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 nonlinear partial differential equation, which is often difficult to approximate since it does not have a closed-form analytical solution. Although attributed to Richards, it is established [2] that this equation was actually discovered 9 years earlier by Lewis Fry Richardson in his book "Weather prediction by numerical process" published in 1922 (p.108).[3]

Darcy's law was developed for saturated flow in porous media; to this Richardson applied a continuity requirement suggested by Edgar 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 writes in one-dimension (vertical):

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

where

K is the hydraulic conductivity,
h is the matric head induced by capillary action,
z is the elevation above a vertical datum,
\( \theta \) is the volumetric water content, and
t is time.

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 \( {\hat {k}}: \)

\({\frac {\partial \theta }{\partial t}}=-{\frac {\partial }{\partial z}}q \)

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

\( {\displaystyle q=-K{\frac {\partial H}{\partial z}}} \)

Substituting q in the equation above, we get:

\( {\displaystyle {\frac {\partial \theta }{\partial t}}={\frac {\partial }{\partial z}}\left[K{\frac {\partial H}{\partial z}}\right]} \)

Substituting for H = h + z:

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

We then get the equation above, which is also called the mixed form [4] of Richards' equation.

Formulations

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. 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

\( {\displaystyle C(h){\frac {\partial h}{\partial t}}=\nabla \cdot \left(K(h)\nabla H\right)} \)

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

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

This function is called 'specific moisture capacity' in the literature, and can 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).[5]
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 the Richards equation is one of the most challenging problems in earth science. [6] Richards' equation has been criticized for being computationally expensive and unpredictable [7][8] 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,[9] and for being in some ways 'overly simplistic' [10] In one dimensional simulations of rainfall infiltration into dry soils, fine spatial discretization less than one cm is required near the land surface.[11], which is due to the small size of the representative elementary volume for multiphase flow in porous media. 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

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.
Knight, John; Raats, Peter. "The contributions of Lewis Fry Richardson to drainage theory, soil physics, and the soil-plant-atmosphere continuum" (PDF). EGU General Assembly 2016.
Richardson, Lewis Fry (1922). Weather prediction by numerical process. Cambridge, The University press. pp. 262.
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.
van Genuchten, M. Th. (1980). "A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils". Soil Science Society of America Journal. 44 (5): 892–898. Bibcode:1980SSASJ..44..892V. doi:10.2136/sssaj1980.03615995004400050002x. hdl:10338.dmlcz/141699.
Farthing, Matthew W., and Fred L. Ogden, (2017). Numerical solution of Richards’ Equation: a review of advances and challenges. Soil Science Society of America Journal, 81(6), pp.1257-1269.
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.
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.
Germann, P. (2010), Comment on “Theory for source-responsive and free-surface film modeling of unsaturated flow”, Vadose Zone J. 9(4), 1000-1101.
Gray, W. G., and S. Hassanizadeh (1991), Paradoxes and realities in unsaturated flow theory, Water Resour. Res., 27(8), 1847-1854.

Downer, Charles W., and Fred L. Ogden (2003), Hydrol. Proc.,18, pp. 1-22. DOI:10.1002/hyp.1306.