Two-Dimensional Solute Transport with Exponential Initial Concentration Distribution and Varying Flow Velocity

Document Type: Original Research Paper


1 Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, Jharkhand, India

2 Ecohydrology, Soil Physics and Land Management, Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands School of Chemistry, Monash University, Melbourne, 3800 VIC, Australia


The transport mechanism of contaminated groundwater has been a problematic issue for many decades, mainly due to the bad impact of the contaminants on the quality of the groundwater system. In this paper, the exact solution of two-dimensional advection-dispersion equation (ADE) is derived for a semi-infinite porous media with spatially dependent initial and uniform/flux boundary conditions. The flow velocity is considered temporally dependent in homogeneous media however, both spatially and temporally dependent is considered in heterogeneous porous media. First-order degradation term is taken into account to obtain a solution using Laplace Transformation Technique (LTT) for both the medium. The solute concentration distribution and breakthrough are depicted graphically. The effect of different transport parameters is studied through proposed analytical investigation. Advection-dispersion theory of contaminant mass transport in porous media is employed. Numerical solution is also obtained using Crank Nicholson method and compared with analytical result. Furthermore, accuracy of the result is discussed with root mean square error (RMSE) for both the medium. This study has developed a transport and prediction 2-D model that allows the early remediation and removal of possible pollutant in both the porous structures. The result may also be used as a preliminary predictive tool for groundwater resource and management.


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. Basha, H. A. and El-Habel, F. S. (1993). Analytical solution of the one-dimensional time-dependent transport equation. Water Resour. Res., 29(9); 3209-3214. Bear, J. (1972). Dynamics of fluids in porous media. (New York: Elsevier). Belyaev, A. Y., Dzhamalov, R. G., Medovar, Y. A. and Yushmanov, I. O. (2007). Assessment of groundwater inflow in urban territories. Water Resour., 34(5); 496-500. Broadbridge, P., Moitsheki, R. J. and Edwards, M. P. (2002). Analytical solutions for two-dimensional solute transport with velocity-dependent dispersion. Environmental mechanics: water, mass and energy transfer in the biosphere, The Philip Vol., 145-153. Carnahan, C. L. and Remer, J. S. (1984). Nonequilibrium and equilibrium sorption with a linear sorption isotherm during mass transport through an infinite porous medium: some analytical solutions. J. Hydrol., 73(3-4); 227-258. Chatterjee, A. and Singh, M. K. (2018). Two-dimensional advection-dispersion equation with depth-dependent variable source concentration. Pollut., 4(1); 1-8. Chen, J. S., Ni, C. F. and Liang, C. P. (2008). Analytical power series solutions to the two‐dimensional advection-dispersion equation with distance-dependent dispersivities. Hydrol. Processes., 22(24); 4670-4678. Chen, J. S., Chen, J. T., Liu, C. W., Liang, C. P. and Lin, C. W. (2011). Analytical solutions to two-dimensional advection-dispersion equation in cylindrical coordinates in finite domain subject to first-and third-type inlet boundary conditions. J. Hydrol., 405; 522-531. Cremer, C. J., Neuweiler, I., Bechtold, M. and Vanderborght, J. (2016). Solute transport in heterogeneous soil with time-dependent boundary conditions. Vadose Zone J., 15(6); 2-17. Hayek, M. (2016). Analytical model for contaminant migration with time-dependent transport parameters. J. Hydrol. Eng., 21(5); 04016009. Kazezyılmaz-Alhan, C. M. (2008). Analytical solutions for contaminant transport in streams. J. Hydrol., 348(3-4); 524-534. Khebchareon, M. (2012). Crank-Nicolson finite element for 2-D groundwater flow, advection-dispersion and interphase mass transfer: 1. Model development. Int. J. Numer. Anal. Model., Series B, 3(2); 109-125. Kumar, A., Jaiswal, D. K. and Kumar, N. (2009). Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain. J. Earth Syst. Sci., 118(5); 539-549. Logan, J. D. (1996). Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions. J. Hydrol., 184(3-4); 261-276. Maraqa, M. A. (2007). Retardation of nonlinearly sorbed solutes in porous media. J. Environ. Eng., 133(12); 1080-1087. Park, E. and Zhan, H. (2001). Analytical solutions of contaminant transport from finite one-, two-, and three-dimensional sources in a finite-thickness aquifer. J. Contam. Hydrol., 53; 41-61.
Pollution, 5(4): 721-737, Autumn 2019
Pollution is licensed under a "Creative Commons Attribution 4.0 International (CC-BY 4.0)"
Sanskrityayn, A. and Kumar, N. (2018). Analytical solutions of ADE with temporal coefficients for continuous source in infinite and semi-infinite media. J. Hydrol. Eng., 23(3); 06017008. Sanskrityayn, A., Singh, V. P., Bharati, V. K. and Kumar, N. (2018). Analytical solution of two-dimensional advection-dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method. Environ. Fluid Mech., 18(3); 739-757. Singh, M. K., Begam, S. Thakur, C. K. and Singh, V. P. (2018). Solute transport in a semi-infinite homogeneous aquifer with a fixed-point source concentration. Environ. Fluid Mech., 18(5); 1121-1142. Singh, M. K., Singh, P. and Singh, V. P. (2010). Analytical solution for two-dimensional solute transport in finite aquifer with time-dependent source concentration. J. Eng. Mech., 136(10); 1309-1315. Singh, M. K., Singh, V. P., Singh, P. and Shukla, D. (2009). Analytical solution for conservative solute transport in one-dimensional homogeneous porous formations with time-dependent velocity. J. Eng. Mech., 135(9); 1015-1021. Tadjeran, C. and Meerschaert, M. M. (2007). A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys., 220(2); 813-823. Tang, D. H., Frind, E. O. and Sudicky, E. A. (1981). Contaminant transport in fractured porous media: Analytical solution for a single fracture. Water Resour. Res., 17(3); 555-564. Van Duijn, C. J. and van der Zee, S. E. A. T. M. (2018). Large time behaviour of oscillatory nonlinear solute transport in porous media. Chem. Eng. Sci., 183; 86-94. Yates, S. R. (1992). An analytical solution for one-dimensional transport in porous media with an exponential dispersion function. Water Resour. Res., 28(8); 2149-2154. Wang, K. and Huang, G. (2011). Effect of permeability variations on solute transport in highly heterogeneous porous media. Adv. Water Resour., 34(6); 671-683. Zhan, H., Wen, Z., Huang, G. and Sun, D. (2009). Analytical solution of two-dimensional solute transport in an aquifer-aquitard system. J. Contam. Hydrol., 107; 162-174.
Zheng, C. and Bennett, G. D. (2002). Applied Contaminant Transport Modeling. Second ed. (Wiley: New York). Zogheib, B. and Tohidi, E. (2016). A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions. Appl. Math. Comput., 291; 1-13.