## 3 Results and Discussion
### Temperature Gradient Evolution in Fractured Granite  
The temperature gradient is a two-dimensional vector whose magnitude equals the maximum rate of temperature change and whose direction is that of increasing temperature. Consequently, the temperature gradient can be utilized to analyze the laws governing heat transfer during heat conduction in fractured granite. The temperature gradient in this study is obtained by taking the limit of the ratio between the temperature difference across adjacent isothermal surfaces and the perpendicular distance separating those surfaces:  
\[T_{\text{Grad}}=\lim_{\Delta x\to 0}\frac{\Delta T}{\Delta x} \tag{3}\]  
where \(T_{\text{Grad}}\) is the temperature gradient, \({}^{\circ}\)C/m; \(\Delta T\) is the temperature difference between adjacent isothermal surfaces, \({}^{\circ}\)C; and \(\Delta x\) is the vertical distance between adjacent isothermal surfaces, m.  
Figure 12 depicts the distribution of the temperature gradient field within the fractured granite at different times under heating by an 80 \({}^{\circ}\)C heat source. During the initial heating stage, the temperature gradient vectors were predominantly vertically downward, indicating upward heat transfer. Concurrently, the temperature gradient exhibited minimal variation in the upper region of the granite, signifying negligible heat transfer. The maximum temperature gradient magnitude across the entire domain was 0.48 \({}^{\circ}\)C/mm, located at the bottom-right corner of the granite (Figure 12a,b). In the middle heating stage, regions of elevated temperature gradient formed annular patterns surrounding the fracture. As heating progressed, the temperature gradient demonstrated minimal change, suggesting that heat transfer was approaching a steady state (Figure 12c,d). During the late heating stage, the temperature gradient within the fractured granite stabilized, the highest and lowest recorded temperature gradients were 0.39 \({}^{\circ}\)C/mm and 0.04 \({}^{\circ}\)C/mm, respectively, and the predominant direction of heat transfer remained essentially unchanged (Figure 12e,f). From Figure 13, it can be seen that the time at which the monitoring point reaches the peak temperature gradient is positively correlated with \(d_{\rm h}\). On the same monitoring line, excluding the monitoring points within the fracture, the peak temperature gradient at the other monitoring points gradually decreases with the increase of the distance from the monitoring point to the heat source. Additionally, the peak temperature gradient at monitoring points 12# and 32# within the fracture attains its maximum value on the monitoring line; however, on the monitoring line D-D, the peak temperature gradient at monitoring point 52# within the fracture is lower than that at monitoring point 7# near the heat source. This is because monitoring point 7# is closer to the heat source, resulting in a greater temperature change rate per unit length compared to monitoring point 52#. Figure 12: Temperature gradient field distribution of granite at different times under a heat source temperature of 80 \({}^{\circ}\)C (In the figure, the pink arrow is the temperature gradient of the monitoring point): (**a**) 5 min; (**b**) 10 min; (**c**) 35 min; (**d**) 60 min; (**e**) 300 min; (**f**) 540 min. The corresponding relationship between the peak temperature gradient at each monitoring point in the fracture and the temperature of the heat source is shown in Figure 14. It can be seen from Figure 14 that the relationships between the sizes of the peak temperature gradients at each monitoring point under different heat source temperatures are 22\(\#\) > 32\(\#\) > 12\(\#\) > 42\(\#\) > 52\(\#\). This indicates that the peak temperature gradient is largest in the middle and lower sections of the fracture and the peak temperature gradient is lowest in the upper right corner.