Darcy-Richards equation

Flow is more difficult to model in the unsaturated zone because hydraulic conductivity changes as water content changes. In unsaturated flow models, the Darcy-Richards equation, which is a more general form of Darcy s law, is used. For isotropic porous media in three dimensions. 

In this equation r) is the volume of water per unit volume of solid, t is time, F is the Darcy flux of water relative to the solid, and m is a space like coordinate defined in terms of the distribution of the solid phase. Substitution of Darcy s law in this equation results in equations similar to the Richards equation and, at least in 1-dimension, these can be solved like their non-swelling analogues. This equation implicitly deals with volume change that might accompany change in water content. 

Work of Richards—L. A. Richards (1931) gave an excellent theoretical presentation of the factors involved in a study of capillary constants. He succeeded in establishing a general capillary equation and gave a method for determining conductivity. The conductivity was defined as the constant contained in Darcy s equation (Eq 13-2)... 

Convection (also known as advection) is the vector, which results from the DARCY or the RICHARDS equations. It describes the flow velocity or the flow distance for a certain time t. In general convection has the major influence on... 

Conventionally used Darcy s law and Richards equation (2 order partial differential equations) are exanq>les of a system with a positive feedback, but witiiout a negative feedback conqmnent. Negative and positive feedback mechanisms are taken into account by using tiie difference-differential equation for soil-moisture balance (75) and the Kuramoto-Sivashinsky equation 74) (see below). 

Though Washburn model [63] is most commonly applied to describe liquid infiltration into textiles, it has got some limitations such as the model cannot be used to model source driven flows without modification and it cannot be applied to describe spreading in two- or three-dimensions. The other equation which is commonly used to model liquid transport in porous material is Richard s equation [64]. Richard s equation is a semi-empirical formulation based on Darcy s law, where the rate of volmnetric liquid flow is proportional to the driving pressure. An essential character of the models based on Richard s equation is that they contain the functional relation between permeability, pressure and moisture. 

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