### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 153-170 |

Number of pages | 18 |

Journal | Acta Mechanica |

Volume | 25 |

Issue number | 3-4 |

DOIs | |

State | Published - Sep 1977 |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics

### Cite this

*Acta Mechanica*,

*25*(3-4), 153-170. https://doi.org/10.1007/BF01376989

**Theoretical derivation of Darcy's law.** / Neuman, Shlomo P.

Research output: Contribution to journal › Article

*Acta Mechanica*, vol. 25, no. 3-4, pp. 153-170. https://doi.org/10.1007/BF01376989

}

TY - JOUR

T1 - Theoretical derivation of Darcy's law

AU - Neuman, Shlomo P

PY - 1977/9

Y1 - 1977/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0017427064&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0017427064&partnerID=8YFLogxK

U2 - 10.1007/BF01376989

DO - 10.1007/BF01376989

M3 - Article

AN - SCOPUS:0017427064

VL - 25

SP - 153

EP - 170

JO - Acta Mechanica

JF - Acta Mechanica

SN - 0001-5970

IS - 3-4

ER -