One Dimensional Solute Transport Originating from a Exponentially Decay Type Point Source Along Unsteady Flow Through Heterogeneous Medium

Author(s)
Premlata Singh

ABSTRACT

One dimensional advection dispersion equation is analytically solved initially in solute free domain by considering uniform exponential decay input condition at origin. Heterogeneous medium of semi infinite extent is considered. Due to heterogeneity velocity and dispersivity coefficient of the advection dispersion equation are considered functions of space variable and time variable. Analytical solution is obtained using Laplace transform technique when dispersivity depended on velocity. The effects of first order decay term and adsorption are studied. The graphical representations are made using MATLAB

One dimensional advection dispersion equation is analytically solved initially in solute free domain by considering uniform exponential decay input condition at origin. Heterogeneous medium of semi infinite extent is considered. Due to heterogeneity velocity and dispersivity coefficient of the advection dispersion equation are considered functions of space variable and time variable. Analytical solution is obtained using Laplace transform technique when dispersivity depended on velocity. The effects of first order decay term and adsorption are studied. The graphical representations are made using MATLAB

Cite this paper

nullP. Singh, "One Dimensional Solute Transport Originating from a Exponentially Decay Type Point Source Along Unsteady Flow Through Heterogeneous Medium,"*Journal of Water Resource and Protection*, Vol. 3 No. 8, 2011, pp. 590-597. doi: 10.4236/jwarp.2011.38068.

nullP. Singh, "One Dimensional Solute Transport Originating from a Exponentially Decay Type Point Source Along Unsteady Flow Through Heterogeneous Medium,"

References

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[3] M. K. Singh, N. K. Mahto and P. Singh, “Longitudinal Dispersion with Time Dependent Source Concentration in Semi Infinite Aquifer,” Journal of Earth System Science, Vol. 117, No. 6, 2008, pp. 945-949. http://dx.doi.org/10.1007/s12040-008-0079-x

[4] M. K. Singh, V. P. Singh, P. Singh, and D. Shukla, “Analytical Solution for Conservative Solute Transport in One Dimensional Homogeneous Porous Formations with Time Dependent Velocity,” Journal of Engineering Mechanics, Vol. 135, No. 9, 2009, pp. 1015-1021. doi:10.1061/(ASCE)EM.1943-7889.0000018

[5] F. B. Smith, “The Diffusion of Smoke from a Continuous Elevated Point Source into a Turbulent Atmosphere,” The Journal of Fluid Mechanics, Vol. 2, No. 01, 1957, pp. 49-76. doi:10.1017/S0022112057000737

[6] A. Ogata, “Theory of Dispersion in Granular Media,” United States Geological Survey professional Paper, No. 411-I, 1970, p. 34.

[7] 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

[8] R. A. Scriven and B. A. Fisher, “The Long Range Transport of Airborne Material and Its Removal by Deposition and Washout—II. The Effect of Turbulent Diffusion,” Atmospheric Environment, Vol. 9, 1975, pp. 59-69.

[9] C. Demuth, “A Contribution to the Analytical Steady Solution of the Diffusion Equation for Line Sources,” Atmospheric Environment, Vol. 12, No. 5, 1978, pp. 1255-1258. doi:10.1016/0004-6981(78)90399-2

[10] V. A. Fry, J. D. Istok and R. B. Guenther, “Analytical Solutions to the Solute Transport Equation with Rate- Limited Desorption and Decay,” Water Resources Research, Vol. 29, No. 9, 1993, pp. 3201-3208.

[11] E. H. Ebach and R. White “Mixing of Fluids Flowing through Beds of Packed Solids,” Journal of American Institute of Chemical Engineers, Vol. 4, 1958, pp. 161-164.

[12] R. B. Banks and S. Jerasate, “Dispersion in Unsteady Porous Media Flow,” Journal of the Hydraulics Division, Vol. 88, No. HY3, 1962, pp. 1-21.

[13] N. Kumar, “Unsteady Flow Against Dispersion in Finite Porous Media,” Journal of Hydrology, Vol. 63, No. 3-4, 1983, pp. 345-358. doi:10.1016/0022-1694(83)90050-1

[14] M. M. Aral and B. Liao, “Analytical Solutions for Two-Dimensional Transport Equations 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)

[15] M. K. Singh, P. Singh and V. P. Singh, “Analytical Solution for Two Dimensional Solute Transport in Finite Aquifer with Time Dependent Source Concentration,” Journal of Engineering Mechanics, Vol. 136, No. 10, 2010, pp. 1309-1315. doi:10.1061/(ASCE)EM.1943-7889.0000177

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[17] C. Zoppou and J. H. Knight, “Analytical Solution of a Spatially Variable Coefficient Advection–Diffusion Equ- ation in up to Three Dimensions,” Applied Mathematical Modelling, Vol. 23, No. 9, 1999, pp. 667-685. doi:10.1016/S0307-904X(99)00005-0

[18] D. K. Jaiswal, A. Kumar and R. R. Yadav, “Analytical Solution to the One Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficient,” Journal of Water Resource and Protection, Vol. 1, 2011, pp. 76-84.

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

[20] D. A. Barry and G. Sposito, “Analytical Solution of a Convection—Dispersion Model with Time Dependent Transport Coefficients,” Water Resources Research, Vol. 25, No. 12, 1989, pp. 2407-2416.

[21] H. A. Basha and F. S. El-Habel, “Analytical Solution of One Dimensional Time-Dependent Transport Equation.” Water Resources Research, Vol. 29, No. 9, 1993, pp. 3209-3214.

[22] G., Marinoschi, U. Jaekel and H. Vereecken, “Analytical Solutions of Three Dimensional Convection-Dispersion Problems with time Dependent Coefficients,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 79, No. 6, 1999, pp. 411-421. doi:10.1002/(SICI)1521-4001(199906)79:6<411::AID-ZAMM411>3.0.CO;2-6

[23] 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

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

[26] J. D. Logan, “Solute Transport in Porous Media with Scale—Dependent Dispersion and Periodic Boundary Conditions,” Journal of Hydrology, Vol. 184, No. 3-4, 1996, pp. 261-276. doi:10.1016/0022-1694(95)02976-1

[27] S. Neelz, “Limitations of an Analytical Solution for Advection—Diffusion Equation with Variable Coefficients,” Communications in Numerical Methods in Engineering, Vol. 22, No. 5, 2006, pp. 387-396.

[28] J. S. P. Guerrero, L. C. G. Pimental, T. H. Skaggs and M. Th. van Genucheten, “Analytical Solution of the Advection-Diffusion Transport Equation Using a Change of Variable and Integral Transform Technique,” International Journal of Heat and Mass Transfer, Vol. 52, No. 13-14, 2009, pp. 3297-3304. doi:10.1016/j.ijheatmasstransfer.2009.02.002

[29] J .S. P. Guerrero and T. H. Skaggs, “Analytical Solution for One-Dimensional Advection—Dispersion Transport Equation with Space-Dependent Coefficients,” Journal of Hydrology, Vol. 390, No. 1, 2010, pp. 57-65. doi:10.1016/j.jhydrol.2010.06.030

[30] V. Srinivasan and T. P. Clement, “Analytical Solutions for Sequentially Coupled One-Dimensional Reactive Transport Problems-Part I: Mathematical Derivations,” Advances in Water Resources, Vol. 31, No. 2, 2008, pp. 203-218. doi:10.1016/j.advwatres.2007.08.002

[31] C. V. Chrysikopoulos, P. K. Kitanidis and P. V. Roberts, “Analysis of One-Dimensional Solute Transport through Porous Media with Spatially Variable Retardation Factor,” Water Resources Research, Vol. 26, No. 3, 1990, pp. 437-446.

[32] K. Huang, M. Th. van Genuchten and R. Zhang, “Exact Solutions for One Dimensional Transport with Asymptotic Scale Dependent Dispersion,” Applied Mathematical Modelling, Vol. 20, 1996, pp. 297-308.

[33] B. Hunt, “Contaminant Source Solutions with scale Dependent Dispersivities,” Journal of Hydrologic Engineering, Vol. 3, No. 4, 1998, pp. 268-275. doi:10.1061/(ASCE)1084-0699(1998)3:4(268)

[34] B. Hunt, “Scale Dependent Dispersion from a Pit,” Journal of the Hydraulics Division, Vol. 104, 2002, pp. 75-85.

[35] L. Pang and B. Hunt, “Solutions and Verification of a Scale—Dependent Dispersion Model,” Journal of Contaminant Hydrology, Vol. 53, No. 1-2, 2001, pp. 21-39.

[36] G. Liu and B. C. Si, “Analytical Modeling of One-Dimensional Diffusion in Layered System with Position-Dependent Diffusion Coefficients,” Advances in Water Resources, Vol. 31, No. 2, 2008, pp.251-268. doi:10.1016/j.advwatres.2007.08.008

[37] J. S. Chen, C. W. Liu and C. M. Liao, “Two-Dimensional Laplace-Transformed Power Series Solution for Solute Transport in a Radially Converget Flow Field,” Advances in Water Resources, Vol. 26, No. 10, 2003, pp. 1113-1124. doi:10.1016/S0309-1708(03)00090-3

[38] J. S. Chen, C. F. Ni, C. P. Liang and C. C. Chiang, “Analytical Power Series Solution for Contaminant Transport with Hyperbolic Asymptotic Distance—Dependent Dispersivity,” Journal of Hydrology, Vol. 362, No. 1-2, 2008, pp. 142-149. doi:10.1016/j.jhydrol.2008.08.020

[39] J. S. Chen, C. S. Chen and C. Y. Chen, “Analysis of Solute Transport in a Divergent Flow Tracer Test with Scale-Dependent Dispersion,” Hydrological Processes, Vol. 21, No. 18, 2007, pp. 2526-2536. doi.org/10.1002/hyp.6496

[40] Y. Nibori, R. Nakata, O. Tochiyama and H Mimura, “Evaluation of Solute Transport through a Fracture by Considering the Spatial Distributions of Retardartion Effect in Granule Scale,” Journal of Hydrologic Engineering, Vol. 14, No. 11, 2009, pp. 1214-1220. doi:10.1061/(ASCE)1084-0699(2009)14:11(1214)

[41] H. Zhan, Z. Wen, G. Huang and D. Sun, “Analytical Solution of Two-Dimensional Solute Transport in an Aquifer-Aquitard System,” Journal of Contaminant Hydrology, Vol. 107, No. 3-4, 2009, pp. 162-174. doi:10.1016/j.jconhyd.2009.04.010

[42] Q. Yang, F. Liu and I. Turner, “Numerical Methods for Fractional Partial Differential Equations with Riesz Space Fractional Derivatives,” Applied Mathematical Modelling, Vol. 34, No. 1, 2010, pp. 200-218. doi:10.1016/j.apm.2009.04.006

[43] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations,” Applied Numerical Mathematics, Vol. 56, No. 1, 2006, pp. 80-90. doi:10.1016/j.apnum.2005.02.008

[44] 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, No. 4, 2009, pp. 254-263. doi:10.1016/j.jher.2009.01.003

[45] J. Crank, “The Mathematics of Diffusion,” Oxford University Press, Oxford, 1975.

[46] 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. doi:10.1007/s12040-009-0049-y

[1] M. Th. Van Genuchten and W. J. Alves, “Analytical Solutions of the One–Dimensional Convective–Dispersive Solute Transport Equation,” Technical Bulletin (United States Department of Agriculture), No. 1661.

[2] I. Javandel, C. Doughty, and C. F. Tsang, “Groundwater Transport: Handbook of Mathematical Models,” American Geophysical Union Water Resources Monograph Series 10, American Geophysical Union, Washington D C, 1984.

[3] M. K. Singh, N. K. Mahto and P. Singh, “Longitudinal Dispersion with Time Dependent Source Concentration in Semi Infinite Aquifer,” Journal of Earth System Science, Vol. 117, No. 6, 2008, pp. 945-949. http://dx.doi.org/10.1007/s12040-008-0079-x

[4] M. K. Singh, V. P. Singh, P. Singh, and D. Shukla, “Analytical Solution for Conservative Solute Transport in One Dimensional Homogeneous Porous Formations with Time Dependent Velocity,” Journal of Engineering Mechanics, Vol. 135, No. 9, 2009, pp. 1015-1021. doi:10.1061/(ASCE)EM.1943-7889.0000018

[5] F. B. Smith, “The Diffusion of Smoke from a Continuous Elevated Point Source into a Turbulent Atmosphere,” The Journal of Fluid Mechanics, Vol. 2, No. 01, 1957, pp. 49-76. doi:10.1017/S0022112057000737

[6] A. Ogata, “Theory of Dispersion in Granular Media,” United States Geological Survey professional Paper, No. 411-I, 1970, p. 34.

[7] 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

[8] R. A. Scriven and B. A. Fisher, “The Long Range Transport of Airborne Material and Its Removal by Deposition and Washout—II. The Effect of Turbulent Diffusion,” Atmospheric Environment, Vol. 9, 1975, pp. 59-69.

[9] C. Demuth, “A Contribution to the Analytical Steady Solution of the Diffusion Equation for Line Sources,” Atmospheric Environment, Vol. 12, No. 5, 1978, pp. 1255-1258. doi:10.1016/0004-6981(78)90399-2

[10] V. A. Fry, J. D. Istok and R. B. Guenther, “Analytical Solutions to the Solute Transport Equation with Rate- Limited Desorption and Decay,” Water Resources Research, Vol. 29, No. 9, 1993, pp. 3201-3208.

[11] E. H. Ebach and R. White “Mixing of Fluids Flowing through Beds of Packed Solids,” Journal of American Institute of Chemical Engineers, Vol. 4, 1958, pp. 161-164.

[12] R. B. Banks and S. Jerasate, “Dispersion in Unsteady Porous Media Flow,” Journal of the Hydraulics Division, Vol. 88, No. HY3, 1962, pp. 1-21.

[13] N. Kumar, “Unsteady Flow Against Dispersion in Finite Porous Media,” Journal of Hydrology, Vol. 63, No. 3-4, 1983, pp. 345-358. doi:10.1016/0022-1694(83)90050-1

[14] M. M. Aral and B. Liao, “Analytical Solutions for Two-Dimensional Transport Equations 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)

[15] M. K. Singh, P. Singh and V. P. Singh, “Analytical Solution for Two Dimensional Solute Transport in Finite Aquifer with Time Dependent Source Concentration,” Journal of Engineering Mechanics, Vol. 136, No. 10, 2010, pp. 1309-1315. doi:10.1061/(ASCE)EM.1943-7889.0000177

[16] A. E. Scheidegger, “The Physics of Flow through Porous Media,” 3rd Edition, University of Toronto Press, 1974.

[17] C. Zoppou and J. H. Knight, “Analytical Solution of a Spatially Variable Coefficient Advection–Diffusion Equ- ation in up to Three Dimensions,” Applied Mathematical Modelling, Vol. 23, No. 9, 1999, pp. 667-685. doi:10.1016/S0307-904X(99)00005-0

[18] D. K. Jaiswal, A. Kumar and R. R. Yadav, “Analytical Solution to the One Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficient,” Journal of Water Resource and Protection, Vol. 1, 2011, pp. 76-84.

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

[20] D. A. Barry and G. Sposito, “Analytical Solution of a Convection—Dispersion Model with Time Dependent Transport Coefficients,” Water Resources Research, Vol. 25, No. 12, 1989, pp. 2407-2416.

[21] H. A. Basha and F. S. El-Habel, “Analytical Solution of One Dimensional Time-Dependent Transport Equation.” Water Resources Research, Vol. 29, No. 9, 1993, pp. 3209-3214.

[22] G., Marinoschi, U. Jaekel and H. Vereecken, “Analytical Solutions of Three Dimensional Convection-Dispersion Problems with time Dependent Coefficients,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 79, No. 6, 1999, pp. 411-421. doi:10.1002/(SICI)1521-4001(199906)79:6<411::AID-ZAMM411>3.0.CO;2-6

[23] 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

[24] S. R. Yates, “An Analytical Solution for One—Dimensional Transport in Heterogeneous Porous Media,” Water Resources Research, Vol. 26, No. 10, 1990, pp. 2331-2338.

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

[26] J. D. Logan, “Solute Transport in Porous Media with Scale—Dependent Dispersion and Periodic Boundary Conditions,” Journal of Hydrology, Vol. 184, No. 3-4, 1996, pp. 261-276. doi:10.1016/0022-1694(95)02976-1

[27] S. Neelz, “Limitations of an Analytical Solution for Advection—Diffusion Equation with Variable Coefficients,” Communications in Numerical Methods in Engineering, Vol. 22, No. 5, 2006, pp. 387-396.

[28] J. S. P. Guerrero, L. C. G. Pimental, T. H. Skaggs and M. Th. van Genucheten, “Analytical Solution of the Advection-Diffusion Transport Equation Using a Change of Variable and Integral Transform Technique,” International Journal of Heat and Mass Transfer, Vol. 52, No. 13-14, 2009, pp. 3297-3304. doi:10.1016/j.ijheatmasstransfer.2009.02.002

[29] J .S. P. Guerrero and T. H. Skaggs, “Analytical Solution for One-Dimensional Advection—Dispersion Transport Equation with Space-Dependent Coefficients,” Journal of Hydrology, Vol. 390, No. 1, 2010, pp. 57-65. doi:10.1016/j.jhydrol.2010.06.030

[30] V. Srinivasan and T. P. Clement, “Analytical Solutions for Sequentially Coupled One-Dimensional Reactive Transport Problems-Part I: Mathematical Derivations,” Advances in Water Resources, Vol. 31, No. 2, 2008, pp. 203-218. doi:10.1016/j.advwatres.2007.08.002

[31] C. V. Chrysikopoulos, P. K. Kitanidis and P. V. Roberts, “Analysis of One-Dimensional Solute Transport through Porous Media with Spatially Variable Retardation Factor,” Water Resources Research, Vol. 26, No. 3, 1990, pp. 437-446.

[32] K. Huang, M. Th. van Genuchten and R. Zhang, “Exact Solutions for One Dimensional Transport with Asymptotic Scale Dependent Dispersion,” Applied Mathematical Modelling, Vol. 20, 1996, pp. 297-308.

[33] B. Hunt, “Contaminant Source Solutions with scale Dependent Dispersivities,” Journal of Hydrologic Engineering, Vol. 3, No. 4, 1998, pp. 268-275. doi:10.1061/(ASCE)1084-0699(1998)3:4(268)

[34] B. Hunt, “Scale Dependent Dispersion from a Pit,” Journal of the Hydraulics Division, Vol. 104, 2002, pp. 75-85.

[35] L. Pang and B. Hunt, “Solutions and Verification of a Scale—Dependent Dispersion Model,” Journal of Contaminant Hydrology, Vol. 53, No. 1-2, 2001, pp. 21-39.

[36] G. Liu and B. C. Si, “Analytical Modeling of One-Dimensional Diffusion in Layered System with Position-Dependent Diffusion Coefficients,” Advances in Water Resources, Vol. 31, No. 2, 2008, pp.251-268. doi:10.1016/j.advwatres.2007.08.008

[37] J. S. Chen, C. W. Liu and C. M. Liao, “Two-Dimensional Laplace-Transformed Power Series Solution for Solute Transport in a Radially Converget Flow Field,” Advances in Water Resources, Vol. 26, No. 10, 2003, pp. 1113-1124. doi:10.1016/S0309-1708(03)00090-3

[38] J. S. Chen, C. F. Ni, C. P. Liang and C. C. Chiang, “Analytical Power Series Solution for Contaminant Transport with Hyperbolic Asymptotic Distance—Dependent Dispersivity,” Journal of Hydrology, Vol. 362, No. 1-2, 2008, pp. 142-149. doi:10.1016/j.jhydrol.2008.08.020

[39] J. S. Chen, C. S. Chen and C. Y. Chen, “Analysis of Solute Transport in a Divergent Flow Tracer Test with Scale-Dependent Dispersion,” Hydrological Processes, Vol. 21, No. 18, 2007, pp. 2526-2536. doi.org/10.1002/hyp.6496

[40] Y. Nibori, R. Nakata, O. Tochiyama and H Mimura, “Evaluation of Solute Transport through a Fracture by Considering the Spatial Distributions of Retardartion Effect in Granule Scale,” Journal of Hydrologic Engineering, Vol. 14, No. 11, 2009, pp. 1214-1220. doi:10.1061/(ASCE)1084-0699(2009)14:11(1214)

[41] H. Zhan, Z. Wen, G. Huang and D. Sun, “Analytical Solution of Two-Dimensional Solute Transport in an Aquifer-Aquitard System,” Journal of Contaminant Hydrology, Vol. 107, No. 3-4, 2009, pp. 162-174. doi:10.1016/j.jconhyd.2009.04.010

[42] Q. Yang, F. Liu and I. Turner, “Numerical Methods for Fractional Partial Differential Equations with Riesz Space Fractional Derivatives,” Applied Mathematical Modelling, Vol. 34, No. 1, 2010, pp. 200-218. doi:10.1016/j.apm.2009.04.006

[43] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations,” Applied Numerical Mathematics, Vol. 56, No. 1, 2006, pp. 80-90. doi:10.1016/j.apnum.2005.02.008

[44] 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, No. 4, 2009, pp. 254-263. doi:10.1016/j.jher.2009.01.003

[45] J. Crank, “The Mathematics of Diffusion,” Oxford University Press, Oxford, 1975.

[46] 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. doi:10.1007/s12040-009-0049-y