GATE 2023 CE (CE2) – Question 19
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 ______, without changing the horizontal scale.
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Correct answer: (A) $\sqrt{k_H/k_V}$
Explanation
**Governing equation.** For steady 2-D seepage in anisotropic soil,
$$k_H\frac{\partial^2h}{\partial x^2}+k_V\frac{\partial^2h}{\partial z^2}=0 .$$
**Transform the vertical coordinate.** Let $z^\prime=z\,\alpha$. Then $\partial^2/\partial z^2=\alpha^2\,\partial^2/\partial z^{\prime 2}$ and the equation becomes
$$k_H\frac{\partial^2h}{\partial x^2}+k_V\alpha^2\frac{\partial^2h}{\partial z^{\prime2}}=0 .$$
For this to be the Laplace equation of an isotropic medium we need $k_V\alpha^2=k_H$, so
$$\alpha=\sqrt{\frac{k_H}{k_V}} .$$
Hence the vertical dimensions (including the embedment depth) are multiplied by $\sqrt{k_H/k_V}$, with the horizontal scale unchanged. Answer **A**.