## 2 Overview of the Mine and Workings
### Determination of Key Parameters for Directional Hydraulic Fracturing
Here, \(N\) can be  
The total number of cracks can be obtained as Equation (9):  
\[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]0  
The number of fracturing segments per borehole is Equation (10):  
\[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]1  
The meanings of the variables involved in the above 10 equations are shown in \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]2  
The effects of the density of drilling holes and the number of fracturing segments on the fracture density and damage variables are shown in \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]3 Drill hole density refers to the number of drill holes per unit volume of coal rock, while fracture density represents the number of fractures per unit volume. As the density of drill holes and the number of fracturing segments increase, the fracture density and damage variables rise steadily. However, the rate of increase slows as the process continues. According to the characteristics of the lithology of the top plate of the Yanghuopan 3\({}^{-1}\) coal seam, the parameters of the directional hydraulic fracturing of the top plate were derived based on the above simulation and the calculation method of the hydraulic fracturing process parameters of the damage variables:  
1. Pre-fracture pressure: Controlled at 10-25 MPa. 2. Pre-fracture penetration radius: 15 m.
3. Fracturing time of single hole: 30 min. 4. Number of fracturing segments per borehole: 5-6. 5. Fracturing method: Transverse fracture. 6. The ratio of coal rock active water was determined according to the actual water quality coefficient of the coal seam at the site. 7. Pre-cracking volume: 0.7 m\({}^{3}\) for single hole, 1-2 m\({}^{3}\) cumulative. \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]4 Effect of drilling density and number of fracturing segments on fracture density and damage variables. Pre-cracking flow rate: according to experience, it should be 0.7-2.0 m\({}^{3}\)/h for dynamic pressure pre-cracking. **4. Directional Hydraulic Fracturing Fracture Evolution Law**  
_4.1. Establishment of RFPA 3D Numerical Simulation_  
Directional hydraulic fracturing is an important technical way to preform artificial fractures in the coal rock body, and then carry out high-pressure water injection fracturing to realize the structural modification and stress transfer of the coal rock body. The directional hydraulic fracturing with preformed cracks is one of the commonly used hydraulic fracturing methods in coal mines [18; 19]. The principle is shown in \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]5  
The fracture is first prefabricated in the direction specified by the borehole, and a stress concentration is generated at the end of the prefabricated fracture. When water is injected into the borehole, the hydraulic fracture crack preferentially expands along the direction of the wedge groove and generates pore water pressure \(P_{1}\) at the front end of the crack A. Combined effects of \(P_{1}\) and the stress field cause the hydraulic fracture crack to initiate and expand. The RFPA 3D numerical model of coal rock body is established to analyze the crack initiation and expansion law of the fracture cracks as shown in \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\]6 The drill hole is first set in the center of the model, and then the drill hole is fractured. The geometry of the numerical model is 20 m \(\times\) 20 m, the model is divided into 240 \(\times\) 240 cells, totaling 57,600. The diameter of the drill holes is 150 mm. The model type is a Moore Coulomb model. This information has been added to the manuscript. Displacement convergence criterion, stress convergence criterion, residual convergence criterion and maximum number of iterations convergence criterion were used during numerical simulation. The finite element method (FEM) and finite difference method (FDM) were used for the numerical simulation of the rock rupture process.