Abstract
Directional hydraulic fracturing (DHF) is more and more widely used in coal mines in China for hard roof and coal burst control. The key to this technology is to determine the crack initiation pressure that affected by the shape of the artificial notch and the stress state around the fracturing hole. Reasonable and simple formula for fracturing pressure calculation is essential since the fracturing pump used in coal mines is usually limited by the harsh conditions and hardly replaced once selected. Based on the superposition principle, the simplified 2D model of DHF was established as the elliptical hole with the internal pressure and solved by using the complex functions method. The analytical solution of tangential stress on the inner surface was obtained meanwhile the corresponding criterion of fracturing pressure can be set up. Considering the characteristics of DHF in coal mines, we further got a simplified formula that controlled by the ratio of major to minor axis of the ellipse-like notch, the ratio of the minimum to the maximum principal stress, as well as the tensile strength of the rock. The formula also gave a guide to the design of the notch that major diameter should be at least twice the minor diameter, and the optimal solution for the ratio is to 2～4 and recommended 4, which can resist the initiation pressure to a large extent affected by the in-situ stress. Once the pressure of the fracturing fluid is high enough to satisfy the equation cracks would arise at the tips of the notch along the major axis which belongs to mode Ⅰ crack and would grow unsteadily and rapidly. A PFC simulation model was used to verify the analysis, the results of which are very consistent with the theoretical solutions.