## 2 Overview of the Mine and Workings
### Determination of Key Parameters for Directional Hydraulic Fracturing  
1. Determination of the extent of the fracturing section Hydraulic fracturing changes the range of influence of the support stress from \(x_{\text{t}}\) to \(x_{\text{td}}\), so it is necessary to fracture all the coal rock body in the range of \(x_{\text{td}}\); thus, the range of fracturing \(L_{\text{h}}\) is determined as \(x_{\text{td}}\). Equation (1) is as follows: \[x_{\text{td}}=\frac{M}{2f}\left(\frac{\left(\frac{1+\sin\varphi}{1-\sin\varphi }\sigma_{3}+\frac{2C\cos\varphi}{1\sin\varphi}\right)(D_{a}-D_{b})}{\sigma_{3 \text{a}}-\sigma_{3\text{b}}}+1\right)\ln\left[\frac{K\gamma\gamma H}{N_{0}} \right]+\frac{M\beta}{2f}\ln K\] (1)
2. Determination of drilling angle The currently adopted prefabricated fracturing technique mainly involves cutting transverse slots in the radial direction of the borehole (the direction of the fracture is perpendicular to the axial direction of the borehole). Therefore, the angle of the drill hole can be determined as (90\({}^{\circ}-\theta\)), where \(\theta\) is the starting angle of the target hydraulic fracturing crack. 3. Determination of drilling spacing On the basis of the highly elliptical morphology of the hydraulic fracture seam, the displacement field relation (2) for the crack in the plane stress state according to the linear elastic fracture mechanics crack is as follows: \[V(x)=\frac{2\sigma}{E}\sqrt{a^{2}-\kappa^{2}}\] (2) where \(\sigma\) is the fluid pressure in the seam, \(E\) is the modulus of elasticity of the coal rock, and \(a\) is the length of the fracture. The established crack geometry is shown in Figure 6.  
Consider the fracturing crack as a synthesis of many I-shaped penetrating cracks arranged continuously in the direction of the seam length. Applying the formula to any crack leads to the seam width in Equation (3):  
\[W(x,z)=\frac{4\sigma_{x}}{E}\sqrt{H^{2}-z^{2}} \tag{3}\]  
Equation (4) is then obtained from the principle of conservation of volume:  
\[Q\cdot t=2\int_{0}^{L}\int_{0}^{H}2W(x,z)dzdx \tag{4}\]  
Figure 6: Geometry of cracks. In the formula, \(H\) is the seam height in the \(Z\) direction, which should be a variable, considering that the model will not be solved if it is analyzed as a variable, so it is considered that the seam heights are equal, and the simplified analysis can be obtained as Equation (5):  
\[Q\cdot t=4\int_{0}^{L}\left[\int_{0}^{H}\frac{4\sigma_{x}}{E}\sqrt{H^{2}-z^{2}} dz\right]dx \tag{5}\]  
Replacing \(\sigma_{(x)}\) with the mean value \(\sigma\) and organizing it gives the crack radius as Equation (6):  
\[a=\frac{QTE}{2\sigma\pi H^{2}} \tag{6}\]  
The spacing of the drilled holes is given in Equation (7):  
\[d=2L=\frac{QTE}{2\sigma\pi H^{2}} \tag{7}\]  
Taking \(Q=7.2\) m\({}^{3}\)/h, \(t=0.5\) h, \(E=5\) GPa, \(\sigma=35\) MPa, and \(H=5\) m, the parameters were substituted into the above equation. The effect of fracturing time \(T\) on the extension range of hydraulic fracturing cracks and damage variables was obtained as shown in Figure 7. It can be observed that as the fracturing time increases, the length of the hydraulic fractures exhibits linear growth, and the damage variable of the coal rock mass similarly increases in a linear manner. 4. Determination of the number of fracturing segments per borehole  
Assume that the volume of a fracturing area is \(V\), the width of the area is \(W\), the height is \(H\), and the depth of fracturing is \(L\). Therefore, \(V=W\times H\times L\). The number of drilled holes is \(m\), the number of fracturing segments per hole is \(n\), and the total number of directional fractures made is \(N\). Then, \(N=m\times n\).