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Routing....................................................................................................................................................... 1
Hydrologic Channel Routing ................................................................................................................. 1
Channel Storage Routing ................................................................................................................... 1
Channel Muskingum Routing ............................................................................................................ 6
Channel Kinematic Routing ............................................................................................................... 7
Hydrologic Reservoir Routing ............................................................................................................... 8
Reservoir Storage Routing ................................................................................................................. 9
Numerical Routing (Power-Law Models) ....................................................................................... 13
References ................................................................................................................................................ 14
Routing
Routing is the name given to the process of determining variations in discharge at a location as it moves
in a conduit (stream, bayou, canal, etc.). The procedures can be as simple as a time lag from input to
output (the reactor equivalent is a plug-flow reactor) or as complex as solving 3-dimensional flow
patterns in the conduit. The procedures based on time lagging and/or conservation of volumes only, are
called hydrologic routing, while those that involve consideration of pressure and momentum are usually
called hydraulic routing. Both methods are useful and appropriate in different situations – hydraulic
routing obviously being more complicated mathematically.
Routing is further subdivided into channel routing (translating hydrographs in a stream channel) or
reservoir routing (translating hydrographs through a storage structure).
Hydrologic Channel Routing
Hydrologic routing involves computing the outflow hydrograph corresponding to an inflow hydrograph
from a conservation of mass (volume) approach. Generally a conduit is divided into sections called
reaches. A section of canal is a good mental model. At the upstream end flow enters and at the
downstream end it exits. The flow distance between the two locations is called the reach length. Once
these terms are defined for a conduit we simply write a storage balance for water in the reach and then
relate storage to outflow by some model.
Channel Storage Routing
The basic model is the discrete continuity equation for the channel reach. The subscripts 1 and 2 are for
times at the beginning and end of a time step of length t. The inflow is located at the upstream end of
the reach, and the outflow at the downstream end of the reach.
S 2  S1 I1  I 2 O1  O2


t
2
2
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The two unknowns in the time evolution of such a model are O2 and S2.
equation is
( S 2  t
A rearrangement of the
O2
O
I  I2
)  ( S1  t 1 )  t 1
2
2
2
The storage in a reach can be estimated as the product of the average cross sectional area for a given
discharge rate and the reach length, as indicated in the sketch below.
Figure 1
A rating equation is used at each cross section to determine the cross section areas (i.e. Manning’s
equation or something similar).
Once the storage-discharge relationship is constructed a table (or characteristic curves) for
O
( S  t ) and
2
O
( S  t ) is prepared.
2
The starting with a known inflow hydrograph and initial storage condition the mass balance can be
propagated forward in time to estimate the outflow hydrograph. The choice of t value should be made
so that it is smaller than the travel time in the reach at the largest likely flow and smaller than about 1/5
the time to peak of the inflow hydrograph, otherwise rapid changes in flow will propagate without
effect in the calculations.
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Example of storage routing:
Consider a channel that is 2500 feet long, with slope of 0.09%, clean sides with straight banks and no
rifts or deep pools. A typical channel cross section is shown below, Manning’s n is 0.030.
Figure 2
The inflow hydrograph is triangular with a time base of 3 hours, and time-to-peak of 1 hour. The peak
inflow rate is 360 cfs.
Figure 3
Routing this inflow hydrograph by storage routing is accomplished as follows.
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Step #1: Construct Depth-Storage-Discharge Table (Rating Table/Characteristic Curves); note how the
time step is specified.
Figure 4
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Step #2: Construct routing table; Use the storage-outflow-depth table to interpolate values as the storage
equation is propagated forward in time. The example spreadsheet has interpolation built-in as an Excel
4.0 Macro sheet.
Figure 5
We are essentially done; a similar table can be constructed using power-law models of the depthdischarge-storage table if one expects to route a lot of different hydrographs through the reach.
Additionally a “unit-transfer” function could also be constructed.
A plot of the result is displayed below.
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Figure 6
Reference: Stream_Storage_Routing.xls : Worksheet with routing tables and interpolation macros.
Channel Muskingum Routing
[description]
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[work same problem]
Channel Kinematic Routing
[description]
[work same problem]
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Hydrologic Reservoir Routing
Hydrologic reservoir routing involves computing the outflow hydrograph corresponding to an inflow
hydrograph from a conservation of mass (volume) approach for a hydraulic storage element such as a
detention pond, reservoir, structural BMP, etc. Generally the element is divided into layers (depths)
that cover different areas of inundation. These layers have a known (computable) area and thickness,
thus a volume can be determined as a function of water surface elevation (or depth). Such a
relationship is the depth-storage characteristic for the reservoir. At the upstream end flow enters and at
the downstream end it exits or is stored. The depth-discharge relationship is determined from overflow
(weir) equations in the case of spillways and similar outlet structures.
The series of figures below illustrates the depth-storage relationship for a hypothetical dam on some
watershed. In the figures, the inundation area changes dramatically as the water surface elevation
increases (such is typical for a reservoir). A graph or table of W.S.E. versus area multiplied by the
contour interval (in this case 10 feet) would produce an elevation-storage relationship.
Figure 7. Innudation area at W.S.E. = 570 feet.
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Figure 8. Innudation area at W.S.E. = 580 feet.
Figure 9. Innudation area at W.S.E. = 590 feet.
Reservoir Storage Routing
In this method geometric considerations are used to construct a storage-discharge table for a reservoir
element, much like the river reach element. The principle difference is that a reservoir can store
significant water before any outflow occurs over a spillway or through a sluice gate. The depth-storagedischarge table will need to reflect this storage capability.
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As an example consider a dam with an uncontrolled spillway (emergency spillway) 10 meters wide with
a crest elevation of 548.0 meters. The weir discharge coefficient is 0.45. The reservoir-elevation
relationship is provided in the table below and was developed from a topographic map in a fashion
similar to the inundation maps above. The elevation-discharge relationship is computed using a weir
equation. The inflow hydrograph is provided and the starting water elevation is 544.0 meters. The
figure below depicts the reservoir situation.
Figure 10
The weir equation used is
Q  K 2g LH 3 / 2  0.45 19.62  10  H 3 / 2  19.9  H 3 / 2
where H is the height of the reservoir water ABOVE the weir crest. If the water is below the crest,
then there is no flow.
The reservoir-area-depth-storage-overflow table is displayed below. First a tabulation that relates
changes in water surface elevation to storage is listed (columns B, C and D). Next a weir equation is
used to relate flow over the spillway to the water elevation (columns E and F). Finally the storage and
outflow relation are tabulated in the sixth column.
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Figure 11
The values in this table are used in the hydrograph routing table to pass the flood hydrograph through
the reservoir and preserve the total mass in the system. The routing model is again simply continunity
expressed as:
( S 2  t
O2
O
I  I2
)  ( S1  t 1 )  t 1
2
2
2
Once this table is built then from any given value of storage+outflow (column F) are interpolated in the
table to determine the associated discharge (column E). These interpolated results are then used to
estimate the S-O relationship for a time-series of inflows starting from known values. Although it
seems complicated it is actually a direct application of a mass balance equation. The next figure shows
the results.
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Figure 12
Lastly, a plot of the inflow and outflow hydrograph is provided to illustrate the effect of storage on
hydrograph behavior.
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Figure 13
Reference: Reservoir_Storage_Routing.xls : Worksheet with routing tables and interpolation macro.
Numerical Routing (Power-Law Models)
Hydraulic Routing
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Hydraulic routing is routing based on solution of hydraulic equations for a conduit. The distinction is
somewhat unclear as hydraulics is considered in hydrologic routing, but in the case of hydraulic routing
pressure and momentum is considered in addition to just storage.
References
Wurbs and James, 2002. Water Resources Engineering. Prentice-Hall, New Jersey. pp 356-408 (Entire
Chapter 6).
Hann, C.T., Barfield, B.J., and Hayes, J.C. 1994. Design Hydrology and Sedimentology for Small
Catchments. Academic Press, San Diego. pp. 182-203 (Entire Chapter 6).
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