magazinelogo

Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 ISSN Print: 2576-0645 CODEN: JAMCEZ
Frequency: quarterly Email: jamc@hillpublisher.com
Total View: 2065847 Downloads: 370160 Citations: 328 (From Dimensions)
ArticleOpen Access http://dx.doi.org/10.26855/jamc.2018.03.001

Analysis of two-dimensional solute transport through heterogeneous porous medium

Atul Kumar1, Lav Kush Kumar2, Shireen3

1Assistant Professor, Department of Mathematics & Computer Science, School of Applied Science, Babu Banarasi Das University, Luck-now, India.

2Assistant Professor, Department of Mathematics, Babu Banarasi Das Northern India Institute of Technology, Lucknow, India.

3Assistant Professor, Department of Mathematics, Babu Banarasi Das National Institute of Technology & Management, Lucknow, India.

*Corresponding author: Atul Kumar

69
Published: April 7,2018

Abstract

In this study, an analytical solution is developed for solute transport through heterogeneous porous medium in two-dimensional horizontal Cartesian plane. In the present problem, the domain is considered semi-infinite extent and of heterogeneous nature along the both perpendicular directions of the plane. In surface water body, the aquifer of shallow depths and lateral component of velocity also considered. Due to heterogeneity of the medium, the velocity components of the flow transporting the solute through the medium along both the directions are considered spatially dependent. The velocity along each direction is supposed to increase linearly while passing through a finite region (into which the solute concentration values are evaluated) starting from the origin, by using interpolating formula. Solute dispersion along both the directions is considered proportional to square of the respective velocity component. The present problem is derived for two cases: first one is uniform pulse type input point source and other one is for varying pulse type input point source. The first input condition is considered initially and second is the far end of the domain. It is considered flux type of homogeneous nature. In the both cases, domain is considered initially not solute free. In the process of analytical solutions, a new space or positional variable is introduced. After that, introducing the Laplace transform to get the analytical solutions of our problems. In both the cases, effects of heterogeneity/inhomogeneity of the medium and solute/solute transport behavior have discussed in the presence and absence of the source. All the possible solute/solute transport behaviors are shown graphically interpreted through MatLab 7.0.

Keywords

Advection-Dispersion Equation, Heterogeneous/Inhomogeneous Medium, Pulse type input point source, Diffusion Processes, Laplace Transformation, Interpolation

References

[1] Van Genuchten, M. Th., & Alves, W. J. (1982). Analytical solutions of the one-dimensional convective-dispersive solute transport equation. Technical Bulletin No 1661, US Department of Agriculture.

[2] Lindstrom, F. T., & Boersma, L. (1989). Analytical solutions for convective-dispersive transport in confined aquifers with different initial and boundary conditions. Water Resources Research, 25, 41-56.

[3] Ogata, A. (1970). Theory of dispersion in granular media. US Geol. Sur. Prof. Paper, 411-434.

[4] vanGenuchten, M. T. (1981). Analytical solutions for chemical transport with simultaneous adsorption zero-order production and first-order decay. Journal of Hydrology, 49, 213-233.

[5] Matheron, G., & deMarsily, G. (1980). Is transport in porous media always diffusive? A counter example. Water Resources Research, 16, 901-917.

[6] Gelhar, L. W., Welty, C., & Rehfeldt, K. R. (1992). A critical review of data on field-scale dispersion in aquifers. WaterResources Research, 28 (7), 1955-1974.

[7] Shan, C., & Javandel, I. (1997). Analytical solutions for solute transport in a vertical aquifer section. Journal of Contaminant Hydrology, 27, 63-82.

[8] 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, 29-38.

[9] Frippiat, C. C., & Holeyman, A. E. (2008). A comparative review of upscaling methods for solute transport in heterogeneous porous media. Journal of Hydrology, 362, 150-176.

[10] Chen, J. S., Ni, C. F., Liang, C. P., & Chiang, C. C. (2008). Analytical power seriessolution for contaminant transport with hyperbolic asymptotic distance dependent dispersivity. Journal of Hydrology, 362, 142-149.

[11] Guerrero, J. S. P., Pimentel, L. C. G., Skaggs, T. H., & van Genuchten, M. Th. (2009). Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique. International Journal of Heat and Mass Transfer, 52, 3297-3304.

[12] Cassol, M., Wortmann, S., & Rizza, U. (2009). Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transform techniques. Environmental Modeling and Software, 24, 144-151.

[13] Moreira, D. M., Vilhena, M. T., Buske, D., & Tirabassi, T. (2009). The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1-17.

[14] Costa, C. P., Vilhena, M. T., Moreira, D. M., & Tirabassi, T. (2006). Semi-analytical solution of the steady threedimensional advection-diffusion equation in the planetary boundary layer. Atmospheric Environment, 40, 5659-5669.

[15] 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-environment Research, 2, 254-263.

[16] Kumar, A., Jaiswal, D. K., & Kumar, N. (2010). Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. Journal of Hydrology, 380, 330-337.

[17] Jaiswal, D. K., Kumar, A., and Yadav, R. R., (2011). Analytical solution to the one-dimensional advection-diffusion equation with temporally dependent coefficients, Journal of Water Resource and Protection, 3, 76-84 

[18] Silva, E. J. G., Tirabassi, T., Vilhena, M. T., & Buske, D. (2013). Solution of the Puff Model for Pollutant Dispersion in the Atmospheric Boundary Layer by the GILTT Method. 22nd International Congress of Mechanical Engineering (COBEM 2013), November 3-7, 2013, Ribeirão Preto, SP, Brazil Copyright 2013 by ABCM, 10384- 10391.

[19] Swami, D., Sharma, P. K. & Ojha, C. S. P. (2014). Simulation of experimental breakthrough curves using multiprocess non-equilibrium model for reactive solutetransport in stratified porous media. Sadhana, 39(6), 1425-1446.

[20] Kumar, P., Ramesh, T. & Sudheendra, S. R. (2014). Mathematical Analysis of Transport of Pollutants through Unsaturated Porous Media with Adsorption and Radioactive Decay. International Journal of Combined Research & Development, 2(4), 1-8.

[21] Karedla, N., Gregor, I. & Enderlein, J. (2014). Analytical approximations of the diffusive dispersion in fluid flows. The Letters Journal Exploring the Frontiers of Physics, 108(4), doi: 10.1209/0295-5075/108/40007.

[22] Pereira, M. F., Pozza, S. A. & Timóteo, V. S. (2014).Numerical Methods for the Evaluation of Pollutant Dispersion Based on Advection-Diffusion Equation Chemical Engineering Transactions, 39, 799-804.

[23] Kumar, A. & Yadav, R. R. (2015). One-dimensional solute transport for uniform and varying pulse type input point source through heterogeneous medium. Environmental Technology, 36(4), 487-495.

[24] Guenther, R. B. & Lee, J. W.(1988). Partial Differential Equations of Mathematical Physics and Integral Equations. Printice-Hall Englewood Califfs., N.J. (ch1).

[25] Logan, J. D. (1994). Introduction to Non-linear Partial Differential Equations. Series in Pure and Applied Mathematics, Wiley-Interscience, New York. (ch1).

[26] Scheidegger, A. E. (1957). The physics of flow through porous media. Toronto, Canada: University of Toronto Press.

How to cite this paper

Analysis of two-dimensional solute transport through heterogeneous porous medium

How to cite this paper: Atul Kumar, Lav Kush Kumar, Shireen. (2018). Analysis of two-dimensional solute transport through heterogeneous porous medium. Journal of Applied Mathematics and Computation, 2(3), 67-83.

DOI: http://dx.doi.org/10.26855/jamc.2018.03.001