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**Floods, Flood Routing and Flood Control**

A flood an unusually high stage in a river, normally the level at which the river overflow its banks and inundates the adjoining area. The design of bridges, culvert waterways and spillways for dams and estimation of 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

**Rational Method**

If t_{p}≥ t_{c}

Q_{p}= 1 / 36.k.P_{c}.A

Where, Q_{p}= Peak discharge in m^{3}/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**Empirical Formulae****(i)****Dickens Formula (1865)**

Where, Q_{p}= Flood peak discharge in m^{3}/sec

A = Catchment area in km^{2}.

C_{D}= Dickens constant, 6 ≤ C_{D}≤ 30.**(ii)****Ryve’s formula (1884)**

Q_{P}= C_{R}.A^{2/3}

Where,

C_{H}= Ryve’s constant

= 8.8 for constant area within 80 km from the cost.

= 8.5 if 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 Km^{2}.

Q_{P}= Peak discharge in m^{3}/sec.**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:**(iv)****Reliability:**(probability of non-occurrence /Assurance) = q^{n}**(v)**Risk = 1 - q^{n}→ Risk = 1(1 - P)^{n}**(vi)**Safety Factory = Design values of hydrologic parameter adopted / Estimated value of hydrological parameter**(vii)**Safety Margin = design value of hydrological parameter – Estimated value of hydrological parameter

**Gumbel’s Method**

The extreme value distribution was 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.

Gunbel 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 probability distribution.

Where, X_{T} = Peak value of hydrologic data

K = Frequency factor

y_{T} = Reduced variate

y_{r }= -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 → ∞

**Confidence Limit**

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 x_{1} and x_{2} given by

X_{2 }/ X_{1} = 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

**Flood Routing**

Flood routing is the technique of determining the flood hydrograph at a section of a river by utilizing the data of flood flow at one or more upstream sections. The hydrologic analysis of problems such a flood forecasting. Flood protection Reservoir design and spillway design invariable includes flood routing.

**Prism Storage:**it is the volume that would exist if the uniform flow occurred at the downstream depth. i.e., the volume formed by an imaginary plane parallel to the channel bottom drawn at the outflow section water surface.**Wedge Storage:**it is the wedge like volume formed between the actual water surface profile and the top surface of the prism storage.**Flood Routing**

**Muskingum Method**

S = S_{p} + S_{w}

Where, S = Total storage in the channel.

S_{p} = Prism storage

= if (Q) = function of outflow discharge.

S_{w} = Wedge storage

= f(I) = function of inflow discharge.

S = f(Q)+f(I)S=k[XI^{m} + (1-X)Q^{m}]

Where, X = Weighting factor

M = Constant = 0.6 for rectangular channels

= 1.0 for nature channels

K = storage time constant

**Method of Channel Routing****Muskingum Method:** Hydrologic channel Routing

where, ΔS → Change in storage in time Δt

Δt → Time interval at which observations are taken. (Routing interval)

i → Avg. in flow rate over the period Δt

Average outflow rate over time period Δt.

- S1 = k[XI
_{1}+ (1 - X)Q_{1}] - S
_{2}= k[XI_{2}+ (1 - X)Q_{2}] - (S
_{2 }- S_{1}) = k[X(I_{2 }- X_{1})+(1 - X)(Q_{2 }- Q_{1})] - Q
_{1}= Q_{1} - Q
_{2}= C_{0}I_{2}+ C_{1}I_{1 }+ C_{2}Q_{1} - Q
_{n}= C_{0}I_{n}+ C_{1}I_{n-1}+ C_{2}Q_{n-1}

Where, C_{0}, C_{1} and C_{2}, are Muskingum constant

- for best result, 2Kx < Δt < k

**Synthetic Hydrograph**

**Synder’s Method:** Synder (1938), based on a study of a large number of catchment in the Appalachian Highlands of eastern United states developed a set of empirical equations for synthetic unit hydrograph in those areas .These equations are in use in the USA. And with some modifications in many other countries, and constitute what is known as Synder’s synthetic unit hydrograph.**(i)** t_{P} = C_{t}[L.L_{C}_{a}]^{0.3}

Where, t_{p} = Time interval between mid-point of unit rainfall excess and peak of unity hydrograph in hour

L = Length of main stream

L_{Ca }= The distance along the main stream from the basin outlet to a point on the stream which is nearest to the centrod of basis (in KM)

C_{t} = Regional constant 0.3 < C_{t} 0.6* S* = Basin slope.

N = Constant = 0.38.**(iii)** t_{r} = t_{p }/ 5.5 t_{r} = Standard duration of U.H in hour**(iv)** Q_{PS} = 2.78C_{P}A / t_{P}

Where, C_{p} = Regional constant = 0.3 to 0.92.

A = Area of catchment in km^{2}.

Q_{PS} = Peak discharge in m^{3}/s.

where, t_{R} = standard rainfall duration.

t_{P} = Basin lag for non-standard U.H.**(vi)** Q_{p} = 2.78C_{P}A / t_{P}**(vii) **t_{B} = (72 + 3t_{P}) hour, for a large catchment.

Where, t_{B} = Base time of synthetic U.H

for small catchment.**(viii) **W_{50} = 5.87 / (q)^{1.08} W_{50} = width of U.H in hour at 50% peak discharge.**(ix)** W_{75} = W_{50 }/ 7.15 W_{75} = Width of U.H in hours at 75% peak discharge.**(x) **q = Q_{P }/ A where, Q_{P} = Peak discharge in m^{3}/sec.

A = Area in km^{2}.

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