## 1. Introduction

[2] 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.

[3] 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].

[4] 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*. [2008] 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.