Time fractional Saint Venant equations reveal the physical basis of hydrograph retardation through model comparison...

River hydrodynamics are influenced by numerous factors that traditional models often fail to fully capture. Simulating complex hydrographs can benefit from parsimonious upscaling models, such as fractional derivative equations, that reduce the need to account for all variables. While fractional Saint-Venant equations (SVEs) have been mathematically explored, they lack clear physical interpretation and have not been applied in practical scenarios. This study introduces novel fractional-order Saint-Venant equations (FSVEs) for simulating river flow dynamics, addressing limitations in conventional modeling. Three models-constant, tempered, and variable time-fractional SVEs (CtFSVE, TtFSVE, and VtFSVE)-are developed to capture peak attenuation and tailing more effectively. Numerical experiments indicate that lower time-fractional derivative values enhance retention, producing a lower peak, delayed peak arrival, and pronounced late-time tailing. TtFSVE models transient tailing in hydrographs, VtFSVE captures transient evolution where inflow and outflow differ, and CtFSVE balances accuracy and simplicity with a single added parameter for various hydrographs. In the simulation of real-world hydrograph data, the fractional SVEs show high predictive accuracy. However, they should be regarded as effective proxy models that require parameter calibration and are not yet fully 'plug-and-play' predictive models. Comparative analysis with the Long Short-Term Memory (LSTM) machine learning model and distributed domain coupling model (DDCM) shows CtFSVE's superior performance in capturing complex flow dynamics with minimal data, while field validation demonstrates its accuracy over traditional SVE, underscoring its practicality for complex river networks. The fractional engine shows promise as an effective tool for upscaling surface flow without the prohibitive burden of mapping detailed system heterogeneity.