# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 5 Discussion
The graphical comparison of the mentioned models, for the last four measurement sessions, together with confidence intervals, is given in Figure 8. In Figure 10, the comparison is made between the measured total water losses, the modeled total water losses using the model suggested in this paper (10), and the modeled total water losses using the model suggested by [PERSON] et al. [4]. Further improvements in the quality of the mathematical model of water loss in the HT of PSHPP \"Bajina Basta\" are described in the following two paragraphs. It should be emphasized that the model developed in this paper also takes into account more input variables (size of crack openings and tunnel aging) than the previous reference model. The sizes of crack openings directly affect the increase or decrease in total water losses, which is important for the quality and accuracy of the proposed model. Consideration of the time component as an input variable gives the possibility of taking into account the irreversible process of concrete and structure aging, but also that, which is most important in the SaFE context, the effect of tunnel reparation on the reduction of total water losses in HT. The advantage of the developed model is also the property of additivity, as there is an implicit assumption that the different model (10) regressors affect water losses in the HT additively. The aggregate and individual effects of PCs and regressors derived from them further shed light upon the process of water losses in the HT (Figure 9). For example, for the measurement sessions in which the temperatures of concrete were 10.7 and 5.4 \({}^{\circ}\)C (Figure 9b), the thermal effect is inversely proportional to water losses and amounts to \(-\)0.73 and 3.31 L/s (Figure 9a). From the SaFE aspect, the exponential influence of time elapsed from last tunnel reparation date in 2005. (predictor t) is particularly interesting. The diagrams shown in Figures 5 and 9a show that the impact of the last tunnel reparation \begin{table}
\begin{tabular}{c c c} \hline \hline  & **LDS** & **TDS** \\ \hline Model (10) adopted in this paper & 1.8842 & 2.0161 \\ Previous reference model & 3.954 & 3.5983 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: RMSE [L/s] of HTs total water loss. Figure 10: Measured values of total water losses, model values of total losses by the model in this paper and by the model from [PERSON] et al. [4], for the period 2005–2015. on water losses decreases by the exponential law and becomes negligible after 8-10 years, i.e., 130 measurements, approximately. The safety of the tunnel cannot be analyzed using the model of total water losses. For this, other methods based on numerical models are needed (FEM, DEM, PDEM, etc. ), which were not the subject of research in this paper. For further research, an integral approach can be applied: the usage of numerical and statistical models. When the tunnel is reopened (upon its inspection), it is necessary to record the condition of the tunnel, conduct mapping of all the cracks in the tunnel, categorize the cracks, and form a three-dimensional numerical model that can analyze the behavior of the tunnel with regards to filtration and stress-strain phenomena. After that, a calibration of the numerical model with the total water loss measurements can be conducted, in order to obtain the best correlation between the calculated losses in the model and the measured total losses. This results in a model that takes into account the realistic field conditions and gives the possibility for a safety analysis of the tunnel. However, the use of statistically based models is an economical method that does not demand complex computing resources and know-how. It can be used daily, and it provides a fast assessment of the conditions in the tunnel and its functionality. For more complex analyses of the conditions in the tunnel and its functionality, more complex numerical models that can take into account significant phenomena of tunnel behavior in interaction with water should be used.