Question

If \( Q \) = depth of runoff in inches, \( P \) = depth of rainfall in inches, \( I_a \) = initial abstraction in inches, and \( S \) = maximum potential retention in inches, then which of the following is the correct equation for the curve number runoff?

• A. \( Q = \frac{(P - I_a)^2}{(P - I_a + S)} \)
• B. \( Q = \frac{(P + I_a)^2}{(P - I_a - S)} \)
• C. \( Q = \frac{(P - I_a)^2}{(P + I_a + S)} \)
• D. \( Q = \frac{(P - I_a)^2}{(P - I_a - S)} \)

Hint:

Use SCS method when rainfall exceeds initial abstraction: \( Q = \frac{(P - I_a)^2}{(P - I_a + S)} \)

Answer 1

The Correct Option is A

To determine the correct equation for the curve number (CN) runoff model, we analyze the given options and compare them with the standard SCS (Soil Conservation Service) runoff equation.

1. Understanding the SCS Runoff Equation:
The standard SCS runoff equation is given by:

\( Q = \frac{(P - I_a)^2}{(P - I_a + S)} \)
where:
- \( Q \) = depth of runoff (inches),
- \( P \) = depth of rainfall (inches),
- \( I_a \) = initial abstraction (inches),
- \( S \) = maximum potential retention (inches).

2. Comparing with Given Options:
We evaluate each option to see which matches the standard equation:

- Option 1: \( Q = \frac{(P - I_a)^2}{(P - I_a + S)} \) → Correct (matches the SCS equation).
- Option 2: \( Q = \frac{(P + I_a)^2}{(P - I_a - S)} \) → Incorrect (signs and structure are wrong).
- Option 3: \( Q = \frac{(P - I_a)^2}{(P + I_a + S)} \) → Incorrect (denominator has incorrect signs).
- Option 4: \( Q = \frac{(P - I_a)^2}{(P - I_a - S)} \) → Incorrect (denominator should have a "+" before \( S \)).

3. Conclusion:
Only Option 1 matches the standard SCS runoff equation.

Final Answer:
The correct equation is \( \boxed{Q = \frac{(P - I_a)^2}{(P - I_a + S)}} \).