## 3 Results and Discussion
### The Average Heating Rate at Monitoring Points  
The heating rate provides a relatively intuitive reflection of the speed of heat transfer within the granite. The heating rates at various monitoring points inside of the granite during the heating process can be calculated according to Equation (2). This study statistically analyzed the heating rates at monitoring points located on lines E-E, F-F, C-C, and D-D during the initial two hours of heating. The results are presented in Figure 7.  
\[\omega=\frac{\Delta T}{\Delta T} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Monitoring Lines**} & **Heat Source** & **Fitting Equation** & **R\({}^{2}\)** \\  & **Temperature(\(T_{\text{h}}\), \({}^{\circ}\)C)** & & \\ \hline \multirow{8}{*}{A-A} & 60 & \(T_{\text{s}}=exp\,\left(4.0-3.3^{-3}\cdot d_{\text{h}}+1.1^{-5}\cdot d_{ \text{h}}^{2}\right)\) & 0.991 \\  & 70 & \(T_{\text{s}}=exp\,\left(4.2-3.9^{-3}\cdot d_{\text{h}}+1.2^{-5}\cdot d_{ \text{h}}^{2}\right)\) & 0.993 \\  & 80 & \(T_{\text{s}}=exp\,\left(4.3-5.2^{-3}\cdot d_{\text{h}}+1.6^{-5}\cdot d_{ \text{h}}^{2}\right)\) & 0.996 \\  & 90 & \(T_{\text{s}}=exp\,\left(4.4-5.6^{-3}\cdot d_{\text{h}}+1.7^{-5}\cdot d_{ \text{h}}^{2}\right)\) & 0.996 \\  & 100 & \(T_{\text{s}}=exp\,\left(4.6-5.1^{-3}\cdot d_{\text{h}}+0.9^{-5}\cdot d_{ \text{h}}^{2}\right)\) & 0.998 \\where \(\omega\) is the average heating rate, \({}^{\circ}\)C/h; \(\Delta T\) is the temperature difference of the monitoring point during a specific period of time, \({}^{\circ}\)C; and \(\Delta\tau\) is the heating time interval, h. In this paper, \(\Delta\tau\) takes 2 min. Figure 7: Variation curves of the average heating rate at monitoring points within fractured granite under a 80 \({}^{\circ}\)C heat source heating during the first 2 h: (**a**) line E-E; (**b**) line F-F; (**c**) line C-C; (**d**) line D-D. Figure 7 reveals that the heating rates at all monitoring points generally exhibited a trend of initial increase followed by a decrease, ultimately stabilizing. Points closer to the heat source demonstrated a more pronounced increasing trend in the heating rate, indicating faster heat transfer and, consequently, higher temperatures at these proximal locations. During the initial heating phase, the significant temperature difference between the granite's initial temperature and the heat source facilitated rapid heat transfer, causing the heating rate to rise gradually. As heat transfer progressed and the granite mass temperature increased, the heating rate declined slowly until it essentially stabilized. For monitoring points along the same line, increasing \(d_{\rm h}\) correlated with a longer duration of the heating rate increase phase and an overall lower heating rate, signifying slower heat transfer. For instance, on line E-E (Figure 7a), the peak heating rate at the point closest to the source (#05) was 111 \({}^{\circ}\)C/h compared to only 9 \({}^{\circ}\)C/h at the farthest point (#59). This represents a reduction of 91.89% in the peak heating rate as the \(d_{\rm h}\) increased from 20 mm to 140 mm. Comparing the heating rates of points #23 and #41 located below and above the fracture, respectively, indicates a significant influence of the fracture on the local heat transfer rate, evidenced by the substantially higher heating rate below the fracture compared to above it. This difference results from the combined effects of \(d_{\rm h}\) and the presence of the fracture. As shown in Figure 7b, monitoring points at similar \(d_{\rm h}\) exhibited essentially identical heating rate trends.