## 1. Introduction

[2] Preferential flow paths in field soils are commonly characterized by macropores, fractures, and cracks. These features create distinctiveness in the transport characteristics of solute migration and can be represented by models based on a dual porous medium approach. Diffusion of solutes from high-permeability features to low-permeability zones have been studied extensively. For instance, diffusion of solutes from fissures and fractures into microfissures and matrix is known to significantly retard radionuclide transport within bedrock surrounding a repository [*Neretnieks*, 1980]. Conceptually similar, but at a much larger scale, is the problem of continuing back diffusion of dissolved DNAPL from a clayey lense into an adjacent aquifer. The issue has received considerable attention, and is recognized as a long-term contamination problem in the field [*Ball et al.*, 1997; *Liu and Ball*, 2002; *Chapman and Parker*, 2005].

[3] Many researchers have developed analytical solutions of the advection-dispersion transport equation (ADE) to describe the movement of adsorbing or nonadsorbing solutes into a soil matrix from a fracture or macropore [*Grisak and Pickens*, 1981; *van Genuchten et al.*, 1984; *Wallach and Parlange*, 1998] or into an aquifer from low-permeability layers [*Sale et al.*, 2008]. However, developing similar solutions for cylindrical coordinates can be difficult. Analytical solutions can serve as important verification tools for the numerical algorithms that are required for complicated systems [*Zyvoloski et al.*, 2008] and can be useful to evaluate the results of laboratory experiments with simple geometry, examples of which are presented below.

[4] *van Genuchten et al.* [1984] obtained closed-form analytical solutions of solute migration from a constant concentration source in a cylindrical macropore with radial matrix diffusion. Most of their analytical solutions were for concentration distributions in the macropore and they presented approximate solutions for radial concentration distributions in the matrix using asymptotic expansions of the modified Bessel functions, while neglecting dispersion in the macropore. Their approximate solutions proved to be valid for short time periods.

[5] *Young and Ball* [1998] used a macropore–matrix system approach to investigate the transport of sorbing and nonsorbing solutes within a sand filled macropore surrounded by a low-permeability soil. Pore diffusion coefficients in the matrix were estimated by fitting the measured breakthrough curves to a numerical model for a time-dependent injection of solutes from the top of the column. Their model included the advective-dispersive transport of the solutes in the sand-filled macropore and only diffusion in the matrix. *Rahman et al.* [2004] also studied the sorption kinetics of organic contaminants migrating from a pulse-type injection within a cylindrical macropore to the soil matrix. *Rahman et al.* [2004] applied the analytical solution for a rectangular system given by *Grisak and Pickens* [1981] to estimate sorption parameters by fitting the model to experimental breakthrough curves. *Allaire et al.* [2002] conducted numerical and experimental studies on the effect of initial and boundary conditions applied to a macropore–matrix system. They obtained numerical solutions in Cartesian coordinates by applying constant and pulse-type injections for a system of macropores with regular or irregular geometries. They concluded that the prediction of the breakthrough curves could be simplified as straight macropores without significant error.

[6] Advection of solutes in high-permeability macropores or fractures along with transverse diffusion in a low-permeability matrix was often assumed to be the most dominant transport processes [*Grisak and Pickens*, 1981; *Rasmuson and Neretnieks*, 1981; *Sudicky and Frind*, 1982; *Wallach and Parlange*, 1998; *Rahman et al.*, 2004]. In this study, we present 2-D radial analytical solutions of solute transport within a macropore–matrix system. Our goals were to obtain analytical solutions and appropriate approximate solutions for three boundary conditions: (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; and then to validate the solutions with previously published data. Like *Grisak and Pickens* [1981] and *Rahman et al.* [2004], our solutions assume that solute transport within the macropore is governed by advection only, while solute transport within the matrix is governed by radial diffusion only. Analytical solutions are compared with the experimental data and also numerical simulations, taking into account dispersion in the macropore and advection in the matrix.