Abstract
The assumption of Vertical Equilibrium (VE) and of parallel flow conditions, in general, is often applied to the modeling of flow and displacement in natural porous media. However, the methodology for the development of the various models is rather intuitive, and no rigorous method is currently available. In this paper, we develop an asymptotic theory using as parameter the variable\(R_{{L}} = L/H\sqrt {k_{{V}} /k_{{H}} } \). It is rigorously shown that the VE model is obtained as the leading order term of an asymptotic expansion with respect to 1/R 2L . Although this was numerically suspected, it is the first time that it is theoretically proved. Using this formulation, a series of special cases are subsequently obtained depending on the relative magnitude of gravity and capillary forces. In the absence of strong gravity effects, they generalize previous works by Zapata and Lake (1981), Yokoyama and Lake (1981) and Lake and Hirasaki (1981), on immiscible and miscible displacements. In the limit of gravity-segregated flow, we prove conditions for the fluids to be segregated and derive the Dupuit and Dietz (1953) approximations. Finally, we also discuss effects of capillarity and transverse dispersion.
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Abbreviations
- C :
-
concentration, dimensionless
- D :
-
dispersion tensor [L 2 T −1]
- f :
-
fractional flow
- g :
-
gravity acceleration [LT −2]
- H :
-
reservoir thickness [L]
- h :
-
dimensionless front location
- k :
-
mean permeability [L 2]
- K :
-
permeability [L 2]
- L :
-
reservoir length [L]
- M :
-
viscosity ratio, dimensionless
- N CT :
-
transverse capillary number
- N G :
-
gravity number
- N TD :
-
transverse dispersion number
- P :
-
dimensionless pressure
- q :
-
flow velocity [LT −1]
- R L :
-
VE parameter
- S :
-
saturation
- T :
-
time [T]
- t :
-
dimensionless time
- u :
-
dimensionless horizontal velocity
- v :
-
dimensionless vertical velocity
- X :
-
horizontal coordinate [L]
- x :
-
dimensionless horizontal coordinate
- y :
-
dimensionless vertical coordinate
- w :
-
dimensionless vertical velocity
- α :
-
dispersivity [L]
- γ :
-
interfacial tension [MT −2]
- δ :
-
permeability ratio, dimensionless
- ε :
-
aspect ratio, dimensionless
- ΘG :
-
gravity number
- κ :
-
dimensionless permeability
- λ :
-
dimensionless mobility
- μ :
-
viscosity [ML −1 T −1]
- Π:
-
dimensionless pressure
- ρ :
-
density [ML −3]
- φ :
-
porosity, dimensionless
- ψ :
-
normalized mobility, dimensionless
- a:
-
air
- c:
-
capillary
- H:
-
horizontal
- L:
-
longitudinal
- o:
-
oil
- or:
-
residual oil
- r:
-
relative
- T:
-
total
- V:
-
vertical
- w:
-
water
- wr:
-
residual water
- O:
-
leading order
References
Bear, J. 1972,Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, Inc., New York.
Beckers, H. L. 1965, The deformation of an interface between two fluids in a porous medium,Appl. Science Res. Annual,14, 101.
Chaoutche, M., Rakotomalala, N., Salin, D. O., Xu, B. and Yortsos, Y. C. 1994, Capillary Effects in Drainage in Heterogeneous Media: Continuum Modelling, Experiments and Pore Network Simulations, Chem. Eng. Sci., Vol. 49(15), 2447–2466.
Coats, K. R., Dempsey, J. R. and Henderson, J. H. 1971, The use of vertical equilibrium in two-dimensional simulation of three-dimensional reservoir performance,Soc. Petroleum Eng. J.,11, 63–71.
Dietz, D. N. 1954,A Theoretical Approach to the Problem of Encroaching and By-Passing Edge Water, Akad. van Wetenschappen, Amsterdam, Proc.56-B, 83.
Fayers, F. J. 1984,An Approximate Model with Physically Interpretable Parameters for Representing Viscous Fingering, paper SPE 13166 presented at the Society of Petroleum Engineers Fall Technical Conference and Exhibition, Houston, Texas.
Fayers, F. J. and Muggeridge, A. H. 1990, Extensions to Dietz theory and behavior of gravity tongues in slightly tilted reservoirs,SPE Reservoir Engineering, 487–494.
Koval, E. J. 1963, A method for predicting the performance of unstable miscible displacement in heterogeneous media,Soc. Petroleum Eng. J.,3, 145–155.
Lake, L. W. 1989,Enhanced Oil Recovery, Chapter 6, Prentice Hall, Englewood Cliffs, NJ.
Lake, L. W. 1991, private communication.
Lake, L. W. and Hirasaki, G. J. 1981, Taylor's dispersion in stratified porous media,Soc. Pet. Eng. J.,21, 459–468.
Lake, L. W., Kasap, E. and Shook, M. 1990,Pseudofunctions — The Key to Practical Use of Reservoir Description, North Sea Oil and Gas Reservoirs-II, The Norwegian Institute of Technology, Graham & Trotman, 297–308.
Le Fur, B. and Sourieau, P. 1963, Etude de l'écoulement diphasique dans une couche inclinée et dans un modéle rectangulaire de milieu poreux,Rev. Inst. Fr. Petrol. 18, 325–343.
Pande, K. K. and Orr, F. M. Jr. 1989,Interaction of Phase Behavior, Reservoir Heterogeneity, and Crossflow in CO 2 Floods, paper SPE 19668 presented at the Society of Petroleum Engineers Fall Technical Conference and Exhibition, San Antonio, Texas.
Taylor, G. I. 1953, Dispersion of soluble matter in solvent flowing slowly through a tube,Proc. Roy. Soc. A,219, 186–203.
Todd, M. R. and Longstaff, W. J. 1972, The development, testing, and application of a numerical simulator for predicting miscible flood performance,JPT, 874–82.
Yang, Z. and Yortsos, Y. C. 1995, in preparation.
Yortsos, Y. C. 1992, Analytical studies for processes at vertical equilibrium,Proc. 3rd European Conference on the Mathematics of Oil Recovery, 183–196, Delft University Press.
Yortsos, Y. C. and Chang, J. 1990, Capillary effects in steady-state flow in heterogeneous cores,Transport in Porous Media,5, 399–420.
Yortsos, Y. C. and Zeybek, M. 1988, Dispersion-driven instability in miscible displacement in porous media,Phys. Fluids,31, 3511–3518.
Yokoyama, Y. and Lake, L. W. 1981,The Effects of Capillary Pressure on Immiscible Displacements in Stratified Porous Media, paper SPE 10109, presented at the Society of Petroleum Engineers Fall Technical Conference and Exhibition, San Antonio, Texas.
Zapata, V. J. and Lake, L. W. 1981,A Theoretical Analysis of Viscous Crossflow, paper SPE 10111 presented at the Society of Petroleum Engineers Fall Technical Conference and Exhibition, San Antonio, Texas.
Zimmerman, W. B. and Homsy, G. M. 1991, Nonlinear viscous fingering in miscible displacement with anisotropic dispersion,Phys. Fluids A, 1959–1871.
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Yortsos, Y.C. A theoretical analysis of vertical flow equilibrium. Transp Porous Med 18, 107–129 (1995). https://doi.org/10.1007/BF01064674
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DOI: https://doi.org/10.1007/BF01064674