Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media

Open access


The two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finitedifference method for the transport of solutes through a homogenous two-dimensional domain that is finite and porous. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the firstorder decay and zero-order production parameters proportional to the seepage velocity, and periodic boundary conditions at the origin and at the end of the domain. Results agree well with analytical solutions that were reported in the literature for special cases. It is shown that the solute concentration profile is influenced strongly by periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in finite media, which is especially important when arbitrary initial and boundary conditions are required.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • Ahmed S.G. 2012. A numerical algorithm for solving advection- diffusion equation with constant and variable coefficients. The Open Num. Meth. J. 4 1-7.

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

  • Anderson J.D. 1995. Computational Fluid Dynamics McGraw-Hill New York.

  • Appadu A.R. 2013. Numerical solution of the 1D advectiondiffusion equation using standard and nonstandard finite difference schemes J. Appl. Math. 2013 1-14.

  • Chatwin P.C. Allen C.M. 1985. Mathematical models of dispersion in rivers and estuaries. Ann. Rev. Fluid Mech. 17 119-149.

  • Chaudhry M.H. Cass D.E. Edinger J.E. 1983. Modelling of unsteady-flow water temperatures. J. Hydraul. Eng. 109 5 657-669.

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

  • Cherry J.A. Gillham R.W. Barker J.F. 1984. Contaminants in Groundwater - Chemical Processes in Groundwater Contamination. National Academy Press Washington DC pp. 46-64.

  • Ciftci E. Avci C.B. Borekci O.S. Sahin A.U. 2012. Assessment of advective-dispersive contaminant transport in heterogeneous aquifers using a meshless method. Environmental Earth Sci. 67 2399−2409.

  • Dehghan M. 2004. Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Appl. Math. Computation 147 307-319.

  • Dhawan S. Kapoor S. Kumar S. 2012. Numerical method for advection-diffusion equation using FEM and B-splines. J. Computat. Sci. 3 429-437.

  • Djordjevich A. Savović S. 2013. Solute transport with longitudinal and transverse diffusion in temporally and spatially dependent flow from a pulse type source. Int. J. Heat Mass Trans. 65 321-326.

  • Fattah Q.N. Hoopes J.A. 1985. Dispersion in anisotropic homogeneous porous media. J. Hydraul. Eng. 111 810-827.

  • Gane C.R. Stephenson P.L. 1979. An explicit numerical method for solving transient combined heat conduction and convection problems. Int. J. Numer. Meth. Eng. 14 1141-1163.

  • Gharehbaghi A. 2016. Explicit and implicit forms of differential quadrature method for advection-diffusion equation with variable coefficients in semi-infinite domain. J. Hydrol. 504(B) 935-940.

  • Gharehbaghi A. 2017. Third- and fifth-order finite volume schemes for advection-diffusion equation with variable coefficients in semi-infinite domain. Water Environm. J. 31 2 184-193.

  • Guvanasen V. Volker R.E. 1983. Numerical solutions for solute transport in unconfined aquifers. Int. J. Numer. Meth. Fluids 3 103-123.

  • Holly F.M. Usseglio-Polatera J.M. 1984. Dispersion simulation in two-dimensional tidal flow. J. Hydraul. Eng. 111 905-926.

  • Huang Q. Huang G. Zhan H. 2008. A finite element solution for the fractional advection-dispersion equation. Adv. Water Resour. 31 1578-1589.

  • Jaiswal D.K. Kumar A. Kumar N. Yadav R.R. 2009. Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in onedimensional semi-infinite medium. J. Hydro-Environm.Res. 2 254-263.

  • Jaiswal D.K. Kumar A. Yadav R.R. 2011. Analytical solution to the one-dimensional advection-diffusion equation with temporally dependent coefficients. J. Water Resource Protect. 3 76-84.

  • Karahan H. 2006. Implicit finite difference techniques for the advection-diffusion equation using spreadsheets. Adv. Eng. Software 37 601-608.

  • Kaya B. Gharehbaghi A. 2014. Implicit solutions of advection diffusion equation by various numerical methods. Aust. J. Basic & Appl. Sci. 8 1 381-391.

  • Kumar N. 1988. Unsteady flow against dispersion in finite porous media. J. Hydrol. 63 345-358.

  • Kumar A. Jaiswal D.K. Kumar N. 2010. Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. J. Hydrol. 380 330−337.

  • Lapidus L. Amundson N.R. 1952. Mathematics of adsorption in beds VI. The effects of longitudinal diffusion in ionexchange and chromatographic columns. J. Phys. Chem. 56 8 984-988.

  • Logan J.D. Zlotnik V. 1995. The convection-diffusion equation with periodic boundary conditions. Appl. Math. Lett. 8 1995 55-61.

  • Marino M.A. 1978. Flow against dispersion in non-adsorbing porous media. J. Hydrol. 37 149-158.

  • Nazir T. Abbas M. Ismail A.I.M. Majid A.A. Rashid A. 2016. The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Appl. Math. Modelling. 40 7-8 4586-4611.

  • Parlange J.-Y. 1980. Water transport in soils. Ann. Rev. Fluid Mech. 12 77-102.

  • Salmon J.R. Liggett J.A. Gallager R.H. 1980. Dispersion analysis in homogeneous lakes. Int. J. Numer. Meth. Eng. 15 1627-1642.

  • Savović S. Caldwell J. 2003. Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. Int. J. Heat Mass Transfer 46 2911−2916.

  • Savović S. Caldwell J. 2009. Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method. Thermal Sci. 13 165−174.

  • Savović S. Djordjevich A. 2012. Finite difference solution of the one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. Int. J. Heat Mass Transfer 55 4291−4294.

  • Savović S. Djordjevich A. 2013. Numerical solution for temporally and spatially dependent solute dispersion of pulse type input concentration in semi-infinite media. Int. J. Heat Mass Transfer 60 291-295.

  • Siemieniuch J.L. Gladwell I. 1978. Analysis of explicit difference methods for a diffusion-convection equation. Int. J. Num. Meth. Eng. 12 6 899-916.

  • Šimůnek J. van Genuchten M.T. Šejna M. Toride N. Leij F.J. 1999. The STANMOD computer software for evaluating solute transport in porous media using analytical solutions of convection-dispersion equation. Versions 1.0 and 2.0. IGWMC-TPS-71. Colorado School of Mines International Ground Water Modeling Center Golden Colorado.

  • Townley L.R. 1995. The response of aquifers to periodic forcing. Adv. Water Resour. 18 125-146.

  • van Genuchten M.T. Šimůnek J. Leij F.J. Toride N. Šejna M. 2012. STANMOD: Model use calibration and validation. Trans. ASABE 55 4 1353-1366.

  • Yadav R.R. Jaiswal D.K. Yadav H.K. Gulrana 2011. Temporally dependent dispersion through semi-infinite homogeneous porous media: an analytical solution. Int. J. Res. Rev. Appl. Sc. 6 158-164.

  • Yadav R.R. Jaiswail D.K. Gulrana 2012. Two-dimensional solute transport for periodic flow in isotropic porous media: an analytical solution. Hydrol. Process. 26 3425−3433.

  • Zhao C. Valliappan S. 1994. Numerical modelling of transient contaminant migration problems in infinite porous fractured media using finite/infinite element technique: theory. Int. J. Num. Anal. Meth. Geomech. 18 523-541.

  • Zlatev Z. Berkowicz R. Prahm L.P. 1984. Implementation of a variable stepsize variable formula in the timeintegration part of a code for treatment of long-range transport of air pollutants. J. Comput. Phys. 55 278-301.

Journal information
Impact Factor

IMPACT FACTOR 2018: 2,023
5-year IMPACT FACTOR: 2,048

CiteScore 2018: 2.07

SCImago Journal Rank (SJR) 2018: 0.713
Source Normalized Impact per Paper (SNIP) 2018: 1.228

Cited By
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 422 273 21
PDF Downloads 207 149 7