Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients

ABSTRACT

In a one-dimensional advection-diffusion equation with temporally dependent coefficients three cases may arise: solute dispersion parameter is time dependent while the flow domain transporting the solutes is uniform, the former is uniform and the latter is time dependent and lastly the both parameters are time dependent. In the present work analytical solutions are obtained for the last case, studying the dispersion of continuous input point sources of uniform and increasing nature in an initially solute free semi-infinite domain. The solutions for the first two cases and for uniform dispersion along uniform flow are derived as particular cases. The dispersion parameter is not proportional to the velocity of the flow. The Laplace transformation technique is used. New space and time variables are introduced to get the solutions. The solutions in all possible combinations of increasing/decreasing temporal dependence are compared with each other with the help of graphs. It has been observed that the concentration attenuation with position and time is the fastest in case of decreasing dispersion in accelerating flow field.

In a one-dimensional advection-diffusion equation with temporally dependent coefficients three cases may arise: solute dispersion parameter is time dependent while the flow domain transporting the solutes is uniform, the former is uniform and the latter is time dependent and lastly the both parameters are time dependent. In the present work analytical solutions are obtained for the last case, studying the dispersion of continuous input point sources of uniform and increasing nature in an initially solute free semi-infinite domain. The solutions for the first two cases and for uniform dispersion along uniform flow are derived as particular cases. The dispersion parameter is not proportional to the velocity of the flow. The Laplace transformation technique is used. New space and time variables are introduced to get the solutions. The solutions in all possible combinations of increasing/decreasing temporal dependence are compared with each other with the help of graphs. It has been observed that the concentration attenuation with position and time is the fastest in case of decreasing dispersion in accelerating flow field.

Cite this paper

nullD. Jaiswal, A. Kumar and R. Yadav, "Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients,"*Journal of Water Resource and Protection*, Vol. 3 No. 1, 2011, pp. 76-84. doi: 10.4236/jwarp.2011.31009.

nullD. Jaiswal, A. Kumar and R. Yadav, "Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients,"

References

[1] A. Ogata and R. B. Banks, “A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media,” US Geological Survey Professional Papers, No. 34, 1961, p. 411-A.

[2] D. R. F. Harleman and R. R. Rumer, “Longitudinal and Lateral Dispersion in an Isotropic Porous Medium,” Journal of Fluid Mechanics, Vol. 16, No. 3, 1963, pp. 385- 394. doi:10.1017/S0022112063000847

[3] V. Guvanasen and R. E. Volker, “Experimental Investigations of Unconfined Aquifer Pollution from Recharge Basins,” Water Resources Research, Vol. 19, No. 3, 1983, pp. 707-717. doi:10.1029/WR019i003p00707

[4] T. J. Marshal, J. W. Holmes and C. W. Rose, “Soil Physics,” 3rd Edition, Cambridge University Press, Cambridge, 1996.

[5] R. B. Banks and J. Ali, “Dispersion and Adsorption in Porous Media Flow,” Journal of Hydraulic Division, Vol. 90, No. 5, 1964, pp. 13-31.

[6] A. Ogata, “Theory of Dispersion in Granular Media,” US Geological Survey Professional Papers, No. 411-1, p. 34, 1970.

[7] S. H. Lai and J. J. Jurinak, “Numerical Approximation of Cation Exchange in Miscible Displacement Through Soil Columns,” Soil Science Society American Proceeding, Vol. 35, No. 6, 1971, pp. 894-899. doi:10.2136/sssaj1971.03615995003500060017x

[8] M. A. Marino, “Distribution of Contaminants in Porous Media Flow,” Water Resources Research, Vol. 10, No. 5, 1974, pp. 1013-1018. doi:10.1029/WR010i005p01013

[9] A. N. S. Al-Niami and K. R. Rushton, “Analysis of Flow against Dispersion in Porous Media,” Journal of Hydrology, Vol. 33, No. 1-2, 1977, pp. 87-97. doi:10.1016/0022-1694(77)90100-7

[10] M. Th. van Genuchten and W. J. Alves, “Analytical Solutions of the One Dimensional Convective-Dispersive Solute Transport Equation,” US Department of Agriculture, Technical Bulletin, No. 1661, 1982.

[11] F. T. Lindstrom and L. Boersma, “Analytical Solutions for Convective Dispersive Transport in Confined Aquifers with Different Initial and Boundary Conditions,” Water Resources Research, Vol. 25, No. 2, 1989, pp. 241-256. doi:10.1029/WR025i002p00241

[12] D. K. Jaiswal, A. Kumar, N. Kumar and R. R. Yadav, “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, Vol. 2, 2009, pp. 254-263. doi:10.1016/j.jher.2009.01.003

[13] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical Solutions to One-Dimensional Advection-Diffusion Equation with Variable Coefficients in Semi-Infinite Media,” Journal of Hydrology, Vol. 380, No. 3-4, 2010, pp. 330-337. doi:10.1016/j.jhydrol.2009.11.008

[14] S. R. Yates, “An Analytical Solution for One Dimensional Transport in Porous Media with an Experimental Dispersion Function,” Water Resources Research, Vol. 28, No. 8, 1992, pp. 2149-2154. doi:10.1029/92WR01006

[15] J. D. Logan and V. Zlotnik, “The convection-Diffusion Equation with Periodic Boundary Conditions,” Applied Mathematics Letter, Vol. 8, No. 3, 1995, pp. 55-61. doi:10.1016/0893-9659(95)00030-T

[16] J. D. Logan, “Solute Transport in Porous Media with Scale-Dependent Dispersion and Periodic Boundary Conditions,” Journal of Hydrology, Vol. 184, No. 3, 1996, pp. 261-276. doi:10.1016/0022-1694(95)02976-1

[17] M. M. Aral and B. Liao, “Analytical Solutions for Two- Dimensional Transport Equation with Time-Dependent Dispersion Coefficients,” Journal of Hydrologic Engineering, Vol. 1, No. 1, 1996, pp. 20-32. doi:10.1061/(ASCE)1084-0699(1996)1:1(20)

[18] R. Haberman, “Elementary Applied Partial Differential Equations,” Prentice-Hall, Englewood Cliffs, 1987.

[19] G. Matheron and G. De Marsily, “Is Transport in Porous Media Always Diffusive, a Counterexample,” Water Resources Research, Vol. 16, No. 5, 1980, pp. 901-917. doi:10.1029/WR016i005p00901

[20] G. Sposito, W. A. Jury and V. K. Gupta, “Fundamental Problems in the Stochastic Convection-Dispersion Model for Solute Transport in Aquifers and Field Soils,” Water Resources Research, Vol. 22, No. 1, 1986, pp. 77-88. doi:10.1029/WR022i001p00077

[21] L. W. Gelhar, C. Welty and K. R. Rehfeldt, “A Critical Review of Data on Field-Scale Dispersion in Aquifers,” Water Resources Research, Vol. 28, No. 7, 1992, pp. 1955-1974. doi:10.1029/92WR00607

[22] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical Solutions of One-Dimensional Advection-Diffusion Equation with Variable Coefficients in a Finite Domain,” Journal of Earth System Science, Vol. 118, No. 5, 2009, pp. 539-549. doi:10.1007/s12040-009-0049-y

[23] H. B. Fischer, E. J. List, R. C. Y. Koh, J. Imberger and N. H. Brooks, “Mixing in Inland and Coastal Waters,” Academic Press, New York, 1979.

[1] A. Ogata and R. B. Banks, “A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media,” US Geological Survey Professional Papers, No. 34, 1961, p. 411-A.

[2] D. R. F. Harleman and R. R. Rumer, “Longitudinal and Lateral Dispersion in an Isotropic Porous Medium,” Journal of Fluid Mechanics, Vol. 16, No. 3, 1963, pp. 385- 394. doi:10.1017/S0022112063000847

[3] V. Guvanasen and R. E. Volker, “Experimental Investigations of Unconfined Aquifer Pollution from Recharge Basins,” Water Resources Research, Vol. 19, No. 3, 1983, pp. 707-717. doi:10.1029/WR019i003p00707

[4] T. J. Marshal, J. W. Holmes and C. W. Rose, “Soil Physics,” 3rd Edition, Cambridge University Press, Cambridge, 1996.

[5] R. B. Banks and J. Ali, “Dispersion and Adsorption in Porous Media Flow,” Journal of Hydraulic Division, Vol. 90, No. 5, 1964, pp. 13-31.

[6] A. Ogata, “Theory of Dispersion in Granular Media,” US Geological Survey Professional Papers, No. 411-1, p. 34, 1970.

[7] S. H. Lai and J. J. Jurinak, “Numerical Approximation of Cation Exchange in Miscible Displacement Through Soil Columns,” Soil Science Society American Proceeding, Vol. 35, No. 6, 1971, pp. 894-899. doi:10.2136/sssaj1971.03615995003500060017x

[8] M. A. Marino, “Distribution of Contaminants in Porous Media Flow,” Water Resources Research, Vol. 10, No. 5, 1974, pp. 1013-1018. doi:10.1029/WR010i005p01013

[9] A. N. S. Al-Niami and K. R. Rushton, “Analysis of Flow against Dispersion in Porous Media,” Journal of Hydrology, Vol. 33, No. 1-2, 1977, pp. 87-97. doi:10.1016/0022-1694(77)90100-7

[10] M. Th. van Genuchten and W. J. Alves, “Analytical Solutions of the One Dimensional Convective-Dispersive Solute Transport Equation,” US Department of Agriculture, Technical Bulletin, No. 1661, 1982.

[11] F. T. Lindstrom and L. Boersma, “Analytical Solutions for Convective Dispersive Transport in Confined Aquifers with Different Initial and Boundary Conditions,” Water Resources Research, Vol. 25, No. 2, 1989, pp. 241-256. doi:10.1029/WR025i002p00241

[12] D. K. Jaiswal, A. Kumar, N. Kumar and R. R. Yadav, “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, Vol. 2, 2009, pp. 254-263. doi:10.1016/j.jher.2009.01.003

[13] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical Solutions to One-Dimensional Advection-Diffusion Equation with Variable Coefficients in Semi-Infinite Media,” Journal of Hydrology, Vol. 380, No. 3-4, 2010, pp. 330-337. doi:10.1016/j.jhydrol.2009.11.008

[14] S. R. Yates, “An Analytical Solution for One Dimensional Transport in Porous Media with an Experimental Dispersion Function,” Water Resources Research, Vol. 28, No. 8, 1992, pp. 2149-2154. doi:10.1029/92WR01006

[15] J. D. Logan and V. Zlotnik, “The convection-Diffusion Equation with Periodic Boundary Conditions,” Applied Mathematics Letter, Vol. 8, No. 3, 1995, pp. 55-61. doi:10.1016/0893-9659(95)00030-T

[16] J. D. Logan, “Solute Transport in Porous Media with Scale-Dependent Dispersion and Periodic Boundary Conditions,” Journal of Hydrology, Vol. 184, No. 3, 1996, pp. 261-276. doi:10.1016/0022-1694(95)02976-1

[17] M. M. Aral and B. Liao, “Analytical Solutions for Two- Dimensional Transport Equation with Time-Dependent Dispersion Coefficients,” Journal of Hydrologic Engineering, Vol. 1, No. 1, 1996, pp. 20-32. doi:10.1061/(ASCE)1084-0699(1996)1:1(20)

[18] R. Haberman, “Elementary Applied Partial Differential Equations,” Prentice-Hall, Englewood Cliffs, 1987.

[19] G. Matheron and G. De Marsily, “Is Transport in Porous Media Always Diffusive, a Counterexample,” Water Resources Research, Vol. 16, No. 5, 1980, pp. 901-917. doi:10.1029/WR016i005p00901

[20] G. Sposito, W. A. Jury and V. K. Gupta, “Fundamental Problems in the Stochastic Convection-Dispersion Model for Solute Transport in Aquifers and Field Soils,” Water Resources Research, Vol. 22, No. 1, 1986, pp. 77-88. doi:10.1029/WR022i001p00077

[21] L. W. Gelhar, C. Welty and K. R. Rehfeldt, “A Critical Review of Data on Field-Scale Dispersion in Aquifers,” Water Resources Research, Vol. 28, No. 7, 1992, pp. 1955-1974. doi:10.1029/92WR00607

[22] A. Kumar, D. K. Jaiswal and N. Kumar, “Analytical Solutions of One-Dimensional Advection-Diffusion Equation with Variable Coefficients in a Finite Domain,” Journal of Earth System Science, Vol. 118, No. 5, 2009, pp. 539-549. doi:10.1007/s12040-009-0049-y

[23] H. B. Fischer, E. J. List, R. C. Y. Koh, J. Imberger and N. H. Brooks, “Mixing in Inland and Coastal Waters,” Academic Press, New York, 1979.