Unit Hydrographs of Different Durations

Lack ofadequate data normally precludes development ofunit hydrographscovering a wide range ofdurations for a given catchment. Undersuch conditions aD hour unit hydrograph is used to develop unit hydrographs of differing durations nD. Two methods are available for this purpose.

 Method of Superposition

If a D-h unit hydrograph is available and it is desired todevelop a unit hydrograph of nDh, where n isan integer, it is easily accomplished by superposing n unit hydrograph with each graph separated from the previous on by D-h.

 

Example 1

The ordinates of a 6-h unit hydrograph are given

Time

(h)

0

6

12

18

24

30

Ordinate of 6-h UH

(m3/s)

0

20

60

150

120

90

Time

(h)

36

42

48

54

60

66

Ordinate of 6-h UH

(m3/s)

66

50

32

20

10

0

 

Derive a 12-h unit hydrograph for the catchment.

Answer 

C1

C2

C3

C4= C2+C3

C5 = (C4/(12/6))

Time

Ordinate of

6-h UH

Ordinates of 6-h UH

lagged by 6-h

 

C5 = (C4/2)

Ordinates of 12-h UH

h

m3/s

m3/s

m3/s

m3/s

0

0

0

0

6

20

0

20

10

12

60

20

80

40

18

150

60

210

105

24

120

150

270

135

30

90

120

210

105

36

66

90

156

78

42

50

66

116

58

48

32

50

82

41

54

20

32

52

26

60

10

20

30

15

66

0

10

10

5

72

0

0

0

 

S-curve

If it is desired to develop a unit hydrograph of durationmD, where m is a fraction, the method of superposition cannot be used. A different technique known as the S-curve method is adopted in such cases, and this method isapplicable forrational values of m. 

The S-curve, also known as S-hydrograph is a hydrograph produced by a continuous effective rainfall at a constant rate for an infinite period. It is a curve obtained by summation of an infinite series of D-h unit hydrographs spaced D-hapart.

 

Fig .1 shows such a series of D-hhydrograph arranged with their starting points D-hapart.

 At any given time the ordinates of the various curves occurring at that time coordinate are summed up to obtain ordinates of the S-curve. A smooth curve through these ordinate results in an S-shaped curve called S-curve.

Description: Description: 261.webp

Fig. .1S-curve.

This S-curve is due to a D-h unit hydrograph. It has an initial steep portion and reaches a maximum equilibrium discharge at a time equal to the first unit hydrograph. The average intensity of ER producing the S-curve is 1/D cm/h and the equilibrium discharge,

Description: Description: 262.webp

Where A is area of catchment in km2 and D is duration in hours of ER of the unit hydrograph used in deriving the S-curve.

 By definition an S-curve is obtained by adding a string of D-h unit hydrographs each lagged by D-hours from one another. Further, if Tb = base period of the unit hydrograph, addition of only Tb/D unit hydrographs are sufficient to obtain the S-curve. However, an easier procedure based on the basic property of the S-curve is available for the construction of S-curves.

Description: Description: 263.webp 

or

Description: Description: 264.webp (26.1)

The term S (t-D)could be called S-curve addition at time t

For all

 Example 2

The ordinate of 2-h unit hydrograph of a basin are given:

Time

(h)

0

2

4

6

8

10

12

2-h UH Ordinates

(m3/s)

0

25

100

160

190

170

110

Time

(h)

14

16

18

20

22

24

26

2-h UH Ordinates

(m3/s)

70

30

20

6

0

0

0

 

Compute a 4-h unit hydrograph ordinate and plot: (i) the S-curve (ii) the 4-h UG



C1

C2

C3

C4

C5

C6 = C4-C5

C7 = C6/ (4/2)

Time

2-h UH Ordinates

S curve addition

S2 curve ordinate

S2 curve lagged by 4 h

DRH of (4/2)= 2 cm

4-h UH Ordinates

h

m3/s

 

 

 

m3/s

m3/s

0

0

0

0

 

0

0.0

2

25

0

25

 

25

12.5

4

100

25

125

0

125

62.5

6

160

125

285

25

260

130.0

8

190

285

475

125

350

175.0

10

170

475

645

285

360

180.0

12

110

645

755

475

280

140.0

14

70

755

825

645

180

90.0

16

30

825

855

755

100

50.0

18

20

855

875

825

50

25.0

20

6

875

881

855

26

13.0

22

0

881

881

875

6

3.0

24

0

881

881

881

0

0.0

26

0

881

881

881

0

0.0

Description: Description: 265.webp