The stress sensitivity of coal seams with different fracture structures affects gas migration characteristics in the fracture zone. In this study, permeability stress experiments for intact and persistent fracture coal samples are carried out. A discrete element method numerical simulation that adopts the changing joint stiffness method is proposed. Based on the experimental data, the corresponding parameters of persistent fractures and elastic joints for fluid-solid coupling simulations are determined. Subsequently, the internal influence mechanisms of fracture morphology on axial and confining stress sensitivities are determined. The simulation results show that sectional fractures with different dip angles in the persistent fracture control the sensitivity to axial and confining stresses. The horizontal segment fracture aperture decreases with an increase in axial stress, which reduces its flow capacity. This results in a decrease in the flow rates in the persistent fracture. When the confining stress increases, the vertical fracture aperture is significantly reduced, considerably reducing the flow rate in the persistent fracture.