An Introduction to

FLOOD

Mrinmoy Majumder

www.baipatra.ws

Definition

• High stage in river when the river overflows and inundates the adjoining area is a situation described as Flood

• Flood peak and frequency of the peak is an important consideration in hydraulic design

• Magnitude and time of the flood varies with change in watershed characteristics

• Peak flood depends on rainfall, discharge and watershed area and type

• Magnitude of flood can be estimated by– Rational method– Empirical method– Unit hydrograph technique– Flood frequency studies

Rational Method

• Assumptions :– Area of watershed must be less than 50 Sq.Km– Rainfall continues beyond time of concentration– Catchment is homogeneous

• The equation for peak flood, according to rational method :

Qp = C A iQp is the peak discharge, C is coefficient of runoff (runoff/rainfall), i is rainfall intensity

which depends on time of concentration and exceedance probability, A is drainage area in Sq.Km

Time of Concentration

• Rainfall intensity is found to be a function of time of concentration (tc) and an exceedance probability P

• tc can be expressed as :

tp = CtL(LLca/(S)1/2)n

where,

C is a constant,L is the distance from farthest point of the catchment to basin divide,Lca is the distance of gage from centroid of the watershed,S is slope between farthest point and outlet,n is the manning’s constant

tc = 0.01947L0.77S-0.385

where,time of concentration (minutes),L is maximum length from farthest point to outlet and S is slope of catchment(from highest point to lowest point)

Rainfall Intensity

• The rainfall intensity is thus represented by :

itc,p = KTx/(tc+a)m

K,a,x and m are constants which can be collected from frequency duration curves

Empirical Formula

Characteristics of Empirical Formulae are :

• Regional formula• Based on correlation• Between flow(Qp) and catchment properties• Almost all the formula represent discharge as a

function of Area• Neglects flood frequency• The reason why empirical formulas are all regional

and gives approximate results when applied to other regions

Dickens Formula

Qp = CDA3/4

CD = Dickens Formula with value between 6 - 30

Ryves Formula

Qp = CRA2/3

CR = Ryves Formula with value between 6.8 – 10.2

Inglis Formula

Qp = (124A)/(A+10.4)1/2

For Western Ghats in Maharashtra

Fullers Formula

QTp = (CfA0.8)/(1+0.8logT)

For USA,T = return period, Cf is a constant =0.18 – 1.88.

Baird and Mcillwraith(1951)

Qmp = (3025A)/(278+A)0.78

From maximum rrecorded floods throughout the world

Estimation of Flood from Frequency Analysis

• In statistics the meaning of Analysis and Synthesis is different

• Analysis is the process of establishing a relation between a variate (extreme events) and its occurrence (probability of the event).

• Synthesis on the other hand can be explained as a process to estimate the magnitude of the variate or probability from its mean and standard deviation i.e. characteristic data set.

• In flood frequency analysis relationship between flood and its occurrence is first estimated and then magnitude of the event or occurrence probability of the event is estimated. To estimate the relationship between flood and its occurrence, following steps are adopted :

– Probable density function (PDF) is determined from the available set of data– Simple moments like mean, standard deviation and variance is calculated– Plotting position of data points are estimated which gives the validation of the

PDF.

Analysis .Step.1 : Probability Density Function

Gumbels Method

Where the distribution has 

mode α, 

mean α+γβ (where γ=0.5772156649... is Euler's constant),

and

variance ⅙β2π2

If f(x) is probability density function, then, Gumbel’s distribution is given by :

Gumbel’s Distribution Explained• There are essentially three types of Fisher-Tippett extreme value

distributions.

• The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions.

• The term "Gumbel distribution" is used to refer to the distribution corresponding to a minimum extreme value distribution (i.e., the distribution of the minimum  ).

• The Gumbel distribution has a location parameter α  and scale parameter  β

Normal Distribution

= variate

= mean

= variance

P(x) and D(x) is respectively the probability density function and distribution function

Log-Normal Distribution

• Normal distribution of the Logarithm of Variate.

• Mean and standard deviation is estimated with the help of logarithm of the variates.

• The probable density function and distribution function is adjusted accordingly.

Pearson Type III Distribution

= gamma function

= variate

= mean =

= variance =

P(x) and Φ(t) is respectively the probability density function and characteristic function

= kurtosis =

= kurtosis =

Log-Pearson Distribution

• Pearson distribution of the Logarithm of the Variates.

• Mean and standard deviation is estimated with the help of logarithm of the variates.

• The probable density function is adjusted accordingly.

Analysis.Step.2 : Validation of the PDF by Estimation of Plotting Position of the Event or Plotting Position Probability

Plotting Position Probability(PPP)

P = m/(N+1)

T= 1/P

Where P is the probability of an extreme event and T is the return period of that event. P rn is the occurrence of the event for r times in n successive years

m is the rank of the event with respect to other events of the year or decade or greater time span.

N = total number of events in the dataset

C =combination of n and r and q = 1-P

Prn = nCrPrqn-r

T= 1/P

Bulletin 17B PPP(InterAgency Advisory Committee on Water Data,1982)

P = (i-a)/(n-a-b+1)

T= 1/P

Where P is the exceedance probability of an extreme event and T is the return period of that event. i is the rank of the event with respect to other events of the year or decade or greater time span.

n = total number of events in the dataset

a and b are constants = f(PDF)

Weibull PPP for Uniformly Distributed Dataset

P = i/(n+1)

T= 1/P

Where P is the exceedance probability of an extreme event and T is the return period of that event. i is the rank of the event with respect to other events of the year or decade or greater time span.

n = total number of events in the dataset

Uniformly distributed dataset means a = b = 0

Hazen PPP for Uniformly Distributed Dataset

P = i-0.5/n

T= 1/P

Where P is the exceedance probability of an extreme event and T is the return period of that event. i is the rank of the event with respect to other events of the year or decade or greater time span.

n = total number of events in the dataset

Cunnane PPP for Uniformly Distributed Dataset

P = (i-0.4)/(n+0.2)

T= 1/P

Where P is the exceedance probability of an extreme event and T is the return period of that event. i is the rank of the event with respect to other events of the year or decade or greater time span.

n = total number of events in the dataset

Synthesis i.e. Determination of the Magnitude

General Equation of Hydrologic Frequency Analysis(Chow,1951)

xT = x’ + K.σ

Where xT is the value of the flood in T return period

x’ is the mean and σ is the standard deviation

K = f (T, frequency distribution) = frequency factor

You can also apply Gumbel’s Method.

Mathematical Model

x = x’ + K.σ

Where x is the value of the flood

x’ is the mean and σ is the standard deviation

K = f (T, frequency distribution) = frequency factor

You can also apply Gumbel’s Method.

K =(x - x’)/σ

Note : If x is known then use Eqn.A and if K is known then use Eqn.B

….A ….B

Note• If only the peak discharge is known, the distribution of the parameter

can be estimated from the probability density function or distribution function.

• The function can be estimated by different methods and in different manner.

• The selection of method will depend on error or difference between actual and estimated curve, location of the basin and time of concentration.

• In hydrologic problems Normal, Log-normal and Pearson and Log-Pearson Type III distribution function is used among which Log-Pearson Type III is recommended by Bulletin 17B of USA.

Adjustment for Urbanization

Rational Equation(modified for urbanization)

Qp = 484.1A0.723(1+U)1.516PE1.113Tr

0.403

A = drainage areaU = fraction of imperviousnessPE = volume of excess rainfallTr = duration of rainfall excess

Adjustment for Urbanization

Adjusted peak discharge due to observed impervious factor

= Observed Peak Discharge multiplied by adjustment factor of the observed imperviousness

Urban Adjustment Factor(Leopold,1968)

Probable Maximum Precipitation

• PMP as the greatest depth of precipitation for a given duration that is meteorologically possible over a given station or a specified area.

• There are two major methods available to estimate value of PMP.

• The first is the meteorological method in which the PMP for different durations over an area is determined by transposition and maximisation of major historical rainstorms.

• The second is a statistical method where the estimates of PMP are derived from frequency analysis of the annual maximum rainfall series for different durations.

• The statistical method is useful when there is insufficient meteorological data to apply storm maximization method.

• A statistical method for estimating PMP for small areas has been developed by Hershfield, 1961 and Hershfield, 1965 based on a general frequency equation given by Chow (1951).

Hershfield Method

where XPMP is the PMP for a given station for a specific duration and X¯n and σn are the mean and standard deviation for a series of n annual maximum rainfall values of a given duration respectively. Km is the frequency factor and is the largest of all the calculated K values for all stations in a given area.

Frequency Factor Calculation

where X1, X¯n−1 and σn−1 are the highest value, mean and standard deviation respectively excluding the X1 value from the series.

Homogeneity of Dataset• The use of a longer series in the estimation of PMP is appropriate only if

they show no significant increasing or decreasing trends. The presence or absence of trends in the annual maximum rainfall series for the rainfall stations used were investigated using the Mann–Kendall rank statistics test (WMO, 1966).

the ratio of τ to the στ gives an indication of trend in the data. For no trend in the data series, this value should lie within the limits of ± 1.96 at the 5% level of significance.