Richards equation
The Richards equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931. It is a nonlinear partial differential equation, which is often difficult to approximate since it does not have a closed-form analytical solution. Although this equation is atttributed to Richards, it is established that this equation was actually discovered 9 years before Lorenzo A. Richards by Lewis Fry Richardson in his book "Weather prediction by numerical process" published in 1922 (p.108).
Darcy's law was developed for saturated flow in porous media; to this Richardson applied a continuity requirement suggested by Edgar Buckingham and obtained a "general partial differential equation describing water movement in unsaturated non-swelling soils". The transient state form of this flow equation, known commonly as Richards equation writes in 1-D:
where
Here we show how to derive the Richards equation for the vertical direction in a very simplistic form. Conservation of mass says the rate of change of saturation in a closed volume is equal to the rate of change of the total sum of fluxes into and out of that volume, put in mathematical language:
Put in the 1D form for the direction :
Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy:
Substituting q in the equation above, we get:
Substituting for H = h + z:
...
Wikipedia