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Advances in analytical solutions for time-dependent solute transport model

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Abstract

This study adopts generalized dispersion theory in one-dimensional advection–dispersion equation (ADE), where time-dependent dispersion and velocity are considered. The generalized dispersion theory allows mechanical dispersion to be directly proportional to seepage velocity with power n, where n is any real number. Homotopy analysis method (HAM) that uses a simple algorithm is adopted to handle the non-linearity that occurred in the ADE under the generalized dispersion. A point source is introduced to the entry boundary and a line source is introduced to the entire model domain. Three time-dependent point sources in the form of (i) exponentially decreasing function, (ii) linear function and (iii) sinusoidal function, at the entry boundary are considered. Two-line sources are considered in the form of (i) linear space-dependent function and (ii) nonlinear space-time-dependent function. Using the HAM, semi-analytical solutions for any power n are derived and semi-analytical solutions for n = 1 and n = 1.5 are discussed in particular. Comparison with the analytical solution is discussed and found good agreement for 6th order of solution obtained by HAM.

Research Highlights

  1. 1.

    Generalized dispersion theory in 1-D ADE

  2. 2.

    Generalized semi-analytical solution using HAM

  3. 3.

    Compared with analytical solution

  4. 4.

    Good agreement for 6th order of semi-analytical solution

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References

  • Aral M M and Liao B 1996 Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients; J. Hydrol. Eng. 1(1) 20–32.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Batu V 2006 Applied Flow and Solute Transport Modelling in Aquifers: Fundamental Principles and Analytical and Numerical Methods; CRC Press, Boca Raton, FL.

  • Bharati V K, Singh V P, Sanskrityayn A and Kumar N 2017 Analytical solution of advection–dispersion equation with spatially dependent dispersivity; J. Eng. Mech. 143(11) 04017126.

    Google Scholar 

  • Bharati V K, Singh V P, Sanskrityayn A and Kumar N 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 72 410–421.

  • Chatterjee A, Singh M K and Singh V P 2020 Groundwater contamination in mega cities with finite sources; J. Earth Syst. Sci. 129(1) 1.

    Article  Google Scholar 

  • Das P, Begam S and Singh M K 2017 Mathematical modeling of groundwater contamination with varying velocity field; J. Hydrol. Hydromech. 65(2) 192–204.

    Article  Google Scholar 

  • Deng B, Li J, Zhang B and Li N 2014 Integral transform solution for solute transport in multi-layered porous media with the implicit treatment of the interface conditions and arbitrary boundary conditions; J. Hydrol. 517 566–573.

    Article  Google Scholar 

  • Ebach E H and White R 1958 Mixing of fluid flowing beds of packed solids; J. Am. Inst. Chem. Eng. 4(2) 161–169.

    Article  Google Scholar 

  • Freeze R A and Cherry J A 1979 Groundwater; Prentice-Hall Inc., Englewood Cliffs, NJ.

  • Gao G, Fu B, Zhan H and Ma Y 2013 Contaminant transport in soil with depth-dependent reaction coefficients and time-dependent boundary conditions; Water Res. 47(7) 2507–2522.

    Article  Google Scholar 

  • Gelhar L W, Welty C and Rehfeldt K R 1992 A critical review of data on field-scale dispersion in aquifers; Water Resour. Res. 28(7) 1955–1974.

    Article  Google Scholar 

  • Ghosh N C and Sharma K D 2006 Groundwater Modelling and Management; Capital Publishing Company, New Delhi.

  • Guerrero J P and Skaggs T H 2010 Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients; J. Hydrol. 390(1–2) 57–65.

    Article  Google Scholar 

  • Guerrero J P, Pimentel L C G and Skaggs T H 2013 Analytical solution for the advection–dispersion transport equation in layered media; Int. J. Heat Mass Transf. 56(1–2) 274–282.

    Article  Google Scholar 

  • Huang K, van Genuchten M T and Zhang R 1996 Exact solutions for one-dimensional transport with asymptotic scale-dependent dispersion; Appl. Math. Model. 20(4) 298–308.

    Article  Google Scholar 

  • Jain S K, Agarwal P K and Singh V P 2007 Hydrology and Water Resources of India; Springer, Netherlands, 87p.

  • Jaiswal D K, Kumar A, Kumar N and Yadav R 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.

    Article  Google Scholar 

  • Jia X, Zeng F and Gu Y 2013 Semi-analytical solutions to one-dimensional advection–diffusion equations with variable diffusion coefficient and variable flow velocity; Appl. Math. Comput. 221 268–281.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Kumar A, Jaiswal D K and Kumar N 2012 One-dimensional solute dispersion along unsteady flow through a heterogeneous medium, dispersion being proportional to the square of velocity; Hydrol. Sci. J. 57(6) 1223–1230.

    Article  Google Scholar 

  • Kumar N 1983 Unsteady flow against dispersion in finite porous media; J. Hydrol. 63(3–4) 345–358.

    Article  Google Scholar 

  • Kumar N and Kumar M 1998 Solute dispersion along unsteady groundwater flow in a semi-infinite aquifer; Hydrol. Earth Syst. Sci. Discuss. 2(1) 93–100.

    Article  Google Scholar 

  • Liao S 2005 A new branch of solutions of boundary-layer flows over an impermeable stretched plate; Int. J. Heat Mass Transf. 48 2529–2539.

    Article  Google Scholar 

  • Liao S 2012 Homotopy Analysis Method in Nonlinear Differential Equations; Higher Education Press, Beijing, pp. 153–165.

  • Liao S J 1992 The proposed homotopy analysis technique for the solution of nonlinear problems; PhD thesis, Shanghai Jiao Tong University.

  • Liao S J 1995 An approximate solution technique not depending on small parameters: A special example; Int. J. Non-Lin. Mech. 30 371–380.

    Article  Google Scholar 

  • Liao S, Su J and Chwang A T 2006 Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body; Int. J. Heat Mass Transf. 49(15–16) 2437–2445.

    Article  Google Scholar 

  • Liu G and Si B C 2008 Analytical modeling of one-dimensional diffusion in layered systems with position-dependent diffusion coefficients; Adv. Water Resour. 31(2) 251–268.

    Article  Google Scholar 

  • Logan J D 1996 Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions; J. Hydrol. 184(3–4) 261–276.

    Article  Google Scholar 

  • Runkel R L 1996 Solution of the advection–dispersion equation: continuous load of finite duration; J. Environ. Eng. 122(9) 830–832.

    Article  Google Scholar 

  • Sander G C and Braddock R D 2005 Analytical solutions to the transient, unsaturated transport of water and contaminants through horizontal porous media; Adv. Water Resour. 28(10) 1102–1111.

    Article  Google Scholar 

  • Sanskrityayn A, Suk H and 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.

    Article  Google Scholar 

  • Scheidegger A E 1957 The Physics of Flow Through Porous Media; University of Toronto Press, Toronto.

  • Scheidegger A E 1961 General theory of dispersion in porous media; J. Geophys. Res. 66(10) 3273–3278.

    Article  Google Scholar 

  • Shi X, Lei T, Yan Y and Zhang F 2016 Determination and impact factor analysis of hydrodynamic dispersion coefficient within a gravel layer using an electrolyte tracer method; Int. Soil Water Conserv. Res. 4(2) 87–92.

    Article  Google Scholar 

  • Singh M K and Chatterjee A 2017 Solution of one-dimensional space-and time-fractional advection–dispersion equation by homotopy perturbation method; Acta Geophys. 65(2) 353–361.

    Article  Google Scholar 

  • Singh M K and Das P 2015 Scale-dependent solute dispersion with linear isotherm in heterogeneous medium; J. Hydrol. 520 289–299.

    Article  Google Scholar 

  • Singh M K and Das P 2018 Response to ‘comment on the paper scale-dependent solute dispersion with linear isotherm in heterogeneous medium (Journal of Hydrology 520 (2015) 289–299); J. Hydrol., https://doi.org/10.1016/j.jhydrol.2018.06.071.

    Article  Google Scholar 

  • Singh M K and Kumari P 2014 Contaminant Concentration Prediction Along Unsteady Groundwater Flow. Modelling and Simulation of Diffusive Processes, Series: Simulation Foundations, Methods and Applications; Springer, XII 257–276.

  • Singh M K, Chatterjee A and Singh V P 2017 Solution of one-dimensional time fractional advection dispersion equation by homotopy analysis method; J. Eng. Mech. 143(9) 04017103.

    Google Scholar 

  • Singh P and Singh V P 2001 Snow and Glacier Hydrology; Kluwer Academic Publishers, Amsterdam, The Netherlands, 78p.

  • 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. 135(9) 1015–1021.

    Google Scholar 

  • Thangarajan M (ed.) 2006 Groundwater: Resource Evaluation, Augmentation, Contamination, Restoration, Modelling and Management; Capital Publishing Company, New Delhi, India, 362p.

  • Todd D K and Mays L W 2007 Groundwater Hydrology; 3rd edn, John Wiley and Sons, Third Reprint. Inc. India, 535p.

  • 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.

    Article  Google Scholar 

  • Wadi A S, Dimian M F and Ibrahim F N 2014 Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration; J. Earth Syst. Sci. 123(6) 1317–1324.

    Article  Google Scholar 

  • 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. 412 193–199.

    Article  Google Scholar 

  • You K and 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.

    Article  Google Scholar 

  • Yu C, Deng A, Ma J, Cai X and Wen C 2018a Semi-analytical solutions for two-dimensional convection–diffusion–reactive equations based on homotopy analysis method; Environ. Sci. Pollut. Res. 25(34) 34,720–34,729.

    Article  Google Scholar 

  • Yu C, Wang H, Fang D, Ma J, Cai X and Yu X 2018b Semi-analytical solution to one-dimensional advective–dispersive–reactive transport equation using homotopy analysis method; J. Hydrol. 565 422–428.

    Article  Google Scholar 

  • Yu C, Zhou M, Ma J, Cai X and Fang D 2019 Application of the homotopy analysis method to multispecies reactive transport equations with general initial conditions; Hydrogeol. J. 27(5) 1779–1790.

    Article  Google Scholar 

  • Zamani K and Bombardelli F A 2014 Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers; Environ. Fluid Mech. 14(4) 711–742.

    Article  Google Scholar 

  • Zoppou C and Knight J H 1997 Analytical solutions for advection and advection–diffusion equations with spatially variable coefficients; J. Hydraul. Eng. 123(2) 144–148.

    Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the Indian Institute of Technology (Indian School of Mines), Dhanbad, India, for providing financial support for PhD studies under the UGC-JRF scheme. This work is partially supported by the DST (SERB) Project EMR/2016/001628.

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Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826 004, India

    Rohit Kumar

  2. Department of Mathematics, The Neotia University, Diamond Harbour, Kolkata, West Bengal, 743 368, India

    Ayan Chatterjee

  3. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826 004, India

    Mritunjay Kumar Singh

  4. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, 70803, USA

    Frank T-C Tsai

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  1. Rohit Kumar
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  2. Ayan Chatterjee
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Contributions

Rohit Kumar: Conceptualization, methodology, writing – original draft preparation, investigation, validation. Ayan Chatterjee: Formal analysis. Mritunjay Kumar Singh: Supervision, writing – reviewing and editing. Frank T-C Tsai: Reviewing and editing.

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Correspondence to Mritunjay Kumar Singh.

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Communicated by Saibal Gupta

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Kumar, R., Chatterjee, A., Singh, M.K. et al. Advances in analytical solutions for time-dependent solute transport model. J Earth Syst Sci 131, 131 (2022). https://doi.org/10.1007/s12040-022-01858-5

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