Operation of the non-linear Muskingum model in the prediction of the pollution breakthrough curves through the river reaches

Document Type : Research Article


Civil engineering department, university of Maragheh


The Muskingum model in both types of the linear and non-linear is one the most common models in the flood routing through the river reaches. The simplicity and being stepwise in calculating the exit flood hydrographs are the advantages of this model. Because of the similarity between the shape of the flood hydrograph and pollution breakthrough curves, it is tried to examine the applicability of the non-linear Muskingum model in the prediction of the contaminant concentration in downstream of the river reaches. The field data series of the MONOCACY and ANTIETAM Creek Rivers which were gathered by USGS have been used. During the tests, Rhodamine was used and the concentration pollute graphs were acquired in the four and eight cross-sections of the mentioned rivers, respectively. The triple sets of the model parameters have been extracted in each reach, then the BC curves have been simulated in each position using them. It is observed that this model can rebuild the dimensions of the exit BC curve properly but, it also has some limitations in the modeling of the convection term of the pollution using average flow velocity. For its solution, the extracted BC curves have been transported along the time axis with the magnitude of  in which L is the reach length and u is the average flow velocity. Also, for a better understanding of the effects of the model parameters in the simulated concentrations, the sensitivity analysis has been performed and it is found that the parameters of the ,  and  are the most to less effective parameters in the concentration calculation, respectively. It was also found that the power parameter of this model (m) for pollution transport fluctuates in the range (0.1-1.4) and has an average value of 0.85. The value of the weighted coefficient (x) was also obtained in the range (-1 to +1), but the frequency of values greater than zero was greater and its average value was reduced to 0.91.


Main Subjects

[1] E.M. Wilson, Engineering hydrology, Macmillan International Higher Education, 1990.
[2] H.M. Samani, S. Jebelifard, Design of circular urban storm sewer systems using multilinear Muskingum flow routing method, Journal of Hydraulic Engineering, 129(11) (2003) 832-838.
[3] G. McCarthy, The unit hydrograph and flood routing, Conference of North Atlantic Division, US Army Corps of Engineers, New London, CT. US Engineering, (1938).
[4] M.A. Gill, Flood routing by the Muskingum method, Journal of hydrology, 36(3-4) (1978) 353-363.
[5] Y.-K. Tung, River flood routing by nonlinear Muskingum method, Journal of hydraulic engineering, 111(12) (1985) 1447-1460.
[6] J. Yoon, G. Padmanabhan, Parameter estimation of linear and nonlinear Muskingum models, Journal of Water Resources Planning and Management, 119(5) (1993) 600-610.
[7] S. Mohan, Parameter estimation of nonlinear Muskingum models using genetic algorithm, Journal of hydraulic engineering, 123(2) (1997) 137-142.
[8] J.H. Kim, Z.W. Geem, E.S. Kim, Parameter estimation of the nonlinear muskingum model using harmony search 1, JAWRA Journal of the American Water Resources Association, 37(5) (2001) 1131-1138.
[9] A. Das, Parameter estimation for Muskingum models, Journal of Irrigation and Drainage Engineering, 130(2) (2004) 140-147.
[10] A. Das, Chance-constrained optimization-based parameter estimation for Muskingum models, Journal of irrigation and drainage engineering, 133(5) (2007) 487-494.
[11] H.-J. Chu, L.-C. Chang, Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model, Journal of Hydrologic Engineering, 14(9) (2009) 1024-1027.
[12] J. Luo, J. Xie, Parameter estimation for nonlinear Muskingum model based on immune clonal selection algorithm, Journal of Hydrologic Engineering, 15(10) (2010) 844-851.
[13] Z.W. Geem, J.H. Kim, G.V. Loganathan, A new heuristic optimization algorithm: harmony search, simulation, 76(2) (2001) 60-68.
[14] S.M. Easa, Closure to “Improved Nonlinear Muskingum Model with Variable Exponent Parameter” by Said M. Easa, December 2013, Vol. 18, No. 12, pp. 1790-1794.
[15] M. Niazkar, S.H. Afzali, Streamline performance of Excel in stepwise implementation of numerical solutions, Computer Applications in Engineering Education, 24(4) (2016) 555-566.
[16] J. Chabokpour, A. Samadi, M. Merikhi, Application of method of temporal moments to the contaminant exit breakthrough curves from rockfill media, Iranian Journal of Soil and Water Research, 49(3) (2018) 629-640. (In Persian)
[17] D.V. Chapman, Water quality assessments: a guide to the use of biota, sediments and water in environmental monitoring, CRC Press, 1996.
[18] J. Chabokpour, Application of hybrid cells in series model in the pollution transport through layered material, Pollution, 5(3) (2019) 473-486.
[19] A.K. Sriwastava, S. Tait, A. Schellart, S. Kroll, M.V. Dorpe, J.V. Assel, J. Shucksmith, Quantifying uncertainty in simulation of sewer overflow volume, Journal of Environmental Engineering, 144(7) (2018) 04018050.
[20] J.C. Refsgaard, J.P. van der Sluijs, A.L. Højberg, P.A. Vanrolleghem, Uncertainty in the environmental modelling process–a framework and guidance, Environmental modelling & software, 22(11) (2007) 1543-1556.
[21] T. Bai, J. Wei, W. Yang, Q. Huang, Multi-Objective Parameter Estimation of Improved Muskingum Model by Wolf Pack Algorithm and Its Application in Upper Hanjiang River, China, Water, 10(10) (2018) 1415.
[22] S. Farzin, V.P. Singh, H. Karami, N. Farahani, M. Ehteram, O. Kisi, M.F. Allawi, N.S. Mohd, A. El-Shafie, Flood routing in river reaches using a three-parameter Muskingum model coupled with an improved bat algorithm, Water, 10(9) (2018) 1130.
[23] G. Tayfur, V.P. Singh, T. Moramarco, S. Barbetta, Flood hydrograph prediction using machine learning methods, Water, 10(8) (2018) 968.
[24] M. Khorashadizadeh, S.A. Hashemimonfared, A. Akbarpour, M. Pourreza-bilondi, Uncertainty assessment of pollution transport model using GLUE method, Iranian Journal of Irrigation & Drainage, 10(3) (2016) 284-293. (In Persian)
[25] D. Barry, K. Bajracharya, On the Muskingum-Cunge flood routing method, Environment International, 21(5) (1995) 485-490.
[26] A.D. Koussis, Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method, Hydrological sciences journal, 54(1) (2009) 43-61.
[27] X.N. Zhang, Y. Shang, L. Wang, Y.B. Song, R.P. Han, Y.Q. Li, Comparison of linear and nonlinear regressive analysis in estimating the Thomas model parameters for anionic dye adsorption onto CPB modified peanut husk in fixed-bed column, in:  Advanced Materials Research, Trans Tech Publ, 2013, pp. 2179-2183.
[28] A.D. Koussis, K. Mazi, Reverse flood and pollution routing with the lag-and-route model, Hydrological Sciences Journal, 61(10) (2016) 1952-1966.
[29] H. Norouzi, J. Bazargan, Efficiency of the Linear Muskingum Method in Flood Routing of Dual Rockfill Detention Dams, Iranian journal of Ecohydrology, 7(4) (2020) 1061-1070. (In Persian)
[30] M. Nasrabadi, A.M. Mazdeh, M.H. Omid, Experimental and Numerical Investigation of Effect of Cadmium Sorption by River Sediments on Longitudinal Dispersion, (2021).