FINITE DIFFERENCE SOLUTION OF TWO-DIMENSIONAL SOLUTE TRANSPORT WITH PERIODIC FLOW IN HOMOGENOUS POROUS MEDIA
DOI:
https://doi.org/10.7251/COMEN1702128DAbstract
Two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finite-difference method for the transport of solutes through a homogeneous, finite, porous, two-dimensional, domain. 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 first-order decay and zero-order production parameters proportional to the seepage velocity, periodic boundary conditions at the origin and the end of the domain. Results are compared to analytical solutions reported in the literature for special cases and a good agreement was found. The solute concentration profile is greatly influenced by the 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 a finite media, which is especially important when arbitrary initial and boundary conditions are required.
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J. D. Logan, V. Zlotnik, The convection–diffusion equation with periodic boundary conditions, Applied Mathematics Letters, Vol. 8 (1995) 55–61.
L. R. Townley, The response of aquifers to periodic forcing, Advances in water Resources, Vol. 18 (1995) 125–146.
D. K. Jaiswal, A. Kumar, N. Kumar, R. R. Yadav, Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite medium, Journal of Hydro-Environment Research, Vol. 2 (2009) 254–263.
D. K. Jaiswal, A. Kumar, R. R. Yadav, Analytical solution to the one-dimensional advec-tion-diffusion equation with temporally dependent coefficients, Journal of Water Resource and Protection, Vol. 3 (2011) 76–84.
A. Kumar, D. K. Jaiswal, N. Kumar, Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media, J. Hydrology, 380 (2010) 330337.
R. R. Yadav, D. K. Jaiswal, H. K. Yadav, Gulrana, Temporally dependent dispersion through semi-infinite homogeneous porous media: an analytical solution, International Journal of Research and Reviews in Applied Sciences, Vol. 6 (2010) 158–164.
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S. Dhawan, S. Kapoor, S. Kumar, Numerical method for advection-diffusion equation using FEM and B-splines, J. Computational Science, Vol. 3 (2012) 429–437.
M. Dehghan, Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Computation, Vol. 147 (2004) 307–319.
H. Karahan, Implicit finite difference techniques for the advection–diffusion equation using spreadsheets, Adv. Eng. Software, Vol. 37 (2006) 601–608.
Q. Huang, G. Huang, H. Zhan, A finite element solution for the fractional advection-dispersion equation, Adv. Water Resources, Vol. 31 (2008) 1578–1589.
C. Zhao, S. Valliappan, Transient infinite element for contaminant transport problems, Int. J. Num. Meth. Eng., Vol. 37 (1994) 113–1158.
M. Srećković, A. A. Ionin, A. Janići-jević, A. Bugarinović, S. Ostojić, M. Janićijević, N. Ratković Kovačević, Formalisms analysis, results and accomplishments with population inversion of material, Contemporary Materials, Vol. VIII−1 (2017) 91−108.
C. Zhao, S. Valliappan, Numerical modelling of transient contaminant migration problems in infinite porous fractured media using finite/infinite element technique: theory, Int. J. Num. Anal. Meth. Geomech., Vol. 18 (1994) 523–541.
S. Savović, A. Djordjevich, Finite difference solution of the one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media, Int. J. Heat Mass Transfer, Vol. 55 (2012) 42914294.
S. Savović, A. Simović, A. Djordje-vich, A. Janićijević, Equilibrium mode distribu-tion in W-type glass optical fibers, Contemporary Materials, Vol. V−1 (2014) 51−58.
S. Savović, A. Djordjevich, Numerical solution for temporally and spatially dependent solute dispersion of pulse type input concentration in semi-infinite media, Int. J. Heat Mass Transfer, Vol. 60 (2013) 291–295.
A. Djordjevich, S. Savović, Solute transport with longitudinal and transverse diffusion in temporally and spatially dependent flow from a pulse type source, International Journal of Heat and Mass Transfer, Vol. 65 (2013) 321−326.
A. N. S. Al-Niami, K. R. Rushton, Analysis of flow against dispersion in porous media, Journal of Hydrology, Vol. 33 (1977) 87–97.
M. A. Marino, Flow against dispersion in non-adsorbing porous media, Journal of Hydrology, Vol. 37 (1978) 149–158.
R. R. Yadav, D. K. Jaiswail, G. Gulrana, Two-dimensional solute transport for periodic flow in isotropic porous media: an analytical solution, Hydrol. Process., Vol. 26 (2012) 34253433.
L. Lapidus, N. R. Amundson, Mathematics of adsorption in beds, VI. The effects of longitudinal diffusion in ion-exchange and chromatographic columns, Vol. 56 (1952) 984–988.
J. A. Cherry, R. W. Gillham, J. F. Barker, Contaminants in groundwater- Chemical processes in Groundwater contamination, National Academy Press: Washington, DC (1984) 46–64.
S. Savović, J. Caldwell, Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions, Int. J. Heat Mass Transfer, Vol. 46 (2003) 29112916.
A. Janićijević, S. Savović, A. Djordje-vich, Numerical solution of the diffusion for oxygen diffusion in soil, Contemporary Materials , Vol. VII−1 (2016) 6−10.
S. Savović, J. Caldwell, Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method, Thermal Sci., Vol. 13 (2009) 165174.
J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, New York, 1995.
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2018-02-14
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