Section 4: Soil and Water Conservation Engineering
Hydrology
Subtopic: Hydrograph analysis, unit hydrograph theory and application
Question Statement:
A watershed has a drainage catchment area of $A = 72.0\text{ km}^2$. Direct runoff from a sequence of continuous unit effective rainfall pulses of duration $D = 4.0\text{ hours}$ produces an S-curve hydrograph. The ultimate equilibrium discharge $Q_e$ of this S-curve in cubic meters per second ($\text{m}^3/\text{s}$) is ________ (round off to one decimal place).
Answer Key:
50.0
Step-by-Step Solution & Explanation
**Method 1: Equilibrium Runoff Discharge Formulation**
1. S-curve equilibrium discharge represents the steady state direct runoff where continuous rainfall excess intensity $i = \frac{1.0\text{ cm}}{D}$ is completely converted to discharge:
$Q_e = \frac{2.778 \times A \times 1.0\text{ cm}}{D}$
Where:
- $A = 72.0\text{ km}^2$
- $D = 4.0\text{ hours}$
$Q_e = \frac{2.778 \times 72.0 \times 1.0}{4.0} = 2.778 \times 18.0 = 50.004\text{ m}^3/\text{s} \approx 50.0\text{ m}^3/\text{s}$
**Method 2: SI First Principles Conversion**
$Q_e = \frac{\text{Volume per duration}}{\text{Duration in seconds}} = \frac{A \times (0.010\text{ m})}{D \times 3600\text{ s}}$
$Q_e = \frac{(72.0 \times 10^6\text{ m}^2) \times 0.010\text{ m}}{4.0 \times 3600\text{ s}} = \frac{720,000\text{ m}^3}{14,400\text{ s}} = 50.0\text{ m}^3/\text{s}$