## 4 Heat Transfer Laws of Linear Heat Sources in Water-Soil Coupled Media  
From Sections 2 and 3, it is known that linear heat sources exhibit different rates of temperature change in the two different media of soil and water. Therefore, Equations (12) and (18)-(20) are used to calculate the variation in the surface temperature of the linear heat source over time in both soil and water media.  
Taking an iron linear heat source as an example, we assume its heat source radius is \(r=0.05\) m, the length of the heat source in the soil body is sufficiently long, and the length in seawater is also \(l=1.00\) m. The heat source density is \(\rho=7850\) kg/m\({}^{3}\), the specific heat capacity of seawater is \(c=460\) J/(kg.\({}^{\circ}\)C), the convective heat transfer coefficients in seawater are \(h=50\) W/(m\({}^{2.\circ}\)C), \(h=500\) W/(m\({}^{2.\circ}\)C), \(h=1500\) W/(m\({}^{2.\circ}\)C) and \(h=3000\) W/(m\({}^{2.\circ}\)C), the ambient temperature of seawater and viscous clay is \(T_{\infty}=25\)\({}^{\circ}\)C, the thermal conductivity of the soil body is \(\lambda=0.25\) J/(m\({}^{\circ}\)C-s), the density of the soil body is \(\rho_{soil}=2000\) kg/m\({}^{3}\) and the specific heat capacity of the soil body is \(c_{solid}=820\) J/(kg.\({}^{\circ}\)C).  
As shown in Figures 10 and 11, there is a significant difference in temperature variation in the iron linear heat source between water and soil media. A distinct temperature gradient is generated at the water-soil interface, which becomes more pronounced with an increase in the convective heat transfer coefficient of the linear heat source in water. When the convective heat transfer coefficient \(h=50\) W/(m\({}^{2.\circ}\)C), the temperature difference in the water-soil media within 10 min is not significant. Subsequently, the temperature difference gradually increases with time, reaching a maximum within 60 min. When the  
Figure 9: The distribution of the soil temperature field during the heating process.  
convective heat transfer coefficient exceeds 500 W/(m\({}^{2.\circ}\)C), the temperature difference pattern of the linear heat source in water-soil media becomes more consistent, with all coefficients reaching a maximum value at 10 min (with a temperature difference of 30.5 \({}^{\circ}\)C at \(h\) = 500 W/(m\({}^{2.\circ}\)C), 31.9 \({}^{\circ}\)C at \(h\) = 1500 W/(m\({}^{2.\circ}\)C) and 31.9 \({}^{\circ}\)C at \(h\) = 3000 W/(m\({}^{2.\circ}\)C); afterwards, it gradually decreases within 60 min. This is because, during the cooling phase, when the linear heat source is in the soil medium, the low thermal diffusivity of the soil leads to a slower heat transfer rate, resulting in a lower rate of surface temperature decrease in the linear heat source. Conversely, when the linear heat source is in the water medium, the good thermal diffusivity of water allows heat to be quickly transferred from the linear heat source to the surrounding environment, leading to a higher rate of surface temperature decrease. Therefore, by monitoring the changes in the surface temperature of the heat source, one can analyze the temperature variation trend of the linear heat source in different media over time, and thus distinguish the different medium interfaces where the linear heat source is located.  
Figure 10: The temperature distribution of the heat source in a water–soil coupled medium under different convective heat transfer coefficients.