Chapter 7, Problem 7.1P

To determine

Find the rate of seepage through the permeable layer.

Answer to Problem 7.1P

The rate of seepage through the permeable layer is $\underset{\_}{14.1\times {10}^{-2}{\text{m}}^{\text{2}}\text{/hr/m}}$.

Explanation of Solution

Given information:

The hydraulic conductivity of the permeable soil layer k is $6\times {10}^{-3}\text{cm/sec}$.

The height of the layer H is 5.4 m.

The angle value $\alpha$ is $7°$.

Calculation:

Draw the free body diagram of the permeable layer as in Figure (1).

Refer Figure (1).

Determine the hydraulic gradient using the relation.

$\begin{array}{c}i=\frac{\Delta h}{L}\\ =\frac{S\tan \alpha }{\frac{S}{\cos \alpha }}\\ =\frac{\sin \alpha }{\cos \alpha }\times \cos \alpha \\ =\sin \alpha \end{array}$ (1)

Here, $\Delta h$ is the head loss and L is the horizontal length of the layer.

Substitute $7°$ for $\alpha$ in Equation (1).

$\begin{array}{c}i=\sin 7°\\ =0.122\end{array}$

Determine the rate of seepage through the permeable layer using the relation.

$\begin{array}{c}q=kiA\\ =ki\left(H\cos \alpha \times 1\right)\end{array}$

Here, A is the area of the permeable layer.

Substitute $6\times {10}^{-3}\text{cm/sec}$ for k, 0.122 for i, 5.4 m for H, and $7°$ for $\alpha$.

$\begin{array}{c}q=6\times {10}^{-3}\text{cm/sec}\times \left(\frac{1\text{m}}{100\text{m}}\right)\times \frac{3,600\sec }{1\text{hr}}\times 0.122\times 5.4\times \cos 7°\times 1\\ =14.1\times {10}^{-2}{\text{m}}^{\text{3}}\text{/hr/m}\end{array}$

Therefore, the rate of seepage through the permeable layer is $\underset{\_}{14.1\times {10}^{-2}{\text{m}}^{\text{2}}\text{/hr/m}}$.