This article is an online, enhanced version of a paper of essentially the same content, entitled
"Analytical verification of Muskingum-Cunge routing".

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Victor M. Ponce



The Muskingum-Cunge method of flood routing has for many years been successfully applied in flood routing, modeling and prediction (Natural Environment Research Council 1975; Ponce and Yevjevich, 1978). 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 with those obtained using an actual numerical application of the Muskingum-Cunge model. The remarkable agreement between analytical and numerical results is an indication of the accuracy of the model.


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 extensive routing in ungaged streams with a reasonable expectation of accuracy, which would otherwise be possible only if the streams were gaged. Since it is impractical to gage more than a few streams, a method that is both accurate and does not rely on stream gaging holds substantial promise.

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 et al., 1950) and the latter is the ratio of physical and numerical diffusivifies (Hayami, 1951; Cunge, 1969):

            c Δt
   C = _______


   D = __________
            So c Δx


in which C = Courant number, D = cell Reynolds number, c = flood wave celerity, Δx = space step, Δt = time step, qo = reference unit-width discharge, and So = bed slope.

With the Muskingum routing parameters defined in this way, the routing coefficients are (Ponce and Yevejevich, 1978):

             -1 + C + D
   C0 = _____________
              1 + C + D


              1 + C - D
   C1 = _____________
              1 + C + D


              1 - C + D
   C2 = _____________
              1 + C + D


and the routing equation is:

   Qj+1n+1 = C0 Qj n+1 + C1 Qj n + C2 Qj+1n


where j = spatial index and n = temporal index.

Cunge's formulation (Cunge, 1969) is based on diffusion wave theory, the applicability of which was confirmed by Ponce et al. (1978). Pioneering analytical work on the diffusion wave is due to Hayami (1951), who reasoned that the large-scale longitudinal mixing caused by channel irregularities could be effectively modeled as a diffusion process. Ponce and Simons (1977) used the theory of linear stability to derive celerity and attenuation functions for sinusoidal perturbations to the steady uniform flow.

In this paper we use the findings of Ponce and Simons (1977) to calculate analytical peak outflow and travel time for a series of sinusoidal hydrographs, and to compare these results with those obtained using the Muskingum-Cunge model (Eqs. 1-6).


The analytical diffusion wave celerity may be approximated as the Seddon speed, or Seddon celerity (Ponce and Simons, 1977):

   c = β u


in which c = flood wave celerity, u = mean velocity, and β = rating exponent in

   Q = α A β


or, for a unit-width channel:

   q = α d β


in which Q = discharge, A = flow area, q = unit-width discharge, and d = flow depth.

The wave attenuation follows an exponential decay function of the following form:

   qpo = qo + (qpi - qo)  exp ( -βI * t*)


in which qpo = peak outflow, qpi = peak inflow, qo = equilibrium flow, βI * = dimensionless amplitude propagation factor, and the dimensionless elapsed time t* = t uo / Lo  [See Eq. 12 for the definition of Lo].

The amplitude propagation factor is the following (Ponce and Simons, 1977):

   βI * = ___ σ*2


in which σ* = dimensionless wavenumber, defined as follows:

   σ* = (2π / L) Lo


in which L = sinusoidal wavelength, and Lo = reference channel length, defined as Lo = do /So, in which do = equilibrium flow depth, and So = channel bed slope.

Since L = cT, where T = wave period, taken as the time base Tb of a sinusoidal hydrograph, and t* = t (uo /Lo), it follows that:

   βI* t* = (______)
2 ν t


in which ν = qo / (2So) is the hydraulic diffusivity (Hayami, 1951).


This paper uses 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) Tb = 96 hr, peak inflow qpi = 200 cfs/ft (cubic feet per second, per foot), and baseflow qb = 50 cfs/ft through a channel of length Lr = 200 miles, slope So = 1 ft/mi, and rating curve:

   q = 0.688 d 5/3


from which the Manning coefficient n = 0.0297 is calculated.

A series of test cases was developed based on Thomas' problem, using: (1) two wave periods Tb: 48 and 96 hr; (2) three peak inflows qpi: 100, 200 and 500 cfs/ft, i.e. three peak inflow-to-base flow ratios qpi /qb: 2, 4, and 10; (3) two channel lengths Lr: 200 and 500 mi. Twelve outflow hydrographs were generated using the constant-parameter Muskingum-Cunge model (Ponce and Yevjevich, 1978), for which the reference flow is defined as follows:

                       qpi - qo
   qo = qb  +  ___________


For Tb = 48 hr, the chosen space and time intervals are: Δx = 6.25 mi, and Δt = 1.5 hr, respectively; for Tb = 96h, the intervals are: Δx = 12.5 mi, and Δt = 3 hr. The chosen wave periods (48 and 96 hr) satisfy diffusion wave applicability criteria (Ponce et al., 1978):

   Tb So (g / do)1/2  ≥  30


in which g = gravitational acceleration. In effect, for the less favorable case (Tb = 48 hr and qpi = 500 cfs/ft), the criterion is:

   Tb So (g / do)1/2  =  31


which guarantees that the routed waves remain within the realm of diffusion waves.


Table 1 summarizes the comparison of analytical and numerical peak outflow qpo and travel time Tt  for the chosen series of sinusoidal hydrographs. The objective is to compare the analytical results of Columns labeled (1) with the numerical results of Columns labeled (2).

Figure 1 shows inflow and oufflow hydrographs for a typical case (Tb = 96 hr, Lr = 500 mi, and qpi /qb = 4). In this case, reference flow is qo = 125 cfs/ft, peak outflow qpo = 176.70 cfs/ft, and travel time Tt = 79.8 hr.

Table 1.  Comparison between analytical and numerical peak outflow qpo (cfs/ft) and travel time Tt (hr)
for a series of sinusoidal hydrographs.
Tb (hr) 48 96
Lr (mi) 200 500 200 500
  qpo Tt qpo Tt qpo Tt qpo Tt
qpi /qb (1)(2) (1)(2) (1)(2) (1)(2) (1)(2) (1)(2) (1)(2) (1)(2)
2 87.9588.0439.1639.00 79.8379.2097.91100.00 96.2196.2039.1639.10 91.5791.6097.9198.00
4 166.41166.7031.9331.90 141.98140.7079.8281.10 189.65189.6031.93 31.90176.74176.7079.8279.80
10 410.46411.8023.2923.00 338.28336.6058.2358.70 473.19473.2023.2923.20 438.85438.8058.2358.20
qb = 50 cfs/ft  (1 cfs/ft = 0.0929 m2/s).
(1) Analytically computed values based on diffusion wave theory.
(2) Numerically generated values using constant-parameter Muskingum-Cunge model.

Fig 1. Inflow and outflow hydrographs for a typical case (Tb = 96 hr, Lr = 500 mi, and qpi / qb = 4).

On the basis of the foregoing analysis, it is concluded 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 findings underscore the utility of the Muskingum-Cunge method of flood routing for practical applications.


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200627 06:20
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