## 2 Materials and Methods
### Selection Equation of Deep Coalbed Directional Hydraulic Splitting Angle
#### 2.1.2 An Angle Selection Equation for the Directional Hydraulic Splitting of Deep Coalbeds  
As can be seen from the analysis of Figure 1, when the splitting piston extends outward, it acts on the coal wall around the borehole and causes the coal to be rigidly displaced. In this process, only the splitting piston plays a role, so the splitting process is simplified as shown in Figure 2. When the hydraulic oil is introduced, the splitting piston extends out, exerting stress on the coal wall. When the stress exceeds the bearing limit, the coal body breaks up. As the crack develops, a coal damage area is generated around the crack, and a disturbed area is formed at the periphery. It can be seen from Figure 2 that the angle between the two cracks \(\alpha\) is related to the width of the damaged area \(R\) and the length of the crack \(H\). Suppose the angle between the two split pistons is the ratio of the width of the damaged area to the length of the crack, then:  
\[\tan\alpha=\frac{R}{H} \tag{1}\]  
Figure 1: Schematic of the cracked coal in the coalbed of the new hydraulic directional drilling splitting device. Figure 2: Schematic diagram of hydraulic directional splitting damage crack propagation model. The analysis of the above equation shows that if the expressions of damaged area width and crack length can be solved, the expressions of splitting pinch angle can also be solved in order to get the selection equation of deep coalbed directional hydraulic splitting pinch angle. First, the damaged area width was explored, and the damaged area width depends on the damaged area boundary range, so the problem was transformed into solving the damaged area boundary range. Damaged area is the pressure exceeding the ultimate bearing capacity of the material, so that the local deformation of the material is not in recoverable yield region. To judge whether the material in the region yields, the criterion often used is the Mises criterion [22], The expression is:  
\[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2}+ \left(\sigma_{3}-\sigma_{1}\right)^{2}=2\sigma_{s}^{2} \tag{2}\]  
Among them, \(\sigma_{s}\)--principal stress and \(\sigma_{1}\sigma_{2}\sigma_{3}\)--component stress. The Mises yield criterion is represented by the principal stress, so the stress at the crack front is represented by the principal stress. If the stress component is known and the principal stress is \(\sigma\), the following determinant is made zero [23]. \[ \begin{bmatrix}\left(\sigma_{x}-\sigma\right)&\tau_{xy}\\ \tau_{yx}&\left(\sigma_{y}-\sigma\right)\end{bmatrix} =0 \tag{3}\]  
Solving this formula, we obtain:  
\[\sigma=\frac{\sigma_{x}+\sigma_{y}}{2}\pm\sqrt{\tau_{xy}^{2}+\frac{1}{2}{ \left(\sigma_{x}+\sigma_{y}\right)}^{2}} \tag{4}\]  
The component stresses are:  
\[\left\{ \begin{array}{l}\sigma_{1}=\frac{\sigma_{x}+\sigma_{y}}{2}+\sqrt{ \tau_{xy}^{2}+\frac{1}{2}{\left(\sigma_{x}-\sigma_{y}\right)}^{2}}\\ \sigma_{2}=\frac{\sigma_{x}+\sigma_{y}}{2}-\sqrt{\tau_{xy}^{2}+\frac{1}{2}{ \left(\sigma_{x}-\sigma_{y}\right)}^{2}}\\ \sigma_{3}=\mu(\sigma_{1}+\sigma_{2})\end{array} \right. \tag{5}\]  
Substitute the stress component formula near the crack tip into Equation (5) to obtain the principal stress at any position near the crack tip as [24]:  
\[\left\{ \begin{array}{l}\sigma_{1}=\frac{K_{\rm I}}{\sqrt{2\pi}}\cos\frac{ \theta}{2}(1+\sin\frac{\theta}{2})\\ \sigma_{2}=\frac{K_{\rm I}}{\sqrt{2\pi}}\cos\frac{\theta}{2}(1-\sin\frac{ \theta}{2})\\ \sigma_{3}=\frac{2\mu K_{\rm I}}{\sqrt{2\pi}}\cos\frac{\theta}{2}\end{array} \right.