WHY IS THE MUSKINGUMCUNGE THE BEST FLOOD ROUTING METHOD?
Professor Emeritus of Civil and Environmental Engineering San Diego State University, San Diego, California
1. INTRODUCTION The subject of flood routing
has preoccupied hydraulic and hydrologic engineers since
the early 1900s. The task is to use an appropriate calculation
to follow
the progression or travel of a flood
wave as it moves downstream along a river channel, from basin headwaters to mouth.
Two practical applications are well established: The subject has important implications for the safety and overall wellbeing of society. Developed societies have an innate tendency to build on the floodplain, and this trend is clearly at the root of the recurrent flood problem. Barring fundamental changes, it is expected that societies around the globe will continue to have to cope with floods, their analysis, design, control, and management. Therefore, to this date, flood wave modeling remains a worthwhile endeavor. In this paper, we review the issues concerning flood routing, including the available methods and their accuracy and practicality to satisfy specific objectives. We examine the related concepts of flood routing, methodologies, design, forecasting, and modeling, in order to clarify the issue for the hydraulic/hydrologic engineer engaged in this field of work. The aim is to help establish the MuskingumCunge method as the method of choice for a variety of flood routing applications, given its demonstrably sound theoretical basis and the substantial body of knowledge that has accrued in the more than 50 years since its original development.
2. FLOODS Floods are perceived to be both good and bad.
Floods are good when they move sediments out of their place of origin, typically at or near basin
headwaters, and move them downstream, along established flow paths, to lower elevations, where they are
subject to deposition. This is the way valleys were formed, and valleys do remain an essential geomorphic
feature of the contemporary societal landscape.
Floods are bad in situations when
rivers overflow their banks, eventually spreading floodwaters in developed urban
settings. Typically, floodwaters carry sediments, which invariably
will have a tendency to settle in unwanted places.
Therefore, the issue of how to best manage floods remains in the eye of the beholder;
it will depend largely on the local situation. In urban settings,
floods are generally considered
bad; therefore, they are to be avoided, managed, and/or controlled.
At this juncture (2023), the stateoftheart regarding the correct approach to floods lies somewhere in between the seemingly contrasting disciplines of flood control and flood management. Flood control is primarily aimed at fighting floods through conventional measures such as reservoirs, canals, and levees. Flood management recognizes that Nature has the advantage of time in her hands, and that solutions that do not heed this fact are eventually destined to fail. Therefore, the matter remains unsetlled. We trust that the future will provide ample experience to help resolve the issue of how to manage floods in the best way.
3. FLOOD ROUTING Flood routing is the process of following, through calculation, the progression or travel of a flood wave as it moves downstream along a channel course, stream, or river. At the outset, two distinct wave properties are recognized: (1) the flood wave celerity, i.e., its rate of speed; and (2) the flood wave's rate of attenuation or dissipation, described by the logarithmic decrement (Wylie, 1966; Ponce and Simons, 1977). The wave celerity is physically related to flow concentration, described by a differential equation of first order. The wave attenuation is physically related to flow diffusion, described by a differential equation of second order. Thirdorder processes (dispersion) are indeed feasible (Ponce, 2020), but at this juncture, they are not utilized in practical flood routing applications. Hydraulic engineers have delved into flood routing issues since the early 1900s. Seddon (1900), working with floods in the Lower Mississippi river, derived an expression for the celerity of a flood wave, among the earliest attempts to focus on the science. The basic elements of Seddon's findings are detailed in Box A.
Progress on flood routing procedures gained considerable
momentum in the 1930s with the
pioneering work of McCarthy (1938), who, working for the U.S. Army Corps of Engineers
on the Muskingum river, in Ohio,
developed a flood routing method which was later referred to as the Muskingum method of McCarthy, or
simply, as the "Muskingum method." The method, described
in the authoritative textbook by McCarthy's Muskingum method was esentially a hydrologic method, relying, for the most part, on actual gage measurements of reach "storage volume" for help in establishing the appropriate values of (reach) routing parameters to define the computation. The original hydrologic basis of the method remained firmly established through the 1960s, until kinematic wave theory came of age, following the comprehensive work of Lighthill and Whitham (1955).
Cunge (1969) is credited with pioneering the interpretation of
the Muskingum method as a hydraulic method,
wherein the "reach storage" could be related to the properties of the underlying
4. MUSKINGUM METHOD The basic elements of the Muskingum method are described in Box B.
Box B. Elements of the Muskingum method (McCarthy, 1938;
Ponce, 2014).
Reach inflow I
Reach outlow O
Reach storage S
Differential equation of storage: I  O =
dS/dt
Storage relation: S = K [ X I + ( 1  X ) O ]
K: Routing parameter, a storage coefficient, or reach travel time
X: Routing parameter,
a dimensionless weighting factor, varying in the range 0 ≤ X ≤ 0.5
Routing equation (fourpoint scheme):
O_{2} = C_{0} I_{2} + C_{1} I_{1} + C_{2} O_{1}
Routing coefficients:
The conventional Muskingum method (of McCarthy) is straightforward and comparatively simple; however, considerable care is required in order to use the method effectively. Its main drawback is the strict requirement for calibration of the routing parameters, which is at best a cumbersome procedure (Ponce, 2014). A typical routing application will necessarily require a calibration of the method's parameters, essentially limiting the predictive stage (of the routing) to flood magnitudes and durations similar to those used in the calibration. This requirement substantially limits the predictive ability of the method, jeopardizing its wider applicability to other floods and/or other reaches located within the same hydrologic system. Thus, we conclude that the conventional Muskingum method is largely ineffective for its use in extensive basin hydrologic modeling, which has become the norm in more recent times.
5. MUSKINGUMCUNGE METHOD
Cunge (1969) substantially improved the conventional Muskingum method,
giving it a decisive hydraulic flavor. He accomplished this feat by
recognizing that the conventional formulation, which related reach storage
to inflow and outflow
through the routing parameters K and X (see Box B),
could be construed as a numerical analog
of the kinematic wave equation (Lighthill and Whitham, 1955).
Cunge's analysis did not limit itself to the kinematic wave, and this remains his greatest contribution. He was able to relate the leading error (the secondorder error) of the numerical analog (of the Muskingum method) to the diffusion term of the kinematicwithdiffusion wave, i.e., the diffusion wave (Ponce and Simons, 1977). This made possible the derivation of an expression for the weighting factor X, substantially enhancing the methodology by giving it a deterministic flavor, since the diffusion wave is based on fundamental principles of mechanics.
Cunge's masterful use of physical and numerical principles provided the enhanced methodology, now
recognized as the MuskingumCunge method (Flood Studies Report, 1975;
Ponce and Yevjevich, 1978),
featuring the important property of grid independence,
which had been unattainable up to that point.
In point of fact, by relating the secondorder error of the numerical analog of the MuskingumCunge method
to the intrinsic properties
of the scheme (the Courant and cell Reynolds numbers),
the pot of gold at the end of the rainbow could be envisioned.
Indeed, the
method has amply demonstrated to be essentially gridindependent,
that is, the calculated flood hydrographs turn out to be the same,
regardless of grid size.
The basic elements of the MuskingumCunge method
are described in
We note that the fundamental tenet of the MuskingumCunge method,
and the reason why it works so well,
is the effective matching of physical and numerical diffusivities.
The physical diffusivity is the (channel) hydraulic diffuvisity ν_{p},
originally due to
Hayami (1951):
The numerical diffusivity ν_{n}
is the diffusivity derived by Cunge (1969):
It is clearly seen that the operational capability of the MuskingumCunge method hinges
on the determination of the values of the two routing parameters: (1)
Courant number C, the ratio
of physical celerity (the flood wave celerity, or Seddon celerity)
to "numerical celerity," Δx/Δt; and
A recommended procedure follows.
In closing, the perceived strengths of the MuskingumCunge method are seen to be twofold:
Note that the method will provide good results when the crosssectional data used in the calculations is truly representative of the reach under consideration. This may require the use of a significant amount of remotesensing data resources, coupled with a healthy dose of field experience.
6. WHY MUSKINGUMCUNGE? In this last section, we answer the question of why it is sensible to use the MuskingumCunge method for flood routing applications in hydraulic and hydrologic engineering. We begin by reiterating the nature of flood routing. Flood routing is the following, by calculation, the travel of a flood wave as it moves downstream through a channel network. The governing
differential equation of
flood routing is the kinematicwithdiffusion wave,
At this juncture, we recognize that computer solutions are the norm in hydraulic engineering.
REFERENCES Chow, V. T. 1959. Openchannel hydraulics. McGrawHill, Inc, New York, NY. Cunge, J. A. 1969. On the Subject of a Flood Propagation Computation Method (Muskingum Method). Journal of Hydraulic Research, Vol. 7, No. 2, 205230. Flood Studies Report. 1975. Vol. III: Flood Routing Studies, Natural Environment Research Council, London, England. Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention Research Institute, No. 1, December.
Lighthill, M. J. and G. B. Whitham. 1955.
On kinematic waves. I. Flood movement in long rivers.
Proceedings, McCarthy, G. T. 1938. The unit hydrograph and flood routing. Unpublished manuscript presented at a Conference of the North Atlantic Division of The U.S. Army Corps of Engineers, June 24, 1938; as cited by V. T. Chow in his Openchannel Hydraulics textbook (1959). Ponce, V. M. and D. B. Simons. 1977. Shallow wave propagation in open channel flow. Journal of Hydraulic Engineering, ASCE, 103(12), December, 14611476. Ponce, V. M., and V. Yevjevich. 1978. MuskingumCunge Method with Variable Parameters. Journal of the Hydraulics Division, ASCE, 104(12), December, 16631667. Ponce, V. M. 1991. New perspective on the Vedernikov number. Water Resources Research, Vol. 27, No. 7, 17771779, July. Ponce, V. M. 2014. Engineering Hydrology, Principles and Practices. Online text. Ponce, V. M. 2014. Fundamentals of Openchannel Hydraulics. Online text. Ponce, V. M. 2020. A dimensionless convectiondiffusiondispersion equation of flood waves. Online publication. Ponce, V. M. 2023. Kinematic and dynamic waves. Online publication. Seddon, J. A. 1900. River hydraulics. Transactions, ASCE, Vol.XLIII, 179243, June. Wylie, C. R. 1966. Advanced Engineering Mathematics, 3rd ed., McGrawHill Book Co., New York, NY.

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