Development and Verification of an Analytical Solution for Forecasting Nonlinear Kinematic Flood Waves
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VIEW THE REPLYPublication: Journal of Hydrologic Engineering
Volume 11, Issue 4
Abstract
A new approximate analytical solution to the nonlinear kinematic wave equation is proposed. The solution has been derived by combining an implicit solution obtained with the method of characteristics with analytical decomposition and successive approximation. The new solution was verified favorably with a finite-difference approximation. A field verification via a comparison with observed storm hydrographs from the Schuylkill River near Philadelphia indicated that with constant lateral flow the nonlinear model reasonably predicted the observed flow rates, except during periods of intense rainfall. An improved nonlinear model that accounts for variable lateral flow due to changing effective precipitation is proposed. Numerical measures of accuracy indicated that the new variable-rate nonlinear model performs better than the constant lateral flow and the linear kinematic wave model. The linear kinematic wave model produced poor forecasts, especially in the flow time and flood-peak time. The new solution is simple to apply and permits the efficient forecast of nonlinear kinematic flood waves without the usual stability restrictions of numerical models. The new formula also permits the analysis of the effect of nonlinear parameters in the area-discharge relationship on hydrograph characteristics. The hydrograph peak time appears to be very sensitive to the magnitude of the nonlinear exponent . For values of , the peak flow occurs at a time earlier than that predicted by a linear hydrograph . For values of , the peak flow occurs at a time later than that predicted by a linear hydrograph.
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Acknowledgments
Streamflow data for the present study was provided online by the U.S. Geological Survey. Rainfall information for the present study was provided online by the U.S. National Oceanic and Atmospheric Administration.
References
Abbaoui, K., and Cherruault, Y. (1999). “Convergence of Adomian’s method applied to differential equations.” Comput. Math. Appl., 28(5), 103–109.
Adomian, G. (1994). Solving frontier problems in physics-The decomposition method, Kluwer Academic, Dordrecht, The Netherlands.
Adomian, G. (1995). “Analytical solution of Navier–Stokes flow of a viscous compressible fluid.” Found. Phys. Lett., 8(4), 389–400.
Ajami, N. K., Gupta, H., Wagener, T., and Sorooshian, S. (2004). “Calibration of a semi-distributed hydrologic model for streamflow estimation along a river system.” J. Hydrol., 298, 112–135.
Akan, A. O. (1989). “Time of concentration formula for previous catchments.” J. Irrig. Drain. Eng., 115(4), 733–735.
Cherruault, Y. (1989). “Convergence of Adomian’s method.” Kybernetik, 18(2), 31–38.
Cherruault, Y., Saccomardi, G., and Some, B. (1992). “New results for convergence of Adomian’s method applied to integral equations.” Math. Comput. Modell., 16(2), 85–93.
Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). Applied hydrology, McGraw–Hill, New York.
Gabet, L. (1992). “Equisse d’une théorie décompositionnelle et application aux equations aux dérivées partialles.” Dissertation, Ecole Centrale De Paris, Paris.
Gabet, L. (1993). “The decomposition method and linear partial differential equations.” Math. Comput. Modell., 17(6), 11–22.
Gabet, L. (1994). “The decomposition method and distributions.” Comput. Math. Appl., 27(3), 41–49.
Hromadka, T. V., and De Vries, J. J. (1988). “Kinematic wave and computational error.” J. Hydraul. Eng., 114(2), 207–217.
Jaber, F. H., and Mohtar, R. H. (2002). “Dynamic time step for one-dimensional overland flow kinematic wave solution.” J. Hydrologic Eng., 7(1), 3–11.
Jayawardena, A. W., and White, J. K. (1977). “A finite element distributed catchment model (1): Analysis basis.” J. Hydrol., 34, 269–286.
Jeffrey, A. (2003). Applied partial differential equations, Academic, New York.
Li, R. M., Simons, D. B., and Stevens, M. A. (1975). “Nonlinear kinematic wave approximation for water routing.” Water Resour. Res., 11, 245–252.
National Oceanic and Atmospheric Administration (NOAA). (2005). “NOAA National Climatic Data Center.” ⟨http://www.ncdc.noaa.gov/oa/climate/climatedata.html⟩
Oden, J. T. (1977). Applied functional analysis, Prentice–Hall, Englewood Cliffs, N.J.
Ponce, V. M. (1991). “Kinematic wave controversy.” J. Hydraul. Eng., 117(4), 511–525.
Ross, B. B., Contractor, D. N., and Shanholtz, V. O. (1979). “A finite element model of overland and channel flow for accessing the hydrologic impact of land use change.” J. Hydrol., 41L, 11–30.
Serrano, S. E. (1995). “Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity.” Water Resour. Res., 31(11), 2733–2742.
Serrano, S. E. (1997). “Hydrology for engineers, geologists, and environmental professionals.” An integrated treatment of surface, subsurface, and contaminant hydrology, HydroScience, Inc., Lexington, Ky.
Serrano, S. E. (2001). “Engineering uncertainty and risk analysis.” A balanced approach to probability, statistics, stochastic models, and stochastic differential equations, HydroScience Inc., Lexington, Ky.
Serrano, S. E. (2003a). “Modeling groundwater flow under transient nonlinear free surface.” J. Hydrologic Eng., 8(3), 123–132.
Serrano, S. E. (2003b). “Propagation of non-linear reactive contaminants in porous media.” Water Resour. Res., 39(8), 1228–1242.
Serrano, S. E. (2004). “Modeling infiltration with approximate solutions to Richard’s equation.” J. Hydrologic Eng., 9(5), 421–432.
Serrano, S. E., and Adomian, G. (1996). “New contributions to the solution of transport equations in porous media.” Math. Comput. Modell., 24(4), 15–25.
Singh, V. P. (1996). Kinematic wave modeling in water resources: Surface water hydrology, Wiley, New York.
Singh, V. P., and Woolhiser, D. A. (2002). “Mathematical modeling of watershed hydrology.” J. Hydrologic Eng., 7(4), 270–292.
Tisdale, T. S., and Hamrick, M. (1999). “Kinematic wave analysis of sheet flow using topography fitted coordinates.” J. Hydrologic Eng., 4(4), 367–370.
U.S. Geological Survey (2005). “NWIS site inventory for the U.S.A.” ⟨http://waterdata.usgs.gov/nwis⟩
Wazwaz, A.-M. (1995). “A new approach to the nonlinear advection problem: An application of the decomposition method.” Appl. Math. Comput., 72, 175–181.
Wazwaz, A.-M. (2000). “A new algorithm for calculating Adomian polynomials for nonlinear operators.” Appl. Math. Comput., 111, 53–69.
Weinmann, P. E., and Laurenson, E. M. (1979). “Approximate flood routing methods: A review.” J. Hydraul. Div., Am. Soc. Civ. Eng., 105(12), 1521–1536.
Wong, T. S. W. (2002). “Generalized formula for time of travel in rectangular channel.” J. Hydrologic Eng., 7(6), 445–448.
Wong, T. S. W., and Chen, C.-N. (1997). “Time of concentration formula for sheet flow of varying flow regime.” J. Hydrologic Eng., 2(3), 136–139.
Xiong, Y., and Melching, C. S. (2005). “Comparison of kinematic-wave and nonlinear reservoir routing of urban watershed runoff.” J. Hydrologic Eng., 10(1), 39–49.
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Published In
Journal of Hydrologic Engineering
Volume 11 • Issue 4 • July 2006
Pages: 347 - 353
Copyright
© 2006 ASCE.
History
Received: Feb 8, 2005
Accepted: Aug 29, 2005
Published online: Jul 1, 2006
Published in print: Jul 2006
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