## 4 Model Implementation and Simulation Using TRNSYS
The results shown in Figure 6 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 readings 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 B (Table 2). The comparison between both datasets (blue and red) suggests limitations in the approximation used. Figure 5: TRNSYS simulation using the simplification of the exponential integral function (term A). Figure 6: TRNSYS simulation using the simplification of the exponential integral function (term B). Simulation in TRNSYS Using the Simplification of the Exponential Integral Function, Employing Term L  
Finally, we tested a more detailed expansion (term L) to assess whether a higher-order approximation yields a closer match to the experimental data, particularly during the long-term phase. The results shown in Figure 7 are as follows:  
* Experimental data (in blue) represent the actual measurements obtained from the Q-Thermie-Unoioi 2 borehole at a depth of 22.62 m. These data correspond to the readings 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 L (Table 2). During the early stages of the simulation (t < 1000 h), a larger discrepancy is observed between the experimental data and the values produced by the model. These differences can be attributed to several factors not considered in the current implementation, such as simplified boundary conditions, which assume an infinite homogeneous medium with no groundwater flow or surface temperature variations. In addition, initial transient effects, thermal inertia of the ground, possible local heterogeneities in the subsurface, and non-uniform initial temperature fields may influence the thermal response in the early phase. The sensor system's response time may also play a role, particularly when temperature gradients are more pronounced. As the system evolves and reaches thermal equilibrium, these discrepancies diminish, as reflected in the high correlation observed for when greater than 1000 h (R\({}^{2}\) = 0.997 corresponding to a 95% goodness of fit), confirming the model's validity in the quasi-steady regime. Additionally, the following indicators have been calculated: Root Mean Square Error (RMSE) = 0.396 and Mean Absolute Error (MAE) = 0.01. As time progresses (t > 1000 h), the divergence between the experimental data and the TRNSYS simulation decreases significantly, suggesting that the model's accuracy improves with the inclusion of additional terms in the series expansion. For t > 1000 h, the high correlation coefficient (R\({}^{2}\) = 0.997) indicates excellent agreement between the simulated and measured experimental values, supporting the validity of the model based on the series expansion of the exponential integral function. This deviation is attributed to model simplifications, including the assumption of a homogeneous, isotropic, and infinite medium, the exclusion of axial heat conduction and  
Figure 7: TRNSYS simulation using the simplification of the exponential integral function (term L). groundwater flow, and a uniform initial ground temperature. The ground may exhibit stratification, anisotropy, or temperature gradients near the borehole. Additional uncertainty may stem from sensor placement or thermal inertia. These effects are more prominent during the early phase of the experiment and gradually diminish as the system stabilizes, which explains the improved match at later times. These results suggest that the methodology implemented in TRNSYS is a reliable approach for predicting thermal behavior under similar conditions. Furthermore, with appropriate adjustments, it could be extended to other scenarios.