Routing

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Routing

• Simulate the movement of water through physical
  components of watershed (e.g., channels)
• Commonly used to predict the magnitudes, volumes,
  and temporal patterns of flow (often a flood
  wave) as it moves down a channel
• Physical/Hydraulic conservation of mass and
  momentum
• Conceptual/Hydrologic some physics
  (continuity), but inexact representations
• Regression/Empirical black box

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)
Presented by Dr. Fritz Fiedler COMET Hydromet
00-1 Monday, 25 October 1999
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Continuity Equation

• The change in storage in a time interval (Dt)
  equals the difference between inflow (I) and
  outflow (O) or in its discrete form

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)

• The continuity equation in differential form is

A the cross-sectional area, Q channel flow,
and q lateral inflow
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Hydrologic Routing

• 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

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)
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Use of Manning Equation

• Stage is also related to the outflow via a
  relationship such as Manning's equation

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)
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Hydraulic Routing

• Hydraulic routing methods combine the continuity
  equation with a more realistic relationship
  describing the actual physics of the movement of
  the water
• The equation used results from conservation of
  momentum, assuming
• uniform velocity distribution (depth averaged)
• hydrostatic pressure
• small bottom slope
• In hydraulic routing analysis, it is intended
  that the dynamics of the water or flood wave
  movement be more accurately described

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)
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Momentum Equation

• Expressed by balancing the external forces acting
  on a control section of water as it moves down a
  channel

Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)

• Henderson (1966) expressed the momentum equation
  as

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Forms of Momentum Equations
Physical (Hydraulic) - continuity -
momentum Conceptual (Hydrologic) -
continuity Regression (Empirical)
Unsteady -Nonuniform
Steady - Nonuniform
Diffusion or non-inertial
Sf So
Kinematic
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Routing Methods

• Modified Puls
• Kinematic Wave
• Muskingum
• Lag and K
• Muskingum-Cunge
• Dynamic

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
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Modified Puls

• The Modified Puls routing method is most often
  applied to reservoir routing
• storage related to outflow
• 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
• As a hydrologic method, the Modified Puls
  equation is described by considering the discrete
  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
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Re-writing (substituting O for Q to follow
convention)
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 (rules) or derived (outlet works).
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Modified Puls Example

• Given the following inflow hydrograph and 2S/Dt
  O curve, find the outflow hydrograph for the
  reservoir assuming it to be completely full at
  the beginning of the storm.
• Inflow hydrograph

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Modified Puls Example

• 2S/Dt O curve

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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 provide (darn hard to see!)

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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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Review of the Method

• What are the critical components?
• Can this method be used for channel routing?
• What effect does the choice of time step have?

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Kinematic Wave

• Kinematic wave channel routing is the most basic
  form of hydraulic routing
• This method combines the continuity equation with
  a simplified form of the momentum equation
• Kinematic wave routing assumes that the friction
  slope is equal to the bed slope
• Additionally, the kinematic wave form of the
  momentum equation assumes a simple
  stage-discharge relationship

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
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Kinematic Wave Basic Equations
Q aAm
Modified Puls Kinematic Wave Muskingum Muskingum-C
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• An explicit finite difference scheme in a
  space-time grid domain is often used for the
  solution of the kinematic wave procedure.

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Working Equation
Modified Puls Kinematic Wave Muskingum Muskingum-C
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Kinematic Wave Assumptions

• The method does not explicitly allow for
  separation of the main channel and the overbanks
• Strictly speaking, the kinematic method does not
  allow for attenuation of a flood wave. Only
  translation is accomplished. There is, however,
  a certain amount of attenuation which results
  from the finite difference approximation used to
  solve the governing equations
• Best when inflow, free-surface slope, and inertia
  terms are small compared to bottom slope and
  friction
• Flow resistance may be estimated via Manning's
  equation or the Chezy equation
• Commonly used in overland flow routing

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Muskingum Method
Modified Puls Kinematic Wave Muskingum Muskingum-C
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Prism Storage
Sp K O
Sw K(I - O)X
Wedge Storage
Combined
S KXI (1-X)O
Obtained by weighting the storage due to inflow
and outflow with X, assuming that discharge and
storage are single-valued functions of depth, and
that storage responds linearly to discharge. K
is a storage factor with units of time.
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Muskingum, cont...
Substitute the storage equation
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
S KXI (1-X)O
into the continuity equation
yields
O2 C0 I2 C1 I1 C2 O1
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Muskingum Notes

• The method assumes a single stage-discharge
  relationship
• However, it is used to handle variable
  storage-discharge relationships
• inflow exceeds outflow positive wedge
• outflow exceeds inflow negative wedge
• constant cross section channel prism storage

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
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Estimating K

• K can be estimated as 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
• Use slope of the XI (1-X)O vs. S plot
• best X is least looped

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
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Estimating X

• The value of X should be between 0.0 and 0.5
• The parameter X is a weighting coefficient for
  inflow and outflow.
• As inflow becomes less important, the value of X
  decreases
• The lower limit of X 0.0 is indicative of a
  situation where inflow, I, has little or no
  effect on the storage
• A reservoir is an example of a situation where
  attenuation would be the dominant process
• 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 obtained for streams with little or no
  flood plains or storage effects
• A value of X 0.5 represents 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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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 caused by downstream dams,
  constrictions, bridges, and tidal influences

Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes
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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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Lag and K Routing

• Derived from Muskingum with X 0
• Combining this with the continuity equation
  yields
• This equation assumes pure reservoir action, and
  the peak of the hydrograph must fall on the
  receding limb of the inflow hydrograph
• Effect of translation re-introduced by lagging
  inflow hydrograph
• Lag and K can be a function of discharge
• Implemented in NWSRFS

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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
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• where
• Q discharge
• t time
• x distance along channel
• qLat 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 derived 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

• A solution can be obtained by discretizing the
  equations

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Calculation of K X
Modified Puls Kinematic Wave Muskingum Muskingum-C
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Q, B, and c are best taken as the average values
over the Dx reach and Dt time step
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Muskingum-Cunge - NOTES

• Muskingum-Cunge formulation is 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
• Rules exist for selecting time and distance steps

Modified Puls Kinematic Wave Muskingum Muskingum-C
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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

• 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 using the peak Q
  taken from the table

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Muskingum-Cunge Example

• Since there is no lateral flow, QL 0. The
  equation can be simplified to

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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 quantities can be determined

10 18 10
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Muskingum-Cunge Example
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Muskingum-Cunge Example
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Dynamic Wave Routing

• The solution of the Saint Venant equations (all
  momentum terms) is known as dynamic wave routing
• They are coupled, non-linear, first-order partial
  differential equations of the hyperbolic type
  that require one initial and two boundary
  conditions to solve
• There is no known general analytical solution -
  must use numerical methods
• characteristics
• finite difference
• finite element
• Useful to describe situations where the
  relationship between stage and discharge is not a
  single-valued function (looped rating curves)

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Dynamic Wave Equations

• NWS dynamic routing models use an extended form
  of the Saint Venant equations
• Conservation form of the equations that account
  for lateral flows, off-channel storage, and
  sinuosity
• Solved with a weighted four-point implicit finite
  difference scheme weighting factor defines
  stability and convergence properties

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Dynamic Wave Solutions

• Characteristics original equations transformed
  into ODEs and solved with simple finite
  difference techniques more commonly used for
  simpler equations (kinematic wave)
• Finite Difference implicit 4-point finite
  difference solutions (Preissmann scheme) are most
  common, but many work well
• Finite Element not advantageous for
  one-dimensional routing, but competitive with
  finite difference methods for two-dimensional
  routing with complex boundaries

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Two-Dimensional Dynamic Routing

• The two-dimensional version of the Saint Venant
  equations form the basis for the shallow water
  equations

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Modified Puls Kinematic Wave Muskingum Muskingum-C
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Some Disadvantages

• 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 - 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.
• Computational limitations - Two dimensional
  routing, and sometimes one-dimensional dynamic
  routing, is not practical for operational
  purposes (yet)

Modified Puls Kinematic Wave Muskingum Muskingum-C
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Routing Review
Modified Puls Kinematic Wave Muskingum Muskingum-C
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• Continuity is the heart of all routing methods
• Modified Puls simple hydrologic method, used
  mostly for reservoir routing (AKA level pool
  routing)
• Kinematic wave simplest hydraulic method, uses
  form of momentum equation where bed slope equals
  the friction slope and single stage-discharge
  relationship

Q aAm
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Routing Review
Modified Puls Kinematic Wave Muskingum Muskingum-C
unge Dynamic Modeling Notes

• Muskingum Method hydrologic, used for
  river-reach routing, accounts for variable
  discharge-storage relationship with wedge and
  prism storage, approximate solution to
  kinematic wave
• Muskingum-Cunge Muskingum with K and X
  functions of channel characteristics and flow
  rate, approximate solution to the diffusion wave,
  hydrologic or hydraulic?
• Dynamic hydraulic, complex but sometimes
  necessary

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Hydrologic Modeling

• Lumped
• Sacramento Soil Moisture Accounting
• Distributed
• SHE
• Semi-Distributed
• Topmodel
• Event
• HEC
• Continuous
• SAC-SMA

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Distributed Modeling
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Distributed Modeling
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Semi-Distributed Modeling