## 3 Dynamic-Static Conversion Relationship
### Relation Establishment
(2019) developed an empirical equation to estimate the static elastic modulus from the dynamic modulus. However, this equation has limitations and is only applicable to microcline and granite. \[E_{\rm s}=1.137E_{\rm d}-9.68\quad\quad\text{with }\mathbb{R}^{2}=0.92\]  
[PERSON] (2017) proposed a linear regression equation (Equation (6)) for igneous and metamorphic rocks in Canada by analyzing the measurement results of 174 specimens. He used sufficient experimental data and analyzed the discrepancy in dynamic and static elastic parameters from the perspective of microfractures or microcracks, but did not consider factors such as the mineral composition, content, pore development degree, stress magnitude, strain amplitude, and testing frequency of the rock sample. Therefore, to establish an effective correlation, it is necessary to carefully control the storage environment of rock samples and the environment for conducting experimental tests. \[E_{\rm s}=1.263E_{\rm d}-29.5\quad\quad\text{with }\mathbb{R}^{2}=0.82\]  
[PERSON] (2017) established the correlation exists between dynamic and static moduli of elasticity by conducting experimental studies on 10 different types of rocks at 4 pressures. His research is based on the general correlation between dynamic and static moduli proposed by [PERSON] (2017) (type I), considering high-stress situations. However, in his research, temperature factors were not considered, which may not be applicable at high temperatures. He believes that in most cases, the dynamic modulus of elasticity exceeds the static modulus of elasticity, while the dynamic Poisson's ratio is lower than the static Poisson's ratio. He proposed a power-law regression equation as shown below:  
\[E_{\rm s}=\alpha E_{\rm d}\,^{\beta},\,\alpha\in 0.097\sim 0.152,\,\beta\in 1.38 \sim 1.48\quad\quad\text{with }\mathbb{R}^{2}=0.96\text{--}0.99\]  
The two parameters \(\alpha\) and \(\beta\) are constants, but their values depend on the stress level. [PERSON] and [PERSON] (2017) obtained the following empirical equations by analyzing a large amount of data. \[E_{\rm s}=0.74E_{\rm d}-0.82\quad\quad\text{with }\mathbb{R}^{2}=0.84\]  
\[\log_{10}E_{\rm s}=0.77\log_{10}(\rho_{bulk}E_{\rm d})+0.02\quad\quad\text{ with }\mathbb{R}^{2}=0.96\] \begin{table}
\begin{tabular}{c c c c} \hline
**Equation Type** & **Relationship** & **Equations** & **Vars.** \\ \hline I & \(E_{\rm s}=\text{a}E_{\rm d}+\text{b}\) & (5), (6), (8), (9), (11), (12), (14), & \(E_{\rm d}\) \\ II & \(\log_{10}E_{\rm s}=c\log_{10}(\rho_{bulk}E_{\rm d})-\text{d}\) & (10), (27), and (29) & \(\rho_{bulk}\);\(E_{\rm d}\) \\ III & \(E_{\rm s}=\alpha E_{\rm d}^{\rm S}\) & (7), (13), (18), (20), (24), (30) and (31) & \(E_{\rm d}\) \\ IV & \(E_{\rm s}=\text{k}E_{\rm d}^{\rm S}+\text{f}E_{\rm d}\) & (15)–(17) & \(E_{\rm d}\) \\ V & \(E_{\rm s}=\text{g}V_{\rm n}^{\rm n}\) & (32) and (33) & \(V_{\rm p}\) \\ VI & \(E_{\rm s}=\rho^{0.00}\frac{\rho_{\rm d}}{\rho_{\rm f}E_{\rm d}}\) & (22) & \(E_{\rm d}\) \\ VII & \(E_{\rm s}=\frac{E_{\rm d}}{3\delta\lambda^{-\alpha\alpha}}\) & (34) & \(\lambda\),\(E_{\rm d}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Relationship models \(E_{\rm s}\) static elastic modulus, \(E_{\rm d}\) dynamic elastic modulus, \(\rho_{\rm bulk}\) bulk density, \(V_{\rm p}\) P-wave velocity, \(\lambda\) as spatial attenuation. These relationships can be used for various rocks. They believe that the correspondence between the two moduli (Equation (9)) is quite low. By incorporating bulk density (g/cm\({}^{3}\)) into the relationship, a better estimate was found (Equation (10)). Therefore, based on the logarithmic relationship, the static elastic modulus can be better predicted. [PERSON] and [PERSON] (2018) investigated rock deformation characteristics using the elastic modulus. They compared dynamic values from geophysical borehole logging with static values from the laboratory testing of rock specimens directly collected from the borehole.