## 3 Results
### Simulation, Hydraulic Model Calibration, and Performance
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Pipe Material** & **Initial Coefficient (C)** & **Calibrated Coefficient (C)** \\ \hline PVC (majority: 34,731 m) & 150 & 100–120 \\ HDPE (significant: 25,044 m) & 155 & 150–155 \\ Ductile iron (minority: 64 m) & 140 & 130–135 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Hazen-Williams coefficient ranges after calibration. Figure 6: Pump performance curve showing the relationship between the flow rate, the head, and the pump efficiency. The additional metrics, beyond the NSE, provide crucial insights into the model's robustness and accuracy. A volumetric efficiency (VE) close to 1 (0.9998 in this case) confirms that the model balances input and output volumes accurately, a critical indicator for mass balance in hydraulic networks. The R\({}^{2}\) value of 0.9002 shows a strong linear correlation between the observed and predicted values. Both mean absolute error (MAE) (0.049) and root mean square error (RMSE) (0.058) are below 0.1, indicating minimal deviation and a close fit between the observed and predicted pressures, consistent with results from [12,29]. Together, these results confirm the model's robustness, despite the data limitations inherent in the Farsala water distribution system. The validated model can serve as a decision support tool for system optimization, leakage detection, and future network upgrades. The summary of model calibration performance metrics is presented in Table 5. This table provides a detailed overview of each evaluation metric, including its mathematical formulation, typical value ranges, accepted benchmarks, and the corresponding values obtained in this study. Formulas for each metric are derived from standard hydrological and statistical modeling references, including [40] for NSE, [43] for MAE and RMSE, and established statistical principles for R\({}^{2}\) and SE. The benchmarks follow common practices for water distribution system modeling (e.g., [12,29]). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Metric** & **Formula** & **Range** & **Benchmark** & **Value** \\ \hline Nash-Sutcliffe & & & & \(\geq\)0.65 (acceptable), & \\   efficiency & \(NSE=1-\frac{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(x_{i}-\overline{x })^{2}}\) & (\(-\infty\), 1) & \(\geq\)0.80 (good), 0.90 & 0.841 \\ Volumetric efficiency & \(VE=1-\frac{\sum_{i=1}^{n}(x_{i}-y_{i})}{\sum_{i=1}^{n}x_{i}}\) & (0, 1) & Close to 1 (ideal) & 0.9998 \\ R\({}^{2}\) (coefficient of & & & \\   determination) & \(R^{2}=\left[\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{ \sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\times\sqrt{\sum_{i=1}^{n}(y_{i} -\overline{y})^{2}}}\right]^{2}\) & (0, 1) & \(>\)0.85 (strong fit) & 0.9002 \\ MAE (mean absolute & & & \\   error) & \(MAE=\frac{\sum_{i=1}^{n}|x_{i}-y_{i}|}{n}\) & (0, +\(\infty\)) & Ideal \(>\) RMSE & 0.049 \\ RMSE (root mean & & & \\   square error) & \(RMSE=\sqrt{\frac{1}{n}{\sum_{i=1}^{n}(x_{i}-y_{i})}^{2}}\) & (0, +\(\infty\)) & \(<\)0.1 (good fit) & 0.058 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model calibration performance metrics. Figure 7: Comparison of the observed vs. simulated elevation of the MBT: (**a**) variations with SCADA data overlaid on the model results for 24 h; (**b**) scatterplot of the observed and simulated values.