## 2-D ANALYTICAL SOLUTION OF SOLUTE TRANSPORT WITH DECAY-TYPE INPUT SOURCE ALONG GROUNDWATER

#### Authors:

Arun Dubey, Dilip Kumar Jaiswal, Gulrana, A. K. Thakur#### DOI NO:

https://doi.org/10.26782/jmcms.2024.04.00001#### Abstract:

The stabilization of groundwater resources in excellent quality is crucial for both the environment and human societies. To examine the contaminant concentration pattern of infinite and semi-infinite aquifers, mathematical models provide accurate descriptions. The two-dimensional model for a semi-infinite heterogeneous porous medium with temporally dependent and space-dependent (degenerate form) dispersion coefficients for longitudinal and transverse directions is derived in this study. The Laplace Integral Transform Techniques (LITT) is used to find analytical solutions. The dispersion coefficient is considered the square of the velocity which represents the seasonal variation of the year in coastal/tropical regions. To demonstrate the solutions, the findings are presented graphically. Figures are drawn for different times for a function and discussed in the result and discussion section. It is also concluded that a two-dimensional model is more useful than a one-dimensional model for assessing aquifer contamination.#### Keywords:

2-D Advection-dispersion equation,Aquifer,Heterogeneity,Pollution,Laplace transform,#### Refference:

I. A. Kumar, D. K. Jaiswal, N. Kumar,: ‘Analytical solutions to one dimensional advection-diffusion equation with variable coefficients in semi—infinite media’. J Hydrol., 380:330–337.(2010)

II. A. Sanskrityayn, N. Kumar,: ‘Analytical solution of ADE with temporal coefficients for continuous source in infinite and semi-infinite media’J. Hydrol. Eng. 23 (3). 06017008.(2018). 10.1061/(ASCE)HE.1943-5584.0001599.

III. C. L. Carnahan, and J S Remer,: ‘Non-equilibrium and equilibrium sorption with a linear sorption isotherm during mass transport through porous medium: some analytical solutions’ J Hydrol.73:227–258.(1984)

IV. C. K. Thakur, M. Chaudhary, van der Zee S.E.A.T.M., and M. K.Singh, : ‘Two-dimensional solute transport with exponential initial concentration distribution and varying flow velocity’, Pollution, 5(4), 721-737. (2019) 10.22059/poll.2019.275005.574

V. D. K. Jaiswal, A. Kumar and N. Kumar, : ‘Discussion on ‘Analytical solutions for advection-dispersion equations with time-dependent coefficients by Baoqing Deng, Fie Long, and Jing Gao.’ J. Hydrol. Eng. 25 (8): 07020012 (1-2).(2020).

VI. D. K. Jaiswal, A. Kumar, N. Kumar and M. K. Singh, : ‘Solute transport along temporally and spatially dependent flows through horizontal semi-infinite media: dispersion being proportional to square of velocity.’ J Hydrol. Eng. 16(3) : 228–238. (2011)

VII. 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.’ J Hydro-environ Res 2: 254-263.(2009).

VIII. D. K. Jaiswal and Gulrana, ; ‘Study of Specially and Temporally Dependent Adsorption Coefficient in Heterogeneous Porous Medium.’ Appli and Applied Mathe: An Int Journal (AMM) 14 (1):485-496.(2019).

IX. D. K. Jaiswal, N. Kumar and R. R. Yadav, : ‘Analytical solution for transport of pollutant from time-dependent locations along groundwater.’ J. Hydro., 610.(2022).

X. J. Crank, : ‘The mathematics of diffusion.’ Oxford University Press, UK. (1975).

XI. J. Crank, (1956). ‘The Mathematics of Diffusion.’ Oxford University Press Inc.: New York; 414.

XII. J. D. Logan, V. Zlotnik, : ‘The convection–diffusion equation with periodic boundary conditions.’ Applied Mathematics Letters 8(3): 55–61.(1995).

XIII. K. Inouchi, Y. Kishi, T. Kakinuma : ‘The motion of coastal groundwater in response to the tide.’ Journal of Hydrology 115: 165–191_8.(1990).

XIV. L.H. Baetsle:‘Migration of radionuclides in Porous media. In Progress in Nuclear energy.’Series XII, Health Physics (ed). A.M.F. Duhamel Pergmon Press: Elmsford, New York; 707–730.(1969).

XV. M. Th. Van Genuchten and W. J. Alves,:‘Analytical solutions of the one-dimensional convective-dispersive solute transport equation.’ USDA ARS Technical Bulletin Number 1661, U.S. Salinity Laboratory.(1982)

XVI. M. Chaudhary, M K Singh,:‘Study of multispecies convection-dispersion transport equation with variable parameters.’J. Hydrol. 591. DOI: 10.1016/j.jhydrol.2020.125562.(2020).

XVII. M. K. Singh, N. K. Mahato, and P. Singh, : ‘Longitudinal dispersion with time dependent source concentration in semi-infinite aquifer.’ J. Earth System Sci 117(6):945-949.(2008).

XVIII. O. Güven, F. J. Molz, J. G. Melville, : ‘An Analysis of Dispersion in a Stratified Aquifer.’ Water Resources Research 20 (10): 1337–135,(1984).

XIX. D. K. Jaiswal, A. Dubey, V. Singh and P. Singh, : ‘Temporally Dependent Solute Transport in One-Dimensional Porous Medium: Analytical and Fuzzy Form Solutions.’ Mathematics in Engineering Science and Aerospace, 14(3), 711-719.(2023).

XX. P. Singh, P. Kumari and D. K Jaiswal,:‘An Analytical model with off diagonal impact on Solute Transport in Two-dimensional Homogeneous Porous Media with Dirichlet and Cauchy type boundary conditions.’GANITA, Vol.72(1), 299-309.(2022).

XXI. P. Singh, S. K. Yadav and N. Kumar, : ‘One-Dimensional Pollutant’s Advective-Diffusive Transport from a Varying Pulse-Type Point Source through a Medium of Linear Heterogeneity.’ J. Hydrol. Eng, 17(9): 1047–1052. (2012).

XXII. P. Singh, S. K. Yadav, O. V. Perig, : ‘Two-dimensional solute transport from a varying pulse type point source Modelling and simulation of diffusive processes.’ 211-232, Springer.(2014).

XXIII. R. A. Freeze, and J. A. Cherry, : ‘Groundwater. Prentice-Hall, New Jersey.(1979)

XXIV. R. Kumar, A. Chatterjee, M. K.Singhand V. P. Singh, : ‘Study of solute dispersion with source/sink impact in semi-infinite porous medium.’ Pollution, 6(1),87-98,(2020). 10.22059/poll.2019.286098.656

XXV. R. R. Rumer, : ’Longitudinal dispersion in steady and unsteady flow.’ J Hydraul. Div. 88:147–173.(1962).

XXVI. R. R. Yadav and L. Kumar, : ‘Solute Transport for Pulse Type Input Point Source along Temporally and Spatially Dependent Flow.’ Pollution, 5(1): 53-70.(2019).

XXVII. R. R. Yadav, D. K. Jaiswal and Gulrana, : ‘Two-Dimensional Solute Transport for Periodic Flow in Isotropic Porous Media: An Analytical Solution.’ Hydrol Process. 26 (12):3425-3433.(2011). DOI: 10.1002/hyp.8398.

XXVIII. R. R. Yadav, D. K. Jaiswal, H. K. Yadav and Gulrana, : ‘Analytical solutions for temporally dependent dispersion through homogeneous porous media.’ Int. J. Hydrology Science and Technology, Vol. 2, No. 1, pp.101–115.(2012).

XXIX. S. E. Serrano, : ‘Hydrologic theory of dispersion in heterogeneous aquifers.’ J Hydrol. Eng. 1:144–151.(1996).

XXX. U. Y. Shamir, D.R.F Harleman, :‘Dispersion in layered porous media.’ J Hydraul. Div. 95:237–260.(1967)

XXXI. Y. Sun, A. S. Jayaraman, and G. S. Chirikjian, : ‘Lie group solutions of advection-diffusion equations.’ Phys. Fluids 33, 046604 (2021); 10.1063/5.0048467.

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