2-D radial analytical solutions for solute transport in a dual-porosity medium

This study presents 2-D analytical solutions for advective solute transport within a macropore with simultaneous radial diffusion into an unbounded soil matrix. Solutions for three conditions are derived: (1) an instantaneous release of solute into a macropore, (2) a constant concentration of solute at the top of a macropore, and (3) a pulse release of solute into a macropore. A system of two governing equations was solved by the Laplace transform method for solute concentration as a function of space and time. Substituting the asymptotic approximations of the modified Bessel functions, we also obtained approximate solutions for all three cases. For instantaneous and pulse-type releases of solutes, the solutes initially diffuse into the soil matrix and then reverse direction away from the matrix as they diminish in the macropore. The matrix behaves as a long-term contaminant source creating long tails in the breakthrough curves. Comparisons between the exact and approximate solutions for all three conditions show that the asymptotic approximations are accurate for relatively short periods of solute movement, with increasing error as time and transport distances increase. The analytical solutions were compared with one set of experimental data and also numerical simulations for contaminant transport in a cylindrical dual-porosity medium. The analytical solutions for case 3 represented the experimental data reported in the literature well. Comparisons with numerical simulations in a two-dimensional cylindrical domain that included dispersion in the macropore and advection in the matrix showed that the error caused by neglecting these two processes was minimal when a relatively low permeability matrix was considered for case 2.