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.
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 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].
 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.
van Genuchten et al.  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.
Young and Ball  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.  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.  applied the analytical solution for a rectangular system given by Grisak and Pickens  to estimate sorption parameters by fitting the model to experimental breakthrough curves. Allaire et al.  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.
 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  and Rahman et al. , 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.
2.1. Governing Equations
 We define the geometry as a vertical, cylindrical macropore surrounded by a semi-infinite soil matrix (Figure 1). Solute within the macropore is distributed uniformly within any cross section and transports vertically at a steady velocity equal to the mean macropore velocity. Assuming only radial diffusion within the matrix and no dispersion within the cylindrical macropore, the ADE provides the following set of equations [Rahman et al., 2004; van Genuchten et al., 1984]:
Equation (1) defines the average solute concentration in the macropore, and equation (2) defines the solute migration in the matrix, where C (M L−3) is solute concentration, (L3L−3) is the volumetric water content, Da is the matrix diffusion coefficient (), rm is the macropore radius (L), r is the radial distance from the center of the macropore (L), t is elapsed time (T), v is the macropore mean velocity (), z is the vertical distance (L), R (dimensionless) is the retardation factor , where Kd is the distribution coefficient for sorption and is the density) [Rahman et al., 2004], and the subscripts m and a denote the macropore and the matrix, respectively.
2.2. Case 1: Instantaneous Release of Solutes
 The initial and boundary conditions for an instantaneous release of solutes are given as
where M is the mass (M) of the contaminants released instantaneously into the cross-sectional area (Am) of the macropore (L2), and is the Dirac delta function (L−1). Taking the Laplace transform of the system above, we obtain
 Likewise, the boundary conditions are transformed to
where K0() and K1() are the modified Bessel functions of the second kind of orders zero and one, respectively. Laplace inversions for equations (8) and (9) are defined by
Equation (10) are evaluated by using the method of contour integration. Equations (8) and (9) have a branch point at the origin (i.e., they are analytical except at s = 0 where they are multivalued). The path of integration can be chosen using the contour given by Carslaw and Jaeger [1959, Figure 40] or by Ozisik [1980, Figure 7-8]. The contour is constructed by drawing a large circle enclosing singularities with a branch cut along the negative real axis and a small circle of radius around the origin so that equations (8) and (9) are single-valued within the contour. Ozisik [1980, equation (7-73)] presented a contour integration formula that can be conveniently applied for this case. Since equations (8) and (9) do not have isolated poles in the s plane, the contour integration formula given by [Ozisik, 1980] becomes
 The Laplace inversion of equations (25) and (26) are obtained by use of equation (11) following the same procedures as in case 1, but in this case the second integral on the right hand side of equation (11) is equal to 1. The general solution for the concentration distribution in the macropore is given by
van Genuchten et al. [1984, equation (92)] provided a solution similar to equation (27) for the concentration distribution in the macropore, however, the form of equation (27) is different. van Genuchten et al. [1984, equation (92)] used Kelvin functions instead of the Bessel functions. We used Mathcad ® to evaluate all numerical integrations. The general solution for the concentration distribution in the matrix is given by
 For short time periods, the approximate solutions for t > Rm(z−z0)/vm, that is,
 The initial and boundary conditions for the pulse-type release of solutes are the same as case 2 except that the macropore boundary condition at z = 0 is replaced with
 The Laplace transform of equation (31) gives . The general solutions for this case were obtained by the same methods given in case 1 and case 2. The general solution for the concentration distribution in the macropore can be written in terms of the solutions in case 2,
where the subscripts 2 denote the case 2 solution. Likewise, the solution for the matrix region is given as
 Applying the contour integration formula (equation (11)) and using the approximations for the Bessel functions (equation (18)), we obtain the approximate solution for the concentration in the macropore as
Equations (34) and (35) are similar to an exact solution developed by Rahman et al. [2004, equations (16) and (18)] for analyses of solute transport in a rectangular macropore surrounded with a matrix. The difference between our equations (34) and (35) and Rahman et al.'s  solution is the exponential terms in front of the erfc functions. The result between the two solutions differs by less than 10% when the argument of the exponential term is less than −0.1. In other words, the difference between the two solutions becomes larger with increasing transport distances, decreasing macropore velocities, or decreasing water content in the matrix.
 Similarly, by use of equation (18), the approximate solution for the matrix is given by
3.1. Comparisons of Exact and Approximate Solutions
 The analytical and approximate solutions for the three cases were calculated for a variety of scenarios. We plotted the resultant output in dimensionless forms to provide a better sense of the potential applicability of the various solutions. Equations (1) and (2) can be rewritten in terms of dimensionless (*) variables by substituting , and , which leads to
where and the equations are defined at 0 < t* < ∞, 1 < r* < ∞, 0 < z* < ∞.
Figure 2a shows the variation of concentration for an instantaneously released unit of solute mass into a macropore (case 1). The solute concentration in the macropore (Cm) is largest near the injection point (z* = 0.1) and decreases as z* increases. Additionally, as z* increases, the duration of t* with significant solute increases. Since there is no dispersion in the modeled macropore, this spreading of the solute pulse through time is because of diffusion into, and then out of, the soil matrix as the concentration gradient increases toward and then reverses direction away from the matrix (back-diffusion phenomenon). Figure 2b shows that as t* increases, the maximum Ca decreases and the solute is spread across a larger range of r*. As t* become increasingly great, the Ca gradient back toward the macropore continually decreases, effectively locking solutes into the matrix for the foreseeable future. Errors for the approximate solution as compared to the exact solution are minimal through the range plotted in Figure 2, and are smallest at small t*, z*, and r*.
 Demonstrations of the case 2 solutions (constant concentration source) for the normalized solute concentration (Cm/C0) are given in Figures 3a and 3b. All approximate solutions underestimated the solute concentrations with increasing error as t* and z* increased. On the basis of these results, the case 2 approximate solutions for both the macropore and the matrix can be applied for t* ≤ 1 with an error of less than 5%.
 Simulations of the case 3 (pulse-type) solutions are presented in Figures 4a and 4b. Cm/C0 increases sharply when z* ≈ t* and then peaks at approximately 2z* (Figure 4a). As z* increases, the length of the Cm/C0 tail increases. As in the case 1 solution, following the transport of solute through the macropore past the z* of interest, solutes from the matrix began to diffuse back into the macropore, creating long tails. These results are also consistent with field observations of long-term dissolved non-aqueous phase liquid (NAPL) plume persistence in aquifers (analogous to macropores) overlying clay-silt layers (analogous to soil matrix) [Chapman and Parker, 2005]. Although the case 3 approximate solutions appear to be in good agreement with the exact solutions for the dimensionless pulse time chosen (t0* = 0.1) (Figures 4a and 4b), the errors grow with increasing t0* as shown in Figure 5.
 The effect of on the concentration at the macropore wall (r* = 1) is demonstrated for cases 2 and 3 in Figure 6, which shows that the concentration decreases at the pore wall with increasing for the constant concentration condition (case 2). For case 3 (a pulse of solute) the peak concentration decreases, and the normalized time to peak increases, with increasing (Figure 6b). The differences between exact and approximate solutions for case 2 increase with increasing dimensionless time.
3.2. Comparison of the Analytical Solutions to Experimental Data
 The case 3 analytical solution (pulse-type injection) was compared to the experimental data from Rahman et al.  to test the analytical model. Rahman et al.  conducted macropore column experiments with four different organic compounds in a cylindrical column (length = 12.1 cm and radius = 1.05 cm) that involved a sand-filled macropore (rm = 0.422 cm) surrounded with a silty loam matrix in the annulus. Measured TCE breakthrough curves from the column were fitted to an analytical solution by Rahman et al.  to estimate the retardation factor and the matrix diffusion coefficient. We used the same parameters given by Rahman et al.  to compute our analytical solutions. Figure 7 shows the comparison of the Rahman et al.  experimental data with the analytical models presented in this study. Both of the two analytical models appear to represent the data equally well. The average increased error of using Rahman et al.'s  solution instead of the case 3 solution is less than 8% for the case given in Figure 7. However, the difference between the two solutions becomes more pronounced as the ratios and (rm/r) increase. For instance, if the column length chosen by Rahman et al.  were 52 cm instead of 12.1 cm, the predicted error would increase to 36%, and similarly, estimation of pore-diffusion coefficient and sorption parameters from the experimental data by fitting Rahman et al.'s analytical solution would also involve larger errors.
3.3. Comparisons With Numerical Simulations
 Numerical simulations were also conducted for a water-filled macropore and soil matrix system that included dispersion in the macropore, and flow and diffusion in radial and vertical directions in the matrix. We used the parameter values given in van Genuchten et al.  based on experimental studies in an artificially constructed 40.4 cm long laboratory soil column (radius = 1.18 cm) with a cylindrical macropore of radius 0.73 mm, where , vm = 0.49 cm/s, and Da is equal to 3.7 × 10−6cm2/s. The goal of these simulations was to evaluate the amount of error created by our analytical solutions because of their negligence of dispersion in the macropore, and vertical and horizontal flow in the matrix.
 We simulated the flow in the macropore–matrix system by applying the modified Navier–Stokes (NS) equation. Using the single-domain approach [Goyeau et al., 2003], where the composite region is considered as a continuum and represented with the following modified NS equation,
where u is the velocity, and is the viscosity. In the macropore, is equal to 1, , and the permeability, k, is infinity and is zero to remove the Darcy term. In the matrix, is equal to porosity, k is equal to the permeability of the matrix, and is 1. For the matrix, all the terms are formally kept, but the Darcy term is dominant. This formulation implicitly imposes continuity of the stresses and velocities at the interface. The two-domain approach requires explicit assignment of the fluid/porous medium stress condition (slip-boundary condition) [Brinkman, 1947; Beavers and Joseph, 1967; Berkowitz, 1989]. The assumption of leads to good agreements between the numerical and experimental results [Neale and Nader, 1974; Song and Viskanta, 1994], and thus we made the same assumption in our numerical simulations. The top boundary in the macropore was set to laminar inflow with a mean velocity of 0.49 cm/s, and the bottom boundary in the macropore was set to vanishing viscous stress and constant pressure equivalent to the height of the water-filled column. In the matrix region, all the boundaries except for the macropore–matrix boundary (continuity of stress) were set as a no-slip boundary condition. The flow was assumed to be at steady state such that the time derivative term on the left-hand side is zero.
 Transport of a conservative contaminant in the macropore–matrix system can also be formulated by using the single-domain approach:
where D is the diffusion coefficient. Equations (40) and (41) are coupled by the fact that the computed steady state velocity distribution from the modified NS equation was used in equation (41) to compute the concentration C. A constant concentration boundary was assumed at the top of the macropore, and at the bottom, the convective flow boundary was set (with diffusive flux equal to zero). In the matrix, all boundaries except for the interface with the macropore were set as no-flux conditions. At the macropore–matrix interface, continuity of the flux was assigned, and the initial concentration distribution was set as zero.
Equation (40) and (41) in 2-D axisymmetrical cylindrical domains were solved using the COMSOL Multiphysics software that is based on the Galerkin finite element method. Equations (40) and (41) were formed in COMSOL by modifying the Incompressible Navier–Stokes and Convection-Diffusion application tools of the Chemical Engineering Module. The domain was discretized using 53,803 triangular elements consisting of 28,031 mesh points. The mesh sizes ranged from 0.24 mm to 0.73 mm and, typically, smaller elements were chosen close to the interface. UMPACK direct matrix solver built into COMSOL was used to solve the models implicitly.
 The accuracy of the numerical methodology was validated by creating an equivalent problem to case 2 where the assumptions of the analytical solution are valid (i.e., advection in the macropore with steady uniform velocity and radial diffusion in the matrix only) (Figure 8a). The numerical solution based on the Galerkin Method becomes unstable when the cell Péclet number, Pe = uh/(2D), where h is the mesh size, is greater than 1 [Zienkiewicz et al., 2005]. The critical mesh size for the stability criterion should be approximately 1.5 nm for transport in the macropore problem stated above. Since this is computationally too expensive, we instead inserted artificial diffusion by using the isotropic diffusion. Using the isotropic diffusion in COMSOL is equivalent to adding a term to the actual diffusion coefficient where is a tuning parameter. A value of 0.1 for was found to result in sufficiently smooth and accurate results. We also tested consistent artificial diffusion methods that do not perturb the original transport equation such as the streamline upwind Petrov–Galerkin and crosswind diffusion built into COMSOL, however, we were unsuccessful at removing the oscillations. The same equivalent problem was also solved numerically by adding dispersion to the flow direction for investigating dispersion (Dm) in the macropore. When dispersion is considered, the solution was stable and there was no need to insert artificial diffusion. Figure 8b shows the effect of different dispersion coefficients to the breakthrough curve. The value of Dm = 7.2 cm2/s was calculated based on the Taylor dispersion formulation, D(1 + 1/48*a2vm2/D2), and the dispersion value of 0.49 cm2/s (numerically equal to mean macropore velocity) was used by van Genuchten et al. . Numerical simulations with dispersion in the macropore were also verified with analytical solutions. van Genuchten et al.  presented analytical solutions for macropore concentration while assuming an infinite macropore length. The effect of finite macropore length becomes significant as the dispersion coefficient increases. The analytical solutions obtained in the Laplace domain for finite macropore length were numerically inverted into the real-time domain using the Stehfest method [Stehfest, 1970] and were then compared with the numerical simulation results. While the error using the unbounded macropore analytical solution instead of the bounded macropore solution for Dm = 0.49 cm2/s and macropore length = 40.4 cm is less than 2%, the solutions diverge for Dm = 7.2 cm2/s.
Figure 8b also shows the breakthrough curve obtained by simulating the modified NS and ADE in both the macropore and the matrix. The dispersion in the macropore occurs naturally because of variation of velocity and concentration. A small permeability of 10−15 m2 was chosen to create nearly immobile matrix conditions. The effect of dispersion in the macropore for this case appears to be small except at early times (<2 min) where the differences could be as much as 30%. The effect of a finite soil matrix in the numerical simulation does not appear to be reflected in the breakthrough curves obtained from the macropore, since the concentrations in the matrix distant from the macropore are small compared to the large concentration in the macropore and its vicinity. This was investigated before by van Genuchten et al. , who found that the boundary effects become more pronounced in the solute concentration of the matrix at longer time periods. Figure 9 shows the effect of matrix permeability on the breakthrough curve obtained by sampling from the end of the macropore. As the permeability of the porous medium surrounding the macropore increases, increasing advective flow results in more mass transfer into the matrix, and this in turn decreases the effluent concentration at the macropore.
4. Concluding Remarks
 Analytical solutions were presented for advective solute transport in a macropore with radial diffusion into an unbounded soil matrix. These analytical solutions were obtained in integral forms for instantaneous, fixed, and pulse-type boundary conditions. We also presented approximate solutions for all three cases. Results suggest that the approximate solutions are sufficient for relatively short solute transport periods. If a priori estimates of key parameters (time, porosities, retardation factors, and problem geometry) are available, the relative error associated with the use of the approximate solution can be gleaned from the figures provided. The analytical solution for case 3 was verified with the experimental data. Quantitative comparisons of the analytical solutions for cyclindrical and planar macropores shows that geometry can have a significant impact on breakthrough curves as the transport distances along and away from the macropore increase. Comparisons of the analytical solutions with the numerical simulations show that neglecting dispersion in the water-filled macropore and flow in the matrix are useful assumptions for studying macropore–matrix transport processes when the matrix is a low-permeability medium. When the permeability of the surrounding soil matrix is less than 10−11 m2, the error using the analytical solutions in case 2 for the conditions tested is less than 3%, except at very early times when the analytical solutions significantly overpredict the concentration. For a coarse soil matrix with a permeability of 10−10 m2, the error increases to as much as 27%.