The following figure (not to scale) depicts a rainfall hyetograph for a storm over a catchment
If the storm produced a direct runoff of 12.5 mm, then the $\phi$-index of the storm for the catchment is ________ mm/hour. (rounded off to two decimal places)
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The $\phi$-index is the average infiltration rate during the period of rainfall excess. It is determined by equating the volume of rainfall excess to the volume of direct runoff.
Step 1: Calculate the total rainfall for the storm.
The hyetograph shows the rainfall intensity (mm/hour) over time (hour). The total rainfall is the area under the hyetograph. We calculate the rainfall in each interval:
0 to 0.5 hour: $10 \times 0.5 = 5$ mm
0.5 to 1.0 hour: $30 \times 0.5 = 15$ mm
1.0 to 1.5 hour: $15 \times 0.5 = 7.5$ mm
1.5 to 2.0 hour: $25 \times 0.5 = 12.5$ mm
2.0 to 2.5 hour: $5 \times 0.5 = 2.5$ mm
2.5 to 3.0 hour: $10 \times 0.5 = 5$ mm
Total rainfall = $5 + 15 + 7.5 + 12.5 + 2.5 + 5 = 47.5$ mm Step 2: Understand the concept of $\phi$-index.
The $\phi$-index is the constant rate of infiltration above which the rainfall volume is equal to the direct runoff volume. Rainfall intensities below the $\phi$-index do not contribute to direct runoff. Step 3: Determine the time intervals with rainfall intensity greater than $\phi$.
Let the $\phi$-index be $\phi$ mm/hour. The duration of each interval is 0.5 hours. The rainfall intensities are 10, 30, 15, 25, 5, and 10 mm/hour.
We need to find a $\phi$ such that the sum of $(i - \phi) \times 0.5$ for all intervals where $i > \phi$ equals the total runoff of 12.5 mm.
Try $\phi = 10$ mm/hour:
$(30-10)0.5 + (15-10)0.5 + (25-10)0.5 + (10-10)0.5 = 10 + 2.5 + 7.5 + 0 = 20 \neq 12.5$
Try $\phi = 15$ mm/hour:
$(30-15)0.5 + (25-15)0.5 = 7.5 + 5 = 12.5$
The intervals with rainfall intensity greater than 15 mm/hour are 0.5-1.0 hour (30 mm/hour) and 1.5-2.0 hour (25 mm/hour).
Excess rainfall in 0.5-1.0 hour = $(30 - 15) \times 0.5 = 7.5$ mm
Excess rainfall in 1.5-2.0 hour = $(25 - 15) \times 0.5 = 5$ mm
Total runoff = $7.5 + 5 = 12.5$ mm.
Thus, the $\phi$-index is 15 mm/hour. Rounded off to two decimal places, it is 15.00 mm/hour.
