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
formula_1
where 
formula_2 is the volumetric flux;
formula_3 is the volumetric water content;
formula_4 is the liquid pressure head, which is negative for unsaturated porous media;
formula_5 is the unsaturated hydraulic conductivity;
formula_6 is the geodetic head gradient, which is assumed as formula_7 for three-dimensional problems.
Considering the law of mass conservation for an incompressible porous medium and constant liquid density, expressed as
formula_8,
where 
formula_9 is the sink term [Tformula_10], typically root water uptake.
Then substituting the fluxes by the Darcy-Buckingham law the following mixed-form Richards equation is obtained:
formula_11.
For modeling of one-dimensional infiltration this divergence form reduces to
formula_12.
Although attributed to L. A. Richards, the equation was originally introduced 9 years earlier by Lewis Fry Richardson in 1922.
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. The above-given mixed formulation involves two unknown variables: formula_3 and formula_4. This can be easily resolved by considering constitutive relation formula_15, which is known as the water retention curve. Applying the chain rule, the Richards equation may be reformulated as either formula_4-form (head based) or formula_3-form (saturation based) Richards equation.
Head-based.
By applying the chain rule on temporal derivative leads to 
 formula_18,
where formula_19 is known as the retention water capacity formula_20. The equation is then stated as
formula_21.
The head-based Richards equation is prone to the following computational issue: the discretized temporal derivative using the implicit Rothe method yields the following approximation:
formula_22
This approximation produces an error formula_23 that affects the mass conservation of the numerical solution, and so special strategies for temporal derivatives treatment are necessary.
Saturation-based.
By applying the chain rule on the spatial derivative leads to
formula_24
where formula_25, which could be further formulated as formula_26, is known as the soil water diffusivity formula_27. The equation is then stated as
formula_28
The saturation-based Richards equation is prone to the following computational issues. Since the limits formula_29 and formula_30, where formula_31 is the saturated (maximal) water content and formula_32 is the residual (minimal) water content a successful numerical solution is restricted just for ranges of water content satisfactory below the full saturation (the saturation should be even lower than air entry value) as well as satisfactory above the residual water content.
Parametrization.
The Richards equation in any of its forms involves soil hydraulic properties, which is a set of five parameters representing soil type. The soil hydraulic properties typically consist of water retention curve parameters by van Genuchten: (formula_33), where formula_34 is the inverse of air entry value [L−1], formula_35 is the pore size distribution parameter [-], and formula_36 is usually assumed as formula_37. Further the saturated hydraulic conductivity formula_38 (which is for non isotropic environment a tensor of second order) should also be provided. Identification of these parameters is often non-trivial and was a subject of numerous publications over several decades.
Limitations.
The numerical solution of the Richards equation is one of the most challenging problems in earth science. Richards' equation has been criticized for being computationally expensive and unpredictable because there is no guarantee that a solver will converge for a particular set of soil constitutive relations. Advanced computational and software solutions are required here to over-come this obstacle. The method has also been criticized for over-emphasizing the role of capillarity, and for being in some ways 'overly simplistic' In one dimensional simulations of rainfall infiltration into dry soils, fine spatial discretization less than one cm is required near the land surface, 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 the 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.
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