Some Contradictions in the Multi-Layer Hele-Shaw Flow
Keywords: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. https://doi.org/10.1098/rspa.1958.0085
Homsy GM. Viscous fingering in porous media, Ann Rev Fluid Mech, 19(1987), 271-311. https://doi.org/10.1146/annurev.fl.19.010187.001415
Saffman PG. Viscous fingering in Hele-Shaw cells, J Fl Mech, 173(1986), 73-94. https://doi.org/10.1017/S0022112086001088
Xu J-J. Interfacial Wave Theory of Pattern Formation in Solidification, Springer Series in Synergetics, Sproner, 2017. https://doi.org/10.1007/978-3-319-52663-8
Al-Housseiny TT, Tsai PA, Stone HA, Control of interfacial instabilities using flow geometry, Nat Phys Lett, 8(2012), 747â€“750. https://doi.org/10.1038/nphys2396
Al-Housseiny TT, Stone HA. Controlling viscous fingering in Hele-Shaw cells, Physics of Fluids, 25(2013), pp. 092102. https://doi.org/10.1063/1.4819317
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. https://doi.org/10.1103/PhysRevE.82.056308
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. https://doi.org/10.1103/PhysRevLett.109.144502
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). https://doi.org/10.2118/71509-MS
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. https://doi.org/10.1137/0143007
Gorell SB, Homsy GM. A theory for the most stable variable viscosity profile in graded mobility displacement process, AIChE J, 31(1985), 1598-1503. https://doi.org/10.1002/aic.690310912
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. https://doi.org/10.2118/4024-PA
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. https://doi.org/10.1016/j.ijengsci.2005.03.006
Daripa P, Pasa G. New bounds for stabilizing Hele-Shaw flows, Appl Math Lett, 18(2005), 12930-1303. https://doi.org/10.1016/j.aml.2005.02.027
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). https://doi.org/10.1088/1742-5468/2006/01/P01014
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. https://doi.org/10.1098/rsta.1993.0049
Tanveer S. Surprises in viscous fingering, J Fluid Mech, 409(2000), 273-368: https://doi.org/10.1017/S0022112099007788
Daripa P. Hydrodynamic stability of multi-layer Hele-Shaw flows, J Stat Mech, Art. No. P12005(2008). https://doi.org/10.1088/1742-5468/2008/12/P12005
Daripa P. Some Useful Upper Bounds for the Selection of Optimal Profiles, Physica A: Statistical Mechanics and its Applications 391(2012). 4065-4069. https://doi.org/10.1016/j.physa.2012.03.041
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. https://doi.org/10.1137/11086046X
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. https://doi.org/10.1007/s11242-012-9977-0
Mungan N. Improved waterflooding through mobility control, Canad J Chem Engr, 49(1971), 32-37. https://doi.org/10.1002/cjce.5450490107
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. https://doi.org/10.1063/1.869943
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. https://doi.org/10.1017/jfm.2013.63
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.