The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited to specific initial and boundary conditions. Proof of the existence and uniqueness of solution was given only in 1983 by Alt and Luckhaus. The equation is based on Darcy-Buckingham law representing flow in porous media under variably saturated conditions, which is stated as

where

Considering the law of mass conservation for an incompressible porous medium and constant liquid density, expressed as

where

Then substituting the fluxes by the Darcy-Buckingham law the following mixed-form Richards equation is obtained:

For modeling of one-dimensional infiltration this divergence form reduces to

Although attributed to L. A. Richards, the equation was originally introduced 9 years earlier by Lewis Fry Richardson in 1922.

The Richards equation appears in many articles in the environmental literature because it describes the flow in the vadose zone between the atmosphere and the aquifer. It also appears in pure mathematical journals because it has non-trivial solutions. The above-given mixed formulation involves two unknown variables: ${\displaystyle \theta }$ and ${\displaystyle h}$. This can be easily resolved by considering constitutive relation ${\displaystyle \theta (h)}$, which is known as the water retention curve. Applying the chain rule, the Richards equation may be reformulated as either ${\displaystyle h}$-form (head based) or ${\displaystyle \theta }$-form (saturation based) Richards equation.