Abstract: This study aims to investigate the crack propagation mechanism of directional hydraulic fracturing of coal mine roof and explore the relationship between fracture criteria. By drawing on fracture mechanics, we propose a hydraulic fracturing mechanical model of drilling and derive the calculation formulas of crack initiation angle and critical water pressure under different fracture criteria to analyze the impact of different variables on crack initiation angle and critical water pressure. Results show that: ① the increase of stress difference leads to a gradual increase of the crack initiation angle; ② the effect of prefab crack length and borehole radius on crack initiation angle is caused by two factors. The crack initiation angle increases with the decrease in the ratio of these two factors; ③ Poisson's ratio has limited effect on crack initiation angle; ④ under high stress difference and low prefab crack inclination angle, the critical water pressure decreases with the increase of stress difference and inclination angle, and vice versa; ⑤ the increase of borehole radius and prefabricated crack length would significantly reduce the critical water pressure. The two criteria yield different calculation results as ① prefab crack length and borehole radius would impact crack initiation angle; ② principal stress difference and prefab crack angle would impact critical water pressure. The theoretical calculation results and the experimental results show a good fit. The present study offers reference to the hydraulic fracturing site construction.