# Evolution of Permeability and Sensitivity Analysis of Gas-Bearing Coal under Cyclic Dynamic Loading
## 5 Sensitivity Analysis of Permeability to Cyclic Loading
### Sensitivity Analysis of Cyclic Load  
The results of the calculation for the variation rate of coal body permeability under different cyclic dynamic loading frequencies are presented in Table 3. When the static loading stage is constant, the permeability variation rate of the coal body decreases as the adsorption amount increases, although it remains positive throughout. This indicates that cyclic dynamic loading increases the permeability of the coal body, but the extent of this increase diminishes as the adsorption amount grows. When the adsorption amount is constant, the variation rate of coal body permeability increases with the increase in the static loading stage, with the maximum increase reaching 75.2%. Hence, it is evident that coal body permeability is highly sensitive to cyclic dynamic loading, with a greater sensitivity associated with larger static loading stages. Conversely, higher adsorption amounts lead to reduced sensitivity. Substitute Equation (1) into (3) to derive the relationship between the coefficient of dynamic sensitivity of permeability \(C_{\text{\tiny{P}}}\) and the dynamic loading frequency \(\
u\). \[C_{\text{\tiny{P}}}=a_{1}e^{b_{1}\
u}\]  
where \(a_{1}\) and \(b_{1}\) are both fitting constants. The value of \(a_{1}\) is related to the magnitude of coal permeability. \(b_{1}\) reflects the rate of change in the dynamic loading sensitivity coefficient, with a greater value of \(b_{1}\) indicating a faster rate of change in the dynamic loading sensitivity factor. The relationship between the dynamic loading sensitivity coefficient and the dynamic loading frequency also follows an exponential function, with a greater dynamic loading frequency resulting in a higher sensitivity coefficient. The calculated results are presented in Table 4.  
The coefficient of the dynamic load sensitivity of permeability exhibits a pattern similar to that of the rate of change in permeability. The permeability of the coal body shows a pronounced sensitivity to dynamic load frequency, with this sensitivity diminishing as the adsorbed gas quantity increases and increasing as the static load phase extends. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Static Load Stage** & **Gas Type** & \(\mathbf{a_{1}}\) & \(\mathbf{b_{1}}\) \\ \hline \multirow{4}{*}{\(0.3\sigma_{c}\)} & He & 0.0682 & 0.0693 \\  & N\({}_{2}\) & 0.0638 & 0.0688 \\  & CH\({}_{4}\) & 0.0585 & 0.0683 \\  & CO\({}_{2}\) & 0.0560 & 0.0667 \\ \hline \multirow{4}{*}{\(0.45\sigma_{c}\)} & He & 0.0728 & 0.0705 \\  & N\({}_{2}\) & 0.0734 & 0.0758 \\ \cline{1-1}  & CH\({}_{4}\) & 0.0637 & 0.0686 \\ \cline{1-1}  & CO\({}_{2}\) & 0.0569 & 0.0629 \\ \hline \multirow{4}{*}{\(0.6\sigma_{c}\)} & He & 0.0866 & 0.0814 \\ \cline{1-1}  & N\({}_{2}\) & 0.0814 & 0.0777 \\ \cline{1-1}  & CH\({}_{4}\) & 0.0706 & 0.0723 \\ \cline{1-1}  & CO\({}_{2}\) & 0.0662 & 0.0695 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation results of the dynamic load sensitivity coefficient. \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multirow{2}{*}{**Static Load Stage**} & \multicolumn{2}{c}{\(\mathbf{D_{\text{\tiny{P}}}}\) (\%)} & & \\ \cline{2-5}  & **He** & \(\mathbf{\text{N}_{2}}\) & \(\mathbf{\text{CH}_{4}}\) & \(\mathbf{\text{CO}_{2}}\) \\ \hline \(0.3\sigma_{c}\) & 48.8 & 45.6 & 29.6 & 25.6 \\ \(0.45\sigma_{c}\) & 57.6 & 52 & 40.8 & 32.8 \\ \(0.6\sigma_{c}\) & 75.2 & 68.8 & 51.2 & 45.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation results of permeability change rate. With the increase in the cyclic dynamic load frequency and amplitude, the working face is damaged and prone to disasters such as coal and gas outbursts. Therefore, in practical engineering, blasting excavation or high-frequency mechanical disturbance should be avoided as much as possible for high-pressure coal seams. When the stress level of the coal seam to be mined is high, such as the coal seam in the yield stage, the change in the permeability of the coal seam and the cracking and damage state of the coal seam should be focused on during the mining process.