Coupled rigid water column–global gradient formulation for leakage assessment in looped water distribution networks

Leakage in water distribution networks remains a major contributor to non-revenue water worldwide, posing significant economic and environmental challenges. Accurate modelling of leakage under transient conditions is therefore critical for improving pressure management and understanding dynamic leakage behaviour in real systems. This study developed an extended hydraulic formulation that integrates the rigid water column model with the global gradient algorithm through a time-dependent valve resistance coefficient to analyse actual water losses in looped water distribution networks. The formulation introduces a generalised hydraulic-loss operator that extends the classical steady-state framework to reproduce inertial effects and the dynamic behaviour of valves under slow transients, while maintaining compatibility with conventional hydraulic solvers. This methodological innovation enables realistic simulation of valve manoeuvres without resorting to full water-hammer models, thereby bridging quasi-steady and transient approaches. Validation in a looped network with pressure-reducing valves was performed by comparing the results against the extended period simulation (EPS). The results demonstrated that considering inertia and the time-dependent valve resistance significantly altered leakage evolution, resulting in variations up to 12.5% in accumulated leakage and non-revenue water. The parameter representing the difference in non-revenue water between the proposed model quantifies the additional transient-induced losses, providing a practical indicator for short-term pressure management and leakage control. This framework supports safer and more efficient operation of pressure-managed systems and contributes to sustainable water-engineering practice in alignment with the United Nations Sustainable Development Goal 6. It also paves the way for future integration with data-driven and machine learning approaches.