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CHAPTER SEVEN

FLOODS

7.1 Introduction

A flood is an unusually high stage in a river – normally the level at which the river

overflows its banks and inundates the adjoining area. The damages caused by floods in terms

of loss of life, property and economic loss due to disruption of economic activity are all too

well known. The hydrograph of extreme floods and stages corresponding to flood peaks

provide valuable data for purposes of hydrologic design. Further, of the various

characteristics of the flood hydrograph, probably the most important and widely used

parameter is the flood peak. At a given location in a stream, flood peaks vary from year to

year and their magnitude constitutes a hydrologic series which enable one to assign a

frequency to a given flood-peak value. In the design of practically all hydrologic structures

the peak flow that can be expected with an assigned frequency (say 1 in 100 years) is of

primary importance to adequately proportion the structure to accommodate its effect. The

design of bridges, culvert waterways and spillways for dams and estimation of scour at a

hydraulic structure are some examples wherein flood-peak values are required.

To estimate the magnitude of a flood peak the following methods are available:

1. Rational method,

2. Empirical method,

3. Unit-hydrograph technique, and

4. Flood-frequency studies.

The use of a particular method depends upon (i) the desired objective, (ii) the

available data and (iii) the importance of the project. Further, the rational method is only

applicable to small-size (<50 km2 ) catchments and the unit-hydrograph method is normally

restricted to moderate-size catchments with areas less than 5000 km2.



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7.2 Rational Method

Consider a rainfall of uniform intensity and very long duration occurring over a basin.

The runoff rate gradually increases from zero to a constant value as indicated in Fig. 7.1. The

runoff increases as more and more flow from remote areas of the catchment reach the outlet.

Designating the time taken for a drop of water from the farthest part of the catchment to reach

the outlet as tc = time of concentration, it is obvious that if the rainfall continues beyond tc,

the runoff will be constant and at the peak value. The peak value of runoff is given by

Q p = C A i ; for t ≥ tc (7.1)

Where C = coefficient of runoff = (runoff/rainfall), A = area of the catchment and i =

intensity of rainfall. This is the basic equation of the rational method. Using the commonly

used units, the above equation is written for field application as

Q p = 1/3.6C(itc,p)A (7.2)

Where Q p = peak discharge (m3/s)

C = coefficient of runoff

(itc,p) = the mean intensity of precipitation (mm/h) for a duration equal to t c and an

exceedence probability P

A = drainage area in km2

The use of this method to compute Q p requires three parameters: tc, (itc,p) and C.

Runoff Coefficient (C)

The coefficient C represents the integrated effect of the catchment losses and hence

depends upon the nature of the surface, surface slope and rainfall intensity. Some typical

values of C are indicated in Table 7.1.



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Table 7.1 Value of the Coefficient C

Type of area Value of C

A Urban area(P = 0.05 to 0.10)Lawns: Sandy soil, flat, 2% 0.05-0.10

Sandy soil, steep, 7% 0.15-0.20

Heavy soil, average, 2.7% 0.18-0.22

Residential areas:

Single family areas 0.30-0.50

Multi units, attached 0.60-0.75

Industrial:

Light 0.50-0.80

Heavy 0.60-0.90

Streets: 0.70-0.95

B Agricultural Area

Flat: Tight clay; cultivated 0.50

woodland 0.40

Sandy loam; cultivated 0.20

woodland 0.10

Hilly: Tight clay; cultivated 0.70

woodland 0.60

Sandy loam: cultivated 0.40

woodland 0.30

The rational formula is found to be suitable for peak flow prediction in small

catchments up to 50 km2 in area. It finds considerable application in urban drainage designs

and in the design of small culverts and bridges.

7.3 Empirical Formulae

The empirical formulae used for the estimation of the flood peak are essentially

regional formulae based on statistical correlation of the observed peak and important

catchment properties. To simplify the form of the equation, only a few of the many

parameters affecting the flood peak are used. For example, almost all formulae use the



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catchment area as a parameter affecting the flood peak and most of them neglect the flood

frequency as a parameter. In view of these, the empirical formulae are applicable only in the

region from which they were developed and when applied to other areas they can at best give

approximate values.

Flood-Peak-Area Relationships

By far the simplest relationships are those which relate the flood peak to the drinage

area. The maximum flood peak Q p from a catchment area A is given by these formulae as

Q p = f(A) (7.3)

While there are a vast number of formulae of this kind proposed for various parts of the

world, only a few popular formulae used are given here.

Dickens Formu;a (1865)

Q p = CD A3/4 (7.4)

Where Q p = maximum flood discharge (m3/s)

A = catchment area (km2)

CD = Dickens constant with value between 6 to 30

Ryves Formula (1884)

Q p = CR A2/3 (7.5)

Where Q p = maximum flood discharge (m3/s)

A = catchment area (km2)

CR = Ryves coefficient

7.4 Unit Hydrograph Method

The unit-hydrograph technique described in the previous chapter can be used to

predict the peak-flood hydrograph, if the rainfall producing the flood, infiltration

characteristics of the catchment and the appropriate unit hydrograph are available. For design

purposes, extreme rainfall situations are used to obtain the design storm (viz., the hyetograph)

of the rainfall excess causing extreme floods. The known or derived unit hydrograph of the

catchment is then operated upon by the design storm to generate the desired flood

hydrograph.



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7.5 Flood-Frequency Studies

Hydrologic processes such as floods are exceedingly complex natural events. They

are the resultants of a number of component parameters of the hydrologic system and are

therefore very difficult to model analytically. For example, the floods in a catchment depend

upon the characteristics of the catchment, rainfall and antecedent conditions, each one of

these factors in turn depend upon a host of constituent parameters. This makes the estimation

of the flood peak a very complex problem leading to many different approaches. The

empirical formulae and unit-hydrograph method presented in the previous sections are some

of them. Another approach to the prediction of flood flows, and also applicable to other

hydrologic process such as rainfall etc. is the statistical method of frequency analysis.

The values of the annual maximum flood from a given catchment area for large

number of successive years constitute a hydrologic data series called the annual series. The

data are then arranged in decreasing order of magnitude and the probability P of each event

being equaled to or exceeded (plotting position) is calculated by the plotting-position formula

P = m/(N+1) (7.6)

where m = order number of the event and N = total number of events in the data.

The recurrence interval, T ( also called the return period or frequency ) is calculated

as T = 1/P (7.7)

Consider, for example, a list of flood magnitudes of a river arranged in descending

order as shown in Table 7.2. The length of the record is 50 years.

Table 7.2 Calculation of Frequency T

Order No.

m

Flood magnitude

Q (m3/s)

T in years

= 51 / m

1

2

3

4

:

:

:

49

50

160

135

128

116

:

:

:

65

63

51.00

25.50

17.00

12.75

:

:

:

1.04

1.02



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The last column shows the return period T of various flood magnitude, Q. A plot of

Q vs T yields the probability distribution. For small return periods ( i.e. for interpolation ) or

where limited extrapolation is required, a simple best-fitting curve through plotted points can

be used as the probability distribution. A logarithmic scale for T is often advantageous.

However, when larger extrapolations of T are involved, theoretical probability distribution

have to be used. In frequency analysis of floods the usual problem is to predict extreme flood

events. Towards this, specific extreme-value distributions are assumed and the required

statistical parameters calculated from the available data. Using these the flood magnitude for

a specific period is estimated.

Chow (1951) has shown that most frequency-distribution functions applicable in

hydraulic studies can be expressed by the following equation known as the general equation

of hydrologic frequency analysis:

xT = x + K σ (7.8)

where xT = value of the variate X of a random hydrologic series with a return period T, x =

mean of the variate, σ = standard deviation of the variate, K = frequency factor which

depends upon the return period, T and the assumed frequency distribution. Some of the

commonly used frequency distribution function for the prediction of extreme flood values

are:

1. Gumbel’s extreme-value distribution,

2. Log-Pearson Type III distribution, and

3. Log normal distribution.

Only the Gumbel distribution is dealt here with emphasis on application.

7.6 Gumbel’s Method

This extreme value distribution was introduced by Gumbel (1941) and is commonly

known as Gumbel’s distribution. It is one of the most widely used probability-distribution

functions of extreme values in hydrological and meteorologic studies for prediction of flood

peaks, maximum rainfalls, maximum wind speed, etc.

Gumbel defined a flood as the largest of the 365 daily flows and the annual series of

flood flows constitute a series of largest values of flows. According to his theory of extreme

events, the probability of occurrence of an event equal to or larger than a value x o is



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in which y is a dimensionless variable given by

y =α

( x – a )a = x – 0.45005 σx (7.10)

α = 1.2825 / σx

Thus

y = (1.2825(x - x )/ σx) + 0.577

Where x = mean and σx = standard deviation of the variate X. In practice it is the value of X

for a given P that is required and as such Eq. (7.9) is transposed as

y p = - ln [ - ln ( 1 – P )] (7.11)

Noting that the return period T = 1/P and designating

yT = the value of y, commonly called the reduced variate, for a given T

yT

= -[ln.ln.(T/(T-1))] (7.12)

or yT = -[0.834 + 2.303 log.log.(T/(T-1))] (7.12a)

Now rearranging Eq. (7.10), the value of the variate X with a return period T is

xT = x + K σx (7.13)

where K = (yT – 0.577)/1.2825 (7.14)

Note that Eq. (7.14) is of the same form as the general equation of hydrologic

frequency analysis, Eq. (7.8). Further, Eqs. (7.13) and (7.14) constitute the basic Gumbel’s

equations and are applicable to an infinite sample size (i.e. N→∝).

Since practical annual data series of extreme events such as floods, maximum rainfall

depths, etc., all have finite lengths of record, Eq. (7.14) is modified to account for finite N as

given below for practical use.

yeo e1)xx(P

−−−=≥ (7.9)



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Gumbel’s Equation for Practical Use

Equation (7.13) giving the value of the variate X with a recurrence interval T is used

as

xT = x + K σn-1 (7.15)

where σn-1= standard deviation of the sample size N

K= frequency factor expressed as

K = (yT - yn)/Sn (7.16)

in which yT = reduced variate, a function of T and is given by

yT = -[ln.ln.(T/(T-1))] (7.17)

or yT = -[0.834 + 2.303 log.log.(T/(T-1))]

yn = reduced mean, a function of sample size N

Sn = reduced standard deviation, a function of sample size N

Table 7.3 Reduced Mean(yn ) and Reduced Standard Deviation ( Sn )

N 10 15 20 25 30 40 50

yn

Sn

0.4952

0.9457

0.5128

1.0206

0.5236

1.0628

0.5309

1.0915

0.5362

1.1124

0.5436

1.1413

0.5485

1.1607

N 60 70 80 90 100 200 500 ∝

yn

Sn

0.5521

1.1747

0.5548

1.1854

0.5569

1.1938

0.5586

1.2007

0.5600

1.2065

0.5672

1.2360

0.5724

1.2588

0.5772

1.2826

These equations are used under the following procedure to estimate the flood

magnitude corresponding to a given return period based on an annual flood series.

1. Assemble the discharge data and note the sample size N. Here the annual flood value is

the variate X. Find x and σn-1 for the given data.

2. Using standard tables determine yn and Sn appropriate to given N.

3. Find yT for a given T by Eq. (7.17).

4. Find K by Eq. (7.16).

5. Determine the required xT by Eq. (7.15).



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The method is illustrated in Example 7.1.

To verify whether the given data follow the assumed Gumbel’s distribution, the

following procedure may be adopted. The value of xT for some return periods T < N are

calculated by using Gumbel’s formula and plotted as xT vs T on a convenient paper such as a

semi-log, log-log or Gumbel probability paper. The use of Gumbel probability paper results

in a straight line for xT vs T plot. Gumble’s distribution has the property which gives T = 2.33

years for the average of the annual series when N is very large. Thus the value of a flood with

T = 2.33 years is called the mean annual flood. In graphical plots this gives a mandatory point

through which the line showing variation of xT with T must pass. For the given data, values

of return periods (plotting positions) for various recorded values, x of the variate are obtained

by the relation T = ( N+1 )/ m and plotted on the graph described above. A good fit of

observed data with the theoretical variation line indicates the applicability of Gumbel’s

distribution to the given data series (Fig.7.1). By extrapolation of the straight line x T vs T,

values of xT for T> N can be determined easily (Example 7.1).

Example 7.1. Annual maximum recorded floods in the river Bhima at Deorgaon, a tributary

of the river Krishna, for the period 1951 to 1977 is given below. Verify whether the Gumbel

extreme-value distribution fit the recorded values. Estimate the flood discharge with

recurrence interval of (i) 100 years and (ii) 150 years by graphical extrapolation.

Year 1951 1952 1953 1954 1955 1956 1957 1958 1959

Max.flood (m3/s) 2947 3521 2399 4124 3496 2947 5060 4903 3757

Year 1960 1961 1962 1963 1964 1965 1966 1967 1968

Max.flood (m3/s) 4798 4290 4652 5050 6900 4366 3380 7826 3320

Year 1969 1970 1971 1972 1973 1977 1978 1979 1980

Max.flood (m3/s) 6599 3700 4175 2988 2709 3873 4593 6761 1971

Solution: The flood discharge values are arranged in descending order and the plotting

position recurrence interval T p for each discharge is obtained as

T p = (N + 1)/m = 28/m

Where m = order number. The discharge magnitude Q are plotted against the corresponding

T p on a Gumbel extreme probability paper (Fig. 7.1).



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The statistics x and σn-1 for the series are next calculated and are shown in Table 7.3.

Using these the discharge xT for some chosen recurrence interval is calculated by using

Gumbel’s formulae [Eqs. (7.17), (7.16) and (7.15)].

From the standard tables of Gumbel’s extreme value distribution, for N = 27, yn =0.5332 and Sn = 1.1004.

Choosing T = 10 years, by Eq. 7. 17

yT = -[ln.ln(10/9)] = 2.25037

K = (2.25037-0.5332)/1.1004 = 1.56

xT = 4263 + (1.56x1432.6) = 6499 m3/s

Table 7.4 Calculation of TP for Observed data – Example 7.1

Order

Number

M

Flood

Discharge

(m3/s)

T p

(year)

Order

Number

M

Flood

Discharge

(m3/s)

T p

(year)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

7826

6900

6761

6599

5060

5050

4903

4798

4652

4593

4366

4290

4175

4124

28.00

14.00

9.33

7.00

5.60

4.67

4.00

3.50

3.11

2.80

2.55

2.33

2.15

2.00

15

16

17

18

19

20

21

22

23

24

25

26

27

3873

3757

3700

3521

3496

3380

3320

2988

3947

2947

2709

2399

1971

1.87

1.7

1.65

1.56

1.47

1.40

1.33

1.27

-

1.17

1.12

11.08

1.04

N = 27 years, x = 4263 m3/s, σn-1 = 1432.6 m3/s



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Fig. 7.1 Flood probability analysis by Gumbel’s distribution

Similarly, values of xT are calculated for two more T values as shown below.

T xT [obtained by Eq. 7.15]

(years) (m3/s)

5.0 5522

10.0 6499

20.0 7436

These values are shown in Fig. 7.1. It is seen that due to the property of Gumbel’s

extreme probability paper these points lie on a straight line. A straight line is drawn through

these points. It is seen that the observed data fit well with the theoretical Gumbel’s extreme-

value distribution.

By extrapolation of the theoretical xT vs T relationship, from Fig. 7.1,

At T = 100 years, xT = 9600 m3/s

At T = 150 yers, xT = 10700 m3/s

[By using Eq. (7.15 ) to (7.17), x100 = 9558 m3/s and x150 = 10088 m3/s.]



90

7.7 Low-Flow Frequency Analysis

Whereas high flows lead to floods, sustained low flows can lead to droughts. A

drought is defined as a lack of rainfall so great and continuing so long as to affect the plant

and animal life of a region adversely and to deplete domestic and industrial water supplies,

especially in those regions where rainfall is normally sufficient for such purposes.

In practice, a drought refers to a period of unusually low water supplies, regardless of

the water demand. The regions most subject to droughts are those with the greatest variability

in annual rainfall. Studies have shown that regions where the variance coefficient of annual

rainfall exceeds 0.35 are more likely to have frequent droughts. Low annual rainfall and high

annual rainfall variability are typical of arid and semiarid regions. Therefore, these regions

are more likely to be prone to droughts.

Studies of tree rings, which document long term trends of rainfall, show clear patterns

of periods of wet and dry weather. While there is no apparent explanation for the cycles of

wet and dry weather, the dry years must be considered in planning water resource projects.

Analysis of long records has shown that there is a tendency for dry years to group together.

This indicates that the sequence of dry years is not random, with dry years tending to follow

other dry years. It is therefore necessary to consider both these severity and duration of a

drought period.

The severity of droughts can be established by measuring (1) the deficiency in

rainfall and runoff, (2) the decline of soil moisture, and (3) the decrease in groundwater

levels. Alternatively, low-flow-frequency analysis can be used in the assessment of the

probability of occurrence of droughts of different durations.

Methods of low-flow frequency analysis are based on an assumption of invariance of

meteorological conditions. The absence of long records, however, imposes a stringent

limitation on low-flow frequency analysis. When records of sufficient length are available,

analysis begins with the identification of the low-flow series. Either the annual minima or the

annual exceedence series are used. In a monthly analysis, the annual minima series is formed

by the lowest monthly flow volumes in each year of record. If the annual exceedence method

is chosen, the lowest monthly flow volumes in the record are selected, regardless of when

they occurred. In the latter method, the number of values in the series need not be equal to the

number of years of record.



91

Figure 7.2 Low-flow frequency curves



92

A flow-duration curve can be used to give an indication of the severity of low flows.

Such a curve, however, does not contain information on the sequence of low flows or the

duration of possible droughts. The analysis is made more meaningful by abstracting the

minimum flows over a period of several consecutive days. For instance, for each year, the 7-

day period with minimum flow volume is abstracted, and the minimum flow is the average

flow rate for that period.. A frequency analysis on the low-flow series, using the Gumbel

method, for instance, results in a function describing the probability of occurrence of low

flows of a certain duration. The same analysis repeated for other durations leads to a family

of curves depicting low-flow frequency, as shown in Fig.7.2.

In reservoir design, the assessment of low flows is aided by a flow-mass curve. The

technique involves the determination of storage volumes required for all low-flow periods.

Although it is practically impossible to provide sufficient storage to meet hydrologic risks of

great rarity, common practice is to provide for a stated risk (i.e., a drought probability) and to

add a suitable percent of the computed storage volume as reserve storage allowance. The

variance coefficient of annual flows is used in determining the risk and storage allowance

levels. Extraordinary drought levels are then met by cutting draft rates.

Regulated rivers may alter natural flow conditions to provide a minimum downstream

flow for specific purposes. In this case, the reservoirs serve as the mechanism to diffuse the

natural flow variability into downstream flows which can be made to be nearly constant in

time. Regulation is necessary for downstream low flow maintenance, usually for the purpose

of meeting agricultural, municipal and industrial water demands, minimum instream flows,

navigation draft, and water pollution control requirements.



93

CHAPTER EIGHT

GROUNDWATER

8.1 Introduction

In the previous chapters various aspects of surface water hydrology have been

discussed. Study of subsurface water is equally important since about 30% of the world’s

fresh water resources exist in the form of groundwater. It is relatively free of pollution and is

especially useful for domestic use in small towns and isolated farms. In arid regions

groundwater is often the only reliable source of water for irrigation.

Aside from its direct use, groundwater is also an important phase of the hydrologic

cycle. Most of the flow of perennial streams originates from subsurface water, while a large

part of the flow of ephemeral streams may percolate beneath the surface.

8.2 Forms of Subsurface Water

Water in the soil mantle is called subsurface water and is considered in two zones

(Fig. 8.1):1. Saturated zone, and

2. Aeration zone.

Fig. 8.1 Classification of subsurface water



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Saturated Zone

This zone, also known as groundwater zone, is the space in which all the pores of the

soil are filled with water. The water table forms its upper limit and marks a free surface, i.e.

a surface having atmospheric pressure.

Zone of Aeration

In this zone the soil pores are only partially saturated with water. The space between

the land surface and the water table marks the extent of this zone. Further, the zone of

aeration has three subzones:

Soil Water Zone

This lies close to the ground surface in the major root band of the vegetation from

which the water is lost to the atmosphere by evapotranspiration.

Capillary Fringe

In this the water is held by capillary action. This zone extends from the water table

upwards to the limit of the capillary rise.

Intermediate Zone

This lies between the soil water and the capillary fringe.

The thickness of the zone of aeration and its constituent subzones depend upon the

soil texture and moisture content and vary from region to region. The soil moisture in the

zone of aeration is of importance in agricultural practice and irrigation engineering. The

present chapter is concerned only with the saturated zone.

All earth materials, from soil to rocks have pore spaces. Although these pores are

completely saturated with water below the water table, from the groundwater utilization

aspect only such material through which water moves easily and hence can be extracted with

ease are significant. On this basis the saturated formations are classified into four categories:

1. Aquifer

2. Aquitard

3. Aquiclude, and

4. Aquifuge.

These are discussed below:



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Aquifer

An aquifer is a saturated formation of earth material which not only stores water but

yields it in sufficient quantity. Thus an aquifer transmits water relatively easily due to its

high premeability. Unconsolidated deposits of sand and gravel form good aquifers.

Aquitard

It is a formation through which only seepage is possible and thus the yield is

insignificant compared to an aquifer. It is partly permeable. A sandy clay unit is an example

of aquitard. Through an aquitard appreciable quantities of water may leak to an aquifer

below it.

Aquiclude

It is a geological formation which is essentially impermeable to the flow of water. It

may be considered as closed to water movement even though it may contain large amounts of

water due to its high porosity. Clay is an example of an aquiclude.

Aquifuge

It is a geological formation which is neither porous nor permeable. There are no

interconnected openings and hence it cannot transmit water. Massive compact rock without

any fractures is an aquifuge.

The definitions of aquifer, aquitard and aquiclude as above are relative. A formation

which may be considered as an aquifer at a place where water is at a premium (e.g. arid

zones) may be classified as an aquitard or even aquiclude in an area where plenty of water is

available.

The availability of groundwater from an aquifer at a place depends upon the rate of

withdrawal and replenishment (recharge). Aquifers play the role of both a transmission

conduit and a storage. Aquifers are classified as unconfined aquifers and confined aquifers

on the basis of their occurrence and field situation. An unconfined aquifer (also known as

water table aquifer ) is one in which a free water surface, i.e. a water table exists (Fig. 8.2).

Only the saturated zone of this aquifer is of importance in groundwater studies. Recharge of

this aquifer takes place through infiltration of precipitation from the ground surface. A well

driven into an unconfined aquifer will indicate a static water level at that location.



96

Fig. 8.2 Confined and unconfined aquifers

A confined aquifer, also known as artesian aquifer , is an aquifer which is confined

between two impervious beds such as aquicludes or aquifuges (Fig. 8.2). Recharge of this

aquifer takes place only in the area where it is exposed at the ground surface. The water in

the confined aquifer will be under pressure and hence the piezometric level will be much

higher than the top level of the aquifer. At some locations: the piezometric level can attain a

level higher than the land surface and a well driven into the aquifer at such a location will

flow freely without the aid of any pump.

A confined aquifer is called a leaky aquifer if either or both of its confining beds are

aquitards.

Water Table

A water table is the free water surface in an unconfined aquifer. The static level of a

well penetrating an unconfined aquifer indicates the level of the water table at that point. The

water table is constantly in motion adjusting its surface to achieve a balance between the

recharge and outflow from the subsurface storage. Fluctuations in the water level in a dug

well during various seasons of the year, lowering of the groundwater table in a region due to

heavy pumping of the wells and the rise in the water table in an irrigated area with poor

drainage, are some common examples of the fluctuation of the water table. In a general

sense, the water table follows the topographic features of the surface. If the water table

intersects the land surface the groundwater comes out to the surface in the form of springs or

seepage.



97

Sometimes a lens or localised path of impervious stream can occur inside an

unconfined aquifer in such a way that it retains the water table above the general water table

(Fig. 8.3). Such a water table retained around the impervious material is known as perched

water table. Usually the perched water table is of limited extent and the yield from such a

situation is very small. In groundwater exploration a perched water table is quite often

confused with a general water table.

Fig. 8.3 Perched water table

The position of the water table relative to the water level in a stream determines

whether the stream contributes water to the groundwater storage or the other way about. If

the bed of the stream is below the groundwater table, during periods of low flows in the

stream, the water surface may go down below the general water table elevation and the

groundwater contributes to the flow in the stream. Such streams which receive groundwater

flow are called effluent streams (Fig. 8.4 (a)). Perennial rivers and streams are of this kind.

If, however, the water table is below the bed of the stream, the stream-water percolates to the

groundwater storage and a hump is formed in the groundwater table (Fig. 8.4 (b)). Such

streams which contribute to the groundwater are known as influent streams. Intermittent

rivers and streams which go dry during long periods of dry spell (i.e. no rain periods) are of

this kind.



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Fig. 8.4 Effluent and influent streams

8.3 Aquifer Properties

The important properties of an aquifer are its capacity to release the water held in its

pores and its ability to transmit the flow easily. These properties essentially depend upon the

completion of the aquifer.

Porosity

The amount of pore space per unit volume of the aquifer material is called porosity. It

is expressed as

n = Vv/Vo (8.1)

where n = porosity, Vv = volume of voids and Vo = volume of the porous medium. In an

unconsolidated material the size distribution, packing and shape of particles determine the

porosity. In hard rocks the porosity is dependent on the extent spacing and the pattern of

fracturing or on the nature of solution channels. In qualitative terms porosity greater than

20% is considered as large, between 5 to 20% as medium and less than 5% as small.

Specific Yield

While porosity gives a measure of the water-storage capacity of a formation, not all

the water held in the pores is available for extraction by pumping or draining by gravity.



99

The pores hold back some water by molecular attraction and surface tension. The actual

volume of water that can be extracted by the force of gravity from a unit volume of aquifer

material is known as the specific yield, S y. The fraction of water held back in the aquifer is

known as specific retention, Sr . Thus porosity

n = Sy + Sr (8.2)

The representative values of porosity and specific yield of some common earth

materials are given in Table 8.1.

Table 8.1 Porosity and SpecificYield of Selected Formations

Formation Porosity

%

Specific yield

%

Clay

Sand

Gravel

Sand stone

Shale

Lime stone

45-55

35-40

30-40

10-20

1-10

1-10

1-10

10-30

15-30

5-15

0.5-5

0.5-5

It is seen from Table 8.1 that although both clay and sand have high porosity the

specific yield of clay is very small compared to that of sand.

Darcy’s Law

In 1856 Henry Darcy, a French hydraulic engineer, on the basis of this experimental

findings proposed a law relating the velocity of flow in a porous medium. This law, known

as Darcy’s law, can be expressed as

V = Ki (8.3)

Where V = Apparent velocity of seepage = Q/A in which Q = discharge and A = cross

sectional area of the porous medium. V is sometimes known as discharge velocity. i = -

dh/ds = hydraulic gradient, in which h = piezometric head and s = distance measured in the

general flow direction; the negative sign emphasizes that the piezometric head drops in the



100

direction of flow. K = a coefficient, called coefficient of permeability, having the unit of

velocity.

Q = KiA (8.4)

= KA (- ∆H/∆s)

where (- ∆H) is the drop in the hydraulic grade line in a length ∆s of the porous medium.

Darcy’s law is a particular case of the general viscous fluid flow. It has been shown

valid for laminar flows only. For practical purposes, the limit of the validity of Darcy’s law

can be taken as Reynolds number of value unity, i.e.

Re = Vda/ν = 1 (8.5)

where Re = Reynolds number

da = representative particle size, usually da = d10 where d10 represents a size such that

10% of the aquifer material is of smaller size.

ν = kinematic viscosity of water

Excepting for flow in fissures and caverns, to a large extend groundwater flow in

nature obeys Darcy’s law. Further, there is no known lower limit for the applicability of

Darcy’s law.

It may be noted that the apparent velocity V used in Darcy’s law is not the actual

velocity of flow through the pores. Owing to irregular pore geometry the actual velocity of

flow varies from point to point and the bulk pore velocity (va) which represents the actual

speed of travel of water in the porous media is expressed as

νa = V/n (8.6)

where n = porosity. The bulk pore velocity va is the velocity that obtained by ticking a tracer

added to the groundwater.

Coefficient of Permeability

The coefficient of permeability, also designated as hydraulic conductivity reflects the

combined effects of the porous medium and fluid properties. From an analogy of laminar

flow through a conduit (Hagen-Poiseuille flow) the coefficient of permeability K can be

expressed as



101

Where dm = mean particle size of the porous medium, γ = ρg = unit weight of fluid, ρ =

density of the fluid, g = acceleration due to gravity, µ = dynamic viscosity of the fluid and

C = a shape factor which depends on the porosity, packing, shape of grains and graiin-size

distribution of the porous medium. Thus for a given porous material

K α 1/ν (8.8)

Where ν = kinematic viscosity = µ/ρ = f(temperature). The laboratory or standard value of

the coefficient of permeability (K s) is taken as that for pure water at a standard temperature of

20° C. The value of K t, the coefficient of permeability at any temperature t can be converted

to K s by the relation

K s = K t(νt/νs) (8.9)

Where νs and νt represent the kinematic viscosity values at 20° C and t°C respectively.

The coefficient of permeability is often considered in two components, one reflecting

the properties of the medium only and the other incorporating the fluid properties. Thus,

referring to the first equation, a term K o is defined as

K = K 0 γ/µ (8.10)

Where K o = Cdm2. The parameter K o is called specific or intrinsic permeability which is a

function of the medium only. Note that K 0 has dimensions of [ L2] it is expressed in units of

cm2 or m2 or in darcys where 1 darcy = 9.87x10-13 m2. Where more than one fluid is involved

in porous media flow or when there is considerable temperature variation,the coefficient K 0

is useful. However, in groundwater flow problems, the temperature variations are rather small

and as such the coefficient of permeability K is more convenient to use. The common units of

K are m/day or cm/s. The conversion factor for these two are

1m/day = 0.0011574 cm/s

or 1cm/s = 864.0 m/day

Consider an aquifer of unit width and thickness B, (i.e. depth of a fully saturated

zone). The disharge through this aquifer under a unit hydraulic gradient is

T = KB (8.11)

µγ

= 2

mCdK (8.7)



102

This discharge is termed transmissibility, T and has the dimension of [L2/T].Its units

are m2/s or litres per day/metre width (l pd/m). Typical value of T lie in the range 1x10 6 l

pd/m to 1x104 l pd/m. A well with a value of T = 1 x 105 l pd/m is considered satisfactory for

irrigation purposes.

8.4 Equilibrium Hydraulics of Wells

Fig. 8.5 shows a well in a homogeneous aquifer of infinite extent with an initially

horizontal water table. For flow to occur to the well there must be a gradient toward the well.

The resulting water-table form is called a cone of depression. If the decrease in water level at

the well (drawdown) is small with respect to the total thickness of the aquifer, if the well

completely penetrates the aquifer, and assuming equilibrium, a formula relating well

discharge and aquifer characteristics can be derived.

Fig 8.5 Definition sketch and flow net for equilibrium flow to a well

Flow toward the well through a cylindrical surface at radius x must equal the

discharge of the well, and from Darcy’s law (Eq.8.3)

dx

dyxyK 2q π= (8.12)



103

Where 2πxy is the area of the cylinder and dy/dx is the slope of the water table.

Integrating with respect to x from r 1 to r 2 and y from h1 to h2 yields

( )

−π

=2

1

2

2

2

1

r r ln

hhK

q (8.13)

Where h is the height of the water table above the base of aquifer at distance r from the

pumped well and ln is the logarithm to the base e. Since we have assumed the drawdown Z to

be small compared with the saturated thickness (h1 ≈ h2 ≈ y), Eq.(8.13) can be written

)ZZ(2

)r /r ln(qT

12

21

−π

= (8.14 )

Equation (8.13) was first derived by Dupuit and subsequently modified by Thiem.

Equations (8.13) and (8.14) can be used to estimate T or K given q and Z, provided that the

assumption of equilibrium is satisfied.

8.5 Non-Equilibrium Hydraulics of Wells

During the initial period of pumping from a new well, much of the discharge is

derived from storage in the portion of the aquifer unwatered as the cone of depression

develops. Equilibrium analysis indicates a permeability which is too high, because only part

of the discharge comes from flow through the aquifer to the well. This leads to an over

estimate of the potential yield of the well.

In 1935 Theis presented a formula based on the heat-flow analogy which accounts for

the effect of time and the storage characteristics of the aquifer. His formula is

duu

e

T4

qZ

u

u

r ∫∞ −

π= (8.15)

where Zr is the drawdown in an observation well at distance r from the pumped well, q is the

flow in cubic feet per day, T is transmissibility in cubic feet per day per foot, and u is given

by



104

Tt4

Sr u c

2

= (8.16)

In Eq.(8.16) t is the time in days since pumping began, and Sc is the storage constant

of the aquifer, or the volume of water removed from a column of aquifer 1 ft square when the

water table or piezometric surface is lowered 1 ft. For water-table aquifers it is essentially the

specific yield. The integral in Eq. (8.15), commonly written W(u) and called the well function

of u, can be evaluated from the series

⋅⋅⋅⋅⋅⋅⋅

+⋅

−+−−=!33

u

!22

uuuln5772.0)u(W

32

(8.17)

Equation (8.15) can be solved graphically by plotting a type curve of u versus W(u)

on logarithmic paper (Fig. 8.5). From Eq. (8.16),

uS

T4

t

r

c

2

= (8.18)

If q is constant, Eq.(8.15) indicates that Zr equals a constant times W(u). Thus a curve

of r 2/t versus Zr should be similar to the type curve of u versus W(u). After the field

observations are plotted, the two curves are superimposed with their axes parallel and

adjusted until some portions of the two curves coincide. The coordinates of a common point

taken from the region where the curves coincide are used to solve for T and Sc, using

Eqs.(8.15) and (8.16). Values of Zr and r 2/t may come from one well with various values of t,



105

from several wells with different values of r, or a combination of both. Metric units may be

used in these equations without change in the constants.

Example 8.1 A 12-in-diameter well is pumped at a uniform rate of 1.5 ft 3 /s while

observations of drawdown are made in a well 100 ft distant. Values of t and Z as observed

and computed values of r 2 /t are given below. Find T and S c for the aquifer and estimate the

drawdown in the observation well at the end of 30 days of pumping .

t, hr 1 2 3 4 5 6 8 10 12 18 24

Z, ft 0.6 1.4 2.4 2.9 3.3 4.0 5.2 6.2 7.5 9.1 10.5

r 2

/t, (ft2

/day)x10-5

2.4 1.2 0.8 0.6 0.5 0.4 0.3 0.24 0.2 0.13 0.1

The relation between r 2/t and Z and between u and W(u) are plotted as shown in Fig.

8.6. The match point coordinates are

Type curve: u = 0.4 W(u) = 0.7

Data curve: Z = 3.4 ft day/ft103.5t

r 242

×=

Substituting these values in Eq. (8.15) and (8.18) and noting that q in ft3 / day is

1.5 × 86,400 = 129,600,

Fig. 8.6 Using Theis method to solve a well problem



106

day/ft21244.356.12

7.0600,129

Z4

)u(qWT 2=

××

=π

=

064.0

103.5

21244.04

t/r

uT4S

42c =

×

××==

From Eq. (8.16) at the end of 30 days

0025.03021244

064.0000,10

Tt4

Sr u c

2

=××

×==

From well function table, W(u) = 5.44 which, substituted in Eq.(6.15) yields

ft4.26212457.12

44.5600,129T4

)u(qWZ =××=

π=

When u is small, the terms of Eq. (8.17) following ln u are small and may be neglected.

Equation (8.16) indicates that u will be small when t is large, and in this case a modified

solution of the Theis method is possible by writing

1

2

t

tlog

Z4

q3.2T =

∆π= (8.19)

where ∆ Z is the change in drawdown between times t1 and t2. The drawdown Z is plotted on

an arithmetic scale against time t on a logarithmic scale (Fig. 8.6). If ∆ Z is taken as the

change in drawdown during one log cycle, log10 (t2/t1) = 1, and T is determined from Eq.

(8.19). When Z = 0,

2

0c

r Tt25.2S = (8.20)

where t0 is the intercept (in days) obtained if the straight-line portion of the curve is extended

to Z = 0.

As in the Theis equation, Theis assumes small drawdown and full penetration of the

well. While Theis adjusts for the effect of storage in the aquifer, he does assume

instantaneous unwatering of the aquifer material as the water table drops. These conditions



107

are reasonably well satisfied in artesian aquifers. However, the procedure should be used

with caution in thin or poorly permeable water-table aquifers.

Example 8.2 Using the modified Theis method, find the transmissibility and storage

constant for the data of Illustrative example 8.1.

The time-drawdown curve for these data is plotted in Fig. 8.7. Between t=3hr and

t=30hr, ∆Z = 11.0 ft. Hence,

T = 2.3 x 129,600 = 2156 ft2

/ day

12.57 x 11

t0 = 2.7 hr = 0.112 day

Sc = 2.25 x 2156 x 0.112 = 0.056

10,000

Fig 8.7 Use of the modified Theis method

8.6 Safe Yield

The safe yield of a groundwater basin is governed by many factors, one of the most

important being the quantity of water available. This hydrologic limitation is often expressed

by the equation.

G = P – Qs – Er + Qg – ∆Sg - ∆Ss (8.21)

where G = safe yield

P = precipitation on the area tributary to the aquifer.



108

Qs = surface streamflow the same area

Er = evapotranspiration

Qs = net groundwater inflow to the area

∆Sg = change in groundwater storage∆Ss = change in surface storage

8.7 Seawater Intrusion

Since seawater (specific gravity about 1.025) is heavier than fresh, the groundwater

under a uniformly permeable circular island would appear as shown in (Fig. 8.8). The lens of

fresh water floating on salt water is known as a Ghyben-Herzbeng lens, after the

codiscoverers of the principle. About 1/40 unit of fresh water is required above seal level for

each unit of fresh water below sea level to maintain hydrostatic equilibrium. True hydrostatic

equilibrium does not exist with a sloping water table since flow must occur. Thus, there is

likely to be seepage face for fresh water flow to the ocean and a zone of mixing along the

saltwater-freshwater interface. Areally variable recharge, pumping of wells, and tidal action

also disturb the equilibrium. A hydrodynamic balance govems the form of the interface. If

velocities are low, the 1/40 ratio may be a reasonable first approximation, but more adequade

methods of analysis are available.

When a cone of depression is formed about a pumping well in the fresh water, an

inverted cone of salt water will rise into the fresh water (Fig. 8.8b). A saltwater rise of

approximately 40m / m (40 ft / ft) of fresh water drawdown may occur, depending on the

local situation. Horizontal skimming wells are commonly used to avoid this effect.

Fig 8.8 Saltwater-freshwater relations adjacent to a coastline



109

CHAPTER NINE

FLOOD ROUTING

Two general approaches to routing are recognized:

(1) Hydrologic routing method -solve the continuity equation and is based on the

storage concept.

(2) Hydraulic routing method -is based on a solution of the energy and momentum

equations.

Two common techniques for flood routing (in hydrologic routing) are:

1. Reservoir routing

2.

Channel routing or streamflow routing.

9.1 Waves in Natural Channels

Experiments in channels of regular cross section confirm the validity of the equations

developed. Also reasonable checks are obtained in natural streams where the effect of local

inflow is negligible. However, simple mathematical treatment of flood waves is possible only

for uniform channels with fairly regular cross sections. Most of the natural channels are non-

uniform with complex cross section and varying slope and roughness and most flood waves

are generated by nonuniform lateral in flow all along the channel. Hence, natural flood waves

are more complex and usually have to be analyzed by the solution of the continuity equation.

9.2 The Storage Equation

The storage concept is well established in routing theory and practice. Routing is the

solution of the storage equation which is an expression of continuity.

Inflow – outflow = change in storage

I - O = ds / dt

t

sOO I I

t

sO I or

∆=

+−

+

∆=−

22

)1.9()(

2121



110

∫∫ −=−=∆2

1

2

1

12

t

t

t

t

dt Odt I S S swhere

Equation (9.1), becomes

Where I = Inflow rate

O = Outflow rate

S = storage

t = routing period = time difference between t1 and t2

1,2 = beginning and end of the period

Most storage-routing methods are based on equation (9.2). It is assumed that I1, I2, O1

and S1 are known and O2 and S2 must be determined. Since there are two unknowns, a second

relation between storage and flow is needed to complete a solution.

Assumption:- I I I

=+

2

21 , which is assumed that the hydrograph is a straight line

during the routing period “t”. To ensure that this assumption is not violated, the routing

period should be short enough.

9.3 Storage Determination

Storage-elevation curves are usually computed by planimetering the area enclosed

within successive contours on a topographic map. The measured area multiplied by the

contour interval given the increment of volume from the mid-point of one contour interval to

the mid-point of the next higher interval.

The common method of determining storage in a reach of natural channel is

calculation of storage from inflow and outflow hydrographs.

Fig (9.1)

)2.9()2

()2

( 212112 t

OOt

I I S S s +−+=−=∆



111

When inflow exceeds outflow, ∆ S is positive, and when outflow exceeds inflow, ∆ S

is negative. Since routing involves only ∆ S, absolute storage volumes are not necessary and

the point of zero storage can be taken arbitrarily.

9.4 Reservoir Routing

A reservoir in which the discharge a function of water-surface elevation offers the

simplest of all routing situations. Such a reservoir may have ungated sluiceways and / or

uncontrolled spillways. Reservoir having sluiceway or spillway gates may be treated as

simple reservoirs if the gates remain a fixed openings.

A known data on the reservoir are the elevation-storage curve and the elevation-

discharge curve (Figure 9.2). Equation(9.2) can be transformed into

)3.9(22

22

11

21

−=

−++ O

t

S O

t

S I I

Fig(9.2) Routing curves for a typical reservoir

All terms on the left-hand side of the equation are known, and a value of( 222

Ot

S + )

can be computed. The corresponding value of O2 can be determined from the routing curve.

The computation is then repeated for succeeding routing periods. [Note :

OOt

S Ot

S 222 −+=− ]



112

Table : Routing with the Ot

S +

2Vs O curve of figure 9.2

Date Hour I (m3 / s)

(given)

Ot

S −

2 (m3 / s)

(Computed)

Ot

S +

2 (m3 / s)

(from routing

curve)

O (m3 / s)

(from routing

curve)

Noon 2.0 5.6 9.0 1.7 (given)1

Midnight 5.2 8.2 12.8 2.3

Noon 10.1 14.9 23.5 4.32

Midnight 12.2 16.2 37.2 10.5

Noon 8.5 16.3 36.9 10.33

Midnight 4.7 16.3 29.5 6.6

Noon 2.3 23.3 4.2

9.5 Channel Routing or Streamflow Routing

The terms stream channel routing and flood routing are often used interchangeably.

This is attributed to the fact that most stream channel routing application are in flood flow

analysis, flood control design, or flood forecasting.Channel routing uses mathematical relations to calculate outflow from a stream

channel inflow. Channel reach refers to a specific length of stream channel possessing certain

translation and storage properties. The hydrograph at the upstream end of the reach is the

inflow hydrograph; the hydrograph at the down stream end is the outflow hydrograph.

Routing in natural river channel is complicated by the fact that storage is not a

function of outflow alone. Routing in stream requires a storage equation including inflow as a

parameter.

9.5.1 Muskingum Method

In the Muskingum method, storage is a linear function of inflow and outflow:

S = k [XI + ( 1 – X) O ] (9.4)

in which S = storage volume

I = inflow



113

O = outflow

k = a time constant or storage coefficient (dimension of time)

(ratio of storage to discharge).

X = a dimensionless weighting factor.

In fact, for X= 0,equation (9.4) reduces to S = k O. In other words, linear reservoir

routing is a special case of Muskingum channel routing for which X = 0.

In the Muskingum method, parameters k and X are determined by calibration using

streamflow records. X is interpreted as a weighting factor and restricted in the range 0.0 to

0.5.

If streamflow records are available, k and X are determined by plotting

S Vs. [XI+(1-X) O ] for various values of X with a trial and error procedure .

Best values of X is that which causes the data to plot most nearly as a straight line

with slope k, units of k depends on the units of flow and storage.

To derive the Muskingum routing equation, the continuity equation is repeated here:

Equation (9.4) is expressed at time levels 1 and 2:

S1 = k [XI1 + ( 1 – X) O1 ] (9.6)

S2 = k [XI2 + ( 1 – X) O2 ] (9.7)

Substituting equations (9.6) and (9.7) into equation (9.5) and solving for O2 the

resulting routing equation is

O2 = C0 I2+ C1 I1 +C2 O1 (9.8)

In which C0 ,C1 and C2 are routing coefficients defined in terms of t, k and x as

follows:

Example (9.1)

An inflow hydrograph to a channel reach is shown in Table . Assume base flow is

352m3/s. Using the Muskingum method, route this hydrograph through a channel reach with

k= 2d and X=0.1 to calculate an outflow hydrograph

)5.9(22

122121

S S t OO

t I I

−=

+−

+

1

5.0

5.0

5.0

5.0,5.0

5.0

210

2

10

=++

+−

−−=

+−

++=+−

−−=

C C C

and t kX k

t kX k C

t kX k

t kX

C t kX k

t kX

C



114

Time (d) 0 1 2 3 4 5 6

Inflow(m3/s) 352 587 1353 2725 4408.5 5987 6704

Time(d) 7 8 9 10 11 12 13

Inflow(m

3

/s) 6951 6839 6207 5346 4560 3861.5 3007Time (d) 14 15 16 17 18 19 20

Inflow(m3/s) 2357.5 1779 1405 1123 952.5 730 605

Time (d) 21 22 23 24 25

Inflow(m3/s) 514 422 352 352 352

Solution: Assume t = 1 day

k = 2d and X=0.1 (given)

therefore, routing coefficients are

C0 = 0.1304 ; C1 = 0.3044 ; and C2 = 0.5652

C0 + C1 + C2 = 1

Partial flows (m3/s)Time Inflow

(m3/s) C0 I2 C1 I1 C2 O1

Outflow

(m3/s)

0 352 352

1 587 76.6 107.1 199.0 382.7

2 1353 176.5 178.6 216.3 571.4

3 2725 355.4 411.8 323.0 1090.2

4 4408.5 575.0 829.4 616.2 2020.6

5 5987 780.9 1341.7 1142.1 3264.7

6 6704 874.4 1822.1 1845.3 4541.8

7 6951 906.7 2040.3 2567.1 5514.1

8 6839 892.0 2115.5 3116.7 6124.2

9 6207 809.6 2081.5 3461.5 6352.6

10 5346 697.3 1889.1 3590.6 6177.0

11 4560 594.8 1627.0 3491.4 5713.2

12 3861.5 503.7 1387.8 3229.2 5120.7

13 3007 392.2 1175.2 2894.3 5561.7

14 2357.5 307.5 915.2 2521.8 3744.5

15 1779 232.0 717.5 2116.5 3066.0



115

16 1405 183.3 514.4 1733.0 2457.7

17 1123 146.5 427.6 1389.1 1963.2

18 952.5 124.2 341.8 1109.6 1575.6

19 730 95.2 289.9 890.6 1275.720 605 78.9 222.2 721.0 1022.1

21 514 67.1 184.1 577.7 828.9

22 422 55.1 156.4 468.5 600.0

23 352 45.9 128.4 384.4 558.7

24 352 45.9 107.1 315.8 468.8

25 352 45.9 107.1 265.0 418.0

The outflow peak is 6352.6 m

3

/sThe inflow peak is 6951 m3/s

The peak outflow occurs at day 9,2 days after the peak inflow, which occurs at day 7. The

time elapsed between the occurrence of peak inflow and peak outflow is generally equal to k,

the travel time.

Example 9.2

Use the outflow hydrograph calculated in the pervious example together with the

given inflow hydrograph to calibrate the Muskingum method, that is, to find the routing

parameters k and X

Solution : Channel storage for S2

S2 = S1 + t/2 [ I1 + I2 – O1 - O2]

For X=0.1,0.2,0.3 trial value, the weighted flows [ X I + (1 – X) 0 ] are calculated, as shown

in table

Time

(day)

Inflow

(m3/s)

Outflow

(m3/s)

Storage

(m3/s) - day

Weighted Flow ( m3/ s )

X = 0.1 X=0.2 X = 0.3

0 952 352 0 - - -

1 587 382.7 102.2 403.0 423.5 443.9

2 1353 571.4 595.2 649.6 727.7 805.9

3 2725 1090.2 1803.4 1253.7 1417.2 1580.6

4 4408.5 2020.6 3814.7 2259.4 2498.2 2737.0

5 5987 3264.7 6369.8 3536.9 3809.2 4081.4

6 6704 4541.8 8812.1 4758.0 4974.2 5190.5



116

7 6951 5514.1 10611.6 5657.8 5801.5 5945.2

8 6839 6124.2 11687.5 6195.7 6267.2 6338.6

9 6207 6352.6 11972.1 6338.0 6323.5 6308.9

10 5346 6177.0 11483.8 6093.9 6010.8 5927.711 4560 5713.2 10491.7 5597.9 5482.6 5367.2

12 3861.5 5120.7 9285.5 4994.8 4868.9 4742.9

13 3007 4461.7 7928.5 4316.2 4170.8 4025.3

14 2357.5 3744.5 6507.7 3605.8 3467.1 3328.4

15 1779 3066.0 5170.7 2937.3 2808.6 2679.9

16 1405 2457.7 4000.8 2352.4 2247.2 2141.9

17 1123 1963.2 3054.4 1879.2 1795.2 1711.118 952.5 1575.6 2322.7 1513.4 1451.1 1388.7

19 730 1275.7 1738.2 1221.1 1166.6 1112.0

20 605 1022.1 1256.8 980.4 938.7 897.0

21 514 828.9 890.8 797.4 765.9 734.4

22 422 680.0 604.4 654.2 628.4 602.6

23 352 558.7 372.0 537.9 517.3 496.6

24 352 468.8 210.3 457.1 445.4 433.825 352 418.0 118.9 411.4 404.8 398.2

Figure (9.3) Calibration of Muskingum routing parameters.

K =2 days and X=0.1 are the Muskingum routing parameters for the given inflow and

chapter seven f l o o d s

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