 21 Oct 2022
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Muskingum Routing
 Updated on 21 Oct 2022
 3 Minutes to read

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The Muskingum Routing unit models the flow of water in open channels based on the continuity equation and the Muskingum storage relationship.
Data
Field in Data Entry Form  Description  Name in Datafile  

Section Label  node label at discharge point  Label1 

Distance to Next  distance to next discharge point or cross section in metres (a zero specifies the end of the reach and for this last node the k and x values will not be used)  dx 

Bed Elevation  bed elevation in m AD (not used in computations)  z 

K Parameter  Muskingum k parameter (s)  k 

X Parameter  Muskingum x parameter  x 

Specify Velocity Using  For “VQ Power Law”, enter the coefficients a, b, V_{0} and Q_{0} to define the flowvelocity relationship; otherwise (VQ Rating), enter the relationship at discrete points in the supplied table 


Constant a  scaling parameter for velocity calculation (>0)  a  
Exponent b  power law parameter for velocity calculation (>0)  b  
Lowest Permitted Velocity  lowest permitted velocity (>0)  V_{0}  
Flow Threshold  flow below which velocity is set to Vo (>0)  Q_{0}  
n/a  number of subsequent V Q data pairs  ndat  
Velocity and Flow Table  a table of velocity and flow values used to interpolate velocity values for subsequent Flood Modeller Quality simulations  V,Q 
 These fields are only used when the results of the simulation are to be used for a subsequent water quality simulation. However, dummy data values must always be specified.
Theory and Guidance
The Muskingum Routing unit models the flow of water in natural and manmade open channels using the Muskingum method to route the flow.
The Muskingum Routing calculates the discharge within a river or channel reach given the inflow hydrograph at the upstream end.
The unit is based on the continuity equation and the Muskingum storage relationship (McCarthy G.T. (1938)).
Cross section details are not required and only the Muskingum parameters k and x are specified. Both parameters are fixed.
A minimum of two Muskingum Routing sections are required for each end of the river or channel reach.
Intermediate discharge points can be specified by using additional Muskingum Routing nodes.
Equations
The equations used in the Muskingum Routing are the continuity equation:

and the storage relationship:
where: I = inflow to the reach [m^{3}/s] O = outflow from the reach [m^{3}/s] S = storage in the reach [m^{3}] t = time [s] x = weighting coefficient k = storage constant [s] 
The basic principle is that storage is a linear function of the weighted average flux, where the latter is defined as the term x * I + (1  x) * O in equation (2).
Combining both equations an explicit equation can be obtained to calculate the outflow at the next time step:


where the coefficients C_{0}, C_{1} and C_{2} are defined as:


The subscripts 1 and 2 on I and O represent the values at time t_{1} and t_{2} respectively.
The values of k and x may be obtained by calibration, with the recommended constraints that
0 < x < ½, and
x < Δt/2k < 1x,
where Δt is the simulation timestep.
Water Quality Simulations
If the flood routing model is to be used to generate output for a future water quality simulation, an approximate method must be used to calculate the nodal velocities required. See Routing Velocity Calculation for more information.
General
A feature of the Muskingum method is the possible generation of undershoots in flow in advance of the rising limb for certain parameter values.
The timestep should never be greater than the value of the k parameter. This value can be approximated by the travel time of the wave through the reach. A value for x between 0.0 and 0.3 is recommended.
The performance of the routing depends upon the values chosen for the routing parameters. When measured flood hydrographs are available calibration of the Muskingum parameters can be derived.
Muskingum units can be preceded or followed by normal Flood Modeller river reaches. However a rating curve (QRATING) needs to be defined at the downstream end of a river reach if it joins a Muskingum unit. The normal Flood Modeller connectivity rules apply for Muskingum units and therefore a HTBDY is required (but not used) at the downstream end of a model coincident with a Muskingum routing reach.
Datafile Format
Line 1  keyword `RIVER' [comment]
Line 2  keyword `MUSKINGUM'
Line 3  Label1
Line 4  dx, z
Line 5  k, x
Line 6  keyword 'VQ POWER LAW' or 'VQ RATING'
Line 7  V_{o}, Q_{o}, a, b (if 'VQ POWER LAW') or ndat (if 'VQ RATING')
Line 8 to Line 7+ndat  V, Q (if 'VQ RATING')