---
title: "Phi-index and rainfall excess duration, 0.340 Mm3 runoff"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:phi-index-500ha-0-340mm3"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-500ha-0-340mm3
site: BCT Engineering
---

# Phi-index and rainfall excess duration, 0.340 Mm3 runoff

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2080 Bhadra, 7 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-500ha-0-340mm3
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

The mass curve of an isolated storm in a 500-ha watershed is as follows:

| Time from start (hr) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|
| Cumulative rainfall (cm) | 0 | 0.8 | 2.6 | 2.8 | 4.1 | 7.3 | 10.8 | 11.8 | 12.4 | 12.6 |

If the direct runoff produced by the storm is measured at the outlet of the watershed as 0.340 Mm$^3$, estimate the $\phi$-index of the storm and duration of the rainfall excess.

## Short answer

φ-index = 0.40 cm/h and duration of rainfall excess = 10 h. Runoff depth was 6.8 cm; five 2-h intervals had rain exceeding the loss rate of 0.8 cm per interval, satisfying total excess runoff.

## Answer

#### Runoff depth
$$
R = \frac{V}{A} = \frac{0.340\times10^{6}\ \text{m}^3}{500\times10^{4}\ \text{m}^2} = 0.068\ \text{m} = 6.8\ \text{cm}
$$

#### Rainfall in each 2-h interval

| Interval (h) | 0-2 | 2-4 | 4-6 | 6-8 | 8-10 | 10-12 | 12-14 | 14-16 | 16-18 |
|---|---|---|---|---|---|---|---|---|---|
| Rain (cm) | 0.8 | 1.8 | 0.2 | 1.3 | 3.2 | 3.5 | 1.0 | 0.6 | 0.2 |

Total rain $P = 12.6$ cm, so total loss $= 12.6 - 6.8 = 5.8$ cm.

#### Trial for $\phi$
Let the loss per 2-h interval be $\phi \times 2$. Try intervals with rain above the loss: 2-4, 6-8, 8-10, 10-12, 12-14 (5 intervals). Their rain $= 1.8 + 1.3 + 3.2 + 3.5 + 1.0 = 10.8$ cm.

$$
10.8 - 5\times(2\phi) = 6.8 \;\Rightarrow\; \phi = 0.40\ \text{cm/h}
$$

Check: loss per interval $= 0.8$ cm. The five intervals have rain $> 0.8$ cm (the smallest is 1.0), and the other intervals (0.8, 0.2, 0.6, 0.2) have rain $\le 0.8$ cm, so they give no excess. The assumption is correct.

| Interval | Excess (cm) |
|---|---|
| 2-4 | 1.0 |
| 6-8 | 0.5 |
| 8-10 | 2.4 |
| 10-12 | 2.7 |
| 12-14 | 0.2 |
| Total | 6.8 |

#### Duration of rainfall excess
Five intervals of 2 h each produce excess: $5\times2 = 10$ h (periods 2-4 h and 6-14 h).

**Answer: $\phi$-index = 0.40 cm/h; duration of rainfall excess = 10 h.**

## Check yourself

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

1. What is the runoff depth from 0.340 Mm3 over 500 ha?
   A. 6.8 cm
   B. 0.68 cm
   C. 3.4 cm
   D. 68 cm
   Answer: A. 500 ha = 5x10^6 m2; 0.34x10^6/5x10^6 = 0.068 m = 6.8 cm. (Key point: Runoff depth)

2. How is phi found?
   A. Assume which intervals give excess, then solve (rain of those intervals - n x loss per interval) = runoff
   B. Use the average rainfall
   C. Use the maximum interval rain
   D. Divide total loss by 18
   Answer: A. 10.8 - 5(2 phi) = 6.8 gives phi = 0.40 cm/h, then verify the assumption. (Key point: Trial method for phi)

3. Why must the assumption be checked?
   A. It need not be checked
   B. To find initial loss
   C. To confirm the total rain
   D. Intervals included must have rain above phi x 2 = 0.8 cm and others must not
   Answer: D. Intervals with rain below the loss produce no excess, so they must be excluded consistently. (Key point: Check of assumption)

4. What is the duration of rainfall excess?
   A. 18 h
   B. 2 h
   C. 10 h (five 2-h intervals)
   D. 6 h
   Answer: C. Five intervals (2-4 and 6-14 h) produce excess, 5 x 2 = 10 h. (Key point: Excess duration)

## Close variants set in other exams

- [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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