Mazaheri, M., M. V. Samani, J., M. V. Samani, H. (2013). Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns. Journal of Agricultural Science and Technology, 15(6), 1231-1245.

M. Mazaheri; J. M. V. Samani; H. M. V. Samani. "Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns". Journal of Agricultural Science and Technology, 15, 6, 2013, 1231-1245.

Mazaheri, M., M. V. Samani, J., M. V. Samani, H. (2013). 'Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns', Journal of Agricultural Science and Technology, 15(6), pp. 1231-1245.

Mazaheri, M., M. V. Samani, J., M. V. Samani, H. Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns. Journal of Agricultural Science and Technology, 2013; 15(6): 1231-1245.

Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns

^{1}Department of Water Structures, Faculty of Agriculture, Tarbiat Modares University, Tehran, Islamic Republic of Iran.

^{2}Department of Civil Engineering, Faculty of Agriculture, Shahid Chamran University, Ahvaz, Islamic Republic of Iran.

Receive Date: 23 February 2012,
Revise Date: 28 July 2012,
Accept Date: 20 October 2012

Abstract

Advection-diffusion equation and its related analytical solutions have gained wide applications in different areas. Compared with numerical solutions, the analytical solutions benefit from some advantages. As such, many analytical solutions have been presented for the advection-diffusion equation. The difference between these solutions is mainly in the type of boundary conditions, e.g. time patterns of the sources. Almost all the existing analytical solutions to this equation involve simple boundary conditions. Most practical problems, however, involve complex boundary conditions where it is very difficult and sometimes impossible to find the corresponding analytical solutions. In this research, first, an analytical solution of advection-diffusion equation was initially derived for a point source with a linear pulse time pattern involving constant-parameters condition (constant velocity and diffusion coefficient). Hence, using the superposition principle, the derived solution can be extended for an arbitrary time pattern involving several point sources. The given analytical solution was verified using four hypothetical test problems for a stream. Three of these test problems have analytical solutions given by previous researchers while the last one involves a complicated case of several point sources, which can only be numerically solved. The results show that the proposed analytical solution can provide an accurate estimation of the concentration; hence it is suitable for other such applications, as verifying the transport codes. Moreover, it can be applied in applications that involve optimization process where estimation of the solution in a finite number of points (e.g. as an objective function) is required. The limitations of the proposed solution are that it is valid only for constant-parameters condition, and is not computationally efficient for problems involving either a high temporal or a high spatial resolution.

1. Abramowitz, M., Stegun, I.A., 1970. Handbook of Mathematical Functions, first ed. Dover Publications Inc., New York. 2. Al-Niami, A.N.S., Rushton, K.R., 1977. Analysis of flow against dispersion in porous media. Journal of Hydrology 33, 87-97. 3. Aral, M.M., Liao, B., 1996. Analytical solutions for two-dimensional transport equation with time-dependent dispersion coefficients. Journal of Hydrologic Engineering 1, 20-32. 4. Banks, R.B., Ali, J., 1964. Dispersion and adsorption in porous media flow. Journal of Hydraulic Division 90, 13-31. 5. Barros, F.P.J., Mills, W.B., Cotta, R.M., 2006. Integral transform solution of a two-dimensional model for contaminant dispersion in rivers and channels with spatially variable coefficients. Environmental Modelling & Software 21, 699-709. 6. Bear, J., 1972. Dynamics of Fluids in Porous Media, first ed. American Elsevier Pub. Co., New York. 7. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, second ed. Oxford University press, Oxford. 8. Chongxuan, L., Szecsody, J.E., Zachara, J.M., Ball, W.P., 2000. Use of the generalized integral transform method for solving equations of solute transport in porous media. Advances in Water Resources 23, 483-492. 9. Cotta, R.M., 1993. Integral Transforms in Computational Heat and Fluid Flow. CRC Press, Boca Raton, FL. 10. Courant, D., Hilbert, D., 1953. Methods of Mathematical Physics, first ed. Wiley Interscience Publications, New York. 11. Fischer, H.B., List, J.E., Robert, K.C., Imberger, J., Brooks, N.H., 1979. Mixing in Inland and Coastal Water, first ed. Academic Press, New York. 12. Govindaraju, R.S., Bhabani S.D., 2007. Moment Analysis for Subsurface Hydrologic Applications, first ed. Springer, Netherlands. 13. Guerrero, P.J.S., Pimentel, L.C.G., Skaggs, T.H., van Genuchten, M.T., 2009. Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique. International Journal of Heat and Mass Transfer 52, 3297-3304. 14. Guerrero, P.J.S., Skaggs, T.H., 2010. Analytical solution for one-dimensional advection-dispersion transport equation with distance-dependent coefficients. Journal of Hydrology 390, 57-65. 15. Guvanasen, V., Volker, R.E., 1983. Experimental investigations of unconfined aquifer pollution from recharge basins. Water Resources Research 19, 707-717. 16. Harleman, D.R.F., Rumer, R.R., 1963. Longitudinal and lateral dispersion in an isotropic porous medium. Journal of Fluid Mechanics 16, 385-394. 17. Jaiswal, D.K., Kumar, A., Kumar, N., Yadav, R.R., 2009. 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 2, 254-263. 18. Jury, W.A., Spencer, W.F., Farmer, W.J., 1983. Behavior assessment model for trace organics in soil: I. Model description. Journal of Environmental Quality 12, 558-564. 19. KazezyIlmaz-Alhan, C.M., 2008. Analytical solutions for contaminant transport in streams. Journal of Hydrology 348, 524-534. 20. Kumar, A., Jaiswal, D.K., Kumar, N., 2010. Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. Journal of Hydrology 380, 330-337. 21. Kumar, A., Jaiswal, D.K., Kumar, N., 2009. Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain. Journal of Earth System Sciences 118, 539-549. 22. Lai, S.H., Jurinak, J.J., 1971. Numerical approximation of Cation exchange in miscible displacement through soil columns. Soil Science Society of America 35, 894-899. 23. Leij, F.J., Skaggs, T.H., van Genuchten, M.T., 1991. Analytical solutions for solute transport in three-dimensional semi-infinite porous media. Water Resources Research 27, 2719-2733. 24. Leij, F.J., van Genuchten, M.T., 2000. Analytical modeling of onaqueous phase liquid dissolution with Green's functions. Transport in Porous Media 38, 141-166. 25. Marino, M.A., 1974. Distribution of contaminants in porous media flow. Water Resources Research 10, 1013-1018. 26. Marshal, T.J., Holmes, J.W., Rose, C.W., 1996. Soil Physics, third ed. Cambridge University Press, Cambridge. 27. Morse, P.M., Feshbach, H., 1953. Methods of Theoretical Physics, first ed. McGraw-Hill, New York. 28. Ogata, A., 1970. Theory of dispersion in granular media. USGS Professional Paper 411-I, 34. 29. Ogata, A., Banks, R.B., 1961. A solution of the differential equation of longitudinal dispersion in porous media. USGS Professional Paper 411-A, 1-9. 30. Ozisik, M.N., 1980. Heat Conduction, first ed. Wiley, New York. 31. Polyanin, A.D., 2002. Handbook of Linear Partial Differential Equations for Engineers and Scientists, first ed. Chapman & Hall/CRC Press, Boca Raton. 32. Quezada, C.R., Clement, T.P., Lee, K., 2004. Generalized solution to multi-dimensional multi-species transport equations coupled with a first-order reaction network involving distinct retardation factors. Advances in Water Resources 27, 507-520. 33. Smedt, D.F., 2007. Analytical solution and analysis of solute transport in rivers affected by diffusive transfer in the hyporheic zone. Journal of Hydrology 339, 29-38. 34. Sneddon, I.N., 1972. The Use of Integral Transforms, first ed. McGraw-Hill, New York. 35. van Genuchten, M.T., 1981. Analytical solutions for chemical transport with simultaneous adsorption, zero-order production and first-order decay. Journal of Hydrology 49, 213-233. 36. van Genuchten, M.T., Alves, W.J., 1982. Analytical solutions of the one-dimensional convective-dispersive solute transport equation. US Department of Agriculture, Technical Bulletin, 1661. 37. Williams, G.P., Tomasko, D., 2008. Analytical Solution to the Advective-Dispersive Equation with a Decaying Source and Contaminant. Journal of Hydrologic Engineering 13, 1193-1196. 38. Zwillinger, D., 1998. Handbook of Differential Equations, third ed. Academic Press, San Diego.