---
title: "Phi-index and W-index"
subject: "Engineering Hydrology"
chapter: 3
key: "semester_5/engineering_hydrology:phi-w-index-explain"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-w-index-explain
site: BCT Engineering
---

# Phi-index and W-index

Engineering Hydrology · Chapter 3: Hydrological Losses

- Asked in: 2066 Magh (old course), 4 marks
- Video: [Infiltration Indices | Engineering Hydrology](https://www.youtube.com/watch?v=L2Noo6H3EJc) — APSEd
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/phi-w-index-explain
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-3

## Question

Explain $\Phi$ (Phi)-index and W-index.

## Short answer

The phi-index is the constant loss rate above which rainfall volume equals direct runoff; losses occur at that rate only in periods with intensity above phi. The W-index is the average infiltration rate over the runoff-producing period, subtracting initial loss and depression storage, so it is slightly less than phi.

## Answer

#### $\phi$-index
The **$\phi$-index** is the constant rate of rainfall (cm/h or mm/h) above which the rainfall volume equals the direct runoff volume. Rain with an intensity below $\phi$ is assumed to be lost completely. When the intensity is above $\phi$, the loss is assumed to occur at the constant rate $\phi$.

$$
\phi = \frac{P - R}{t_e}
$$

where $P$ is the total rainfall, $R$ is the direct runoff depth and $t_e$ is the duration of rainfall in which intensity exceeds $\phi$. It is found by trial from the hyetograph so that the area above the $\phi$ line equals $R$. It is simple, but it is only an average and ignores the fact that infiltration decreases with time.

```text
 i
 |        oooo
 |        oooo        oooo
 |  oooo  oooo        oooo
 |- - - - - - - - - - - - - - - - -  phi line
 |  ####  ####        ####
 |  ####  ####  ####  ####
 |  ####  ####  ####  ####  ####
 +--------------------------------- t
   oooo = rainfall excess (runoff R), #### = loss
```

#### W-index
The **W-index** is the average infiltration rate during the time that rainfall intensity exceeds the infiltration capacity. It corrects the $\phi$-index for depression storage and initial losses:

$$
W = \frac{P - R - S}{t_r}
$$

where $S$ is the initial loss plus depression storage (interception) and $t_r$ is the time in which the rainfall intensity is actually larger than the infiltration capacity. Because it removes the storage, $W$ is slightly less than $\phi$.

Both indices are used to estimate rainfall excess when direct infiltration data are missing.

## Check yourself

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

1. φ-index loss assumption
   A. Nothing lost
   B. Everything lost
   C. Loss grows
   D. Rate above φ lost at constant φ; below lost fully
   Answer: D. That is the definition. (Key point: φ)

2. φ = 
   A. R/te
   B. (P - R)/te
   C. (P+R)/te
   D. P/R
   Answer: B. Loss over duration where i > φ. (Key point: Formula)

3. φ-index weakness
   A. Too complex
   B. Ignores decreasing infiltration with time
   C. Needs lysimeter
   D. Ignores rain
   Answer: B. It is an average. (Key point: Drawback)

4. W is less than φ because
   A. Initial loss and storage removed
   B. Longer time
   C. Rain larger
   D. Runoff larger
   Answer: A. W = (P - R - S)/tr. (Key point: W)

## Related in Engineering Hydrology

- [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×
- [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×
- [Phi-index and cumulative infiltration curve](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-cum-infil-0-2km2) — 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
- [Phi-index of an 8-hour storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-15cm-8-7) — Engineering Hydrology, chapter 3
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