Transport in Porous Media

, Volume 1, Issue 2, pp 105–125

Flow in porous media II: The governing equations for immiscible, two-phase flow

  • Stephen Whitaker
Article

Abstract

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important whenμβ/μγ is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.

Key words

Volume averaging interfacial phenomena closure 

Nomenclature

Roman Letters (ω, η = β, γ, σ and ω ≠ η)

Aωη

interfacial area of theω-η interface contained within the macroscopic system, m2

Aωe

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

Aωη

interfacial area of theω-η interface contained within the averaging volume, m2

Aωη*

interfacial area of theω-η interface contained within a unit cell, m2

Aωe*

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

g

gravity vector, m2/s

H

mean curvature of theβ-γ interface, m−1

Hβγ

area average of the mean curvature, m−1

\(\tilde H\)

H − 〈Hβγ, deviation of the mean curvature, m−1

I

unit tensor

K

Darcy's law permeability tensor, m2

Kω

permeability tensor for theω-phase, m2

Kβγ

viscous drag tensor for theβ-phase equation of motion

Kγβ

viscous drag tensor for theγ-phase equation of motion

L

characteristic length scale for volume averaged quantities, m

ω

characteristic length scale for theω-phase, m

nωη

unit normal vector pointing from theω-phase toward theη-phase (nωη = −nηω)

pc

pηη − 〈Pββ, capillary pressure, N/m2

pω

pressure in theω-phase, N/m2

pωω

intrinsic phase average pressure for theω-phase, N/m2

\(\tilde p\)ω

pω − 〈pωω, spatial deviation of the pressure in theω-phase, N/m2

r0

radius of the averaging volume, m

t

time, s

vω

velocity vector for theω-phase, m/s

vω

phase average velocity vector for theω-phase, m/s

vωω

intrinsic phase average velocity vector for theω-phase, m/s

\(\tilde v_\omega \)

vω − 〈vωω, spatial deviation of the velocity vector for theω-phase, m/s

V

averaging volume, m3

Vω

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

Greek Letters

ω

Vω/V, volume fraction of theω-phase

ρω

mass density of theω-phase, kg/m3

μω

viscosity of theω-phase, Nt/m2

σ

surface tension of theβ-γ interface, N/m

τω

viscous stress tensor for theω-phase, N/m2

μ/ϱ

kinematic viscosity, m2/s

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Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Stephen Whitaker
    • 1
  1. 1.Department of Chemical EngineeringUniversity of CaliforniaDavisUSA

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