Abstract
An analytical solution to the one-dimensional solute advection-dispersion equation in multi-layer porous media is derived using a generalized integral transform method. The solution was derived under conditions of steady-state flow and arbitrary initial and inlet boundary conditions. The results obtained by this solution agree well with the results obtained by numerically inverting Laplace transform-generated solutions previously published in the literature. The analytical solution presented in this paper provides more flexibility with regard to the inlet conditions. The numerical evaluation of eigenvalues and matrix exponentials required in this solution technique can be accurately and efficiently computed using the sign-count method and eigenvalue evaluation methods commonly available. The illustrative calculations presented herein have shown how an analytical solution can provide insight into contaminant distribution and breakthrough in transport through well defined layered column systems. We also note that the method described here is readily adaptable to two and three-dimensional transport problems.
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Al-Niami, A. N. S.: 1977, Analytical and numerical solutions for miscible two phase flow in porous media, PhD dissertation, University of Birmingham, Birmingham, U.K.
Al-Niami, A. N. S. and Rushton, K. R.: 1979, Dispersion in stratified porous media, Water Resour. Res. 15, 1044-1048.
Bear, J.: 1970, Dynamics of Fluid in Porous Media, Elsevier, New York.
Brenner, H.: 1962, The diffusion model of longitudinal mixing in beds of finite length, numerical values, Chem. Eng. Sci 17, 229-243.
Cotta, R. M.: 1993, Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Inc, Florida.
Dagan, G.: 1988, Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifer, Water Resour. Res. 24, 1491-1500.
Freeze, R. A. and Cherry, J. A.: 1979, Groundwater, Prentice-Hall, Englewood Cliffs, New Jersey.
Gelhar, L. W.: 1986, Stochastic subsurface hydrology — from theory to applications, Water Resour. Res. 22, 135S-145S.
Jacobson, O. H., Leij, F. J. and van Genuchten, M. T.: 1992, Lysimeter study of anion transport during steady flow through layered coarse-textured soil profiles, Soil Sci. 154, 196-205.
Leij, F. J., Dane, J. H. and van Genuchten, M. T.: 1991, Mathematical analysis of one-dimensional solute transport in a layered soil profile, Soil Sci. Soc. Am. J. 55, 944-953.
Leij, F. J. and van Genuchten, M. T.: 1995, Approximate analytical solutions for solute transport in two-layer porous media, Transport in Porous Media 18, 65-85.
Mikhailov, M. D., Ozisik, M. N. and Vulchanov, N. L.: 1983, Diffusion in composite layers with automatic solution of the eigenvalue problem, Int. J. Heat Transfer 26, 1131-1141.
Mikhailov, M. D. and Ozisik, M. N.: 1984, Unified Analysis and Solution of Heat and Mass Diffusion, Wiley, New York.
Mikhailov, M. D. and Vulchanov, N. L.: 1983, A computational procedure for Sturm-Liouville problem, J. Comput. Phys. 50, 323-336.
Moler, C. and van Loan, C.: 1978, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev. 20, 801-837.
Nakayama, S., Takagi, I., Nakai, K. and Higashi, K.: 1984, Migration of radionuclide through two-layered geologic media, J. Nuclear Sci. Technol. 21, 139-147.
Ozisik, M. N.: 1993, Heat Conduction, Johs Wiley & Son, Inc.
Parker, J. C. and van Genuchten, M. T.: 1984, Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resour. Res. 20, 866-872.
Rowe, R. K., Caers, C. J. and Chan, C.: 1993, Evaluation of a compacted till liner test pad constructed over a granular subliner contingency layer, Canad. Geotech. J. 30, 667-689.
Rugh, W. J.: 1996, Linear system theory, Prentice-Hall, Upper Saddle River, New Jersey.
Selim, H. M., Davison, J. M. and Rao, P. S. C.: 1977, Transport of reactive solutes through multilayered soils, Soil Sci. Soc. Am. J. 41, 3-10.
Thoma, G. J., Reible, D. D., Valsaraj, K. T. and Thibodeaux, L. J.: 1993, Efficiency of capping contaminated sediments in situ. 2. Mathematics of diffusion-adsorption in the capping layer, Environ. Sci. Technol. 27, 2412-2419.
van Genuchten, M. T. and Alves, W. J.: 1982, Analytical solutions of the one-dimensional convective-dispersive solute transport equation, USDA Tech. Bull. 1661.
van Genuchten, M. T. and Parker, J. C.: 1984, Boundary conditions for displacement experiments through short laboratory soil, Soil Sci. Soc. Am. J. 48, 703-708.
Wang, X. Q., Thibodeaux, L. J., Valsaraj, K. T. and Reible, D. D.: 1991, Efficiency of capping contaminated bed sediments in situ. 1. Laboratory-scale experiments on diffusion-adsorption in the capping layer, Environ. Sci. Technol. 25, 1578-1584.
Wittrick, W. H. and William, F. W.: 1974a, Buckling and vibrations of anisotropic or isotropic plate assemblies under combined loading, Int. J. Mech. Sci. 16, 209-239.
Wittrick, W. H. and William, F. W.: 1974b, A general algorithm for computing natural frequencies of elastic structure, Quart. J. Mech. Appl. Math. 24, 263-284.
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Liu, C., Ball, W.P. & Ellis, J.H. An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media. Transport in Porous Media 30, 25–43 (1998). https://doi.org/10.1023/A:1006596904771
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DOI: https://doi.org/10.1023/A:1006596904771