# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 4 Discussion--Single-Pipe Model
### Analysis of the Seepage Field  
Similarly, the change in groundwater seepage velocity also reflects the freezing effect of the frozen pipes. The seepage field maps of different pipes after freezing were studied, taking the straight pipe and the curved pipe with s = 7 as an example. Figure 5 shows the final seepage-velocity maps of the straight pipe and the curved pipe with s = 7 after freezing for 58 d, and a series of seepage-velocity contours are plotted to analyze the seepage velocity. As shown in the seepage-velocity cloud diagram, the seepage velocity on the left side of the soil body decreases from the model boundary to the freezing area, and the seepage velocity on the right side gradually increases from the freezing area to the downstream boundary. The flow is nearly absent in the freezing area, and seepage velocity  
Figure 4: Schematic representations that indicate surface positions of frozen wall observation in both hydrostatic and dynamic conditions for each pipe type, with histograms depicting variations in thickness with changing positions. is almost 0; i.e., the freezing wall blocks the hydraulic connection between the inside and outside of the wall. This is due to the decreasing temperature of the freezing tubes, which causes the hydric soil in the surrounding area to completely transform into ice and causes the groundwater to flow downstream from the upper and lower sides around the freezing walls. Considering the seepage-velocity contour, the upper and lower ends have higher seepage velocities that are several orders of magnitude higher than those of the left and right ends. This can be attributed to the large heat flux and low heat-exchange capacity at the upper and lower ends. In addition, during freezing, more and more moisture due to the generation of the freezing wall is blocked from being gathered at the upper and lower ends, which accelerates the seepage velocity at the upper and lower ends. Various tube-freezing methods exhibit similar trends in their impact on the seepage field, albeit with some distinctions in their outcomes. When plotting seepage-velocity contours, it becomes evident that identical isovelocity lines assume varying positions on the cloud diagrams of different frozen tubes. In this study, the primary focus is on examining the obstructive influence of the frozen wall on the water flow. As a result, our analysis concentrates on the coverage of isovelocity lines at the left and right extremities. A seepage velocity of 9.43 \(\times\) 10\({}^{-6}\) m/s was selected for the study, i.e., the part circled in red in Figure 5, where the water velocity is lower than 9.43 \(\times\) 10\({}^{-6}\) m/s within the range and higher outside it. The reference position is taken to be the geometric center of the straight pipe, and the furthest distance of this flow velocity value is measured both upstream and downstream from the geometric center. This measurement is conducted to facilitate a more effective comparison of the freezing and waterproofing effects, as illustrated in Figure 6.  
As shown in Figure 6, the change in the s-value has a negligible effect on the farthest downstream distance; the smaller the s, the larger the arc, and the farthest upstream distance gradually increases; i.e., the same isovelocity line is farther away from the freezing area, and the seepage velocity within the isovelocity line is smaller, which indicates that the freezing wall produces a better interception of the flow of groundwater. At the position with the same seepage velocity, the smaller the value of s, the closer the isovelocity line of the curved pipe is to the upstream head boundary, reflecting better water insulation. The reason may be that due to the influence of seepage, the efficiency of heat exchange both upstream and downstream is increased, resulting in a rapid drop in stratum temperature and an accelerated phase-change rate.