Following assumptions are made to derive the Laplace equation:

- Darey's law is valid
- The soil is fully saturated
- The soil is isotropic and homogeneous
- The flow is steady
- The flow is two-dimensional
- Water and soil are incompressible

Let,

Consider, an element of soil $\rightarrow dx$

through which flow is taking place $\rightarrow dz$

The third dimension of an element is taken as unity

$x \rightarrow direction$

Inlet velocity = $V_x$

outlet velocity = ($V_x + \frac{\delta V_x}{\delta x} dx$)

$z \rightarrow direction$

Inlet velocity = $V_z$

outlet velocity = ($V_z + \frac{\delta V_z}{\delta z} dz$)

The discharge is entering the element equal to that leaving the element

$V_x.dz + V_z.dx $= ($V_x + \frac{\delta V_x}{\delta x} dx$) + ($V_z + \frac{\delta V_z}{\delta z} dz$)

$[(\frac{\delta V_x}{\delta x}) + (\frac{\delta V_z}{\delta z})] = 0 ...................................(1)$

equation (1) is continuity equation for 2-dimentional flow

$h \rightarrow $ Total head at any point

$i_x = -\frac{\delta h}{\delta x} $ and $i_z = -\frac{\delta h}{\delta z} $

-ve sign indicates that the head decreases in the direction of flow

From Darely law,

$V_x = -K_x \frac{\delta h}{\delta x}$

$V_z = -K_z \frac{\delta h}{\delta z}$

putting the above value in equation (1) we have

$\frac{\delta^2 h}{\delta x^2} + \frac{\delta^2 h}{\delta z^2} = 0$...............................(2)

Equation (2) is laplace equation in terms of head (h) If velocity potential $\phi = -kh$

$\frac{\delta^2 \phi}{\delta x^2} + \frac{\delta^2 \phi}{\delta z^2} = 0$...............................(3)

Equation (3) is laplace equation in terms of velocity potential $(\phi)$