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## Channel Routing

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Title: Channel Routing

1
Channel Routing
• Simulate the movement of water through a channel
• Used to predict the magnitudes, volumes, and
temporal patterns of the flow (often a flood
wave) as it translates down a channel.
• 2 types of routing hydrologic and hydraulic.
• both of these methods use some form of the
continuity equation.

Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
Presented by Dennis Johnson COMET Hydromet
99-2 Monday, 28 June 1999
2
Continuity Equation
Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
• The change in storage (dS) equals the difference
between inflow (I) and outflow (O) or
• For open channel flow, the continuity equation is
also often written as

A the cross-sectional area, Q channel flow,
and q lateral inflow
3
Hydrologic Routing
Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
• Methods combine the continuity equation with some
relationship between storage, outflow, and
possibly inflow.
• These relationships are usually assumed,
empirical, or analytical in nature.
• An of example of such a relationship might be a
stage-discharge relationship.

4
Use of Manning Equation
Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
• Stage is also related to the outflow via a
relationship such as Manning's equation

5
Hydraulic Routing
• Hydraulic routing methods combine the continuity
equation with some more physical relationship
describing the actual physics of the movement of
the water.
• The momentum equation is the common relationship
employed.
• In hydraulic routing analysis, it is intended
that the dynamics of the water or flood wave
movement be more accurately described

Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
6
Momentum Equation
Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
• Expressed by considering the external forces
acting on a control section of water as it moves
down a channel
• Henderson (1966) expressed the momentum equation
as

7
Combinations of Equations
• Simplified Versions

Continuity equation Hydrologic Routing Hydraulic
Routing Momentum Equation
Diffusion or noninertial
Kinematic
Sf So
8
Routing Methods
• Modified Puls
• Kinematic Wave
• Muskingum
• Muskingum-Cunge
• Dynamic

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
9
Modified Puls
• The modified puls routing method is probably most
often applied to reservoir routing
• The method may also be applied to river routing
for certain channel situations.
• The modified puls method is also referred to as
the storage-indication method.
• The heart of the modified puls equation is found
by considering the finite difference form of the
continuity equation.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
10
Modified Puls
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
Continuity Equation
Rewritten
• The solution to the modified puls method is
accomplished by developing a graph (or table) of
O -vs- 2S/?t O. In order to do this, a
stage-discharge-storage relationship must be
known, assumed, or derived.

11
Modified Puls Example
• Given the following hydrograph and the 2S/Dt O
curve, find the outflow hydrograph for the
reservoir assuming it to be completely full at
the beginning of the storm.
• The following hydrograph is given

12
Modified Puls Example
• The following 2S/Dt O curve is also given

13
Modified Puls Example
• A table may be created as follows

14
Modified Puls Example
• Next, using the hydrograph and interpolation,
insert the Inflow (discharge) values.
• For example at 1 hour, the inflow is 30 cfs.

15
Modified Puls Example
• The next step is to add the inflow to the inflow
in the next time step.
• For the first blank the inflow at 0 is added to
the inflow at 1 hour to obtain a value of 30.

16
Modified Puls Example
• This is then repeated for the rest of the values
in the column.

17
Modified Puls Example
• The 2Sn/Dt On1 column can then be calculated
using the following equation

Note that 2Sn/Dt - On and On1 are set to zero.
30 0 2Sn/Dt On1
18
Modified Puls Example
• Then using the curve provided outflow can be
determined.
• In this case, since 2Sn/Dt On1 30, outflow
5 based on the graph provide (darn hard to see!)

19
Modified Puls Example
• To obtain the final column, 2Sn/Dt - On, two
times the outflow is subtracted from 2Sn/Dt
On1.
• In this example 30 - 25 20

20
Modified Puls Example
• The same steps are repeated for the next line.
• First 90 20 110.
• From the graph, 110 equals an outflow value of
18.
• Finally 110 - 218 74

21
Modified Puls Example
• This process can then be repeated for the rest of
the columns.
• Now a list of the outflow values have been
calculated and the problem is complete.

22
Review of the Method
• What are the critical issues/parameters/variables?
• Can this method be used for channel routing?
• What would you need to create?

23
Muskingum Method
Sp K O
Prism Storage
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
Sw K(I - O)X
Wedge Storage
Combined
S KXI (1-X)O
24
Muskingum, cont...
Substitute storage equation, S into the S in
the continuity equation yields
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
S KXI (1-X)O
O2 C0 I2 C1 I1 C2 O1
25
Muskingum Notes
• The method assumes a single stage-discharge
relationship.
• In other words, for any given discharge, Q, there
can be only one stage height.
• This assumption may not be entirely valid for
certain flow situations.
• For instance, the friction slope on the rising
side of a hydrograph for a given flow, Q, may be
quite different than for the recession side of
the hydrograph for the same given flow, Q - or
• This causes an effect known as hysteresis, which
can introduce errors into the storage assumptions
of this method.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
26
Estimating K
• K is estimated to be the travel time through the
reach.
• This may pose somewhat of a difficulty, as the
travel time will obviously change with flow.
• The question may arise as to whether the travel
time should be estimated using the average flow,
the peak flow, or some other flow.
• The travel time may be estimated using the
kinematic travel time or a travel time based on
Manning's equation.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
27
Estimating X
• The value of X must be between 0.0 and 0.5.
• The parameter X may be thought of as a weighting
coefficient for inflow and outflow.
• As inflow becomes less important, the value of X
decreases.
• The lower limit of X is 0.0 and this would be
indicative of a situation where inflow, I, has
little or no effect on the storage.
• A reservoir is an example of this situation and
it should be noted that attenuation would be the
dominant process compared to translation.
• Values of X 0.2 to 0.3 are the most common for
natural streams however, values of 0.4 to 0.5
may be calibrated for streams with little or no
flood plains or storage effects.
• A value of X 0.5 would represent equal
weighting between inflow and outflow and would
produce translation with little or no attenuation.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
28
More Notes - Muskingum
• The Handbook of Hydrology (Maidment, 1992)
includes additional cautions or limitations in
the Muskingum method.
• The method may produce negative flows in the
initial portion of the hydrograph.
• Additionally, it is recommended that the method
be limited to moderate to slow rising hydrographs
being routed through mild to steep sloping
channels.
• The method is not applicable to steeply rising
hydrographs such as dam breaks.
• Finally, this method also neglects variable
backwater effects such as downstream dams,
constrictions, bridges, and tidal influences.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
29
Muskingum Example Problem
• A portion of the inflow hydrograph to a reach of
channel is given below. If the travel time is
K1 unit and the weighting factor is X0.30, then
find the outflow from the reach for the period
shown below

30
Muskingum Example Problem
• The first step is to determine the coefficients
in this problem.
• The calculations for each of the coefficients is
given below

C0 - ((10.30) - (0.51)) / ((1-(10.30)
(0.51)) 0.167
C1 ((10.30) (0.51)) / ((1-(10.30)
(0.51)) 0.667
31
Muskingum Example Problem
C2 (1- (10.30) - (0.51)) / ((1-(10.30)
(0.51)) 0.167
• Therefore the coefficients in this problem are
• C0 0.167
• C1 0.667
• C2 0.167

32
Muskingum Example Problem
• The three columns now can be calculated.
• C0I2 0.167 5 0.835
• C1I1 0.667 3 2.00
• C2O1 0.167 3 0.501

33
Muskingum Example Problem
• Next the three columns are added to determine the
outflow at time equal 1 hour.
• 0.835 2.00 0.501 3.34

34
Muskingum Example Problem
• This can be repeated until the table is complete
and the outflow at each time step is known.

35
Muskingum-Cunge
• Muskingum-Cunge formulation is similar to the
Muskingum type formulation
• The Muskingum-Cunge derivation begins with the
continuity equation and includes the diffusion
form of the momentum equation.
• These equations are combined and linearized,

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
36
Muskingum-Cungeworking equation
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
• where
• Q discharge
• t time
• x distance along channel
• qx lateral inflow
• c wave celerity
• m hydraulic diffusivity

37
Muskingum-Cunge, cont...
• Method attempts to account for diffusion by
taking into account channel and flow
characteristics.
• Hydraulic diffusivity is found to be

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
• The Wave celerity in the x-direction is

38
Solution of Muskingum-Cunge
• Solution of the Muskingum is accomplished by
discretizing the equations on an x-t plane.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
39
Calculation of K X
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
Estimation of K X is more physically based
and should be able to reflect the changing
conditions - better.
40
Muskingum-Cunge - NOTES
• Muskingum-Cunge formulation is actually
considered an approximate solution of the
convective diffusion equation.
• As such it may account for wave attenuation, but
not for reverse flow and backwater effects and
not for fast rising hydrographs.
• Properly applied, the method is non-linear in
that the flow properties and routing coefficients
are re-calculated at each time and distance step
• Often, an iterative 4-point scheme is used for
the solution.
• Care should be taken when choosing the
computation interval, as the computation interval
may be longer than the time it takes for the wave
to travel the reach distance.
• Internal computational times are used to account
for the possibility of this occurring.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
41
Muskingum-Cunge Example
• The hydrograph at the upstream end of a river is
given in the following table. The reach of
interest is 18 km long. Using a subreach length
Dx of 6 km, determine the hydrograph at the end
of the reach using the Muskingum-Cunge method.
Assume c 2m/s, B 25.3 m, So 0.001m and no
lateral flow.

42
Muskingum-Cunge Example
• First, K must be determined.
• K is equal to
• Dx 6 km, while c 2 m/s

43
Muskingum-Cunge Example
• The next step is to determine x.
• All the variables are known, with B 25.3 m, So
0.001 and Dx 6000 m, and the peak Q taken from
the table.

44
Muskingum-Cunge Example
• Then a simplification of the original formula can
• Since there is not lateral flow, QL 0. The
simplified formula is the following

45
Full Dynamic Wave Equations
• The solution of the St. Venant equations is known
as dynamic routing.
• Dynamic routing is generally the standard to
which other methods are measured or compared.
• The solution of the St. Venant equations is
generally accomplished via one of two methods
1) the method of characteristics and 2) direct
methods (implicit and explicit).
• It may be fair to say that regardless of the
method of solution, a computer is absolutely
necessary as the solutions are quite time
consuming.
• J. J. Stoker (1953, 1957) is generally credited
for initially attempting to solve the St. Venant
equations using a high speed computer.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
46
Dynamic Wave Solutions
• Characteristics, Explicit, Implicit
• The most popular method of applying the implicit
technique is to use a four point weighted finite
difference scheme.
• Some computer programs utilize a finite element
solution technique however, these tend to be
more complex in nature and thus a finite
difference technique is most often employed.
• It should be noted that most of the models using
the finite difference technique are
one-dimensional and that two and
three-dimensional solution schemes often revert
to a finite element solution.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
47
Dynamic Wave Solutions
• Dynamic routing allows for a higher degree of
accuracy when modeling flood situations because
it includes parameters that other methods
neglect.
• Dynamic routing, when compared to other modeling
techniques, relies less on previous flood data
and more on the physical properties of the storm.
This is extremely important when record
rainfalls occur or other extreme events.
• Dynamic routing also provides more hydraulic
information about the event, which can be used to
determine the transportation of sediment along
the waterway.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
48
Courant Condition?
• If the wave or hydrograph can travel through the
subreach (of length ?x) in a time less than the
computational interval, ?t, then computational
instabilities may evolve.
• The condition to satisfy here is known as the
Courant condition and is expressed as

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
49
• Geometric simplification - some models are
designed to use very simplistic representations
of the cross-sectional geometry. This may be
valid for large dam breaks where very large flows
are encountered and width to depth ratios are
large however, this may not be applicable to
smaller dam breaks where channel geometry would
be more critical.
• Model simulation input requirements - dynamic
routing techniques generally require boundary
conditions at one or more locations in the
domain, such as the upstream and downstream
sections. These boundary conditions may in the
form of known or constant water surfaces,
hydrographs, or assumed stage-discharge
relationships.
• Stability - As previously noted, the very complex
nature of these methods often leads to numeric
instability. Also, convergence may be a problem
in some solution schemes. For these reasons as
well as others, there tends to be a stability
problem in some programs. Often times it is very
difficult to obtain a "clean" model run in a cost
efficient manner.

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
50
Get to the Important Stuff!
• What causes the greatest errors?
• Muskingum-Cunge The minimum rise time of the
hydrograph to maintain an error of less than 10
increases as the slope decreases.
• i.e. fast rising hydrographs on flat slopes
cause problems.
• Level-pool has larger errors for fast rising
outflows (dam breaks) and very long narrow
reservoirs and flashy inflow hydrographs.
• Worst case for level pool is a long, shallow
reservoir with a small fast rising inflow
hydrograph
• In the above cases - expect errors gt 10.

51
More tidbits
• Dam break wave will travel approximately _at_ 2
(slope).5, where slope is in feet/mile and
resulting velocity is in mph!
• Remember that a dam break wave/flood is 10-100
times greater than has ever been seen before by
that river ecosystem!