Richards equation

The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931.^[1] It is a quasilinear partial differential equation; its analytical solution is often limited to specific initial and boundary conditions.^[2] Proof of the existence and uniqueness of solution was given only in 1983 by Alt and Luckhaus.^[3] The equation is based on Darcy-Buckingham law^[1] representing flow in porous media under variably saturated conditions, which is stated as

${\displaystyle {\vec {q}}=-\mathbf {K} (\theta )(\nabla h+\nabla z),}$

where

${\displaystyle {\vec {q}}}$ is the volumetric flux;

${\displaystyle \theta }$ is the volumetric water content;

${\displaystyle h}$ is the liquid pressure head, which is negative for unsaturated porous media;

${\displaystyle \mathbf {K} (h)}$ is the unsaturated hydraulic conductivity;

${\displaystyle \nabla z}$ is the geodetic head gradient, which is assumed as ${\displaystyle \nabla z=\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)}$ for three-dimensional problems.

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

${\displaystyle {\frac {\partial \theta }{\partial t}}+\nabla \cdot {\vec {q}}+S=0}$,

where

${\displaystyle S}$ is the sink term [T${\displaystyle ^{-1}}$], typically root water uptake.^[4]

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

${\displaystyle {\frac {\partial \theta }{\partial t}}=\nabla \cdot \mathbf {K} (h)(\nabla h+\nabla z)-S}$.

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

${\displaystyle {\frac {\partial \theta }{\partial t}}={\frac {\partial }{\partial z}}\left(\mathbf {K} (\theta )\left({\frac {\partial h}{\partial z}}+1\right)\right)-S}$.

Although attributed to L. A. Richards, the equation was originally introduced 9 years earlier by Lewis Fry Richardson in 1922.^[5][6]