Two-dimensional advection-dispersion equation with depth- dependent variable source concentration

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.