Abstract
An idealized model of a porous rock consisting of a bundle of capillary tubes whose cross-sections are regular polygons is used to assess the importance of viscous coupling or lubrication during simultaneous oil-water flow. Fluids are nonuniformly distributed over tubes of different characteristic dimension because of the requirements of capillary equilibrium and the effect of interfacial viscosity at oil-water interfaces is considered. With these assumptions, we find that the importance of viscous coupling depends on the rheology of the oil-water interface. Where the interfacial shear viscosity is zero, viscous coupling leading to a dependence of oil relative permeability on oil-water viscosity ratio for viscosity ratios greater than one is important for a range of pore cross-section shapes and pore size distributions. For nonzero interfacial shear viscosity, viscous coupling is reduced. Using values reported in the literature for crude oil-brine systems, we find no viscous coupling.
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Abbreviations
- a, b, c :
-
Undertermined parameter sets in velocity distribution function
- k :
-
Permeability
- n :
-
Unit normal vector too-w interface
- r :
-
Coordinate in cylindrical system
- s :
-
Number of sides of polygon tube cross section
- v :
-
Fluid velocity
- z :
-
Coordinate in cylindrical system
- F, F o, Fw :
-
Coefficients in tube flow equation
- Gp :
-
Pressure Gradient
- H :
-
Interface mean curvature
- J,J 1,J 2,J 3 :
-
Integrals defined by (A3) and (A11)–(A13)
- M :
-
Number of tube sizes
- N :
-
Defines number of undetermined parameter in velocity distribution
- N ca :
-
Capillary number
- N ε :
-
Dimensionless interfacial shear viscosity
- P :
-
Pressure
- Q :
-
Volume flow rate
- R :
-
Characteristic tube dimension (Figure A1)
- S o, Sw :
-
Oil and water saturations
- S t, Si :
-
Tube wall and oil-water interface area
- V (m) :
-
Pore volume of tubes of dimensionR(m) in bundle
- ε :
-
Interfacial shear viscosity
- Μ :
-
Viscosity
- σ :
-
Interfacial tension
- θ:
-
Coordinate in cylindrical system
- Φ:
-
Coordinate in second cylindrical system
- Τ:
-
Stress tensor
- o :
-
Oil phase quantity
- w :
-
Water phase quantity
- r :
-
r-component of vector
- θ :
-
θ-component of vector
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Ehrlich, R. Viscous coupling in two-phase flow in porous media and its effect on relative permeabilities. Transp Porous Med 11, 201–218 (1993). https://doi.org/10.1007/BF00614812
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DOI: https://doi.org/10.1007/BF00614812