## 4 Model Implementation and Simulation Using TRNSYS  
To analyze the thermal interference between shallow geothermal boreholes, the geothermal heat pump (GHP) package of TRNSYS was employed. Using this package, the influence of the simplification of the exponential integral was studied, and the following approach was adopted. Starting from Table 2, instead of using the value of the exponential integral function, terms from the series expansion shown in Equation (8) were calculated for the temperature sensor located at a depth of 22.62 m from the surface, whose thermal conductivity and diffusivity are known. Figure 1: Graphical representation of exponential integral function for the data shown in Table 1 (Equation (7)). Figure 2: Representation of the experimental data at a depth of 22.62 m in the Q-Thermie 2 borehole during the thermal heating process in the Q-Thermie 1 borehole. The number of terms in the series expansion of the exponential integral function was determined empirically through an iterative process. We started with a minimal number of terms (e.g., 1-2) and increased them gradually until the simulated temperature curve showed strong agreement with the experimental data, particularly after 1000 h, where transient effects are more dominant. To evaluate convergence, we monitored the coefficient of determination (R\({}^{2}\)) between the simulated and measured temperatures at the sensor depth (22.62 m). When adding more terms yielded a negligible improvement (\(\Delta\)R\({}^{2}\) < 0.003), the expansion was considered to be sufficiently accurate. This process ensures both numerical efficiency and model fidelity and can be replicated in other scenarios using the same strategy: iteratively increasing the number of terms and monitoring the impact on model performance. The model is implemented using three modules:  
* Type 9e represents the temperature data recorded at the Q-Thermie-Uniovi-2 borehole at a depth of 22.62 m over a period of 3 months. * The element labeled 1410 evaluates the thermal behavior of the borehole through a series expansion of the exponential integral function. This requires defining the power applied during the TRT, the depth of the Q-Thermie-Uniovi-2 borehole, the thermal conductivity and diffusivity at a depth of 22.62 m, and the distance between boreholes 1 and 2 (Figure 3). \begin{table}
\begin{tabular}{c c} \hline \hline
**Equation** & **Term** \\ \hline \(-\gamma-Ln\,\left(\frac{r^{2}}{4\cdot 4\cdot t}\right)\) & \(A\) \\ \hline \(-  
* Type65c allows for a graphical representation of the solution to the proposed problem. Figure 4 illustrates the implementation of the model in the TRNSYS graphical environment. Simulation in TRNSYS Using the Simplification of the Exponential Integral Function, Employing Term A  
The results shown in Figure 5 are as follows:  
* Experimental data (in blue) represent the actual measurements obtained from the Q-Thermie-Uniovi 2 borehole at a depth of 22.62 m. These data correspond to the recordings of the sensor laid at this precise location. * TRNSYS simulation (in red) displays the simulation performed using the TRNSYS software, which employs a series expansion designated as term A (Table 2). This simplification, with a fixed term of the exponential integral function, aims to model the behavior observed in the experimental data. Figure 4: Graphical environment of the TRNSYS software created to carry out the study of thermal interference between boreholes. Figure 3: User-defined features in Type 1410. Simulation in TRNSYS Using the Simplification of the Exponential Integral Function, Employing Term B  
To further evaluate the model's accuracy, we increased the number of terms in the series expansion (term B) and repeated the simulation, aiming to reduce the discrepancies observed in the initial approach.