Machine learning-based discharge coefficient estimation in trapezoidal-arched labyrinth weirs

Abstract

Weirs represent a frequently employed mechanism for regulating water surface elevations and managing flow within canals and hydraulic infrastructures. Among these, labyrinth weirs constitute a distinctive variant capable of accommodating a specific discharge while maintaining a reduced upstream water level compared to conventional linear weirs. The present investigation delved into the evaluation of the effectiveness of multilayer perceptron (MLP) networks, support vector machine (SVM), gene expression programming (GEP), and multivariate adaptive regression splines (MARS), aiming to predict the discharge coefficient (C_d) of a trapezoidal-arched labyrinth weir with an expanded central cycle. A dataset including 108 laboratory observations was utilized. The dimensionless parameters were obtained from the parameters including inside apex width of the middle cycle (w₁), inside apex width of the end cycles (w₂), weir height on the upstream side (B), unsubmerged total upstream head on the weir (H_d), and gravitational acceleration (g). The model was developed with the dimensionless parameters and C_d. Root mean square error (RMSE), determination coefficient (R²), mean absolute error (MAE), and developed discrepancy ratio (DDR) were used as performance assessment criteria. Based on these metrics, all four models exhibited the latent capacity to predict the C_d value. However, the MLP model demonstrated superior performance among the models during both training (RMSE = 0.024, MAE = 0.020, R² = 0.816, and C_d[DDRmax] = 8.07) and testing (RMSE = 0.011, MAE = 0.006, R² = 0.688, and C_d[DDRmax] = 11.32) phases. Sequentially, the subsequent standings were secured by the SVM, GEP, and MARS. MLP outperformed SVM, GEP, and MARS models in predicting C_d, achieving the highest R² and lowest RMSE/MAE values.

Keywords

Discharge coefficient / Laboratory observations / Machine learning models / Prediction / Trapezoidal-arched labyrinth weir

Funding

None.

Conflict of interest

Upaka Rathnayake is an Editorial Board Member of this journal but was not in any way involved in the editorial and peer-review process conducted for this paper, directly or indirectly. Separately, other authors declared that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

References

[1] Afaridegan E, Amanian N, Parsaie A, Gharehbaghi A. Hydraulic investigation of modified semi-cylindrical weirs. Flow Meas Instrum. 2023; 93:102405. doi: 10.1016/j.flowmeasinst.2023.102405

[2] Bekheet AA, AboulAtta NM, Saad NY, El-Molla DA. Effect of the shape and type of piano key weirs on the flow efficiency. Ain Shams Eng J. 2022; 13(3):101620. doi: 10.1016/j.asej.2021.10.015

[3] Nourani B, Arvanaghi H, Pourhosseini FA, Javidnia M, Abraham J. Enhanced support vector machine with particle swarm optimization and genetic algorithm for estimating discharge coefficients of circular-crested oblique weirs. Iran J Sci Technol Trans Civil Eng. 2023; 47(5):3185-3198. doi: 10.1007/s40996-023-01110-0

[4] Roushangar K, Alami MT, Shiri J, Asl MM. Determining discharge coefficient of labyrinth and arced labyrinth weirs using support vector machine. Hydrol Res. 2018; 49(3):924-938. doi: 10.2166/nh.2017.214

[5] Taylor G. The Performance of Labyrinth Weirs. PhD Thesis. Nottingham, UK: University of Nottingham; 1968.

[6] Hay N, Taylor G. Performance and design of labyrinth weirs. J Hydraul Div. 1970; 96(11):2337-2357. doi: 10.1061/JYCEAJ.0002766

[7] Lux F. Design and construction of labyrinth spillways. In: 15th Commission Internationale des Grands Barrages. Ontario: Canadian Dam Association; 1985. p. 249-274.

[8] Kocahan H, Taylor G. Rehabilitation of Black Rock Dam (Seepage & Inadequate Spillway). In: Association of State Dam Safety Officials Conference. Arlington, VA, USA; 2002.

[9] Crookston BM, Tullis B. Labyrinth Weirs. Hydraulic Structures. Vol. 59. United States: Utah States University; 2010.

[10] Christensen NA, Tullis BP. Arced Labyrinth Weir Flow Characteristics. United States: Utah States University; 2012.

[11] Seamons TR. Labyrinth Weirs: A Look into Geometric Variation and its Effect on Efficiency and Design Method Predictions. United States: Utah State University; 2014.

[12] Bijankhan M, Kouchakzadeh S. Unified discharge coefficient formula for free and submerged triangular labyrinth weirs. Flow Meas Instrum. 2017; 57:46-56. doi: 10.1016/j.flowmeasinst.2017.08.007

[13] Monjezi R, Heidarnejad M, Masjedi A, Purmohammadi MH, Kamanbedast A. Laboratory investigation of the discharge coefficient of flow in arced labyrinth weirs with triangular plans. Flow Meas Instrum. 2018; 64:64-70. doi: 10.1016/j.flowmeasinst.2018.10.011

[14] Seyedian SM, Haghiabi A, Parsaie A. Reliable prediction of the discharge coefficient of triangular labyrinth weir based on soft computing techniques. Flow Meas Instrum. 2023; 92:102403. doi: 10.1016/j.flowmeasinst.2023.102403

[15] Zare H, Vaghefi M, Mahmoudi A, Behroozi AM. Experimental exploration of flow hydraulics and discharge coefficient for an inclined circular labyrinth weir. Water Resour Manag. 2023; 37(11):4521-4536. doi: 10.1007/s11269-023-03577-y

[16] Nematollahi S, Izadinia E, Khoshfetrat A. Investigation of discharge coefficient and flow energy loss in sinusoidal-labyrinth weirs. Iran J Sci Technol Trans Civil Eng. 2025:1-10.doi: 10.1007/s40996-025-01820-7

[17] Ahmed AM, Altalib AN. Determination of discharge coefficient of arced labyrinth weir with semicircular openings. Iran J Sci Technol Trans Civil Eng. 2025:1-12. doi: 10.1007/s40996-025-01747-z

[18] Masoudi MH, Yari A, Sadeghian J, Norouzi H. Experimental investigation of the discharge coefficient of the rectangular and trapezoidal labyrinth weirs considering variable congress lengths. Model Earth Syst Environ. 2024; 10(2):2819-2832. doi: 10.1007/s40808-023-01925-w

[19] Hadi ZM, Majeed HQ. Experimental and Numerical Study of the Discharge Capacity of a Labyrinth Side Weir in a Straight Channel. In: AIP Conference Proceedings. Vol. 3091. Melville, NY: AIP Publishing; 2024.

[20] Darvas LA. Discussion of performance and design of labyrinth weirs, by Hay and Taylor. J Hydraul Eng. 1971; 97:1246-51.

[21] Lux F, Hinchliff D. Design and Construction of Labyrinth Spillways. In: 15th Congress International Commission on Large Dams ICOLD. Vol. 4. Lausanne, Switzerland:1985. p. 249-274.

[22] Melo JF, Ramos CM, Magalhães AP. Descarregadores Com Soleira em Labirinto de um Ciclo em Canais Convergentes. In: Determinação da Capacidade de Vazão Proceedings 6th Congresso da Água. 2002.

[23] Crookston BM. Labyrinth Weirs. Doctor of Philosophy, Utah State University [PhD Dissertation]; 2010.

[24] Emiroglu ME, Kaya N, Agaccioglu H. Discharge capacity of labyrinth side weir located on a straight channel. J Irrig Drain Eng. 2009; 136(1):37-46. doi: 10.1061/(ASCE)IR.1943-4774.0000112

[25] Karami H, Karimi S, Bonakdari H, Shamshirband S. Predicting discharge coefficient of triangular labyrinth weir using extreme learning machine, artificial neural network and genetic programming. Neural Comput Appl. 2018; 29(11):983-989. doi: 10.1007/s00521-016-2588-x

[26] Bonakdari H, Ebtehaj I, Gharabaghi B, Sharifi A, Mosavi A. Prediction of discharge capacity of labyrinth weir with gene expression programming. In: Proceedings of SAI Intelligent Systems Conference. Cham: Springer International Publishing; 2020. p. 202-217. doi: 10.1007/978-3-030-55180-3_17

[27] Khani M, Shabanlou S. Assessment of discharge coefficient in trapezoidal and rectangular canals through regularized extreme learning machine. Measurement. 2021; 180:109493. doi: 10.1016/j.measurement.2021.109493

[28] Mahmoud A, Hu T, Zeng X, Jing P, Li X, Ribeiro EDC. Hydraulic informed multi-layer perceptron for estimating discharge coefficient of labyrinth weir. Eng Appl Artif Intell. 2023; 123:106435. doi: 10.1016/j.engappai.2023.106435

[29] Emami H, Emami S, Parsa J. A walnut optimization algorithm applied to discharge coefficient prediction on labyrinth weirs. Soft Comput. 2022; 26:12197-12215. doi: 10.1007/s00500-022-07041-8

[30] Majedi-Asl M, Fuladipanah M, Arun V, Tripathi RP. Using data mining methods to improve discharge coefficient prediction in piano key and labyrinth weirs. Water Supply. 2022; 22(2):1964-1982. doi: 10.2166/ws.2021.304

[31] Mustafa M, Mansoor T, Muzzammil M. Support vector machine (SVM) approach to develop the discharge prediction model for triangular labyrinth weir. Water Supply. 2022; 22:8942-8956. doi: 10.2166/ws.2022.393

[32] Wang F, Zheng S, Ren Y, Liu W, Wu C. Application of hybrid neural network in discharge coefficient prediction of triangular labyrinth weir. Flow Measure Instrument. 2022; 83:102108. doi: 10.1016/j.flowmeasinst.2021.102108

[33] Hu Z, Karami H, Rezaei A, et al. Using soft computing and machine learning algorithms to predict the discharge coefficient of curved labyrinth overflows. Eng Appl Comput Fluid Mech. 2021; 15(1):1002-1015. doi: 10.1080/19942060.2021.1934546

[34] Mahmoud A, Yuan X, Hajilounezhad T, Yuan Y. Investigation on labyrinth spillway multi-objective optimization with an emphasis on predicting discharge coefficient through different artificial neural networks. Measurement. 2021; 174:109036. doi: 10.1016/j.measurement.2021.109036

[35] Shafiei S, Najarchi M, Shabanlou SA. Novel approach using CFD and neuro-fuzzy-firefly algorithm in predicting labyrinth weir discharge coefficient. J Braz Soc Mech Sci Eng. 2020; 42:44. doi: 10.1007/s40430-019-2109-9

[36] Norouzi P, Rajabi A, Izadbakhsh MA, Shabanlou S, Yosefvand F, Yaghoubi B. A new non‐tuned self‐adaptive machine‐learning approach for simulating the discharge coefficient of labyrinth weirs. Irrig Drainag. 2020; 69(3):398-416. doi: 10.1002/ird.2423

[37] Olyaie E, Banejad H, Heydari M. Estimating discharge coefficient of PK-weir under subcritical conditions based on high-accuracy machine learning approaches. Iran J Sci Technol Trans Civil Eng. 2019; 43:89-101. doi: 10.1007/s40996-018-0150-z

[38] Roushangar K, Alami MT, Shiri J, Asl MM. Determining discharge coefficient of labyrinth and arced labyrinth weirs using support vector machine. Hydrol Res. 2018; 49(3):924-938. doi: 10.2166/nh.2017.214

[39] Uyumaz A, Danandeh Mehr A, Kahya E, Erdem H. Rectangular side weirs discharge coefficient estimation in circular channels using linear genetic programming approach. J Hydroinform. 2014; 16(6):1318-1330. doi: 10.2166/hydro.2014.112

[40] Emiroglu ME, Kisi O. Prediction of discharge coefficient for trapezoidal labyrinth side weir using a neuro-fuzzy approach. Water Resour Manag. 2013; 27:1473-1488. doi: 10.1007/s11269-012-0249-0

[41] Cortes C, Vapnik V. Support-vector networks. Mach Learn. 1995; 20:273-297. doi: 10.1007/BF00994018

[42] Sandamal K, Shashiprabha S, Muttil N, Rathnayake U. Pavement roughness prediction using explainable and supervised machine learning technique for long-term performance. Sustainability. 2023; 15(12):9617. doi: 10.3390/su15129617

[43] Ferreira C. Gene expression programming: A new adaptive algorithm for solving problems. Complex Syst. 2001; 13:87-129. doi: 10.48550/arXiv.cs/0102027

[44] Fuladipanah M, Shahhosseini A, Rathnayake N, et al. In-depth simulation of rainfall-runoff relationships using machine learning methods. Water Pract Technol. 2024; 19(6):2442-2459. doi: 10.2166/wpt.2024.147

[45] Perera A, Hazi MD, Upaka R. Comparison of different artificial neural network (ANN) training algorithms to predict the atmospheric temperature in Tabuk, Saudi Arabia. MAUSAM. 2021; 71(2):233-244. doi: 10.54302/mausam.v71i2.22

[46] Agbasi JC, Egbueri JC. Prediction of potentially toxic elements in water resources using MLP-NN, RBF-NN, and ANFIS: A comprehensive review. Environ Sci Pollut Res Int. 2024; 31:30370-30398. doi: 10.1007/s11356-024-33350-6

[47] Lukas P, Melesse AM, Kenea TT. Predicting reservoir sedimentation using multilayer perceptron - artificial neural network model with measured and forecasted hydrometeorological data in gibe-III reservoir, omo-gibe river basin, Ethiopia. J Environ Manag. 2024; 359:121018. doi: 10.1016/j.jenvman.2024.121018

[48] Friedman JH.Multivariate adaptive regression splines. Ann Statis. 1991; 19:1-67. doi: 10.1214/aos/1176347963

[49] Garai S, Paul RK, Yeasin M, Paul AK. CEEMDAN-based hybrid machine learning models for time series forecasting using MARS algorithm and PSO-optimization. Neural Process Lett. 2024; 56(2):92. doi: 10.1007/s11063-024-11552-w

[50] Fuladipanah M, Azamathulla HM, Kisi O, Kouhdaragh M, Mandala V. Quantitative forecasting of bed sediment load in river engineering: An investigation into machine learning methodologies for complex phenomena. Water Supply. 2024; 24(2):585-600. doi: 10.2166/ws.2024.017

[51] Noori R, Khakpour A, Omidvar B, Farokhnia A. Comparison of ANN and principal component analysis-multivariate linear regression models for predicting the river flow based on developed discrepancy ratio statistic. Expert Syst Appl. 2010; 37(8):5856-5862. doi: 10.1016/j.eswa.2010.02.020

[52] Koskinas A, Tegos A, Tsira P, et al. Insights into the Oroville dam 2017 spillway incident. Geosciences. 2019; 9(1):37. doi: 10.3390/geosciences9010037