Question

An anisotropic soil deposit has coefficient of permeability in vertical and horizontal directions as $k_{z}and{k}_x$, respectively. For constructing a flow net, the horizontal dimension of the problem's geometry is transformed by a multiplying factor of

• A. $\frac{k_z}{k_x}$
• B. $\frac{k_x}{k_z}$
• C. $\sqrt{\frac{k_z}{k_x}}$
• D. $\sqrt{\frac{k_x}{k_z}}$

Answer 1

The correct option is C $\sqrt{\frac{k_z}{k_x}}$

Flow net for anisotropic soil,

$k_x\frac{{\partial }^{2}h}{\partial x^2}+k_z\frac{{\partial }^{2}h}{\partial z^2}=0$

$\Rightarrow \frac{k_x}{k_z}\frac{{\partial }^{2}h}{\partial x^2}+\frac{{\partial }^{2}h}{\partial z^2}=0...\left(i\right)$

If we transformed geometry in x direction, let x coordinate be transformed into the new coordinate $x_t$.

By the transformation,

$x_t=x\sqrt{\frac{k_z}{k_x}}$

$\Rightarrow x=x_t\sqrt{\frac{k_x}{k_z}}....\left(ii\right)$

From eq. (i) & (ii),

$\Rightarrow \frac{k_x}{k_z}\frac{{\partial }^{2}h}{\partial {\left(x_t\sqrt{\frac{k_x}{k_z}}\right)}^2}+\frac{{\partial }^{2}h}{\partial z^2}=0$

$\Rightarrow \frac{{\partial }^{2}h}{\partial x_t^2}+\frac{{\partial }^{2}h}{\partial z^2}=0$ $\left(Henceproved\right)$

$So,multipliedfactor=\sqrt{\frac{k_z}{k_x}}$