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Theoretical derivation of Darcy's law

Eine theoretische Ableitung des Darcyschen Gesetzes

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Summary

Darcy's law for anisotropic porous media is derived from the Navier-Stokes equation by using a formal averaging procedure. Particular emphasis is placed upon the proof that the permeability tensor is symmetric. In addition, it is shown that there is a one-to-one relationship between the local and macroscopic velocity fields. This leads to the interesting phenomenological observation that the local velocity vector at any given point must always lie either on a fixed line or in a fixed plane. All of this holds true for an incompressible homogeneous Newtonian fluid moving slowly through a rigid porous medium with uniform porosity under isothermal and steady state conditions. The question whether Darcy's law is applicable under nonsteady or compressible flow conditions, or when the medium has nonuniform porosity, is also discussed. Finally, it is shown that the Hagen-Poiseuille equation, as well as the expression describing Couette flow between parallel plates, can be derived from the equations presented in this work and may thus be viewed as special cases of Darcy's law.

Zusammenfassung

Das Darcysche Gesetz für anisotrope poröse Werkstoffe wird von den Navier-Stokes-Gleichungen durch formale Mittelwertbildung abgeleitet. Insbesondere betont wird der Beweis der Symmetrie des Permeabilitätstensors. Weiter wird gezeigt, daß eine eineindeutige Beziehung zwischen lokalen und makroskopischen Geschwindigkeitsfeldern existiert. Dies führt zur interessanten phänomenologischen Beobachtung, daß in jedem Punkt der lokale Geschwindigkeitsvektor entweder auf einer festen Geraden oder in einer festen Ebene liegt. All dies gilt für inkompressible homogene Newtonsche Flüssigkeiten, die sich langsam, stationär unter isothermen Bedingungen durch einen starren porösen Körper gleichförmiger Porösität bewegen. Die Frage, ob das Darcysche Gesetz für instationäre Strömungen oder kompressible Fälle oder für ungleichförmige Porösität gilt, wird ebenfalls diskutiert. Abschließend wird gezeigt, daß die Hagen-Poiseuille-Gleichung und der Ausdruck für die Couette-Strömung zwischen parallelen Platten von der in dieser Arbeit angegebenen Gleichung abgeleitet und daher als Spezialfälle des Darcyschen Gesetzes betrachtet werden können.

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Abbreviations

a :

scalar ratio betweenv i and projection of <v i >* at centroid ofR in the direction ofv i

b :

half distance between parallel plates,L

d :

diameter of straight tube,L

f :

any local property of the fluid defined in the pores and vanishing in the solids

<f>:

bulk-volume average off

<f>* :

pore-volume average off

g :

acceleration due to gravity,LT −2

g i :

(0,0,−g),LT −2

G :

Green's function of porous space

h(f) :

probability density off onГ sf

k ij :

permeability tensor,L 2

l i :

unit vector lying in the plane ofv i (orm i ) and <v i >* and perpendicular tom i

m i :

unit vector parallel tov i

n i :

unit vector normal toГ sf and pointing into the solids

p :

local fluid pressure,ML −1 T −2

<p>* :

macroscopic (pore-volume average) pressure,ML −1 T −2

r :

radius of straight tube,L

R :

representative elementary volume of porous medium,L 3

t :

time,T

v i :

local fluid velocity vector,LT −1

<v i >:

bulk-volume average ofv i (darcy velocity or specific flux vector),LT −1

x i :

vector of Cartesian coordinates,x 3 being the vertical,L

α ij :

−am i m j or −(δ ij −l i l j )

Г m :

Dirichlet boundary segment of Ω,L 2

Г sf :

solid-fluid interfaces inR, L 2

δ; ij :

Kronecker delta

μ:

dynamic fluid viscosity,ML −1 T −1

ϱ:

fluid density,ML −3

ϕ:

porosity ofR

Ф:

local force potential, defined as\(---x_i g_i + \frac{p}{\varrho }\),L 2 T −2

〈Φ〉* :

macroscopic (pore-volume average) force potential, defined as\(---x_i g_i + \frac{{\left\langle p \right\rangle *}}{\varrho }\),L 2 T −2

Φ m :

prescribed values of Ф at external boundaries of porous medium, L2T−2

Δ 2 :

Laplacian operator,L −2

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  1. Department of Hydrology and Water Resources, The University of Arizona, 85721, Tucson, AZ, USA

    S. P. Neuman

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Neuman, S.P. Theoretical derivation of Darcy's law. Acta Mechanica 25, 153–170 (1977). https://doi.org/10.1007/BF01376989

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