Abstract
The derivation of the governing equations for immiscible, two-phase flow through porous media by Whitaker (Transport in Porous Media 1, 105–125 (1986)) contains an error which is corrected in the present work. The modified equations contain terms not present in the original equations, but their presence does not cause any fundamental changes from the conclusions reached in the original work. However, these extra terms may be important in computations associated with the closure problem.
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Abbreviations
- A αω :
-
interfacial area of the α−ω interface contained within the averaging volume, m2
- A *αω :
-
interfacial area of the α−ω interface contained within a unit cell, m2
- A α i :
-
tensor defined in part by Equations (16) and (18), i = 1, 2
- a α i :
-
vector defined in part by Equations (17) and (19), i = 1, 2, m-1
- H :
-
mean curvature of the β−γ interface, m-1
- 〈H〉βγ :
-
area average of the mean curvature, m-1
- ≈H :
-
deviation of the mean curvature, m-1
- h i :
-
vector used in the decomposition given by equation 14 (i = 1, 2), s/m2
- L :
-
characteristic length scale for volume averaged quantities, m
- ℓα :
-
characteristic pore-scale length of the α-phase, m
- n αω :
-
unit normal at the α−ω interface pointing from the α-phase
- p c :
-
capillary pressure, N/m2
- \(\tilde p_\alpha \) :
-
spatial deviation of the pressure in the α-phase, N/m2
- 〈p α〉α :
-
intrinsic phase average pressure of the α-phase, N/m2
- r o :
-
characteristic length of the averaging volume, m
- V :
-
averaging volume, m3
- V α :
-
volume of the α-phase contained within the averaging volume, m3
- v α :
-
velocity of the α-phase, m/s
- \(\tilde v_\alpha \) :
-
spatial deviation of the velocity for the α-phase, m/s
- 〈v α〉α :
-
intrinsic phase average velocity of the α-phase, m/s
- w :
-
velocity of the β−γ interface, m/s
- ε:
-
(V β+V γ)/V, total void fraction
- εα :
-
V α/V, volume fraction of the α-phase
- μα :
-
viscosity of the α-phase, N s/m2
- ξα :
-
scalar defined in part by equations (17) and (19), s-1
- σ:
-
interfacial tension of the β−γ interface, N/m
- ψα :
-
represents any α-phase quantity
- ψ α :
-
vector defined in part by Equations (16) and (18), m/s
- σ:
-
solid phase
- β:
-
β-phase fluid
- γ:
-
γ-phase fluid
- 〈ψα〉α :
-
intrinsic phase average for the α-phase
- \(\tilde \psi _\alpha \) :
-
spatial deviation for the α-phase
- 〈 〉βγ :
-
interfacial average over the β−γ interface
References
Anderson, T. B. and Jackson, R., 1967, A fluid mechanical description of fluidized beds, Ind. Eng. Chem. Fundam. 6, 527–538.
Carbonell, R. G. and Whitaker, S., 1984, Heat and mass transfer in porous media, in J. Bear and M. Y. Corapcioglu (eds.), Fundamentals of Transport Phenomena in Porous Media, Martinus Nijhoff, Dordrecht, pp. 121–198.
Marle, C. M., 1967, Ecoulements monophasiques en mileu poreux, Rev. Inst. Francais du Petrole 22, 1471–1509.
Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media, AIChE J J. 13, 1066–1071.
Whitaker, S., 1967, Diffusion and dispersion in porous media, AIChE J. 13, 420–427.
Whitaker, S., 1986a, Flow in porous media I: A theoretical derivation of Darcy's law, Transport in Porous Media 1, 3–25.
Whitaker, S., 1986b, Flow in porous media II: The governing equations for immiscible, two-phase flow, Transport in Porous Media 1, 105–125.
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Torres, F.E. Closure of the governing equations for immiscible, two-phase flow: A research comment. Transp Porous Med 2, 383–393 (1987). https://doi.org/10.1007/BF00136443
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DOI: https://doi.org/10.1007/BF00136443