Wei et al. (2025) Time fractional Saint Venant equations reveal the physical basis of hydrograph retardation through model comparison and field data

Identification

• Journal: Scientific Reports
• Year: 2025
• Date: 2025-11-10
• Authors: Hui Wei, Song Wei, Qi Wang, HongGuang Sun, Jonathan Frame, Yong Zhang
• DOI: 10.1038/s41598-025-23061-4

Research Groups

• School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China
• Center for Applied Mathematics of Guangxi (GUET), Guilin, China
• College of Environmental Science and Engineering, Guilin University of Technology, Guilin, China
• Collaborative Innovation Center for Water Pollution Control and Water Safety in Karst Area, Guilin University of Technology, Guilin, China
• State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing, China
• Department of Geological Sciences, University of Alabama, Tuscaloosa, AL, USA

Short Summary

This study introduces novel fractional-order Saint-Venant equations (FSVEs) for simulating river flow dynamics, demonstrating their superior accuracy in capturing hydrograph retardation, peak attenuation, and tailing behavior compared to traditional models and machine learning approaches, particularly in data-sparse conditions.

Objective

• To develop three novel time-fractional derivative Saint-Venant equations (Constant, Tempered, and Variable) to characterize flow processes in rivers.
• To explore the physical basis of fractional derivative parameters (fractional order and truncation coefficient) by comparing FSVEs with a process-based Distributed Domain Coupling Model (DDCM).
• To validate the effectiveness of the proposed fractional derivative models using real-world field hydrograph data and compare their performance with popular machine learning models.

Study Configuration

• Spatial Scale:
  • Synthetic case: Rectangular river channel, 6000 meters long, 40 meters wide.
  • River Wyre, UK: Observation points 70 kilometers apart, river width ranging from 39 meters to 53 meters (average 44.8 meters).
  • Songzi River, China: River network divided into 10 channels, with lengths from 905 meters to 12,548 meters and widths from 35.3 meters to 673.3 meters.
• Temporal Scale:
  • Synthetic case: Peak flood time of 3600 seconds. Total modeling period up to 4 × 10^7 seconds.
  • River Wyre, UK: Two flood events (December 1960 and January 1969), each lasting less than 200 hours.
  • Songzi River, China: Daily flow data from April 1 to September 30, 2021 (calibration: April 1 to May 30, 2021; validation: June 1 to September 30, 2021). LSTM extended training to October 31, 2022. Numerical simulations used a time step of 10 minutes.

Methodology and Data

• Models used:
  • Constant time-fractional Saint-Venant equations (CtFSVE)
  • Tempered time-fractional Saint-Venant equations (TtFSVE)
  • Variable time-fractional Saint-Venant equations (VtFSVE)
  • Classical Saint-Venant equations (SVE)
  • Distributed Domain Coupling Model (DDCM)
  • Long Short-Term Memory (LSTM) machine learning model (standard and multiplier variants)
  • Numerical solvers: Implicit finite difference scheme for FSVEs; Preissmann difference scheme and alternating direction implicit method for DDCM.
• Data sources:
  • Synthetic data for numerical experiments.
  • Field flood data from the River Wyre, UK (O’Donnell, 1985).
  • Daily flow data from Songzi River, China, provided by the Changjiang Water Resources Commission of the Ministry of Water Resources.
  • River channel geometry (length, width, slope) obtained from Google Earth.

Main Results

• Lower time-fractional derivative (α) values in FSVEs lead to reduced peak discharge, delayed peak arrival, and more pronounced late-time tailing in hydrographs, effectively capturing retardation effects.
• The truncation parameter (λ) in TtFSVE primarily influences the hydrograph tail, shifting it from power-law to exponential decay as λ increases, with minimal impact on the peak.
• VtFSVE captures dynamic river systems where riparian zone properties evolve over time, showing increased cumulative flow with decreasing α(t) (representing increased porosity/permeability) and greater water loss with increasing α(t) (representing decreased porosity/permeability).
• Comparison with the physical-based DDCM model revealed that lower α values in TtFSVE correspond to higher hydraulic conductivity (K) in the riparian zone, and smaller λ values indicate greater riparian zone heterogeneity.
• Field validation on the River Wyre, UK, demonstrated TtFSVE's high accuracy (e.g., R² = 0.9663, RMSE = 6.31 m³/s for Event Dec. 1960) over SVE and DDCM in capturing flood magnitude and pattern.
• For the Songzi River, China, CtFSVE (R² > 0.94, NSE > 0.91) outperformed SVE (R² > 0.91, NSE > 0.89) in simulating successive flood events in a complex river network, particularly in flood peak magnitude and timing.
• CtFSVE also outperformed LSTM models, especially in data-sparse conditions, due to its strong physical basis and direct parameter calibration.
• FSVEs conserve mass, but lower α values indicate a longer duration for full discharge downstream due to increased retention.

Contributions

• Developed and systematically evaluated three novel time-fractional Saint-Venant formulations (CtFSVE, TtFSVE, VtFSVE) within a unified modeling framework.
• Presented the first known application and validation of fractional SVEs using both synthetic benchmarks and real-world field hydrograph data (River Wyre, UK, and Songzi River, China).
• Provided a physical interpretation of fractional derivative parameters (fractional order and truncation coefficient) by comparing FSVEs with a process-based Distributed Domain Coupling Model (DDCM), linking them to riparian zone hydraulic conductivity and heterogeneity.
• Demonstrated the superior performance of FSVEs over traditional SVE and data-driven LSTM models in capturing complex flow dynamics and retardation effects with minimal data, highlighting their parsimony and physical relevance.
• Recommended CtFSVE for practical applications due to its balance of accuracy, simplicity, and the addition of only one parameter (time-fractional derivative) to the classical SVE.

Funding

• National Natural Science Foundation of China (No. 42207063)
• Science and Technology Project of Guangxi (Guike AD23023002)
• Anhui Natural Science Foundation (2108085MA14)
• National Science Foundation, United States (Grant 2412673)

Citation

@article{Wei2025Time,
  author = {Wei, Hui and Wei, Song and Wang, Qi and Sun, HongGuang and Frame, Jonathan and Zhang, Yong},
  title = {Time fractional Saint Venant equations reveal the physical basis of hydrograph retardation through model comparison and field data},
  journal = {Scientific Reports},
  year = {2025},
  doi = {10.1038/s41598-025-23061-4},
  url = {https://doi.org/10.1038/s41598-025-23061-4}
}