How to Compute Runoff Using Unit Hydrograph? | Geography

In this article we will discuss about how to compute runoff caused due to rainfall using unit hydrograph.

Introduction to Unit Hydrograph:

If two identical rainfalls regarding their characteristics take place in a drainage basin having the same conditions prior to the rainfall, the runoff hydrographs from the two storms would be the same. This is the basic concept used to derive the unit hydrograph from a storm hydrograph.

In actual speaking, the two identical storms are rarely occur in basin. They vary in their duration, amount and aerial distribution.

Sherman (1932) investigated the unit hydrograph theory, which is widely used for computing the flood or runoff volume for various purposes. Unit hydrograph is defined as the direct runoff hydrograph, produced by a rain/storm of unit duration resulting the effective rainfall depth as 1 cm which is uniformly distributed over the entire watershed area.

Elements of Unit Hydrograph:

1. Base Width:

It is the total time period of direct runoff due to any storm. Sometimes, it is also known as time base.

2. Unit Storm:

A storm of unit duration regardless of its intensity is known as unit storm.

3. Unit Period:

The time period of unit storm is called unit period.

4. Lag Time (t_p):

It is the time interval between the centre of a unit storm to the peak of corresponding hydrograph.

5. Recession Time (T_r):

The time period of direct runoff, after the end of excess or net rainfall is noted as recession time.

Assumptions of Unit Hydrograph Theory:

The derivation of unit hydrograph is based on the following assumptions:

1. The effective rainfall is uniformly distributed throughout the watershed area.

2. The base or time duration of hydrograph of the direct runoff due to an effective rainfall of unit duration, is constant.

3. The effective rainfall is uniformly distributed within its duration.

4. The ordinates of direct runoff hydrograph of common base time are directly proportional to the amount of effective rainfall. Sometimes, this assumption is also called principle of linearity.

5. For a given watershed the hydrograph of direct runoff of a given rainfall duration, reflects all physical characteristics of the watershed.

Criteria for Derivation of Unit Hydrograph:

Unit hydrograph is derived mainly on the following three criterion:

1. The runoff producing storms of equal duration will produce the direct runoff hydrograph of approximately equivalent time, regardless of their intensity.

2. The magnitude of unit hydrograph ordinate is proportional to the effective rainfall depth.

3. For a given watershed the time distribution of direct runoff of a specified storm is independent of precipitation from antecedent or subsequent storm.

Derivation of Unit Hydrograph:

The following steps are used for deriving the unit hydrograph, from a storm hydrograph of unit duration (Fig. 2.8).

Step-1:

From the past rainfall record, select a unit period iso­lated storm that is uniformly distributed over the entire water­shed area.

Step-2:

Develop runoff hydrograph using discharge Vs time data of selected storm.

Step-3:

Separate the base flow from the hydrograph using most accurate method.

Step-4:

Find the ordinates of direct runoff. It is the ordinate after deducting the base flow.

Step-5:

Calculate the total depth of effective rainfall.

This is obtained by using the following formula:

Effective rainfall depth = 0.36(∑0) t/A cm

Where,

∑0 = sum of all direct runoff ordinates (cumec)

t = time interval between successive ordinates, hour

A = watershed area, km²

Step-6:

Find out the ordinates of unit hydrograph, using the following formula:

Step-7:

Plot the unit hydrograph ordinates and corresponding time.

In this way, the obtained hydrograph is the unit hydrograph, which should have unit area below the curve.

Duration of Unit Hydrograph:

The duration of unit hydrograph mainly depends on the rainfall. However, a rough idea can be have about the duration of unit hydrograph, that it should not exceed the followings:

1. The time of rise

2. The basin lag, and

3. The time of concentration.

It is reported that the duration of unit hydrograph as 1/4 of the basin lag provides a best choice. In addition, 12-hours duration is also preferred for the watersheds having more than 1200 km² area.

Limitations of Unit Hydrograph:

The following limitations of unit hydrograph are the main:

1. Unit hydrograph assumes that the rainfall is uniformly distributed throughout the watershed, but it is never satisfied. Similarly, there is also assumed that the rainfall intensity is constant during excess rainfall. In real since, it also never satisfied in field.

2. Precipitation must be in the form of rainfall only. The unit hydrograph cannot be derived for snow fall; and also cannot be applied for computing the runoff, when an appreciable portion of storm occurs as snow fall.

3. This method is not suitable for estimating the surface runoff from the watershed of area less than 25 km².

4. Theoretically, the unit hydrograph is applicable for any size of watershed. However, to fulfill the basic assumption that the hydrographs resulting from the storms of same duration have similar shapes with the ordinates proportional to the amount of effective rainfall and storm should be uniformly distributed over the area, the size of watershed should always be small, because in larger watersheds it is not possible to get satisfy the assumptions. In real since such storms rarely occur in a big size watershed.

Therefore, large size watershed is divided into sub-watersheds; and for each of them an individual unit hydrograph is derived. The total runoff yield from the entire watershed is obtained by adding the runoff obtained from U.H.G. of each sub-watershed. The unit hydrograph used to compute the direct runoff is limited to the watershed area of about 5000 km².

5. Unit hydrograph from one-day storm cannot be derived for that watershed in which infiltration capacity of soil is greater than the rainfall intensity.

6. This method is not technically feasible for the odd shaped watersheds, especially long and narrow watersheds, as they have very uneven rainfall distribution.

7. This method requires a large number of observed data for which more number of gauging stations are required to install in the watershed.

Unit Hydrograph from a Complex Storm:

When suitable and simple isolated storms are not available for deriving the unit hydrograph, then a complex storm with varying intensity is used for the purpose. In case of double peaked complex hydrograph resulting from sufficiently spaced storms, the base flow is separated by using the direct-runoff depletion curve. The unit hydrograph is then developed for each storm.

Procedure:

From a complex storm, to derive the unit hydrograph, the following steps are adopted:

1. Let, the direct runoff ordinates of a given complex storm are indicated by Q₁, Q₂, Q₃,… Q_n in successive periods of time.

2. Let, the ordinates of unit hydrograph are designated as u₁, u_2, u₃….u_n at successive time intervals.

3. To derive the unit hydrograph, assume that the entire period of complex storm has three storms of unit duration, for which unit hydrograph is to be derived. Now, the hydrograph is superimposed on the infiltration curve and excess rainfall is obtained. Let R₁, R₂ and R₃ are the depth of respective rainfall excess for the assumed storm. Thus, during entire storm duration, there will be three similar unit hydrographs, lagged by unit duration.

Unit Hydrograph of Different Durations:

Ideally, a unit hydrograph is derived from a simple isolated storm. If duration of various storms do not differ very much from each other, then they can be kept under one average duration. For practical applications, a unit hydrograph of a specified duration is required to charge into of different durations.

The following two methods are commonly used for this purpose:

1. Method of Super­position:

Method of superposition is based on the principle, that if there a number of continuous and/or isolated storms of uniform intensity, are available, then they are divided into several unit storms; and hydrographs for each storm are derived and their ordinates are added at the time lag to get the combined hydrograph.

In Fig. 2.10 there are three 4-hour unit hydrographs A, B, and C are shown, in which the hydrograph B begins after 4-hour from A, and hydrograph C begins after 4-hour from B. Combining all three hydrographs, gives the direct runoff hydrograph for 3-cm depth of effective rainfall. If the ordinates of this direct runoff hydrograph are divided by 3, then the resulting unit hydrograph will be for 12-hour duration. Method of superposition is applied to convert a small duration U.H.G. in the multiple of given duration.

2. S-Hydrograph:

It is also known as summation curve, is defined as the hydrograph of direct runoff resulted from a continuous succession of unit storms producing unit depth of effective rainfall in the unit duration. In other words, a summation curve represents the cumulative unit hydrograph, resulting from several unit depth of effective rainfalls occurring in a close succession for an infinite period. Because, the shape of summation curve is obtained as ‘S’, hence it is nomenclatured as S curve.

Since, 1 cm of net rainfall on the watershed is being supplied and removed at every t_R hour, so only T/t_R unit graphs are necessary to produce the S-hydrograph, in which T is denoted as the time base of hydrograph. The S curve oscillates at the top about equilibrium value of discharge. The oscillation is removed by drawing an average smooth curve, passing through the constant discharge.

The constant discharge is given by:

Q_c = 2.78A/t_R ……. (2.25)

In which, Q_c is the constant discharge, A is the watershed area (km²) and t_R is the duration of unit hydrograph. S-curve methods is used for converting the U.H.G. of specified duration to a large and small duration U.H.G., both.

The S-curve is constructed by adding together a number of unit hydrographs of unit duration, spaced at unit time duration, i.e., the duration of effective rainfall (Fig. 2.11).

Synthetic Unit Hydrograph:

Synthetic unit hydrographs are derived for ungauged watersheds by computing various coefficients based on the physical features of the watershed. The coefficients are computed with the help of data obtained from the gauged watershed, proved that the gauged and ungauged watershed are hydrologically same.

In other words, the synthetic unit hydrographs are developed on the basis of known physical characteristics of a gauged watershed which is identical, both hydrologically and meteorologically to that watershed which rainfall and runoff data are not available; and it is desired to derive S.U.H. for it.

A number of methods have been investigated for developing the synthetic unit hydrograph. However, all are based on empirical relations, which are applicable only to that specified region for which they are derived. They should not be considered as a general relationship for all the regions. Among all the methods, the Synder’s method is most common, and is widely used.

Snyder’s Method:

Snyder (1938) developed a set of empirical relations on the basis of analysis of large number of hydrographs resulting from several watersheds ranging from 25 to 2500 km² in the size of United States for deriving the synthetic unit hydrograph.

These equations are given as under:

t_p^, = basin lag for storm duration of t_r^,

tp = lag time (basin lag), hour

C_t, C_p = empirical constants, depend on the watershed characteristics, in which C_t approximately varies from 0.3 to 6.5; and C_p from 0.56 to 0.69.

A = watershed area, km²

L = longest length of water course, km

L_ca = length of the main stream from the observation point to a point on the stream opposite to the centre of gravity of the watershed, km

Linsley et al. (1958) reported that the watershed lag time (t_p) is better correlated with the watershed Para meter (L.L_ca/√S)ⁿ, in which S is the watershed slope.

Accordingly, the modified form of lag time is’ given as under:

Where,

C_t = constant, depends on the topographical features of the watershed area. For mountainous region it is 1.715; for foot hill regions 1.03 and for valley area it is 0.5, The L and L_ca are measured in miles.

To derive the synthetic unit hydrograph for the watershed of which the runoff data is not available, the data of gauged watershed such as, area (A), longest length of stream flow path (L) and length along the main stream from the gauging station to a point opposite the centre of gravity of the watershed (L_ca) are determined.

The values of constant C_p and C_t are then determined, using the above data. After that the parameters for deriving the synthetic unit hydrograph such as t_pk, Q_p are also calculated. And having all these data bases, the synthetic unit hydrograph is sketched, in such a way that the area enclosed by the hydrograph should represent unit depth of rainfall excess.

In order to assist the sketching of unit hydrograph, the width of hydrograph at different heights is also required. U.S. Army Corps of Engineers (1959) developed the following relationships for the same for two different durations, i.e. at 50% and 75% of the peak.

These are given as:

Where,

W₅₀ = width of unit hydrograph at 50% peak (hour)

W₇₅ = width of unit hydrograph at 75% peak (hour)

and q = peak discharge per unit area of watershed.

Procedure for deriving the Synthetic Unit Hydrograph:

The procedure followed for deriving the synthetic unit hydrograph is given below:

1. Collect the rainfall-runoff data of gauged watershed to calculate the parameters, required to develop the synthetic unit hydrograph, such as t_r, t_p and the peak discharge Q_p.

2. Find out the values of L and L_ca from the topo map of the gauged and ungauged watersheds. To determine the value of L_ca the centroid of the watershed is located by cutting stiff pattern of the watershed map and intersecting the plumb lines for different rotations of the pattern.

3. Compute the values of constants C_t and C_p, as follows:

Assume t_p‘ is equal to t_p, and compute the value of t_r using equation (2.27). If computed value of t_r is equal to or close to t_R, then further assume q_pr=q_p and find C_t and C_p, using the equation 2.26 and 2.28, res­pectively. If computed t_r does not equal to t_R then use t_R = t_p‘ and calculate t_r using the equation (2.29) and also compute t_p. Again find the values of C_p and C_t by using the equation (2.26) and (2.28), respectively.

4. Calculate the value of t_p‘ and Q_p by using the equation (2.29) and (2.28) respectively. Similarly, the peak discharge for the required unit hydrograph is obtained by multiplying q_pr to the watershed area.

Dimensionless Unit Hydrograph:

Derivation of this type of unit hydrograph is based on the study of a large number of unit hydrographs.

Dimensionless unit hydrographs are used for following purposes:

1. These are used for comparing different unit hydro- graphs of different watersheds varying in their size as well as shape.

2. Such unit hydrographs are useful for transposing the unit hydrograph of a gauged watershed over an ungauged watershed having the same hydro-meteorological conditions.

3. Used for synthesization of representative dimensionless unit hydrograph for a given watershed. This is done simply by averaging several dimen­sionless unit hydrographs.

A unit hydrograph for an ungauged watershed can be obtained from this type of unit hydrograph, by substituting app­ropriate value of lag runoff volume and its duration. The lag time for ungauged watershed can be predicted by using the Snyder’s formula, i.e. t_p = C_t (L. L_ca)^0.3

A typical dimension less unit hydrograph is shown in Fig. 2.12.

Derivation Procedure:

The derivation of dimensionless unit hydrograph can be done by using the following steps:

Step (1):

Reduce its time scale by dividing with the factor called lag plus semi duration and multiply it by 100. The lag plus semi-duration factor represents the time from start of effective rainfall to the centre of mass of direct runoff.

Step (2):

Reduce the instantaneous discharge scale of unit hydrograph by multiplying a factor equal to lag plus semi duration, divided by the total amount of direct runoff, obtained from the graph.

The above two changes in the unit hydrograph largely eliminate the effects of watershed size, areal pattern and duration of effective rainfall.