GATE AG 2025 • Q51
NAT • 2 Marks
Section 4: Soil and Water Conservation Engineering
Hydrology
Subtopic: Infiltration – indices and equations
Question Statement:
An 8 ha watershed receives rainfall intensities 2.5, 3.6, 5.4, 3.3, 2.6, 1.2 cm/h at successive 30-min intervals. Surface runoff volume = 4800 m³. Neglecting initial abstraction, find the W-index (cm/h). (Rounded to 2 decimal places)
Official Answer Key:
1.08 to 1.12
Step-by-Step Solution & Explanation
• **Concept & Given Data:**
- Watershed area $A = 8\text{ ha} = 80,000\text{ m}^2$.
- Total surface runoff volume $V = 4800\text{ m}^3$.
- Runoff depth $R = \frac{V}{A} = \frac{4800\text{ m}^3}{80,000\text{ m}^2} = 0.06\text{ m} = 6.0\text{ cm}$.
- Time step $\Delta t = 30\text{ min} = 0.5\text{ h}$.
- Rainfall intensities: $I = [2.5, 3.6, 5.4, 3.3, 2.6, 1.2]\text{ cm/h}$.
• **Total Rainfall Depth ($P$):**
$P = \sum (I_i \times \Delta t) = (2.5 + 3.6 + 5.4 + 3.3 + 2.6 + 1.2) \times 0.5 = 18.6 \times 0.5 = 9.3\text{ cm}$
• **$\phi$-Index Determination:**
- Assuming all 6 intervals produce runoff ($t_e = 6 \times 0.5 = 3.0\text{ h}$):
$\phi = \frac{P - R}{t_e} = \frac{9.3 - 6.0}{3.0} = \frac{3.3}{3.0} = 1.10\text{ cm/h}$
- Verify threshold: Lowest intensity is $1.2\text{ cm/h} > 1.10\text{ cm/h}$, so rainfall exceeds $\phi$ in every single time block. Thus, the effective rainfall duration is indeed $3.0\text{ h}$.
• **Conclusion:** $\phi$-index $= 1.10\text{ cm/h}$ (Official range: 1.08 to 1.12).
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