Empirical Modeling of Flow Behavior over Stepped Weirs Using Brink Depth for Sustainable Irrigation Applications

Very Dermawan; Edijatno; Runi Asmaranto; Evi Nur Cahya; Nenny Roostrianawaty

doi:10.48084/etasr.16662

Authors

Very Dermawan Department of Water Resources Engineering, Faculty of Engineering, Universitas Brawijaya, Indonesia
Edijatno Department of Civil Engineering, Faculty of Civil Planning and Geo Engineering, Institut Teknologi Sepuluh Nopember, Indonesia
Runi Asmaranto Department of Water Resources Engineering, Faculty of Engineering, Universitas Brawijaya, Indonesia
Evi Nur Cahya Department of Water Resources Engineering, Faculty of Engineering, Universitas Brawijaya, Indonesia
Nenny Roostrianawaty Department of Water Resources Engineering, Faculty of Engineering, Universitas Brawijaya, Indonesia

Volume: 16 | Issue: 3 | Pages: 35620-35629 | June 2026 | https://doi.org/10.48084/etasr.16662

Received: 2 December 2025 | Revised: 23 December 2025, 16 January 2026, 2 February 2026, 24 February 2026, and 18 March 2026 | Accepted: 4 April 2026 | Online: 6 June 2026

Corresponding author: Very Dermawan

Abstract

Accurate and efficient discharge estimation is significant for optimizing irrigation system performance, particularly in resource-constrained regions such as Indonesia. Conventional discharge measurement techniques often rely on auxiliary equipment that is vulnerable to damage and requires extensive maintenance, especially in stepped-weir structures. This study introduces brink depth (y_b) as a practical and reliable parameter for predicting discharge and energy dissipation over inclined smooth and stepped weirs. Laboratory experiments were conducted with six slope angles (26.57°-72°), five step numbers (N = 2-32), and discharge rates ranging from 1.73 to 6.15 L/s. The results demonstrated a strong correlation between brink depth and discharge, enabling accurate prediction of energy dissipation. The maximum average relative energy loss reached 55.64% for the inclined smooth weir and 92.58% for the stepped weir. The proposed empirical equations yielded a Mean Absolute Percentage Error (MAPE) of 10.41% for the smooth weir and 12.96% for the stepped weir, confirming their robustness and applicability. This study provides a cost-effective, field-oriented methodology for discharge estimation and supports sustainable irrigation management.

Keywords:

brink depth, stepped weir, discharge estimation, energy dissipation, irrigation, hydraulic modeling

References

N. K. Alomari, M. S. Khaleel, A. Y. Mohammed, and I. A. Juma, "Discharge formula based on brink depth over sharp-crested weirs," Water Supply, vol. 24, no. 2, pp. 615–624, Feb. 2024.

R. H. Irzooki and N. A. Saoud, "Experimental Investigation for Free Overfall of Flow in Semi-circular Channels," IOP Conference Series: Earth and Environmental Science, vol. 1120, Dec. 2022, Art. no. 012010.

S. Hussein and M. Shamkhi, "Experimental Study of Flow Regimes of Stepped Weir," Wasit Journal of Engineering Sciences, vol. 11, no. 2, pp. 82–93, Aug. 2023.

U. A. Jahad et al., "Dissolved Oxygen Variation on the Steps with a Quarter Circle End Sill for Flows over the Stepped Spillways," International Journal of Design & Nature and Ecodynamics, vol. 17, no. 5, pp. 639–648, Oct. 2022.

F. Gorgin and A. R. Vatankhah, "Rectangular top-hinged plate as portable flow measuring device," Water Supply, vol. 22, no. 12, pp. 8637–8658, Dec. 2022.

A. H. Azimi and N. Rajaratnam, "A Note on the Discharge over Full-width Rectangular Sharp-‎crested Weirs and Weirs of Finite Crest Length‎," Journal of Applied and Computational Mechanics, vol. 9, no. 2, pp. 458–463, Oct. 2022.

F. Alizadeh Sanami, M. H. Afshar, and M. Saneie, "Experimental study on the discharge coefficient of triangular piano key weir*," Irrigation and Drainage, vol. 71, no. 2, pp. 333–348, Apr. 2022.

S. Katuwal, G. M. Johnson, A. J. Craig, N. P. Rogovska, T. M. Isenhart, and R. W. Malone, "Calibration of V-Notch and Compound Weirs for Subsurface Drainage Water Level Control Structures," Applied Engineering in Agriculture, vol. 40, no. 4, pp. 453–463, 2024.

M. O. A. Alsaydalani, "Discharge Coefficient of a Two-Rectangle Compound Weir combined with a Semicircular Gate beneath it under Various Hydraulic and Geometric Conditions," Engineering, Technology & Applied Science Research, vol. 14, no. 1, pp. 12587–12594, Feb. 2024.

M. Ibrahim, "A Hydraulic Study of Sharp Crested Weir with Orifices," Mansoura Engineering Journal, vol. 39, no. 3, July 2020, Art. no. 5.

M. Hamed, "An Experimental Study of Flow over Rectangular Broad Crested Weir," ERJ. Engineering Research Journal, vol. 46, no. 1, Nov. 2022.

S. M. H. Al-Mehmdy and A. T. Fal-Issawi, "Effect of Interval and Depth Irrigation on Water Use Efficiency, Cucumber Productivity under Green House Conditions and Drip Irrigation," IOP Conference Series: Earth and Environmental Science, vol. 1252, Dec. 2023, Art. no. 012052.

S. M. Seyedian, A. Haghiabi, and A. Parsaie, "Reliable prediction of the discharge coefficient of triangular labyrinth weir based on soft computing techniques," Flow Measurement and Instrumentation, vol. 92, Aug. 2023, Art. no. 102403.

S. Saffar, M. S. Babarsad, M. M. Shooshtari, M. Hosein Poormohammadi, and R. Riazi, "Prediction of the discharge of side weir in the converge channels using artificial neural networks," Flow Measurement and Instrumentation, vol. 78, Apr. 2021, Art. no. 101889.

M. S. B. Shokrana and E. Ghane, "An Empirical V-notch Weir Equation and Standard Procedure to Accurately Estimate Drainage Discharge," Applied Engineering in Agriculture, vol. 37, no. 6, pp. 1097–1105, 2021.

O. J. Gericke and V. H. Williams, "Could a one-size-fits-all approach apply to the extension of stage-discharge relationships at flow-gauging weirs?," Journal of the South African Institution of Civil Engineering, vol. 65, no. 2, pp. 17–27, June 2023.

J. H. Sahib, L. K. Al-Waeli, and A. H. Jaber Al Rammahi, "Utilization of ANN technique to estimate the discharge coefficient for trapezoidal weir-gate," Open Engineering, vol. 12, no. 1, pp. 142–150, Mar. 2022.

A. B. Amsie, A. T. Ayalew, Z. M. Mada, and M. M. Finsa, "Acclimatize experimental approach to adjudicate hydraulic coefficients under different bed material configurations and slopes with and without weir," Heliyon, vol. 10, no. 11, June 2024, Art. no. e32162.

A. N. Altalib, "Discharge coefficient of flow over Al-Shalalat stepped weir on Al-Khusr River," Applied Water Science, vol. 11, no. 2, Feb. 2021, Art. no. 16.

N. Rajaratnam, "Skimming Flow in Stepped Spillways," Journal of Hydraulic Engineering, vol. 116, no. 4, pp. 587–591, Apr. 1990.

U. A. Jahad et al., "Flow characteristics and energy dissipation over stepped spillway with various step geometries: case study (steps with curve end sill)," Applied Water Science, vol. 14, no. 3, Mar. 2024, Art. no. 60.

N. Aein, M. Najarchi, S. M. Mirhosseini Hezaveh, M. M. Najafizadeh, and E. Zeighami, "Simulation and prediction of discharge coefficient of combined weir–gate structure," Proceedings of the Institution of Civil Engineers - Water Management, vol. 173, no. 5, pp. 238–248, Oct. 2020.

R. Gharib, M. Heydari, S. Kardar, and S. Shabanlou, "Simulation of discharge coefficient of side weirs placed on convergent canals using modern self-adaptive extreme learning machine," Applied Water Science, vol. 10, no. 1, Jan. 2020, Art. no. 50.

M. M. Hameed, M. K. AlOmar, F. Khaleel, and N. Al-Ansari, "An Extra Tree Regression Model for Discharge Coefficient Prediction: Novel, Practical Applications in the Hydraulic Sector and Future Research Directions," Mathematical Problems in Engineering, vol. 2021, Sept. 2021, Art. no. 7001710.

M. S. Saleh, S. A. M. Al-Hashimi, and A.-S. T. Al-Madhhachi, "Effect of Tail Water Depth on Characteristics of Hydraulic Jump Formed Downstream of Stepped Weir," Mathematical Modelling of Engineering Problems, vol. 11, no. 11, pp. 2919–2928, Nov. 2024.

M. Napierała, "Application of Simple Crested Weirs to Control Outflows from Tiles Drainage," Water, vol. 15, no. 18, Sept. 2023, Art. no. 3248.

K. R. Gubashi, S. Mulahasan, Z. A. Hacheem, and A. Q. Rdhaiwi, "Effect of the Stepped Spillway Geometry on the Flow Energy Dissipation," Civil Engineering Journal, vol. 10, no. 1, pp. 145–158, Jan. 2024.

A. Ghaderi and S. Abbasi, "Experimental and Numerical Study of the Effects of Geometric Appendance Elements on Energy Dissipation over Stepped Spillway," Water, vol. 13, no. 7, Mar. 2021, Art. no. 957.

H. H. Albank and S. I. Khassaf, "An Experimental Investigation of Energy Dissipation for Stepped Spillways with Different Flow Conditions," Mathematical Modelling of Engineering Problems, vol. 10, no. 1, pp. 340–346, Feb. 2023.

M. S. Jomaa, and A. Y. Mohammed, "Flow and Energy Dissipation Over a Cylindrical Stepped Weir," Advanced Engineering Letters, vol. 1, no. 2, pp. 57–64, July 2022.

R. Norouzi, P. Sihag, R. Daneshfaraz, J. Abraham, and V. Hasannia, "Predicting relative energy dissipation for vertical drops equipped with a horizontal screen using soft computing techniques," Water Supply, vol. 21, no. 8, pp. 4493–4513, Dec. 2021.

R. Daneshfaraz, S. Sadeghfam, V. Hasanni̇Ya, J. Abraham, and R. Norouzi̇, "Experimental Investigation on Hydraulic Efficiency of Vertical Drop Equipped with Vertical Screens," Teknik Dergi, vol. 33, no. 5, pp. 12379–12399, Sept. 2022.