Comparative Study of Analytical Solutions for Time-Dependent Solute Transport Along Unsteady Groundwater Flow in Semi-infinite Aquifer

ABSTRACT

A comparative study is made among Laplace Transform Technique (LTT) and Fourier Transform Technique (FTT) to obtain one-dimensional analytical solution for conservative solute transport along unsteady groundwater flow in semi-infinite aquifer. The time-dependent source of contaminant concentration is considered at the origin and at the other end of the aquifer is supposed to be zero. Initially, aquifer is not solute free which means that the solute concentration exits in groundwater system and it is assumed as a uniform concentration. The aquifer is considered homogeneous and semi-infinite. The time-dependent velocity expressions are considered. The result may be used as preliminary predictive tools in groundwater management and benchmark the numerical code and solutions.

A comparative study is made among Laplace Transform Technique (LTT) and Fourier Transform Technique (FTT) to obtain one-dimensional analytical solution for conservative solute transport along unsteady groundwater flow in semi-infinite aquifer. The time-dependent source of contaminant concentration is considered at the origin and at the other end of the aquifer is supposed to be zero. Initially, aquifer is not solute free which means that the solute concentration exits in groundwater system and it is assumed as a uniform concentration. The aquifer is considered homogeneous and semi-infinite. The time-dependent velocity expressions are considered. The result may be used as preliminary predictive tools in groundwater management and benchmark the numerical code and solutions.

Cite this paper

nullM. Kumar Singh, N. Kumar Mahato and P. Kumar, "Comparative Study of Analytical Solutions for Time-Dependent Solute Transport Along Unsteady Groundwater Flow in Semi-infinite Aquifer,"*International Journal of Geosciences*, Vol. 2 No. 4, 2011, pp. 457-467. doi: 10.4236/ijg.2011.24048.

nullM. Kumar Singh, N. Kumar Mahato and P. Kumar, "Comparative Study of Analytical Solutions for Time-Dependent Solute Transport Along Unsteady Groundwater Flow in Semi-infinite Aquifer,"

References

[1] S. Mohan and M. Muthukumaran, “Modeling of pollutant transport in groundwater,” Journal of Institution of Engineers (India): Environmental Division, Vol. 85, No. 1, 2004, pp. 22-32.

[2] H. D. Sharma and K. R. Reddy, “Geo-environmental Engi- neering,” John Willy & sons Inc., Hoboken, 2004.

[3] R. Rausch, W. Schafer, R. Therrien and C. Wagner, “Solute Transport Modelling: An Introduction to Models and Solu on Strategies,” Gebrüder Borntrager Verlagsbuchhandlung Science Publishers, Berlin, 2005.

[4] M. Thangarajan, “Groundwater: Resource Evaluation, Augmentation, Contamination, Restoration, Modeling and Management,” Capital Publishing Company, New Delhi, 2006.

[5] F. J. Leij, N. Toride and T. V. Genuchten, “Analytical Solution for Non-Equilibrium Solute Transport in Three- Dimension Porous Media,” Journal of Hydrology, Vol. 151, No. 2-4, 1993, pp. 193-228. doi:10.1016/0022-1694(93)90236-3

[6] J. J. Fried, “Groundwater Pollution,” Elsevier Scientific Publishing Company, Amsterdam, 1975.

[7] I. C. Javendel and C. F. Tsang, “Groundwater Transport: Hand book of Mathematics Models,” Water Resource Monogrser, AGU, Washington, Vol. 10, 1984.

[8] J. Bear and A. Verruijt, “Modeling Groundwater Flow and Pollution,” D. Reidel Pubulising Company, Dor- drecht, 1987. doi:10.1007/978-94-009-3379-8

[9] M. P. Anderson and W. W. Woessner, “Applied Ground- water Modeling: Simulation of Flow and Advective Transport,” Academic Press, London, 1992.

[10] S. N. Rai, “Role of Mathematical Modeling in Ground- water Resources Management,” NGRI, Hyderabad, 2004.

[11] N. C. Ghosh and K. D. Sharma, “Groundwater Modelling and Management,” Capital Publishing Company, New- Delhi, 2006

[12] G. S. Kumar, M. Sekher and D. Mishra, “Time-Dependent Dispersivity Behaviour of Non Reactive Solutes in a System of Parallel Structures,” Hydrology and Earth System Sciences, Vol. 3, No. 3, 2006, pp. 895-923. doi:10.5194/hessd-3-895-2006

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

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

[15] R. P. Banks and S. T. Jerasate, “Dispersion in Unsteady Porous Media Flow,” Journal of Hydraulic Division, Vol. 88, No. HY3, 1962, pp. 1-21.

[16] E. H. Ebach and R. White “Mixing of Fluid Flowing through Beds of Packed Solids,” Journal of American Institute of Chemical Engineers, Vol. 4, No. 2, 1958, pp. 161-164. doi:10.1002/aic.690040209

[17] R. R. Rumer, “Longitudinal Dispersion in Steady and Un- Steady Flow,” Journal of Hydraulic Division, Vol. 88, No. HY4, 1962, pp. 147-172.

[18] J. K. Mitchell, “Fundamentals of Soil Behavior,” Wiley, New York, 1976.

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

[20] A. Ogata, R. B. Banks, “A Solution of the Differential Equation of Longitudinal Dispersion in Porous Medium,” US Geological Survey, Washington D.C., 1961.

[21] M. Th. Van Genucheten and W. J. Alves, “Analytical Solutions for Chemical Transport with Simultaneous Adsorption, Zero-Order Production and finite Order Decay,” Journal of Hydrology, Vol. 49, No. 3-4, 1981, pp. 213-233. doi:10.1016/0022-1694(81)90214-6

[22] M. Th. Van Genucheten and W. J. Alves, “Analytical So- lution of One Dimensional Convective-Dispersion Solute Transport Equation,” Technical Bulletin, 1982, pp. 1-51.

[23] N. Kumar, “Dispersion of Pollutants in Semi-Infinite Porous Media with Unsteady Velocity Distribution,” Nordic Hydrology, Vol. 14, No. 3, 1983, pp. 167-178.

[24] 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

[25] D. A. Barry and G. Sposito, “Analytical Solution of a Convection-Dispersion Model with Time-Dependent Trans- port Coefficients,” Water Recourses Research, Vol. 25, No. 12, 1989, pp. 2407-2416. doi:10.1029/WR025i012p02407

[26] H. A. Basha and F. S. El-Habel, “Analytical Solution of One Dimensional Time-Dependent Transport Equation,” Water Recourses Research, Vol. 29, No. 9, 1993, pp. 3209- 3214. doi:10.1029/93WR01038

[27] V. A. Fry, J. D. Istok and R. B. Guenther, “An Analytical Solution of the Solute Transport Equation with Rate- Limited Desorption and Decay,” Water Recourses Research, Vol. 29, No. 9, 1993, pp. 3201-3208. doi:10.1029/93WR01394

[28] C. V. Chrysikopoulos and Y. Sim, “One-Dimensional Virus Transport Homogeneous Porous Media with Time Dependent Distribution Coefficient,” Journal of Hydrology, Vol. 185, No. 1-4, 1996, pp. 199-219. doi:10.1016/0022-1694(95)02990-7

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

[30] M. Flury, Q. J. Wu, L. Wu and L. Xu, “Analytical Solu- tion for Solute Transport with Depth Dependent Trans- formation or Sorption Coefficient,” Water Recourses Re- search, Vol. 34, No. 11, 1998, pp. 2931-2937. doi:10.1029/98WR02299

[31] N. Kumar and M. Kumar, “Solute Dispersion along Unsteady Groundwater Flow in a Semi-Infinite Aquifer,” Hydrology and Earth System Sciences, Vol. 2, No. 1, 1998, pp. 93-100. doi:10.5194/hess-2-93-1998

[32] Y. Sim and C. V. Chrysikopoulos, “Analytical Solutions for Solute Transport In Saturated Porous Media with Semi-Infinite Or Finite Thickness,” Advances in Water Recourses, Vol. 22, No. 5, 1999, pp. 507-519. doi:10.1016/S0309-1708(98)00027-X

[33] A. Alshawabkeh and N. Rahbar, “A Parametric Study of One Dimensional Solute Transport in Deformable Porous Medium,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 132, No. 8, 2006, pp. 100-1010. doi:10.1061/(ASCE)1090-0241(2006)132:8(1001)

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

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

[36] J. Lee, P. J. Fox and J. J. Lenhart, “Investigation of Consolidation-Induced Solute Transport. I. Effect of Consolidation on solute Transport Parameters,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 9, 2009, pp. 1228-1238. doi:10.1061/(ASCE)GT.1943-5606.0000047

[37] J. Lee and P. J. Fox, “Investigation of Consolidation- Induced Solute Transport. II. Experimental and Numerical Results,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 9, 2009, pp. 1239- 1253. doi:10.1061/(ASCE)GT.1943-5606.0000048

[38] Y. C. Li and P. J. Cleall, “Analytical solutions for Contaminant Diffusion in Double-Layered Porous Media,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 11, 2010, pp. 1542-1554. doi:10.1061/(ASCE)GT.1943-5606.0000365

[1] S. Mohan and M. Muthukumaran, “Modeling of pollutant transport in groundwater,” Journal of Institution of Engineers (India): Environmental Division, Vol. 85, No. 1, 2004, pp. 22-32.

[2] H. D. Sharma and K. R. Reddy, “Geo-environmental Engi- neering,” John Willy & sons Inc., Hoboken, 2004.

[3] R. Rausch, W. Schafer, R. Therrien and C. Wagner, “Solute Transport Modelling: An Introduction to Models and Solu on Strategies,” Gebrüder Borntrager Verlagsbuchhandlung Science Publishers, Berlin, 2005.

[4] M. Thangarajan, “Groundwater: Resource Evaluation, Augmentation, Contamination, Restoration, Modeling and Management,” Capital Publishing Company, New Delhi, 2006.

[5] F. J. Leij, N. Toride and T. V. Genuchten, “Analytical Solution for Non-Equilibrium Solute Transport in Three- Dimension Porous Media,” Journal of Hydrology, Vol. 151, No. 2-4, 1993, pp. 193-228. doi:10.1016/0022-1694(93)90236-3

[6] J. J. Fried, “Groundwater Pollution,” Elsevier Scientific Publishing Company, Amsterdam, 1975.

[7] I. C. Javendel and C. F. Tsang, “Groundwater Transport: Hand book of Mathematics Models,” Water Resource Monogrser, AGU, Washington, Vol. 10, 1984.

[8] J. Bear and A. Verruijt, “Modeling Groundwater Flow and Pollution,” D. Reidel Pubulising Company, Dor- drecht, 1987. doi:10.1007/978-94-009-3379-8

[9] M. P. Anderson and W. W. Woessner, “Applied Ground- water Modeling: Simulation of Flow and Advective Transport,” Academic Press, London, 1992.

[10] S. N. Rai, “Role of Mathematical Modeling in Ground- water Resources Management,” NGRI, Hyderabad, 2004.

[11] N. C. Ghosh and K. D. Sharma, “Groundwater Modelling and Management,” Capital Publishing Company, New- Delhi, 2006

[12] G. S. Kumar, M. Sekher and D. Mishra, “Time-Dependent Dispersivity Behaviour of Non Reactive Solutes in a System of Parallel Structures,” Hydrology and Earth System Sciences, Vol. 3, No. 3, 2006, pp. 895-923. doi:10.5194/hessd-3-895-2006

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

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

[15] R. P. Banks and S. T. Jerasate, “Dispersion in Unsteady Porous Media Flow,” Journal of Hydraulic Division, Vol. 88, No. HY3, 1962, pp. 1-21.

[16] E. H. Ebach and R. White “Mixing of Fluid Flowing through Beds of Packed Solids,” Journal of American Institute of Chemical Engineers, Vol. 4, No. 2, 1958, pp. 161-164. doi:10.1002/aic.690040209

[17] R. R. Rumer, “Longitudinal Dispersion in Steady and Un- Steady Flow,” Journal of Hydraulic Division, Vol. 88, No. HY4, 1962, pp. 147-172.

[18] J. K. Mitchell, “Fundamentals of Soil Behavior,” Wiley, New York, 1976.

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

[20] A. Ogata, R. B. Banks, “A Solution of the Differential Equation of Longitudinal Dispersion in Porous Medium,” US Geological Survey, Washington D.C., 1961.

[21] M. Th. Van Genucheten and W. J. Alves, “Analytical Solutions for Chemical Transport with Simultaneous Adsorption, Zero-Order Production and finite Order Decay,” Journal of Hydrology, Vol. 49, No. 3-4, 1981, pp. 213-233. doi:10.1016/0022-1694(81)90214-6

[22] M. Th. Van Genucheten and W. J. Alves, “Analytical So- lution of One Dimensional Convective-Dispersion Solute Transport Equation,” Technical Bulletin, 1982, pp. 1-51.

[23] N. Kumar, “Dispersion of Pollutants in Semi-Infinite Porous Media with Unsteady Velocity Distribution,” Nordic Hydrology, Vol. 14, No. 3, 1983, pp. 167-178.

[24] 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

[25] D. A. Barry and G. Sposito, “Analytical Solution of a Convection-Dispersion Model with Time-Dependent Trans- port Coefficients,” Water Recourses Research, Vol. 25, No. 12, 1989, pp. 2407-2416. doi:10.1029/WR025i012p02407

[26] H. A. Basha and F. S. El-Habel, “Analytical Solution of One Dimensional Time-Dependent Transport Equation,” Water Recourses Research, Vol. 29, No. 9, 1993, pp. 3209- 3214. doi:10.1029/93WR01038

[27] V. A. Fry, J. D. Istok and R. B. Guenther, “An Analytical Solution of the Solute Transport Equation with Rate- Limited Desorption and Decay,” Water Recourses Research, Vol. 29, No. 9, 1993, pp. 3201-3208. doi:10.1029/93WR01394

[28] C. V. Chrysikopoulos and Y. Sim, “One-Dimensional Virus Transport Homogeneous Porous Media with Time Dependent Distribution Coefficient,” Journal of Hydrology, Vol. 185, No. 1-4, 1996, pp. 199-219. doi:10.1016/0022-1694(95)02990-7

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

[30] M. Flury, Q. J. Wu, L. Wu and L. Xu, “Analytical Solu- tion for Solute Transport with Depth Dependent Trans- formation or Sorption Coefficient,” Water Recourses Re- search, Vol. 34, No. 11, 1998, pp. 2931-2937. doi:10.1029/98WR02299

[31] N. Kumar and M. Kumar, “Solute Dispersion along Unsteady Groundwater Flow in a Semi-Infinite Aquifer,” Hydrology and Earth System Sciences, Vol. 2, No. 1, 1998, pp. 93-100. doi:10.5194/hess-2-93-1998

[32] Y. Sim and C. V. Chrysikopoulos, “Analytical Solutions for Solute Transport In Saturated Porous Media with Semi-Infinite Or Finite Thickness,” Advances in Water Recourses, Vol. 22, No. 5, 1999, pp. 507-519. doi:10.1016/S0309-1708(98)00027-X

[33] A. Alshawabkeh and N. Rahbar, “A Parametric Study of One Dimensional Solute Transport in Deformable Porous Medium,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 132, No. 8, 2006, pp. 100-1010. doi:10.1061/(ASCE)1090-0241(2006)132:8(1001)

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

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

[36] J. Lee, P. J. Fox and J. J. Lenhart, “Investigation of Consolidation-Induced Solute Transport. I. Effect of Consolidation on solute Transport Parameters,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 9, 2009, pp. 1228-1238. doi:10.1061/(ASCE)GT.1943-5606.0000047

[37] J. Lee and P. J. Fox, “Investigation of Consolidation- Induced Solute Transport. II. Experimental and Numerical Results,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 9, 2009, pp. 1239- 1253. doi:10.1061/(ASCE)GT.1943-5606.0000048

[38] Y. C. Li and P. J. Cleall, “Analytical solutions for Contaminant Diffusion in Double-Layered Porous Media,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 11, 2010, pp. 1542-1554. doi:10.1061/(ASCE)GT.1943-5606.0000365