In this article we will discuss about:- 1. Definition of Unit Hydrograph 2. Definitions involved in Unit Hydrograph Theory 3. Specifications of Unit Hydrograph 4. Assumptions made in the Unit Hydrograph Theory 5. Use of Unit Hydrograph to Derive Flood Hydrograph 6. S Hydrograph 7. Limitations of Unit Hydrograph Theory 8. Derivation of Unit Hydrograph and Other Details.
Contents:
- Definition of Unit Hydrograph
- Definitions involved in Unit Hydrograph
- Specifications of Unit Hydrograph
- Assumptions made in the Unit Hydrograph Theory
- Use of Unit Hydrograph to Derive Flood Hydrograph
- S Hydrograph
- Limitations of Unit Hydrograph Theory
- Derivation of Unit Hydrograph
- Averaging of Unit Hydrograph
- Synthetic Unit Hydrograph
- Distribution Graph
- Instantaneous Unit Hydrograph
- Derivation of Instantaneous Unit Hydrograph
1. Definition of Unit Hydrograph:
For identical precipitation, hydrographs observed for different catchments will have different shapes. This is because the shape of the hydrograph of every catchment depends on the characteristics of the catchment.
An attempt is made to correlate the hydrograph of each catchment with the precipitation. This is done by the unit hydrograph theory. The theory of unit hydrograph was first presented by L.K. Sherman in Engineering News Record in April 1932. Originally, this theory was known as unit graph. However, title unit graph was misinterpreted, and hence it was modified as unit hydrograph.
Subsequent to the introduction of this theory by Sherman, It underwent a number of modifications, but the basic principle as presented by Sherman remained the same.
When a unit excess rainfall occurs uniformly distributed over a catchment area, then the resultant hydrograph is known as unit hydrograph (Fig. 2.22).
2. Definitions involved in Unit Hydrograph Theory:
The theory of unit hydrograph involves the following definitions:
1. The excess rainfall means the precipitation after all the abstractions, such as interception, evaporation, depression storage, infiltration, etc., are met with.
2. The excess rainfall is 1 unit. This unit may be 1 cm or 5 cm. If the unit is 1 cm, then the excess rainfall may be 1 cm/h for 1 h, or 1/2 cm/h for 2 h, or 2 cm/h for 1/2 h. So, that the total excess rainfall will be 1 cm, i.e. 1 unit.
3. The duration of the excess rainfall should be sufficiently less than the time of concentration.
Preferably, t = t0/4.
4. The base period T is the total time of the flood hydrograph at the gauging site.
3. Specifications of Unit Hydrograph:
The unit of precipitation as well as intensity of the excess precipitation are the controlling parameters. The unit hydrograph is, therefore, specified as ‘1 cm-1h unit hydrograph’. Here the unit precipitation is 1 cm and the period of precipitation is 1 h. (Naturally, the intensity of precipitation is 1 cm/h).
The surface runoff, in case of a 1 cm -1 h unit hydrograph from a catchment area A km2, will be as below:
The surface runoff = (1 cm/100) × A × 106 m3
= (A/100) × 106 m3
= A × 104 m3.
4. Assumptions made in the Unit Hydrograph Theory:
The following assumptions are made in the unit hydrograph theory:
1. The excess rainfall is uniformly distributed over the entire catchment area.
2. The base period T of the unit hydrograph depends on the duration of rainfall and the basic characteristics of the catchment and not on the intensity of rainfall.
3. The combined effect of all the physical characteristics of the catchment, viz., its slope, shape, Manning’s coefficient, and so on, is reflected in the shape of the hydrograph.
4. The unit hydrograph coordinates are time invariant. This means that the coordinates do not change with respect to period. This means that the hydrograph coordinates will remain unchanged for any season, any month, any day or even any year.
5. Use of Unit Hydrograph to Derive Flood Hydrograph:
The flood hydrograph can be derived from a unit hydrograph if the excess rainfall storm is known.
6. S Hydrograph:
Consider a 1 cm – 1h unit hydrograph as shown in Fig. 2.23.
Now imagine that the unit excess rainfall of the same intensity occurs for a period that tends to infinity means the excess rainfall of that intensity occurs continuously up to infinite time.
This excess precipitation tending to infinity can be divided into time intervals equal to unit hydrograph excess precipitation of 1 cm/h for 1 h one after another, tending to infinity.
Naturally, each slab of excess precipitation will have one hydrograph, which is equivalent to unit hydrograph one after another with a time lag of 1 h as follows:
The resulting hydrograph coordinates will be as shown in Table 2.7.
The S hydrograph coordinates will be as follows:
The graphical representation will be as shown in Fig, 2.24.
The resulting hydrograph resembles the letter S. Hence, it is called S hydrograph.
Thus, an S hydrograph may be .defined as a hydrograph observed at a catchment outlet, when the excess precipitation uniformly distributed over the entire catchment occurs for a period which tends to infinity.
Note the following:
1. The S hydrograph is a mass curve.
2. The S hydrograph discharge is constant (82) after the base period T (10 h).
3. The constant value of discharge is the summation of all the unit hydrograph coordinates (82 m3/s).
4. Being a mass curve, it will not have negative slope at any point. The slope is horizontal after the base period T.
It may also be noted here that, in practice, there are so many errors committed in observing and as such the S curve plotted from the field observation is normally not a smooth curve as shown in Fig 2.25.
In such case, a smooth curve is drawn and used for further calculations.
7. Limitations of Unit Hydrograph Theory:
The limitations of the unit hydrograph theory are as follows:
a. The basic assumption that the excess rainfall is uniformly distributed over the entire area may not be practicable. It is very difficult to have this condition fulfilled in practice.
b. The principle of linearly assumed is not correct.
c. The unit hydrograph theory is not applicable for surface runoff originated from snow and ice.
d. The unit hydrograph theory is applicable to in-bank floods only. If the flood water overtops the bank, this theory will not be applicable.
e. The unit hydrograph theory is applicable for catchments less than 5000 km2.
f. The theory is not applicable to narrow elongated catchments because it is not possible to have a uniform precipitation over the entire catchment.
g. The theory is not applicable if there are storage on the channel or on its tributaries in the catchment upstream of the gauging station.
8. Derivation of Unit Hydrograph:
Unit hydrograph may be derived from the observed rainfall and its resultant hydrograph. For this, the available data will have to be scanned before it is used.
The step-by-step procedure is as follows:
i. From the observed data, select an isolated storm and its resultant hydrograph.
ii. Check the precipitation data for the accuracy of any rain gauge station. If required, evaluate the missing data. Find the average hour1 precipitation over the catchment and plot the hyetograph.
If the unit hydrograph required to for t hours, then the hyetograph may be divided into slabs of t hours. Assuming a ф indeed for infiltration, evaluate the average hourly excess rainfall hyetograph.
iii. Check the resultant hydrograph for its accuracy. Separate the base flow. Then the correct observed flood hydrograph will be arrived at.
Then the observed corrected data of excess precipitation and its resultant storm hydrograph can be used for further analysis.
Consider a Specific Case:
After scanning the data, the excess rainfall hyetograph and the resultant storm hydrograph are as follows:
Excess precipitation- 2 cm/h for 1 h followed by 4 cm/h for 1 h followed by 3cm/h for 1 h.
The resultant flood hydrograph is below:
This scanned data can be represented graphically as shown in Fig. 2.30.
A1 h – 1 cm unit hydrograph is to be derived. After examining the observed data.
Following observations can be made:
1. The storm is of 3 h.
2. The base period of the combined flood hydrograph is 12 h.
3. For the 1 h -h cm unit hydrograph, the base period Twill be 12 – 3 + 1 = 10 h.
4. The total resultant flood hydrograph is the summation of three hydrographs, viz.
(a) Flood hydrograph due to 2 cm/h precipitation.
(b) Flood hydrograph due to 4 cm/h precipitation with a time lag of 1 h from the beginning.
(c) Flood hydrograph due to 3 cm/h precipitation with a time lag of 2 h from the beginning.
Since the unit hydrograph base period is 10 h, assume the unit hydrograph coordinates for 10 h to be U0, U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, out of which U0 and U10 are equal to zero.
The unit hydrograph can be derived by the following methods:
i. Ven Te Chow’s Forward Substitution Method:
The total resultant flood hydrograph is the summation of the three hydrographs.
Thus, the total flood hydrograph can be expressed in terms of the assumed hourly unit hydrograph coordinates as follows:
From the first equation, we get Q0 = 0. Substitute its value in the next equation and we get Q0 = 6. Thus, by substituting the unit hydrograph coordinate, calculated in the previous step. In next step, all the unit hydrograph coordinates can be evaluated.
The procedure was first suggested by Prof. Dr. Ven Te Chow, hence this method is known as Ven Te Chow’s forward substitution method.
ii. Ven Te Chow’s Backward Substitution Method:
The same analogy as followed for the forward substitution method can be applied here and the unit hydrograph coordinates can be worked out by backward substitution as follows:
Consider the last equation. Solve it. We get U9 = 1. Substitute this value in the previous equation. We get U8 = 2. Thus, by substituting the values in the previous equation, all the coordinates can be evaluated.
This backward substitution method was also suggested by Prof. Dr Ven Te Chow, and hence it is known as Ven Te Chow’s backward substitution method.
For both the methods, the unit hydrograph is checked so that the area of the hydrograph is one unit for unit excess precipitation uniformly distributed over the basin. The unit hydrograph thus derived by both the methods is shown in Fig. 2.31.
iii. Collin’s Method:
This is a trial and error method, very commonly used for deriving unit hydrograph from a complex storm and resultant hydrograph.
The procedure followed is as follows:
1. The numerical values of the unit hydrograph coordinates are assumed. The flood hydrograph coordinates for the excess rainfall calculated are worked out and then compared with the observed ones. These two coordinates may not tally initially.
2. The assumed unit hydrograph coordinates are modified suitably and the flood hydrographs for these modified coordinates and the excess precipitation are again worked out and compared with the observed ones.
3. Again, the modified coordinates of the unit hydrograph are modified, and this trial and error is continued till the calculated hydrograph tallies with the observed hydrograph, with an acceptable error.
Then the unit hydrograph coordinates are finalized.
9. Averaging of Unit Hydrograph:
There are a number of assumptions made in the unit hydrograph derivation. So also there might be some errors in the observed precipitation and the discharge data, and hence unit hydrographs derived from the different observed data for the same catchment may not tally with each other. Thus, an average of all the unit hydrographs derived is worked out for further studies.
The procedure followed is as stated below:
Plot all the hydrographs on a simple graph paper
where t1, t2, t3, = Time in hours of the maximum discharge from the beginning of the different unit hydrographs derived.
tp = Time in hours of the maximum discharge from the start of the final average unit hydrograph
Q1, Q2, Q3 = Maximum flood discharge of the various unit hydrographs in m3/s
Q = Maximum flood discharge of the final average unit hydrograph
T1, T2, T = Base period of the various unit hydrograph in hours
T = Base period of the final average unit hydrograph in hours
The average unit hydrograph may be plotted taking into consideration the parameters Q5 T and tp.
An average unit hydrograph developed in a specific case is shown in Fig 2.32. It should be checked the total area under the average unit hydrograph is unity.
10. Synthetic Unit Hydrograph:
When sufficient observed data are not available for the derivation of a unit h hydrograph, then the unit hydrograph is framed based on the catchment characteristics.
a. Snyder Method:
F.F. Snyder analysed a number of unit hydrograph in the Appalachian mountain region in USA and presented a set of equations for the synthetic unit hydrograph based on the following three catchment area characteristics viz. A, L & Lc (Fig 2.33).
A = Catchment area in km2
L = Length of main stream in km
Lc – Distance in km along the main stream from the outlet up to a point nearest to the centre of gravity (CG) of the catchment area
t = Duration of unit hydrograph in hours
tp = Time between the CG of effective rainfall to the peak discharge in hours (basin lag in hours)
Qp = Peak discharge in m3/s
T = Base period in hours
W75 = Width of unit hydrograph for 75 % of Qp
W50 = Width of unit hydrograph for 50 % of Qp
Ct = A constant depending upon the slope and SD of the catchment area. (Value is normally between 1.35 to 1.65).
Cp = A coefficient ranging from 0.56 to 0.69
The equations suggested are:
tp = Ct(L.Lc)0.3
Qp = 2.778 (Cp/tp) A
T = 5.455 tp
W50 =2.14 (Q/A)-1.08
W75 = 1.22 (Q/A)-1.08
t = 2tp/11
The widths of unit hydrographs W50 and W75, respectively, may be divided into two parts such that 1/3 lies in the rising curve and 2/3 in the recession curve.
The unit hydrograph can be framed, based on the above set of equations. As a check, the area of the unit hydrograph may be checked and corrected for a value of unit precipitation (Fig. 2.34).
b. Central Water Commission Method for Indian Catchments:
The Central Water Commission (CWC), India, has recommended the following procedure for the construction of a synthetic unit hydrograph. This is based on the number of catchments in India of various shapes and sizes.
The recommended formulae are:
Qpd = 4.44 A3/4 for S > 1 in 360
Qpd = 2.22 A3/4 for 5 < 1 in 360
Tp = 3.95 / (Qpd / A)0.9
D = 1.1 tp
Where, Qpd = Peak discharge in m3/s
A = Catchment area in km2
S = Weighted mean slope of the catchment
tp = Time in hours from the centroid of the effective rainfall to the peak discharge
D = Duration of effective rainfall
c. Dimensionless Unit Hydrograph:
The United States, Soil Conservation Service (SCS) proposed a dimensionless unit hydrograph on the basis of the unit hydrographs derived for different basins having different catchment areas. This unit hydrograph is in the dimensionless form.
On the Y-axis, Q1/QP is expressed and on the x-axis, the ratio of t/tp is expressed. The base period t/tp is taken to be equal to 5.
Where, Qt = Discharge at time t
11. Distribution Graph:
Barnard proposed a unit hydrograph in this distribution form. It is a dimensionless representation of a unit hydrograph in the form of a bar chart. The base period of a unit hydrograph is divided into a number of equal time intervals. The runoff in each of this interval is first calculated and then its percentage with respect to the total runoff is worked out. The area during that interval is divided by the total area of the unit hydrograph. It is then represented on y-axis for that interval. Figure 2.36 below shows a distribution graph in a specific case.
This unit hydrograph is in the form of percentage for each time interval. It is known as a distributing graph. The total of all percentages will be 100%. All unit storms irrespective of their intensity for a catchment have practically the same distribution graphs.
12. Instantaneous Unit Hydrograph:
The characteristics of a unit hydrograph such as base time, maximum discharge, etc., depend upon the following factors:
i. Catching characteristic.
ii. Intensity of the unit precipitation.
iii. Time of unit precipitation.
In order to simplify the unit hydrograph, it is assumed that the unit precipitation occurs unit formally over the entire catchment instantaneously, i.e. t = 0 or t → 0.
It is a conceptual fictitious unit hydrograph.
This is a hypothetical case. It is incorrect to assume that the unit precipitation occurs instantly. However, further analysis of unit hydrograph theory is simplified. With this assumption, one factor (time of unit precipitation) influencing unit hydrograph is eliminated. Thus, a unit hydrograph of a catchment area, resulting from a unit precipitation occurring instantly, uniformly distributed over the entire catchment, is known as instantaneous unit hydrograph as shown in Fig. 2.37.
It is normally denoted as IUH and its shape is similar to a single peak hydrograph.
It may be noted that the basic principles of linearity and time invariance, assumed in the unit hydrograph theory, hold good for IUH also.
Indirectly, a unit hydrograph of time of precipitation of t → 0 is an IUH.
13. Derivation of Instantaneous Unit Hydrograph:
A lot of work and research is being done for the derivation of an instaneous unit hydrograph. It can be derived from the unit hydrograph or from and S hydrograph. A number of conceptual models have been proposed.
The most important are the following:
i. Clark model.
ii. Nash cascade model.
iii. Chow-Kulandaiswami mode.
iv. Dooge model.
v. Clark model
vi. Nash cascade model, and
v. Dooge model.
