CIVE 445 - ENGINEERING HYDROLOGY
CHAPTER 9A: STREAM CHANNEL ROUTING, MUSKINGUM METHOD

Stream channel routing uses mathematical relations to calculate outflow from a stream channel, once inflow, lateral contributions, and channel characteristics are known.
Stream channel routing implies open-channel flow conditions.
Channel reach is a specific length of stream channel possessing certain translation and storage properties.
The terms stream channel routing and flood routing are used interchangeably.
This is because most channel routing applications are in flood hydrology.
Two general approaches to stream channel routing are recognized:

Hydrologic, and
Hydraulic.

Hydrologic channel routing is based on the storage concept (Chapter 4).
Hydraulic channel routing is based on the principles of mass and momentum conservation.
Hydraulic channel routing techniques are of three types:

Kinematic,
Diffusion,
Dynamic.

The dynamic wave is the most complete model of unsteady flow.
Kinematic and diffusion waves are useful approximations to the dynamic wave.
In recent years, a new approach has emerged.
This approach is similar to hydrologic routing but contains enough physical information to compare favorably with the hydraulic routing methods.
This is the basis of the Muskingum-Cunge method.

A typical model consists of:

system
input
output.

There are three types of problems:

Prediction: system and input are known; output is sought.
Calibration: input and output are known; system characterization is sought.
Inversion: system and output are known; input is sought.

Prediction is the most typical application.

9.1 MUSKINGUM METHOD

The Muskingum method of flood routing was developed by McCarthy in the 1930's in connection with the design of flood protection schemes in the Muskingum river basin, Ohio.
It is the most widely used method of hydrologic stream channel routing.
The Muskingum method is based in the differential equation of storage:

I - O = ΔS/Δt

In an ideal channel, storage is a function of inflow and outflow.
This should be contrasted with reservoirs, where storage is only a function of outflow.
In the Muskingum method, storage is a linear function of inflow and outflow:

S = K [ X I + (1 - X) O ]

in which K and X are routing parameters.
K is a time constant, or travel time through the reach.
X is a weighting factor.
The Muskingum storage equation is a generalization of the linear reservoir equation.
For X = 0, it reduces to a linear reservoir.
The discretization of the differential equation of storage on the x-t plane leads to:

(I₁ + I₂) / 2 - (O₁ + O₂) / 2 = (S₂ - S₁) / Δt

Fig. 8-2

At time level 1:

S₁ = K [ X I₁ + (1 - X) O₁ ]

At time level 2:

S₂ = K [ X I₂ + (1 - X) O₂ ]

Substituting these equations into the discretized differential equation of storage:

O₂ = C₀I₂ + C₁I₁ + C₂O₁

in which the routing coefficients are defined as follows:

C₀ = [(Δt/K) - 2X] / [2(1-X) + (Δt/K)]

C₁ = [(Δt/K) + 2X] / [2(1-X) + (Δt/K)]

C₂ = [2(1-X) - (Δt/K)] / [2(1-X) + (Δt/K)]

Since C₀ + C₁ + C₂ = 1, the routing coefficients are interpreted as weighting coefficients.
These coefficients are a function of Δt/K and X.
Given and inflow hydrograph, an initial condition, a time interval Δt and parameters K and X, the routing proceeds to calculate the outflow hydrograph.
The parameters K and X are related to flow and channel characteristics.
K is interpreted as the travel time through the reach (translation).
K is a function of reach length and flood wave speed.
X accounts for the storage or diffusion portion of the routing.
The effect of storage is to reduce the peak flow and spread the hydrograph in time.
X is a function of the flow and channel characteristics that cause runoff diffusion.
In the Muskingum method, X is interpreted as a weighting factor restricted in the range 0.0-0.5.
Values of X greater than 0.5 produce unrealistic hydrograph amplification and should be avoided.
For K = Δt and X = 0.5, the hydrograph is translated downstream without diffusion (kinematic wave).

In practice, parameters K and X are determined by calibration using streamflow records (inflow and outflow).
This is a trial-and-error procedure, which can be readily computerized.
The procedure lacks predictive capability.
Values of K and X determined in this way are valid only for the reach and flood event used in the calibration.
Extrapolation to other reaches or other flood events (within the same reach) is usually unwarranted.
When sufficient data is available, a calibration can be performed for several flood levels.
In this way, the variation of K as a function of flood level can be ascertained.
X is sensitive to time-of-rise of the inflow hydrograph.
Example 9-1 and Figure 9-2, showing sketch of the variation of K with stage.
Example 9-1 (continuation).

Unlike reservoir routing:
stream channel routing exhibits a definite lag between the start of inflow and the start of outflow
maximum outflow does not occur at the time that inflow and outflow coincide.

The calibration problem is shown in
Example 9-2.
Example 9-2 (continuation 1).
Example 9-2 (continuation 2).
The parameters K and X tend to vary with flow rate and time-of-rise (nonlinear effect).
In the Muskingum-Cunge method, the parameters are related to flow and channel characteristics.
This eliminates the need for trial-and-error calibration.
Parameter K is related to reach length and flood wave velocity (celerity).
Parameter X is related to the diffusivity characteristics of the flow and channel.

Go to Chapter 9B.

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