Contents preface (VII) introduction 1—37

Method of Superposition

If n number of D-hour unit hydrographs, each one separated from the previous one by D hour, are added, one would obtain a characteristic hydrograph for n units of rainfall excess and nD-hour duration. Dividing the ordinates of this characteristic hydrograph by n would, obviously, yield a unit hydrograph (with unit rainfall excess) of duration equal to nD hours. Figure 2.23 illustrates the method of superposition which, obviously, requires n to be an integer. When n is not an integer, the summation curve (or S-curve) method is to be used.

74

Dischargeincumecs

IRRIGATION AND WATER RESOURCES ENGINEERING

30

Sum of 4-hr unit hydrographs

25
4-hr unit hydrograph
20
8-hr unit hydrograph (derived)
15

10
5
0
	0	4	8	12	16	20	24	28	30
									Time in hours

Fig. 2.23 Derivation of unit hydrograph of duration 2t from t-h UH

2.7.4.2. Summation Curve (or S-Curve) Method
The S-curve (or S-hydrograph) is a direct runoff hydrograph resulting from a continuous effective rainfall of uniform intensity. The S- curve is obtained by adding together a series of unit hydrographs of, say, D-hour duration each lagged by D hour in relation to the preceding one (Fig. 2.24). The intensity of effective rainfall for this S-hydrograph would, therefore, be 1/D cm/hr. This means that each S-curve applies to a specific duration D within which one unit of direct runoff is generated. If the time base of the D-hour unit hydrograph is T hour (Fig. 2.20), then a continuous rainfall producing one unit of runoff in every period of D hours would yield a constant outflow at the end of T hours. Therefore, one needs to combine only T/ D unit hydrographs to produce an S-curve whose equilibrium flow rate Q_e (in m³/s) would be the product of area of the catchment basin and the intensity of the effective rainfall (or rainfall

F		1			I
excess) i.e., 1/D cm/hr G	=			m / hrJ .
100 D
	H				K		F	1 I
∴			Q = (A × 10^6) G		J	m^3/hr
				e			H 100 DK
				F	A		4 I
				= G		× 10	J	m^3/hr	(2.22)
				D
					H		K

where, A = area of the catchment basin in km², and

D = duration in hours of the effective rainfall of the unit hydrograph.

HYDROLOGY

3	/s
	Dischargeinm

75
Rainfall excess 1/D cm/hr

60

50 S-Curve

40

30

20

10

0

0	4	8	12	16	20	24	24	32
			Time in hours

Fig. 2.24 Illustration of S-hydrograph (or S-curve)

6

F

1

I

F

1

1

I

3

Alternatively,

Qe = (A × 10 ) G

100

J

G

×

J

m /s

H

DK

H 60

60K

Q = 2.78

A

m^3/s

...(2.23)

e _D

The S -curve ordinates sometimes oscillate at and around the equilibrium discharge. This may be due to the unit hydrograph (used for the derivation of the S-curve) which might have been derived from storms not satisfying the requirements of an ideal storm for the derivation of the unit hydrograph. An average S-curve can, however, be still drawn so as to attain the equilibrium discharge rather smoothly. The S-curve, obtained from a D-hour unit hydrograph of a catchment basin, can be used to obtain unit hydrograph (for the same catchment, of course) of another duration, say, D′ hour as explained in the following steps :

1. 
   Draw two S-curves (obtained from a D-hour unit hydrograph) with their initial points displaced on time axis by D′ hour, Fig. 2.25.
   

2. 
   The effective rainfall hyetographs (ERH) producing these two S-curves are also drawn in the same figure. The two ERH are also displaced by D′ hour. The difference between these two effective rainfall hyetographs represents a storm of duration D′ with an intensity of 1/D cm/hr and, hence, a rainfall of magnitude D′/D cm.
   

3. 
   The difference between the ordinates of the two S-curves at any time [i.e. S(t)-S (t – D′)] gives the ordinate of a direct runoff hydrograph at that time, Fig. 2.25. This hydrograph is, obviously, for the storm of duration D′ with an intensity of 1/D cm/hr having rainfall excess of
   

D′/D cm which is the runoff volume.

76 IRRIGATION AND WATER RESOURCES ENGINEERING

		Excess rainfall at	1	cm/hr
		D
	D¢
Discharge	S-Curve
		Lagged s-curve
	S( t )	S( t – D¢ )
	t
	D¢
Discharge		Difference graph
	[S( t ) – S( t – D¢ )]

Time in hours

Fig. 2.25 Derivation of D′ hour UH by S-curve method
4. Compute the ordinates of the D′-hour unit hydrograph by multiplying the S-curve differences [i.e., S(t) – S(t – D′)] with the ratio D/D′.
Earlier, it has been stated that one requires combining T/D unit hydrographs suitably for obtaining a S -curve. However, one can construct S-curve without requiring to tabulate and adding T/D unit hydrographs with successive time lags of D hours. The difference of two-S-curves (derived from a D-hour unit hydrograph) lagged by D hour itself is nothing but the D-hour unit hydrograph itself. Therefore, the ordinate U(t) of a D-hour unit hydrograph at any time t is given as
U(t) = S(t) – S (t – D)
or S(t) = U(t) + S(t – D) (2.24)
The term S (t – D) is called the S-curve addition which is an ordinate of S-curve itself but at time (t – D). It may be noted that for t ≤ D.

S(t) = U(t)

(2.25)

And for t > D, one has to use Eq. (2.24) for constructing S-curve.