Application of Homotopy Analysis Method in One-Dimensional Instability Phenomenon Arising in Inclined Porous Media

Kajal K. Patel^1, , M. N. Mehta¹ and Twinkle R. Singh¹

¹Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat, India

Cite this paper:
Kajal K. Patel, M. N. Mehta and Twinkle R. Singh. Application of Homotopy Analysis Method in One-Dimensional Instability Phenomenon Arising in Inclined Porous Media. American Journal of Applied Mathematics and Statistics. 2014; 2(3):106-114. doi: 10.12691/ajams-2-3-4

Abstract

During secondary oil recovery process when water is injected in inclined oil formatted area then phenomenon of instability occurs due to viscosity difference of water and oil. The non-linear partial differential equation for this instability phenomenon have been obtained. The Homotopy analysis method has been applied to this governing equation by using appropriate initial and boundary conditions. The guess value of saturation of injected water has been satisfying its initial and boundary conditions. The numerical value and graphical presentation are given by using Maple software and it is concluded that the saturation of injected water is increasing during instability phenomenon in inclined porous media when length of fingers and time increases.

Keywords:
instability phenomenon homotopy analysis method porous matrix secondary oil recovery process

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Figures

References:

[1]	Bear, J., Dynamics of Fluids in Porous Media, American Elsevier Publishing Co., New York, pp. 687-701, 1972.
[2]	Booth, R.J., Miscible flow through porous media .PhD thesis, University of Oxford page-5-10, 2008.
[3]	Chuoke, R.L., van Meurs, P. and van der Poel, C., The Instability of Slow, Immiscible Viscous Liquid-Liquid Displacements in Permeable Media, Trans., AIME 216, 188-94, 1959.
[4]	Dake. L. P., The Practice of Reservoir Engineering, Elsevier Science B. V., Amsterdam, 1994.
[5]	Darcy H., Les Fontaines publiques de lo ville de Dijon, Dalmont, Paris, 1856.
[6]	Joshi M.S. and Mehta M.N., Solution by Group Invariant method of Instability phenomenon arising in fluid flow through porous media. International Journal of Engineering Research and Industrial Applications, Vol. 2, No. 1, 35-48, 2009.
[7]	Kinjal R. P., Mehta M. N. and Patel T. R., An approximate solution of Instability phenomenon in heterogeneous porous media with mean pressure., Gen. Math. Notes., Vol. 10 (2), 9-21, 2012.
[8]	Liao S. J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int. J. Nonlinear. Mech, 32-815, 1997.
[9]	Liao, S. J., An explicit, totally analytic approximation of Blasius viscous flow problems. Intl J. Non-Linear Mech. 34, 759-778, 1999 a.
[10]	Liao S. J., On the Homotopy analysis method for nonlinear problems. Appl. Math. Comput. Vol. 147, pp. 499-513, 2004.
[11]	Liao, S. J., The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992.
[12]	Meher R. K. and Mehta M. N. Meher S. K., Instability phenomenon arising in double phase flow through porous media with capillary pressure., Int. J. of Appl. Math and Mech. 7 (15): 97-112, 2011.
[13]	Mehta M. N., Asymptotic expansion of fluid flow through porous media, Ph.D. thesis South Gujarat University, Surat, India, 1977.
[14]	Muskat M., Physical principles of oil production, Mc-graw-hill Book co., Inc New York, 1949.
[15]	Oroveanu T., Scurgerea Fluiidelor Prin Medii Poroase Neomogene, Editura Academiei Republicii Populare Romine 92, 328, 1963.
[16]	Patel T. R. and Mehta M. N., A classical solution of the Burger's equation arising into the instability phenomenon in double phase flow through homogenous porous media, International Journal of Applied Mathematics, Bulgaria, Vol. 25 (1), 2008.
[17]	Saffman, P.G. and Taylor, G., The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a more Viscous Liquid, Proc. Roy. Soc. London A 245, 312-329, 1958.
[18]	Scheidegger A. E., The physics of flow through Porous Media, University of Toronto Press, 1960.
[19]	Scheidegger A. E. and Johnson E. F., The statistical behaviour of instabilities in displacement process in porous media., Canadian J. Physics, 39-326, 1961.
[20]	Vyas N. B., Patel K. R., Mehta M. N. and Patel T. R., Power series solution of instability phenomena in double phase flow through porous media under magnetic field effect, Int. J. of Appl. Math and Mech. 7 (16): 1-10, 2011.
[21]	Verma A. P., Statistical behavior of fingering in a displacement process in heterogeneous porous medium with capillary pressure. Can. J. Phys., 47 (3), 319-324, 1969.