## 4 Theoretical Solutions of the Surround Rock Plastic Zone of the Borehole
### The Influence of Coal Strength on borehole Surrounding Rock Permeability-Increasing Circle  
The distribution pattern and range of plastic zone in the borehole surrounding rock are directly related to the strength of the coal body. As the main parameters to characterize its strength, cohesion and internal friction angle of the coal body must become the key factors affecting the permeability radius of surrounding rock. The relationship between the distribution of the plastic zone of the borehole surrounding rock and the cohesion and internal friction angle of the coal body is shown in Figures 6 and 7. The relationship between permeability-increasing radius and coal cohesion and internal friction angle is shown in Figures 8 and 9.  
Figure 4: Plastic zone shape and gas permeability-increasing circle of surrounding rock of boreholes under different confining pressure ratios. (**a**) Round (**b**) Oval (**c**) Butterfly.  
Figure 5: Confining pressure ratio and permeability-increasing radius of borehole surrounding Rock.  
Figure 8: Permeability-increasing radius and cohesive force of borehole surrounding rock.  
Figure 6: Distribution patterns of plastic zones in surrounding rock of boreholes under different cohesion. (**a**) Round (**b**) Oval (**c**) Butterfly.  
Figure 7: Distribution patterns of plastic zones in surrounding rock of boreholes under different internal friction angles. (**a**) Round (**b**) Oval (**c**) Butterfly.  
When the plastic zone is circular or elliptical, the impact of cohesion or internal friction angle on its distribution is generally small. The capacity of the plastic zone to keep its original form rapidly reduces with increasing cohesion or internal friction angle, and it eventually approaches the empty surface of the borehole's surrounding rock. As illustrated in Figure 6a,b and Figure 7a,b. The four butterfly leaves deteriorate fast when the plastic zone is butterfly-shaped. The smaller the cohesiveness or internal friction angle, the more sensitive to the degradation rate of the four butterfly leaves in the plastic zone. As illustrated in Figures 6c and 7c.  
From the analysis of Figures 8 and 9, it is found that when the plastic zone is circular or elliptical, the sensitivity of the permeability-increasing radius to the cohesion or internal friction angle is relatively weak. Additionally, it decreases approximately linearly with the increase of the two. When the cohesion and internal friction angle increase to a certain value, the permeability-increasing effect of the plastic zone is almost zero. For the butterfly plastic zone, when the cohesion and internal friction angle is small, the permeability-increasing radius is very sensitive to the two and increases nonlinearly with the decrease of the two. It's very beneficial to gas drainage. As the cohesion and internal friction angle continue to increase, the sensitivity of the permeability-increasing radius to the two is gradually weakened. The permeability-increasing radius is continuously reduced. The range of gas flow channels is gradually reduced. The surrounding rock gradually loses the permeability-increasing effect.  
Therefore, the increase in coal strength has a certain inhibitory effect on the formation of gas flow channels. Theoretically proves the fact that a hard coal body is extremely unfavorable to efficient gas extraction.