Engineering Hydrology Lecture Note
TABLE OF CONTENTS
4 Flood Routing....................................................................................................................2
4.1 General.......................................................................................................................2
4.2 Simple Non-storage Routing.....................................................................................3
4.3 Storage Routing.........................................................................................................4
4.4 Reservoir or level pool routing..................................................................................5
4.5 Channel routing.........................................................................................................7
4.5.1 Muskingum Method of Routing........................................................................8
4.5.2 Application of the Muskingum Method:...........................................................9
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4 Flood Routing
4.1 General
At a river gauging station, the stage and discharge hydrographs represent the passage of waves of
river depth and stream flow during flood, respectively. As this wave moves down the river, the
shape of the wave gets modified due to various factors, such as channel storage, resistance,
lateral addition or withdrawal of flows etc. when a flood wave passes through a reservoir, its peak
is attenuated and the time base is enlarged (translated) due to the effect of storage. Flood waves
passing down a river have their peaks attenuated due to friction if there is no lateral inflow. In
both reservoir and channel conditions the time to peak is delayed, and hence the peak discharge is
translated.
Flood routing is the technique of determining the flood hydrograph at a section of a river by
utilizing the data of flood flow at one or more upstream sections. The hydrologic analysis of
problems such as flood forecasting, flood protection, reservoir and spillway design invariably
include flood routing. In these applications two broad categories of routing can be recognized.
These are:
i) Reservoir routing and
ii) Channel routing
In reservoir routing the effect of a flood wave entering a reservoir is studied. Knowing the
volume-elevation characteristics of the reservoir and the out flow elevation relationship for
spillways and other outlet structures in the reservoir; the effect of a flood wave entering the
reservoir is studied to predict the variation of reservoir elevation and out flow discharge with
time. This form of routing is essential (i) in the design of the capacity of spillways and other
reservoir outlet structures and (ii) in the location and sizing of the capacity of reservoirs to meet
specific requirements.
In channel routing the changes in the shape of a hydrograph as it travels down a channel is
studied. By considering a channel reach and an input hydrograph at the upstream end, this form
of routing aims to predict the flood hydrograph at a various sections of the reach. Information on
the flood-peak attenuation and the duration of high-water levels obtained by channel routing is
utmost importance in flood forecasting operations and flood protection works.
A variety of flood routing methods are available and they can be broadly classified in to two
categories as: (i) hydraulic routing and (ii) hydrologic routing. Hydrologic routing methods
employ essentially the equation of continuity and a storage function, indicated as lumped routing.
Hydraulic methods, on the other hand, employ the continuity equation together with the equation
of motion of unsteady flow. The basic differential equations used in the hydraulic routing, known
as St. Venant equations afford a better description of unsteady flow than hydrologic methods.
A flood hydrograph is modified in two ways as the storm water flows downstream. Firstly, and
obviously, the time of the peak rate of flow occurs later at downstream points. This is known as
translation. Secondly, the magnitude of the peak rate of flow is diminished at downstream
points, the shape of the hydrograph flattens out, and the volume at the floodwater takes longer to
pass a lower section. This modification of the hydrograph is called attenuation.
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Figure 4.1: Flood translation and attenuation
4.2 Simple Non-storage Routing
Relationship between flood events and stages at upstream and downstream points in a single river
reach can be established by correlating known floods and stages at certain conditions. The
information could be obtained from flood marks on river banks and bridge sides.
Measurements/estimates of floods can then be related to known the level of the flood at the
upstream and downstream locations. With such curves it is possible to give satisfactory forecasts
of the downstream peak stage from an upstream peak stage measurement.
Figure 4.2: Peak stage relationship
The time of travel of the hydrograph crest (peak flow) also need to be determined to know the
complete trace of modification of the wave. Curves of upstream stage plotted against time travel
to the required downstream point can be compiled from the experience of several flood events.
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Figure 4.3: Flood peak travel time
The complexities of rainfall-runoff relationships are such that these simple methods allow only
for average conditions. Flood events can have very many different causes that produce flood
hydrographs of different shapes. The principal advantages of these simple methods are that they
can be developed for stations with only stage measurements and no rating curve, and they are
quick and easy to apply especially for warning of impending flood inundations when the required
answers are immediately given in stage heights.
4.3 Storage Routing
When a storm event occurs, an increased amount of water flows down the river and in any one
short reach of the channel there is a greater volume of water than usual contained in temporary
storage. If at the beginning of the reach the flood hydrograph is (above normal flow) is given as I,
the inflow, then during the period of the flood, T1, the channel reach has received the flood
volume given by the area under the inflow hydrograph. Similarly, at the lower end of the reach,
with an outflow hydrograph O, the flood is given by the area under the curve. In a flood situation
relative quantities may be such that lateral and tributary inflows can be neglected, and thus by the
principle of continuity, the volume of inflow equals the volume of outflow, i.e. the flood
The
principle of hydrologic flood routings (both reservoir and channel) uses the continuity equation
in the form of “Inflow minus outflow equals rate of change of storage”.
(3.1)
Where:
It = Inflow in to the reach
Ot= Outflow from the reach
dS/dt =Rate of change of storage within the reach.
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Alternatively, the continuity (storage) equation can be stated as in a small time
interval Δt the difference between the total inflow volume and total outflow
volume in a reach is equal to the change in storage in that reach, i.e.,
(3.2)
(3.3)
The time interval Δt should be sufficiently short so that the inflow and out flow hydrographs can
be assumed to be straight line in that interval. As a rule of thumb Δt ≤ 1/6 of the time to peak of
the inflow hydrograph is required.
The continuity equation (I-Q = dS/dt), forms basis for all the storage routing methods. The
routing problem consists of finding Q as a function of time, given I as a function of time, and
having information or making assumptions about storage, S.
4.4 Reservoir or level pool routing
A flood wave I(t) enters a reservoir provided with an outlet such as a spillway. The outflow is a
function of the reservoir elevation only, i.e., O = O (h). The storage in the reservoir is a function
of the flow reservoir elevation, S = S(h). Further, the water level in the reservoir changes with
time, h = h(t) and hence the storage and discharge change with time. It is required to find the
variation of S, h and O with time, i.e., find S=S (t), O = O (t) and h = h (t), given I =I (t)
Figure 4.4: Storage routing (schematic)
Depending on the forms of the outlet relations for O (h) will be available. For reservoir routing,
the following data have to be known:
1. Storage volume versus elevation for the reservoir
2. Water surface elevation versus out flow and hence storage versus outflow discharge
3. Inflow hydrograph, I= I(t); and
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4. Initial values of S, I and O at time t = 0
The finite difference form of the continuity equation (Equation. 3.4) can be rewritten as:
(3.4)
Where, (I1+I2)/2= I ; (O1+ O2)/2 = O and S2-S1=ΔS and suffixes 1 and 2 to denote the beginning
and end of the time interval Δt Rearranging Equation (3.4) to get the unknowns S 2 and O2 on one
side of the equation and to adjust the O 1 term to produce:
(3.5)
Since S is a function of O, [(S/Δt) + (O/2)] is also a specific function of O (for a given Δt).
Replacing {(S/Δt) + (O/2)} by G, for simplification, equation (3.5) can be written:
G2 = G1 + Im –O1 or more generally
Gi+1 = Gi + Im,i – Oi (3.6)
Where:
Im = (I1 + I2)/2
To apply this method we need beside I t also the G-O relation. The latter is easily established from
S-H and O-H relations, where for equal values of H, S and O are determined; after which the
proper interval Δt the G-O relation is established. Note that G is dependent on the chosen routing
interval Δt. The routing period, Δt, has to be chosen small enough such that the assumption of a
linear change of flow rates, I and O, during Δt is acceptable (as a guide, Δt should be less than
1/6 of the time of rise of the inflow hydrograph). So, in short, the method consists of three steps:
1. Inspect the inflow hydrograph and select the routing interval: Δt ≤ 1/6 time to peak
2. Establish the G-O relation
3. Carry out the routing according to equation (3.6)
A useful check on the validity of any level pool routing calculation is that the peak of the outflow
hydrograph should occur at the intersection of the inflow and out flow hydrograph on the same
plot. At that point, I = O, so ds/dt = 0, i.e. storage is a maximum and therefore O is a maximum.
Therefore, the temporary storage is depleted.
Figure 4.5: Storage routing
Table 4.1: Tabular computation of level pool routing
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4.5 Channel routing
In reservoir routing presented in the previous section, the storage was a unique function of the
outflow discharge, S=f(O). However in channel routing the storage is a function of both outflow
and inflow discharges and hence a different routing method is needed. The flow in a river during
a flood belongs to the category of gradually varied unsteady flow. For a river reach where the
water surface cannot be assumed horizontal to the river bottom during the passage of a flood
wave, the storage in the reach may be split up in two parts: (i) prism storage and (ii) wedge
storage
Prism Storage is the volume that would exist if uniform flow occurred at the downstream depth,
i.e. the volume formed by an imaginary plane parallel to the channel bottom drawn at a direct
function of the stage at the downstream end of the reach. The surface is taken parallel to the river
bottom ignoring the variation in the surface in the reach relative to the bottom. Both this storage
and the outflow can be described as a single function of the downstream water level and the
storage is a single function of the out flow alone.
Wedge Storage is the wedge-like volume formed b/n the actual water surface profile and the top
surface of the prism storage. It exists because the inflow, I, differs from O (out flow) and so may
be assumed to be a function of the difference between inflow and outflow, (I-O).
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Figure 4.6: Determining storage in a river reach
At a fixed depth at a downstream section of river reach, prism storage is constant while the
wedge storage changes from a positive value at the advancing flood wave to a negative value
during a receding flood. The total storage in the channel reach can be generally represented by:
S = f1(O)+f2(I-O) (4.7)
And this can then be expressed as:
S = K (x Im + (1-x)Om) (4.8)
Where K and x are coefficients and m is a constant exponent. It has been found that the value of
m varies from 0.6 for rectangular channels to value of about 1.0 for natural channels.
4.5.1 Muskingum Method of Routing
Using m =1 for natural channels, equation (2.8), reduces to a linear relationship for S in terms of
I and Q as
S= K (x I+ (1-x)O (4.9)
This relationship is known as the Muskingum Equation. In this the parameter x is known as
weighing factor and take a value between 0 and 0.5. When x=0, obviously the storage is a
function of discharge only and equation (4.9) reduces to:
S = KQ (4.10)
Such storage is known as linear storage or linear reservoir. When x= 0.5 both the inflow and out
flow are equally important in determining the storage.
The coefficient K is known as storage-time constant and has dimensions of time. K is
approximately equal to the time of travel of a flood wave through the channel reach.
As before, writing the continuity equation in finite difference form, we can write
S2 - S1 = {(I1+I2)Δt}/2 - {(O1+O2) Δt}/2 (4.11)
For a given channel reach by selecting a routing interval Δt and using the Muskingum equation,
the change in storage can be determined.
S1 = K(xI1 + (1-x) O1) (4.12)
S2 = K(xI2 + (1-x) O2) (4.13)
Substituting equations (4.12) and (4.13) in equation (4.11) and after rearrangements gives:
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(4.14)
Note that ΣC=1 and thus when C1 and C2 have been found C3=1-C1-C2. Thus the outflow at the
end of a time step is the weighted sum of the starting inflow and outflow and the ending inflow. It
has been found that best results will be obtained when routing interval should be so chosen that
K>Δt>2kx. If Δt < 2kx, the coefficient C2 will be negative.
4.5.2 Application of the Muskingum Method:
In order to use equation (2.14) for O i+1, it is necessary to know K and x for calculating the
coefficients, C. Using recorded hydrographs of a flood at the beginning and end of the river
reach, trial values of x are taken, and for each trial the weighted flows in the reach, [xI+(1-x)O],
are plotted against the actual storages determined from the inflow and out flow hydrographs as
indicated in the following figure.
Figure 4.7: Trial plots for Muskingum X values
When the looping plots of the weighted discharge against storages have been narrowed down so
that the values for the rising stage and the falling stage for a particular value of x merge together
to form the best approximation to a straight line, then that x value is used, and the slope of the
straight line gives the required value of K. for natural channels, the best plot is often curved,
making a straight line slope difficult to estimate.
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