---
title: "Total rainfall, runoff and peak discharge with phi-index"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:phi-index-50ha-2-5"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-50ha-2-5
site: BCT Engineering
---

# Total rainfall, runoff and peak discharge with phi-index

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2073 Shrawan, 6 marks
- Video: [Type 1: Phi index sum| when Precipitation & Runoff is given| find phi index & W index| Phi & W index](https://www.youtube.com/watch?v=IPbekfEYSH4) — Professor Nidhi Trivedi
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-50ha-2-5
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

For a storm of 3 hours on a 50 ha catchment, the rainfall rates are as follows:

| Time of rain from beginning (min) | 0 | 30 | 45 | 75 | 100 | 125 | 150 | 180 |
|---|---|---|---|---|---|---|---|---|
| Rainfall rate (cm/hour) | 0 | 2.5 | 3.5 | 2.0 | 4.8 | 5.2 | 1.8 | 5.3 |

If the $\phi$ index of this basin is 2.5 cm/hour, calculate total rainfall, runoff in (cm) and peak discharge.

## Short answer

Total rainfall = 10.69 cm, runoff = 3.73 cm, peak discharge = 3.89 m³/s.

## Answer

Each rate is taken as the rainfall rate during the interval ending at the stated time. Area $= 50$ ha $= 5\times10^5$ m².

| Interval (min) | Duration (h) | Rate (cm/h) | Rain (cm) | Excess rate (rate - 2.5) | Runoff (cm) |
|---|---|---|---|---|---|
| 0-30 | 0.5 | 2.5 | 1.25 | 0 | 0 |
| 30-45 | 0.25 | 3.5 | 0.875 | 1.0 | 0.25 |
| 45-75 | 0.5 | 2.0 | 1.0 | 0 | 0 |
| 75-100 | 0.4167 | 4.8 | 2.0 | 2.3 | 0.958 |
| 100-125 | 0.4167 | 5.2 | 2.167 | 2.7 | 1.125 |
| 125-150 | 0.4167 | 1.8 | 0.75 | 0 | 0 |
| 150-180 | 0.5 | 5.3 | 2.65 | 2.8 | 1.4 |
| **Total** |  |  | **10.69** |  | **3.733** |

**Total rainfall** $= 10.69$ cm.

**Runoff** $= \sum (i - \phi)\Delta t$ for $i > \phi$ $= 0.25 + 0.958 + 1.125 + 1.40 = 3.73$ cm.

**Peak discharge:** the maximum excess rate is $5.3 - 2.5 = 2.8$ cm/h (last interval):

$$
Q_{p} = \frac{0.028\ \text{m/h}\times5\times10^5\ \text{m}^2}{3600} = 3.89\ \text{m}^3/\text{s}
$$

**Answer: Total rainfall = 10.69 cm; runoff = 3.73 cm; peak discharge = 3.89 m³/s (for the direct runoff rate, ignoring catchment lag).**

## Check yourself

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

1. How is runoff found from a hyetograph with φ = 2.5 cm/h?
   A. Sum of all rainfall x φ
   B. Sum of (i - φ)Δt over intervals where i > φ
   C. Total rainfall minus φ x total time
   D. Average of all rates minus φ
   Answer: B. Only intervals with rate above φ give excess, which is (i - φ)Δt. (Key point: Runoff with φ-index)

2. Which intervals give no runoff (φ = 2.5 cm/h)?
   A. 75-100 and 100-125 min
   B. Only the last interval
   C. 30-45 and 150-180 min
   D. 0-30 (2.5), 45-75 (2.0) and 125-150 (1.8) min
   Answer: D. Rates equal to or below 2.5 cm/h have no excess: 2.5, 2.0 and 1.8. (Key point: Which intervals)

3. How is peak discharge found?
   A. Total runoff x area / 3 h
   B. φ x area
   C. Maximum rainfall rate (5.3 cm/h) x area
   D. Maximum excess rate (2.8 cm/h) x area, converted to m3/s
   Answer: D. Qp = 0.028 m/h x 5x10^5 m2 / 3600 = 3.89 m3/s, using the max excess rate 5.3 - 2.5. (Key point: Peak discharge)

4. Total rainfall of the storm is the sum of rate x duration. Its value is
   A. 8.2 cm
   B. 3.73 cm
   C. 15.9 cm
   D. 10.69 cm
   Answer: D. Summing the interval depths gives 10.69 cm; 3.73 cm is the runoff. (Key point: Total rainfall)

## Related in Engineering Hydrology

- [Surface runoff with initial loss and phi-index, 100-min storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Asurface-runoff-initial-loss-0-6) — Engineering Hydrology, chapter 3
- [Total rainfall, net runoff, W-index of 140-min storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-w-index-140min-storm) — Engineering Hydrology, chapter 3, asked 2×
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- [Phi-index of storm, 20 cm rain, 12 cm runoff](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-20cm-12cm) — Engineering Hydrology, chapter 3
- [Total rainfall and phi-index, 100 km2 basin](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Atotal-rainfall-phi-100km2) — Engineering Hydrology, chapter 3
- [Phi-index and W-index of isolated storm, 500 ha](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-w-index-500ha) — Engineering Hydrology, chapter 3, asked 2×

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