Abstract
This study deals with the 1-D pollutant transport model in homogeneous and heterogeneous semi-infinite groundwater reservoir. The pollutant concentration is considered in liquid as well as in solid phases. The Laplace transform technique is adopted to solve the 1-D ADE and that has been contributing significantly in the field of pollutant transport modelling. Dirichlet-type and Neumann-type boundary conditions are considered in the modelled domain which is not solute free initially. Analytical solutions are investigated for different geological formations such as sandstone, shale and gravel to set the physical insight of the problem. Hydrodynamic dispersion theory is employed in this model. An impact of pollutant existing in liquid and solid phases is shown in the modelled domain and accordingly pollutant concentrations are graphically depicted. The objective of this study is to provide a development of solution for the contaminated groundwater transport with major focus on solute dynamics with reactive species in the different geological reservoir. In addition, it is also important to observe the diffusion effects on the solute transport for both sites. The Crank–Nicolson approach was applied for numerical simulation of governing transport equation. The corroboration of transport parameter demonstrates the good applicability of the proposed mathematical model in more realistic problems. This study may be useful as one of the preliminary predictive tools to groundwater resource and remediation project planning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahsan M (2012) Numerical solution of the advection-diffusion equation using Laplace transform finite analytical method. Int J River Basin Manag 10(2):177–188. https://doi.org/10.1080/15715124.2012.679736
Antonopoulos VZ, Papazafiriou ZG (1990) Simultaneous one-dimensional water flow and mass transport of conservative solutes in porous media. Water Resour Manag 4(1):47–62. https://doi.org/10.1007/BF00429924
Atangana A (2014) A generalized advection dispersion equation. J Earth Syst Sci 123(1):101–108. https://doi.org/10.1007/s12040-013-0389-5
Basha HA, El-Habel F (1993) Analytical solution of the one-dimensional time-dependent transport equation. Water Resour Res 29(9):3209–3214. https://doi.org/10.1029/93WR01038
Bouteligier R, Vaes G, Berlamont J, Flamink C, Langeveld JG, Clemens FHLR (2005) Advection-dispersion modelling tools: what about numerical dispersion? Water Sci Technol 52(3):19–27. https://doi.org/10.2166/wst.2005.0057
Brusseau ML (1994) Transport of reactive contaminants in heterogeneous porous media. Rev Geophys 32(3):285–313. https://doi.org/10.1029/94RG00624
Chen CM, Liu F, Burrage K (2008a) Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation. Appl Math Comput 198(2):754–769. https://doi.org/10.1016/j.amc.2007.09.020
Chen JS, Ni CF, Liang CP, Chiang CC (2008b) Analytical power series solution for contaminant transport with hyperbolic asymptotic distance-dependent dispersivity. J Hydrol 362(1):142–149. https://doi.org/10.1016/j.jhydrol.2008.08.020
Chen JS, Lai KH, Liu CW, Ni CF (2012a) A novel method for analytically solving multi-species advective-dispersive transport equations sequentially coupled with first-order decay reactions. J Hydrol 420:191–204. https://doi.org/10.1016/j.jhydrol.2011.12.001
Chen JS, Liu CW, Liang CP, Lai KH (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 456:101–109. https://doi.org/10.1016/j.jhydrol.2012.06.017
Ciftci E, Avci CB, Borekci OS, Sahin AU (2012) Assessment of advective–dispersive contaminant transport in heterogeneous aquifers using a meshless method. Environ Earth Sci 67(8):2399–2409. https://doi.org/10.1007/s12665-012-1686-z
Clement TP (2001) Generalized solution to multispecies transport equations coupled with a first-order reaction network. Water Resour Res 37(1):157–163. https://doi.org/10.1029/2000WR900239
Crank J (1975) The mathematics of diffusion, 2nd edn. Oxford University Press, London
Das P, Begam S, Singh MK (2017) Mathematical modeling of groundwater contamination with varying velocity field. J Hydrol Hydromech 65(2):192–204. https://doi.org/10.1515/johh-2017-0013
De Smedt F (2006) Analytical solutions for transport of decaying solutes in rivers with transient storage. J Hydrol 330(3):672–680. https://doi.org/10.1016/j.jhydrol.2006.04.042
Delay F, Porel G, de Marsily G (1997) Predicting solute transport in heterogeneous media from results obtained in homogeneous ones: an experimental approach. J Contam Hydrol 25(1–2):63–84. https://doi.org/10.1016/S0169-7722(96)00020-4
Freeze RA, Cherry JA (1979) Groundwater. Prentice-Hall International, Englewood Cliffs, New Jersey
Gharehbaghi A (2016) Explicit and implicit forms of differential quadrature method for advection-diffusion equation with variable coefficients in semi-infinite domain. J Hydrol 541:935–940. https://doi.org/10.1016/j.jhydrol.2016.08.002
Guerrero JP, Skaggs TH (2010) Analytical solution for one-dimensional advection-dispersion transport equation with distance-dependent coefficients. J Hydrol 390(1):57–65. https://doi.org/10.1016/j.jhydrol.2010.06.030
Guerrero JP, Pontedeiro EM, van Genuchten MT, Skaggs TH (2013) Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions. Chem Eng J 221:487–491. https://doi.org/10.1016/j.cej.2013.01.095
Hayek M (2016) Analytical model for contaminant migration with time-dependent transport parameters. J Hydrol Eng 21(5):04016009. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001360
Hill DJ, Minsker BS, Valocchi AJ, Babovic V, Keijzer M (2007) Upscaling models of solute transport in porous media through genetic programming. J Hydroinform 9(4):251–266. https://doi.org/10.2166/hydro.2007.028
Jhamnani B, Singh SK (2009) Groundwater contamination due to Bhalaswa landfill site in New Delhi. Int J Environ Sci Eng 1(3):121–125
Leij FJ, van Genuchten MT, Dane JH (1991) Mathematical analysis of one-dimensional solute transport in a layered soil profile. Soil Sci Soc Am J 55(4):944–953. https://doi.org/10.2136/sssaj1991.03615995005500040008x
Liu C, Szecsody JE, Zachara JM, Ball WP (2000) Use of the generalized integral transform method for solving equations of solute transport in porous media. Adv Water Resour 23(5):483–492. https://doi.org/10.1016/S0309-1708(99)00048-2
Manger GE (1963) Porosity and bulk density of sedimentary rocks. U.S. Atomic Energy Commission USGPO, Washington, D.C.
Mazaheri M, Samani JMV, Samani HMV (2013) Analytical solution to one-dimensional advection-diffusion equation with several point sources through arbitrary time-dependent emission rate patterns. J Agric Sci Technol 15(6):1231–1245
Mishra S, Parker JC (1990) Analysis of solute transport with a hyperbolic scale-dependent dispersion model. Hydrol Process 4(1):45–57. https://doi.org/10.1002/hyp.3360040105
Ogata A (1970) Theory of dispersion in a granular medium. US Government Printing Office, Washington
Rezaei A, Zhan H, Zare M (2013) Impact of thin aquitards on two-dimensional solute transport in an aquifer. J Contam Hydrol 152:117–136. https://doi.org/10.1016/j.jconhyd.2013.06.008
Sanskrityayn A, Suk H, Kumar N (2017) Analytical solutions for solute transport in groundwater and riverine flow using Green’s Function Method and pertinent coordinate transformation method. J Hydrol 547:517–533. https://doi.org/10.1016/j.jhydrol.2017.02.014
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 Transf 55(15):4291–4294. https://doi.org/10.1016/j.ijheatmasstransfer.2012.03.073
Sharma PK, Ojha CSP, Swami D, Joshi N, Shukla SK (2015) Semi-analytical solutions of multiprocessing non-equilibrium transport equations with linear and exponential distance-dependent dispersivity. Water Resour Manag 29(14):5255–5273. https://doi.org/10.1007/s11269-015-1116-6
Sheng F, Wang K, Zhang R, Liu H (2011) Modeling preferential water flow and solute transport in unsaturated soil using the active region model. Environ Earth Sci 62(7):1491–1501. https://doi.org/10.1007/s12665-010-0633-0
Sim Y, Chrysikopoulos CV (1995) Analytical models for one-dimensional virus transport in saturated porous media. Water Resour Res 31(5):1429–1437. https://doi.org/10.1029/95WR00199
Sim Y, Chrysikopoulos CV (1996) One-dimensional virus transport in porous media with time-dependent inactivation rate coefficients. Water Resour Res 32(8):2607–2611. https://doi.org/10.1029/96WR01496
Singh MK, Das P (2015) Scale dependent solute dispersion with linear isotherm in heterogeneous medium. J Hydrol 520:289–299. https://doi.org/10.1016/j.jhydrol.2014.11.061
Singh MK, Kumari P (2014) Contaminant concentration prediction along unsteady groundwater flow. In: Basu SK, Kumar N (eds) Modelling and simulation of diffusive processes. Springer, Cham, pp 257–275. https://doi.org/10.1007/978-3-319-05657-9_12
Singh MK, Singh P, Singh VP (2010) Analytical solution for two-dimensional solute transport in finite aquifer with time-dependent source concentration. J Eng Mech 136(10):1309–1315. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000177
Singh MK, Singh VP, Das P (2015) Mathematical modeling for solute transport in aquifer. J Hydroinform 18(3):481–499. https://doi.org/10.2166/hydro.2015.034
Singh MK, Chatterjee A, Kumari P (2018) Mathematical modeling of one-dimensional advection dispersion equation in groundwater contamination using different velocity and dispersion for different zones. In: Singh MK, Kushvah BS, Seth G, Prakash J (eds) Applications of fluid dynamics. Springer, Singapore, pp 585–592. https://doi.org/10.1007/978-981-10-5329-0_44
Su N, Sander GC, Liu F, Anh V, Barry DA (2005) Similarity solutions for solute transport in fractal porous media using a time-and scale-dependent dispersivity. Appl Math Model 29(9):852–870. https://doi.org/10.1016/j.apm.2004.11.006
Van Genuchten MT (1982) Analytical solutions of the one-dimensional convective-dispersive solute transport equation (No. 1661). US Department of Agriculture, Agricultural Research Service
Yates SR (1990) An analytical solution for one-dimensional transport in heterogeneous porous media. Water Resour Res 26(10):2331–2338. https://doi.org/10.1029/WR026i031
You K, Zhan H (2013) New solutions for solute transport in a finite column with distance-dependent dispersivities and time-dependent solute sources. J Hydrol 487:87–97. https://doi.org/10.1016/j.jhydrol.2013.02.027
Yuste SB, Acedo L (2005) An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J Numer Anal 42(5):1862–1874. https://doi.org/10.1137/030602666
Zhao P, Zhang X, Sun C, Wu J, Wu Y (2017) Experimental study of conservative solute transport in heterogeneous aquifers. Environ Earth Sci 76(12):421. https://doi.org/10.1007/s12665-017-6734-2
Zhuang P, Liu F (2006) Implicit difference approximation for the time fractional diffusion equation. J Appl Math Comput 22(3):87–99. https://doi.org/10.1007/BF02832039
Acknowledgements
Authors would like to acknowledge IIT(ISM) Dhanbad for supporting this work through JRF scheme. This work is also partially supported by Council of Scientific and Industrial Research, New Delhi under the Project No. 25(0251)/16/EMR-II.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chaudhary, M., Thakur, C.K. & Singh, M.K. Analysis of 1-D pollutant transport in semi-infinite groundwater reservoir. Environ Earth Sci 79, 24 (2020). https://doi.org/10.1007/s12665-019-8748-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12665-019-8748-4