## 1 Introduction
### Evolution Behaviors of Mohr-Coulomb Strength Parameters
#### 2.2.2 Empirical Formula for Cohesion and the Internal Friction Angle  
It is necessary to provide an empirical formula to describe the behaviors of cohesion and the internal friction angle of different granite samples after high-temperature and water cooling treatment. As the different groups of experiments do not have the same maximum treatment temperature, considering that the temperature of 600 \({}^{\circ}\)C is high enough for most of the dry hot rock reservoirs, here in this study the experimental data in the range from room temperature to 600 \({}^{\circ}\)C are used for consistency to obtain the best fitted curves and equations. Although the granite samples show different behaviors, it has been found that unified empirical Equations (4) and (5) can be applied for the different types of cohesion and the internal friction angle. The fitting curves for the normalized cohesion and internal friction angle are presented in Figures 4 and 5, respectively, and the fitting coefficients are listed in Table 4.  
\[c_{\mathrm{N}}=\frac{1}{1+a\Big{(}\frac{T}{100}-0.2\Big{)}+b\Big{(}\frac{T}{100 }-0.2\Big{)}^{2}} \tag{4}\]  
\[\varphi_{\mathrm{N}}=\frac{1-A}{1+((\frac{T}{100})/x_{0})^{p}}+A \tag{5}\]  
where \(a\), \(b\), \(A\), \(x_{0}\), \(P\) are the fitting coefficients.  
The physical meanings of the fitting coefficients can be discussed by observing their influences on the fitting curves. Figure 6 shows the variation of normalized cohesion by the effects of coefficients \(a\) and \(b\). It can be found that \(a\) dominates the two types of cohesion behaviors, i.e., cohesion shows type I behavior if \(a<0\), while it has type II behavior for \(a\geq 0\). According to Figure 6b, for both type I and II behaviors of cohesion, \(b\) value has an influence on the decreasing slope of the curve. The higher \(b\) value will result in the steeper decreasing trend.  
Figure 4: _Cont._  
Figure 4: Fitting curves for the normalized cohesion of (**a**) Fangshan granite, (**b**) Dabie granite, (**c**) Gonghe granite-1, (**d**) Gonghe granite-2, (**e**) Gonghe granite-3, (**f**) Sichuan granite, (**g**) Wuyi granite, (**h**) Rizhao granite-1, and (**i**) Rizhao granite-2.  
Figure 5: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Samples**} & \multicolumn{2}{c}{**Parameters in Equation (4)**} & \multicolumn{4}{c}{**Parameters in Equation (5) for the**} \\ \cline{2-7}  & **for Cohesion** & \multirow{2}{*}{\(R^{2}\)} & \multicolumn{2}{c}{**Internal Friction Angle**} & \multirow{2}{*}{\(R^{2}\)} \\ \cline{2-7}  & \(a\) & \(b\) & & \(A\) & \(x_{0}\) & \(p\) \\ \hline Fangshan granite & \(0.0198\) & \(0.0008\) & \(0.19\) & \(1.02\) & \(0.24\) & \(8.92\) & \(0.07\) \\ Dabie granite & \(-0.2104\) & \(0.0351\) & \(0.61\) & \(0.99\) & \(0.87\) & \(54.19\) & \(0.11\) \\ Gonghe granite-1 & \(-0.3681\) & \(0.1069\) & \(0.93\) & \(1.26\) & \(4.02\) & \(20.00\) & \(0.69\) \\ Gonghe granite-2 & \(-0.2326\) & \(0.0750\) & \(0.52\) & \(1.28\) & \(3.79\) & \(35.00\) & \(0.43\) \\ Gonghe granite-3 & \(-0.4848\) & \(0.1818\) & \(0.80\) & \(1.40\) & \(3.13\) & \(15.00\) & \(0.74\) \\ Sichuan granite & \(0.1012\) & \(-0.0069\) &Figure 7 illustrates the variation of the normalized internal friction angle by the effects of coefficients \(A\), \(x_{0}\) and \(p\). It is apparent to see that \(A\) dominates the peak and residual value of the internal friction angle, while \(x_{0}\) is the central point of the temperature range for the increasing phase of the internal friction angle, and \(p\) affects the range of the increasing stage as well as the increasing slope.  
Figure 6: Variation of the normalized cohesion with fitting coefficients: (**a**) \(a\) as variable with \(b=0.1\); and (**b**) \(b\) as variable with \(a=-0.3\) and \(a=0.3\).  
Figure 7: Variation of the normalized internal friction angle with fitting coefficients: (**a**) \(A\) as a variable with \(x_{0}=3\) and \(p=20\); (**b**) \(x_{0}\) as a variable with \(A=1.3\) and \(p=20\); and (**c**) \(p\) as a variable with \(A=1.3\) and \(x_{0}=3\).