Analytical Model for Contaminant Migration with Time-Dependent Transport Parameters
Publication: Journal of Hydrologic Engineering
Volume 21, Issue 5
Abstract
A general analytical model for one-dimensional (1D) contaminant transport in infinite domain with time-dependent transport parameters is presented in this paper. The model is based on the advection-dispersion equation with a time-dependent flow velocity, a time-dependent dispersion coefficient, and a time-dependent distribution coefficient due to sorption. It takes into account a first-order irreversible reaction (decay), an arbitrary initial distribution of the contaminant, and an arbitrary space- and time-dependent sink/source term. The model can handle any time-dependent transport parameter. Analytical solutions are provided for both the advection dominant and the advection-dispersion transport equations. The proposed analytical solutions are general and they can be used for problems where one or more parameters are time-dependent. It is shown that the presented solutions can be reduced to other existent solutions where one transport parameter is assumed to be time-dependent. The general analytical solutions are obtained by using the Fourier transform, and they are presented in integral forms. Several closed-form solutions can be derived from the general integral form. Such closed-form solutions are of great interest since they allow the dependence of the solutions on the underlying physical parameters to be studied in an analytical manner. In particular, the author presents some closed-form solutions for the case of an initial step function and discusses through the numerical examples some insights about the errors that can be made by assuming constant parameters instead of time-dependent ones. These assumptions may lead to overestimated or underestimated concentrations. The proposed analytical solutions are useful for benchmarking numerical solutions to problems in hydrogeology and chemical engineering. They are also of great importance to the investigation of quantitative accuracy assessment. One of the presented closed-form solutions is used to compare with numerical solutions.
Get full access to this article
View all available purchase options and get full access to this article.
References
Abramowitz, M., and Stegun, I. A. (1972). “Handbook of mathematical functions with formulas, graphs, and mathematical tables.” Government Printing Office, Washington, DC.
Aral, M. M., and Liao, B. (1996). “Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients.” J. Hydrol. Eng., 20–32.
Barry, D. A., and Sposito, G. (1989). “Analytical solution of a convection-dispersion model with time-dependent transport coefficients.” Water Resour. Res., 25(12), 2407–2416.
Basha, H. A. (1997). “Analytical model of two-dimensional dispersion in laterally nonuniform axial velocity distributions.” J. Hydraul. Eng., 853–862.
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, Dover, Mineola, NY.
Chrysikopoulos, C. V., and Sim, Y. (1996). “One-dimensional virus transport in homogeneous porous media with time-dependent distribution coefficient.” J. Hydrol., 185(1–4), 199–219.
Corey, J. C., Hawkins, R. H., Overman, R. F., and Green, R. E. (1970). “Miscible displacement measurements within laboratory columns using the gamma photoneutron method.” Soil Sci. Soc. Am. Proc., 34(6), 854–858.
Deng, B., and Qiu, Y. (2012). “Comment on “Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media” by Kumar, A., Jaiswal, D. K., Kumar, N., J. Hydrol., 2010, 380, 330–337.” J. Hydrol., 424–425, 278–279.
Fenske, P. R. (1979). “Time-dependent sorption on geological materials.” J. Hydrol., 43(1–4), 415–425.
Freeze, R. A., and Cherry, J. A. (1979). Groundwater, Prentice Hall, Englewood Cliffs, NJ.
Fried, J. J. (1975). Groundwater pollution, Elsevier Applied Science, New York.
Gelhar, L. W., Gutjahr, A. L., and Naff, R. L. (1979). “Stochastic analysis of macrodispersion in a stratified aquifer.” Water Resour. Res., 15(6), 1387–1397.
Hashimoto, I., Deshpande, K. B., and Thomas, H. C. (1964). “Peclet numbers and retardation factors for ion exchange columns.” Ind. Eng. Chem. Res., 3(3), 213–217.
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. Eng., 228–238.
Jaiswal, D. K., Kumar, A., Kumar, N., and Yadav, R. (2009). “Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media.” J. Hydro-Environ. Res., 2(4), 254–263.
Knupp, P. M., and Salari, K. (2003). Verification of computer codes in computational science and engineering, Chapman & Hall/CRC, London.
Kookana, R. S., Aylmore, L. A. G., and Gerritse, R. G. (1992a). “Time-dependent sorption of pesticides during transport in soils.” Soil Sci., 154(3), 214–225.
Kookana, R. S., Gerritse, R. G., and, Aylmore, L. A. G. (1992b). “A method for studying nonequilibrium sorption during transport of pesticides in soil.” Soil Sci., 154(5), 344–349.
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.
Kumar, A., Jaiswal, D. K., and Kumar, N. (2010). “Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media.” J. Hydrol., 380(3–4), 330–337.
Moody, J. B. (1982). “Radionuclide migration/retardation: Research and development technology status report.”, Office of Nuclear Waste Isolation, Battelle Memorial Inst., Springfield, VA, 61.
Perko, J., Mallants, D., Jacques, D., and Wang, L. (2009). “Coupling time-dependent sorption values of degrading concrete with a radionuclide migration model.” Proc., ASME 2009 12th Int. Conf. on Environmental Remediation and Radioactive Waste Management, Liverpool, U.K., 9–17.
Pickens, J. F., and Grisak, G. E. (1981). “Modeling of scale-dependent dispersion in hydrogeologic systems.” Water Resour. Res., 17(6), 1701–1711.
Selvadurai, A. P. S. (2004). “On the advective-diffusive transport in porous media in the presence of time-dependent velocities.” Geophys. Res. Lett., 31(13), L13505.
Selvadurai, A. P. S. (2008). “On nonclassical analytical solutions for advective transport problems.” Water Resour. Res., 44, W00C03.
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., 1015–1021.
Smith, D. W., Rowe, R. K., and Booker, J. R. (1993). “The analysis of pollutant migration through soil with linear hereditary time-dependent sorption.” Int. J. Numer. Anal. Meth. Geomech., 17(4), 255–274.
Sneddon, I. H. (1972). The use of integral transforms, McGraw-Hill, New York.
van Beinum, W., Beulke, S., and Brown, C. D. (2005). “Pesticide sorption and diffusion in natural clay loam aggregates.” J. Agric. Food Chem., 53(23), 9146–9154.
Villaverde, J., van Beinum, W., Beulke, S., and Brown, C. D. (2009). “The kinetics of sorption by retarded diffusion into soil aggregate pores.” Environ. Sci. Technol., 43(21), 8227–8232.
Yadav, R. R., Jaiswal, D. K., Yadav, H. K., and Rana, G. U. L. (2010). “One-dimensional temporally dependent advection–dispersion equation in porous media: Analytical solution.” Nat. Resour. Model., 23(4), 521–539.
Yadav, S. K., Kumar, A., and Kumar, N. (2012). “Horizontal solute transport from a pulse type source along temporally and spatially dependent flow: Analytical solution.” J. Hydrol., 412–413, 193–199.
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.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 21 • Issue 5 • May 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Jul 2, 2015
Accepted: Dec 7, 2015
Published online: Feb 23, 2016
Published in print: May 1, 2016
Discussion open until: Jul 23, 2016
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.