## 3 Materials and Methods
### Rock Ultrasonic Testing  
Figure 1 shows the non-metallic ultrasonic instrument RS-ST02C. The experimental set-up includes a waveform generator, two piezoelectric transducers with a maximum resonant frequency of 1 MHz mounted on the sample holder, and a USB interface connected to a computer. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mineral Composition** & **Percentage Weight (\%)** \\ \hline \multirow{4}{*}{Component} & calcite & 92 \\  & dolomite & 3 \\  & irony & 2 \\  & clay & 2 \\  & others & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main mineral compositions of limestone samples. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Average** & **Range** \\ \hline Initial density (g/cm\({}^{3}\)) & 2.76 & 2.68–2.81 \\ Natural moisture content (\%) & 1.9 & 1.1–2.7 \\ Cohesive force (MPa) & 27 & 21–35 \\ Frictional angle (\({}^{\circ}\)) & 33 & 28–37 \\ Poisson’s ratio & 0.27 & 0.25–0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic physical and mechanical properties of limestone. To ensure the conduction of the longitudinal wave ability of the coupling medium, vaseline was selected as the coupling agent, which is applied to the top and bottom of the limestone specimen. After installing the connection, checks were performed to ensure the sensitivity of the P-wave velocity measurement. Five sampling depths A, B, C, D, E were selected; the measurement results of longitudinal wave velocities of 12 standard samples at each depth are shown in Table 3. The initial damage degrees of the longitudinal wave velocity for five different drilling locations are shown in Figure 2. Figure 2 shows that:  
1. The rock samples' longitudinal wave velocities are similar at the same drilling locations, and there is a small longitudinal wave velocity change of rock samples at each drilling position, except point D.
2. With an increasing drilling depth, the longitudinal wave velocity increases, and the maximum value is approximately 6000 m/s; the smaller the drilling depth, the greater the degree of damage disturbance, and the smaller the wave velocity. 3. The initial damage varies non-linearly with respect to the longitudinal wave velocity of the five drilling locations. In order to quantify the extent of nonlinearity in the initial damage behavior, an exponential relationship is applied here and expressed as: \[D=1.52-0.39\text{e}^{(\
u/4402.95)}\] (3) where \(D\) is the initial damage value, and \(\
u\) is the longitudinal wave velocity of the five drilling locations. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
u}^{\prime}\) & **Average** & **No** & \(\mathbf{\
u}^{\prime}\) & **Average** \\ \hline A1 & 2823 & & B1 & 3230 & & C1 & 3564 & & D1 & 3952 & & E1 & 5599 & \\ A2 & 2846 & & B2 & 3250 & & C2 & 3602 & & D2 & 4150 & & E2 & 5645 & \\ A3 & 2875 & & B3 & 3298 & & C3 & 3624 & & D3 & 4221 & & E3 & 5742 & \\ A4 & 2902 & & B4 & 3311 & & C4 & 3649 & & D4 & 4465 & & E4 & 5778 & \\ A5 & 2940 & & B5 & 3326 & & C5 & 3672 & & D5 & 4723 & & E5 & 5891 & \\ A6 & 2961 & & B6 & 3348 & & C6 & 3688 & & D6 & 4904 & & E6 & 5911 & \\ A7 & 2986 & & B7 & 3369 & & C7 & 3693 & & D7 & 5021 & & E7 & 5946 & \\ A8 & 3010 & & B8 & 3389 & & C8 & 3702 & & D8 & 5086 & & E8 & 5979 & \\ A9 & 3054 & & B9 & 3426 & & C9 & 3721 & & D9 & 5126 & & E9 & 5982 & \\ A10 & 3097 & & B10 & 3475 & & C10 & 3740 & & D10 & 5242 & & E10 & 5990 & \\ A11 & 3104 & & B11 & 3480 & & C11 & 3780 & & D11 & 5367 & & E11 & 5992 & \\ A12 & 3114 & & B12 & 3485 & & C12 & 3797 & & D12 & 5478 & & E12 & 5997 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Vertical wave velocities of five limestone specimen borehole locations. Figure 1: Measuring instrument and schematic diagram of RS-ST02C: (**a**) RS-ST02C non-metallic ultrasonic instrument; (**b**) Schematic diagram of the RS-ST02C measurement.