Calculating the Phi Index ($\phi$-index) in Hydrology

The $\phi$-index is a crucial concept in hydrology used to estimate the average rate of infiltration above which the rainfall becomes surface runoff. It represents the average rainfall intensity above which rainfall excess occurs over a given area and duration.

To calculate the $\phi$-index, we need the total rainfall, the resulting surface runoff, and the storm duration. The difference between the total rainfall and the surface runoff represents the total losses, primarily due to infiltration, evaporation, and initial abstraction, assumed to be uniform throughout the storm above the $\phi$-index rate.

The formula for $\phi$-index is given by:

$\phi\text{-index} = \frac{\text{Total Rainfall Depth} - \text{Runoff Depth}}{\text{Storm Duration}}$

Given Information about the Catchment Area and Storm

• Catchment Area: 1200 km²
• Storm Duration: 6 hours
• Total Rainfall: 16 cm
• Surface Run-off Volume: $1.2 \times 10^8$ m³

Step-by-Step Calculation of the Phi Index

First, we need to ensure all measurements are in consistent units. The surface runoff is given in volume (m³) and the area in km². We will convert the runoff volume into an equivalent depth over the catchment area, expressed in centimeters, to match the total rainfall unit.

1. Convert Catchment Area to Square Meters:

   1 km = 1000 m
   1 km² = $(1000 \text{ m})^2 = 10^6$ m²
   Catchment Area (A) = 1200 km² = $1200 \times 10^6$ m² = $1.2 \times 10^9$ m²

2. Calculate Runoff Depth:

   Runoff Depth (R) is the volume of runoff spread over the catchment area.
   Runoff Depth = $\frac{\text{Runoff Volume}}{\text{Catchment Area}}$
   R = $\frac{1.2 \times 10^8 \text{ m}^3}{1.2 \times 10^9 \text{ m}^2} = 0.1$ m

3. Convert Runoff Depth from Meters to Centimeters:

   1 m = 100 cm
   R = $0.1 \text{ m} \times 100 \text{ cm/m} = 10$ cm

4. Calculate Total Losses:

   Total losses represent the portion of rainfall that did not become surface runoff.
   Total Losses = Total Rainfall Depth - Runoff Depth
   Total Losses = 16 cm - 10 cm = 6 cm

5. Calculate the Phi Index:

   Now, we use the formula for the $\phi$-index.
   Storm Duration (t) = 6 hours
   $\phi\text{-index} = \frac{\text{Total Losses}}{\text{Storm Duration}}$
   $\phi\text{-index} = \frac{6 \text{ cm}}{6 \text{ hours}} = 1.0 \text{ cm/hr}$

Summary of Results

We calculated the runoff depth from the given runoff volume and catchment area. Then, we subtracted this runoff depth from the total rainfall depth to find the total losses during the storm. Finally, we divided the total losses by the storm duration to determine the $\phi$-index.

The calculated $\phi$-index is 1.0 cm/hr. This means that, on average, 1.0 cm/hr of rainfall intensity was lost to infiltration and other abstractions during the 6-hour storm before runoff began or while it was occurring. Any rainfall intensity above this rate contributed to the surface runoff.

Revision Table: Key Concepts in Hydrology Calculations

Reviewing the essential formulas and conversions helps in solving hydrology problems like calculating the $\phi$-index.

Additional Information on Rainfall-Runoff Analysis and Phi Index

Understanding the relationship between rainfall and runoff is fundamental in hydrology. Not all rainfall becomes runoff; a significant portion is lost due to various processes collectively known as abstractions. The $\phi$-index is a simplified way to account for these losses.

• Infiltration: The process by which water on the ground surface enters the soil. The rate of infiltration depends on soil type, moisture content, and ground cover.
• Initial Abstraction: Losses that occur before the start of surface runoff. These include interception by vegetation, depression storage (water filling small surface depressions), and initial infiltration into dry soil.
• W-index: Another measure of infiltration, similar to the $\phi$-index, but it attempts to exclude initial abstraction. The W-index is typically lower than the $\phi$-index because the $\phi$-index includes all losses, including initial abstraction.
• Rainfall Excess: The portion of rainfall that becomes surface runoff after all losses (like infiltration and initial abstraction) have been satisfied.
• Unit Hydrograph: A concept often used in conjunction with rainfall excess to predict the shape and timing of the resulting hydrograph (flow rate over time) at a catchment outlet. The rainfall excess is often calculated using methods involving the $\phi$-index or W-index.

The $\phi$-index method assumes a constant loss rate throughout the duration when rainfall intensity exceeds this rate. While a simplification, it is widely used for practical hydrological calculations, especially when detailed infiltration data is unavailable. The calculated $\phi$-index of 1.0 cm/hr for this specific storm provides insight into the average loss rate experienced by the catchment area during the 6-hour storm event.