---
title: "Rational and empirical flood equations"
subject: "Engineering Hydrology"
chapter: 6
key: "semester_5/engineering_hydrology:rational-empirical-eqns"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rational-empirical-eqns
site: BCT Engineering
---

# Rational and empirical flood equations

Engineering Hydrology · Chapter 6: Flood Hydrology

- Asked in: 2078 Kartik, 6 marks
- Video: [Flood: Rational method and empirical formula](https://www.youtube.com/watch?v=WAGe8-HCFMU) — RUBIA KHAN
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rational-empirical-eqns
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6

## Question

Write the equations of flood prediction by rational and empirical methods. Also write the limitations and appropriate uses of these equations.

## Short answer

Rational: Qp = 0.278 C i A (A in km²). Empirical: Dickens Q = CD A^(3/4), Ryves Q = CR A^(2/3), Inglis Q = 124A/√(A+10.4), Fuller Q = Cf A^0.8(1+0.8 log T), WECS for Nepal. Rational limited to small catchments (<50 km²), gives only peak; empirical methods are regional, ignore rainfall intensity and catchment characteristics.

## Answer

#### Rational method
$$
Q_p = \frac{C\,i\,A}{360}\quad(\text{A in ha, i in mm/h, Q in m}^3/\text{s})\qquad\text{or}\qquad Q_p=0.278\,C\,i\,A\ (A\text{ in km}^2)
$$

i = intensity for duration $t_c$ and return period T; C = runoff coefficient.

- **Limitations:** only small catchments (up to about 50 km²); uniform rainfall and constant C assumed; gives only the peak; ignores storage and antecedent moisture; needs IDF curves.
- **Appropriate use:** design of urban drainage, culverts, small bridges, storm sewers and spillways of small catchments.

#### Empirical methods
Based on regional observations, the flood is a function of catchment area (and sometimes return period):

1. **Dickens (north India):** $Q = C_D\,A^{3/4}$ (A in km²; $C_D$ = 11 – 14 for the plains of north India, 14 – 28 for the hilly regions of north India, and 22 – 28 for central India).
2. **Ryves (south India):** $Q = C_R\,A^{2/3}$ ($C_R$ = 6.8 – 10.1 depending on distance from the coast).
3. **Inglis (Maharashtra, ghat areas):** $Q = \dfrac{124A}{\sqrt{A+10.4}}$
4. **Fuller (USA):** $Q_T = C_f\,A^{0.8}(1+0.8\log T)$
5. **Nepal (WECS/DHM method):** $Q_2 = 1.8767\,(A_{<3000}+1)^{0.8783}$ and $Q_{100}=14.63\,(A_{<3000}+1)^{0.7342}$, where $A_{<3000}$ is the catchment area below 3000 m elevation (km²), used for rivers without records.

- **Limitations:** valid only for the region and range of area where the constants were derived; they ignore rainfall intensity, slope, shape and land use of the particular catchment; the constants are subjective; the return period is not known (except Fuller and WECS).
- **Appropriate use:** quick, approximate estimates for preliminary design or for checking other methods, mainly in ungauged basins; not for final design of major structures.

Final design of large projects should use flood-frequency analysis (Gumbel, log-Pearson) or the unit hydrograph method with the observed data.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Dickens' empirical formula is:
   A. Q = C_D A^(2/3)
   B. Q = C A^2
   C. Q = 124A/sqrt(A+10.4)
   D. Q = C_D A^(3/4)
   Answer: D. Dickens: A^(3/4); Ryves: A^(2/3); Inglis: 124A/sqrt(A+10.4). (Key point: Dickens formula)

2. A main limitation of empirical flood formulae is:
   A. They are valid only for the region where the constants were derived
   B. They need IDF curves
   C. They give full hydrographs
   D. They are accurate for any catchment
   Answer: A. They ignore rainfall intensity, slope and land use and are regional. (Key point: Empirical limitation)

3. Empirical formulae are best used for:
   A. Final design of major dams
   B. Quick approximate estimates in ungauged basins and preliminary design
   C. Estimating flood volume
   D. Replacing frequency analysis for gauged basins
   Answer: B. Final design of large projects should use frequency analysis or the unit hydrograph. (Key point: Appropriate use)

4. The rational method is appropriate for:
   A. Large river basins
   B. Reservoir routing
   C. Urban drainage, culverts and small catchments
   D. Snowmelt floods
   Answer: C. It suits small catchments up to about 50 km2. (Key point: Rational use)

## Related in Engineering Hydrology

- [Different methods of flood prediction](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aflood-prediction-methods) — Engineering Hydrology, chapter 6
- [Rational method and its limitations](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arational-method-limits) — Engineering Hydrology, chapter 6, asked 3×
- [Empirical flood formula, 100 sq km catchment](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aempirical-flood-100sqkm) — Engineering Hydrology, chapter 6
- [Rational method peak flow, Melamchi storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arational-melamchi) — Engineering Hydrology, chapter 6
- [Slope-area flood discharge, 6 km reach](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-60-45) — Engineering Hydrology, chapter 4
- [Gumbel flood at given risk, 50-year project life](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-risk-63-58-4200) — Engineering Hydrology, chapter 6

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