AN APPROXIMATE ANALYTICAL SOLUTION OF ONE-DIMENSIONAL GROUNDWATER RECHARGE BY SPREADING

Year 2019, Volume: 9 Issue: 4, 838 - 850, 01.12.2019

M. A. Patel N. B. Desai

Abstract

The present paper discusses the problem of one dimensional groundwater recharge in the vertical direction. The groundwater is recharged by spreading of water in vertical direction and the moisture content of soil increases. On the basis of linear and nonlinear conductivity and diusivity functions, three cases are considered for Brooks- Corey model. The governing nonlinear partial dierential equations has been solved by homotopy analysis method. The proper value of convergence control parameter for con- vergent solution has been chosen from c0-curve. The numerical and graphical solutions are presented.

Keywords

Groundwater, Moisture content, HAM.

References

• [1] Allan, M. B., (1985), Numerical modeling of multiphase flow in porous media, Advances in Water Resources, 8(4), pp. 162-187.
• [2] Bear, J., (1972), Dynamics of fluids in porous media, American Elsevier Publishing Company, New York.
• [3] Bear, J. and Cheng, A. H. D., (2010), Modeling groundwater flow and contaminant transport: Theory and applications of transport in porous media, Springer Dordrecht, Heidelberg, London, New York.
• [4] Corey, A. T., (1994), Mechanics of immiscible fluids in porous media, Water Resources Publications, Littleton, Colorado.
• [5] Darvishi, M. T. and Khani, F., (2009), A series solution of the foam drainage equation, Computers and Mathematics with Applications, 58, pp. 360-368.
• [6] Desai, N. B., (2002), The study of problems arises in single phase and multiphase flow through porous media, Ph.D. Thesis, South Gujarat University, Surat, India.
• [7] Ghotbi, A. R., Omidvar, M. and Barari, A., (2011), Infiltration in unsaturated soils - An analytical approach, Computers and Geotechnics, 38, pp. 777-782.
• [8] Joshi, M. S., Desai, N. B. and Mehta, M. N., (2010), One dimensional and unsaturated fluid flow through porous media, Int. J. Appl. Math. and Mech., 6(18), pp. 66-79.
• [9] Kheiri, H., Alipour, N. and Dehghani, R., (2011), Homotopy analysis and homotopy pade methods for the modified Burgers-Korteweg-de Vries and the Newell -Whitehead equations, Mathematical Sciences, 5(1), pp. 33-50.
• [10] Klute, A., (1952), A numerical method for solving the flow equation for water in unsaturated materials, Soil Science, 73(2), pp. 105-116.
• [11] Liao, S. J., (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China.
• [12] Liao, S. J., (2003), Beyond perturbation: Introduction to the homotopy analysis method, Chapman and Hall/CRC Press, Boca Raton.
• [13] Liao, S. J., (2012), Homotopy analysis method in nonlinear differential equations, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg.
• [14] Liao, S. J., (2013), Advances in the homotopy analysis method, World Scientific, Singapore.
• [15] Mehta, M. N., (1975), A singular perturbation solution of one-dimensional flow in unsaturated porous media with small diffusivity coefficient, Proceeding of the National conference on Fluid Mechanics and Fluid Power, pp. E1-E4.
• [16] Mehta, M. N. and Patel, T. R., (2006), A solution of the Burger’s equation type one dimensional groundwater recharge by spreading in porous media, J. Indian Academy of Mathematics, 28(1), pp. 25-32.
• [17] Nasseri, M., Daneshbod, Y., Pirouz, M. D., Rakhshandehroo, G. R. and Shirzad, A., (2012), New analytical solution to water content simulation in porous media, J. Irrigation and Drainage Engineering, 138(4), pp. 328-335.
• [18] Odibat, Z. M., (2010), A study on the convergence of homotopy analysis method, Applied Mathematics and Computation, 217, pp. 782-789.
• [19] Patel, M. A. and Desai, N. B., (2016), Homotopy analysis solution of countercurrent imbibition phenomenon in inclined homogeneous porous medium, Global Journal of Pure and Appl. Math., 12(1), pp. 1035-1052.
• [20] Patel, M. A. and Desai, N. B., (2017), Homotopy analysis method for fingero-imbibition phenomenon in heterogeneous porous medium, Nonlinear Science Letters A: Math., Phy. and Mech., 8(1), pp. 90-100.
• [21] Patel, M. A. and Desai, N. B., (2017), A mathematical model of cocurrent imbibition phenomenon in inclined homogeneous porous medium, Kalpa Publications in Computing, ICRISET2017, Selecting Papers in Computing, 2, pp. 51-61.
• [22] Patel, M. A. and Desai, N. B., (2017), Mathematical modelling and analysis of cocurrent imbibition phenomenon in inclined heterogeneous porous medium, Int. J. Compu. and Appl. Math., 12(3), pp. 639-652.
• [23] Patel, M. A. and Desai, N. B., (2017), An approximate analytical solution of the Burger’s equation for longitudinal dispersion phenomenon arising in fluid flow through porous medium, Int. J. on Recent and Innovation Trends in Computing and Communication, 5(5), pp. 1103-1107.
• [24] Prasad, K.H., Kumar, M. M. and Sekhar, M., (2001), Modeling flow through unsaturated zone: Sensitivity to unsaturated soil properties, Sadhana, Indian Academy of Sciences, 26(6), pp. 517-528.
• [25] Richards, L. A., (1931), Capillary conduction of liquids through porous mediums, Physics, 1, pp. 318-333.
• [26] Vajravelu, K. and Van Gorder, R. A., (2012), Nonlinear flow phenomena and homotopy analysis: Fluid flow and Heat transfer, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg.
• [27] Verma, A. P., (1969), The laplace transform solution of a one dimensional groundwater recharge by spreading, Annals of Geophysics, 22(1), pp. 25-31.
• [28] Witelski, T. P., (2005), Motion of wetting fronts moving into partially pre-wet soil, Advances in Water Resources, 28, pp. 1133-1141.