Abstract
In the presence of diffusion, stability of three-layer Hele-Shaw flows which models enhanced oil recovery processes by polymer flooding is studied for the case of variable viscosity in the middle layer. This leads to the coupling of the momentum equation and the species advection-diffusion equation the hydrodynamic stability study of which is presented in this paper.
Linear stability analysis of a potentially unstable three-layer rectilinear Hele-Shaw flow is used to examine the effects of species diffusion on the stability of the flow. Using a weak formulation of the disturbance equations, upper bounds on the growth rate of individual disturbances and on the maximal growth rate over all possible disturbances are found. Analytically, it is shown that a short-wave disturbance if unstable can be stabilized by mild diffusion of species, where as an unstable long-wave disturbance can always be stabilized by strong diffusion of species. Thus, an otherwise unstable three-layer Hele-Shaw flow can be completely stabilized by a large enough diffusion, i.e., by increasing enough the magnitude of the species diffusion coefficient. The magnitude of this diffusion coefficient required to completely stabilize the flow will depend on the magnitude of interfacial viscosity jumps and the viscosity gradient of the basic viscous profile of the middle layer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almgren R.F. (1995). Crystalline Saffman–Taylor fingers. SIAM J. Appl. Math. 55: 1511–1535
Chouke R.L., Meurs P. and Poel C. (1959). The stability of a slow, immiscible, viscous liquid-liquid displacements in a permeable media. Petrol. Trans. AIME 216: 188–194
Darcy, H.: Les Fontaines Publiques de la Ville de Dijon. Victor Dalmond (1856)
Daripa P., Glimm J., Lindquist B. and McBryan O. (1988). Polymer floods: A case study of nonlinear wave analysis and instability control in tertiary oil recovery. SIAM J. Appl. Math. 49: 353–373
Daripa P. and Pasa G. (2005). New bounds for stabilizing Hele-Shaw flows. Appl. Math. Lett. 18(11): 1293–1303
Daripa, P., Pasa, G.: A simple derivation of an upper bound in the presence of viscosity gradient in three-layer Hele-Shaw flows, J. Stat. Mech. 11 pp, P01014. doi:10.1088/1742-5468/2006/01/P01014 (2006)
Duchon J. and Roberts R. (1984). Evolution dune interface par capillarite et diffusion de volume I. Existence locale en temps. Ann. Inst. H. Poincare, Anal. Non Lineaire 1: 361–378
Escher J. and Simonett G. (1996). On Hele-Shaw models with surface tension. Math. Res. Lett. 3: 467–474
Gorell S.B. and Homsy G.M. (1983). A theory of the optimal policy of oil recovery by the secondary displacement process. SIAM J. Appl. Math. 43: 79–98
Hickernell F.J. and Yortsos Y.C. (1986). stability of miscible displacement processes in porous media in the absence of dispersion. Stud. Appl. Math. 74: 93–115
Hill S. (1952). Channelling in packed columns. Chem. Eng. Sci. 1: 247–253
Homsy G.M. (1987). Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19: 271–311
Howison S.D. (1986). Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46: 20–26
Kadanoff L.P. (1990). Exact solutions for the Saffman-Taylor problem with surface tension. Phys. Rev. Letts. 65: 2986–2988
Kessler D.A., Koplik J. and Levine H. (1988). Pattern selection in fingered growth phenomena. Adv. Phys. 37: 255–329
Littman W. (1988). Polymer Flooding: Developments in Petroleum Science Elsevier, Amsterdam, p. 24
Loggia D., Rakotomalala N., Salin D. and Yortsos Y.C. (1999). The effect of mobility gradients on viscous instabilities in miscible flows in porous media. Phys. Fluids 11(3): 740–742
McLean J.W. and Saffman P.G. (1981). Effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102: 445–469
Needham R.B. and Doe P.H. (1987). Polymer flooding review. J. Pet. Technol. 12: 1503–1507
Nie Q. and Tian F.R. (1998). in Hele-Shaw flows. SIAM J. Appl. Math. 58: 34–54
Otto F. and Weinan E. (1997). Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys. 107: 10177–10184
Oztekin A., Cumbo L.J. and Liakopoulos A. (1999). Temporal stability of boundary-free shear flows: The effects of diffusion, Theor.. comp. fl. dyn. 13(2): 77–90
Pearson, H.J.: The stability of some variable viscosity flow with application in oil extraction, Cambridge Univ. Report (1977)
Saffman P.G. (1986). Viscous Fingering in Hele-Shaw Cells. J. Fluid Mech. 173: 73–94
Saffman P.G. and Taylor G.I. (1958). The penetration of a fluid in a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. Roy. Soc. A 245: 312–329
Shah D. and Schecter R. (1977). Improved Oil Recovery by Surfactants and Polymer Flooding. Academic Press, New York
Shariati M. and Yortsos Y.C. (2001). Stability of miscible displacements across stratified porous media. Phys. Fluids 13(8): 2245–2257
Slobod R.L. and Lestz J.S. (1960). Use of a graded viscosity zone to reduce fingering in miscible phase displacements. Producers Monthly 24: 12–19
Sorbie K.S. (1991). Polymer-Improved Oil Recovery. CRC Press, Boca Raton Florida
Tanveer S. (2000). Surprises in viscous fingering. J. Fluid Mech. 409: 273–308
Tanveer S. and Xie X. (2003). Analyticity and nonexistence of classical steady Hele-Shaw fingers. Commun. Pure Appl. Math. 56: 353–402
Tian F.R. (1996). A Cauchy integral approach to Hele-Shaw problems with a free boundary: the case of zero surface tension. Arch. Ration. Mech. Anal. 135: 175–196
Uzoigwe A.C., Scanlon F.C. and Jewett R.L. (1974). Improvement in polymer flooding: The programmed slug and the polymer-conserving agent. J. Petrol. Tech. 26: 33–41
Vasconceles G.L. and Kadanoff L.P. (1991). Stationary solutions for the Saffman-Taylor problem with surface tension. Phys. Rev. A. 44: 6490–6495
Xie X. and Tanveer S. (2003). Rigorous results in steady finger selection in viscous fingering. Arch. Ration. Mech. Anal. 166: 219–286
Yortsos Y.C. and Hickernell F.J. (1989). Linear stability of immiscible displacement in porous media. SIAM J. Appl. Math. 49(3): 730–748
Yortsos Y.C. and Zeybek M. (1988). Dispersion driven instability in miscible displacement in porous-media. Phys. Fluid 31(12): 3511–3518
Zhijun Z. and Yongmei H. (2002). Economic evaluation shows polymer flooding effectiveness. Oil & Gas Journal. 100(43): 56–59
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Daripa, P., Paşa, G. Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient. Transp Porous Med 70, 11–23 (2007). https://doi.org/10.1007/s11242-007-9122-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-007-9122-7