Document Type : Original Research Paper

**Authors**

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

**Abstract**

In the present study, analytical solutions are obtained for two-dimensional advection dispersion equation for conservative solute transport in a semi-infinite heterogeneous porous medium with pulse type input point source of uniform nature. The change in dispersion parameter due to heterogeneity is considered as linear multiple of spatially dependent function and seepage velocity whereas seepage velocity is n^{th} power of spatially dependent function. Two forms of the seepage velocity namely exponentially decreasing and sinusoidal form are considered. First order decay and zero order production are also considered. The geological formation of the porous medium is considered of heterogeneous and adsorbing nature. Domain of the medium is uniformly polluted initially. Concentration gradient is considered zero at infinity. Certain new transformations are introduced to transform the variable coefficients of the advection diffusion equation into constant coefficients. Laplace Transform Technique (LTT) is used to obtain analytical solutions of advection-diffusion equation. The solutions in all possible combinations of temporally and spatially dependence dispersion are demonstrated with the help of graphs.

**Keywords**

Al-Niami, A. N. S. and Rushton, K. R. (1977). Analysis of flow against dispersion in porous media. J. Hydrol., 33; 87-97.

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

Banks, R. B. and Ali, J. (1964). Dispersion and adsorption in porous media flow. J. Hydrol. Div., 90; 13-31.

Bing, B., Huawei, L., Tao, X. and Xingxin, C. (2015). Analytical solutions for contaminant transport in a semi-infinite porous medium using the source function method. Comp. and Geotech., 69; 114-123.

Carnahan, C. L. and Remer, J. S. (1984). Non-equilibrium and equilibrium sorption with a linear sorption isotherm during mass transport through porous medium: Some analytical solutions. J. Hydrology (Amsterdam, Neth), 73; 227-258.

Chen, J. S. and Liu, C. W. (2011). Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition. Hydrol. Earth Sys. Sci., 15; 2471-2479.

Chen, J. S., Liu, C. W., Hsu, H. T. and Liao, C. M. (2003). A Laplace transformed power series solution for solute transport in convergent flow field with scale-dependent dispersion. Water Resou. Res., 39(8); 14-1–14-10.

Chen, J. S., Ni, C. F. and Liang, C. P. (2008). Analytical power series solution to the two-dimensional advection-dispersion equation with distance-dependent dispersivity. Hydrol. Process., 22(24); 4670-4678.

Crank, J. (1975). The Mathematics of Diffusion. Oxford Univ. Press London, 2nd ed.

Das, P., Akhter, A. And Singh, M. K. (2018). Solute transport modelling with the variable temporally dependent Boundary. Sådhanå; 43:12

Djordjevich, A. and Savovic, S. (2017). Finite difference solution of two-dimensional solute transport with periodic flow in homogeneous porous media. J. Hydrol. and Hydromech., 65(4); 426-432.

Hunt, B. (1998). Contaminant source solutions with scale-dependent dispersivities. J. Hydrol. Eng., 3(4); 268-275.

Jaiswal, D. K., Kumar, A., Kumar, N. and Singh, M. K. (2011). Solute Transport along Temporally and Spatially Dependent Flows through Horizontal Semi-Infinite Media: Dispersion Proportional to Square of Velocity. J. Hydrol. Eng, 16(3); 228-238.

Kumar, A. and Yadav, R. R. (2015). One-dimensional solute transport for uniform and varying pulse type input point source through heterogeneous medium. Environ. Tech., 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. Hydrol., 380; 330-337.

Kumar, N. and Kumar, M. (1998). Solute dispersion along unsteady groundwater flow in a semi-infinite aquifer. Hydro. and Earth System Sciences, 2; 93-100.

Majdalani, S., Chazarin, J. P., Delenne, C. and Guinot, V. (2015). Solute transport in periodical heterogeneous porous media: Importance of observation scale and experimental sampling. J. Hydrol., 520; 52-60.

Marino, M. A. (1974). Distribution of contaminants in porous media flow. Water Resour. Res., 10; 1013-1018.

Ogata, A. (1970). Theory of dispersion in granular media. US Geological Survey Professional Papers, 411-I: 34.

Sanskrityayn, A., Bharati , V. K. and Kumar, N. (2018). Solute transport due to spatio-temporally dependent dispersion coefficient and velocity: analytical solutions. J. Hydrol. Eng., 23(4); 04018009.

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. J. Groundwater Res., 5(1); 24-31.

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

Smedt, F. D. (2006). Analytical solutions for transport of decaying solutes in rivers with transient storage. J. Hydrol., 330(3-4); 672-680.

Su, N., Sander, G. C., Liu, F., Anh, V. and Barry, D. A. (2005). Similarity solutions for solute transport in fractal porous media using a time and scale-dependent dispersivity. App. Mathematical Model., 29; 852-870.

Sudicky, E. A. (1986). A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process. Water Resour. Res., 22(13); 2069-2082.

Sykes, J. F., Pahwa, S. B., Lantz, R. B. and Ward, D. S. (1982). Numerical simulation of flow and contaminant migration at an extensively monitored landfill. Water Resour. Res., 18(6); 1687-1704.

Van-Genuchten, M. T. and Alves, W. J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. U S Dept. Ag. Tech. Bull. No. 1661; 1-51.

Yadav, R. R. and Kumar, L. K. (2018). Two-dimensional conservative solute transport with temporal and scale-dependent dispersion: Analytical solution. Int. J. Adv. in Math., 2018(2); 90-111.

Yadav, R. R. and Jaiswal, D. K. (2011). Two-dimensional analytical solutions for point source contaminants transport in semi- infinite homogeneous porous medium. J. Eng. Sci. and Tech., 6(4); 459-468.

Yadav, R. R., Jaiswal, D. K., Yadav, H. K. and Gulrana. (2011). Temporally dependent dispersion through semi-infinite homogeneous porous media: an analytical solution. IJRRAS, 6 (2); 158-164.

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

Yates, S. R. (1990). An analytical solution for one-dimension transport in heterogeneous porous media. Water Resour. Res., 26(10); 2331-2338.

Yates, S. R. (1992). An analytical solution for one-dimension transport in porous media with an exponential dispersion function. Water Resour. Res., 28(8); 2149-2154.

Zhan, H., Wen, Z., Huang, G. and Sun, D. (2009). Analytical solution of two dimensional solute transports in an aquifer-aquitard system. J. Contaminant Hydrol., 107; 162-174.

January 2019

Pages 53-70