## 3 Heat Transfer Theory of Linear Heat Sources in Soil Media  
### Derivation of the Heat Transfer Formula for Linear Heat Sources  
Porous soils and the seabed surfaces are characterized by fluid motion between solid particles, typically induced by tides and waves. Additionally, sediment transport plays a significant role, which in turn impacts heat transfer [23; 24]. In solid media, due to the close contact between molecules, heat is conducted through the vibration and transfer of molecules [25; 26]. The main mode of heat transfer is thermal conduction, while the heat transferred through convection and radiation is negligible and can be disregarded. Thermal conduction refers to the mode of heat transfer where heat is passed through the collisions and transfers between molecules within a material [27; 28]. The speed and intensity of this conduction depend on the material's thermal conductivity and the temperature gradient [29]. Thermal conductivity is used to measure a material's ability to conduct heat, which is the relationship between the amount of heat transferred per unit area per unit time and the temperature gradient; the temperature gradient refers to the change in temperature over a unit length [30]. Heat transfer is the process of heat moving from one object to another, which is divided into transient heat transfer and steady-state heat transfer based on the changes in the temperature field [31; 32]. In transient heat transfer, both the internal temperature and the speed of heat transfer in an object change over time; in steady-state heat transfer, the internal temperature distribution and magnitude do not change over time, and the speed of heat transfer remains constant [33]. For active heating temperature measurement systems, they can be considered models of heating rods buried in soil. When analyzing them, transient heat transfer theory is used to simplify them into a slender, linear heat source in a uniform and isotropic infinite solid environment, and they absorb heat at a constant power, meaning that the amount of heat absorbed per unit length per unit time remains unchanged [34; 35; 36]. The common order of magnitude for the heat diffusion rate in soil media is between \(10^{-6}\) and \(10^{-7}\), indicating that the speed of heat propagation in soil media is very slow. It can be assumed that the length of the linear heat source is sufficiently long, much greater than the distance over which heat propagates in the soil per unit time. Therefore, within the plane along the rod direction, the model can be further simplified to a non-steady-state heat conduction problem for a semi-infinite body [37; 38]. According to the formula for calculating the heat transfer [39; 40],  
\[Q=A\int_{0}^{t}qdt=A\int_{0}^{t}\frac{\lambda(T-T_{\infty})}{\sqrt{\pi\alpha t} }dt=2A\sqrt{\frac{\rho_{solid}c_{solid}\lambda t}{\pi}}(T-T_{\infty}) \tag{13}\]  
In the formula, \(Q\) represents the heating power; \(A\) is the contact area between the linear heat source per unit length and the soil; \(q\) is the heat flux density perpendicular to the direction of the linear heat source; \(\lambda\) is the thermal conductivity of the soil medium; \(T\) is the surface temperature of the linear heat source; \(T_{\infty}\) is the initial temperature of the soil medium; \(\rho_{solid}\) is the density of the soil medium; \(c_{solid}\) is the specific heat capacity of the soil medium. The condition for this equation to hold is that \(T\) does not change with time. However, in actual situations, the surface temperature of the linear heat source varies over time. Here, the concept of differentiation is used, taking an extremely short time element \(\Delta t\), within which the surface temperature of the linear heat source remains constant [41]. Equation (13) is transformed into the following:  
\[\Delta Q=2A\sqrt{\frac{\rho_{solid}c_{solid}\lambda t}{\pi}}(T-T_{\infty}) \tag{14}\]  
In Equation (14), \(\Delta Q\) represents the amount of heat conducted over the time interval \(\Delta t\).