Environmental Earth Sciences

, Volume 65, Issue 3, pp 849–859

Three-dimensional temporally dependent dispersion through porous media: analytical solution

Original Article

Abstract

A three-dimensional model for non-reactive solute transport in physically homogeneous subsurface porous media is presented. The model involves solution of the advection-dispersion equation, which additionally considered temporally dependent dispersion. The model also account for a uniform flow field, first-order decay which is inversely proportional to the dispersion coefficient and retardation factor. Porous media with semi-infinite domain is considered. Initially, the space domain is not solute free. Analytical solutions are obtained for uniform and varying pulse-type input source conditions. The governing solute transport equation is solved analytically by employing Laplace transformation technique (LTT). The solutions are illustrated and the behavior of solute transport may be observed for different values of retardation factor, for which simpler models that account for solute adsorption through a retardation factor may yield a misleading assessment of solute transport in ‘‘hydrologically sensitive’’ subsurface environments.

Keywords

Advection Dispersion Retardation Groundwater 

References

  1. Al-Niami ANS, Rushton KR (1977) Analysis of flow against dispersion in porous media. J Hydrol 33:87–97CrossRefGoogle Scholar
  2. Aral MM, Liao B (1996) Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients. J Hydrol Eng 1(1):20–32CrossRefGoogle Scholar
  3. Banks, Robert B, Jerasate S (1962) Dispersion in unsteady porous media flow. J Hydraul Div HY3:1–21Google Scholar
  4. Batu V (1987) Introduction of the stream function concept to the analysis of hydrodynamic dispersion in porous media. Water Resour Res 23(7):1175–1184CrossRefGoogle Scholar
  5. Bruch JC (1970) Two dimensional dispersion experiments in a porous medium. Water Resour Res 6:791–800CrossRefGoogle Scholar
  6. Chen JS, Liu CW, Liao CM (2003) Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field. Adv Water Resour 26:1113–1124CrossRefGoogle Scholar
  7. Chrysikopoulos CV, Voudrias EA, Fyrillas MM (1994) Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media. Transp Porous Med 16:125–145CrossRefGoogle Scholar
  8. Cirpka OA, Valocchi AJ (2009) Reply to comments on “Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state” by H. Shao et al. Adv Water Resour 32(2):298–301CrossRefGoogle Scholar
  9. Costa CP, Vilhena MT, Moreira DM, Tirabassi T (2006) Semi-analytical solution of the steady three-dimensional advection-diffusion equation in the planetary boundary layer. Atmos Environ 40:5659–5669CrossRefGoogle Scholar
  10. Crank J (1975) The mathematics of diffusion, 2nd edn. Oxford Univ Press, LondonGoogle Scholar
  11. Diersch HJ, Prochnow D, Thiele M (1984) Finite-element analysis of dispersion––affected saltwater upconing below a pumping well. Appl Math Model 8(5):305–312CrossRefGoogle Scholar
  12. Gershon ND, Nir A (1969) Effects of boundary conditions of models on tracer distribution in flow through porous mediums. Water Resour Res 5(4):830–839CrossRefGoogle Scholar
  13. Harleman DRF, Rumer RR (1963) Longitudinal and lateral dispersion in an isotropic porous medium. J Fluid Mech 16(3):385–394CrossRefGoogle Scholar
  14. Jaiswal DK, Kumar A, Kumar N, Yadava RR (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 2:254–263CrossRefGoogle Scholar
  15. Kim KY, Kim T, Kim Y, Woo NC (2007) A semi-analytical solution for groundwater responses to stream-stage variations and tidal fluctuations in a coastal aquifer. Hydrol Process 21:665–674CrossRefGoogle Scholar
  16. Kumar A, Jaiswal DK, Kumar N (2010) Analytical solutions to one-dimensional advection-diffusion with variable coefficients in semi-infinite media. J Hydrol 380(3–4):330–337CrossRefGoogle Scholar
  17. Lapidus L, Amundson NR (1952) Mathematics of adsorption in beds, VI. The effects of longitudinal diffusion in ion-exchange and chromatographic columns. J Phys Chem 56:984–988CrossRefGoogle Scholar
  18. Liao B, Aral MM (2000) Semi-analytical solution of two-dimensional sharp interface LNAPL transport models. J Contam Hydrol 44:203–221CrossRefGoogle Scholar
  19. Lin SH (1977) Non-linear adsorption in porous media with variable porosity. J Hydrol 35:235–243CrossRefGoogle Scholar
  20. Lin JS, Hildemann LM (1996) Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities. Atmos Environ 30(2):239–254CrossRefGoogle Scholar
  21. Liu SH, Liedl R, Grathwohl P (2010) Simple analytical solutions for oxygen transfer into anaerobic groundwater, Water Resources Research Volume: 46 Article Number: W10542 Published: OCT 29 2010Google Scholar
  22. Moreira DM, Vilhena MT, Buske DE, Tirabassi T (2006) The GILTT solution of the advection-diffusion equation for an inhomogeneous and nonstationary PBL. Atmos Environ 40:3186–3194CrossRefGoogle Scholar
  23. Ogata A, Banks RB (1961) A solution of differential equation of longitudinal dispersion in porous media. US Geol Surv Prof Pap 411:A1–A7Google Scholar
  24. Shao H, Centler F, Biase CD, Thullner M, Kolditz O (2009) Comments on “Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady-state” by OA Cirpka and AJ Valocchi. Adv Water Resour 32(2):293–297CrossRefGoogle Scholar
  25. Sirin H (2006) Ground water contaminant transport by nondivergence-free, unsteady and nonstationary velocity fields. J Hydrol 330:564–572CrossRefGoogle Scholar
  26. Todd DK (1980) Groundwater hydrology, 2nd edn. Wiley, New YorkGoogle Scholar
  27. Tracy FT (1995) 1-D, 2-D, and 3-D analytical solutions of unsaturated flow in groundwater. J Hydrol 170:199–214CrossRefGoogle Scholar
  28. van Genuchten M Th, Alves WJ (1982) Analytical solutions of one dimensional convective-dispersive solute transport equations, United State Dept. of Agriculture, Technical Bulletin No. 1661Google Scholar
  29. Zheng C, Bennett GD (2002) Applied contaminant transport modeling, 2nd edn. Wiley, New York, pp 56–57Google Scholar
  30. Zoppou C, Knight JH (1999) Analytical solution of a spatially variable coefficient advection-diffusion equation in up to three dimensions. Appl Math Model 23:667–685CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and AstronomyLucknow UniversityLucknowIndia

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