M A Patel and N B Desai
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 www.chitkara.edu.in/publications.|
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.
|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.