The behavior of hydraulic fractures initiated along the radial and axial directions of borehole has been investigated by the author in previous hydraulic fracturing experimental and engineering applications. Based on this, and considering the stress fields at the tips of fractures, a mechanical model of the progressive initiation and propagation of radial and axial hydraulic fractures in the borehole is established in this paper. Subsequently, based on the mechanical model, criteria for the progressive initiation and propagation of radial and axial hydraulic fractures are postulated. According to the criteria, the axial fracture begins at the midpoint of the radial fracture. When the second and third principal stresses near the axial fracture tip are equal, the axial fracture begins to turn. Finally, through an actual triaxial fracturing experiment, the injection pressure curve and internal fracture shape of a test block were obtained, which are the discrimination factors of the criteria. The fracture initiation pressure of the radial and axial fractures of the test block satisfied the fracture initiation condition of the criteria. The accuracy of the criteria in predicting the turning position of the axial fracture was 89.6% in comparison to the experimental data. Radial and axial fractures can propagate horizontally and longitudinally on the hard roof to fully break it, which provides a new idea and essential theoretical basis for the control of the end hanging roof of coal mining face. [ABSTRACT FROM AUTHOR]