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Flow in porous media III: Deformable media

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Abstract

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.

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Abbreviations

A βσ :

interfacial area of theβ-σ interface contained within the macroscopic system, m2

A βσ :

interfacial area of theβ-σ interface contained within the averaging volume, m2

A σe :

area of entrances and exits for theσ-phase contained within the macroscopic system, m2

A * βσ :

interfacial area of theβ-σ interface contained within a unit cell, m2

A * βe :

area of entrances and exits for theσ-phase contained within a unit cell, m2

E σ :

Young's modulus for theσ-phase, N/m2

e i :

unit base vectors (i = 1, 2, 3)

g :

gravity vector, m2/s

H :

height of elastic, porous bed, m

k :

unit base vector (=e 3)

ℓ ω :

characteristic length scale for theω-phase, m

L :

characteristic length scale for volume-averaged quantities, m

n βσ :

unit normal vector pointing from theβ-phase toward theσ-phase (n βσ = -n σβ )

p β :

pressure in theβ-phase, N/m2

P β :

p β −ρ β g·r, N/m2

r 0 :

radius of the averaging volume, m

r :

position vector, m

t :

time, s

T ω :

total stress tensor in theσ-phase, N/m2

T 0σ :

hydrostatic stress tensor for theσ-phase, N/m2

u σ :

displacement vector for theσ-phase, m

V :

averaging volume, m3

V ω :

volume of theω-phase contained within the averaging volume, m3

v ω :

velocity vector for theω-phase, m/s

∈ ω :

V ω /V, volume fraction of theσ-phase

ρ ω :

mass density of theω-phase, kg/m3

μ β :

shear coefficient of viscosity for theβ-phase, Nt/m2

μ σ :

first Lamé coefficient for theσ-phase, N/m2

λ σ :

second Lamé coefficient for theσ-phase, N/m2

κ β :

bulk coefficient of viscosity for theβ-phase, Nt/m2

τ σ :

T σ −T 0 σ , a deviatoric stress tensor for theσ-phase, N/m2

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Authors and Affiliations

  1. Department of Chemical Engineering, University of California, 95616, Davis, CA, USA

    Stephen Whitaker

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  1. Stephen Whitaker
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Whitaker, S. Flow in porous media III: Deformable media. Transp Porous Med 1, 127–154 (1986). https://doi.org/10.1007/BF00714689

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  • DOI: https://doi.org/10.1007/BF00714689

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