Enhanced geothermal system (EGS) is an effective method for developing and utilizing hot dry rock (HDR). The key to the effectiveness of EGS is the construction of an artificial fracture network. The permeability of fractures has severe effects on the heat transfer efficiency and sustainability of geothermal energy. However, the evolution characteristics of hydraulic conductivity under different failure modes have not been adequately studied for HDR. To clarify this, rocks with different failure modes were investigated by conducting thermal triaxial compression experiments, and the fluid seepage related to different rock failure modes was comprehensively investigated. The results showed that the characteristic stresses and crack surface roughness of the rock increased as the confining pressure increased. The permeability in the composited failure mode was the largest (11.4 μm2), followed by that in the Y-shaped shear failure mode (9.7 μm2), and that in the single-shear failure mode was the smallest (7.2 μm2). The confining pressures had an inhibitory effect on permeability. As the confining pressure increased from 5 to 30 MPa, the permeability decreased by 88.8%, 88.4%, and 89.9%, respectively. In contrast, the permeability was significantly enhanced by 128.3%, 94.6%, and 131% as the flow rate increased from 3 to 7 mL/min.