
A streamflow hydrograph at any point on a stream is a graph of the time distribution of water discharge at that point. Generally, the hydrograph is a continuous curve, whereas a rainfall hyetograph is discrete.

A hydrograph for a given storm reflects the influences of both (1) all the physical characteristics of the drainage basin and (2) the characteristics of the storm causing the hydrograph.

The actual shape of a hydrograph is determined by the rate at which water is transmitted from the various parts of drainage basin to the gage. Most of this water is carried by the channels, but small amount of water flows overland directly to the gage.

No two drainage basins would produce identical hydrographs for the same storm.
Similarly, no two storms would produce identical hydrographs from the same basin.

Streamflow is those parts of runoff eventually realized in the stream channels. Runoff may infiltrate into soils or evaporate into atmosphere before traveling to the stream channels.
A.
COMPONENTS OF RUNOFF

SURFACE RUNOFF (= OVERLAND FLOW) – Fast component

GROUNDWATER RUNOFF – Slow component
including base flow (never intercept with ground surface) and spring

INTERFLOW (THROUGHFLOW) – Intermediate
Horizontal flow occurred in the unsaturated zone, e.g., macro-pore flow
B.
ELEMENTS OF HYDROGRAPH

Rising limb (concentration curve)

The rising limb extends from the time of the beginning of surface runoff to the first inflexion point on the hydrograph. It depends upon storm and catchment characteristics.

Crest segment
1


Crest segment refers to the part of the hydrograph between the two inflexion points (one on the rising segment and the other on the recession segment). The peak always occurs within this segment.

Recession limb (depletion curve)

Recession limb (or called receding limb) refers to the remaining part of the hydrograph which may or may not reduce to zero. It represents the withdrawal of water from storage after excess rainfall has ceased. The shape of the recession curve depends on the catchment characteristics.

The recession curve can be represented by an exponentially decaying type equation of the form
Q t
= Q
0
e -Kt where Q t
- discharge at time 't'
Q
0
- initial discharge (at t = 0)
K - recession constant

This equation plots as a straight line on semi-log scale:
ln(Q t
) = ln (Q
0
) - K t
2

C.
HYDROGRAPH TIME CHARACTERISTIC

Time to Peak
–
the time elapsed from the beginning of the rising limb to the
peak discharge.

Time of Concentration
– the time required for a drop of water falling on the most remote part of the drainage basin to reach the basin outlet or gage.
–
It includes the time required for all portions of the drainage basin to contribute runoff to the hydrograph, therefore if it is shorter than the rain duration, it represents the maximum
3

discharge that can occur from a given storm intensity over the drainage basin.

Lag Time
– the elapsed time between the center of mass of effective rainfall and the center of mass of the direct runoff hydrograph.
–
Assume uniform effective rainfall over the entire basin
–
Due to the difficulty in determining the center of mass of the direct-runoff hydrograph, lag time sometimes is also defined as the elapsed time between the center of mass of the effective rainfall and the peak of the hydrograph.
A. RAINFALL CHARACTERISTICS – Rainfall characteristics predominate in determining the shape of the rising segment of the hydrograph. The main climatic factors are

Temporal Variability of Rainfall (i.e, intensity, intermittency, and duration)

The intensity governs the time to peak and the peak value whereas the duration governs the time base of the hydrograph.

Spatial Distribution of Rainfall in the catchment

Depending upon where the high intensity occurs, the base length of the hydrograph may increase or decrease. For example, if the high intensity rainfall occurs near the outlet, the time base will be short. In comparison, if the high intensity rainfall occurs at the far end of the catchment, the time base would be relatively longer.

Direction of storm movement

The direction of storm movement tends to increase or decrease the peak value by decreasing or increasing the time base of the hydrograph depending upon whether the storm is moving downstream or upstream.

Type of precipitation

If the intensity of rainfall excess is high, the hydrograph would be rapidly rising.
In contrast, intermittent rainfall tends to give rather flat, multi-peak hydrographs.
4

B. DRAINAGE BASIN CHARACTERISTICS

Catchment area and shape - Increase in the catchment area tends to smooth the fluctuations of hydrograph

Stream network

Shape of main stream and valleys

Depression storage - reduces the peak
C. SOIL, GEOLOGY, AND LANDS USE

Soil and geological factors govern the flow of sub-surface runoff and therefore mainly affect the recession curve.

Most changes in land use, like urbanization, farming, forest removal, and building roads, all will increase surface runoff.
Depth of flow is measured as stage, which is the elevation of the water surface level above a known datum.
A.
STAGE MEASUREMENTS
Depth of flow is measured as stage, which is the elevation of the water surface level above a known datum.

Staff gauges ( 水標尺 )

Crest gauges

Stage recorders

Rating curve
B.
VELOCITY MEASUREMENTS

Floats ( 浮標 )

Current meters ( 流速儀 )
C.
DISCHARGE MEASUREMENTS
Direct Method - By flow measuring devices
(1) Velocity-Area Station Method
Using the current meter method to determine the velocity, then the discharge can be determined by
5

Q

AV
 i
N 

1 q i where A and V indicate the cross sectional area and velocity. The q i
is the discharge at sub-section i, and N is the total number of sub-section.
(2) Chemical Gauging (Dilution Method)

The principle of flow measurement is that when a tracer of known concentration is introduced into a river, its dilution, measured at a downstream point is a function of the discharge. One important requirement for using this method is that the tracer must be completely mixed before reaching the sampling point. Mixing is usually achieved by the turbulence of the flow.

Especially useful in very small streams

The tracer used should be stable, soluble in water, capable of detection in small concentrations, not affected by sediments, and non-toxic. Commonly used tracers are common salt, fluorescent dyes, and radioactive materials such as Tritium and Bromium-82.
(3) Electromagnetic Method (4) Ultrasonic Method
Indirect Method - By measuring depth and velocity

Flow-measuring structures: broad-crest weirs, flumes. V-notches.

Q = f (H); For a weir:
Q

CBH X in which B is the width of the weir crest, C is the discharge coefficient, and X is an exponent. Both C and X are specific for a weir, and are obtained by calibration. X =1.5 for broad-crest weirs and 2.5 for triangular weirs. C takes into account the channel geometry, the friction loss due to the weir, the form of the weir, etc.
6


Rational formula method (Empirical)

First introduced by Mulvaney, an Irish Engineer in 1851.
Q p
= CIA where Q p
- peak runoff
C - runoff coefficient (0 < C < 1)
I - rainfall intensity of a storm whose duration is greater than or equal to the time of concentration of the catchment
A - catchment area

When Q p
is expressed in m
3
/s; I in mm/hr; A in km
2
,
Q p
= 0.278CIA

The formula assumes that the rainfall intensity is uniform over the entire basin throughout the duration of the storm.

Time of concentration - The time required for a particle of water to move from the furthest point in the catchment to its outlet. There are several empirical equations to estimate it.

Kirpich (1940) equation t c
= 0.0195 L 0.77
S -0.385
where t c
- time of concentration (minutes)
L - maximum length of travel (meters)
S - slope = (H/L) where H is the difference in elevation between the furthest point in the catchment and the outlet.

The time of concentration can also be expressed as a sum of the time of entry t e
and a time of travel t t
as follows: t c
= t e
+ t t

Mockus (1957) equation t e
= t
L
/ 0.6 where t e
- time of entry (overland flow concentration time or time of overland flow) t
L
- lag time
7


Ragan and Duru (1972) equation t e
 6 .
917 (
I
0 .
4 nL )
0 .
6
S
0 .
3
min. where n - Manning's roughness coefficient
(= 0.2 for paved areas and 0.5 for grass)

Time of travel is simply the length of channel divided by the velocity.

There are other equations of the form
Q p

A
N

Runoff coefficient - Depends upon
 the nature of the surface
 the slope
 the surface storage
 the degree of saturation
 the rainfall intensity.
For non-homogeneous areas, a weighted runoff coefficient can be obtained by the following equation:
C w
=


A C
1 1
A
1

PEAK Discharge as a significant design parameter
TIME - AREA METHOD - AN EXTENSION OF RATIONAL METHOD

If the catchment is large, the times of concentration are best determined by dividing the catchment into different zones. The zoning is such that from any point on any given “isochrone” (definition: A set of points with the property that a given process or trajectory will take the same length of time to complete starting from any point in the isochrone) the runoff reaches the outlet at the same time. With the time - area distribution and assuming different runoff coefficients for different regions and times, it is possible to determine the hydrograph for a catchment.

When the catchment is non-homogeneous, it should be subdivided into a number of homogeneous sub-catchments, then the Rational Method can be applied to each of the homogeneous sub catchments. The practice follows two rules:

For each inlet area, the Rational Method is used to compute the peak discharge
8


For inlets where the runoff is arriving from two or more sub areas, the longest time of concentration is used to determine the design rainfall intensity.
Runoff is calculated using a weighted Runoff Coefficient.
Example
In a catchment the equal-interval isochrone defines the following four zones from downstream to upstream: A
1
= 500 ha; A
2
= 600 ha; A
3
= 800 ha and A
4
= 200 ha. The rainfall distribution is as shown in the Table. The runoff coefficients are
0 - 1 hr, C = 0.5 1 - 2 hr, C = 0.7 2 - 3 hr, C = 0.8 3 - 4 hr, C = 0.85
Assuming that the rainfall and runoff coefficients are average values for the duration of each hour, determine the resulting direct runoff hydrograph.
Time
(hr)
1
0 - 1
1 - 2
2 - 3
3 - 4
4 - 5
5 - 6
6 - 7
Rainfall
(mm)
2
15
30
30
15
Contribution from each area Q = CIA Total
Area A
1
Area A
2
Area A
3
Area A
4
Runoff
3
10.4
29.2
33.3
17.7
4
-
12.5
35.0
40.0
20.8
5
- -
6 7
10.4
- -
16.67 -
41.7
85.0
46.67 41.67 146.0
53.3 116.67 191.0
27.78 133.33 161.0
69.44 69.4
Computations for columns 3 - 7 are done as follows:
For A
1
, hour 1, Q
1
= 0.5 x 15 x 500 (mm.ha/hr) = 10.4 m 3 /sec hour 2, Q hour 3, Q
2
3
= 0.7 x 30 x 500
= 0.8 x 30 x 500
= 29.2
= 33.3 hour 4, Q
4
= 0.85 x 15 x 500 = 17.7
For A
2
, hour 1, Q
1
= 0.5 x 15 x 600 (mm.ha/hr) = 12.5 m
3
/sec hour 2, Q
2
= 0.7 x 30 x 600 (mm.ha/hr) = 35.0 m
3
/sec hour 3, Q
3
= 0.8 x 30 x 600 (mm.ha/hr) = 40.0 m
3
/sec hour 4, Q
4
= 0.85 x 15 x 600 (mm.ha/hr) = 20.8 m 3 /sec
(For the second hour discharges are delayed by one hour)
Similarly, for A
3
and A
4
, the corresponding discharges are computed and are summed up to obtain the total effect.
9

STRAIGHT LINE METHOD
CONCAVE METHOD
AREA METHOD (FIXED BASE LENGTH SEPARATION)
VARIABLE SLOPE SEPARATION

First, the existing recession curve is extended to a point in time just below the peak
(AB). Point D represents the end of surface runoff and is obtained by a semi-log plot.
The semi-log part is extended backwards (DC) to a point C which is just below the second point of inflexion (on the recession curve) of the hydrograph. The method of joining BC is rather arbitrary.
MASTER RECESSION (DEPLETION) CURVE METHOD

The procedure is based on the recession data from a number of hydrographs, which will cover a wide range of discharges and seasons. The recession curves are plotted using semi-log scale on tracing paper with one graph for each event. On a separate master sheet, also using semi-log scale, the recession parts are transferred from the individual sheets starting from the graph having the lowest magnitude such that all the recession parts fall upon the straight segment of the master depletion curve. An empirical equation can then be fitted to the master depletion curve.
10

11

Effective Rainfall (ER)
ER is defined as that amount of rainfall producing direct (surface) runoff.
Rainfall excess
Equal to effective rainfall.
Direct runoff (Surface Runoff)
Direct runoff is the runoff component directly resulting from rainfall excess. It is the difference between the total runoff and base flow.
(1) The Unit hydrograph method is one of the basic tools in hydrological computations. It was originally developed by L.K. Sherman in 1932.
(2) A unit hydrograph (UH) of a watershed is defined as the direct runoff hydrograph (DRH) resulting from one unit (1 inch or 1cm) of effective rainfall (ER) occurring uniformly over the watershed at a uniform rate during a specified period of time .
The specified period of time is called the unit storm duration or simply unit duration .
The assumption of “ER occurring uniformly over the watershed” is more suitable for small watershed. Rainfall storms that produce intense rainfall usually do not extend over large areas.
The assumption of “at a uniform rate” is more suitable for large watersheds.
(3) The specified period of time, i.e., unit storm duration, is not necessarily equal to unity. It can be any finite duration up to the time of concentration. It is the period for which the UH is determined; that is, as soon as this period changes, so does the UH for a watershed.
(4) Thus, there can be as many UHs for a watershed as many period of rainfall, for example, 1hour UH, 6-hour UH, or 12-hour UH. The 1, 6, 12 here are not the duration for which the UH occurs, but it is the duration of the ER for which the UH is defined.
(5) Because effective rainfall is assumed to occur uniformly, its duration defines its intensity.
(6) When ER is 1 unit / hr, DRH and 1-hour UH are numerically identical.
(7) Notice that the dimension of UH is either
L 2
T
1
or
T
.
12

The UH theory assumes that the rainfall excess - direct runoff process is
(1) Linear - DRH is derived from the Effective Rainfall Hyetograph (ERH) by a liner operation; that is, the principles of superposition and proportionality .
(2) Time-invariant – ignore the influence of antecedent moisture condition of a catchment; allow the translation in time.
1. For a given basin, the duration of surface runoff is constant for all uniform-intensity storms of the same length, regardless of differences in the total volume of surface runoff. However, for two uniform-intensity storms of the same length, the rates of surface runoff are in the same proportion to the total volume of surface runoff. (The principle of proportionality)
2. The time distribution of surface runoff from a given storm period is independent of concurrent runoff from antecedent storm periods. (The principle of superposition)
3. The ER is uniformly distributed within its duration. This means that the rainfall intensity is uniform throughout the watershed during the time rain falls. This steady rainfall intensity must occur even if the unit time is 1 hour or 2 hours or whatever.
4.
Hydrologic losses must be uniform over the entire watershed.
The principles of proportionality and superposition define linearity of the UH theory. They lead to very useful practical applications of the unit hydrograph concept. For example, a hydrograph of discharge resulting from a series of rainfall excesses may be constructed by summing up the hydrographs due to each single unit of rainfall excess. They also imply that the time base of direct runoff hydrographs resulting from rainfall excesses of same unit duration is the same regardless of the intensity.
The above assumptions are all based on the assumption that the watershed is linear. Remember in mind, in fact, all watersheds in nature are non-linear; some are more non-linear and some less.
13

14

D.
Let the ERH (effective rainfall hyetograph) of D-hour duration be denoted as:
I ( t )

I ,

0 , t
0
 t

D

D
(D is the duration of ER)
The resulting DRH (direct runoff hydrograph) is:
Q ( t )
 h ( D , t ) I ( t ) D
 h ( D , t ) ID when ID is 1, the DRH and UH are numerically the same.
In the right side of the figure below, let each pulse be of duration D hours, intensity I
1
, I
2
, I
3
, the
DRH Q
1
( t ), Q
2
( t ), Q
3
( t ) . If the D-hour UH is h ( D , t ) . Then the DRH is:
Q ( t )

Q
1
( t )

Q
2
( t )

Q
3
( t )
15

Notice that since Q
2
( t ) does not start until t = D , and Q
3
( t ) does not start until t = 2 D . Therefore, by using D-hr UH h ( D , t ) , each individual DRH can be expressed as
Q
1
( t )

Q
2
( t )

Q
3
( t )
 h ( D , t ) I
1
D h ( D , t

D ) I
2
D ; h ( D , t

2 D ) I
3
D ;
Q
2
( t )

0 ,
Q
3
( t )

0 , t

D t

D
Q ( t )

 h ( D ,

3 j

1 h t

)
D
I
1
, t
D

 h ( D , t
 j

1

D


I
D ) j
D
I
2
D
 h ( D , t

2 D ) I
3
D
Q ( t ) or Q ( t )

  n j
 t j h

D , t

1

1 h

D , t

 j

1

D

I j
D

 j

1

D

I j
D
When the duration of ER is infinitely small, the UH theory approaches the reality. The limiting case is D

0 :
IUH ( t )
 h ( t )

( t )
 lim
D

0
 lim
D

0 h ( D , t )
 h ( 0 , t )
I ( t , D ) D  Dirac Delta function

-

 
( t ) dt

1 ;

( t )

0 , for t
Q ( t )
 
0 t h ( t
 
) I (

) d

;

0 h (

)

0 , for
 
0 or by simple change of variables
Q ( t )
 
0 t h (

) I ( t with

0
 h ( s ) ds lim t


1 ; h ( t )

0 ; h ( t )

0 ,
 
) d
 convolutio nal integral for any t

0
16

When the duration of ER is infinitely long, and its intensity is one unit per unit storm duration
(say, D -hour), the hydrograph so obtained is called S-hydrograph or summation hydrograph.
If a ERH is broken into pulses each of D -hour duration: Q [ D , t

( j

1 ) D ] represents the DRH due to the j -th pulse of rain: if
S ( t )
D


1
D
0 ,
 t j

1
Q

D , t h

D , t


 j

1

D


 j

1

D


 t j

1 h ( t ) h

D , t

 j

1

D


S ( t )
  t
0 h ( s ) ds
17

RELATION BETWEEN IUH and SH: as D

0
IUH ( t )
 h ( t )
 dS ( t ) dt
(1) Procedure

Select a storm which is isolated, intense and uniform over the catchment and time by scanning through rainfall and runoff records. The duration of the storm should not be greater than the period of rise (time of concentration).

Plot the observed discharge hydrograph and separate the base flow.

Determine the area under the direct runoff hydrograph. The area may be expressed as a volume of runoff (discharge x time = volume) or as a depth of runoff by dividing by the catchment area.

Divide the direct runoff hydrograph ordinates by the depth of runoff referred to above to obtain the unit hydrograph ordinates. The area under the unit hydrograph must be unity.
(2) Determination of the unit storm duration
A catchment can have an infinite number of unit hydrographs because each duration of rainfall excess produces its own unit hydrograph. The unit storm duration t0 is determined by assuming that the volume (or depth) of direct runoff is equal to the volume (or depth) of rainfall excess.
The difference between the recorded rainfall and the direct runoff is taken as losses due to infiltration and other causes.
The recorded rainfall is therefore adjusted to give a depth equal to the depth of direct runoff. The effective duration of the adjusted rainfall is the unit duration.
(3) Conversion of a unit hydrograph of one unit duration to that of a different unit duration

When the required unit duration is an integer multiple of the given unit duration
If the given unit duration is t0 hours, then superposition of n ‘t0’ hour unit hydrographs will give a ‘nt0’ hour hydrograph (not a unit hydrograph). The ‘nt0’ hour unit hydrograph can be obtained by dividing the ordinates of the hydrograph by ‘n’.
18


When the required unit duration is not an integer multiple of the given unit duration (S-Curve method)
The S-Curve is the hydrograph resulting from an infinite series of runoff increments of unit magnitude with a unit duration of t0 hours. In other words it is the hydrograph for a continuous storm with an intensity of l/t0.
With such a continuous storm, an equilibrium stage will be attained after some time. Thus, the name S-Curve. It is constructed by adding the unit hydrographs of t0 hour duration, each displaced by t0 hours.
The difference between two S-Curves displaced by t1 hours (where t1 is the required unit storm duration) gives a hydrograph of duration t1 hours.
19

Examples:
1.
A rainfall-runoff event is selected to derive a 2-hour unit hydrograph for a watershed.
The corresponding effective rainfall hyetograph (ERH) and direct runoff hydrograph
(DRH) has been computed as following:
I ( t )

5 cm / hr

0
0
 t

2 t

2
Q ( t )

2 .
5 t

10

2 .
5 t
0
 t
2
 t

2

4
where t is measured in hours.
Determine (a) the 2-hour unit hydrograph (UH)
(b) the S-hydrograph for the watershed
(c) Compute the 3-hour UH
(d) Compute the 4-hour UH
20

21