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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:

$\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial z}[K(\theta )(\frac{\partial \psi }{\partial z}+1)]$

where

$K$ is the hydraulic conductivity,

$\psi$ is the pressure head,

$z$ is the elevation above a vertical datum,

$\theta$ 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).

Derivation

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:

$\frac{\partial \theta }{\partial t}=\overrightarrow{\mathrm{\nabla }}\cdot (\sum _{i=1}^n{\overrightarrow{q}}_{i,\text{in}}-\sum _{j=1}^m{\overrightarrow{q}}_{j,\text{out}})$

Put in the 1D form for the direction $\hat{k}$:

$\frac{\partial \theta }{\partial t}=-\frac{\partial }{\partial z}q$

Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy:

$q=-K\frac{\partial h}{\partial z}$

Substituting q in the equation above, we get:

$\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial z}\left[K\frac{\partial h}{\partial z}\right]$

Substituting for h = ψ + z:

$\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial z}\left[K(\frac{\partial \psi }{\partial z}+\frac{\partial z}{\partial z})\right]=\frac{\partial }{\partial z}\left[K(\frac{\partial \psi }{\partial z}+1)\right]$

We then get the equation above, which is also called the mixed form of Richard equation.

Formulations

The Richards Equation appears in many articles in the environmental literature due to the fact that it describes the flow in the interface between fully saturated aquifers and surface water and/or the atmosphere. It also appears in pure mathematical journals due to the fact that it has non-trivial solutions. Usually, it is presented in one of three forms. The mixed form containing the pressure and the saturation is discussed above. It can also appear in two other formulations: head-based and saturation-based.

Head-based

$C(h)\frac{\partial h}{\partial t}=\mathrm{\nabla }\cdot K(h)\mathrm{\nabla }h$

Where C(h) [1/L] is a function describing the rate of change of saturation with respect the hydraulic head:

$C(h)\equiv \frac{\partial \theta }{\partial h}$

This function is called 'specific moisture capacity' in the literature, and could be determined for different soil types using curve fitting and laboratory experiments measuring the rate of infiltration of water into soil column, as described for example in Van Genuchten, 1980.

Saturation-based

$\frac{\partial \theta }{\partial t}=\mathrm{\nabla }\cdot D(\theta )\mathrm{\nabla }\theta$

Where D(θ) [L²/T] is 'the soil water diffusivity':

$D(\theta )\equiv \frac{K(\theta )}{C(\theta )}\equiv \frac{\partial h}{\partial \theta }$

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