Mathematical Modelling of Transport of Pollutants of Longitudinal Dispersion in Unsaturated Porous Media

References

[1] Verruijt (1990): Modelling Groundwater Flow and Pollution. D Reidel Publishing Co., Tokyo.

[2] Al-Niami A N S and Rushton K R 1977 Analysis of flow against dispersion in porous media; J. Hydrol. 33 87–97.

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

[4] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical Solutions to One-Dimensional Advection-Diffusion Equation with Variable Coefficients in Finite Domain,” Journal of Earth System Science, Vol. 118, No. 5, 2009, pp. 539-549

[5] Banks R B and Ali J 1964 Dispersion and adsorption in porous media flow; J. Hydraul. Div. 90 13-31.

[6] Barry, D.A., Parlange, J.Y., Sander, G.C., and M. Sivaplan (1993): A class of exact solutions for Richards equation. J. of Hydrology. 142: 29-46.

[7] Basak, P., and V.V.N. Murthy (1981): Groundwater quality improvement through nonlinear diffusion. J. of Hydrology. 53: 151-159.

[8] Bear, J., and A. Verruijt (1990): Modelling Groundwater Flow and Pollution. D Reidel Publishing Co., Tokyo.

[9] Branes, C.J (1989): Solute and water movement in unsaturated soils. Water Resour. Res., 25(1): 38-42.

[10] Bruce, J.C.; and Street, R.L. (1966). Studies of free surface flow and twodimensional dispersion in porous media. Report No. 63, Civil Eng. Dept. Stanford Univivrsity.

[11] Celia, M.A., E.T. Bouloutas and R.L. Zarba (1990): A general mass conservative numerical solution of the unsaturated flow equation, Water Resour. Res. 26 (7): 1483-1496.

[12] Crank J 1975 the Mathematics of Diffusion (London: Oxford Univ. Press).

[13] Chen, J.S., Liu, C.W., Liao, C.M., 2003. Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field. Advances in Water Resources 26, 1113–1124.

[14] De Smedt, F. and Wierenga, P.J., 1979b. Mass transfer in porous media with immobile water. J. Hydrol., 41: 59967.

[15] Domenico, P.A. and G.A. Robbins 1984: A Dispersion Scale Effect in Model calibrations & Field Tracer Expts. J. of Hydrology 70: 123-132.

[16] Eungyer Park & Hongbin Zhan (2001): Analytical solutions of contaminant transport from finite one-, two- and three-dimensional sources in a finite-thickness aquifer. J. of Contaminant Hydrology. 53: 41-61

[17] 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.

[18] Guerrero, J. S. Pérez., Pontedeiro, E.M., Van Genuchten, M.Th., and Skaggs, T.H., “Analytical solutions of the one-, two- and three-dimensional advection-dispersion solute transport equation subject to time-dependent boundary conditions”, Chemical Engineering Journal, Vol. 221, pp. 487-491, 2013.

[19] Harleman, D.R.F., and R.R., Rumer, Jr (1963): Longitudinal and lateral dispersion in an isotropic porous medium. J. Fluid Mech., 16: 1-12.

[20] Hanks, R.J., Klute A., and E.Bresler (1969): A numerical methods for estimating infiltration, Redistribution drainage, and evaporation of water from soil. Water Resour. Res., 5(5): 1064-1069.

[21] Jaiswal, D.K., Kumar, A., Kumar, N., Yadav, R.R., 2009. Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media. Journal of Hydro environmental Research 2, 254–263.

[22] Kim Kue-Young, Kim T, Kim Y and Woo Nam-Chil 2007 A semi-analytical solution for ground water responses to stream–stage variations and tidal fluctuations in a coastal aquifers; Hydrol. Process. 21 665–674.

[23] Kumar, A. and Yadav, R. R., “One-dimensional solute transport for uniform and varying pulse type input point source through heterogeneous medium”, Environmental Technology, Vol.36, No.4, pp.487-495, 2015.

[24] Leij F J, Toride N and van Genutchen M Th 1993 Analytical solutions for non-equilibrium solute transport in three-dimensional porous media; J. Hydrol. 151 193-228.

[25] Liping Pang & Bruce Hunt (2001): Solutions and verification of a scale-dependent dispersion model. J. Contam. Hydrol., in press.

[26] Oberhettinger and Baddi(1973) tables of Laplace transforms Springer- verlag, Newyork

[27] Ogata, A. and R.B. Banks (1961): A solution of the differential equation of longitudinal dispersion in porous media, U.S. Geol. Surv. Prof. Pap. 411-A, A1-A9.

[28] Pang L, Gottz M and Close M 2003 Application of the method of temporal moment to interpret solute transport with sorption and degradation; J. Contam. Hydrol. 60 123–134.

[29] Paniconi, C., Aldama, A.A and Wood, E.F (1991): Numerical evaluation of iterative and noniterative methods for the solution of nonlinear Richards equation. Water Resour. Res., 27(6): 1147-1163.

[30] Pullan A.J. (1990): The quasilinear approximation for unsaturated porous media flow. Water Resour. Res. 26(6): 1219-1234.

[31] Sanskrityayn, A., Bharati, V. K. and Kumar, N., “Analytical solution of advection dispersion equation with spatiotemporal dependence of dispersion coefficient and velocity using green’s function method”, Journal of Groundwater Research, Vol. 5, No. 1, pp.24-31, 2016.

[32] Severino, G., Tartakovsky, D., Srinivasan, G., & Viswanathan, H. (2012). Lagrangian models of reactive transport in heterogeneous porous media with uncertain properties. In, Vol. 468. Proceedings of the royal society of london A: Mathematical, physical and engineering sciences (pp. 1154e1174).

[33] Sirin, H. (2006). Ground water contaminant transport by non-divergence free, unsteady and non-stationary velocity fields. Journal of Hydrology, 330(3-4), 564-572.

[34] Smedt, F.D. (2007). Analytical solution and analysis of solute transport in rivers affected by diffusive transfer in the hyporheic zone. Journal of Hydrology, 339(1-2), 29-38.

[35] Sudheendra S.R., 2010 A solution of the differential equation of longitudinal dispersion with variable coefficients in a finite domain, Int. J. of Applied Mathematics & Physics, Vol.2, No. 2, 193-204.

[36] Sudheendra S.R., 2011. A solution of the differential equation of dependent dispersion along uniform and non-uniform flow with variable coefficients in a finite domain, Int. J. of Mathematical Analysis, Vol.3, No. 2, 89-105.

[37] Sudheendra S.R. 2012. An analytical solution of one-dimensional advection-diffusion equation in a porous media in presence of radioactive decay, Global Journal of Pure and Applied Mathematics, Vol.8, No. 2, 113-124.

[38] Sudheendra S.R., Raji J, & Niranjan CM, 2014. Mathematical Solutions of transport of pollutants through unsaturated porous media with adsorption in a finite domain, Int. J. of Combined Research & Development, Vol. 2, No. 2, 32-40.

[39] van Genuchten M Th and Alves W J 1982 Analytical solutions of the one dimensional convective-dispersive solute transport equation; US Dept. Agriculture Tech. Bull. No. 1661 151p. van Kooten J J A 1996 A method to solve the advection dispersion equation with a kinetic adsorption isotherm; Adv. Water Res. 19 193–206.

[40] Wang, F.C. and V. Lakshminarayana (1968): Mathematical simulation of water movement through unsaturated non-homogeneous soils. Soil Science Soc. Am. Proc. 32: 329-334.

[41] Warrick, A.J., Biggar, J.W., and D.R. Nielsen (1971): Simultaneous solute and water transfer for an unsaturated soil. Water Resour. Res. 7(2); 1216-1225.

[42] Wortmanna, S.; Vilhena, M.T.; Moreirab, D.M.; and Buske, D. (2005). A new analytical approach to simulate the pollutant dispersion in the PBL. Atmospheric Environment, 39(12), 2171–2178.

[43] Yates S R 1990 An analytical solution for one-dimensional transport in heterogeneous porous media; Water Resour. Res. 26 2331–2338

[44] Yates S R 1992 An analytical solution for one-dimensional transport in porous media with an exponential dispersion function; Water Resour. Res. 28 2149–2154.

[45] Zyvolski, G., Bruch, J.C., and James M. Sloss (1976): Solution of equation for two-dimensional infiltration problems. Soil Science 122 (2): 65-70.

@article{1703775,
      author = {Niranjan C M, Sudheendra S R},
      title = {Mathematical Modelling of Transport of Pollutants of Longitudinal Dispersion in Unsaturated Porous Media},
      journal = {Iconic Research And Engineering Journals},
      year = {2022},
      volume = {6},
      number = {2},
      pages = {146-152},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1703775.pdf},
      abstract = {In order to understand the behavior of contaminant transport through different types of media, several researchers are carrying out experimental investigations through laboratory and field studies. Many of them are working on the analytical and numerical studies to simulate the movement of contaminants in soil and groundwater of the contaminant transport. A key to the management of groundwater is the ability to model the movement of fluids and contaminants in the subsurface environment. It is obvious that the contaminant source activities cannot be completely eliminated and perhaps our water bodies will continue to serve as receptors of vast quantities of waste. The solution is obtained for the given mathematical model in a finite length initially solute free domain. The input condition is considered continuous of uniform and of increasing nature both. The solution has been obtained using Laplace transform, moving coordinates and Duhamel?s theorem is used to get the solution in terms of complementary error function.},
      keywords = {Advection, dispersion, adsorption, Integral transforms, Fick?s law, Moving coordinates, Duhamel?s theorem},
      month = {August},
  }