Calculating W-index from Rainfall Data and $\phi$-index

This problem involves calculating the W-index for a given storm using the provided rainfall rates and the $\phi$-index. The W-index is a measure of the overall infiltration rate during a storm event.

Here's the given information:

• Duration of storm: 4 hours
• Duration of each rainfall period: 30 minutes (0.5 hours)
• Rainfall rates (cm/hr) for successive 30-minute periods: 3.5, 4.5, 5.6, 8.5, 7.8, 6.4, 4.0, 3.8
• $\phi$-index: 5.0 cm/hr

The $\phi$-index represents the average rate of infiltration and other losses (like interception and depression storage) such that the volume of rainfall at intensities greater than the $\phi$-index equals the volume of direct runoff.

To calculate the W-index, we first need to determine the total rainfall and the total runoff based on the given $\phi$-index.

Step 1: Calculate Total Rainfall Depth

The total rainfall depth is the sum of the rainfall depths from each 30-minute period. Rainfall depth for a period is the rate multiplied by the duration (0.5 hours).

Let $R_i$ be the rainfall rate for period $i$, and $\Delta t = 0.5$ hr be the duration of each period.

Total Rainfall Depth $= \sum (R_i \times \Delta t)$

Total Rainfall Depth $= (3.5 \times 0.5) + (4.5 \times 0.5) + (5.6 \times 0.5) + (8.5 \times 0.5) + (7.8 \times 0.5) + (6.4 \times 0.5) + (4.0 \times 0.5) + (3.8 \times 0.5)$

Total Rainfall Depth $= 0.5 \times (3.5 + 4.5 + 5.6 + 8.5 + 7.8 + 6.4 + 4.0 + 3.8)$

Total Rainfall Depth $= 0.5 \times 44.1$

Total Rainfall Depth $= 22.05$ cm

Step 2: Identify Runoff-Producing Rainfall and Calculate Runoff Depth

Runoff occurs only when the rainfall intensity exceeds the $\phi$-index. The excess rainfall intensity ($R_i - \phi$) contributes to runoff during those periods.

Given $\phi$-index = 5.0 cm/hr.

Let's check which rainfall rates are greater than 5.0 cm/hr:

• 3.5 cm/hr < 5.0 cm/hr (No runoff from this period)
• 4.5 cm/hr < 5.0 cm/hr (No runoff from this period)
• 5.6 cm/hr > 5.0 cm/hr (Runoff from this period)
• 8.5 cm/hr > 5.0 cm/hr (Runoff from this period)
• 7.8 cm/hr > 5.0 cm/hr (Runoff from this period)
• 6.4 cm/hr > 5.0 cm/hr (Runoff from this period)
• 4.0 cm/hr < 5.0 cm/hr (No runoff from this period)
• 3.8 cm/hr < 5.0 cm/hr (No runoff from this period)

The rainfall rates that contribute to runoff are 5.6, 8.5, 7.8, and 6.4 cm/hr.

The runoff depth for each of these periods is $(R_i - \phi) \times \Delta t$.

Total Runoff Depth $= ((5.6 - 5.0) \times 0.5) + ((8.5 - 5.0) \times 0.5) + ((7.8 - 5.0) \times 0.5) + ((6.4 - 5.0) \times 0.5)$

Total Runoff Depth $= (0.6 \times 0.5) + (3.5 \times 0.5) + (2.8 \times 0.5) + (1.4 \times 0.5)$

Total Runoff Depth $= 0.3 + 1.75 + 1.4 + 0.7$

Total Runoff Depth $= 4.15$ cm

Step 3: Calculate Total Losses

Total Losses (infiltration, etc.) can be calculated as the difference between the Total Rainfall Depth and the Total Runoff Depth.

Total Losses = Total Rainfall Depth - Total Runoff Depth

Total Losses $= 22.05$ cm $- 4.15$ cm

Total Losses $= 17.9$ cm

Step 4: Calculate W-index

The W-index is defined as the total infiltration (total losses) during the storm divided by the total duration of the storm.

Total storm duration = 4 hours.

W-index $= \frac{\text{Total Losses}}{\text{Total Duration}}$

W-index $= \frac{17.9 \text{ cm}}{4 \text{ hr}}$

W-index $= 4.475$ cm/hr

The calculated W-index is 4.475 cm/hr.