NUMERICAL SOLUTION OF BOUSSINESQ EQUATION ARISING IN ONE-DIMENSIONAL INFILTRATION PHENOMENON BY USING FINITE DIFFERENCE METHOD

Ravi N. Borana, V. H. Pradhan, M.N. Mehta

Abstract: Infiltration is a gradual flow or movement of groundwater into and through the pores of an unsaturated porous medium (soil). The fluid infiltered in porous medium (unsaturated soil), its velocity decreases as soil becomes saturated, and such phenomena is called infiltration. The present model deals with the filtration of an incompressible fluid (typically, water) through a porous stratum, the main problem in groundwater infiltration. The present model was developed first by Boussinesq in 1903 and is related to original motivation of Darcy. The mathematical formulation of the infiltration phenomenon leads to a non-linear Boussinesq equation. In the present paper, a numerical solution of Boussinesq equation has been obtained by using finite difference method. The numerical results for a specific set of initial and boundary conditions are obtained for determining the height of the free surface or water mound. The moment infiltrated water enters in unsaturated soil; the infiltered water will start developing a curve between saturated porous medium and unsaturated porous medium, which is called water table or water mound. Crank-Nicolson finite difference scheme has been applied to obtain the required results for various values of time. The obtained numerical results resemble well with the physical phenomena. When water is infiltered through the vertical permeable wall in unsaturated porous medium the height of the free surface steadily and uniformly decreases due to the saturation of infiltered water as time increases. Forward finite difference scheme is conditionally stable. In the present paper, the graphical representation shows that Crank-Nicolson finite difference scheme is unconditionally stable. Numerical solution of the governing equation and graphical presentation has been obtained by using MATLAB coding

Keywords: Infiltration, porous media, Darcy’s law, Crank-Nicolson finite difference scheme

DOI: https://doi.org/10.15623/ijret.2013.0208034