Yadav, R., Kumar, L. (2018). Analytical solutions of one-dimensional Advection equation with Dispersion coefficient as function of Space in a semi-infinite porous media. Pollution, 4(4), 745-758. doi: 10.22059/poll.2018.253350.410

R. R. Yadav; L. K. Kumar. "Analytical solutions of one-dimensional Advection equation with Dispersion coefficient as function of Space in a semi-infinite porous media". Pollution, 4, 4, 2018, 745-758. doi: 10.22059/poll.2018.253350.410

Yadav, R., Kumar, L. (2018). 'Analytical solutions of one-dimensional Advection equation with Dispersion coefficient as function of Space in a semi-infinite porous media', Pollution, 4(4), pp. 745-758. doi: 10.22059/poll.2018.253350.410

Yadav, R., Kumar, L. Analytical solutions of one-dimensional Advection equation with Dispersion coefficient as function of Space in a semi-infinite porous media. Pollution, 2018; 4(4): 745-758. doi: 10.22059/poll.2018.253350.410

Analytical solutions of one-dimensional Advection equation with Dispersion coefficient as function of Space in a semi-infinite porous media

^{}Department of Mathematics & Astronomy, Lucknow University, Lucknow-226007, U.P, India

Abstract

The aim of this study is to develop analytical solutions for one-dimensional advection-dispersion equation in a semi-infinite heterogeneous porous medium. The geological formation is initially not solute free. The nature of pollutants and porous medium are considered non-reactive. Dispersion coefficient is considered squarely proportional to the seepage velocity where as seepage velocity is considered linearly spatially dependent. Varying type input condition for multiple point sources of arbitrary time-dependent emission rate pattern is considered at origin. Concentration gradient is considered zero at infinity. A new space variable is introduced by a transformation to reduce the variable coefficients of the advection-dispersion equation into constant coefficients. Laplace Transform Technique is applied to obtain the analytical solutions of governing transport equation. Obtain results are shown graphically for various parameter and value on the dispersion coefficient and seepage velocity. The developed analytical solutions may help as a useful tool for evaluating the aquifer concentration at any position and time.

Abramowitz, M. and Stegun, I. A. (1970). Handbook of mathematical functions. First Edition, Dover Publications Inc., New York, 1019.

Aral, M. M. and Liao, B. (1996). Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients. J. of Hydrol. Eng., 1(1); 20-32.

Bear, J. (1972). Dynamics of fluids in porous media. New York: Amr. Elsev. Co.

Bharati, V. K., Sanskrityayn, A. and Kumar, N. (2015). Analytical solution of ADE with linear spatial dependence of dispersion coefficient and velocity using GITT. J. of Groundwater Res., 3(4); 13-26.

Chen, J. S., Ni, C. F., Liang, C. P. and Chiang, C. C. (2008). Analytical power series solution for contaminant transport with hyperbolic asymptotic distance-dependent dispersivity. J. of Hydrology, 362(1-2); 142-149.

Chen, J. S., Liu, C. W. and Liao, C. M. (2003). Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field. Advances in Water Resources, 26; 1113-1124.

Das, P., Begam, S. and Singh, M. K. (2017). Mathematical modeling of groundwater contamination with varying velocity field. J. of Hydrology and Hydromechanics, 65 (2); 192-204.

DeSmedt, F. and Wierenga, P. J. (1978). Solute transport through soil with non-uniform water content. Soil Sci. Soc. Am. J., 42(1); 7-10.

Elfeki, A. M. M., Uffink, G. and Lebreton, S. (2011). Influence of temporal fluctuations and spatial heterogeneity on pollution transport in porous media. Hydrogeology Journal, 20; 283-297.

Flury, M., Wu, Q. J., Wu, L. and Xu, L. (1998). Analytical solution for solute transport with depth-dependent transformation or sorption coefficient. Water Resour. Res., 34(11); 2931-2937.

Freeze, R. A. and Cherry, J. A. (1979). Groundwater. Prentice-Hall, New Jersey.

Guerrero, J. S. P., Pimentel, L. C. G., Skaggs, T. H. and Van Genuchten, M.Th., (2009). Analytical solution of the advection-diffusion transport equation using a change-of- variable and integral transform technique. Int. J. of Heat and Mass Transfer, 52; 3297-3304.

Harleman, D. R. F. and Rumer, R. R. (1963). Longitudinal and lateral dispersion in an isotropic porous medium. J. of Fluid Mechanics, 16(3); 385-394.

Huang, K., Van Genuchten, M. Th. and Zhang, R. (1996). Exact solutions for one dimensional transport with asymptotic scale-dependent dispersion. Applied Mathematical Modeling, 20; 298-308.

Jaiswal, D. K., Kumar, A., Kumar, N. and Yadav, R. R., (2009). Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media. J. of Hydro-environment Research, 2; 254-263.

Kumar, A. and Yadav, R. R. (2015). One-dimensional solute transport for uniform and varying pulse type input point source through heterogeneous medium. Environmental Technology, 36(4); 487-495.

Kumar, A., Jaiswal, D. K. and Kumar, N. (2010). Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. J. of Hydrology, 380; 330-337.

Massabo, M., Cianci, R. and Paladino, O. (2006). Some analytical solutions for two- dimensional Convection-dispersion equation in cylindrical geometry. Environmental Modelling and Software, 21; 681-688.

Moghaddam, M. B., Mazaheri, M. and Vali Samani, J. M. (2017). A comprehensive onedimensional numerical model for solute transport in rivers, Hydrol. Earth Syst. Sci, 21; 99-116.

Ogata, A. and Bank, R. B. (1961). A solution of differential equation of longitudinal dispersion in porous media. U. S. Geol. Surv. Prof. Pap. 411, A1-A7.

Pang, L. and Hunt, B. (2001). Solutions and verification of a scale-dependent dispersion model. Journal of Contaminant Hydrology, 53; 21-39.

Pickens, J. F. and Grisak, G. E. (1981). Scale-dependent dispersion in a stratified granular aquifer. Water Resources Res., 17(4); 1191-1211.

Rumer, R. R. (1962). Longitudinal dispersion in steady and unsteady flow. J. of Hydraulic Division, 88; 147-173.

Sanskrityayn, A., Bharati, V. K. and Kumar, N. (2016). Analytical solution of ADE with spatiotemporal dependence of dispersion coefficient and velocity using green’s function method, Journal of Groundwater Research, 5(1); 24-31.

Sauty, J. P. (1980). An analysis of hydro-dispersive transfer in aquifers. Water Resources Research, 16(1); 145-158.

Singh, M. K., Ahamad, S. and Singh, V. P., (2014). One-dimensional uniform and time varying solute dispersion along transient groundwater flow in a semi-infinite aquifer. Acta geophysica, 62 (4); 872-892.

Singh, M. K., Kumari, P. and Mahato, N. K. (2013). Two-dimensional solute transport in finite homogeneous porous formations. Int. J. of Geology, Earth & Environmental Sciences, 3(2); 35-48.

Sudicky, E.A. and Cherry, J.A. (1979). Field observations of tracer dispersion under natural flow conditions in an unconfined sandy aquifer. Fourteenth Canadian Symposium on Water Pollution Research, University of Toronto, Feb. 22, Water Pollution Research, Canada, 14, 1-17.

Todd, D. K., (1980). Groundwater Hydrology. 2nd edn., John Wiley & Sons.

Van Genuchten, M. Th. and Alves, W. J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. Technical Bulletin No 1661, US Department of Agriculture.

Volocchi, A. J. (1989). Spatial movement analysis of the transport of kinetically adsorbing solute through stratified aquifers. Water Resources Res., 25; 273-279.

Wierenga, P. J. (1977). Solute distribution profiles computed with steady state and transient. Water Movement Models. Soil Sci. Soc. Amer. J., 41; 1050-1055.

Yadav, S. K., Kumar, K. and Kumar, N. (2012). Horizontal solute transport from a pulse type source along temporally and spatially dependent flow: Analytical solution. Journal of Hydrology, (412-413); 193-199.