Question

In curve number method if \(Q\) is actual runoff, \(P\) is rainfall, \(I_a\) is initial abstraction and \(S\) is storage capacity, then the relationship is shown by:

• A. \( Q = \dfrac{(P-0.8S)^2}{P+0.2S} \)
• B. \( Q = \dfrac{(P-0.2S)^2}{P+0.8S} \)
• C. \( Q = \dfrac{(P+0.2S)^2}{P-0.8S} \)
• D. \( Q = \dfrac{(P+0.8S)^2}{P-0.2S} \)

Hint:

In SCS Curve Number Method, always remember: \[ I_a = 0.2S \] and the standard runoff equation: \[ \boxed{ Q = \frac{(P-0.2S)^2}{P+0.8S} } \] Memory Trick: \[ \text{“0.2 below and 0.8 below denominator”} \] which helps remember the coefficients correctly.

Answer 1

The Correct Option is B

Concept: The Curve Number (CN) Method is one of the most widely used hydrological methods developed by the Soil Conservation Service (SCS) of the United States Department of Agriculture (USDA) for estimating direct runoff from rainfall events. This method establishes a relationship between:
• Rainfall depth
• Soil characteristics
• Land use
• Antecedent moisture condition
• Surface runoff The method is extremely important in:
• Watershed management
• Runoff estimation
• Flood analysis
• Stormwater drainage design According to the SCS Curve Number theory: \[ Q = \frac{(P-I_a)^2}{P-I_a+S} \] where:
• \(Q\) = direct runoff depth
• \(P\) = rainfall depth
• \(I_a\) = initial abstraction
• \(S\) = potential maximum retention after runoff begins The initial abstraction \(I_a\) includes:
• Interception
• Surface storage
• Infiltration before runoff starts Empirically: \[ I_a = 0.2S \] Substituting this value gives the standard runoff equation.

Step 1: Writing the general SCS runoff equation. The basic SCS runoff relationship is: \[ Q = \frac{(P-I_a)^2}{P-I_a+S} \] This equation is valid when: \[ P>I_a \] Otherwise, runoff does not occur.

Step 2: Using the empirical relation for initial abstraction. According to the SCS assumption: \[ I_a = 0.2S \] Substituting into the runoff equation: \[ Q = \frac{(P-0.2S)^2}{P-0.2S+S} \]

Step 3: Simplifying the denominator carefully. Now simplify: \[ P-0.2S+S \] Since: \[ -0.2S+S = 0.8S \] Therefore: \[ P-0.2S+S = P+0.8S \] Hence the runoff equation becomes: \[ Q = \frac{(P-0.2S)^2}{P+0.8S} \]

Step 4: Comparing with the given options. Now examine all options carefully. Option (A): \[ Q = \frac{(P-0.8S)^2}{P+0.2S} \] Incorrect because the coefficient of \(S\) in numerator and denominator is reversed. Hence: \[ \boxed{\text{Option (A) is incorrect}} \] Option (B): \[ Q = \frac{(P-0.2S)^2}{P+0.8S} \] This exactly matches the standard SCS Curve Number runoff equation. Hence: \[ \boxed{\text{Option (B) is correct}} \] Option (C): \[ Q = \frac{(P+0.2S)^2}{P-0.8S} \] Incorrect because signs are wrong in both numerator and denominator. Hence: \[ \boxed{\text{Option (C) is incorrect}} \] Option (D): \[ Q = \frac{(P+0.8S)^2}{P-0.2S} \] Incorrect formulation. Hence: \[ \boxed{\text{Option (D) is incorrect}} \] Final Conclusion: The correct runoff equation in the SCS Curve Number method is: \[ \boxed{ Q = \frac{(P-0.2S)^2}{P+0.8S} } \] Therefore, the correct answer is: \[ \boxed{(B)\ Q = \dfrac{(P-0.2S)^2}{P+0.8S}} \]