Subtopic: Planning, design and layout of surface and sub-surface drainage systems
Question Statement:
Parallel relief tile drains are to be installed in a homogeneous soil overlying an impermeable layer. The hydraulic conductivity of the soil is $k = 0.8\text{ m/day}$. The depth of the impermeable barrier below the drain center is $d = 3.0\text{ m}$. The steady design drainage discharge rate is $q = 0.004\text{ m/day}$ ($4\text{ mm/day}$) and the maximum permissible water table height midway between drains above drain level is $h = 1.0\text{ m}$. Using Hooghoudt's equation ($q = \frac{8 k d h + 4 k h^2}{L^2}$), the required drain spacing $L$ is ________ $\text{m}$ (round off to two decimal places).
Answer Key:
74.83
Step-by-Step Solution & Explanation
Applying Hooghoudt's steady-state drainage equation:
$q = \frac{8 k d h + 4 k h^2}{L^2} \implies L = \sqrt{\frac{8 k d h + 4 k h^2}{q}}$
Given values:
- $k = 0.8\text{ m/day}$
- $d = 3.0\text{ m}$
- $h = 1.0\text{ m}$
- $q = 0.004\text{ m/day}$
Evaluating the numerator:
$\text{Term 1} = 8 \times 0.8 \times 3.0 \times 1.0 = 19.2$
$\text{Term 2} = 4 \times 0.8 \times (1.0)^2 = 3.2$
$\text{Numerator} = 19.2 + 3.2 = 22.4$
Calculating drain spacing $L$:
$L^2 = \frac{22.4}{0.004} = 5600$
$L = \sqrt{5600} = 74.833\text{ m} \approx 74.83\text{ m}$