Solution of Fingering Phenomenon in Double Phase Flow through Heterogeneous Porous Media for Vertically Downward Direction

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2021 by IJMTT Journal
Volume-67 Issue-9
Year of Publication : 2021
Authors : Pratiksha A. More, Priti V. Tandel
  10.14445/22315373/IJMTT-V67I9P514

MLA

MLA Style: Pratiksha A. More, Priti V. Tandel"Solution of Fingering Phenomenon in Double Phase Flow through Heterogeneous Porous Media for Vertically Downward Direction" International Journal of Mathematics Trends and Technology 67.9 (2021):118-129. 

APA Style: Pratiksha A. More, Priti V. Tandel(2021). Solution of Fingering Phenomenon in Double Phase Flow through Heterogeneous Porous Media for Vertically Downward Direction  International Journal of Mathematics Trends and Technology, 67(9), 118-129.

Abstract
The present paper discusses the fingering phenomenon in the vertical direction via heterogeneous porous media. Governing equation of this phenomenon is a nonlinear second order partial differential equation. It is analysed with suitable initial condition by Reduced differential transform method (RDTM). The obtained solutions are represented numerically as well as graphically. 

Reference

[1] A. E. Scheidegger and Johnson, The statistical behaviour of instabilities in displacement process in porous media, Canadian Journal of Physics. 39(2) (1961) 326–334.
[2] A. E. Scheidegger, The physics of flow through Porous Media. University of Toronto Press, (1960).
[3] A.P. Verma, Statistical behaviour of fingering in a displacement in heterogeneous porous medium with capillary pressure, Canadian Journal of Physics. 47(3) (1969) 319–324.
[4] A. P. Verma and S. K. Mishra, Similarity solution for instabilities in double-phase flow through porous media, Journal of Applied Physics. 44(4) (1973) 1622–1624.
[5] A. Kumar, R. Arora, Solutions of the coupled system of Burgers’ equations and coupled Klein-Gordon equation by RDT Method, International Journal of Advances in Applied Mathematics and Mechanics. 1(2) (2013) 133-145.
[6] A.K. Parikh, Mathematical model of instability phenomenon in homogeneous porous medium in vertical downward direction, International Journal for Innovative Research in Multidisciplinary Field. 3(1) (2017) 211–218.
[7] D.A. Shah, A.K. Parikh, Mathematical solution of fingering phenomenon in vertical downward direction through heterogeneous porous medium, Advances in Mathematics: Scientific Journal. 10(1) (2021) 483–496.
[8] D. J. Prajapati, N. B. Desai, Application of the Basic Optimal Homotopy Analysis Method to Fingering Phenomenon, Global Journal of Pure and Applied Mathematics. 12(3) (2016) 2011–2022.
[9] D. J. Prajapati, N. B. Desai, Analytic Analysis for Oil Recovery During Cocurrent Imbibition in Inclined Homogeneous Porous Medium, International Journal on Recent and Innovation Trends in Computing and Communication. 5(7) (2017) 189 – 194.
[10] H. Patel, R. Maher, Simulation of Fingering Phenomena in Fluid Flow through Fracture Porous Media with Inclination and Gravitational Effect, Journal of Applied Fluid Mechanics. 9(6) (2016) 3135-3145.
[11] J. Bear, A.H. Chang, Modelling groundwater flow and contaminant transport, Dynamics of fluids in porous media, Springer Science Business, Media B. V., (2010).
[12] M. Rawashdeh, Using the Reduced Differential Transform Method to Solve Nonlinear PDEs Arises in Biology and Physics, World Applied Sciences Journal. 23 (8) (2013) 1037-1043.
[13] M. Rawashdeh, N. A. Obeidat, On Finding Exact and Approximate Solutions to Some PDEs Using the Reduced Differential Transform Method, Applied Mathematics & Information Sciences. 8(5) (2014) 2171-2176.
[14] M. S. Mohamed, K. A. Gepreel, Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations, Journal of the Egyptian Mathematical Society, Journal of the Egyptian Mathematical Society. 25 (2017) 1–7.
[15] P. G. Saffman, G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Show cell containing a more viscous fluid, Proc. R. Soc. London Series A.,245 (1985) 312–329.
[16] P. R. Mistry, V. H. Pradhan, K. R. Desai, Mathematical Model and Solution for Fingering Phenomenon in Double Phase Flow through Homogeneous Porous Media, The Scientific World Journal. 2013(2013).
[17] R. Borana, V. Pradhan, M. Mehta, Numerical solution of instability phenomenon arising in double phase flow through inclined homogeneous porous media, Perspectives in Science. 8 (2016) 225–227.
[18] R. Meher, M. N. Mehta, S. K. Meher, Instability phenomenon arising in double phase flow through porous medium with capillary pressure, International Journal of Applied Mathematics and Mechanics. 7 (15) (2011) 97-112.
[19] S. Mohmoud, M. Gubara, Reduced differential transform method for solving linear and nonlinear Goursat problem, Applied Mathematics. 7(2016) 1049-1056.
[20] S.S. Chen, C.K. Chen, Application to differential transformation method for solving systems of differential equations, Nonlinear analysis: Real world applications. 10 (2) (2009) 881-888.
[21] Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation.10 (6) (2009) 741-749.
[22] Y. Keskin, G. Oturanc, Application of Reduced differential transformation method for solving gas dynamics equation, International Journal of Contemporary Mathematical Sciences. 5(22) (2010) 1091-1096.
[23] Y. Keskin, G. Oturanc, Reduced differential transform method for generalized kdv equations, Mathematical and Computational Applications. 15 (3) (2010) 382-393.
[24] Z. Cheng, Reservoir Simulation: Mathematical techniques in oil recovery, Society for Industrial and Applied Mathematics, Philadelphia, (2007) 1-25.

Keywords : Fingering phenomenon, Secondary oil recovery process, Heterogeneous porous medium, Reduced Differential Transform Method.