Nonlinear evolution of viscous and gravitational instability in two-phase immiscible displacements is analyzed with a high-accuracy numerical method. We compare our results with linear stability theory and find good agreement at small times. The fundamental physical mechanisms of finger evolution and interaction are described in terms of the competing viscous, capillary, and gravitational forces. For the parameter range considered, immiscible viscous fingers are found to undergo considerably weak interaction as compared to miscible fingers. The wave number of nonlinear fingers decreases rapidly due to the shielding mechanism and scales uniformly as at long times. We have observed that even a small amount of density contrast can eliminate viscous fingers. The dominant feature for these flows is the gravity tongue, which develops a “ridge instability” when capillary and gravity effects are of similar magnitude.
REFERENCES
1.
G. M.
Homsy
, “Viscous fingering in porous media
,” Annu. Rev. Fluid Mech.
19
, 271
(1987
).2.
3.
Y. C.
Yortsos
and F. J.
Hickernell
, “Linear stability of immiscible displacement in porous media
,” SIAM J. Appl. Math.
49
, 730
(1989
).4.
5.
A. E.
Scheidegger
, The Physics of Flow through Porous Media
, 3rd ed. (University of Toronto Press
, Toronto
, 1974
).6.
M.
Sahimi
, “Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automate, and simulated annealing
,” Rev. Mod. Phys.
65
, 1393
(1993
).7.
M.
Blunt
and M. J.
King
, “Simulation and theory of two-phase flow in porous media
,” Phys. Rev. A
46
, 7680
(1992
).8.
Y. C.
Yortsos
, B.
Xu
, and D.
Salin
, “Delineation of microscale regimes of fully-developed drainage and implications for continuum models
,” Comput. Geosci.
5
, 257
(2001
).9.
W. F.
Engelberts
and L. J.
Klinkenberg
, “Laboratory experiments on the displacement of oil by water from packs of granular material
,” in Proceedings of the Third World Petroleum Congress
, The Hague, 1951
, Vol. II
, p. 544
.10.
P.
van Meurs
, “Use of transparent three-dimensional models for studying the mechanism of flow processes in oil reservoirs
,” Trans. AIME
210
, 295
(1957
).11.
R. L.
Chouke
, P.
van Meurs
, and C.
van Der Poel
, “The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media
,” Trans. AIME
216
, 188
(1959
).12.
E. J.
Peters
and D. L.
Flock
, “The onset of instability during two-phase immiscible displacements in porous media
,” SPE J.
21
, 249
(1981
).13.
D.
Pavone
, “Observations and correlations for immiscible viscous-fingering experiments
,” SPE Reservoir Eng.
7
, 187
(1992
); see also SPE paper 19670.14.
M. J.
King
and V. A.
Dunayevsky
, “Why waterflood works: A linearized stability analysis
,” Presented at the SPE annual Technical Conference and Exhibition, SPE Paper 19648 (San Antonio, TX, October 1989
).15.
A.
Riaz
and H. A.
Tchelepi
, “Linear stability analysis of immiscible two-phase flow in porous media with capillary dispersion and density variation
,” Phys. Fluids
16
, 4727
(2004
).16.
J.
Hagoort
, “Displacement stability of water drives in water-wet connate-water-bearing reservoirs
,” Soc. Pet. Eng. J.
14
, 63
(1974
).17.
A. J.
Chorin
, “The instability of fronts in a porous medium
,” Commun. Math. Phys.
91
, 103
(1983
).18.
M. T.
Vives
, Y.-C.
Chang
, and K. K.
Mohanty
, “Effect of wettability on adverse-mobility immiscible floods
,” SPEJ
4
, 260
(1999
).19.
M. J.
King
and H.
Scher
, “Probabilistic stability analysis of multiphase flow in porous media
,” presented at the SPE Annual Technical Conference and Exhibition, SPE Paper 14366 (Las Vegas, NV, September 1985
).20.
K. D.
Stephen
, G. E.
Pickup
, and K. S.
Sorbie
, “The local analysis of changing force balances in immiscible incompressible two-phase flow
,” Transp. Porous Media
45
, 63
(2001
).21.
M. J.
Blunt
, J. W.
Barker
, B.
Rubin
, M.
Mansfield
, I. D.
Culverwell
, and M. A.
Christie
, “Predictive theory for viscous fingering in compositional displacement
,” SPE Reservoir Eng.
9
, 73
(1994
).22.
D. S.
Hughes
and P.
Murphy
, “Use of a Monte Carlo method to simulate unstable miscible and immiscible flow through porous media
,” SPE Reservoir Eng.
3
, 1129
(1988
); see also SPE paper 17474.23.
D.
Gottlieb
and S. A.
Orszag
, Numerical Analysis of Spectral Methods: Theory and Applications
(Society for Industrial and Applied and Mathematics
, Philadelphia
, 1977
).24.
C.
Canuto
, M. Y.
Hussaini
, A.
Quarteroni
, and T. A.
Zang
, Spectral Methods in Fluid Dynamics
, in Springer Series in Computational Dynamics
(Springer
, New York
, 1986
).25.
R.
Peyret
, Spectral Methods for Incompressible Viscous Flow
, in Applied Mathematical Sciences
(Springer
, New York
, 2001
).26.
S. K.
Lele
, “Compact finite differences with spectral-like resolution
,” J. Comput. Phys.
103
, 16
(1992
).27.
M.
Ruith
and E.
Meiburg
, “Miscible rectilinear displacements with gravity override. Part 1. Homogeneous porous medium
,” J. Fluid Mech.
420
, 225
(2000
).28.
A.
Riaz
and E.
Meiburg
, “Three-dimensional vorticity dynamics of miscible porous media flows
,” J. Turbul.
3
, 61
(2002
).29.
W. B.
Zimmerman
and G. M.
Homsy
, “Nonlinear viscous fingering in miscible displacement with anisotropic dispersion
,” Phys. Fluids
3
, 1859
(1991
).30.
D. W.
Peaceman
, Fundamentals of Numerical Reservoir Simulation
(Elsevier Scientific
, North-Holland
, 1977
).31.
K.
Aziz
and A.
Settari
, Petroleum Reservoir Simulation
(Elsevier Applied Science
, London
, 1979
).32.
A. T.
Corey
, C. H.
Rathjens
, J. H.
Henderson
, and M. R. J.
Wyllie
, “Three-phase relative permeability
,” Trans. AIME
207
, 348
(1956
).33.
M.
Honarpour
, L.
Koederitz
, and A. H.
Harvey
, Relative Permeability of Petroleum Reservoirs
(CRC Press
, Boca Raton, FL
, 1986
).34.
E. J.
Peters
and S.
Khataniar
, “The effect of instability on relative permeability curves obtained by the dynamic-displacement method
,” SPE Form. Eval.
2
, 469
(1987
).35.
A.
Riaz
and E.
Meiburg
, “Vorticity interaction mechanisms in variable-viscosity heterogeneous miscible displacements with and without density contrast
,” J. Fluid Mech.
517
, 1
(2004
).36.
A.
Riaz
and E.
Meiburg
, “Miscible, porous media displacements with density stratification
,” Ann. N.Y. Acad. Sci.
1027
, 342
(2004
).37.
S. E.
Buckley
and M. C.
Leverett
, “Mechanisms of fluid displacements in sands
,” Trans. AIME
146
, 107
(1942
).38.
G. I.
Barenblatt
, Similarity, Self-similarity, and Intermediate Asymptotics
, edited by N.
Stein
; translation editor, M. Van
Dyke
(Consultants Bureau
, New York
, 1979
) (translated from Russian).39.
G. I.
Barenblatt
, V. M.
Entov
, and V. M.
Ryzhik
, Theory of Fluid Flows through Natural Rocks
, in Theory and Applications of Transport in Porous Media Series
(Kluwer Academic
, Boston
, 1990
).40.
Z.-X.
Chen
, “Some invariant solutions to two-phase fluid displacement problems including capillary effects
,” SPE Reservoir Eng.
3
, 691
(1988
).41.
Y. C.
Yortsos
and A. S.
Fokas
, “An analytical solution for linear waterflood including the effects of capillary pressure
,” SPE J.
23
, 115
(1983
); see also SPE paper 9407.42.
E. D.
Chikhliwala
, A. B.
Huang
, and Y. C.
Yortsos
, “Numerical study of the linear stability of immiscible displacements in porous media
,” Transp. Porous Media
3
, 257
(1988
).43.
C. T.
Tan
and G. M.
Homsy
, “Simulation of nonlinear viscous fingering in miscible displacement
,” Phys. Fluids
31
, 1330
(1988
).44.
H. A.
Tchelepi
and F. M.
Orr
, Jr., “Interaction of viscous fingering, permeability inhomogeneity and gravity segregation in three dimensions
,” SPE Reservoir Eng.
9
, 266
(1994
).45.
H. A.
Tchelepi
, F. M.
Orr
, Jr., N.
Rakotomalala
, D.
Salin
, and R.
Woumeni
, “Dispersion, permeability heterogeneity and viscous fingering: Acoustic experimental observations and particle tracking simulations
,” Phys. Fluids
5
, 1558
(1993
).46.
E. J.
Peters
, J. A.
Broman
, and W. H.
Broman
, Jr., “Computer image processing: A new tool for studying viscous fingering in core floods
,” SPE Reservoir Eng.
2
, 720
(1987
).47.
“
Visualization of immiscible displacements in a three-dismensional transparent porous medium
,” Exp. Fluids
4
, 336
(1986
).48.
J. P.
Stokes
, D. A.
Weitz
, J. P.
Gollub
, A.
Dougherty
, M. O.
Robbins
, P. M.
Chaikin
, and H. M.
Lindsay
, “Interfacial instability of immiscible displacements in a porous medium
,” Phys. Rev. Lett.
57
, 1718
(1986
).49.
G.
Lovoll
, Y.
Meheust
, R.
Toussaint
, J.
Schmittbuhl
, and K. J.
Maloy
, “Growth activity during fingering in a porous Hele-Shaw cell
,” Phys. Rev. E
70
, 026301
(2004
).50.
R.
Toussaint
, G.
Lovoll
, Y.
Meheust
, K. J.
Maloy
, and J.
Schmittbuhl
, “Influence of pore-scale disorder on viscous fingering during drainage
,” Europhys. Lett.
71
, 583
(2005
).51.
V.
Frette
, K. J.
Maloy
, F.
Boger
, J.
Feder
, and T.
Jossang
, “Diffusion-limited-aggregation-like displacement structures in a three-dimensional porous medium
,” Phys. Rev. A
42
, 3432
(1990
).52.
A.
Riaz
and E.
Meiburg
, “Three-dimensional miscible displacement simulations in homogeneous porous media with gravity override
,” J. Fluid Mech.
494
, 95
(2003
).53.
F. J.
Fayers
and A. H.
Muggeridge
, “Extensions to Dietz theory and behavior of gravity tongues in slightly tilted reservoirs
,” SPE Reservoir Eng.
5
487
(1990
); see also SPE paper 18438.© 2006 American Institute of Physics.
2006
American Institute of Physics
You do not currently have access to this content.