Darcy Law

The Darcy law (also Darcy equation ), named after the French engineer Henry Darcy , is an empirically (i.e. through tests) determined law of fluid mechanics . It was published in 1856 in connection with the water extraction system designed by Darcy for the city of Dijon . For a long time it was not clear why Darcy's law worked and what it was derived from. Today we know that it is a special solution of the Navier-Stokes equation .

definition

Darcy's law states that the amount of water ( flow rate in m³ / s) that flows through an entire cross-sectional area ( pore space + matrix) of a porous medium (e.g. sand ) in a laminar manner is directly proportional to the hydraulic gradient : ${\ displaystyle Q}$ ${\ displaystyle A}$ ${\ displaystyle i}$

{\ displaystyle {\ frac {Q} {A}} = v _ {\ mathrm {f}} = - k _ {\ mathrm {f}} \ cdot i}

The term filter speed has grown over time; In fact, it is an area-related flow ( English specific discharge ), which has the unit of a speed: It is also called volume flow density . ${\ displaystyle v _ {\ mathrm {f}}}$ ${\ displaystyle \ mathrm {{\ frac {m ^ {3}} {s \ cdot m ^ {2}}} = {\ frac {m} {s}}}.}$

The proportionality factor is the permeability coefficient.

{\ displaystyle k _ {\ mathrm {f}}}

The minus sign expresses that the flow is in the direction of falling standpipe levels .

The remaining two quantities of the Darcy equation are explained in the following subsections.

Permeability coefficient

The proportionality of Darcy's law, the permeability coefficient is a dimensions afflicted characteristic value ( unit m / s), which can be determined by laboratory tests (permeability test). It is not only dependent on the pore geometry, but also on the density (in kg / m³) and the dynamic viscosity (in Ns / m²) of the flowing fluid , e.g. B. Water at 10 ° C or petroleum in the soil ( petrochemicals ): ${\ displaystyle k _ {\ mathrm {f}}}$ ${\ displaystyle \ rho _ {\ mathrm {F}}}$ ${\ displaystyle \ eta _ {\ mathrm {F}}}$

{\ displaystyle k _ {\ mathrm {f}} = K {\ frac {\ rho _ {\ mathrm {F}}} {\ eta _ {\ mathrm {F}}}} \, g}

Inside is:

${\ displaystyle K}$ the (intrinsic) permeability (unit m²), a characteristic value for the permeability of a porous medium that is independent of the medium flowing through; often also given in the unit Darcy .
${\ displaystyle g}$ the acceleration due to gravity

Hydraulic gradient

The dimensionless hydraulic gradient (also called hydraulic or potential gradient ) is generally, like the filter speed v _f , a vector quantity and is therefore directed. It results from the local derivative of the standpipe mirror height (piezometer height) h ( x ) in the individual coordinate directions x : ${\ displaystyle i}$

{\ displaystyle i (x) = - \ operatorname {grad} (h (x)) = - {\ frac {\ mathrm {d} h} {\ mathrm {d} x}}.}

In groundwater hydrology , the hydraulic gradient between two points B and C with the distance L from each other along the flow path is often assumed to be linear :

{\ displaystyle i = {\ frac {h_ {B} -h_ {C}} {L}}}

${\ displaystyle \ Rightarrow {\ frac {Q} {A}} = - k _ {\ mathrm {f}} \, {\ frac {h_ {B} -h_ {C}} {L}} \ ;.}$

Transport speed

The transport speed of water particles (or completely dissolved substances in water) is described by the distance speed, which is formed as the quotient of the filter speed and the effective porosity : ${\ displaystyle v _ {\ textrm {a}}}$ ${\ displaystyle \ Phi _ {f}}$

{\ displaystyle v _ {\ textrm {a}} = {\ frac {1} {\ Phi _ {f}}} \, v _ {\ textrm {f}} \ ;.}

Since is less than one, the distance speed and thus also the transport speed is greater than the filter speed. ${\ displaystyle \ Phi _ {f}}$

Non-linear areas

The proportionality of speed and hydraulic gradient determined by Darcy cannot always be observed in experiments.

If, for example, the velocities in the pores become so great that there is no laminar but rather a turbulent flow , a greater potential reduction occurs due to increased dissipation , and a plot between flow velocity and gradient becomes more non-linear in this area. In order to take the turbulent effects into account, Philipp Forchheimer's Darcy equation was expanded to include a term for the Forchheimer equation .

There are similar non-linear effects with very small gradients. Then surface forces can dominate, so that a non-linear decrease in the filter speed with falling gradient can be observed.

Flow of immiscible fluids

Strictly speaking , Darcy's law does not apply if several fluids can reside and move in the pores. How strong the influence is depends on the viscosity of the fluids involved. This can e.g. B. occur when immiscible liquids ( LNAPL or DNAPL ) move in the groundwater .

In the infiltration of rainfall in the soil can often assume that the air can escape quickly enough, and always atmospheric pressure in the gas phase prevails. This flow process is often described in analogy to Darcy's law, but with a k _f value that depends on the water saturation (partially saturated flow).

literature

Christoph Adam, Walter Gläßer, Bernward Hölting: Hydrogeological dictionary. Enke Verlag, Stuttgart / New York 2000, ISBN 3-13-118271-7 .
Jacob Bear: Dynamics of fluids in Porous Media. Dover Publications, New York 1972, ISBN 0-444-00114-X .
Karl-Franz Busch , Ludwig Luckner, Klaus Tiemer: Geohydraulik. (= Textbook of Hydrogeology. Volume 3). 3. Edition. Borntraeger brothers, Berlin / Stuttgart 1993, ISBN 3-443-01004-0 .
W. Kinzelbach, R. Rausch: Groundwater modeling. Media combination. Bornträger, Berlin / Stuttgart 1995, ISBN 3-443-01032-6 .
R. Allan Freeze, John A. Cherry: Groundwater. Prentice-Hall, Englewood Cliffs, NJ 1979, ISBN 0-13-365312-9 .
Hanspeter Jordan, Hans-Jörg Weder: Hydrogeology. Basics and methods. Enke, Stuttgart 1995, ISBN 3-432-26882-3 .
Amin F. Zarandi, Krishna M. Pillai, Adam S. Kimmel: Spontaneous imbibition of liquids in glass-fiber wicks. Part I: Usefulness of a sharp-front approach. In: American Institute of Chemical Engineers AIChE Journal. Volume 63, 2018, pp. 294-305. doi: 10.1002 / aic.15965