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
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
Unsteady -Nonuniform
Steady - Nonuniform
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
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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.

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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

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Modified Puls Example

• The following 2S/Dt O curve is also given

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Modified Puls Example

• A table may be created as follows

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Modified Puls Example

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

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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.

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Modified Puls Example

• This is then repeated for the rest of the values
  in the column.

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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
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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 provided.

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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

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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

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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.

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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
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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
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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.
• 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
25
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
26
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
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Did you know?

• Lag and K is a special case of Muskingum -gt X0!

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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

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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
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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

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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

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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

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Muskingum Example Problem

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

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Muskingum Example - Tenkiller
Look at R-5
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Reach at Outlet
K 2 hrs X 0.25
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K 2 4 Hrs.X 0.25
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K 2, 4, 6 Hrs.X 0.25
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Lets Alter XK 2 hrs.
K 2 hrs X 0.25
K 2 hrs X 0.5
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And Again....
K 2 hrs X 0.5
K 2 hrs X 0.05
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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
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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

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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

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Solution of Muskingum-Cunge

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

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
45
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.
46
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
47
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.

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Muskingum-Cunge Example

• First, K must be determined.
• K is equal to

Is c a constant?

• Dx 6 km, while c 2 m/s

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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.

50
Muskingum-Cunge Example

• The coefficients of the Muskingum-Cunge method
  can now be determined.

Using 7200 seconds or 2 hrs for timestep - this
may need to be changed
51
Muskingum-Cunge Example

• The coefficients of the Muskingum-Cunge method
  can now be determined.

52
Muskingum-Cunge Example

• The coefficients of the Muskingum-Cunge method
  can now be determined.

53
Muskingum-Cunge Example

• The coefficients of the Muskingum-Cunge method
  can now be determined.

54
Muskingum-Cunge Example

• Then a simplification of the original formula can
  be made.

• Since there is not lateral flow, QL 0. The
  simplified formula is the following

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Muskingum-Cunge Example

• A table can then be created in 2 hour time steps
  similar to the one below

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Muskingum-Cunge Example

• It is assumed at time zero, the flow is 10 m3/s
  at each distance.

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Muskingum-Cunge Example

• Next, zero is substituted into for each letter to
  solve the equation.

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Muskingum-Cunge Example

• Using the table, the variables can be determined.

10 18 10
59
Muskingum-Cunge Example

• Therefore, the equation can be solved.

60
Muskingum-Cunge Example

• Therefore, the equation can be solved.

61
Muskingum-Cunge Example

• This is repeated for the rest of the columns and
  the subsequent columns to produce the following
  table. Note that when you change rows, n
  changes. When you change columns, j changes.

62
Look again at the Discretization
t
X
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Things to Consider

• As the wave comes into a reach
• the flow and depth go up and then down
• the wave speed and ALL coefficients would then
  change...
• What effect would this have?

64
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
65
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
66
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
67
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
68
Some DISadvantages

• 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