## 4 Establishment and Validation of Numerical Simulation Model for a Linear Heat Source
### Analysis and Validation of Numerical Simulation Model  
The surface heat transfer characteristics of a ferrous linear heat source in a soil medium were calculated, along with its temperature rise characteristics and temperature distribution cloud map in the soil medium, as shown in Figure 3.  
Figure 3 shows that with the increase in time, the increase in the temperature of the linear heat source in the soil medium is very rapid, while it is relatively slow in the water medium, which is consistent with the non-steady-state heat transfer theory of linear heat sources in water-soil media [12]. This is because the linear heat source exchanges heat with the water medium through thermal convection [26], and seawater has a high convective heat transfer coefficient, so the heat of the linear heat source is absorbed and carried away by the seawater in a short time. In contrast, the linear heat source exchanges heat with the soil medium through thermal conduction, and the soil has a lower thermal conductivity. Therefore, the heat loss from the linear heat source due to soil thermal conduction is much lower than that due to seawater thermal convection, indicating that seawater has good thermal conductivity characteristics, while the soil medium has good thermal insulation characteristics. In addition, the transfer of heat from the linear heat source to the soil medium results in a small range of temperature rise in the soil, with the temperature impact range limited to the vicinity of the linear heat source. This is also related to the low thermal conductivity of the soil, which prevents heat from spreading rapidly within the soil. In order to validate the effectiveness of the numerical model calculations, we established a finite element numerical model with the same parameters as the indoor test described in reference [12] and compared the simulation results with the experimental outcomes. According to the experimental parameters provided in the aforementioned reference, the model sensor had a diameter of 5 cm and a length of 70 cm, with the linear heat source configured to have a heating power of 500 watts for a duration of 30 min. Figure 2: Solid model and finite element mesh model. Additionally, to further contrast the differences between numerical calculations and both theoretical calculations and indoor experimental results, Table 1 compares the theoretical, experimental, and numerical calculation results of the temperature differences in the linear heat source in the water-soil coupled media. Figure 4 illustrates the discrepancies between the numerical values (solid line) and the experimental values (dashed line). The results show that the numerical simulation value can capture the temperature change trend and soil-water interface, and the error of the numerical simulation calculation of the surface temperature of the linear heat source is much smaller than that of the theoretical calculation. The transition section of temperature change at the interface of water and soil is well demonstrated in the numerical simulation results. This is because some details were simplified during the theoretical research; for instance, the change in the radial temperature of the linear heat source in the heat transfer process was ignored, and the linear heat source was assumed to be located in a semi-infinite medium. This confirms