Diffusive mass transfer into and out of intragranular micropores (“intragranular diffusion”) plays an important role in the transport of some groundwater contaminants. We are interested in understanding the combined effect of pore-scale advection and intragranular diffusion on solute transport at the effective porous medium scale. We have developed a 3-D pore-scale numerical model of fluid flow and solute transport that incorporates diffusion into and out of intragranular pore spaces. A series of numerical experiments allow us to draw comparisons between macroscopic measures computed from the pore-scale simulations (such as breakthrough curves) and those predicted by multirate mass transfer formulations that assume complete local mixing at the pore scale. In this paper we present results for two model systems, one with randomly packed uniform spherical grains and a second with randomly packed spheres drawn from a binary grain size distribution. Non-Fickian behavior was observed at all scales considered, and most cases were better represented by a multirate mass transfer model even when there was no distinct secondary porosity (i.e., no intragranular diffusion). This suggests that pore-scale diffusive mass transfer processes between preferential flow paths and relatively immobile zones within the primary porosity may have significant impact on transport, particular in low-concentration tails. The application of independently determined mass transfer rate parameters based on an assumption of well-mixed concentrations at the pore scale tends to overestimate the amount of mass transfer that occurs in heterogeneous pore geometries in which preferential flow leads to incomplete pore-scale lateral mixing.
 In heterogeneous groundwater aquifers, transported solutes commonly experience a mixture of relatively fast-moving (advection-dominated or mobile) and relatively slow-moving (diffusion-dominated or immobile) regions. In saturated groundwater systems, exchange of solute mass between mobile and relatively immobile states (mass transfer) can result from a number of different processes including (1) chemical processes such as surface complexation that lead to intermittent immobilization on solid surfaces [Brusseau et al., 1991]; (2) slow advective processes that drive solute through low-permeability regions [Guswa and Freyberg, 2000]; and (3) diffusion into low-permeability regions in porous media [Zheng and Gorelick, 2003], into the porous matrix in fractured systems [Haggerty et al., 2001; Neretnieks, 1980], or into pore space within grains or grain aggregates [Ball and Roberts, 1991; Ewing et al., 2010; Wood et al., 1990]. In cases where rates of mass exchange between mobile and immobile states are slow relative to groundwater flow rates, this results in nonequilibrium conditions that are manifested as strongly asymmetric breakthrough curves with enhanced late tails relative to those predicted by standard (Fickian) dispersion models. It has also been shown that local diffusive mass transfer leads to an apparent scale dependence of desorption rates of sorbing contaminants because of implicit incorporation of the time required to diffuse to and from the reactive surface into the sorption/desorption rate [Liu et al., 2008]. These processes can significantly lengthen the time required for aquifer restoration (either by active remediation or monitored natural attenuation), because the removal of dissolved contaminants trapped in immobile regions is limited by the relatively slow mass transfer rate.
 It is therefore important to properly account for these processes in numerical simulations of contaminant transport and remediation. The kinetically controlled exchange of mass between mobile and immobile states can be mathematically modeled by adding a source-sink term to the classical advection-dispersion equation (ADE). Early modeling studies [Coats and Smith, 1964; VanGenuchten and Wierenga, 1976] utilized a first-order single-rate mass transfer model; subsequent works expanded the approach to incorporate a distribution of (multiple) first-order mass transfer rates [Carrera et al., 1998; Haggerty and Gorelick, 1995], referred to here as the multirate model or MRM. Related mathematical formulations that are under certain conditions equivalent to the MRM include nonlocal formalisms [Cushman et al., 1994], fractional ADEs [Benson et al., 2000; Schumer et al., 2003], continuous-time random walks [Dentz and Berkowitz, 2003], correlated random walks [Scheibe and Cole, 1994], and memory function models [Ginn, 2000, 2009].
 In this paper we focus on diffusive mass transfer of nonreacting solutes at the pore scale, with exchanges of solute occurring between advecting intergranular pore spaces (i.e., between solid grains) and nonadvecting intragranular pore spaces (i.e., within solid grains) as illustrated in Figure 1. We refer to this process here as “intragranular diffusion” (IGD). Multirate mass transfer, including diffusive mass transfer into intragranular pore spaces, has been identified as a significant control on the rate of radionuclide transport at the U.S. Department of Energy (DOE) Hanford Site in southeastern Washington State. While there have been observations made and model applications at grain, column, and field scales [Greskowiak et al., 2010; Liu et al., 2006, 2008, 2009; Ma et al., 2010], we currently lack the ability to reliably predict macroscopic parameters (i.e., mass transfer rate distributions in a MRM) from microscopic (grain-scale) information and must typically instead estimate parameters through a fitting exercise. Recent advances in pore-scale simulation provide us the capability to directly simulate fundamental processes (in this case, intragranular diffusion) at the microscale and to thereby observe directly the relationship between microscale properties and macroscopic behavior of the models. Gouze et al.  point out that the multiple rate formulation introduces many parameters into the model, and that it is not clear a priori whether these parameters have physical meaning or simply provide extra degrees of freedom in fitting observations. They utilized a pore-scale approach based on X-ray tomographic observations to characterize the memory function of rock matrix in a fractured rock system and showed that the memory function thus determined is consistent with that fitted to field observations, indicating that there is a direct physical relationship between pore-scale geometry and effective mass transfer rates at larger scales. In a comparable manner, we present here 3-D pore-scale simulations of flow and solute transport in granular porous media, incorporating intragranular diffusion, in computer-generated synthetic pore geometries. The results of the pore-scale simulations are analyzed in terms of 1-D effective (macroscopic) models of transport with multirate mass transfer. In this manner we aim to identify the critical controls on macroscopic parameter distributions and link them directly to measurable grain-scale (microscopic) properties. We focus, in particular, on the effects of incomplete mixing at the pore scale on the apparent distribution of mass transfer rates at the Darcy scale.
 Granular porous media are typically conceptualized as comprising solid grains of varying size and shape packed in specific configurations with the intervening pore spaces filled with some fluid (e.g., water, air, and oil). However, the solid grains often themselves contain microporous structures such as microfractures, weathering rinds, and/or porous grain coatings or may actually be aggregates of multiple smaller grains with intervening micropore spaces (see Figure 1). This pore space within solid grains, referred to here as intragranular porosity (IGP), if connected with the surface of the grains, can serve as a secondary (immobile) porous domain with which solute from the primary (mobile) porous domain can exchange by diffusion.
 The simplest model of intragranular pore space topology assumes that solid grains are an internally homogeneous porous medium with an internal porosity (θIG) and an effective internal diffusivity (dIG). This assumption has been widely used in a number of different applications involving intragranular diffusion, including packed bed reactors in chemical engineering, chromatographic separation, soil science, atmospheric science, and transport in porous media [Haggerty and Gorelick, 1995; Rounds and Pankow, 1990; Wu and Gschwend, 1986]. For particles of spherical shape, this leads to a radial diffusion model that is amenable to mathematical analysis. We recognize that the actual topology of intragranular pore space is likely to deviate from this assumption and to significantly impact the character of grain-scale mass transfer [Basagaoglu et al., 2004; Ewing et al., 2010]. However, for simplicity and consistency with a wide range of previous studies, we adopt the radial diffusion assumption in the initial work reported here. Because the pore-scale simulation framework utilized here can be easily modified to incorporate nonuniform IGD, we expect that future expansions of this work will explore the effects of other forms of IGD on effective mass transfer rate distributions.
 Our theoretical starting point is the work of Haggerty and Gorelick , who applied the MRM to intragranular diffusion and sorption, and demonstrated mathematical equivalence between the MRM with specific rate distributions and the radial diffusion model. Following their notation and neglecting the source-sink term and sorption (neither of which are included in the simulations reported here), the mobile-immobile model can be written as:
where cm and cim are the solute concentrations in the mobile and immobile domains [units M L−3], respectively, D is the hydrodynamic dispersion tensor [L2T−1], and v is the average pore water velocity vector in the mobile domain [L T−1]. L() is used here to denote a general advection-dispersion operator, β is the dimensionless “capacity ratio,” representing the ratio of mass in the immobile domain to mass in the mobile domain at equilibrium. In the absence of sorption, the equilibrium state is equal concentration, so the capacity ratio is simply equal to the ratio of immobile porosity (θim) to mobile porosity (θm):
 The single-rate linear mass transfer model utilizes a mass transfer term of the form:
where α is the mass transfer rate coefficient [T−1]. Extending this approach to incorporate mass transfer into multiple (N) immobile domains, each with a different first-order rate coefficient (e.g., the MRM) results in
in which (cim)j,αj, and βj are the concentration, the linear mass transfer rate coefficient, and the capacity ratio of the jth immobile domain, respectively. The sum of capacity ratios of the individual immobile domains must equal the total capacity ratio:
Haggerty and Gorelick  demonstrated equivalence between the radial diffusion model into uniform spheres and the MRM above for the case where the rate coefficients and capacity ratios are given by the following formulae [Haggerty and Gorelick, 1995, Table 1]:
where a is the radius of the sphere and dm is the effective molecular diffusion coefficient in the intragranular pore space. In practice, this infinite series of parameters must be truncated. For 1-D advection-dispersion-reaction systems, Haggerty and Gorelick  recommend using the first N terms of the series with N chosen such that the mass transfer Damkohler number meets the following criterion:
where Lx is the length scale of the 1-D domain, and v is the magnitude of the average pore water velocity. For terms with j ≥ N, the mass transfer is fast enough that equilibrium is a good approximation and these terms can be neglected. Since the sum of individual capacity ratios must equal the total capacity ratio, βN should be increased as necessary to meet this condition (that is, βN should incorporate all the truncated βj, j > N). For details of methods for selecting the last term of the αj and βj series, see Haggerty and Gorelick  and Table 2 of Haggerty .
Table 1. Parameters Used in Pore-Scale Simulations, Short Domain (Figure 4)a
Other than the length of the domain, the parameters used for the longer domain (Figure 5) are very similar and are not shown here. Units for the third column are in cgs (centimeters, grams, seconds) units, and the fourth column shows the equivalent values used in the SPH simulations (in SPH model units).
Case 1: Uniform Grain Size
Intergranular (mobile) porosity
2.833 × 10−3
Average pore velocity
7.092 × 10−3
Case 2: Mixture of Two Grain Sizes
Intergranular (mobile) porosity
1.507 × 10−3
Average pore velocity
4.302 × 10−3
Parameters Applicable to Both Cases
SPH weighting function radius
Number of SPH particles
3.44 × 10−3
Width/depth of domain
Height of domain
SPH particle mass
2.37 × 10−9
SPH repulsive force
Solute diffusivity in water
1.0 × 10−4
3.7 × 10−4
Effective intragranular solute diffusivity
1.0 × 10−5
3.7 × 10−5
Effective intragranular (immobile) porosity
0.0 or 0.1
0.0 or 0.1
Table 2. Fitted Longitudinal Dispersivity Values as a Function of Transport Distance for Case 3A
Transport Distance (Model Units)
Dispersivity α (Model Units)
 For mixtures of spheres with variable radial dimension (grain size), the mass transfer rate series for each grain size can simply be combined, with the capacity ratios βj of a particular grain size scaled by the volume fraction (or mass fraction if all grains are assumed to be of equal density) of that grain size. For example, an equal-volume mixture of two grain sizes, each represented by the first 25 terms of the spherical rate series, would result in a total of 50 αj and 50 βj with each of the original βj values multiplied by a factor of 0.5 (50%) such that the total capacity ratio equals that of the overall mixture.
 Since the MRM for spheres as defined earlier is equivalent to the radial diffusion model, one would expect that the macroscopic outcomes (e.g., breakthrough curves) of a pore-scale model in which diffusion into spherical grains is explicitly simulated should be well represented by an effective MRM with the above rate series. However, an assumption implicit to the rate series model described earlier is that the solute is well mixed in the intergrain (mobile) pore space. That is, each point on the surface of every grain at a given transport distance experiences the same concentration exposure history, represented by the macroscopic average concentration. In fact, pore-scale variations in advective velocity in the primary pore space lead to local variations in solute concentration and in some cases, may give rise to strong preferential flow and transport pathways at the pore scale. Depending on the ratio of advective and diffusive time scales (e.g., the Peclet number (Pe)), pore-scale mixing may be more or less complete, and this may be expected to have an effect on the apparent mass transfer rate distribution, perhaps leading to discrepancies between the pore-scale simulation and the predicted macroscopic behavior. Here we perform comparisons between the outcomes of pore-scale simulators and effective 1-D MRMs to identify the magnitude of discrepancies caused by incomplete pore-scale mixing and the conditions under which such discrepancies occur.
3.1. Pore-Scale Simulation Approach
 Pore-scale simulations were conducted in 3-D domains with explicit specification of pore geometry. Processes simulated were single-phase (water) flow, solute advection, molecular diffusion in the bulk water phase in the pore space ΩP, and molecular diffusion in water within the solid phase ΩS (intragranular diffusion). The fluid flow in ΩP is governed by the combination of the Navier-Stokes (NS) and continuity equations. The fluid in ΩS is assumed to be immobile. The diffusion of solute in is described by the advection-diffusion equation with space-dependent diffusion coefficient d (x) and porosity ε (x):
where the total derivative is defined as
 ( for ). The porosity and diffusion coefficient in the advection-diffusion equation is defined as
where εeff is the effective porosity of the soil grains, dm is the molecular diffusion coefficient in bulk water, and deff is the effective diffusion coefficient in the soil grains. A similar diffusion model was used previously in hybrid pore-scale/Darcy-scale models for diffusion-reaction [Tartakovsky et al., 2008] and advection-diffusion [Ryan and Tartakovsky, 2011] systems. Ramirez et al.  provide further theoretical background on dispersion in systems where the velocity and diffusion coefficient vary discontinuously and demonstrate that the probabilistic motion of solute molecules in such systems converges to Brownian motion with zero drift and an effective dispersion coefficient.
 The numerical method used is smoothed particle hydrodynamics (SPH) [Tartakovsky et al., 2007a, 2007b]. SPH is a Lagrangian mesh-free method that discretizes the fluid domain using movable nodes and the solid domain using immobile nodes. Each node (referred to as a “particle”) has associated mass , solute mass mi, location ri, velocity vi, diffusion coefficient di, and porosity εi (for fluid particles εi is set to unity). For fluid particles, is the mass of solution carried by the particles. For the solid particles, without loss of generality we assume that the density of the solid phase is the same as the density of the solution and the mass of the solution contained in the solid particles is . This assumption does not affect the simulation results because the solid particles do not move and the solution of the governing equations does not depend on the density of the solid phase.
 Here we use a standard SPH discretization of the Navier-Stokes equation [Tartakovsky et al., 2007a, 2007b]:
where , , is the fluid pressure, µ is the fluid viscosity, is the particle number density, is the fluid density, and is any external force (per unit mass) acting on particle i (e.g., gravity). W is the SPH Gaussian-like smoothing function with compact support h [Tartakovsky et al., 2007a, 2007b]. The SPH NS equation is written for the SPH fluid particles, the velocities of the solid particles are constant and set to zero, and the summations in the SPH NS equation are over all fluid and solid particles. To close the system of SPH equations we use the equation of state, , and the equation for the particle density:
 The SPH discretization of advection-diffusion equations is usually written in terms of the mass fraction of the solute . The mass fraction Xi does not change due to the advection of SPH particles, and the numerical dispersion can be avoided completely. The SPH discretization of the advection-diffusion equation is given by
 The details of the derivation of the advection-diffusion equation in media with variable porosity can be found in Ryan and Tartakovsky  and Tartakovsky et al. . The concentration of the solute (mass of the solute per unit volume of solution) is related to the mass fraction as
where is the volume of particle i.
 For simplicity, in the SPH NS equation we assume that the solute concentration is small and the total mass of the solution carried by particle i is assumed to be constant and equal to . Extension to concentration-dependent flow is described by Tartakovsky and Meakin .
 Since 3-D pore-scale simulation is computationally demanding, the SPH method has been implemented in a modular parallel code that is scalable to thousands of cores on massively parallel computers as described by Palmer et al. .
3.2. Pore-Scale Intragranular Diffusion Model Verification
 The implementation of intragranular diffusion in the SPH code is a new extension of the simulator beyond that reported by Palmer et al. . Here we present results of verification tests of the diffusive mass transfer process as implemented in the code, which demonstrate its ability to accurately represent the pore-scale intragranular diffusion process.
 The intragranular diffusion model developed for this study was verified against a one-dimensional continuum diffusion model solved using standard finite volume techniques. The model consists of a periodic system of length L broken up into three regions as shown schematically in Figure 2. The intervals [−L/2,−L/4] and [L/4,L/2] are characterized by a diffusion coefficient DS while the center interval [−L/4,L/4] is characterized by a diffusion coefficient DL. The coefficients DS and DL are representative of diffusion in the intragranular and liquid regions, respectively. The initial concentration of the diffusing species is set to a constant value in the “solid” regions at either end of the system and to zero in the “liquid” region in the center. The corresponding SPH calculation consisted of a rectangular 3-D simulation that is elongated along the z axis. The system is divided up into three segments along the z axis consisting of two solid caps at either end and a section of liquid particles in the middle. The proportions are chosen to match the continuum model. Again, the initial concentration of the diffusing species is chosen so that the concentration in the solid is uniform and the concentration in the liquid is zero. The SPH simulations are uniform in the xy plane, so the systems simulated are effectively 1-D. The body force term in these simulations (gravity) is set to zero, so there is no fluid flow.
 Because the diffusion constant is discontinuous in the continuum model, special care was required in formulating the finite volume equations for the grid points near the discontinuity. This discontinuity is assumed to occur halfway between adjacent, evenly spaced cell centers. A schematic of the grid in the region of the discontinuity is shown in Figure 2. The indexing shown in Figure 2 is used in the following discussion. To correctly evaluate the fluxes near the discontinuity in the diffusion coefficient, which occurs at the cell face between the concentration values c−1 and c1 in Figure 2, we assume that there is a concentration “c0” defined on this face. The diffusive fluxes on either side of the dividing surface can be defined to a first-order approximation as
where Δx is the grid spacing. In order to conserve mass, we must require that the fluxes on either side of the surface of discontinuity be equal (f− = f+). This allows us to eliminate c0 entirely and results in the following expressions for the fluxes
 The remaining fluxes can be approximated using the standard formula
 The subscript i + 1/2 indicates the face between centers i and i + 1. The finite volume equations for the time rate of change of the concentrations are then
 Using this formula, the finite volume equations for the cells on either side of the discontinuity are
 The remaining concentrations obey the usual finite volume expressions. The time-dependent equations are solved using a standard, fully implicit first-order Euler approach.
 To compare the SPH and finite volume solution an SPH simulation consisting of a box 10 × 10 × 40 in size was constructed (SPH model units are defined in terms of the SPH weighting function). The first and last 10 units in the z direction represent solid, and the remaining particles in the center are liquid. The density of SPH particles is 27 particles per unit volume. The initial concentration in the solid is set to 1. The diffusion coefficients in the solid and liquid have values of 0.005 and 0.1, respectively. For these simulations the effect of porosity was ignored (it essentially modifies the effective value of the diffusion coefficient). The same dimensions and diffusion coefficients were used in the finite volume simulations. Both simulations were run for 500 time units. A comparison is shown in Figure 3. The two methods give virtually identical results.
3.3. Pore-Scale Model Domain and Parameters
 Simulations described here utilized pore-scale domains with dimensions of approximately 1 mm in each of the two directions transverse to flow. In the direction parallel to flow, we ran two cases, one with a length of approximately 2 mm and a second with a length of approximately 4 mm. Running these two different cases provided an opportunity to explore both differences in behavior between two different realizations of the synthetic pore geometries and the character of transport behavior at a range of transport distances. Pore geometries were specified either by computer generation of synthetic random packings of spherical grains with specified size distributions (Figure 4). We considered two cases: (1) a monodisperse (uniform) grain size packing (grain diameter of 200 µm) and (2) a binary mixture of two uniform grain sizes (grain diameters of 280 and 120 µm). The synthetic packings were generated by a zero-temperature Monte Carlo algorithm [Allen and Tildesley, 1989]. The algorithm starts with the desired number of spherical grains placed on a regular lattice in a large box. The box dimensions are selected so that the relative proportions of the box match the desired proportions of the final configuration (e.g., if the box is twice as long in the z direction as the x and y directions, then the proportions of the box should be 1:1:2). The initial grain spacing is relatively large so that the particles can move about freely at the start of the simulation. The simulation consists of two parts, repeated multiple times. The first part serves to randomize the configuration of grains. Each of the grains is moved successively by a small, random increment. After each grain is moved, the configuration is checked to see if the grain overlaps with another grain. If there is an overlap, the move is rejected and the grain is returned to its old position, and if there is no overlap, the move is accepted. This is repeated for all grains. These moves serve to randomize the configuration of grains. Each pass over the complete set of grains is referred to as a sweep. After 10 sweeps, the second part of the algorithm is to try and reduce the volume by a small random increment. A fractional volume decrement is generated randomly and used to reduce the system volume. The cube root of the ratio of the new volume to the old volume can be used to scale all grain locations. The new configuration is then checked to see if any grains overlap. If there are any overlaps, then the new configuration and system size are rejected and the system is reset to the old values, otherwise the new coordinates and system dimensions are used (the simulation is zero-temperature because the volume adjustments always decrease the system size, they never increase it). After every volume move, the porosity of the system is calculated. When it falls below a certain threshold, the simulation is terminated and the final configuration is written out. Since simulations of uniform sized spheres can result in regular packing if the cutoff porosity is set too low (e.g., around 0.4), the simulations were terminated at a higher target porosity (e.g., 0.5) and then the size of the spheres was increased until the final target porosity was closely matched. This resulted in some overlap between spheres but prevented them from “freezing” into a relatively ordered lattice structure. The target porosity for the monodisperse case was 0.400 [Woronow, 1986]. The target porosity for the binary mixture was 0.35 based on Equation (13) of [Mota et al., 2001], which relates the porosity of a binary mixture to the volume fraction of large particles and the size ratio of the two particle sizes. In this case, we consider spheres with a size ratio of 0.43 and a mixture with 12.5% by number larger spheres and 87.5% by number smaller spheres. This mixture corresponds to a volume fraction of large spheres of 0.65, which in turn approximately corresponds to the optimal packing (minimum porosity) for this size ratio [see Mota et al., 2001, Figure 1]. The binary mixture naturally has a lower porosity than the uniform spheres because the smaller spheres are able to fit within the pore spaces between the larger spheres. The uniform and binary mixture geometries for the shorter domain are shown in Figure 4; those for the longer domain are shown in Figure 5. In our simulations, advective flow was allowed only in the open pores, and diffusion was allowed both within the bulk water phase and within the solid grains.
 The SPH simulations discretized the pore-scale domains shown in Figure 4 with approximately 750,000 particles each, and those in Figure 5 with approximately 1.5 million particles each. The head gradient driving flow (specified in SPH as a body force) was specified such that the resulting Reynolds number was close to but less than unity (still well within the Stokes flow regime but with high enough velocity to allow simulation of full breakthrough curves in a reasonable amount of computational time). The Reynolds number for this system is defined as where ρ is the fluid density, v is the average pore velocity, l is the characteristic length (taken as the mean grain size), and µ is the fluid dynamic viscosity. The bulk water diffusion coefficient was specified so as to obtain a Peclet number where dm is the molecular diffusion coefficient between 10 and 100, indicating an advection-dominated system and consistent with experimental observations in sandpack [e.g., Bijeljic et al., 2011, Figure 4]. Published values of measured intragranular porosity of geologic materials range from as low as 1% [Ball et al., 1990; Hay et al., 2011] to as large as 17% [Wood et al., 1990; Wu and Gschwend, 1986], and intragranular porosity of grain aggregates in soils can be much larger [Nkedi-Kizza et al., 1982; VanGenuchten and Wierenga, 1977]. For this study, we assumed an intragranular porosity of 10% (0.1) for the solid grains. Reported values of effective diffusivity in intragranular pore spaces range widely, from 2 to 3 orders of magnitude smaller than bulk diffusivity [Ball and Roberts, 1991; Hay et al., 2011] to a factor of 4–10 [Jenck, 1991; Liu et al., 2011]. Here we assume effective intragranular diffusivity to be a factor of 10 smaller than bulk diffusivity in water. A summary of parameters used in the pore-scale simulations is given in Table 1.
3.4. Darcy-Scale Simulation
 For both pore-scale and macroscale simulations, steady flow was assumed (for the pore-scale simulations, for which the simulation method is inherently transient, a runup period was simulated until a quasi steady flow rate was achieved prior to solute injection), and the initial solute concentration everywhere in the domain was set to zero. A pulse of solute of specified time length and normalized concentration of unity was then injected at the inflow face of the domain. The results of the pore-scale simulations were compared to an equivalent 1-D Darcy-scale or macroscopic representation. For each pore-scale simulation, breakthrough curves (flux-weighted average concentration as a function of time) were computed at multiple planes perpendicular to the mean flow direction, at distances of 15, 30, and 45 model units (for the shorter system with total length 48 units), and additionally at 60, 75, and 90 units (for the longer system with total length 96 units). The breakthrough curves serve as the primary means of comparing pore-scale and Darcy-scale (macroscale) simulations. Whereas at the pore scale, diffusion processes were explicitly simulated, at the macroscale they were represented using an effective MRM. The parameters of the model (αj and βj, j = 1, 2, …, N) were specified using the theory of Haggerty and Gorelick  as described earlier based on an assumption of spherical immobile domains. Macroscopic simulations were performed using the public-domain code STAMMT-L [Haggerty, 2009], which simulates 1-D advective-dispersive transport with single-rate or multirate mass transfer. The mean advective velocity was fixed to match that computed directly from the pore-scale simulations, and the longitudinal dispersivity was specified by first performing a pore-scale simulation with no intragranular diffusion and fitting the resulting breakthrough curve with a 1-D advection-dispersion model (no mass transfer). It should be noted that the downstream (exit) boundary condition in STAMMT-L (concentration gradient of zero) was applied at a distance much larger than the actual domain length (we used 1000 units based on a series of comparative simulation tests), in order to accurately represent the periodic nature of the 3-D SPH simulations. The resulting dispersivity was then used for subsequent simulations incorporating multirate mass transfer. Comparison of estimated dispersivities at the three breakthrough planes provides insight as to whether the travel distance is sufficient to achieve asymptotic dispersion behavior. Comparison of breakthrough curves computed using the MRM and theoretically based parameters with those computed from direct pore-scale simulation provides insight regarding the effects of incomplete mixing on mass transfer.
4.1. Simulation Set 1: Short Domain
 The first set of simulation results presented is for the shorter simulation domains shown in Figure 4. After an initial simulation period to spin up the flow model to steady state, a conservative solute pulse of dimensionless concentration of unity and length of 300 model time units was introduced at the inflow (bottom) boundary. A finite pulse allows us to study both early arrival and late tails of the breakthrough curve; the length of the pulse was selected by trial-and-error to provide sufficient resolution of the breakthrough curve while minimizing computational requirements. Solute breakthrough curves were output from the SPH simulation at distances of x = 15, 30, and 45 SPH model units (total system length 48 SPH units). The results are presented first for the cases with no intragranular diffusion (no IGD), as these are used to independently estimate longitudinal dispersivity parameters for use in the subsequent multirate mass transfer (MRM) simulations (cases with intragranular diffusion).
4.1.1. Case 1A: Uniform Grain Size, Short Domain, No IGD
 Figure 6 presents visualizations of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) at two selected model times, for this case. Breakthrough curves computed from the pore-scale simulations are bimodal at short distances, apparently indicative of a strong preferential flow path (visually evident in Figure 6), which becomes subdued by x = 45 (symbols in Figure 7). Notwithstanding this, all three breakthrough curves can be approximated by the advection-dispersion equation (ADE) with velocity v =7.091e−3 and dispersivity α = 4.5 (solid curves in Figure 7). This result (no apparent scale dependence of dispersivity) suggests that the asymptotic value of longitudinal dispersivity has already been achieved by the distance of 15 model units (0.6 mm) in this case. We note that the time evolution of dispersivity is related to that of the second spatial moment of concentration; strictly speaking, dispersivity estimates from ADE fits may not capture actual changes of dispersivity with time. When viewed on a logarithmic scale (Figure 7b), it can be seen that the pore-scale simulations exhibit enhanced tailing relative to the fitted ADE models, particularly at the shorter distances. This behavior is indicative of non-Fickian transport at shorter distances, despite the fact that the dispersivity appears to have converged to a stable value. Of course, the clear bimodal structure of the breakthrough curves is also reflective of non-Fickian transport (persistent preferential flow) at small scales, and the heavy tails observed may be attributed partly to diffusive mass transfer between the preferential flow path (high advective velocity) and relatively low-advection or even immobile regions within the primary pore space. This behavior was characteristic of many of the simulated cases and will be explored further in the ensuing sections. Note that here and throughout this paper, breakthrough curves are presented in terms of nondimensional time, normalized by the characteristic diffusion time where l is the characteristic length scale (mean grain size) and dm is the molecular diffusion coefficient.
4.1.2. Case 1B: Uniform Grain Size, Short Domain With IGD
 Figure 8 presents visualizations of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) at two selected model times, for this case. Although there is little visual distinction at the early time (t = 2500) as compared to the case without IGD (Figure 6), the differences at the later time (t = 8000) are quite evident, as there remains low concentrations of solute that have diffused into the outer portions of the solid grains near the inlet end (bottom). These trapped solutes give rise to the reduced peak concentrations and heavy tails observed in the breakthrough curves as discussed later.
 For analysis of the breakthrough curves, we focus on the x = 45 distance, for which the transport behavior in the no-IGD case was most Fickian in character. Figure 9 shows a comparison of the 3-D pore-scale simulation results with IGD against the 1-D MRM with mass transfer rate parameters specified based on the spherical diffusion model of Haggerty and Gorelick  and dispersivity based on Case 1A. Also shown for reference are the no-IGD 3-D simulations and ADE fit from Case 1A. We note the following observations from these plots: (1) the theoretical MRM provides a good match to the enhanced tailing of the case with IGD (logarithmic axis, Figure 9b); and (2) the MRM predicts a peak concentration that is slightly too low compared to the 3-D pore-scale results (arithmetic axis, Figure 9a).
4.1.3. Case 2A: Binary Grain Size Mixture, Short Domain, No IGD
 Figure 10 presents visualizations of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) at two selected model times, for this case. In this case, there is no strong evidence for preferential flow in the breakthrough curves (no bimodal character as observed in Case 1A), and the solute distributions shown in Figure 10 also visually exhibit more uniform plug-like solute movement (compare to Figure 6). Manual fits of the ADE to the pore-scale breakthrough curves are shown in Figure 11. The fitted dispersivity at x = 15 (α = 1.9) is smaller than that at x = 30 and 45 (α = 3.0 and α = 2.7, respectively), indicating that despite the unimodal breakthrough it takes longer to achieve asymptotic dispersion in the binary mixture system than in the uniform case. We also note that the ADE fit at x = 45 is poorer than in the uniform case, and in particular, the peak arrival is delayed relative to the pore-scale simulation. The plot in logarithmic axes again illustrates heavy tails suggestive of diffusive mass transfer, particularly at shorter distances, despite the fact that no IGD is included in these simulations.
 In this case, because of the poorer fit of the ADE at x = 45 and the strong tailing effects observed, we explored further the representation of the breakthrough using an MRM. Since the pore-scale simulation in this case does not include intragranular diffusion (IGD), the application of an MRM to this case is conceptualized as diffusive mass transfer between advection-dominated (high-velocity) and diffusion-dominated (low-velocity) regions of the primary pore space. This conceptualization is supported by the solute distribution shown in Figure 10b (later time), in which small pockets of solute trapped in the primary pore space can be observed near the inlet end of the domain, lagging far behind the main solute pulse. Figure 12 shows the simulated pore-scale breakthrough curves fitted by both the ADE (the same fit as shown in Figure 11) and an MRM assuming a small immobile porosity of 0.04 with an effective radius of 4 SPH units. This reduces the mobile pore space to 0.3103 and therefore increases the average pore velocity (given a fixed Darcy flux) to 4.857 × 10−3. The increase in velocity shifts the MRM breakthrough curve earlier in time, and the corresponding increase in peak concentration is offset by the diffusive mass transfer and a slight increase in dispersivity (to 2.5), thus providing a much closer match to the 3-D simulation results. The MRM also provides a slightly improved fit to the tail of the breakthrough curve as compared to the ADE, although the differences are small in this case (logarithmic plot not shown).
4.1.4. Case 2B: Binary Grain Size Mixture, Short Domain With IGD
 Figure 13 presents visualizations of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) at two selected model times, for this case. Figure 14 shows a comparison of the macroscopic 1-D MRM (using dispersivities from Case 2A and mass transfer rate parameters based on the spherical diffusion model of Haggerty and Gorelick ) with the pore-scale 3-D simulation results. The MRM prediction matches best at the intermediate distance (x = 30) but overpredicts the amount of mass transfer (peak breakthrough concentration is too low) at short and long distances. However, it is difficult to assess whether the observed discrepancies (and for that matter the good match at x = 30) are associated primarily with non-ADE behavior observed in Case 2A or with incorrect representation of the IGD mass transfer process.
 To attempt to separate out these effects, we consider in more detail the x = 45 breakthrough curve. As described earlier in Case 2A, we previously fitted the no-IGD case effectively with an MRM reflecting mass transfer within the primary pore space. To simulate this case, we can combine three mass transfer rate series, two representing the spherical diffusion into the two different grain sizes (IGD) and the third taken from the fit achieved in Case 2A (non-IGD mass transfer). By combining all three rate series, we achieve an improved fit to the pore-scale breakthrough curve timing as shown in Figure 15. However, we note that this combined MRM still underestimates the peak concentration of the breakthrough, or differently stated overestimates the amount of mass transfer. As will be discussed further later, this suggests that incomplete solute mixing at the pore scale limits the rate of mass transfer relative to the model of Haggerty and Gorelick , which assumes complete pore-scale mixing (all solid grains experience the same macroscopic concentration history).
4.2. Simulation Set 2: Long Domain
 In the first set of simulations presented earlier we observed non-Fickian attributes of the solute transport breakthrough predicted by the 3-D pore-scale simulations, even at the longest transport distances. In this section we present the same analyses performed on pore domains that are twice as long in the direction of transport (Figure 5). The geometries are statistically the same as those in the first set, but we generated new realizations of the pore geometry for this case. This provides us two interesting opportunities: (1) to explore whether Fickian behavior is achieved at longer distances and (2) to examine differences between alternative realizations of the same statistical pore structure. The simulation conditions and parameters were the same as in the previously presented simulations. In addition to the distances of x = 15, 30, and 45 model units presented for the first case, solute breakthrough curves were also output from the pore-scale simulations and analyzed at distances of x = 60, 75, and 90 units (total system length 96 units). The various cases (with and without IGD, uniform, and binary grain size distributions) are presented in the same order as in the previous section.
 Because of the different realization of pore geometry used, some of the model parameters differed slightly from those listed in Table 1. The porosity of the uniform grain size pore domains was 0.3997 rather than 0.3995. The same body force was applied in all cases, but the resulting fluxes were slightly different. For the uniform grain size case, the nondimensional Darcy flux was 2.952 × 10−3 (average pore velocity 7.386 × 10−3). For the binary mixture case, the Darcy flux was 1.313 × 10−3 (average pore velocity 3.748 × 10−3). Given the comparable values for the shorter case (see Table 1), the effective permeability of the longer domains is 4% larger (uniform grain size) and 13% smaller (binary mixture of grains), reflective of a small level of variability between different realizations of the same statistical pore geometry.
4.2.1. Case 3A: Uniform Grain Size, Long Domain, No IGD
 Figure 16a presents a visualization of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) for a single selected model time, for this case. In contrast to the shorter realization, this realization of the uniform distribution does not exhibit bimodal breakthrough behavior at any distance. As shown in Figure 17, the ADE model provides a reasonable fit to all the breakthrough curves. The apparent dispersivity (fitted) ranges from 1.5 at the shortest distance to 3.5 at the longest distance but the actual values waver inconsistently. Fitted dispersivities at all distances are given in Table 2 (compare to fitted α = 4.5 at x = 15, 30, and 45 for the shorter realization). Despite the generally good match to the ADE, there are some systematic discrepancies, specifically a slightly delayed peak arrival in the ADE model and heavier tailing in the 3-D pore-scale simulation results, again suggesting that a slightly improved fit could be achieved by adding a mass transfer mechanism. The heavy tails (indicative of non-Fickian behavior) are evident even at the longest transport distance, as indicated in the logarithmic plot of the breakthrough curves at x = 90 (Figure 17b).
4.2.2. Case 3B: Uniform Grain Size, Long Domain With IGD
 Figure 16b presents a visualization of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) for a single selected model time, for this case. Here we focus on analysis of the x = 90 breakthrough curve, for which the corresponding non-IGD transport was best represented by the ADE. Figure 18 shows a comparison of the 3-D pore-scale IGD simulation results against the MRM with mass transfer rate parameters specified based on the spherical diffusion model of Haggerty and Gorelick  and dispersivity based on Case 3A. Also shown for reference are the no-IGD simulation results and fitted ADE model. As observed in the previous cases, the MRM overestimates the amount of mass transfer (the peak is a little too low and delayed), suggesting impacts of incomplete mixing in the pore-scale simulation. The logarithmic plot provides evidence that the tailing behavior is captured well by the MRM.
4.2.3. Case 4A: Binary Grain Mixture, Long Domain, No IGD
 Figure 19a presents a visualization of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) for a single selected model time, for this case. Unlike any of the prior cases, the ADE with the specified velocity is completely unable to match the observed breakthrough curves at shorter distances (Figure 20); the peaks of the ADE breakthrough curves lag far behind those of the pore-scale simulations. Although this system does not exhibit bimodal breakthrough, these results are suggestive of very strong preferential flow with an advective porosity [Tartakovsky and Neuman, 2008] much smaller than the actual porosity. The fitted dispersivities increase monotonically from 0.5 at x = 15 to 5.5 at x = 90, and the match between the pore-scale simulation results and the ADE improves with distance until at the longest transport distance of x = 90 the ADE fit appears to be quite good.
 We examine more closely the breakthrough at x = 45. Similar to the Case 2A analysis, we introduce an MRM to represent the effects of an immobile porosity of 0.105, conceptualized as diffusion-dominated (low-advection) regions of the primary pore space. Again we emphasize that this case has no IGD, so any mass transfer processes must occur within the primary pore space. The assignment of nonadvective pore space reduces the mobile pore space by the same amount (to 0.2453) and therefore increases the pore velocity (given the same Darcy flux) to 5.353 × 10−3. Using a dispersivity of 1.9, a much closer match to the SPH simulated breakthrough curve is obtained, both in the peak and in the tails, as shown in Figure 21. Closer examination of the x = 90 breakthrough curve (Figure 22) shows that while the fit of the ADE model is by far the best at this distance, there remains a slight lag in peak arrival. However, little or no enhanced tailing is observed in this case for the latest time simulated (although it may become evident if longer simulation times were performed).
4.2.4. Case 2B: Binary Grain Mixture, Long Domain With IGD
 Figure 19b presents a visualization of the simulated pore-scale tracer distribution (with local velocity vectors superimposed) for a single selected model time, for this case. Since the only breakthrough curve for which the no-IGD case was matched reasonably well by the ADE was at the distance x = 90, we focus on that distance in this analysis. Applying the MRM with two grain sizes, we obtain the macroscopic model prediction shown in Figure 23, compared with both the pore-scale simulation results and the ADE fitted to the non-IGD case.
 Consistent with the other cases studied, the MRM predicts a peak concentration that is too low in comparison to the pore-scale simulation results, indicating that the mass transfer is too large in the MRM which we attribute to the effect of incomplete mixing at the pore scale. The misfit in the timing of the peaks between the MRM and pore-scale simulations is a residual effect of the corresponding delay in the ADE peak as shown in Figure 22 and is attributed to mass transfer between advection-dominated and stagnant regions within the primary pore space.
 For all pore geometries and transport distances considered, evidence was observed of varying degrees of non-Fickian behavior in the cases with no IGD, in particular, earlier peak arrival and heavier tails of the pore-scale breakthrough curves relative to the 1-D ADE. While these effects were expected for the case with IGD, observation of these behaviors in simulations with no IGD was somewhat surprising. In some cases, these effects were quite pronounced (bimodal breakthrough, much earlier peak arrivals, much heavier tails). We demonstrated that these cases could be represented effectively by a multirate mass transfer model with parameters representative of a secondary porosity within the primary pore space (mass transfer between preferential flow paths and relatively nonadvective regions). One might ask whether this improved model is representative of an actual physical process or simply reflects additional degrees of freedom in model fitting. While this is a valid concern, physical meaning can be attributed to the resulting parameters (e.g., the average size of diffusive domains should correspond approximately to the mean spacing of preferential flow paths), although they cannot be rigorously defined. Moreover, the process of solute trapping in low-velocity regions of the primary pore space can clearly be seen in Figure 10, as well as in an animated visualization of a similar simulation (see supporting information), indicating that this process is feasible even in relatively simple pore geometries. A potential implication is that a non-Fickian model such as the MRM (or other similar models) is applicable to all porous media transport problems, and not only those in which there is an obvious secondary porosity, particularly in cases where low-concentration tails are of importance (such as transport of contaminants for which very low concentrations are of concern, such as radionuclides or pathogens). Fitting of upscaled model parameters to results of pore-scale simulations may serve as an effective approach to rigorously parameterize MRMs or related models for use at larger scales, along the lines previously proposed by Rhodes et al. [2008, 2009]. Combining multiple MRMs to represent jointly the effects of intragranular diffusion and mass transfer between subdomains within the primary porosity may be more effective than applying an intragranular diffusion model on top of an ADE that poorly represents the underlying non-IGD transport.
 Another consistent observation is that the MRM, parameterized using the spherical diffusion model of Haggerty and Gorelick , tended to overestimate the amount of mass transfer in the IGD process. This was reflected primarily in predicted peak breakthrough concentrations that were consistently too low relative to the full 3-D simulation results. We interpret this result as reflecting the effects of incomplete mixing at the pore scale. The theory of Haggerty and Gorelick  assumes that every grain at a given transport distance is exposed to the same concentration history at its surface, which implies a high degree of transverse mixing. However, in these simulations there is strong evidence of preferential flow and mass exchange between advection- and diffusion-dominated regions of the primary porosity, indicating that the system is not well mixed laterally. Because of this, some grains are not exposed to solute as much as they would be in a fully mixed situation, thus posing a limitation on intragranular mass transfer. This effect, while not large in terms of the observed magnitudes of discrepancies in the simulated breakthrough curves, may be an important consideration in application of mass transfer models to heterogeneous porous media at multiple scales.
 We conclude with a brief consideration of the significance of the observed simulation results in terms of the existence and definition of a representative elementary volume (REV) in porous media. Bear [1972, pp. 20–21] presents the classic definition of the REV in terms of porosity but notes that “Sometimes one must define REVs of the medium on the basis of parameters other than porosity.” The results presented here clearly demonstrate that the definition of the REV is process- and/or property-specific. In terms of porosity, there was essentially no variation between different realizations of the same statistical pore geometry, indicating that the simulated domain size was sufficient to form an REV. However, the effective permeabilities of the different domains varied mildly and fitted dispersivity values, and the observed character of breakthrough curves varied widely. Therefore, while one might infer based on one parameter (e.g., porosity or permeability) that the volume considered appropriately represents an REV, application of that REV to another process/property (e.g., dispersivity) may be inappropriate. Also, for most of the systems studied here, the transport behavior became more Fickian in character with increasing transport distance as expected, but some elements of non-Fickian behavior were expressed in all simulations, even at transport distances of nearly 20 mean grain diameters. These results indicate that to the degree that a general REV for transport can be well defined, it is likely to be larger than the O(mm3) volumes considered here, and additional numerical simulation with larger systems is needed. Fortunately, pore-scale simulation capabilities have advanced rapidly in recent years, and computational capabilities are such that we expect that larger domains will be simulated in the near future to provide further insight into this issue. Such advances should also facilitate simulation of more complex geometries representative of natural porous media (e.g., as measured by X-ray microtomographic methods) as well as examination of alternative formulations to the multirate mass transfer model [e.g., Dentz et al., 2012].
 This research was supported by the U.S. Department of Energy (DOE) Office of Biological and Environmental Research (BER), Subsurface Biogeochemical Research program, through the Scientific Focus Area project at Pacific Northwest National Laboratory. Computations described here were performed using computational facilities of the Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by DOE-BER and located at PNNL; computational facilities of the National Energy Research Supercomputing Center (NERSC), which is supported by the DOE Office of Science under contract DE-AC02-05CH11231; and PNNL's Institutional Computing facilities. PNNL is operated for the DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. We thank Roy Haggerty for helpful discussions and for making available the STAMMT-L code.