Question Bank • QB_SWCE_ADV_121 NAT • 2 Marks

Section 4: Soil and Water Conservation Engineering

Hydrology

Subtopic: Infiltration – indices and equations

Question Statement:

The infiltration capacity of a catchment soil is described by Horton's equation: $f_p = 10.0 + (40.0 - 10.0) e^{-0.80 t}$, where $f_p$ is in $\text{mm/h}$ and $t$ is in hours. During a storm with continuous rainfall exceeding the infiltration capacity, the cumulative total infiltration depth $F$ in $\text{mm}$ during the first $t = 3.0\text{ hours}$ is ________ (round off to one decimal place).

Answer Key:

64.1

Step-by-Step Solution & Explanation

1. Horton's infiltration equation: $f_p = f_c + (f_0 - f_c) e^{-k t}$ Here: - $f_c = 10.0\text{ mm/h}$ - $f_0 = 40.0\text{ mm/h}$ - $f_0 - f_c = 30.0\text{ mm/h}$ - $k = 0.80\text{ h}^{-1}$ 2. Cumulative infiltration depth $F(t) = \int_0^t f_p dt$: $F(t) = f_c t + \frac{f_0 - f_c}{k} (1 - e^{-k t})$ 3. Evaluating at $t = 3.0\text{ h}$: $f_c t = 10.0 \times 3.0 = 30.0\text{ mm}$ $\frac{f_0 - f_c}{k} = \frac{30.0}{0.80} = 37.5\text{ mm}$ $e^{-k t} = e^{-0.80 \times 3.0} = e^{-2.40} \approx 0.090718$ $1 - e^{-2.40} = 1 - 0.090718 = 0.90928$ $F(3) = 30.0 + 37.5 \times 0.90928 = 30.0 + 34.098 = 64.098\text{ mm} \approx 64.1\text{ mm}$
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