## 1. Introduction

[2] Mass transfer processes occur in almost all porous media and over various scales ranging from pore diffusion at the grain scale and diffusion into low hydraulic conductivity inclusions at the centimeter to meter scale to matrix diffusion into fractured rocks. The significance and diversity of these processes has been recognized already for a long time [e.g., *van Genuchten and Wierenga*, 1976; *Neretnieks*, 1980] due to its important effect on solute transport observed in laboratory and field studies. More recent experimental findings showed that, for example, the sorption/desorption mechanism is often limited by the diffusive transport within the fluid phase of the intraparticle pores of the sediment grains [e.g., *Ball and Roberts*, 1991; *Pignatello and Xing*, 1996; *Luthy et al.*, 1997; *Rügner et al.*, 1999]. Furthermore, various authors [e.g., *Guswa and Freyberg*, 2002; *Zinn and Harvey*, 2003; *Liu et al.*, 2004] have demonstrated, that solute transport through heterogeneous aquifers with connected high-conductivity pathways and/or lenses of low-conductivity material is often better upscaled using an advection-dispersion model including a mass transfer component. This is believed to be the reason, why a successful modeling of the tracer test at the Macrodispersion Experiment (MADE) site using the classical advection-dispersion model and the hydraulic conductivity data obtained in the field has still not been achieved [*Harvey and Gorelick*, 2000; *Feehley et al.*, 2000]. In fact, various authors [*Berkowitz and Scher*, 1998; *Schumer et al.*, 2003] demonstrated that a certain form of a multirate mobile/immobile model reproduces the anomalous tracer plume spreading at the MADE site better than a classical single rate mass transfer model. Thus the proper modeling of these processes for groundwater risk assessment is critical, especially when assessing the persistence of contaminants in aquifers and the long-term performance of remediation technologies.

[3] Unfortunately, mass transfer processes complicate solute transport simulations and therefore many of the studies performed up to now consider only a simple first-order mass transfer model. This is from a mathematical and numerical point of view a very convenient approach. Yet it does not represent correctly the underlying physical process in many cases, which can lead to a misinterpretation of the fitted values [*Young and Ball*, 1995; *Haggerty et al.*, 2004] and therefore lead to noticeable different results when used for prediction. Moreover, mass transfer processes might not be correctly described by using a spatially homogeneous coefficient and only a small number of authors have investigated the effects when mass transfer rates are treated as a spatial random variable [e.g., *Li and Brusseau*, 2000; *Huang and Hu*, 2000; *Huang et al.*, 2003]. Hence modeling all these different mass transfer processes requires the implementation of a possibly spatially heterogeneous, small-scale physical process into a coarse-scale numerical model without loosing computational efficiency, but yet approximating reality which is often being limited by these small-scale diffusion processes.

[4] In order to encompass all these different mass transfer processes within one mathematical model which is still based on the classical advection-dispersion equation, two formulations have been presented in literature. The first approach was introduced by *Carrera et al.* [1998] and consists of using a source-sink term in the advection-dispersion equation to represent the rate of loss or gain of concentration to or from the immobile domain. The source-sink term is expressed as a convolution product of a memory function. Choosing the adequate memory function simple first-order mass transfer, diffusion using a distribution function of mass transfer rates, e.g., gamma or power law distribution, and diffusion into different geometries, e.g., sphere, layer, or cylinder, can be modeled [*Carrera et al.*, 1998; *Haggerty et al.*, 2000].

[5] The second approach was presented by *Haggerty and Gorelick* [1995] and is based on the idea of using the conventional first-order mass transfer equation [*Nkedi-Kizza et al.*, 1984] but superimposing multiple first-order mass transfer rates to represent various diffusion processes. In fact, by choosing a certain infinite sum of first-order rates the model permits the simulation of diffusion into different geometries. It is interesting to note that the memory function used by *Carrera et al.* [1998], necessary when using the convolution approach, has also the form of an infinite sum with the same mass transfer rates as required by the multirate approach. The parallelism between these two formulations was already pointed out by *Carrera et al.* [1998].

[6] Note that similar models trying to capture the anomalous behavior of solute plumes in heterogeneous aquifers caused by mass transfer processes have been developed recently. For example, continuous time random walk models particle transport in a heterogeneous medium as a random walk in space and time [e.g., *Berkowitz and Scher*, 1998; *Dentz and Berkowitz*, 2003; *Dentz et al.*, 2004] and can be formulated to be generally equivalent to the multirate mass transfer approach of *Haggerty and Gorelick* [1995]. Furthermore, fractal models have also been expanded to include a mobile/immobile domain for the purpose of a better description of anomalous transport [*Schumer et al.*, 2003]. Although both approaches have proven to be flexible and have been applied successfully to a variety of field studies [e.g., *Berkowitz and Scher*, 1998], we do not discuss them here in detail as the method presented is based on the classical advection-disperison equation as opposed to the continuous time random walk and the fractal models.

[7] Because of the mathematical complexity of the formulations presented by *Haggerty and Gorelick* [1995] and *Carrera et al.* [1998] only a few numerical solutions exist, which do not impose any restrictive assumptions on the spatial variability of advection, dispersion, and mass transfer rates in a three-dimensional numerical model [e.g., *Carrera et al.*, 1998; *Wang et al.*, 2005]. In this paper we will present a new numerical method to implement the multirate mass transfer approach into random walk particle tracking. Random walk particle tracking has been used to model solute transport in aquifers for a long time [e.g., *Prickett et al.*, 1981; *Kinzelbach*, 1987]. It consists in moving a cloud of particles advectively according to the flow path lines and adding a random displacement for each time step to simulate dispersion. Its main advantages, i.e., the nonexistence of numerical dispersion, computational efficiency, and local as well as global mass conservation, have turned random walk particle tracking into a valuable tool for inverse modeling, uncertainty assessment, and solute transport in highly discretized, heterogeneous aquifers [e.g., *Tompson*, 1993; *Salamon et al.*, 2006].

[8] Various methods to include simple linear mass transfer into random walk particle tracking have been presented in literature. *Valocchi and Quinodoz* [1989] compared three techniques for modeling kinetic sorption which is mathematically equivalent to first-order mass transfer as pointed out by *Haggerty and Gorelick* [1995]: (1) the continuous time step method which simulates the history of phase changes of a particle for each time step, (2) the arbitrary time step method which uses probability density functions to calculate the time spent in the aqueous/sorbed phase of a particle during a time step, and (3) the small time step method originally introduced by *Kinzelbach* [1987], which performs a Bernoulli trial using transition probabilities to determine if the particle will be in the aqueous or sorbed phase for the next time step. *Andricevic and Foufoula-Georgiu* [1991] present a method similar to the arbitrary time step method, however using a different approach to calculate the fraction of time a particle spends in a certain phase. *Michalak and Kitanidis* [2000] again review the methods presented by *Valocchi and Quinodoz* [1989] and add the semianalytical moment method, which performs a Bernoulli trial using phase transition probabilities, obtained by the method of moments to determine the phase of a particle for the next time step. *Huang et al.* [2003] develop a first-order mass transfer model based on the phase transition probabilities of *Valocchi and Quinodoz* [1989]. Finally, a method for the diffusion into fractures using particle tracking was presented by *Tsang and Tsang* [2001]. They use analytical solutions to calculate the residence time caused by matrix diffusion into homogeneous finite or infinite matrix blocks.

[9] In this work we will present the implementation of the multirate mass transfer approach with the random walk particle tracking method. Section 2 first outlines the general mathematical framework for the multirate mass transfer approach and the calculation of the zeroth spatial moment, required to determine the phase transition probabilities. In the following subsections examples for first-order mass transfer, multirate mass transfer, and diffusion into various geometries are given. Section 3 outlines some numerical implementation details with respect to the choice of the time step size for particle tracking, the approximation of a matrix exponential using Taylor series, and the truncation of the series for diffusion into different geometries. Section 4 illustrates a numerical example for the influence of a heterogeneous distribution of intraparticle pore diffusion on contaminant transport. Finally, section 5 summarizes the main results and conclusions from this paper.