## 3 Results and Discussion
### Empirical Correlations for the Rock Mechanical Properties
#### 3.3.1 Strength Parameters  
The correlation between the uniaxial compressive strength and the P-wave interval transit time is shown in Figure 7. From Figure 7, the _UCS_ of rocks decreased with an increasing P-wave interval transit time. The finding was consistent with previous studies [17; 18; 19; 20; 21; 22; 23; 24; 29; 30; 31; 32; 33; 34; 35], suggesting that the _UCS_ of rocks increased as the P-wave velocity increased (the wave velocity and wave interval transit time are reciprocal). From Table 2, empirical equations for estimating the _UCS_ of rocks include linear, exponential, and power forms. Therefore, linear, logarithmic, exponential, and power curves were used to fit experimental data points, and then equations with the highest correlation coefficient (\(R^{2}\)) were obtained. As shown in Figure 7, the curve based on the exponential function had the best fit, with an \(R^{2}\) of 0.9028. The equation of the curve is:  
\[UCS=925.2\rm{e}^{-0.031\times\Delta t_{c}}\ \ R^{2}=0.9029 \tag{23}\]  
Figure 5: The P-wave interval transit time of the samples.  
The correlation between the cohesive strength and the P-wave interval transit time is shown in Figure S6 (within the Supplementary Data). From Figure S6 (within the Supplementary Data), the \(C\) of the rocks is negatively correlated with the P-wave interval transit time. The fitting effect between the \(C\) and the P-wave interval transit time was not good. Therefore, a combination of parameters considering the two parameters of the P-wave interval transit time and density were adopted to develop the equation for estimating the \(C\) of the rocks. The relationship between \(C\) and \(\Delta t_{\rm c}\)/DEN is described in  
Figure 6: The rock mechanical properties of the samples ((**a**) uniaxial compressive strength; (**b**) Young’s modulus; (**c**) Poisson’s ratio; (**d**) cohesive strength; (**e**) internal friction angle).  
Figure 8, indicating that the curve based on the power function had the best fit, with an \(R^{2}\) of 0.8898. The empirical equation for estimating the \(C\) of the rocks is as follows:  
\[C=2976.9\times(\Delta t_{\rm c}/DEN)^{-1.402}\;\;R^{2}=0.8898 \tag{24}\]  
The correlation between the internal friction angle and the P-wave interval transit time is shown in Figure S7 (within the Supplementary Data), where the \(\varphi\) of the rocks was negatively correlated with the P-wave interval transit time, consistent with previous studies [16], indicating that the \(\varphi\) of rocks increased with the increasing P-wave velocity. The \(\varphi\) can also be estimated from the P-wave interval transit time or a combination of parameters considering the two parameters of the P-wave interval transit time and density. Simultaneously, the Mohr-Coulomb criterion indicates that there is a correlation among uniaxial compressive strength, cohesion, and the angle of internal friction [36]. Two of the three parameters have been established and the third parameter can be obtained via calculation. In other words, if the \(UCS\) and \(C\) are known, the \(\varphi\) can be calculated form the equation, as follows:  
\[\varphi=\left(2\arctan\left(\frac{UCS}{2C}\right)-\frac{\pi}{2}\right)\times \left(\frac{180}{\pi}\right) \tag{25}\]