Solute Transport Phenomena in a Heterogeneous Semi-infinite Porous Media: An Analytical Solution

  • R. R. YadavEmail author
  • Joy Roy
Original Paper


The aim of this study is to develop an analytical solution for one-dimensional advection dispersion equation in semi-infinite heterogeneous porous medium. The pollutants are considered to be of non-reactive and emitted from a time-dependent two-stage point source. Dispersion coefficient is considered proportional to the square of the groundwater velocity while groundwater velocity is proportional to spatially dependent linear function. Initially medium is not solute free. The solute presence is linear function of space. First-order decay and zero-order production are also considered. Flux type boundary condition is assumed at the other end of the domain. A new transformation is used to reduce variable coefficient into a constant coefficient. Laplace Transformation Technique is employed to get the solution of the proposed problem. The obtained results are compared with published result to check its validity and illustrated graphically for parameters and value caused on concentration behaviour.


Advection Dispersion Aquifer Porous medium Laplace transformation 


  1. 1.
    Banks, R.B., Ali, J.: Dispersion and adsorption in porous media flow. J. Hydraul. Div. 90, 13–31 (1964)Google Scholar
  2. 2.
    Bear, J.: Dynamics of Fluid in Porous Media. Elsevier Publication Co, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Chen, JSh, Ni, ChF, Liang, ChP, Chiang, ChCh.: Analytical power series solution for contaminant transport with hyperbolic asymptotic distance-dependent dispersivity. J. Hydrol. 362(1–2), 142–149 (2008)CrossRefGoogle Scholar
  4. 4.
    Chen, J.S., Li, L.Y., Lai, K.H., Liang, C.P.: Analytical model for advective-dispersive transport involving flexible boundary inputs, initial distributions and zero-order productions. J. Hydrol. 554, 187–199 (2017)CrossRefGoogle Scholar
  5. 5.
    Djordjevich, A., Savoic, S.: Aco Janicijevic (2017) “Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media”. J. Hydrol. Hydromech. 65(4), 426–432 (2017)CrossRefGoogle Scholar
  6. 6.
    Freeze, R. A., Cherry, J. A.: Groundwater. Prentice-Hall, Englewood Cliffs, NJ (1979)Google Scholar
  7. 7.
    Güven, O., Molz, F.J., Melville, J.G.: An analysis of dispersion in a stratified aquifer. Water Resour. Res. 20(10), 1337–1354 (1984)CrossRefGoogle Scholar
  8. 8.
    Huang, K., Van Genuchten, MTh, Zhang, R.: Exact solutions for one dimensional transport with asymptotic scale-dependent dispersion. Appl. Math. Model. 20, 298–308 (1996)CrossRefGoogle Scholar
  9. 9.
    Jaiswal, D.K., Kumar, A., Kumar, N., Yadav, R.R.: 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, 254e263 (2009)CrossRefGoogle Scholar
  10. 10.
    Jaiswal, D.K., Kumar, A.: Analytical solutions of advection-dispersion equation for varying pulse type input point source in one-dimension. Int. J. Eng. Sci. Technol. 3(1), 22–29 (2011)CrossRefGoogle Scholar
  11. 11.
    Jaiswal, D.K., Kumar, A.: Analytical solutions of time and spatially dependent one-dimensional advection-diffusion equation. Elixir Poll. 32, 2078–2083 (2011)Google Scholar
  12. 12.
    Jaiswal, D.K., Kumar, A., Kumar, N., Singh, M.K.: Solute transport along temporally and spatially dependent flows through horizontal semi-infinte media: dispersion being proportional tosquare of velocity. J. Hydrol. Eng. (ASCE) 16(3), 228–238 (2011)CrossRefGoogle Scholar
  13. 13.
    Jaiswal, D.K., Yadav, R.R.: Contaminant Diffusion along uniform flow velocity with pulse type input sources in finite porous medium. Int. J. Appl. Math. Electron. Comput. 2(4), 19–25 (2014)CrossRefGoogle Scholar
  14. 14.
    Kumar, A., Jaiswal, D.K., Kumar, N.: Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. J. Hydrol. 380, 330–337 (2010)CrossRefGoogle Scholar
  15. 15.
    Kumar, A., Jaiswal, D.K., Kumar, N.: 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 (2012)CrossRefGoogle Scholar
  16. 16.
    Lai, K.H., Liu, C.W., Liang, C.P., Chen, J.S., Sie, B.R.: A novel method for analytically solving a radial advection-dispersion equation. J. Hydrol. 542, 532–540 (2016)CrossRefGoogle Scholar
  17. 17.
    Liang, C.P., Hsu, S.Y., Chen, J.S.: An analytical model for solute transport in an infiltration tracer test in soil with a shallow groundwater table. J. Hydrol. 540, 129–141 (2016)CrossRefGoogle Scholar
  18. 18.
    Marino, M.A.: Flow against dispersion in non adsorbing porous media. J. Hydrol. 37, 149–158 (1978)CrossRefGoogle Scholar
  19. 19.
    Massabo, M., Cianci, R., Paladino, O.: Some analytical solutions for two dimensional convection-dispersion equation in cylindrical geometry. Environ. Model Softw. 21(5), 681–688 (2006)CrossRefGoogle Scholar
  20. 20.
    Matheron, G., de Marsily, G.: Is transport in porous media always diffusive, a counter example. Water Resour. Res. 16, 901–917 (1980)CrossRefGoogle Scholar
  21. 21.
    Ogata, A.: Theory of dispersion in granular media, U.S. Geol. Sur. Prof. Paper 4111I (1970), 34Google Scholar
  22. 22.
    Pickens, J.F., Grisak, G.E.: Modeling of scale-dependent dispersion in hydro-geologic systems. Water Resour. Res. 17, 1701–1711 (1981)CrossRefGoogle Scholar
  23. 23.
    Scheidegger, A.E.: The Physics of Flow Through Porous Media. University of Toronto Press, Toronto (1957)zbMATHGoogle Scholar
  24. 24.
    Singh, M.K., Das, P., Singh, V.P.: Solute transport in a semi-infinite geological formation with variable porosity. J. Eng. Mech. ASCE 141(11), 1–13 (2015). CrossRefGoogle Scholar
  25. 25.
    Sposito, G.W., Jury, W.A., Gupta, V.K.: Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and field soils. Water Resour. Res. 22, 77–78 (1986)CrossRefGoogle Scholar
  26. 26.
    Todd, D.K.: Groundwater Hydrology. Wiley, New York (1980)Google Scholar
  27. 27.
    Van Genuchten, M.T., Leij, F.J., Skaggs, T.H., Toride, N., Bradford, S.A., Pontedeiro, E.M.: Exact analytical solutions for contaminant transport in rivers: 1. The equilibrium advection–dispersion equation. J. Hydrol. Hydromech. 61(2), 146–160 (2013)CrossRefGoogle Scholar
  28. 28.
    Van Genuchten, M.T., Leij, F.J., Skaggs, T.H., Toride, N., Bradford, S.A., Pontedeiro, E.M.: Exact analytical solutions for contaminant transport in rivers: 2, Transient storage and decay chain solutions. J. Hydrol. Hydromech. 61(3), 250–259 (2013)CrossRefGoogle Scholar
  29. 29.
    Yadav, R.R., Kumar, N.: One dimensional dispersion in unsteady flow in an adsorbing porous media: an analytical solution. Hydrol. Process. 4, 189–196 (1990)CrossRefGoogle Scholar
  30. 30.
    Yim, C.S., Mohsen, M.F.N.: Simulation of tidal effects on contaminant transport in porous media. Ground Water 30(1), 78–86 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Department of Mathematics and AstronomyUniversity of LucknowLucknowIndia

Personalised recommendations