Math. J. Interdiscip. Sci.

Homotopy Analysis Approach of Boussinesq Equation for Infiltration Phenomenon in Unsaturated Porous Medium

M A Patel and N B Desai

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Groundwater, In iltration, Unsaturated soil, Homotopy analysis method

PUBLISHED DATE September, 2018
PUBLISHER The Author(s) 2018. This article is published with open access at

Boussinesq’s equation is one-dimensional nonlinear partial differential equation which represents the infiltration phenomenon. This equation is frequently used to study the infiltration phenomenon in unsaturated porous medium. Infiltration is the process in which the groundwater of the water reservoir has entered in the unsaturated soil through vertical permeable wall. An approximate analytical solution of nonlinear partial differential equation is presented by homotopy analysis method. The convergence of homotopy analysis solution is discussed by choosing proper value of convergence control parameter. The solution represents the height of free surface of infiltrated water.


The fluid flow in porous media has great importance in various fields of engineering and science. In particular, the groundwater flow is very important part of fluid mechanics, hydrology, water resources engineering, irrigation engineering, etc. (Bear et al. 1972, PolubarinovaKochina et al. 1962, Scheidegger et al. 1960, Vazquez et al. 2007, Desai et al. 2002). In this work, we examined the fluid flow problem in groundwater infiltration. Infiltration is the process in which water on the ground surface enters into unsaturated soils and pass into rocks through cracks and interstices. If the storage for the additional water had been available then the infiltration process can continue for a long time. The availability of additional water into the soil is dependent on the porosity of the soil. Once water has infiltrated into the soil it may stay in soil until it gradually evaporated, absorbed by plant roots and later transpired. The rate of infiltration process is dependent on different factors like as texture and structure of soil, storage capacity of soil, the depth of water reservoirs, the amount of plant over the region, etc.

Many researchers have been discussed various problems of groundwater infiltration like as (Troch et al. 1993) have derived an expression for mean water table height on the basis of hydraulic groundwater theory by means of Boussinesq equation, (Govindaraju and Koelliker, 1994) have developed the expression for the flow rate from the stream to the aquifer, (Hogarth et al. 1997) have discussed an analytical approach for Boussinesq equation with constant and time dependent boundary conditions, (Hogarth et al. 1999) have obtained the approximate analytical solution of Boussinesq equation which is accurate solution by comparison with the numerical solution when the boundary conditions is a power to time, (Wojnar, 2010) has discussed the Boussinesq equation for flow in the aquifer with time dependent porosity, (Moutsopoulos, et al, 2013) has discussed Boussinesq equation with nonlinear robin boundary condition, (Basha, et al. 2013) has discussed the traveling wave solution of the groundwater flow in horizontal aquifers.

The aim of current work is to obtain the solution of Boussinesq equation for infiltration phenomenon. The mathematical form of the infiltration phenomenon gives the nonlinear partial differential equation in the form of Boussinesq equation. This equation is solved using homotopy analysis method. The BVPh package for nonlinear equations is employed to interpret numerically and graphically solution. (Liao, et al. 1992) has employed the homotopy analysis method to solve nonlinear equations. It has been successfully employed to solve many nonlinear equations. The homotopy analysis solution is strongly dependent on convergence control parameter and its proper value chosen from the valid region of c0. The valid region of c0 is obtained from the c0-curve. The line segment almost parallel to horizontal line in c0-curve gives us the admissible range of c0.

Page(s) 21-28
ISSN Print: 2278-9561, Online: 2278-957X

The Boussinesq equation is discussed for infiltration phenomenon in unsaturated soil. The homotopy analysis solution of the governing equation is obtained with boundary condition. The convergence of homotopy analysis solution is discussed by c0-curve. The solution represents the height of free surface which is discussed graphically and numerically.

  • Basha H. A. (2013). Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers, Water Resources Research, 49, 1668–1679.
  • Boussinesq J. (1904). Recherches theoriques sur l’ecoulement des nappes d’eau infiltrees dans le sol et sur le debit des sources, Journal de Mathematiques Pures et Appliquees, 10, 5–78.
  • Govindaraju R. S. & Koelliker J. K. (1994). Applicability of linearized Boussinesq equation for modeling bank storage under uncertain aquifer parameters, Journal of Hydrology, 157, 349–366.
  • Hogarth W. L., Govindaraju R. S., Parlange J. Y. & Koelliker J. K. (1997). Linearised Boussinesq equation for modelling bank storage - a correction, Journal of Hydrology, 198, 377–385.
  • Hogarth W. L., Parlange J. Y., Parlange M. B. & Lockington D. (1999). Approximate analytical solution of the Boussinesq equation with numerical validation, Water Resources Research, 35(10), 3193–3197.
  • Kheiri H., Alipour N., 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), 33–50.
  • Moutsopoulos K. N. (2013). Solutions of the Boussinesq equation subject to a nonlinear Robin boundary condition, Water Resources Research, 49, 7–18.
  • Patel M. A. & Desai N. B. (2016). Homotopy analysis solution of countercurrent imbibition phenomenon in inclined homogeneous porous medium, Global Journal of Pure and Applied Mathematics, 12(1), 1035–1052.
  • Patel M. A. & Desai N. B. (2017). Homotopy analysis method for fingero-imbibition phenomenon in heterogeneous porous medium, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 8(1), 90–100.
  • Patel M. A. & Desai N. B. (2017). Mathematical modelling of fingero-imbibition phenomenon in heterogeneous porous medium with magnetic field effect, PRAJNA - Journal of Pure and Applied Sciences, 24-25, 15–22.
  • Patel M. A. & Desai N. B. (2017). Homotopy analysis method for nonlinear partial differential equation arising in fluid flow through porous medium, International Journal of Computer & Mathematical Sciences, 6(5), 14–18.
  • Rashidi M. M., Domairry G. & Dinarvand S. (2009). Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14, 708–717.
  • Troch P. A., De Troch F. P. & Brutsaert W. (1993). Effective water table depth to describe initial conditions prior to storm rainfall in humid regions, Water Resources Research, 29(2), 427–434.
  • Wojnar R. (2010). Boussinesq equation for flow in an aquifer with time dependent porosity, Bulletin of Polish Academy of Sciences: Technical Sciences, 58(1), 165–170.
  • Bear J. (1972). Dynamics of fluids in porous media, American Elsevier Publishing Company, Inc., New York.
  • Darcy H. (1856). Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris.
  • Liao S. J. (2003). Beyond perturbation: Introduction to the homotopy analysis method, Chapman and Hall/CRC Press, Boca Raton.
  • Liao S. J. (2012). Homotopy analysis method in nonlinear differential equations, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg.
  • Polubarinova-Kochina P. Y. (1962). Theory of Groundwater Movement, Princeton Univ. Press, Princeton.
  • Scheidegger A. E. (1960). The Physics of flow through porous media, Revised edition, University of Toronto Press, Toronto.
  • Vajravelu K. & 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.
  • Vazquez J. L. The porous medium equation: mathematical theory, Oxford University Press (2007).
  • Desai N. B. The study of problems arises in single phase and multiphase ow through porous media, Ph.D. Thesis, South Gujarat University, Surat, India (2002).
  • Liao S. J. The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China (1992).