## 5 Results and Discussion
### Local Heat Transfer Coefficient  
Figure 17 depicts the local heat transfer coefficients of the rock fracture surfaces. The local convection heat transfer coefficient is observed to gradually decrease along the outlet direction, which is consistent with the previous study [22,42]. This is attributed to the gradual decrease in the temperature difference between the fluid and fracture wall along the outlet direction. The fracture model shows that the fracture roughness of \(\alpha=30^{\circ}\) and \(\alpha=60^{\circ}\) are different in the three axes, while that of \(\alpha=0^{\circ}\) and \(\alpha=90^{\circ}\) are different in the two axes. The flow direction on the fractured surface is slightly deflection, and consequently the local convection heat transfer coefficient in the negative direction of the \(x\)-axis exceeds that of the positive direction (Figures 11 and 12). In addition, the distribution of the local heat transfer coefficient is strongly correlated with the rough fracture profile [45]. Figure 18 shows the distribution of the local heat transfer coefficients on the fracture profile at \(x=0\) m. Figure 18 shows that the model of fracture roughness from large to small at \(x=0\) m is \(\alpha=0^{\circ}\), \(\alpha=30^{\circ}\), \(\alpha=60^{\circ}\), and \(\alpha=90^{\circ}\). The fluctuation of local convection heat transfer coefficient increases with the increase of fracture roughness. At \(\alpha=90^{\circ}\), the local heat transfer coefficients exhibit a logarithmic distribution in the smooth section along the \(z\)-axis. Similar to the fracture profile, the distribution curve of the local heat transfer  
Figure 16: Variations in the total heat transfer coefficient with time for various initial conditions. Top: \(T_{0}=90\)\({}^{\circ}\)C. Below: \(V_{0}=10\) mL/min. coefficient is characterized by peaks and troughs. The local heat transfer coefficient curves at \(\alpha=0^{\circ}\), \(\alpha=30^{\circ}\), and \(\alpha=60^{\circ}\) reveal that the local heat transfer coefficient at the peaks of the fracture profile is consistently greater than that at the troughs. Moreover, increasing the injection flow rate not only improves the local heat transfer coefficient, but also has an enhancing effect that makes the difference between the peaks and troughs of the local heat transfer coefficient more evident. The enhanced effect can be observed via the increased slope of the local heat transfer coefficient curve. Figure 19 shows the enhancement effect of increasing the flow rate on the convective heat transfer intensity. Figure 17: Distribution of local heat transfer coefficients on rock fractures as water flows through the fracture (\(V_{0}=10\) mL/min. \(T_{0}=90\)\({}^{\circ}\)C. \(t=10\) min). Figure 18: _Cont._  
Figure 20 presents the distribution of the local heat transfer coefficient, wall temperature, and average flow rate on the fracture profile at \(z=0.05\) m. The curves of the average flow rate and local heat exchange coefficient become increasingly complex as \(\alpha\) increases from \(0^{\circ}\) to \(90^{\circ}\). Since the fracture model is constructed by translating along the \(y\)-axis, the distribution of fracture aperture in each direction of the model is not uniform. Therefore, a position with a higher absolute gradient results in a smaller fracture aperture, which results in a low flow rate as the water flows through the path of the small aperture. In contrast, fractures with larger aperture will form a dominant channel where the flow rate is greater. Compared with the temperature, the difference of flow rates at fractures with different apertures is the main factor causing uneven distribution of local heat transfer coefficient. In addition, a strong positive correlation is observed between the local heat exchange coefficient and the flow rate distribution, which indicates that the flow rate is the main factor affecting the heat transfer intensity. Figure 19: Variation of \(h_{x}\) with the flow rate at adjacent peak and troughs of the fracture profile.