## 4 Leak Detection Technologies
### Software-Based Methods
#### 4.2.4 Numerical Methods  
Numerical modeling techniques for detecting water leakage primarily involve the conventional hydraulic modeling approach, used through specialized software tools, and the methods based on transient events. Hydraulic modeling is another software-based tool for monitoring and managing water supply networks in the present and the future. This technique is used to anticipate problems in planned networks, design effective interventions, and operate and manage water supply systems. The hydraulic modeling involves organizing data into a hierarchical structure and calibrating the model until it closely reflects reality. However, model calibration remains a challenging issue [63]. The calibration process aims to reduce the discrepancies between the model results and measured values by adjusting model parameters, such as roughness, water consumption, and water losses [64]. The genetic algorithm is a commonly used method for calibrating hydraulic models [65; 66; 67]. The accuracy of the hydraulic model can be influenced by various factors, such as the quality of the database used (e.g., topography, pipe diameter, and length), uncertainty in valve status, and the accuracy of measurement systems (such as sensors and data loggers). The hydraulic model can estimate the amount of leakage within specific areas by reducing the differences between the measured and numerically estimated pressure and flow rates [68; 69]. Additionally, it can serve various purposes for managing leakage, such as modeling leakage as pressure-dependent demand [70; 71; 72] and creating pressure management plans to control leakage [73]. As hydraulic modeling continues to play a significant role in water network management, future research should focus on enhancing model calibration techniques, improving data accuracy, and developing innovative strategies for more proactive and precise water supply network management. Inverse transient analysis (ITA) is a sophisticated method used for detecting leaks in water distribution systems by analyzing pressure transients that propagate through the pipeline due to disturbances, such as valve closures or pump failures. This method operates by comparing the measured transient pressure data from the pipeline with simulated data obtained from a mathematical model of the system. The goal is to minimize the difference between the actual and simulated data, treating the leak detection problem as an inverse problem solved using optimization algorithms. ITA has been particularly successful in identifying leaks by exploiting pressure waves' sensitivity to anomalies, such as leaks, within the pipeline [74]. [PERSON] et al. [75] applied ITA in controlled laboratory conditions and quasi-field environments, demonstrating its efficacy in identifying leak locations and sizes when the system's physical and hydraulic characteristics are well known. Their findings showed that the accuracy of ITA depends significantly on accurate system modeling, making it highly suitable for controlled environments. [PERSON] and [PERSON] [76] further investigated ITA by applying it to polyethylene pipe systems at Imperial College London and Thames Water Utilities. The results of their study confirmed that ITA could successfully detect and locate leaks, provided that the pipeline's boundary conditions and physical characteristics were well understood and controlled. However, they also highlighted the challenges of extending ITA to real-world applications, where noise and system uncertainties can interfere with the accuracy of leak detection. To address the computational and noise-related challenges of ITA, recent studies have incorporated advanced optimization techniques. [PERSON] et al. [77] introduced a genetic-algorithm-based ITA framework, which significantly reduced the computation time and enhanced the leak localization accuracy, even in systems affected by noise. Despite its advantages, ITA faces limitations when applied to large-scale systems with complex topologies, noise interference, and uncertain boundary conditions.