Introduction
This section presents two ways of routing flood hydrographs: storage (or reservoir) routing and channel routing:
♦ Use storage routing to account for inflow and outflow rates and significant water storage characteristics associated with reservoirs and detention.
♦ Use channel routing when known hydrographic data are located somewhere other than the point of interest or the channel profile or plan is changed to alter the natural velocity or channel storage characteristics.
Storage Routing
As a flood hydrograph approaches and passes through a reservoir or detention facility, the characteristics of unsteady flow become significant. You must make an accounting of inflow and outflow rates and water storage characteristics by routing a flood hydrograph through the storage facility.
Reservoir or detention pond storage routing also applies when outflow depends only upon the volume of flood storage. Use storage routing techniques to do the following:
♦ determine peak discharges from watersheds containing reservoir flood water detention basins and other flow retardation structures
♦ analyze pump station performance
♦ specify overtopping flood magnitudes
♦ evaluate traffic interruption due to roadway overtopping and the associated economic losses
Hydrograph Storage Routing Method Components
Several analytical and graphical methods route flood hydrographs through reservoirs or other detention facilities. All of the methods require reliable descriptions of the following three items:
♦ an inflow runoff hydrograph for the subject flood
♦ the storage capacity versus water elevation within the facility
♦ the performance characteristics of outlet facilities associated with the operation of the facility
By definition, when inflow and outflow from a reservoir (or any type of storage facility) are equal, a steady-state condition exists. If the inflow exceeds the outflow, the additional discharge is stored in the system. Conversely, when the outflow exceeds the inflow, water is taken from storage.
The basic reservoir routing equation is as follows:
Average Inflow - Average Outflow = Rate of Change In Storage
In numerical form, this statement of flow continuity can be written in the form of Equation 5-23.
Equation 5-23: It + It+1 Ot+ Ot+1 St+1 −−−− St
− =
2 2 ∆Τ∆Τ∆Τ∆Τ
where:
It = inflow at time step number t It+1 = inflow at time step number t + 1 Ot = outflow at time step number t Ot+1 = outflow at time step number t + 1
St = storage in the reservoir at time step number t St+1 = storage in the reservoir at time step number t + 1
T = the time increment t = time step number
Various routing methods are useful in specific instances. Some of the more prominent and effective methods are storage-indication, ripple mass curve, and Sorenson graphical.
Storage Indication Routing Method
Of the many methods for routing floods through reservoirs, the Storage-Indication Method is a relatively simple procedure suitable for most highway drainage applications. Since the outflow discharge (O) is a function of storage alone, it is convenient to rewrite the routing equation as Equation 5-24.
Equation 5-24: 2St+1
Ot+1 It + It+1 Ot
+ = −
∆Τ∆Τ∆Τ
∆Τ + 2St
∆Τ
∆Τ
∆Τ
∆Τ
Relationship Determination
The use of the Storage-Indication Method requires that you determine the relationships among stage, storage, and discharge. This information is in addition to a description of the inflow hydrograph.
The stage-storage relation is simply the volume of water held by the reservoir or storage facility as a function of the water surface elevation or depth. This information is often available from the reservoir sponsor or owner. Where the stage-storage relation is not available, you may need to develop one by successive calculations of storage vs. associated stages in the storage facility.
The stage-outflow relation is based on the association of the reservoir stage (head) and the resulting outflow from the storage facility. This description of performance characteristics may be the following:
♦ ratings of the primary and/or emergency spillway of a reservoir
♦ pump flow characteristics in a pump station
♦ hydraulic performance curve of a culvert or bridge on a highway
♦ hydraulic performance curve of a weir and orifice outlet of a detention pond
The stage-outflow relation of the outlet works of a reservoir often is available through the reservoir sponsor or owner. In some cases, the highway designer may have developed it.
With stage-storage and stage-outflow relations established, storage and outflow can be related at each stage. The relationship is described in the form of O vs 2S
T O
ổ∆
ốỗ ử
ứữ+ You can plot this relation over the range of anticipated stages. Figures 5-12 (English measurement) and 5-13 (metric) illustrate sample relationships.
Figure 5-14. Storage Outflow Relation (English)
The form of Equation 5-24 is especially useful because the terms on the left side of the equation are known. With the relation between the outflow and storage determined (Figure 5-12), the ordinates on the outflow hydrograph can be determined directly.
Storage-Indication Routing Procedure
Use the following steps to route an inflow flood runoff hydrograph through a storage system such as a reservoir or detention pond:
1. Acquire or develop a design flood runoff hydrograph.
2. Acquire or develop a stage-storage relation.
3. Acquire or develop a stage-outflow relationship.
4. Develop a storage-outflow relation curve.
5. Assume an initial value for Ot as equal to It. At time step one (t = 1), assume an initial value for Ot as equal to It. Usually, at time step one, inflow equals zero, so outflow will be zero and 2S1/ T - O1 equals zero. Note that to start, t + 1 in the next step is 2.
6. Compute 2St+1/ T + Ot+1 using Equation 5-24.
7. Interpolate to find the value of outflow. From the storage-outflow relation, interpolate to find the value of outflow (Ot+1) at (2St+1)/(∆T)+Ot+1 from step 6.
8. Determine the value of (2St+1)/(∆T)-Ot+1. Use the relation (2St+1)/(∆T)-Ot+1 = (2St+1)/(∆T)+Ot+1 - 2Ot+1.
9. Assign the next time step to the value of t., e.g., for the first run through set t = 2.
10. Repeat steps 6 through 9 until the outflow value (Ot+1) approaches zero.
11. Plot the inflow and outflow hydrographs. The peak outflow value should always coincide with a point on the receding limb of the inflow hydrograph.
12. Check conservation of mass to help identify success of the process. Use Equation 5-25 to compare the inflow volume to the sum of retained and outflow volumes.
Equation 5-25: ∆T⋅ồIt =Sr +∆T⋅ồOt where:
Sr = volume of runoff completely retained (cu. ft. or m3)
t = sum of inflow hydrograph ordinates (cfs or m3/s)
t = sum of outflow hydrograph ordinates (cfs or m3/s)
There will be no retention volume if the outflow structure is at the flow line of the pond.
You can expect a degree of imbalance due to the discretization process. If the difference is large yet the calculations are correct, reduce the time increment ( T); determine the inflow hydrograph values for the new time steps, and repeat the routing process. (See Storage Indication Routing Example.)
Channel Routing
Routing of flood hydrographs by means of channel routing procedures is useful in instances where known hydrographic data are at a point other than the point of interest. This is also true in those instances where the channel profile or plan is changed in such a way as to alter the natural velocity or channel storage characteristics. Routing analysis estimates the effect of a channel reach on an inflow hydrograph. This section describes the
Muskingum Method Equations, a lumped flow routing technique that approximates storage effects in the form of a prism and wedge component (Chow, 1988).
Total Storage Equation. The Muskingum Method combines a prism component of storage, KO, and a wedge component, KX(I-O), to describe the total storage in the reach as Equation 5-26:
Equation 5-26: S = K [XI + (1-X) O]
where:
S = total storage (cu. ft. or m3)
K = a proportionality constant representing the time of travel of a flood wave to traverse the reach (s). Oftentimes, this is set to the average travel time through the reach.
X = a weighting factor describing the backwater storage effects approximated as a wedge.
I = inflow (cfs or m3/s) O = outflow (cfs or m3/s)
The value of X depends on the amount of wedge storage; when X = 0, there is no backwater (reservoir type storage), and when X = 0.5, the storage is described as a full wedge. The weighting factor, X, ranges from 0 to 0.3 in natural streams. A value of 0.2 is typical.
Time Rate of Change Equation. Equation 5-27 represents the time rate of change of storage as the following:
Equation 5-27: S S T
K XI X O XI X O
T
t+1− t = t+1+ −1 t+1 − t + −1 t
∆ ∆
{[ ( ) ] [ ( ) ]}
where:
∆T = time interval usually ranging from 0.3⋅K to K t = time step number
Flow-Routing Equation. Applying continuity to Equation 5-28 produces the Muskingum flow routing equation as follows:
Equation 5-28: Ot+1 = C1⋅It+1 + C2⋅It + C3⋅Ot
where:
Equation 5-29: C T KX
K X T
1
2
=2 1 −
− +
∆
∆
( )
Equation 5-30: C T KX
K X T
2
2
= 2 1 +
− +
∆
∆ ( ) Equation 5-31: C K X T
K X T
3
2 1
= 2 1− −
− +
( )
( )
∆
∆
By definition, the sum of C1, C2, and C3 should be 1. If measured inflow and outflow hydrographs are available, you may approximate K and X using Equation 5-33. Calculate X by plotting the numerator on the vertical axis and the denominator on the horizontal axis, and adjusting X until the loop collapses into a single line. The slope of the line equals K.
Equation 5-32: K T I I O O
X I I X O O
t t t t
t t t t
= + − +
⋅ + − + + − + + − 0 5
1
1 1
1 1
. [( ) ( )]
( ) ( )( )
∆
You may also approximate K and X using the Muskingum-Cunge Method described in Chow, 1988; or Fread, 1993.
Section 10