One–Dimensional Solute Transport for an Input against the Flow in a Homogeneous Finite Porous Media

Year 2019, Volume: 5 Issue: 2, 69 - 77, 30.06.2019

Raja Ram Yadav , Joy Roy Dilip Kumar Jaiswal

Abstract

A theoretical model comprising advection-dispersion equation with temporal seepage velocity,
dispersion coefficient and time dependent pulse type input of uniform nature
applied against the flow in a finite porous domain. Input concentration is any
continuous smooth function of time acts up to some finite time and then
eliminated. Concentration gradient at other boundary is proportional to
concentration. Dispersion is proportional to seepage velocity. Interpolation method is applied to
reduce the input function into a polynomial. Certain transformations are
utilized to reduce the variable coefficient into constant coefficient in the
advection dispersion equation. The Laplace transform technique is applied to
get the solution of advection dispersion equation. Two different functions of
input are discussed to understand the utility of the present study. Obtained
result is demonstrated graphically with the help of numerical example.

Keywords

Advection, Dispersion, Porous Medium, Interpolation, Laplace Transformation Technique

References

• L.F. Konikow, D.J. Goode, “Apparent dispersion in transient groundwater flow”, Water Resources Research, 1990 , Vol 26, Issue 10,October Pages 2339–2351.
• K. Huang, M.Th. van Genuchten, and R. Zhang, “Exact solutions for one dimensional transport with asymptotic scale-dependent dispersion” Appl. Math. Model., 1996, 20, 298-308.
• S.J. Watson, D.A. Barry, R.J. Schotting, S.M. Hassanizadeh,“Validation of classical density-dependent solute transport theory for stable, high-concentration-gradient brine displacements in coarse and medium sands” Advances in Water Resources, 2002,Volume 25, Issue 6, Pages 611–635.
• V. Batu, “A generalized two-dimensional analytical solution for hydrodynamic dispersion in bounded media with the first-type boundary condition at the source”, Water Resour. Res., 1989, Vol. 25, 1125-1132.
• Verma, A.K., Bhallamudi, S.M., and Eswaran, V.. “Overlapping Control Volume Method for Solute Transport.” J. Hydrol. Eng., 2000 5(3), 308-316.
• M. Jang, J. Lee, J. Choe, and J. M. Kang, “Modeling of solute transport in a single fracture using streamline simulation and experimental validation.” J. Hydrol., 2002. 261, 74-85.
• M. Massabo, R. Cianci, and O. Paladino, “Some analytical solutions for two dimensional convection-dispersion equation in cylindrical geometry.” Environ. Modell. Softw., 2006, Vol.. 21 (5), 681-688.
• D.K. Jaiswal, A. Kumar, N. Kumar, 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, 2009,Vol. 2, 254–263.
• D. K. Jaiswal, A. Kumar, “Analytical solutions of advection-dispersion equation for varying pulse type input point source in one-dimension” , International Journal of Engineering, Science and Technology , 2011,Vol. 3, No. 1, pp. 22-29 .
• R. R. Yadav and L. K. Kumar, (2017) “One-dimensional spatially dependent solute transport in semi-infinite porous media: an analytical solution”, International Journal of Engineering, Science and Technology, Vol. 9, No. 4, pp. 20-27
• M. K. Singh and P. Das 2015. Scale Dependent Solute Dispersion with Linear Isotherm in Heterogeneous Medium. J. Hydrology, Elsevier, Vol. 520, PP.289-299
• V. K. Bharati, A. Sanskrityayn, N. Kumar, “Analytical Solution of ADE with Linear Spatial Dependence of Dispersion Coefficient and Velocity using GITT” Journal of Groundwater Research, 2015,Vol.3, 4/1.
• C. Liu, W. P. Ball and J. HughEllis . “An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media.”, Transp.in Porous Med., 1998, Vol. 30, 25-43.
• N. K. Mahato, S. Begam, P. Das, and M. K. Singh, “Two-dimensional Solute Dispersion Along and Against the Unsteady Groundwater Flow in Aquifer” ,Journal of Groundwater Research, 2015, Vol.3, 4/1.
• A. Sanskrityayn, H. Suk, N. Kumar, “Analytical solutions for solute transport in groundwater and riverine flow using Green’s Function Method and pertinent coordinate transformation method”, Journal of Hydrology, 2017, Volume 547, Pages 517-533.
• K.I.,Hamz, “Numerical Solution of the Two-Dimensional Time-Dependent Transport Equation,Second International Conference on Saltwater Intrusion and Coastal Aquifers— Monitoring, Modeling, and Management” Mérida, México, 2003,March 30-April 2.
• K. Rajsekhar , P. K. Sharma and S. K. Shukla “Numerical modeling of virus transport through unsaturated porous media”, Cogent Geoscience 2016, Vol. 2.
• R.R. Yadav, D. K. Jaiswal and and Gulrana “Analytical Solution for Solute Transport in One- Dimensional Porous Media with a Periodic Boundary Condition” International Journal of Pure and Applied Mathematical Sciences , (2011), Vol. 5, Number 1-2, pp. 1-13
• D.K. Jaiswal, R. R. Yadav, and Gulrana, “Solute-Transport under Fluctuating Groundwater Flow in Homogeneous Finite Porous Domain”, J Hydrogeol Hydrol Eng 2013, 2:1
• J. Bear, “Dynamics of fluids in porous media”, American Elsevier, New York, 1972.
• J. Crank, “The Mathematics of Diffusion” Oxford Univ. Press, London, 2nd ed. ,1975
• D.K., Todd, “Groundwater Hydrology”, John Wiley, 1980, New York, USA.