# Richards equation

The Richards' equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931.[1] 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} \left[ K(\theta) \left (\frac{\partial \psi}{\partial z} + 1 \right) \right]\$

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:

$\frac{\partial \theta}{\partial t}= \vec{\nabla} \cdot \left(\sum_{i=1}^n{\vec{q}_{i,\,\text{in}}} - \sum_{j=1}^m{\vec{q}_{j,\,\text{out}}} \right)$

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:

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

We then get the equation above, which is also called the mixed form [2] of Richards' 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.

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

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).[3]

### Saturation-based

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

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

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

## Limitations

The numerical solution of Richards equation has been criticized for being computationally expensive and unpredictable [4][5] because there is no guarantee that a solver will converge for a particular set of soil constitutive relations. This prevents use of the method in general applications where the risk of non-convergence is high. The method has also been criticized for over-emphasizing the role of capillarity,[6] and for being in some ways 'overly simplistic' [7] In one dimensional simulations of rainfall infiltration into dry soils, fine spatial discretization less than one cm is required near the land surface. In three-dimensional applications the numerical solution of Richards' equation is subject to aspect ratio constraints where the ratio of horizontal to vertical resolution in the solution domain should be less than about 7.

## References

1. ^ Richards, L.A. (1931). "Capillary conduction of liquids through porous mediums". Physics 1 (5): 318–333. Bibcode:1931Physi...1..318R. doi:10.1063/1.1745010.
2. ^ Celia et al. (1990). "A general Mass-Conservative Numerical Solution for the Unsaturated Flow Equation". Water Resources Research 26 (7): 1483–1496. Bibcode:1990WRR....26.1483C. doi:10.1029/WR026i007p01483.
3. ^ van Genuchten, M. Th. (1980). "A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils" (PDF). Soil Science Society of America Journal 44 (5): 892–898. doi:10.2136/sssaj1980.03615995004400050002x.
4. ^ Short, D., W.R. Dawes, and I. White, 1995. The practicability of using Richards' equation for general purpose soil-water dynamics models. Envir. Int'l. 21(5):723-730.
5. ^ Tocci, M. D., C. T. Kelley, and C. T. Miller (1997), Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines, Adv. Wat. Resour., 20(1), 1–14.
6. ^ Germann, P. (2010), Comment on “Theory for source-responsive and free-surface film modeling of unsaturated flow”, Vadose Zone J. 9(4), 1000-1101.
7. ^ Gray, W. G., and S. Hassanizadeh (1991), Paradoxes and realities in unsaturated flow theory, Water Resour. Res., 27(8), 1847-1854.