Richard's equation

The Richards equation , after Lorenzo A. Richards (1904-1993), describes the seepage flow of a fluid (e.g. water or oil) in a porous medium (e.g. the ground ).

The fluid can be in an unsaturated state; H. In addition to the fluid, air can also be present in the pores of the medium. This is based on a simplified view of the pore structure of the medium: the medium is reduced to the proportion of pore volume to solid and the hydraulic conductivity of the porous medium. In this sense, fluid, air and medium are at the same time in a point in space (this is also called a representative element volume). A local averaging process thus takes place. The proportion of the fluid in the pore volume of a point in space is called saturation .

The Richards equation can be derived by combining the continuity equation and Darcy's law . The continuity equation describes the mass retention of the fluid, Darcy's law forms the basis for the flow behavior of the fluid in the porous medium.

The Richards equation is a 2nd order partial differential equation of parabolic type :

${\displaystyle {\frac {\partial }{\partial t}}S(p)-\nabla \cdot K(p)\,\nabla p=0}$

With

• the ( partial ) time derivative ${\displaystyle {\frac {\partial }{\partial t}}}$
• the saturation of the phase ${\displaystyle S}$
• the pressure ${\displaystyle p}$
• the hydraulic conductivity of the medium ${\displaystyle K}$
• the Nabla operator . ${\displaystyle \nabla }$

The conductivity of the medium is determined experimentally and depends on the saturation of the fluid: . Since these can be determined again by the pressure of the fluid ( ), also depends on the conductivity of the fluid pressure from: . The pressure is created by capillary forces . ${\displaystyle K(S)}$ ${\displaystyle S(p)}$ ${\displaystyle K(p)}$

The Richards equation can be solved numerically with the help of the finite element method .

See also

• Surface tension

literature

• Lorenzo Adolph Richards: Capillary conduction of liquids through porous mediums . In: Physics. 1, 1931, ZDB -ID 220641-9 , pp. 318-333 (Ithaca NY, Cornell Univ., PhD Thesis, 1931), doi : 10.1063 / 1.1745010 .
• M. Amin F. Zarandi, Krishna M. Pillai, Spontaneous imbibition of liquid in glass fiber wicks, Part II: Validation of a diffuse-front model , American Institute of Chemical Engineers AIChE Journal. Volume 64, Issue 1, January 2018, Pages 306–315. DOI: 10.1002 / aic.15856