## 5 Results and Discussion  
### Temperature Field Distribution  
Table 5 lists the outlet water temperature determined from the 32 simulations. The outlet water temperature is maximized at \(\alpha=90^{\circ}\), followed by models \(\alpha=0^{\circ}\), \(\alpha=30^{\circ}\), and \(\alpha=60^{\circ}\). Figure 9 depicts the distribution characteristics of the temperature fields on the fracture surfaces of four different model cases at several injection flow rates. With the injection of low temperature water, the temperature inside the fracture begins to decrease and the cold front pushes towards the outlet. This phenomenon is consistent with previous studies [37; 43; 44]. The temperature at the center of the fracture typically exceeds that on both sides of the fracture. This is attributed to the greater thermal replenishment at the center of the fracture by the cylinder model compared to the two sides. Increasing the flow rate enhances the rate of the temperature reduction in the fracture. The temperature on the fractured surface of model \(\alpha=0^{\circ}\) is distributed symmetrically along the \(z\)-axis due to the symmetry of the model. At \(\alpha=30^{\circ}\) and \(\alpha=60^{\circ}\), the fracture surface temperature in the positive direction of the \(x\)-axis exceeds that in the negative direction. This indicates that the angle between the fracture morphology and flow direction has an important influence on the convection heat transfer process. Figure 10 presents the local temperature distribution along the \(z\)-axis at \(x=0.02\) m and \(x=-0.02\) m. At \(\alpha=30^{\circ}\) and \(\alpha=60^{\circ}\), the local wall temperature at \(x=0.02\) m is lower than that at \(x=-0.02\) m. This is related to the migration of the channel formed by the fracture profile towards the \(x\)-axis, which makes the fracture geometry profile no longer perpendicular to the flow direction. The fluid preferentially flows through the fractures in an easily accessible path where the flow direction is deflected. In addition, Figure 10 shows that the temperature curve at low flow rates (\(10\) mL/min) reaches its peak value, while the temperature curve at high flow rates (\(30\) mL/min) does not. This indicates that the cryogenic fluid injected at high flow rates penetrates the fractures more quickly and results in a thermal breakthrough in a short time, which causes rapid wall temperature reduction and no timely thermal replenishment. Fluids with low flow rates are more likely to utilize matrix heat transfer to compensate for the decrease in wall temperature caused by convection heat transfer. Figure 11 illustrates the streamline distribution characteristics of the models with \(\alpha=30^{\circ}\) and \(\alpha=60^{\circ}\). The flow direction is observed to slightly deflect in the fracture towards the \(x\)-axis, which results in the temperature difference shown in Figures 9 and 10. This deflection is attributed to the presence of a gently inclined microfractured surface in front of the flow field, preventing the flow from passing. Figure 12 simplifies fracture surface morphology to account for the deflection of the fluid flow direction. A fractured surface consists of a limited number of microfractured surfaces. When the direction of the fluid injection is not parallel to the microfracture dip, the fluid will slightly deflect towards a blunt angle between the flow direction and the microfracture surface strike. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(T_{0}\) (\({}^{\circ}\)C) & \(V_{0}\) (mL/min) & \multicolumn{3}{c}{\(T_{outlet}\) (\({}^{\circ}\)C)} \\ \hline  & & \(\alpha=0^{\circ}\) & \(\alpha=30^{\circ}\) & \(\alpha=60^{\circ}\) & \(\alpha=90^{\circ}\) \\ \hline  & 10 & 54.08 & 53.93 & 53.87 & 54.19 \\  & 20 & 36.47 & 36.36 & 36.23  
Figure 9: Temperature field distribution on the fracture surface (\(t=10\) min). Figure 11: Streamline distribution characteristics of models with \(\alpha\) = 30\({}^{\circ}\) and \(\alpha\) = 60\({}^{\circ}\) (\(V_{0}\) = 10 mL/min).