Document Type : Original Research Paper

**Authors**

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

**Abstract**

The present work solves two-dimensional Advection-Dispersion Equation (ADE) in a semi-infinite domain. A variable source concentration is regarded as the monotonic decreasing function at the source boundary (x=0). Depth-dependent variables are considered to incorporate real life situations in this modeling study, with zero flux condition assumed to occur at the exit boundary of the domain, i.e. its semi-infinite part. Without losing any generality, one can consider that the aquifer is initially contamination-free. Thus, the current study explores variations of two-dimensional contaminant concentration with depth throughout the domain, showing them graphically. Non-point source problem, i.e. the line source problem, can be discussed by solving two-dimensional depth-dependent variable source problem, as x=0 is a 2D line. A new transformation has been used to transform the time-dependent ADE to one with constant coefficients, with Matlab (pdetool) being employed in order to solve the problem, numerically, using finite element method.

**Keywords**

Ahmad, Z., Kothyari, U.C. and Ranga Raju, K.G. (1999). Finite difference scheme for longitudinal dispersion in open channels. J. Hydraul. Research, 37(3): 389-406.

Arias-Estévez, M., López-Periago, E., Martínez-Carballo, E., Simal-Gándara, J., Mejuto, J.C. and García-Río, L. (2008). The mobility and degradation of pesticides in soils and the pollution of groundwater resources. Agriculture, Ecosystems & Environment, 123(4): 247-260.

Ataie-Ashtiani, B., Lockington, D.A. and Volker, R.E. (1996). Numerical correction for finite difference solution of the advection dispersion equation with reaction. J. Contam. Hydrol., 23: 149-156.

Benson, A.D., Wheatcraft, S.W. and Meerschaert, M.M. (2000). Application of a fractional advection-dispersion equation. Water Resour. Res., 36(6): 1403-1412.

Bijeljic, B. and Blunt, M.J. (2007). Pore-scale modeling of transverse dispersion in porous media. Water Resour. Res., 43; W12S11, doi: 10.1029/2006WR005700.

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., Lai, K.H., Liu, C.W. and Ni, C.F. (2012a).A novel method for analytically solving multi-species advective-dispersive transport equations sequentially coupled with first-order decay reactions. J. Hydrol., 420-421(14): 191-204.

Chen, J.S., Liu, C.W., Liang, C.P. and Lai, K.H. (2012b). Generalized analytical solutions to sequentially coupled multi-species advective-dispersive transport equations in a finite domain subject to an arbitrary time-dependent source boundary condition. J. Hydrol., 456-457(16): 101-109.

Chrysikopoulos, C.V., Kitanidis, P.K. and Roberts, P.V. (1990).Analysis of one dimensional solute transport through porous media with spatially variable retardation factor. Water Resour. Res., 26(3): 437-446.

Diwa, E.B., Lehmann, F. and Ackerer, Ph. (2001).One dimensional simulation of solute transfer in saturated-unsaturated porous media using the discontinuous finite element method. J. Contam. Hydrol., 51(3-4): 197-213.10

Gelher, L.W. and Collins, M.A. (1971). General analysis of longitudinal dispersion in non uniform flow. Water Resour. Res., 7: 1511-1521.

Gilliom, R.J. (2007). Pesticides in US streams and groundwater. Environmental Science & Technology, 41(10): 3408-3414.

Gilliom, R.J., Barbash, J.E., Kolpin, D.W. and Larson, S.J. (1999). Peer reviewed: testing water quality for pesticide pollution. Environmental Science & Technology, 33(7): 164A-169A.

Huang, Q., Huang, G. and Zhan, H. (2008).A finite element solution for the fractional advection dispersion equation. Advances in Water Resources, 31: 1578-1589.

Hunt, A.G. and Ghanbarian, B. (2016). Percolation theory for solute transport in porous media: Geochemistry, geomorphology, and carbon cycling. Water Resour. Res., 52(9): 7444-7459.

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. Engg., 16(3): 228-238.

Khalifa, M.E. (2003).Some analytic solutions for the advection-dispersion equation. Appl. Math comput., 139: 299-310.

Kumar, R.P., Dodagoudar, G.R. and Rao, B.N. (2007).Meshfree modeling of one dimensional contaminant transport in unsaturated porous media. Geomechanics and Geoengineering: An International Journal, 2(2): 129-136.

Lai, K.H., Liu, C.W., Liang, C.P., Chen, J.S., & Sie, B.R. (2016). A novel method for analytically solving a radial advection-dispersion equation. J. Hydrol., 542: 532-540.

Leij, Feike J., Bradford, Scott A., Sciortino, Antonella (2016). Analytic solutions for colloid transport with time- and depth-dependent retention in porous media. J. Contam. Hydrol., doi: 10.1016/j.jconhyd.2016.10.006.

Li, Y., Yuan, D., Lin, B. and Teo, F.Y. (2016). A fully coupled depth-integrated model for surface water and groundwater flows. J. Hydrol., 542: 172-184.

Logan, J.D. (1996). Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions. J. Hydrol., 184(3-4): 261-276.

Saied, E.A. and Khalifa, M.E. (2002). Analytical solutions for groundwater flow and transport equation. Transp. Porous Media., 47: 295-308.

Sander, G.C. and Braddock, R.D. (2005). Analytical solutions to the transient, unsaturated transport of water and contaminants through horizontal porous media. Adv. Water Res., 28: 1102-1111.

Sim, Y. and Chrysikopoulos, C.V. (1999).Analytic solution for solute transport in saturated porous media with semi-infinite or finite thickness. Adv. Water Res., 22(5): 507-519.

Singh, M.K. and Kumari, P. (2014). Contaminant concentration prediction along unsteady groundwater flow. Modelling and Simulation of Diffusive Processes, Series: Simulation Foundations, Methods and Applications. Springer, XII; 257-276.

Singh, M.K., Mahato, N.K. and Kumar, N. (2015). Pollutant’s Horizontal Dispersion Along and Against Sinusoidally Varying Velocity from a Pulse Type Point Source. Acta Geophysica, 63(1): 214-231.

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. Engg. Mech., 135(9): 1015-1021.

Singh, M.K., Mahato, N.K. and Singh, P. (2008).Longitudinal dispersion with time dependent source concentration in semi-infinite aquifer. J. Earth Syst. Sci., 117(6): 945-949.

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

Srinivasan, V. and Clement, T.P. (2008). An analytical solution for sequentially coupled one-dimensional reactive transport problems Part-I: Mathematical Derivations. Water Resour. Res., 31: 203.

Van Genuchten, M.Th. (1982). A comparison of numerical solutions of the one dimensional unsaturated-saturated flow and mass transport equations. Adv. Water Resour., 5: 47-55.

Zamani, K. and Bombardelli, F.A. (2013). Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers. Environ. Fluid Mech., doi: 10.1007/s10652-013-9325-0.

Zhang, J., Clare, J. and Guo, J. (2012). A semi-analytical solution based on a numerical solution of the solute transport of a conservative and non reactive trace. Groundwater, 50(4): 633-638.

Winter 2018

Pages 1-8