## 2 Materials and Methods  
### Mathematical Model for the Study of Thermal Behavior in a Geothermal Heating Process  
The Fourier Equation is a fundamental tool in the study of heat conduction and the thermal behavior of materials. This equation describes how temperature varies across a medium as a function of time and position, allowing for the modeling of heat transfer processes. Its versatility and accuracy make the General Fourier Equation an essential mathematical model for thermal analysis in geothermal heating processes (1). \[\frac{\partial T}{\partial t}=\alpha\cdot\left(\frac{\partial^{2}T}{\partial x ^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{ 2}}\right) \tag{1}\]  
This expression can also be written as follows:  
\[\frac{\partial T}{\partial t}=\alpha\cdot\
abla^{2}\;T \tag{2}\]  
where  
\(\bullet\quad\
abla^{2}\)T is the divergence of the temperature gradient. The analytical model used in this study is based on the solution of the Fourier heat conduction equation for a line source in an infinite, homogeneous medium. The following assumptions apply: the soil is homogeneous and isotropic, the thermal properties are constant, there is no internal heat generation, and the heat transfer is purely radial. These conditions justify the use of the exponential integral function to represent temperature evolution in the ground. However, this approach does not account for advective transport (e.g., groundwater flow), stratification, or anisotropic thermal conductivity. As such, it is best suited for steady geological formations with limited hydrological variability. The solution to this equation is provided by expressions (3) and (4) [18], which describes the temperature at a distance \(r\) from the heat source, for a given time \(t\) (with \(t=0\) being the instant when heat transfer begins), assuming that the initial temperature is uniform \(T_{0}\) and that \(q\) is constant. \[T\;(r,t)-T_{0}=\frac{q}{4\cdot\pi\cdot\lambda}\cdot\int_{r^{2}/4+\alpha\cdot t }^{\infty}\frac{e^{-u}}{u}du=-\frac{q}{4\cdot\pi\cdot\lambda}\cdot Ei\left[- \frac{r^{2}}{4\cdot\alpha\cdot t}\right] \tag{3}\]  
\[T\;(r,t)-T_{0}=-\frac{q}{4\cdot\pi\cdot\lambda}\cdot Ei\left[-\frac{r^{2}}{4 \cdot\alpha\cdot t}\right] \tag{4}\]  
where  
\(\bullet\quad T(r,t)\) [\({}^{\circ}\)C] is the temperature at a distance \(r\) and at a given time \(t\). \(T_{0}\) [\({}^{\circ}\)C] is the initial or reference temperature. * \(q\) [W/m] is the heat injection rate per meter, also referred to as the linear heat rate. * \(r\) [m] is the radial distance from the measurement point to the heat source. * \(\alpha\) [m\({}^{2}\)/s] represents the apparent thermal diffusivity. * \(Ei(x)\) is the exponential integral function given by the following expression: \[Ei(x)=\int\limits_{-\infty}^{x}\frac{e^{u}}{u}du\] (5)  
In this case \(x=\frac{r^{2}}{4\cdot\alpha\cdot t}\) and it is always positive, meaning that the argument of the exponential integral function is always negative. In such cases, to avoid the discontinuity of \(\frac{e^{u}}{u}\) at zero, it is advisable to use the \(E_{1}\) function. \[E_{1}(x)=\int_{x}^{\infty}\frac{e^{-u}}{u}du \tag{6}\]  
This is related to the previous one through the following expression: \(-Ei(-x)=E_{1}(x)\). Therefore, Equation (4) is given by the following:  
\[T\left(r,t\right)=T_{0}+\frac{q}{4\cdot\pi\cdot\lambda}\cdot E_{1}\left[\frac {r^{2}}{4\cdot\alpha\cdot t}\right] \tag{7}\]  
Equation (7) is employed, enabling the evaluation of thermal behavior in one borehole while a geothermal heating process occurs in a nearby borehole. The application of the exponential integral function \(E_{1}(x)\) is complex; therefore, it is commonly replaced by its series expansion:  
\[E_{1}\left[\frac{r^{2}}{4\cdot\alpha\cdot t}\right]=-\gamma-ln\left[\frac{r^{ 2}}{4\cdot\alpha\cdot t}\right]-\sum\limits_{n=1}^{\infty}(-1)^{n}\frac{\left( r^{2}/4\cdot\alpha\cdot t\right)^{n}}{n\;n!} \tag{8}\]  
where  
* \(\gamma\) is [PERSON]'s constant, which has a value of approximately 0.5772.