# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 4 Case Study
### Correlation Analysis and PCA of Total Water Losses  
4.4.1. Correlation Analysis  
A correlation analysis was performed for the available dataset of measuring sessions total water loss of 50-79. The values of Pearson's correlation coefficient are given in Table 2.  
The correlation coefficient values in Table 2 indicate that most of the input variables can be considered linearly independent since most correlation coefficient values are less than 0.7 [17]. The correlation coefficient between variables \(H_{p3}\) and \(H_{p4}\) is 0.873, which indicates a strong correlation. This is to be expected because the piezometers are at a close distance and under the same geological conditions. The concrete temperature on the extrados of the concrete lining \(T_{c}\) is in a strong correlation (0.840) with the water temperature \(T_{w}\), which is reasonable because the temperature is measured in the interior and exterior of the concrete lining, so these temperatures are interdependent. Variables \(T_{c}\) and \(T_{w}\) are in a very strong negative linear correlation with the sizes of crack openings \(cr_{3}\) and \(cr_{4}\). This can be explained by the fact that at high temperatures there is an increase in the expansion of concrete, and consequently the cracks are closed, and in the case of lower temperatures, the concrete contracts and the cracks open up. The average water level during a single measurement step, \(H_{st}\) has a strong influence on total water loss, the coefficient of correlation being 0.758. Variables \(cr_{3}\) and \(cr_{4}\) are in positive correlation with variable \(q\), because the increase in crack openings leads to an increase in total water losses. The increase in water and concrete temperature leads to expansion of the tunnel's concrete lining, thereby reducing the size of crack openings which leads to a reduction of water losses from the tunnel. For these reasons, concrete and water temperatures are in negative correlation with the variable \(q\), with the correlation coefficients being \(-0.257\) and \(-0.291\), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(H_{r}\) & \(H_{p3}\) & \(H_{p4}\) & \(T_{c}\) & \(T_{w}\) & \(H_{st}\) & \(cr_{3}\) & \(cr_{4}\) & \(t\) & \(q\) \\ \hline  & **[m a.s.l.]** & **[m a.s.l.]** & **[m a.s.l.]** & [\({}^{\circ}\)C] & [\({}^{\circ}\)C] & **[m a.s.l.]** & **[mm]** & **[mm]** & **[months]** & **[L/s]** \\ \hline Min. & 854.380 & 862.950 & 862.780 & 2.520 & 3.190 & 831.825 & \(-0.722\) & -0.033 & 4.0 & \(-0.190\) \\ Max. & 880.260 & 879.650 & 879.550 & 14.560 & 21.000 & 879.996 & 0.129 & 0.424 & 139.0 & 25.511 \\ Average & 874.660 & 874.379 & 873.697 & 9.100 & 11.730 & 860.617 & \(-0.480\) & 0.103 & 72.0 & 9.845 \\ St. dev. & 5.116 & 3.793 & 3.888 & 3.105 & 4.377 & 10.640 & 0.224 & 0.122 & 41.3 & 5.176 \\ \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of measured values for measurement sessions 50–79. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \(H_{r}\) & \(H_{p3}\) & \(H_{p4}\) & \(T_{c}\) & \(T_{w}\) & \(H_{st}\) & \(cr_{3}\) & \(cr_{4}\) & \(t\) & \(q\) \\ \hline  & **[m a.s.l. ]** & **[m a.s.l. ]** & **[m a.s.l. ]** & [\({}^{\circ}\)C] & [\({}^{\circ}\)C] & **[m a.s.l. ]** & **[mm]** & **[mm]**Since the measurement of total losses is carried out at a constant water level in the upper reservoir, in piezometers, and under variable conditions in the surge tank, it is therefore expected that the coefficient of correlation between \(H_{r}\), \(H_{p3}\), and \(H_{p4}\) on \(q\) is small. Although these variables individually have little linear effect on the output (the correlation coefficients are significantly lower than 0.3), in order to initially include a more comprehensive set of measured quantities, their interaction effects, and derived variables, the mentioned variables are taken into account for further consideration.