4. Estimation of Peak discharge

Estimating the magnitude of a flood peak, or design discharge, is a crucial process for adequately designing hydraulic structures such as dams, spillways, bridges, and culverts to safely accommodate extreme flow conditions. The choice of estimation method depends on the project's importance, the desired objective, the availability of data, and the size of the catchment area.

The primary methods for peak discharge estimation include:

1. Rational Method

The Rational Method is a simple and widely used approach for estimating peak runoff, primarily restricted to small-size catchments (typically less than 50 km²) where the time of concentration is relatively short.  It relies on the formula $Q_p=CAi$, where:

• $Q_p$ is the peak design discharge.
• $C$ is a dimensionless runoff coefficient that represents the integrated effect of catchment losses, surface slope, and land use.
• $i$ is the rainfall intensity for a duration equal to the catchment's time of concentration.
• $A$ is the area of the watershed.

2. Empirical Formulae

Empirical formulae are regional equations based on the statistical correlation of observed flood peaks with catchment properties, most commonly just the drainage area. Because they often omit flood frequency as a parameter, they are specific to the regions where they were developed and give approximate values. Common empirical formulae used in India include:

• Dickens Formula ($Q_p=C_D{A}^{3/4}$): Used primarily in the central and northern parts of India.
• Ryves Formula ($Q_p=C_R{A}^{2/3}$): Originally developed for the Tamil Nadu region and also used in parts of Karnataka and Andhra Pradesh.
• Inglis Formula ($Q_p=\frac{124A}{\sqrt{A+10.4}}$): Based on flood data from catchments in the Western Ghats of Maharashtra.
• Envelope Curves: In regions with meager data, recorded maximum floods are plotted against catchment areas on a log-log scale. An enveloping curve is drawn to encompass all points, providing a quick, rough estimation of peak values.

3. Unit-Hydrograph Method

This method is normally restricted to moderate-size catchments (areas less than 5000 km²). It predicts the peak-flood hydrograph by operating a "design storm" (such as a Standard Project Storm or Probable Maximum Precipitation) on the known or derived unit hydrograph of the catchment. The analysis takes into account the rainfall producing the flood as well as the specific infiltration characteristics of the area.

4. Flood-Frequency Studies

Hydrologic processes like floods are exceedingly complex, making them difficult to model purely analytically. Flood-frequency studies overcome this by applying statistical analysis to historical flow records. The annual maximum flood values over many successive years are compiled into a hydrologic data series. By using specific probability distributions—such as Gumbel's extreme value distribution, log-Pearson type III, or log-normal distribution—engineers can estimate the magnitude of a flood peak that corresponds to a specific return period (e.g., a 1-in-100-year flood).

5. SCS Curve Number Method

For small to medium-sized watersheds, the Soil Conservation Service (SCS) Curve Number method is often used to estimate peak discharge and direct runoff. It evaluates runoff potential based on a dimensionless "Curve Number" (ranging from 30 to 100) that is determined by classifying the watershed's land use, hydrologic soil group (A, B, C, or D based on infiltration rates), and the antecedent moisture conditions prior to the rainfall event.

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Example A: Simple Direct Calculation

• Problem: Calculate the design discharge for a watershed with an area ($A$) of $100{\text{km}}^2$, a rainfall intensity ($i$) of $5\text{cm/hour}$, and a runoff coefficient ($C$) of $0.5$.
• Step 1: Convert rainfall intensity from cm/hour to m/s. $i=\frac{5}{100\times 3600}=1.388\times 10^{-5}\text{ m/s}$.
• Step 2: Convert the area to ${\text{m}}^2$ ($100{\text{km}}^2=100\times 10^6{\text{ m}}^2$) and apply the formula $Q=C\times i\times A$. $Q=0.5\times (1.388\times 10^{-5})\times (100\times 10^6)$.
• Answer: $Q=694{\text{m}}^3\text{/s}$.

Example B: Incorporating Time of Concentration ($t_c$) and Intensity Interpolation

• Problem: An urban catchment has an area of $85\text{ha}$ ($0.85{\text{km}}^2$), a slope of $0.006$, and a maximum travel length of $950\text{m}$. Assume a runoff coefficient ($C$) of $0.3$. Calculate the 25-year return period peak flow, given that the 25-year maximum rainfall depth is $40\text{mm}$ for $20\text{min}$ and $50\text{mm}$ for $30\text{min}$.
• Step 1: Find the time of concentration ($t_c$) using the Kirpich equation ($t_c=0.01947L^{0.77}{S}^{-0.385}$). $t_c=0.01947\times (950{)}^{0.77}\times (0.006{)}^{-0.385}=27.4\text{minutes}$.
• Step 2: Interpolate the maximum rainfall depth for the calculated $t_c$ of $27.4\text{minutes}$ using the provided $20\text{min}$ and $30\text{min}$ depth data. $\text{Depth}=40+\frac{(50-40)}{10}\times 7.4=47.4\text{mm}$.
• Step 3: Calculate the average rainfall intensity ($i$). $i=\frac{47.4\text{mm}}{27.4\text{min}}\times 60\text{min/h}=103.8\text{mm/h}$.
• Step 4: Calculate the peak discharge using the modified formula $Q_p=\frac{CiA}{3.6}$. $Q_p=\frac{0.30\times 103.8\times 0.85}{3.6}$.
• Answer: $Q_p=7.35{\text{m}}^3\text{/s}$.

Example C: Calculating an Equivalent Runoff Coefficient ($C_e$)

• Problem: If the $85\text{ha}$ catchment from Example B is non-homogeneous, calculate the equivalent runoff coefficient ($C_e$) given the following land uses: Roads ($8\text{ha},C=0.70$), Lawn ($17\text{ha},C=0.10$), Residential area ($50\text{ha},C=0.30$), and Industrial area ($10\text{ha},C=0.80$).
• Step 1: Apply the weighted average formula $C_e=\frac{\sum C_i{A}_i}{A}$. $C_e=\frac{(0.70\times 8)+(0.10\times 17)+(0.30\times 50)+(0.80\times 10)}{8+17+50+10}$.
• Step 2: Solve the equation. $C_e=\frac{30.3}{85}$.
• Answer: $C_e=0.36$.

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2. Empirical Formulae Numericals

Empirical formulas are regional equations that use catchment area ($A$, in ${\text{km}}^2$) and regionally specific constants to estimate peak flow.

Problem: Estimate the maximum flood flow for a catchment area of $40.5{\text{km}}^2$ across different geographical regions using the appropriate empirical formulas.

• Scenario 1: Western Ghat area, Maharashtra (Inglis Formula)
  • Formula: $Q_p=\frac{124A}{\sqrt{A+10.4}}$.
  • Calculation: $Q_p=\frac{124\times 40.5}{\sqrt{40.5+10.4}}$.
  • Answer: $Q_p=704{\text{m}}^3\text{/s}$.
• Scenario 2: Gangetic Plain (Dickens Formula)
  • Formula: $Q_p=C_D{A}^{3/4}$ (where the coefficient $C_D=6.0$ is recommended for Northern plains).
  • Calculation: $Q_p=6.0\times (40.5{)}^{0.75}$.
  • Answer: $Q_p=96.3{\text{m}}^3\text{/s}$.
• Scenario 3: Cauvery Delta, Tamil Nadu (Ryves Formula)
  • Formula: $Q_p=C_R{A}^{2/3}$ (where $C_R=6.8$ is used for areas within $80\text{km}$ from the east coast).
  • Calculation: $Q_p=6.8\times (40.5{)}^{2/3}$.
  • Answer: $Q_p=80.2{\text{m}}^3\text{/s}$.
• Scenario 4: Maximum World Flood Experience (Baird & McIllwraith Envelope Curve)
  • Formula: $Q_{mp}=\frac{3025A}{(278+A{)}^{0.78}}$.
  • Calculation: $Q_{mp}=\frac{3025\times 40.5}{(278+40.5{)}^{0.78}}$.
  • Answer: $Q_{mp}=1367{\text{m}}^3\text{/s}$.