Question

A vertical sheet pile wall is installed in an anisotropic soil having coefficient of horizontal permeability, $k_H$, and coefficient of vertical permeability, $k_V$. In order to draw the flow net for the isotropic condition, the embedment depth of the wall should be scaled by a factor of \underline{\hspace{1cm}, without changing the horizontal scale.}

• A. $\sqrt{\dfrac{k_H}{k_V}}$
• B. $\sqrt{\dfrac{k_V}{k_H}}$
• C. 1.0
• D. $\dfrac{k_H}{k_V}$

Hint:

For anisotropic soils: scale vertical distances by $\sqrt{\tfrac{k_H}{k_V}}$ to construct isotropic flow nets. This transformation ensures equipotential and flow lines remain perpendicular.

Answer 1

The Correct Option is A

Step 1: Concept of anisotropic soil.
In anisotropic soils, the permeability in the horizontal direction ($k_H$) and in the vertical direction ($k_V$) are not equal. To construct a flow net, we transform the anisotropic medium into an equivalent isotropic medium using a scale transformation.

Step 2: Transformation rule.
The vertical dimension is scaled by the factor: \[ \sqrt{\frac{k_H}{k_V}} \] This makes the flow lines and equipotential lines orthogonal, allowing a valid isotropic flow net to be drawn.

Step 3: Apply to the sheet pile embedment.
Since only the vertical direction is scaled, the embedment depth of the wall must also be scaled by this factor: \[ \text{Embedment depth (scaled)} = \text{Embedment depth (actual)} \times \sqrt{\frac{k_H}{k_V}} \]

Step 4: Conclusion.
Thus, the required scaling factor is $\sqrt{\dfrac{k_H}{k_V}}$.
\[ \boxed{\sqrt{\dfrac{k_H}{k_V}}} \]