## 3 Influence Factor Analysis for Water Jet Slotting Depth
### Rotation Speed and Slotting Time  
The water jet slotting in a coal seam is different from conventional moving water jet cutting. First, the nozzles are fixed to the drill bit with their axes perpendicular to the drill bit axis. Then, in the process of water jet slotting, the nozzles rotate along with the drill pipe, and water jets generated by the nozzle impact coal with a rotating trajectory. Finally, a disc-shaped slot is formed. Therefore, the effects of the rotation speed and slotting time on the slotting depth are essentially the effects of the translation  
Figure 4: Comparison of cutting depth subjected to high-pressure water jet with different jet pressures.  
Figure 5: The effect of jet pressure on cutting depth.  
speed and cutting times on the cutting depth, respectively. The relationship between the rotation speed and the translation speed is expressed as follows:  
\[n=\frac{30v_{T}}{\pi r} \tag{9}\]  
\[t=\frac{J}{n}, \tag{10}\]  
where \(n\) and \(t\) are the rotation speed of the drill pipe and slotting time, respectively; \(v_{T}\) and \(J\) are the translation speed and cutting times of the water jet, respectively; and \(r\) is the initial standoff distance of the water jet; that is \(r=(D_{0}-d_{0})/2\), where \(D_{0}\) and \(d_{0}\) are the borehole diameter and outer diameter of the slotting device, respectively.  
In the cutting experiments at different translation speeds, the nozzle diameter, jet pressure, and standoff distance were 3 mm, 15 MPa, and 165 mm, respectively. The experimental results are presented in Figure 6. The cutting depth decreased as a power function with the increase of the jet translation speed. Additionally, the decrease became progressively smaller. When the translation speed was 60 mm/s, the sample was completely cut through and the cutting depth was more than 200 mm, as shown in Figure 7. This is attributed to the fact that the initial damage of the rock sample occurred in a very short period of a few milliseconds, when the water jet impacted the rock. Subsequently, the impact depth gradually increased. However, when the impact time was excessively long, the high-pressure water accumulated in the slot and formed a water cushion, which weakened the jet impacting capacity. Hence, the cutting depth growth rate decreased as the impact time increased. Equation (11) describes the relationship between the cutting depth and the jet translation speed, as follows:  
\[h_{v_{T}}=k_{2}v_{T}{}^{\alpha_{1}}, \tag{11}\]  
where \(h_{vT}\) is the cutting depth at different translation speeds; \(k_{2}\) is the proportionality coefficient; \(\alpha_{1}\) is the index.  
Figure 6: The effect of translation speed on cutting depth.  
The deep slot resulted from multiple high-pressure water jet cuts. Therefore, it was necessary to investigate the effect of cutting times on the total cutting depth. With regard to the cutting times, previous studies have found that the first few cuttings play a major role in jet cutting, and the number of cuttings has a significant impact on the total depth [31]. When the number of cuts increases, the cutting depth slowly increases, but the increment is small. When the cutting number reaches a certain value, the cutting depth is no longer affected. The relationship between the total cutting depth and the cutting times is given in Figure 8. Thus, the relationship between the total cutting depth and the first cutting depth can be expressed as follows:  
\[h_{J-n}=h_{1-nJ}\alpha_{2}, \tag{12}\]  
where \(h_{J-n}\) is the total cutting depth at different cutting times; \(h_{J-1}\) is the cutting depth when the water jet cuts the sample once; \(\alpha_{2}\) is the index.  
Figure 8: The effect of cutting times on cutting depth [31].  
Figure 7: Cutting depths from a high-pressure water jet with different translation speeds.