## 4 Numerical Simulation Analysis of the Hole-by-Hole blasting and Three-Hole Simultaneous Blasting
### Simulation Calculation and Analysis  
#### 4.4.1 Reliability Analysis of Numerical Simulation Results  
To verify the reliability of the numerical simulation results, a unit of the numerical model was selected for investigation. The considered unit was located at a certain distance directly behind the borehole, as depicted in Figure 10. In this investigation, the value of the vibration velocity of this unit is extracted. By comparing the vibration data monitored at similar locations to the onsite blasting test results, as shown in Figure 15, the reliability of numerical simulations was analyzed.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{C_{0}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{C_{5}}\) & \(\mathbf{C_{6}}\) & \(\mathbf{E_{0}}\) & \(\mathbf{V_{0}}\) \\ \hline
0.0 & 0.0 & 0.0 & 0.0 & 0.4 & 0.4 & 0.0 & \(2.5\times 10^{-6}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Air state equation parameters.  
Figure 14: Delay time setting for numerical simulations.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \begin{tabular}{c} **Density** \\ \(\rho\)/**g\(\cdot\)cm\({}^{-3}\)** \\ \end{tabular} & \begin{tabular}{c} **Detonation** **Velocity/cm\(\cdot\)us\({}^{-1}\)** Explosion Pressure \(\mathbf{Pcj}\)/\(\mathbf{GPa}\)** \\ \end{tabular} & \begin{tabular}{c} **\(A\)/GPa** \\ \end{tabular} & \begin{tabular}{c} **\(B\)/GPa** \\ \end{tabular} & \begin{tabular}{c} **\(R_{1}\) \\ \end{tabular} & \begin{tabular}{c} **\(\mathbf{R_{2}}\)** \\ \end{tabular} & \
\begin{tabular}{c} **\(\mathbf{\omega}\)** \\ \end{tabular} &
\begin{tabular}{c} **\(E\)/GPa** \\ \end{tabular} \\ \hline
1.08 & 0.380 & 9.9 & 214.4 & 182 & 4.2 & 0.9 & 0.15 & 4.19 \\ \hline \hline \end{tabular} \
\begin{tabular}{c} Air Material Model \\ \end{tabular}
\begin{tabular}{c} In this paper, an empty material constitutive model was used for air simulation, namely the \({}^{\star}\)MAT\(\_\)NULL model. Meanwhile, the state equation adopts a Linear\(\_\)Polynomial as follows:  
\[p=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+(C_{4}+C_{5}\mu+C_{6}\mu^{2})E \tag{23}\]  
\[\mu=\frac{1}{V}-1E_{0}V_{0} \tag{24}\]  
where \(C_{0}\), \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\), \(C_{6}\) is a user-defined constant, \(p\) is pressure, and \(E\) is the initial internal energy per unit.  
The density of air is considered as 1.29 kg/m\({}^{3}\) in the calculation, and the parameters of the state equation are from reference [34] as shown in Table 4.  
\end{table}
Table 3: The parameters of the explosive material model.  
As shown in Figure 15, it was noted that there were multiple peak values of surface vibration velocity caused by the onsite blasting tests. These were related to the blasting method of the onsite blast holes. During the onsite blasting, multiple segments of blast holes were sequentially initiated. Considering the numerical simulation results, it was noted that there was only one vibration velocity peak of 10.51 cm/s for the three-hole simultaneous blasting, whereas the peak surface vibration velocity caused by the onsite blasting tests was 15.63 cm/s. On this basis, it was shown that the peak vibration velocities generated by the onsite blasting tests and numerical simulations were of the same order of magnitude. At the same time, the blasting vibration velocities generated by both the onsite blasting tests and numerical simulations ultimately decayed to 0 cm/s over time. Based on these findings, it was highlighted that the different material models and parameters selected in the numerical simulation process were feasible. As a result, it can be claimed that the simulation results of the three-hole simultaneous blasting method and the hole-by-hole blasting were reliable, considering the established model.