroleum engineering,researchers have been using numerical simulation methods to probe the failure mechanisms of reservoir rocks subject to hydraulic stimulation for exploitation of unconventional oil and gas resources.Prediction of hydraulic fracturing initiation and propagation in low-permeability reservoirs is of great significance to enhancing recovery of unconventional gas resources.In this paper,we incorporated finite element methods(FEM)and linear elastic fracture mechanics(LEFM)to investigate the initiation and propagation behaviors of hydraulic fractures in reservoir rocks subject to various perforation conditions and geo stresses.The stress intensity factors(SIFs),definition of kink angle and fitting of crack front were taken into account in the computation.The stress intensity factors around crack front were calculated by M-integral which is the evolution of the J-integral.To accurately compute the SIFs,three basic measures,i.e.,singular element embedded in crack tip,sufficient mesh density in crack area,and symmetric mesh in crack front,were adopted to reduce discrete errors.The criteria,such as,max tensile stress,max energy release rate,principle of local symmetry and min strain energy density which tend to minimize KII can be used for the definition of the kink angle at crack front.Various polynomial and spline fitting curves were adopted to fit the predicted extension points at crack tip.The simulation results suggest that the three-dimensional fracture initiation and propagation of reservoir rocks subject to complex geo stresses and hydraulic perforation can be well simulated by FRANC3D and ANSYS.Increase of horizontal stress ratio leads to the decrease of initiation fracture pressure.Perforation angle parallel to the maximum horizontal stress has the smallest initiation fracture pressure while the propagation plane is perpendicular to the direction of the minimum horizontal stress.The study indicates that linear perforation distribution is superior to symmetric distribution and staggered distribution as it contributes the minimum initiation fracture pressure.