Section 4: Soil and Water Conservation Engineering
Soil and Water Erosion
Subtopic: Earthen dams
Question Statement:
A homogeneous earth dam of height $15\text{ m}$ has a horizontal filter at its downstream base. A parabolic phreatic line is formed with focal distance $x_0 = 3.0\text{ m}$ measured from the downstream toe. The equation of the basic parabola is $y^2 = 4 x_0 x + 4 x_0^2$. If the total seepage head is $H = 12\text{ m}$ and soil permeability is $k = 4 \times 10^{-6}\text{ m/s}$, the rate of seepage per meter length of the dam in $\text{m}^3\text{/(day}\cdot\text{m)}$ is ________ (round off to 2 decimal places).
Answer Key:
1.04
Step-by-Step Solution & Explanation
According to Schaffernak and Casagrande's solution for a homogeneous dam with a horizontal downstream filter:
Seepage discharge per unit width is given by:
$q = k \times x_0$
where $x_0$ is the focal distance of the base parabola.
Given:
• $k = 4 \times 10^{-6}\text{ m/s}$
• $x_0 = 3.0\text{ m}$
$q = (4 \times 10^{-6}) \times 3.0 = 1.20 \times 10^{-5}\text{ m}^3\text{/(s}\cdot\text{m)}$
Converting to $\text{m}^3\text{/(day}\cdot\text{m)}$:
$q = 1.20 \times 10^{-5} \times 86400 = 1.0368\text{ m}^3\text{/(day}\cdot\text{m)} \approx 1.04\text{ m}^3\text{/(day}\cdot\text{m)}$