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 whenμ β /μ γ 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.
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Abbreviations
- A ωη :
-
interfacial area of theω-η interface contained within the macroscopic system, m2
- A ωe :
-
area of entrances and exits for the ω-phase contained within the macroscopic system, m2
- A ωη :
-
interfacial area of theω-η interface contained within the averaging volume, m2
- A * ωη :
-
interfacial area of theω-η interface contained within a unit cell, m2
- A * ωe :
-
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
- 〈H〉 βγ :
-
area average of the mean curvature, m−1
- \(\tilde H\) :
-
H − 〈H〉 βγ , deviation of the mean curvature, m−1
- I :
-
unit tensor
- K :
-
Darcy's law permeability tensor, m2
- K ω :
-
permeability tensor for theω-phase, m2
- K βγ :
-
viscous drag tensor for theβ-phase equation of motion
- K γβ :
-
viscous drag tensor for theγ-phase equation of motion
- L :
-
characteristic length scale for volume averaged quantities, m
- ℓ ω :
-
characteristic length scale for theω-phase, m
- n ωη :
-
unit normal vector pointing from theω-phase toward theη-phase (n ωη = −n ηω )
- p c :
-
〈p η〉η − 〈P β〉β, capillary pressure, N/m2
- p ω :
-
pressure in theω-phase, N/m2
- 〈p ω〉ω :
-
intrinsic phase average pressure for theω-phase, N/m2
- \(\tilde p\) ω :
-
p ω − 〈p ω〉ω, spatial deviation of the pressure in theω-phase, N/m2
- r 0 :
-
radius of the averaging volume, m
- t :
-
time, s
- v ω :
-
velocity vector for theω-phase, m/s
- 〈v ω 〉:
-
phase average velocity vector for theω-phase, m/s
- 〈v ω 〉ω :
-
intrinsic phase average velocity vector for theω-phase, m/s
- \(\tilde v_\omega \) :
-
v ω − 〈v ω〉ω, spatial deviation of the velocity vector for theω-phase, m/s
- V :
-
averaging volume, m3
- V ω :
-
volume of theω-phase contained within the averaging volume, m3
- ∈ω :
-
V ω/V, volume fraction of theω-phase
- ρ ω :
-
mass density of theω-phase, kg/m3
- μ ω :
-
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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Whitaker, S. Flow in porous media II: The governing equations for immiscible, two-phase flow. Transp Porous Med 1, 105–125 (1986). https://doi.org/10.1007/BF00714688
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DOI: https://doi.org/10.1007/BF00714688