In this study, the viscous fingering instability of miscible displacement involving a viscoelastic fluid is investigated using both linear stability analysis and computational fluid dynamics for the first time. The Oldroyd-B model is used as the constitutive equation of a viscoelastic fluid. Here, it is assumed that one of the displacing fluids or the displaced one is viscoelastic. In linear stability analysis, the quasi-steady state approximation and six order shooting method are used to predict the growth rate of the disturbance in the flow. It is shown that the flow is more stabilized when the elasticity (Weissenberg number) of the displaced or displacing viscoelastic fluid is increased. In the nonlinear simulation, using the spectral method based on Hartley transforms and the fourth-order Adams-Bashforth technique, the effect of the viscoelastic fluid on this instability has been studied. Evaluation of concentration contours, mixing length, sweep efficiency, and transversely average concentration show that the elasticity has a significant effect on the fingering instability and the flow becomes more stable by increasing the Weissenberg number.
REFERENCES
1.
M. A.
Nilsson
, R.
Kulkarni
, L.
Gerberich
, R.
Hammond
, R.
Singh
, E.
Baumhoff
, and J. P.
Rothstein
, “Effect of fluid rheology on enhanced oil recovery in a microfluidic sandstone device
,” J. Non-Newtonian Fluid Mech.
202
, 112
(2013
).2.
S.
Hill
and F.
Inst. P
, “Channeling in packed columns
,” Chem. Eng. Sci.
1
, 247
(1952
).3.
P. G.
Saffman
and G.
Taylor
, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid
,” Proc. R. Soc. A
245
, 312
(1958
).4.
E.
Koval
, “A method for predicting the performance of unstable miscible displacement in heterogeneous media
,” Soc. Pet. Eng. J.
3
, 145
(1963
).5.
G. M.
Homsy
, “Viscous fingering in porous media
,” Annu. Rev. Fluid Mech.
19
, 271
(1987
).6.
C. T.
Tan
and G. M.
Homsy
, “Simulation of nonlinear viscous fingering in miscible displacement
,” Phys. Fluids
31
, 1330
(1988
).7.
C. T.
Tan
and G. M.
Homsy
, “Stability of miscible displacements in porous media: Rectilinear flow
,” Phys. Fluids
29
, 3549
(1986
).8.
D. W.
Peaceman
and H. H.
Rachford
, Jr., “Numerical calculation of multidimensional miscible displacement
,” Soc. Pet. Eng. J.
2
, 327
(1962
).9.
M.
Christie
and D.
Bond
, “Detailed simulation of unstable processes in miscible flooding
,” SPE Reservoir Eng.
2
, 514
(1987
).10.
W. B.
Zimmerman
and G. M.
Homsy
, “Nonlinear viscous fingering in miscible displacement with anisotropic dispersion
,” Phys. Fluids A
3
, 1859
(1991
).11.
W. B.
Zimmerman
and G. M.
Homsy
, “Three-dimensional viscous fingering: A numerical study
,” Phys. Fluids A
4
, 1901
(1992
).12.
C. T.
Tan
and G. M.
Homsy
, “Viscous fingering with permeability heterogeneity
,” Phys. Fluids A
4
, 1099
(1992
).13.
O.
Manickam
and G. M.
Homsy
, “Stability of miscible displacements in porous media with nonmonotonic viscosity profiles
,” Phys. Fluids A
5
, 1356
(1993
).14.
A.
De Wit
and G. M.
Homsy
, “Viscous fingering in periodically heterogeneous porous media. I. Formulation and linear instability
,” J. Chem. Phys.
107
, 9609
(1997
).15.
A.
De Wit
and G. M.
Homsy
, “Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations
,” J. Chem. Phys.
107
, 9619
(1997
).16.
K.
Ghesmat
and J.
Azaiez
, “Viscous fingering instability in porous media: Effect of anisotropic velocity-dependent dispersion tensor
,” Transp. Porous Media
73
, 297
(2008
).17.
M.
Islam
and J.
Azaiez
, “Miscible thermo-viscous fingering instability in porous media. Part 1: Linear stability analysis
,” Transp. Porous Media
84
, 821
(2010
).18.
M.
Islam
and J.
Azaiez
, “Miscible thermo-viscous fingering instability in porous media. Part 2: Numerical simulations
,” Transp. Porous Media
84
, 845
(2010
).19.
M.
Norouzi
and M.
Shoghi
, “A numerical study on miscible viscous fingering instability in anisotropic porous media
,” Phys. Fluids
26
, 084102
(2014
).20.
J.
Nittmann
, G.
Daccord
, and H. E.
Stanley
, “Fractal growth of viscous fingers: Quantitative characterization of a fluid instability phenomenon
,” Nature
314
, 141
(1985
).21.
G.
Daccord
, J.
Nittmann
, and H. E.
Stanley
, “Radial viscous fingers and diffusion-limited aggregation: Fractal dimension and growth sites
,” Physical Rev. Lett.
56
, 336
(1986
).22.
H.
Pascal
, “Stability of a moving interface in porous medium for non-Newtonian displacing fluids and its applications in oil displacement mechanism
,” Acta Mech.
58
, 81
(1986
).23.
H.
Pascal
, “Stability of non-Newtonian fluid interfaces in a porous medium and its applications in an oil displacement mechanism
,” J. Colloid Interface Sci.
123
, 14
(1988
).24.
S.
Wilson
, “The Taylor–Saffman problem for a non-Newtonian liquid
,” J. Fluid Mech.
220
, 413
(1990
).25.
D. E.
Smith
, X. Z.
Wu
, A.
Libchaber
, E.
Moses
, and T.
Witten
, “Viscous finger narrowing at the coil-stretch transition in a dilute polymer solution
,” Phys. Rev. A
45
, 2165
(1992
).26.
J. E.
Sader
, D. Y.
Chan
, and B. D.
Hughes
, “Non-Newtonian effects on immiscible viscous fingering in a radial Hele-Shaw cell
,” Phys. Rev. E
49
, 420
(1994
).27.
Y. S.
Wu
and K.
Pruess
, “A numerical method for simulating non-Newtonian fluid flow and displacement in porous media
,” Adv. Water Resour.
21
, 351
(1998
).28.
B. K.
Singh
and J.
Azaiez
, “Numerical simulation of viscous fingering of shear-thinning fluids
,” Can. J. Chem. Eng.
79
, 961
(2001
).29.
J.
Azaiez
and B. K.
Singh
, “Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell
,” Phys. Fluids
14
, 1557
(2002
).30.
H.
Li
, B.
Maini
, and J.
Azaiez
, “Experimental and numerical analysis of the viscous fingering instability of shear-thinning fluids
,” Can. J. Chem. Eng.
84
, 52
(2006
).31.
M. C.
Kim
and C. K.
Choi
, “Linear analysis on the stability of miscible dispersion of shear-thinning fluids in porous media
,” J. Non-Newtonian Fluid Mech.
166
, 1211
(2011
).32.
J.
Azaiez
, “Fingering instabilities in miscible displacement flows of non-Newtonian fluids
,” J. Porous Media
7
, 29
(2004
).33.
M.
Norouzi
and M. R.
Shoghi
, “Nonlinear simulation of non-Newtonian viscous fingering instability in anisotropic porous media
,” Modares Mech. Eng.
15
, 415
(2015
).34.
M. R.
Shoghi
and M.
Norouzi
, “Linear stability analysis and nonlinear simulation of non-Newtonian viscous fingering instability in heterogeneous porous media
,” Rheol. Acta
54
, 973
(2015
).35.
S.
Mora
and M.
Manna
, “From viscous fingering to elastic instabilities
,” J. Non-Newtonian Fluid Mech.
173
, 30
(2012
).36.
M. H.
Kayhani
, H.
Shokri
, and M.
Norouzi
, “Nonlinear simulation of viscoelastic fingering instability
,” Modares Mech. Eng.
16
, 47
(2016
).37.
D. J.
Pye
, “Improved secondary recovery by control of water mobility
,” J. Pet. Technol.
16
, 911
(1964
).38.
B.
Sandiford
, “Laboratory and field studies of water floods using polymer solutions to increase oil recoveries
,” J. Pet. Technol.
16
, 917
(1964
).39.
J.
Wang
and M.
Dong
, “Optimum effective viscosity of polymer solution for improving heavy oil recovery
,” J. Pet. Sci. Eng.
67
, 155
(2009
).40.
Z.
Guo
, M.
Dong
, Z.
Chen
, and J.
Yao
, “A fast and effective method to evaluate the polymer flooding potential for heavy oil reservoirs in Western Canada
,” J. Pet. Sci. Eng.
112
, 335
(2013
).41.
H.
Pei
, G.
Zhang
, J.
Ge
, L.
Zhang
, and H.
Wang
, “Effect of polymer on the interaction of alkali with heavy oil and its use in improving oil recovery
,” Colloids Surf., A
446
, 57
(2014
).42.
R. B.
Bird
, R.
Armstrong
, and O.
Hassager
, Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics
(Wiley
, 1987
).43.
M. G.
Alishaev
and A. K.
Mirzadjanzade
, “For the calculation of delay phenomenon in filtration theory
,” Izvestiia VUZov Neft i Gaz
6
, 71
(1975
).44.
Z.
Zhang
, C.
Fu
, and W.
Tan
, “Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below
,” Phys. Fluids
20
, 084103
(2008
).© 2017 Author(s).
2017
Author(s)
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