Flood-Peak Estimation - Engineering Hydrology - Civil Engineering (CE)

Introduction

A flood is an unusual high stage in a river, normally the level at which the river overflows its banks and inundates the adjoining area. The design of bridges, culvert waterways and spillways for dams and estimation of the score at a hydraulic structure are some examples wherein flood-peak values are required.
To estimate the magnitude of a flood peak the following alternative methods are available:

• Rational method
• Empirical method
• Unit-hydrograph technique
• Flood-frequency studies

1. Rational Method
   The most realistic way to use the Rational Method is to consider it as a statistical link between the frequency distribution of rainfall and runoff. As such, it provides a means of estimating the design flood of a certain return period, with the rainfall duration equal to the time of concentration
   If t_p ≥ t_c
   Q_P = 1.36.k.P_c.A
   Where,
   Q_p = Peak discharge in m3/sec
   P_C = Critical design rainfall in cm/hr
   A = Area catchment in hectares
   K = Coefficient of runoff.
   t_D = Duration of rainfall
   t_C = Time of concentration
2. Empirical Formulae
   (i) Dickens Formula (1865)
   Q_V = C_D.A^3/4
   Where,
   Q_p = Flood peak discharge in m³/sec
   A = Catchment area in km².
   C_D = Dickens constant, 6 ≤ C_D ≤ 30.
   (ii) Ryes formula (1884)
   Q_P = C_R.A^2/3
   Where,
   C_H = Ryes constant
   = 8.8 for the constant area within 80 km from the cost.
   = 8.5 if the distance of area is 80 km to 160 km from the cost.
   = 10.2 if area is Hilley and away from the cost.
   (iii) Inglis Formula (1930)
   
   Where, A = Catchment area in Km².
   Q_P = Peak discharge in m³/sec.
3. Flood Frequency Studies
   (i) Recurrence interval or return Period:
   T = 1 / P where, P = Probability of occurrence
   (ii) Probability if non-occurrence: q = 1 - P
   (iii) Probability of an event occurring r times in ‘n’ successive years: = ⁿC_r x p^r x q^n-r
   (iv) Reliability: (probability of non-occurrence / Assurance) = qⁿ
   (v) Risk = 1 - qⁿ
   = 1 - (1 - P)ⁿ
   (vi) Safety Factor = Design values of hydrologic parameter adopted / Estimated value of hydrological parameter
   (vii) Safety Margin = Design value of the hydrological parameter – Estimated value of the hydrological parameter

Gumbel’s Method

The extreme value distribution was an introduction by Gumbel (1941) and is commonly known as Gumbel’s distribution. it is one of the most widely used probability distribution functions for extreme values in hydrologic and meteorologic studies for prediction of flood peaks, maximum rainfall, maximum wind speed.
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.
Based on the probability distribution.
P(x ≥ x₀) =

Where, X_T = Peak value of hydrologic data
K = Frequency factor

y_T = Reduced variate
y_T = -log_∈log_∈(T/T-1)
T = Recurrence interval in year
y_n = Reduced mean = 0.577
S_n = Reduced standard deviation.
S_n = 1.2825 for N → ∞

Since the value of the variate for a given return period, xT determined by Gumbel’s method can have errors due to the limited sample data used. An estimate of the confidence limits of the estimates is desirable the confidence interval indicates the limits about the calculated value between which the true value can be said to lie with specific probability based on sampling errors only.
For a confidence probability c, the confidence interval of the variate x_T is bounded by value x₁ and x₂ given by
X₂ / X₁ = X_T ± f(c) . S_∈
Where, f(c) is a function of confidence probability ‘C’.

S_e = Probability error
Where, N = Sample size
B = factor
σ = Standard deviation