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 non-linear partial differential equation, which is often difficult to approximate since it does not have a closed-form analytical solution.

Darcy's law was developed for saturated flow in porous media; to this Richards applied a continuity requirement suggested by 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:

:$frac\{partial\; heta\}\{partial\; t\}=\; frac\{partial\}\{partial\; z\}\; left\; [\; K(psi)\; left\; (frac\{partial\; psi\}\{partial\; z\}\; +\; 1\; ight)\; ight]$where:$K$ is the hydraulic conductivity,:$psi$ is the pressure head,:$z$ is the elevation above a vertical datum,:$heta$ is the water content, and:$t$ is time

Richards equation is equivalent to the groundwater flow equation, which is in terms of hydraulic head ("h"), by substituting "h"="ψ"+"z", and changing the storage mechanism to dewatering. The reason for writing it in the form above is for convenience with boundary conditions (often expressed in terms of pressure head, for example atmospheric conditions are "ψ"=0).