## 3 Results  
### Thermal Characteristic  
The evolution of temperature in the freezing model is the main thermal characteristic during natural cold gas injection, which reflects the development of the frozen range. Figure 3 depicts the change of temperature at each monitoring point. Obviously, the natural cold gas used in the experiments was in the range of -15 to -10 \({}^{\circ}\)C, and the temperature of each point experienced a slow decrease, and then a sharp drop in temperature occurred at approximately 0 \({}^{\circ}\)C. Before the temperature in the freezing model dropped below 0 \({}^{\circ}\)C, the water had not been frozen yet, and the thermal conductivity of the water, 0.613 W/(m\(\cdot\)K), is much lower than that of ice, which is 2.31 W/(m\(\cdot\)K). Therefore, the temperature reduction rate at each point was slower, and the farther away it was from the inlet, the slower it would be. This stage can be regarded as the initial cooling stage. When the temperature decreased to about 0 \({}^{\circ}\)C, the sand pores were in the state of water-ice two-phase coexistence. The phase transition of water into ice released a lot of latent heat [26], which was absorbed by cold gas, and the temperature was almost unchanged; this is the phase transition stage. As the temperature continued to decrease, water in the pores gradually turned into ice and latent heat was released completely. The surface of sand changed from water-encapsulated to ice-encapsulated, accordingly, and the thermal conductivity was greatly increased. Therefore, it can be seen from Figure 3 that the temperature dropped sharply below 0 \({}^{\circ}\)C, and rapidly to the same level as the cold gas. This area can be regarded as completely frozen, and this stage is the subcooled stage. It is noteworthy that the throttling effect may occur at the outlet of the cylinder due to the sudden drop of pressure, which made the gas temperature lower than the measured ambient temperature. Therefore, it can be seen from Figure 3b,d that the temperature (T1) at some points would be lower than the ambient temperature (T7), which was not obvious in the low gas flow rate. In order to compare the differences between the three stages more intuitively under different flow rates and initial water contents, we took the temperature at the T2 position (20 cm away from the inlet) as an example to analyze the freezing characteristics. Figure 4 depicts the comparison of temperatures at the T2 position in each experiment. In the initial cooling stage, it can be seen from Figure 4c,d that the initial water content changed from 0.2 to 0.4, and there was little difference in the time needed to reduce the temperature to 0 \({}^{\circ}\)C. It means that the cooling rate was the same under different water contents. However, that could be greatly improved by increasing the flow rate of the cold gas (Figure 4a,b), because the increase in water content in the pores did not increase the internal energy per unit volume significantly, while the increase in the flow rate could directly enhance the convective heat transfer intensity; thus, the flow rate of cold gas played a leading role in the initial \begin{table}
\begin{tabular}{c c c} \hline
**Experiment** & **Initial Water** & **Gas Flow Rate** \\
**Number** & **Content** & **(L/min)** \\ \hline
1 & 0.2 & 15 \(\pm\) 0.2 \\
2 & 0.2 & 25 \(\pm\) 0.2 \\
3 & 0.4 & 15 \(\pm\) 0.2 \\
4 & 0.4 & 25 \(\pm\) 0.2 \\ \hline \end{tabular}  
* The initial temperature of the freezing pipe is 5 \({}^{\circ}\)C.  
\end{table} 
Table 2: Experimental design \({}^{\mathrm{a}}\). cooling rate. In the phase transition stage, it can be seen from Figure 4c,d that the time required for the phase transition increased by, essentially, the same proportion as the initial water content as it doubled, while it could also be reduced by increasing the flow rate of the cold gas. The evolution characteristics of temperature in the initial cooling stage and phase transition stage directly affected the freezing rate.