The well-known Muskingum-Cunge method of flood routing (Natural Environment Research Council 1975; Ponce and Yevjevich, 1978) has for many years been successfully applied in flood modeling and prediction. The Muskingum routing parameters were theoretically derived by Cunge (1969) based on the approximation error obtained by a Taylor series expansion of the grid function. Although the parameters so derived have been shown to work reasonably well in a variety of practical applications (Hydrologic Engineering Center, US Army Corps of Engineers, 1990), their link to the analytical expressions for wave celerity (Seddon, 1900) and hydraulic diffusivity (Hayami, 1951) has yet to be demonstrated in an actual routing application. This paper endeavors to test the Muskingum-Cunge model by comparing the analytically derived flood wave celerity and peak attenuation amounts with those obtained using an actual numerical application of the Muskingum-Cunge model. Agreement between analytical and numerical results is taken as an indication of the potential accuracy of the model, when used to route actual floods in a practical setting.
The Muskingum-Cunge model of flood routing (Cunge, 1969; Natural Environment Research Council, 1975) is an improvement of the classical Muskingum model (Chow, 1959), in which the routing parameters are based on readily measurable hydraulic data (stage-discharge rating, channel slope, channel width, and reach length), instead of being based on historic flood discharge data. This allows routing in ungaged streams with a reasonable expectation of accuracy, which would otherwise be possible only if the streams were gaged. As it is impractical to gage more than a few streams, a method that is both accurate and does not rely on stream gaging has to be acknowledged.
Ponce and Yevjevich (1978) improved the Muskingum-Cunge model by expressing the routing parameters as the dimensionless Courant and cell Reynolds numbers, where the former is the ratio of physical and numerical celerities (Seddon, 1900; O'Brien
in which With the Muskingum routing parameters defined in this way, the routing coefficients are (Ponce and Yevejevich, 1978):
and the routing equation is:
where
Cunge's formulation (Cunge, 1969) is based on diffusion wave theory, the applicability of which was confirmed by Ponce In this paper we use the findings of Ponce and Simons (1977) to calculate theoretical (analytical) peak outflow and travel time for a series of sinusoidal hydrographs, and to compare these results with those obtained using the (numerical) Muskingum-Cunge model (Eqs. 1-6). The use and limitations of the Muskingum-Cunge model have been given by Ponce (1989).
The analytical diffusion wave celerity can be approximated as the Seddon speed (Ponce and Simons, 1977):
in which
or
in which The wave attenuation follows an exponential decay function of the following form:
in which q is peak inflow, _{pi}q_{0} is reference flow, β_{l}_{*} is dimensionless amplitude propagation factor, and t_{ *} dimensionless elapsed time.
The amplitude propagation factor is (Ponce and Simons, 1977)
in which
in which d_{0} / S_{0}.
As t_{ *} = t (u_{0}/L_{0}), it follows that:
in which
We used an extension of Thomas' (1934) classical flood routing problem to test the agreement between analytically and numerically derived peak outflow and travel time. The Thomas problem routes a sinusoidal flood hydrograph of time base (i.e. wave period) q = 200 cfs_{pi}^{-1} (cubic feet per second per foot), and basellow q = 50 cfs ft_{b}^{-1} through a channel of length L = 200 miles, slope _{r}S_{0} = 1 ft mile^{-1}, and rating curve
from which the Manning coefficent
A series of test cases was developed based on Thomas' problem, using: (1) two wave periods q: 100, 200 and 500 cfs ft_{pi}^{-1} , i.e. three peak inflow-to-base flow ratios q / _{pi}q: two, four and ten; (3) two channel lengths _{b}L: 200 and 500 miles. Twelve outflow hydrographs were generated using the constant-parameter Muskingum-Cunge model (Ponce and Yevjevich, 1978), where the reference flow is taken as:
_{r}
For x = 6 .25 miles, and Δt = 1.5 h, respectively; for T = 96h_{b}x = 12.5 miles, and Δt = 3 h. The chosen wave periods (48 and 96 h) satisfy diffusion wave applicability criteria (Ponce et al., 1978):
where q = 500 cfs ft_{pi}^{-1}), the criterion is:
which guarantees that the routed waves remain within the realm of diffusion waves.
Table 1 summarizes the comparison of analytical and numerical peak outflow T, for the chosen series of sinusoidal hydrographs. Figure 1 shows inflow and oufflow hydrographs for a typical case _{t}T = 96 h_{b}L = 500 miles, and _{r}q /_{pi}q = 4). In this case, reference flow is _{b}q_{0} = 125 cfs ft^{-1}, peak outflow q = 176.70 cfs ft_{po}^{-1}, and travel time T = 79.8 h.
_{t}
On the basis of the foregoing analysis, we conclude that Muskingum-Cunge routing accurately simulates flood wave propagation. In twelve (12) tests encompassing a wide range of conditions likely to be encountered in practice, the ratio of numerical to analytical peak outflows varied within the range 0.991-1.003, and the ratio of numerical to analytical travel times varied within the range 0.987-1.021. The agreement in indeed remarkable, considering that a constant reference flow was used in the computation of the analytical values.
The present study was performed in autumn 1994, while the second author was at San Diego State University, on leave from the Ganga Plains Regional Centre, National Institute of Hydrology, Patna (Bihar), India. His leave was funded by the United Nations Development Programme.
Chow, V.T., 1950. Open-channel Hydraulics. McGraw-Hill, New York.
Cunge, J.A., 1969. On the subject of a flood propagation computation method (Muskingum method). Hayami,S., 1951.On the Propagation of flood waves. Disaster Prey. Res. Inst, Kyoto Univ. Bull., 1: 1-16 Hydrologic Engineering Center, US Army Corps of Engineers, HEC-1, Flood hydrograph package. user's manual. Version 4.0, September, Davis, CA. Natural Environment Research Council. 1975. Flood Studies Report, Vol. 5: Flood Routing Studies. NERC, London.
O'Brien,G.G., Hyman, M.A. and Kaplan, S., 1950. A study of the numerical solution of partial differential equations.
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Ponce, V.M. and Yevjevich, V., 1978. Muskingum -Cunge method with variable parameter.
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Thomas, H.A., 1934. |

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