Seepage in anisotropic soil

In anisotropic soil the permeability differs in the two directions: $k_x$ (horizontal, usually larger) and $k_z$ (vertical). The flow net cannot be drawn directly with square fields, so the section is transformed.

Governing equation

Continuity for steady flow, $\dfrac{\partial v_x}{\partial x} + \dfrac{\partial v_z}{\partial z} = 0$, with Darcy's law $v_x = k_x\dfrac{\partial h}{\partial x}$ and $v_z = k_z\dfrac{\partial h}{\partial z}$ (sign neglected), gives

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

Transformation

Put $X = x\sqrt{\dfrac{k_z}{k_x}}$. Then $\dfrac{\partial}{\partial x} = \sqrt{\dfrac{k_z}{k_x}}\dfrac{\partial}{\partial X}$ and the equation becomes

$$k_z\frac{\partial^2 h}{\partial X^2} + k_z\frac{\partial^2 h}{\partial z^2} = 0 \;\Rightarrow\; \frac{\partial^2 h}{\partial X^2} + \frac{\partial^2 h}{\partial z^2} = 0$$

which is the Laplace equation for an isotropic medium. So the section is redrawn with all horizontal dimensions multiplied by $\sqrt{k_z/k_x}$ (vertical dimensions unchanged), and the flow net with square fields is drawn on this transformed section.

Discharge

The flow across a vertical plane in the real section, through a vertical height $dz$:

$$\Delta q = v_x\,dz = k_x\frac{\partial h}{\partial x}dz = k_x\sqrt{\frac{k_z}{k_x}}\frac{\partial h}{\partial X}dz = \sqrt{k_xk_z}\,\frac{\partial h}{\partial X}\,dz$$

This is Darcy's law in the transformed section with the equivalent permeability

$$k' = \sqrt{k_x\,k_z}$$

For a flow net of square fields with total head loss $h$:

$$\boxed{q = \sqrt{k_x k_z}\;h\,\frac{N_f}{N_d}}$$

The same expression follows by considering the flow across a horizontal plane, so the result is independent of the direction.