Some Contradictions in the Multi-Layer Hele-Shaw Flow


  • Gelu PASA Simion Stoilow Institute of Mathematics of Romanian Academy, Calea Grivitei 21, Bucuresti Sector 1, Romania



Hele-Shaw immiscible displacement, Porous media flow, Linear stability.


 An important problem concerning the Hele-Shaw displacements is to minimize the Saffman - Taylor instability. To this end, some constant viscosity fluid layers can be introduced in an intermediate region ( ) between the displacing fluids. However, we prove that very small (positive) values of the growth rates can be obtained only for a very large (unrealistic) On the contrary, when the length is constrained by certain conditions (for instance, geological), then the maximum value of the growth constants can not fall below a certain value, not depending on the number of layers. This maximum value is not so small.


Bear J. Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.

Hele-Shaw HS. Investigations of the nature of surface resistence of water and of streamline motion under certain experimental conditions, Inst. Naval Architects Transactions 40(1898), 21-46.

Lamb H. Hydrodynamics, Dower Publications, New York, 1933.

Saffman PG, Taylor GI. The penetration of a fluid in a porous medium or Helle-Shaw cell containing a more viscous fluid, Proc Roy Soc A, 245(1958), 312-329.

Homsy GM. Viscous fingering in porous media, Ann Rev Fluid Mech, 19(1987), 271-311.

Saffman PG. Viscous fingering in Hele-Shaw cells, J Fl Mech, 173(1986), 73-94.

Xu J-J. Interfacial Wave Theory of Pattern Formation in Solidification, Springer Series in Synergetics, Sproner, 2017.

Al-Housseiny TT, Tsai PA, Stone HA, Control of interfacial instabilities using flow geometry, Nat Phys Lett, 8(2012), 747–750.

Al-Housseiny TT, Stone HA. Controlling viscous fingering in Hele-Shaw cells, Physics of Fluids, 25(2013), pp. 092102.

Chen C-Yo, Huang C-W, Wang L-C, Miranda JA. Controlling radial fingering patterns in miscible confined flows, Phys Rev E, 82(2010), pp. 056308.

Diaz EO, Alvaez-Lacalle A, Carvalho MS, Miranda JA. Minimization of viscous fluid fingering: a variational scheme for optimal flow rates, Phys Rev Lett, 109(2012), pp. 144502.

Sudaryanto B, Yortsos YC. Optimization of Displacements in Porous Media Using Rate Control, Society of Petroleum Engineers, Annual Technical Conference and Exhibition, 30 September-3 October, New Orleans, Louisiana (2001).

Gilje E, Simulations of viscous instabilities in miscible and immiscible displacement, Master Thesis in Petroleum Technology, University of Bergen, 2008.

Gorell SB, Homsy GM. A theory of the optimal policy of oil recovery by secondary displacement process, SIAM J Appl Math, 43(1983), 79-98.

Gorell SB, Homsy GM. A theory for the most stable variable viscosity profile in graded mobility displacement process, AIChE J, 31(1985), 1598-1503.

Shah G, Schecter R, eds., Improved Oil Recovery by Surfactants and Polymer Flooding, Academic Press, New York, 1977.

Slobod RL, Lestz SJ. Use of a graded viscosity zone to reduce fingering in miscible phase displacements, Producers Monthly, 24(1960), 12-19.

Uzoigwe AC, Scanlon FC, Jewett RL. Improvement in polymer flooding: The programmed slug and the polymerconserving agent, J Petrol Tech, 26(1974), 33-41.

Daripa P, Pasa G. On the growth rate of three-layer Hele- Shaw flows - variable and constant viscosity cases, Int J Engng Sci, 43(2004), 877-884.

Daripa P, Pasa G. New bounds for stabilizing Hele-Shaw flows, Appl Math Lett, 18(2005), 12930-1303.

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, P 01014(2006).

Tanveer S. Evolution of Hele-Shaw interface for small surface tension, Philosophical Trans Roy Soc A, Published 15 May 1993.DOI: 10.1098/rsta.1993.0049.

Tanveer S. Surprises in viscous fingering, J Fluid Mech, 409(2000), 273-368:

Daripa P. Hydrodynamic stability of multi-layer Hele-Shaw flows, J Stat Mech, Art. No. P12005(2008).

Daripa P. Some Useful Upper Bounds for the Selection of Optimal Profiles, Physica A: Statistical Mechanics and its Applications 391(2012). 4065-4069.

Daripa P, Ding X, Universal stability properties for Multi-layer Hele-Shaw flows and Applications to Instability Control, SIAM J Appl Math, 72(2012), 1667-1685.

Daripa P, Ding X. A Numerical Study of Instability Control for the Design of an Optimal Policy of Enhanced Oil Recovery by Tertiary Displacement Processes, Tran. Porous Media 93(2012), 675-703.

Mungan N. Improved waterflooding through mobility control, Canad J Chem Engr, 49(1971), 32-37.

Loggia D, Rakotomalala N, Salin D, Yortsos YC. The effect of mobility gradients on viscous instabilities in miscible flows in porous media, Ì? Physics of Fluids, 11(1999), 740-742.

Talon L, Goyal N, Meiburg E. Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis, J Fluid Mech, 721(2013), 268-294.

US Geological Survey, Applications of SWEAT to select Variable-Density and Viscosity Problems, U. S Department of the Interior, Specific Investigations Report 5028 (2009).

Flory PJ. Principles of Polymer Chemistry, Ithaca, New York, Cornell University Press, 1953.




How to Cite

Gelu PASA. Some Contradictions in the Multi-Layer Hele-Shaw Flow. Int. J. Petrol. Technol. [Internet]. 2019May12 [cited 2021Sep.25];6(1):41-8. Available from: