Fractal Dispersion Pollutant Transport Modeling with Spatially Varying Sorption and Degradation Effect
Publication: Journal of Environmental Engineering
Volume 148, Issue 8
Abstract
The present study deals with the generalized one-dimensional (1-D) advection fractal dispersion (AFD) equation. The numerical simulation is carried out to simulate the plume’s fate and transport phenomenon in a finite porous medium. The present model is developed for chemical substitute migration in soil with spatial varying advection and its corresponding fractal dispersion. Moreover, it is adequate for deliberating spatial varying sorption and first-order biological degradation due to reaction processes in the soil. As in generalized dispersion theory, the dispersion term is expressed to be directly proportional to seepage velocity by some power , where varies from 1 to 2. The present numerical study is capable of dealing with various types of advection and its corresponding fractal dispersion. Initially, we have assumed that the aquifer was pollutant-free. At the inlet of the domain, some pollutant source has been considered, and at the outlet boundary, it is assumed that the pollutant concentration disappears to zero. The pollutant concentration distribution has been observed in different types of porous medium as well as for various hydrological parameters. The current generalized AFD model is based on the implicit finite difference technique, which can illustrate chemical migration in the soil to the groundwater, and further, the MATLAB version R2013a(8.1.0.604) Pdepe tool technique validates it. A special case study is also discussed regarding the migration of Iron (Fe) and Zinc (Zn) in the soil. Afterwards, the numerical model is used to illustrate the breakthrough curves (BTCs) for the various transport parameters, and the accuracy of the model problem is also discussed using root mean square error (RMSE).
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions. Code available upon request includes the numerical code developed to solve the generalized fractal dispersion model.
Acknowledgments
The authors are thankful to the Indian Institute of Technology (Indian School of Mines) and Dhanbad for providing financial assistance to the Ph.D. Scholar. The authors are also thankful to the editor and reviewers for their constructive comments which helped to improve the quality of the paper.
References
Aral, M. M., and B. Liao. 1996. “Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients.” J. Hydrol. Eng. 1 (1): 20–32. https://doi.org/10.1061/(ASCE)1084-0699(1996)1:1(20).
Baeumer, B., D. A. Benson, and M. M. Meerschaert. 2005. “Advection and dispersion in time and space.” Physica A 350 (2–4): 245–262. https://doi.org/10.1016/j.physa.2004.11.008.
Banaei, S. M. A., A. H. Javid, and A. H. Hassani. 2021. “Numerical simulation of groundwater contaminant transport in porous media.” Int. J. Environ. Sci. Technol. 18 (1): 151–162. https://doi.org/10.1007/s13762-020-02825-7.
Basha, H. A., and F. S. El-Habel. 1993. “Analytical solution of the one-dimensional time-dependent transport equation.” Water Resour. Res. 29 (9): 3209–3214. https://doi.org/10.1029/93WR01038.
Batu, V. 2005. Applied flow and solute transport modeling in aquifers: Fundamental principles and analytical and numerical methods. London: CRC Press.
Bear, J. 1969. “Hydrodynamic dispersion.” In Flow through porous media. San Diego, CA: Academic Press.
Bharati, V. K., V. P. Singh, A. Sanskrityayn, and N. Kumar. 2017. “Analytical solution of advection-dispersion equation with spatially dependent dispersivity.” J. Eng. Mech. 143 (11): 04017126. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001346.
Bharati, V. K., V. P. Singh, A. Sanskrityayn, and N. Kumar. 2018. “Analytical solutions for solute transport from varying pulse source along porous media flow with spatial dispersivity in fractal & Euclidean framework.” Eur. J. Mech. B Fluids 72 (Nov): 410–421. https://doi.org/10.1016/j.euromechflu.2018.07.008.
Chai, T., and R. R. Draxler. 2014. “Root mean square error (RMSE) or mean absolute error (MAE)?—Arguments against avoiding RMSE in the literature.” Geosci. Model Dev. 7 (3): 1247–1250. https://doi.org/10.5194/gmd-7-1247-2014.
Charbeneau, R. J. 2000. Groundwater hydraulics and pollutant transport. Englewood Cliffs, NJ: Prentice Hall.
Chaudhary, M., C. K. Thakur, and M. K. Singh. 2020. “Analysis of 1-D pollutant transport in semi-infinite groundwater reservoir.” Environ. Earth Sci. 79 (1): 1–23. https://doi.org/10.1007/s12665-019-8748-4.
Chen, J. S., C. S. Chen, and C. Y. Chen. 2007. “Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion.” Hydrol. Processes 21 (18): 2526–2536. https://doi.org/10.1002/hyp.6496.
Chen, J. S., and C. W. Liu. 2011. “Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition.” Hydrol. Earth Syst. Sci. 15 (8): 2471. https://doi.org/10.5194/hess-15-2471-2011.
Clement, T. P. 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.
Cotta, R. M., M. J. Ungs, and M. D. Mikhailov. 2003. “Contaminant transport in finite fractured porous medium: Integral transforms and lumped-differential formulations.” Ann. Nucl. Energy 30 (3): 261–285. https://doi.org/10.1016/S0306-4549(02)00060-9.
Das, P., and M. K. Singh. 2019. “One-dimensional solute transport in porous formations with time-varying dispersion.” J. Porous Media 22 (10): 1–19. https://doi.org/10.1615/JPorMedia.2019025964.
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.
Ebach, E. H., and R. White. 1958. “The mixing of fluids flowing through packed solids.” Am. Inst. Chem. Eng. 4 (2): 161–169. https://doi.org/10.1002/aic.690040209.
Flury, M., Q. J. Wu, L. Wu, and L. Xu. 1998. “Analytical solution for solute transport with depth-dependent transformation or sorption coefficients.” Water Resour. Res. 34 (11): 2931–2937. https://doi.org/10.1029/98WR02299.
Freeze, R. A., and J. A. Cherry. 1979. Groundwater. Englewood Cliffs, NJ: Prentice-Hall.
Ganjiani, M. 2010. “Solution of non-linear fractional differential equations using homotopy analysis method.” Appl. Math. Modell. 34 (6): 1634–1641. https://doi.org/10.1016/j.apm.2009.09.011.
Gao, G., B. Fu, H. Zhan, and Y. Ma. 2013. “Contaminant transport in soil with depth-dependent reaction coefficients and time-dependent boundary conditions.” Water Res. 47 (7): 2507–2522. https://doi.org/10.1016/j.watres.2013.02.021.
Gelhar, L. W., C. Welty, and K. R. Rehfeldt. 1992. “A critical review of data on field-scale dispersion in aquifers.” Water Resour. Res. 28 (7): 1955–1974. https://doi.org/10.1029/92WR00607.
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 (Oct): 935–940. https://doi.org/10.1016/j.jhydrol.2016.08.002.
Guerrero, J. P., L. C. G. Pimentel, T. H. Skaggs, and M. T. Van Genuchten. 2009. “Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique.” Int. J. Heat Mass Trans. 52 (13–14): 3297–3304. https://doi.org/10.1016/j.ijheatmasstransfer.2009.02.002.
Guerrero, J. P., E. M. Pontedeiro, M. T. van Genuchten, and T. H. Skaggs. 2013. “Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions.” Chem. Eng. J. 221 (Apr): 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.
Hoffman, J. D. 1992. Numerical methods for engineering and scientists. New York: Marcel Dekker, Inc.
Kangle, H., M. T. van Genuchten, and Z. Renduo. 1996. “Exact solutions for one-dimensional transport with asymptotic scale-dependent dispersion.” Appl. Math. Modell. 20 (4): 298–308. https://doi.org/10.1016/0307-904X(95)00123-2.
Kim, S., and M. L. Kavvas. 2006. “Generalized Fick’s law and fractional ADE for pollution transport in a river: Detailed derivation.” J. Hydrol. Eng. 11 (1): 80–83. https://doi.org/10.1061/(ASCE)1084-0699(2006)11:1(80).
Kim, S., M. L. Kavvas, and A. Ercan. 2015. “Fractional ensemble average governing equations of transport by time-space nonstationary stochastic fractional advective velocity and fractional dispersion. II: Numerical investigation.” J. Hydrol. Eng. 20 (2): 04014040. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000994.
Kookana, R. S., L. A. G. Aylmore, and R. G. Gerritse. 1992a. “Time-dependent sorption of pesticides during transport in soils.” Soil Sci. 154 (3): 214–225. https://doi.org/10.1097/00010694-199209000-00005.
Kookana, R. S., R. G. Gerritse, and L. A. G. Aylmore. 1992b. “A method for studying non equilibrium sorption during transport of pesticides in soil.” Soil Sci. 154 (5): 344–349. https://doi.org/10.1097/00010694-199211000-00002.
Kumar, A., and R. R. Yadav. 2015. “One-dimensional solute transport for uniform and varying pulse type input point source through heterogeneous medium.” Environ. Technol. 36 (4): 487–495. https://doi.org/10.1080/09593330.2014.952675.
Latinopoulos, P., D. Tolikas, and Y. Mylopoulos. 1988. “Analytical solutions for two-dimensional chemical transport in aquifers.” J. Hydrol. 98 (1–2): 11–19. https://doi.org/10.1016/0022-1694(88)90202-8.
Liu, C., J. E. Szecsody, J. M. Zachara, and W. P. Ball. 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.
Logan, J. D. 1996. “Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions.” J. Hydrol. 184 (3–4): 261–276. https://doi.org/10.1016/0022-1694(95)02976-1.
Naveen, B. P., J. Sumalatha, and R. K. Malik. 2018. “A study on contamination of ground and surface water bodies by leachate leakage from a landfill in Bangalore, India.” Int. J. Geo-Eng. 9 (1): 27. https://doi.org/10.1186/s40703-018-0095-x.
Neuman, S. P. 1990. “Universal scaling of hydraulic conductivities and dispersivities in geologic media.” Water Resour. Res. 26 (8): 1749–1758. https://doi.org/10.1029/WR026i008p01749.
Pang, L., and B. Hunt. 2001. “Solutions and verification of a scale-dependent dispersion model.” J. Contamin. Hydrol. 53 (1–2): 21–39. https://doi.org/10.1016/S0169-7722(01)00134-6.
Paul, E. A. 2007. “Soil microbiology, ecology, and biochemistry in perspective.” In Soil microbiology, ecology and biochemistry, 3–24. Cambridge, MA: Academic Press.
Pickens, J. F., and G. E. Grisak. 1981. “Scale-dependent dispersion in a stratified granular aquifer.” Water Resour. Res. 17 (4): 1191–1211. https://doi.org/10.1029/WR017i004p01191.
Rajput, S., and M. K. Singh. 2021. “Off-diagonal dispersion effect with pollutant migration in groundwater system.” J. Eng. Mech. 147 (12): 04021114. https://doi.org/10.1061/(ASCE)EM.1943-7889.0002009.
Ramos, C., L. Molbak, and S. Molin. 2000. “Bacterial activity in the rhizosphere analyzed at the single-cell level by monitoring ribosome contents and synthesis rates.” Appl. Environ. Microbiol. 66 (2): 801–809. https://doi.org/10.1128/AEM.66.2.801-809.2000.
Rubio, A. D., A. Zalts, and C. D. El Hasi. 2008. “Numerical solution of the advection-reaction-diffusion equation at different scales.” Environ. Modell. Software 23 (1): 90–95. https://doi.org/10.1016/j.envsoft.2007.05.009.
Rumer, R. R. 1962. “Longitudinal dispersion in steady and unsteady flow.” J. Hydraul. Div. 88 (4): 147–172. https://doi.org/10.1061/JYCEAJ.0000740.
Schumer, R., D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft. 2001. “Eulerian derivation of the fractional advection–dispersion equation.” J. Contam. Hydrol. 48 (1–2): 69–88. https://doi.org/10.1016/S0169-7722(00)00170-4.
Sciortino, A., F. J. Leij, M. C. Caputo, and N. Toride. 2015. “Modeling transport in dual-permeability media with unequal dispersivity and velocity.” J. Hydrol. Eng. 20 (7): 04014075. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001073.
Serrano, S. E. 2001. “Solute transport under non-linear sorption and decay.” Water Res. 35 (6): 1525–1533. https://doi.org/10.1016/S0043-1354(00)00390-0.
Si, Y., K. Takagi, A. Iwasaki, and D. Zhou. 2009. “Adsorption, desorption and dissipation of metolachlor in surface and subsurface soils.” Pest Manage. Sci. Formerly Pesticide Sci. 65 (9): 956–962. https://doi.org/10.1002/ps.1779.
Singh, M. K., A. Chatterjee, and V. P. Singh. 2017. “Solution of one-dimensional time fractional advection dispersion equation by homotopy analysis method.” J. Eng. Mech. 143 (9): 04017103. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001318.
Singh, M. K., and P. Das. 2015. “Scale dependent solute dispersion with linear isotherm in heterogeneous medium.” J. Hydrol. 520 (Jan): 289–299. https://doi.org/10.1016/j.jhydrol.2014.11.061.
Singh, M. K., and P. Kumari. 2014. “Contaminant concentration prediction along unsteady groundwater flow.” In Modelling and simulation of diffusive processes. Simulation foundations, methods and applications. Berlin: Springer.
Singh, P., S. K. Yadav and N. Kumar. 2012. “One-dimensional pollutant’s advective-diffusive transport from a varying pulse-type point source through a medium of linear heterogeneity.” J. Hydrol. Eng. 17 (9): 1047–1052. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000553.
Su, N., G. C. Sander, F. Liu, V. Anh, and D. A. Barry. 2005. “Similarity solutions for solute transport in fractal porous media using a time-and scale-dependent dispersivity.” Appl. Math. Modell. 29 (9): 852–870. https://doi.org/10.1016/j.apm.2004.11.006.
Todd, D. K., and L. W. Mays. 1980. Groundwater hydrology. New York: Wiley.
Vanderborght, J., and H. Vereecken. 2007. “One-dimensional modeling of transport in soils with depth-dependent dispersion, sorption and decay.” Vadose Zone J. 6 (1): 140–148. https://doi.org/10.2136/vzj2006.0103.
van Genuchten, M., F. J. Leij, T. H. Skaggs, N. Toride, S. A. Bradford, E. M. Pontedeiro. 2013. “Exact analytical solutions for contaminant transport in rivers 1. The equilibrium advection–dispersion equation.” J. Hydrol. Hydromech. 61 (2): 146–160. https://doi.org/10.2478/johh-2013-0020.
van Genuchten, M. T. 1981. “Analytical solutions for chemical transport with simultaneous adsorption, zero-order production and first-order decay.” J. Hydrol. 49 (3–4): 213–233. https://doi.org/10.1016/0022-1694(81)90214-6.
Yadav, S. K., A. Kumar, D. K. Jaiswal, and N. Kumar. 2011. “One-dimensional unsteady solute transport along unsteady flow through inhomogeneous medium.” J. Earth Syst. Sci. 120 (2): 205–213. https://doi.org/10.1007/s12040-011-0048-7.
Yang, W. Y., W. T. S. CaoChung, T. S. Chung, and J. Morris. 2005. Applied numerical methods using MATLAB. Hoboken, NJ: Willey.
Yates, S. R. 1990. “An analytical solution for one-dimensional transport in heterogeneous porous media.” Water Resour. Res. 26 (10): 2331–2338. https://doi.org/10.1029/WR026i010p02331.
You, K., and H. Zhan. 2013. “New solutions for solute transport in a finite column with distance-dependent dispersivities and time-dependent solute sources.” J. Hydrol. 487 (Apr): 87–97. https://doi.org/10.1016/j.jhydrol.2013.02.027.
Zhan, H., Z. Wen, G. Huang, and D. Sun. 2009. “Analytical solution of two-dimensional solute transport in an aquifer-aquitard system.” J. Contamin. Hydrol. 107 (3–4): 162–174. https://doi.org/10.1016/j.jconhyd.2009.04.010.
Zheng, C., and G. D. Bennett. 2002. Applied contaminant transport modeling. New York: Wiley.
Information & Authors
Information
Published In
Journal of Environmental Engineering
Volume 148 • Issue 8 • August 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Nov 11, 2021
Accepted: Mar 27, 2022
Published online: Jun 3, 2022
Published in print: Aug 1, 2022
Discussion open until: Nov 3, 2022
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.