---
title: "Phi-index of storm, 20 cm rain, 12 cm runoff"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:phi-index-20cm-12cm"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-20cm-12cm
site: BCT Engineering
---

# Phi-index of storm, 20 cm rain, 12 cm runoff

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2068 Chaitra, 4 marks
- Video: [5. Calculate Phi-Index Value from Storm Rainfall and Runoff](https://www.youtube.com/watch?v=t3mY6wGOzZk) — Dhrubo Raj Roy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-index-20cm-12cm
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

A storm with 20 cm of precipitation produced a surface runoff of 12 cm. Estimate the $\Phi$ index of the storm.

| Storm time (hr) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Rainfall (cm/hr) | 0.7 | 2.2 | 3.0 | 4.6 | 3.6 | 3.2 | 2.0 | 0.7 |

## Short answer

Total precipitation is 20 cm, runoff 12 cm, so loss is 8 cm. Excluding the two 0.7 cm/h hours, trial gives φ-index = 1.1 cm/h.

## Answer

Total rainfall $= 0.7+2.2+3.0+4.6+3.6+3.2+2.0+0.7 = 20.0$ cm; runoff $R = 12$ cm, so total loss $= 8$ cm over 8 h.

**Trial 1:** if all 8 hours exceed $\phi$: $20 - 8\phi = 12 \Rightarrow \phi = 1.0$ cm/h. But the 1st and last hours (0.7 cm) are below 1.0, so they do not produce runoff.

**Trial 2:** exclude the two 0.7 cm hours (their full 1.4 cm is lost) and use 6 hours:

$$
(20 - 1.4) - 6\phi = 12 \Rightarrow 18.6 - 6\phi = 12 \Rightarrow \phi = 1.1\ \text{cm/h}
$$

**Check:** hourly rain 2.2, 3.0, 4.6, 3.6, 3.2 and 2.0 are all above 1.1, and 0.7 is below.
Excess $= (2.2+3.0+4.6+3.6+3.2+2.0) - 6\times1.1 = 18.6 - 6.6 = 12.0$ cm. Valid.

**Answer: $\phi$-index = 1.1 cm/h.**

## Check yourself

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

1. Loss is
   A. 8 cm
   B. 12 cm
   C. 32 cm
   D. 20 cm
   Answer: A. 20 - 12 = 8. (Key point: Total loss)

2. Trial 1 gives φ = 1.0 but fails because
   A. 4.6 is above
   B. Runoff is 12
   C. 0.7 cm/h hours are below 1.0
   D. Hours are 8
   Answer: C. Hours with rain below φ give no excess. (Key point: Trial 1)

3. Excluding 0.7 hours: 18.6 - 6φ = 12 gives
   A. 1.1 cm/h
   B. 2.2 cm/h
   C. 1.0 cm/h
   D. 0.7 cm/h
   Answer: A. φ = 6.6/6 = 1.1. (Key point: Trial 2)

## 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
- [Phi-index and rainfall excess duration, 0.340 Mm3 runoff](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-500ha-0-340mm3) — Engineering Hydrology, chapter 3, asked 2×
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- [Total rainfall, runoff and peak discharge with phi-index](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-50ha-2-5) — Engineering Hydrology, chapter 3
- [Phi-index, excess rainfall hyetograph, runoff volume](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-hyetograph-5km2) — Engineering Hydrology, chapter 3
- [Total rainfall, net runoff, W-index of 120-min storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-w-index-120min-storm) — Engineering Hydrology, chapter 3, asked 2×

## Related in other subjects

- [Runoff from a 20 ha built-up area](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Arunoff-builtup-area-20ha) — Sanitary Engineering, chapter 2