Darcy's law
The Darcy's law (also Darcy equation ), named after the French engineer Henry Darcy, is an empirically (ie, by way of trials ) determined laws of fluid mechanics. It was published in 1856 in connection with the designed by Darcy water recovery plant for the city of Dijon. For a long time it was not clear why the Darcy's law works and what it is derived. Today we know that there is a special solution of the Navier -Stokes equation.
Definition
The Darcy's law states that the amount of water (flow rate in m³ / s), a total cross- sectional area ( pore space matrix ) of a porous medium ( eg sand) laminar flow is directly proportional to the hydraulic gradient:
- The term filtration rate has grown over time; in fact, it is a surface-related flow (english specific discharge), which specify the unit of speed comprising: It is also called volume flux density.
- The minus sign expresses the fact that the flow is in the direction of decreasing hydraulic heads.
The remaining two variables of the Darcy equation are discussed in the following subsections.
Coefficient of permeability
The proportionality of the Darcy's law, the permeability coefficient, is a dimensionless lossy characteristic value ( unit m / s), which can be determined by laboratory tests ( permeability test). It is dependent not only on the pore geometry, but also on the density (kg / m³), and dynamic viscosity (in Ns / m² ) of fluid flowing through, for example, water at 10 ° C or oil in the bottom ( petrochemical ):
Therein:
- K is the ( intrinsic) permeability ( unit m² ), an independent from the medium flowing through characteristic value for the permeability of a porous medium; often given in units of Darcy.
- G is the gravitational acceleration ( 9.81 m / s ² on Earth's surface ).
Hydraulic gradient
The dimensionless hydraulic gradient (also called hydraulic or potential gradient ) is generally how the filter velocity vf also a vector quantity and thus directed. It results from the local derivative of the piezometric head ( piezometric head ) h ( x) in the individual coordinate directions x:
In the groundwater hydrology of the hydraulic gradient between two points B and C is often assumed by the distance L from each other along the path of flow linear:
Transport speed
The transport velocity of water particles ( or completely dissolved substances in water ) is described by the distance velocity va [L / T], which is formed as quotient of the filtering speed and the effective flow area of the perfused medium, through the porosity expressed:
Non- linear regions
The time calculated by Darcy proportionality of velocity and hydraulic gradient can not always be observed in experiments.
For example, if the velocities of the pores are so large that no laminar but turbulent flow prevails, as a result of increased dissipation occurs a stronger potential loss, a plot of flow rate and between the gradient in the field of non-linear. To account for the turbulent effects, the Darcy equation by Philipp Forchheimer was extended by a term for Forchheimer equation.
Similar non-linear effects, it is also with very small gradients. Then surface forces dominate so that a non-linear decrease in the filtration rate can be observed with a falling gradient.
Flow of immiscible fluids
The Darcy's law is valid, strictly speaking, not when there are more fluids in the pores and can move. How strong is the influence depends on the viscosity of the fluids involved. This can occur, for example, in shifting of immiscible liquids ( LNAPL or DNAPL ) in groundwater.
With the infiltration of precipitation in the soil can often assume that the air can escape quickly enough and always is at atmospheric pressure in the gas phase. This flow process is often described analogous to Darcy's law, but with one dependent on the water saturation kf ( partially saturated flow ).