Document Type : Original Article

Author

• Kamran Mohammadi

Assistant Professor Department of Water Engineering, College of Agriculture and Natural Resources, Razi University, Kermanshah, Iran

Abstract

Introduction

Accurate prediction of backwater profiles in subcritical open-channel flows is of fundamental importance for hydraulic structure design, flood management, and operational planning. Both analytical and numerical approaches have been widely applied in this context, yet their relative performance, computational efficiency, and robustness under practical conditions require systematic evaluation. This study investigates and compares three step-by-step numerical methods—fourth-order Runge–Kutta, Kutta–Merson, and Trapezoidal—with three integral analytical approaches—Bresse, Bakhmeteff, and Bakhmeteff–Chow—in simulating backwater curves. The analytical formulations were implemented in Wolfram Mathematica using numerical quadrature, while the numerical schemes were coded in Visual Basic. The case study is based on a trapezoidal-section concrete channel within the Dez Dam irrigation network, where accurate backwater prediction is critical for operational safety and water delivery efficiency.

Method

The analysis employed a combination of numerical integration for the analytical solutions and explicit/implicit marching schemes for the numerical methods. Statistical performance indicators included Relative Root Mean Square Error (Rel.RMSE), Relative Bias (Rel.Bias), and relative error compared to a high-resolution reference solution. Computational performance was evaluated through execution time measurements in both Mathematica and Visual Basic. Sensitivity analysis was carried out for key hydraulic parameters—bed slope, Manning’s coefficient, channel width, and discharge—by applying ±20% variations and assessing their impact on backwater length and prediction accuracy. Stability and convergence behavior were examined by varying step sizes (numerical methods) and integration tolerances (analytical methods).

Results

Numerical methods, particularly Kutta–Merson, demonstrated superior accuracy with Rel.RMSE below 0.5% and Rel.Bias below 0.3%, while maintaining very short computation times between 4.1–6.1 ms. Analytical methods, despite delivering acceptable accuracy (Rel.RMSE < 1%), required significantly longer runtimes (21.3–24.1 ms) due to multiple quadrature iterations and showed greater sensitivity to integration tolerance. Sensitivity analysis revealed that a 20% reduction in bed slope increased backwater length by over 30%, while a 20% increase in Manning’s coefficient nearly doubled the prediction error. Variations in discharge and channel width had smaller effects on both profile length and statistical errors. In terms of stability, all step-by-step numerical methods operated without oscillations or divergence across all test scenarios and exhibited uniform convergence with smaller step sizes. Analytical methods required stricter tolerance settings to prevent accuracy degradation, highlighting their dependence on quadrature precision.

Conclusions

The findings indicate that adaptive numerical methods—particularly Kutta–Merson—offer the most favorable balance between accuracy, computational speed, and stability, making them well-suited for real-time applications and operational decision-making in hydraulic engineering. Analytical methods, implemented via Mathematica, provide a valuable reference for baseline studies, calibration, and verification of numerical models, but their higher computational demand and sensitivity to integration tolerance limit their practicality in time-critical simulations. Overall, the complementary use of adaptive numerical schemes for routine operations and analytical methods for validation purposes can yield a robust and efficient modeling framework for backwater prediction in subcritical open-channel flows.

Author Contributions

All authors participated in writing and revising the article.

Conflict of Interest

Authors declared no conflict of interest.

Keywords

• Backwater profile
• Wolfram Mathematica
• Numerical and analytical methods
• Sensitivity analysis
• Stability and convergence analysis

Main Subjects

• Water and hydraulic structures