## 3 Heat Transfer Theory of Linear Heat Sources in Soil Media
### Analysis of the Heat Conduction Laws during the Heating Process  
To intuitively observe the derivation results, Equations (18)-(20) are used to calculate and plot the heating curve of the linear heat source in the soil medium. The parameters of the soil medium are shown in Tables 2 and 3 (data source: \"_Engineering Geology Handbook_\" [42]), and the parameters of the linear heat source are shown in Table 4. The results are shown in Figures 5-8.  
From Figures 5 and 6, it can be observed that in cohesive soil, when heating is applied with the same power, the corresponding soil parameters have a significant impact on the rate of temperature increase. In the same soil medium and with the same heating power, the temperature rise rate of the PVC linear heat source is higher than that of the iron linear heat source. This is because, although the specific heat capacity of iron is only 38.3% of that of PVC material, its thermal conductivity is 273.6 times that of PVC. With the same amount of heat input, iron material will transfer heat to the surrounding soil more rapidly. A similar trend is also found in silty soil, as shown in Figures 7 and 8.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**MQ**} & \multicolumn{1}{c}{\(\mathbf{T_{1}}\)**(\({}^{\circ}\)C)} & \multicolumn{1}{c}{**Q (J/s)**} & \multicolumn{1}{c}{\(\mathbf{r}\) (m)} & \multicolumn{1}{c}{\(\mathbf{\rho}\) (kg/m\({}^{3}\))} & \multicolumn{1}{c}{\(\mathbf{c}\)} & \multicolumn{1}{c}{\(\mathbf{\lambda}\) [J/(kg\({}^{\circ}\)C)]} & \multicolumn{1}{c}{**[J/(m\({}^{\circ}\)C-s)]**} \\ \hline IRON & 70 & 2000 & 0.05 & 7850 & 460 & 46.52 \\ PVC & 55 & 2000 & 0.05 & 1380 & 1200 & 0.17 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Corresponding calculation parameter table for linear heat source materials.  
Figure 5: Temperature change law during heating process of claye soil.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & _MC (\%)_ & _\(\mathbf{\rho}\)_ **(kg/m\({}^{3}\))** & \(c\) **[J/(kg\({}^{\circ}\)C)]** & _\(\mathbf{\lambda}\)_ **[J/(m\({}^{\circ}\)C-s)]** \\ \hline N1 & 5 & 2000 & 820 & 0.25 \\ N2 & 15 (upper bound) & 1950 & 1050 & 1.08 \\ N3 & 15 (upper bound) & 1900 & 1350 & 1.25 \\ N4 & 25 (lower bound) & 1850 & 1250 & 1.15 \\ N5 & 25 (upper bound) & 1850 & 1650 & 1.85 \\ N6 & 35 (lower bound) & 1800 & 1550 & 1.25 \\ N7 & 35 (upper bound) & 1750 & 1850 & 1.95 \\ N8 & 45 & 1700 & 2350 & 2.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Thermodynamic parameter table for clay medium.  
To investigate the distance over which heat from a linear heat source is transferred into the soil medium, the temperature distribution of two types of linear heat sources after heating in cohesive and silty soils with different water contents is presented in Figure 9.  
Figure 8: Final temperature of silty soil at 3600 s of heating.  
Figure 6: Final temperature of clayey soil at 3600 s of heating.  
Figure 7: Temperature change law during heating process of silty soil.  
From Figure 9, it can be seen that the heat transfer distance of the linear heat source in the soil medium is very short, only about 0.2 m at 3600 s. This is due to the low thermal conductivity of the soil, which limits the propagation of heat within it.