Calculating the $\phi$-Index from Rainfall and Runoff

The $\phi$-index is a crucial concept in hydrology. It represents the average rate of water loss due to infiltration, evaporation, and depression storage during a rainfall event. In simpler terms, it's the minimum rainfall intensity required for surface runoff to begin. Only the portion of rainfall intensity that exceeds the $\phi$-index contributes to the total surface runoff.

Given data for this problem:

• Storm duration: 3 hours
• Rainfall intensity in the 1st hour: \(I_1 = 3.5 \text{ cm/hr}\)
• Rainfall intensity in the 2nd hour: \(I_2 = 4.2 \text{ cm/hr}\)
• Rainfall intensity in the 3rd hour: \(I_3 = 2.9 \text{ cm/hr}\)
• Total surface runoff: \(R = 3 \text{ cm}\)

The total surface runoff is the sum of the "excess rainfall" over the duration of the storm. Excess rainfall for a given hour is the rainfall intensity minus the $\phi$-index, but only if the intensity is greater than the $\phi$-index. If the intensity is less than or equal to the $\phi$-index, the excess rainfall for that hour is zero.

The formula for total runoff is:

\( R = \sum_{i=1}^{n} \max(I_i - \phi, 0) \times \Delta t \)

Where:

• \(R\) is the total surface runoff.
• \(I_i\) is the rainfall intensity in the \(i\)-th time interval.
• \(\phi\) is the $\phi$-index.
• \(\Delta t\) is the duration of each time interval (here, 1 hour).
• \(n\) is the total number of time intervals (here, 3 hours).

Substituting the given values and \(\Delta t = 1\) hour:

\( 3 \text{ cm} = \max(3.5 - \phi, 0) \times 1 + \max(4.2 - \phi, 0) \times 1 + \max(2.9 - \phi, 0) \times 1 \)

We need to find a value of \(\phi\) that satisfies this equation. The value of \(\phi\) must be less than or equal to the minimum rainfall intensity that contributes to runoff.

Let's test scenarios based on which intensities are greater than \(\phi\):

Scenario 1: Assume all three intensities (3.5, 4.2, and 2.9 cm/hr) are greater than \(\phi\).

If this is true, then \(\max(I_i - \phi, 0) = I_i - \phi\) for all \(i\).

\( 3 = (3.5 - \phi) + (4.2 - \phi) + (2.9 - \phi) \)

\( 3 = 3.5 + 4.2 + 2.9 - 3\phi \)

\( 3 = 10.6 - 3\phi \)

\( 3\phi = 10.6 - 3 \)

\( 3\phi = 7.6 \)

\( \phi = \frac{7.6}{3} \approx 2.533 \text{ cm/hr} \)

Let's check if this value of \(\phi\) is consistent with our assumption that all intensities are greater than \(\phi\):

• Is \(3.5 > 2.533\)? Yes.
• Is \(4.2 > 2.533\)? Yes.
• Is \(2.9 > 2.533\)? Yes.

Since the calculated \(\phi = 2.533 \text{ cm/hr}\) is less than all the given rainfall intensities, our assumption is consistent. This value represents the average loss rate, and all rainfall above this rate contributed to runoff.

Let's briefly consider other scenarios to ensure this is the correct one:

Scenario 2: Assume only intensities 4.2 and 3.5 cm/hr are greater than \(\phi\), and 2.9 cm/hr is less than or equal to \(\phi\).

\( 3 = (4.2 - \phi) + (3.5 - \phi) + 0 \)

\( 3 = 7.7 - 2\phi \)

\( 2\phi = 7.7 - 3 = 4.7 \)

\( \phi = \frac{4.7}{2} = 2.35 \text{ cm/hr} \)

Check assumption: Is \(2.9 \le 2.35\)? No, \(2.9 > 2.35\). This scenario is inconsistent.

Since Scenario 1 yielded a consistent result matching one of the options, we can conclude that \(\phi = 2.533 \text{ cm/hr}\) is the correct value.

The calculated value of the $\phi$-index is approximately \(2.533 \text{ cm/hr}\).

Comparing this with the given options:

• 2.212 cm/hr
• 2.331 cm/hr
• 2.412 cm/hr
• 2.533 cm/hr

The calculated value matches the fourth option.

Total calculated runoff = \(0.967 + 1.667 + 0.367 = 3.001 \text{ cm}\). This is very close to the given total runoff of 3 cm, accounting for minor rounding.

Revision Table: Key Hydrology Concepts

Additional Information on $\phi$-Index and Runoff

The $\phi$-index method is a simple way to estimate the volume of runoff from a rainfall event. It assumes a constant rate of water loss once the initial losses are met. In reality, infiltration rates usually decrease over time during a continuous rainfall event as the soil becomes saturated. More complex methods, like the Horton's equation or the Soil Conservation Service (SCS) Curve Number method, provide more detailed ways to model infiltration and runoff generation.

Key points about the $\phi$-index:

• It is a single value representing the combined effect of all losses (infiltration, depression storage, interception) that occur *before* and *during* the storm at intensities below the $\phi$-index.
• Rainfall intensities below the $\phi$-index do not contribute to surface runoff.
• The $\phi$-index is calculated from observed rainfall and runoff data for a specific storm over a specific area.
• It is an average value and may not perfectly represent the varying loss rates throughout the storm.

Understanding the $\phi$-index helps hydrologists estimate how much rainfall will become runoff, which is crucial for flood forecasting, water resource management, and designing drainage structures in a drainage basin.