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 The complexity of mass transfer processes often complicates solute transport simulations. We present a new approach for the implementation of the multirate mass transfer model into random walk particle tracking. This novel method allows for a spatially heterogeneous distribution of mass transfer coefficients as well as hydrodynamic parameters in three dimensions, and it is well suited for avoiding numerical dispersion and solving computationally demanding transport simulations. For this purpose the normalized zeroth spatial moments of the multirate transport equations are derived and used as phase transition probabilities. Performing a simple Bernoulli trial on the appropriate phase transition probabilities the particle distribution between the mobile domain and any immobile domain can be determined. The approach is compared satisfactorily to analytical and semianalytical solutions for one-dimensional, advective-dispersive transport with different types of mass transfer. Aspects of the numerical implementation of this approach are presented, and it is demonstrated that two restrictive criteria for the time step size have to be considered. By adjusting the time step size for each grid cell on the basis of the cell specific velocity field and mass transfer rate a correct simulation of solute transport is assured, while at the same time computational efficiency is preserved. Finally, an example is presented evaluating the effect of a heterogeneous intraparticle pore diffusion in a synthetic aquifer. The results demonstrate that for this specific case the heterogeneous distribution of mass transfer rates does not have a significant influence on mean solute transport behavior but that at low concentration ranges, differences between the different mass transfer models become visible.
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 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.
 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.
 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.  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].
 The second approach was presented by Haggerty and Gorelick  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. , 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. .
 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 . 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.
 Because of the mathematical complexity of the formulations presented by Haggerty and Gorelick  and Carrera et al.  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].
 Various methods to include simple linear mass transfer into random walk particle tracking have been presented in literature. Valocchi and Quinodoz  compared three techniques for modeling kinetic sorption which is mathematically equivalent to first-order mass transfer as pointed out by Haggerty and Gorelick : (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 , 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  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  again review the methods presented by Valocchi and Quinodoz  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.  develop a first-order mass transfer model based on the phase transition probabilities of Valocchi and Quinodoz . Finally, a method for the diffusion into fractures using particle tracking was presented by Tsang and Tsang . They use analytical solutions to calculate the residence time caused by matrix diffusion into homogeneous finite or infinite matrix blocks.
 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.
2. Mathematical Framework
2.1. Multirate Model
 The multirate model describes mass transfer between a mobile domain and any number of immobile domains with varying properties. The advection-dispersion equation for this model can be written as follows (according to Haggerty and Gorelick ):
where cm [M/L3] is the aqueous concentration in the mobile domain; cim,j [M/L3] is the aqueous concentration in the jth immobile domain; D [L2/T] is the hydrodynamic dispersion tensor; v [L/T] is the velocity vector; θm and θim,j (dimensionless) are the porosities of the mobile and the jth immobile domain, respectively; Rm and Rim,j (dimensionless) are the retardation factors for the mobile and the jth immobile domain, respectively; and N (dimensionless) is the number of distinct immobile phases. Here, changes in porosities with time are assumed negligible as typically used in subsurface hydrology. The retardation factors are given as
where ρb [M/L3] is the bulk density of the porous medium; Kd [L3/M] is the distribution coefficient; and fm and fim,j (dimensionless) are the mass fractions of the sorbed phase in sorption equilibrium with the mobile domain and the jth immobile domain, respectively. The sum of all fm and fim,j is 1. The mass transfer equations for the multirate model are given as
where α′j [T−1] is the first-order mass transfer rate associated with the jth immobile zone.
 As can be seen from equations (1) and (3) the model permits not only to use a spatial heterogeneous distribution of mass transfer rates, but also a distribution of various diffusion processes within one grid cell. Haggerty and Gorelick  argue that at the grain scale a contaminant may diffuse into stagnant zones of water, intraparticle pores, and larger aggregates of grains and therefore a numerical model should be able to account for these different processes.
2.2. Development of Phase Transition Probabilities
 The term transition probability has its origin in the context of continuous time Markov chains and denotes the probability that a process presently in state i will be in state j a time t later [e.g., Ross, 2003]. Using this principle in particle tracking and provided that we know the transition probability function we can determine if a particle is in the mobile phase and thus susceptible to advection and dispersion or in the immobile phase after a time step Δt by simply performing a Bernoulli trial on the appropriate phase transition probability. Valocchi and Quinodoz  demonstrated the analogy between the first-order, reversible, linear rate expression for kinetic sorption and a homogeneous, continuous time, two-state Markov chain and used the corresponding phase transition probability functions to simulate kinetic sorption.
 A second idea important for the development of the following method was introduced by Michalak and Kitanidis . They state, that when the spatial moments of concentrations are interpreted as those of particles, the normalized zeroth spatial moment, which describes the distribution of mass in the different phases, can be used as a phase transition probability function. Applying this idea, Michalak and Kitanidis  obtained the same phase transition probability functions for the kinetic sorption case as Valocchi and Quinodoz .
 In this section we will outline the derivation of the zeroth spatial moment for the multirate model. The spatial moments are generally calculated in three dimensions [Aris, 1956]. For the sake of simplicity we will present them here only in one dimension:
where μn and νn,j are the nth mobile and immobile phase moments, respectively.
 The concentrations for the mobile and immobile phases are now substituted with their Fourier transforms in order to be able to calculate the zeroth moment in the Fourier domain. The following definition of the Fourier transform and its inverse will be used here
 We note that our objective here is to find the mechanism by which the particle is exchanged between the mobile/immobile domain in the random walk method. Thus, according to Kitanidis  we view each particle as a very small plume compared to the grid block so that the effects of boundaries are negligible in equation (9).
 The spatial moments of the concentration in the real domain can be evaluated from the concentration Fourier coefficients [Goltz and Roberts, 1987]:
 In this work we only require the zeroth moments, hence from equations (11) and (12) follows
where βj (dimensionless) is the so-called capacity ratio associated with the jth immobile zone. Equations (15) and (16) represent a system of linear differential equations, which can be written in matrix form as
Δt[T] is the time step, and M0 is a vector containing the initial distribution of mass in the different phases.
 A general solution to the exponential matrix of equation (19) and thus to the vector M containing the zeroth spatial moments for each domain is not easy to find. This is especially the case when a large number of mass transfer rates is modeled and therefore the corresponding matrices are of a high order. The evaluation of the exponential of a matrix has been subject to intensive research over the last decades and in the work by Moler and van Loan  a comprehensive overview of the existing approaches is presented. However, computing the whole exponential matrix numerically for every time step and each particle would decrease computational efficiency of the particle tracking significantly. Fortunately, using particle tracking to simulate contaminant transport provides two important advantages: First, we do not always need to compute the entire matrix (i.e., when no phase transition occurs). Secondly, we can use a low-order Taylor series to approximate the exponential matrix, as the time steps used in particle tracking are usually small.
 The reader should keep in mind that using a polynomial series as shown in the following sections is not the only way to compute the exponential of a matrix. Methods employing the matrix eigenvalues, differential equations, or approximation theory have been proposed in literature. However, in practice computational stability and accuracy indicate that some of the methods are preferable to others but that none are completely satisfactory [Moler and van Loan, 2003]. We chose the Taylor series method due to the above mentioned advantages and its simplicity regarding the mathematical formulation and the numerical implementation. Other methods might lead to the same results requiring however different stability and accuracy criteria as given in sections 3.2 and 3.3.
2.2.1. Simple First-Order Mass Transfer
 The simplest case of mass transfer is described by a single first-order rate coefficient. When using the multirate approach this is done by setting N = 1. This results in square matrices of second-order in equation (18). For this case an analytical solution for the exponential matrix of equation (19) can be found
 On the basis of this analytical solution for the zeroth spatial moments the particle phase transition probabilities can be derived easily. Assuming that at the beginning of the time step all the solute mass M is in the mobile domain and setting the solute mass M = 1 (μ0(0) = 1 and ν0(0) = 0) the phase transition probabilities can be written as
where the superscripts m represent solute originating in the mobile domain, and Pm→m and Pm→im refer to the probability of a particle starting in the mobile phase and ending in the mobile/immobile phase, respectively.
 Conversely, assuming that at the beginning of the time step all the solute mass is in the immobile domain (μ0(0) = 0 and ν0(0) = 1) the phase transition probabilities are
where the superscript im represents solute originating in the immobile domain, and Pim→m and Pim→im refer to the probability of a particle starting in the immobile phase and ending in the mobile/immobile phase, respectively. It can be seen from equations (22) to (25) that Pm→m = 1 − Pm→im and Pim→im = 1 − Pim→m. Valocchi and Quinodoz  and Michalak and Kitanidis  obtained equivalent expressions for the case of kinetic sorption, as mentioned above.
 Having calculated the phase transition probabilities, numerical implementation into particle tracking is done easily. For each time step a uniform [0, 1] random number Y is drawn for each particle and is compared to the corresponding probability. The state of a particle being in the mobile phase is adjusted according to
where Xp is the position of the particle at time t + Δt. For a particle being in the immobile domain the final state is adjusted according to
Having finished the trial a particle is only allowed to move when being in the mobile phase.
Figure 1 and Table 1 show the breakthrough curve and input parameters obtained for a one-dimensional system with simple, first-order mass transfer using the random walk model. The curve is compared with the results obtained using the well-known CXTFIT Code [Toride et al., 1995].
Table 1. Input Parameters for One-Dimensional Solute Transport in Figures 1 and 2
First-Order Mass Transfer
For spherical diffusion, α = is the diffusion rate coefficient.
2.2.2. Multiple Mass Transfer and Diffusion Into Various Geometries
 One of the main advantages of the multirate model is the possibility not only to simulate a certain number of linear mass transfer processes, but to model diffusion into spheres, cylinders, and layers by choosing appropriate values for the first-order rates and capacity coefficients [Haggerty and Gorelick, 1995]. The series of these coefficients for the different geometries are shown in Tables 2 and 3. However, modeling these processes usually requires a relatively large number of mass transfer rates resulting into high-order matrices in equation (18). Analytical solutions, as presented in section 2.2.1, for equation (19) do not exist and thus the exponential matrix has to be calculated using a numerical approach.
 In this work we will employ the following Taylor series approximation of exponential matrices
where I is the identity matrix.
 Because of the fact that in random walk particle tracking the time step chosen is normally small, as already mentioned above, we do not require to use a lot of terms in equation (28) to approximate the exponential of the matrix [(A−1B)Δt]. Instead, using only the terms up to the third-order the zeroth spatial moments for each domain are sufficiently well approximated.
 Using these spatial moments the phase transition probabilities for a particle are calculated with a similar procedure as outlined in section 2.2.1. Assuming that at the beginning of the time step all the solute mass M is in the mobile domain and setting the solute mass M = 1 (μ0(0) = 1 and ν0,j(0) = 0) the probability of a particle originating in the mobile phase and being in the mobile phase Pm→m at the elapsed time Δt can be approximated as follows
 The probability of a particle originating in the mobile phase and changing into the ith immobile phase Pm→im,i can be written as
 Conversely, assuming that at the beginning of the time step all the solute mass is in the ith immobile domain (μ0(0) = 0, ν0,i(0) = 1, and ν0,j(0) = 0 for all j ≠ i) the phase transition probability for a particle to move from the ith immobile domain to the mobile domain Pim,i→m is
 The probability of a particle to stay in the ith immobile domain Pim,i→im,i can be approximated as
 Although the different immobile domains are not connected with each other there exists still the possibility that a particle moves first from the ith immobile domain to the mobile domain and from there into the kth immobile domain Pim,i→im,k within one time step, as transition probabilities only describe the initial and the final state during the elapsed time Δt, but do not account for state changes within Δt. The probability for this case can be calculated as
 It can be observed from equation (33) that the probability for this case is very small, as we would expect, because the first-order term is canceled out. Therefore particles will move from one immobile zone to another only for considerably large time steps.
 Numerical implementation is, again, similar to the simple first-order mass transfer case. However, we will now have to keep track of in which immobile domain the particle is currently located. The state of a particle being located in the mobile phase is adjusted according to
For a particle being located in the ith immobile phase the final state is being adjusted as follows
 A breakthrough curve obtained using random walk and a multirate series for spherical diffusion is presented in Figure 2. The input parameters are shown in Table 1. The curve is compared with the results obtained using STAMMT-L [Haggerty and Reeves, 2002]. STAMMT-L is a code that provides a semianalytical solution to one-dimensional, dual porosity, advective-dispersive transport, where mass transfer between the mobile and immobile domains is generalized to include multiple immobile domains.
3. Numerical Implementation Details
3.1. Random Walk Particle Tracking
 Random walk particle tracking simulates solute transport by partitioning the solute mass into a large number of representative particles. The evolution in time of a particle is driven by a drift term that relates to the advective movement and a superposed Brownian motion responsible for dispersion. The displacement of a particle is calculated as follows [e.g., Tompson, 1993]
where B1 is a “drift” vector, B2, the displacement matrix, is a tensor defining the strength of dispersion, and ξ(t) is a vector of independent, normally distributed random variables with zero mean and unit variance. In expression (36)B1 corresponds to
and the displacement matrix B2 is related to the dispersion tensor as
 The velocity vector is computed using linear interpolation of interface velocities. The dispersion tensor field is obtained by first extrapolating interface velocities to surrounding nodes. This gives all three components of the vector pore velocity to each grid node, which is then used to estimate the dispersive component of the random walk using trilinear interpolation. Various authors have demonstrated [LaBolle et al., 1996; Salamon et al., 2006], that this hybrid scheme yields local as well as global divergence-free velocity fields within the solution domain and a continuous dispersion tensor field that approximates well mass balance at grid interfaces of adjacent cells with contrasting hydraulic conductivities. Furthermore, a constant displacement scheme [Wen and Gómez-Hernández, 1996] which modifies automatically the time step size for each particle according to the local velocity is employed in order to decrease computational effort.
 For the examples presented in this article the numerical random walk particle tracking code developed by Fernàndez-Garcia et al.  was extended to simulate mass transfer according to the procedure outlined in section 2.
3.2. Choice of Time Step Sizes
 One important problem, when simulating mass transfer processes using particle tracking is, that the phase transition probabilities do not describe the number of phase changes occurring within one time step, but only determine the initial and final state of a particle during the elapsed time Δt [Parzen, 1962]. Thus the time a particle actually spends in the mobile/immobile phase can differ significantly within a time step for the case of high mass transfer rate in relation to the time step size. This problem is complicated further when using a constant displacement scheme which adjusts automatically the time step size according to the particle displacement in the mobile phase. For the case of having various linear mass transfer processes acting at the same time it is therefore possible that a particle moves from the immobile phase i to the mobile phase and from there into the immobile phase k as shown in section 2.2.2.
 One simple way to overcome this problem is to choose a time step size small enough so that the probability of various phase changes within one time step is negligible [Valocchi and Quinodoz, 1989; Kinzelbach, 1987]. In order to determine the proper time step size we will look at the case of advective solute transport with first-order mass transfer in a one-dimensional, homogeneous column. Particles will be released at a point on the left border of the column and arrival times will be measured at the right column border. Δt is given here as a function of the Courant number (Cr = (vΔt)/L, where L is the column length). By varying Cr and comparing the cumulative breakthrough curves with a reference curve obtained using the random walk approach with a very small time step (Cr = 0.00001) the root mean of squared residual errors (RMS) is calculated
where N is the total number of observations, cali [M/L3] and refi [M/L3] are the calculated and the reference concentration values of the breakthrough curves at a certain time. An overview of the input parameters for the following examples is given in Table 4.
Table 4. Input Parameters for Solute Transport Examples in Figures 3, 4, and 6a
 To characterize the ratio of the mass transfer timescale to the advection timescale the Damköhler number is used. In the case of one-dimensional flow and transport and using the multirate model, each mass transfer reaction has a Damköhler number associated, which can be expressed as follows
where L [L] is the length scale, which for this case corresponds to the column length.
Figure 3 shows the RMS for different Damköhler numbers. As expected, increasing the first-order mass transfer rate requires a smaller time step, in order to represent solute transport correctly. Figure 4 illustrates the effects of large time steps on the breakthrough curves for the case of a high mass transfer-advection ratio (DaI,1 = 200). If time step size is not sufficiently small the detention of particles in the immobile compartments is too large and artificial dispersion and an increased tailing in the breakthrough curve is introduced.
 However, Bahr and Rubin  stated that for a Damköhler value greater than approximately 100 the mass transfer relationship is effectively at equilibrium and therefore practical problems usually have Damköhler numbers smaller than 100. Thus, for most cases a Cr of 0.01 is sufficient for a correct simulation of mass transfer using random walk.
3.3. Approximation Criteria for the Exponential Matrix
 A further difficulty concerns the fact that equations (29) to (33) only represent an approximation to the matrix exponential exp[(A−1B)Δt]. Choosing a very large time step or increasing the number of multirate parameters can result in an erroneous approximation of the phase transition probabilities when only three terms are employed. A variety of restrictive criteria have been suggested in literature concerning the truncation of Taylor series [e.g., Everling, 1967; Bickart, 1968]. Here, the criterion established by Liou  will be used
where D (dimensionless) is the number of terms used for the approximation, and δ and δmax (dimensionless) are the truncation error bound function and the prescribed absolute error tolerance, respectively. The matrix norm used here is a 1 norm or also called the maximum absolute column sum norm:
 In case of exceeding δmax either the number of multirate parameters or the time steps have to be adjusted. Figure 5 shows the relation between the matrix norm and the error tolerance for a matrix exponential approximation with terms up to the third order. Experience has shown, that when choosing δmax = 0.1 the matrix exponential is well approximated and for most cases the matrix norm does not exceed the restriction criterion.
3.4. Truncation of the Multirate Series
 A final issue concerns the truncation of the multirate series when simulating diffusion into different geometries, which was already addressed by Haggerty and Reeves . They found that very precise results can be obtained with truncated series as long as appropriate expressions for the final term of the series are used (see Table 3). According to Haggerty and Reeves  usually less than 30 terms are sufficient for the representation of a diffusion process into a specific geometry.
 Unfortunately, when using a multirate series to simulate diffusion, some immobile domain compartments will always have relatively high Damköhler numbers, hence requiring a small Courant number for the random walk simulations. Figure 6 shows the root-mean-square error (see equation (39)) for comparing cumulative breakthrough curves obtained using STAMMT-L and the method presented here for one-dimensional, advective-dispersive solute transport with diffusion into spherical grains (input parameters are illustrated in Table 4). The curves simulated with STAMMT-L employ 30 terms for the multirate series. It should be noted that the DaI values presented in Figure 6 represent the ratio of mass transfer and advective timescale of the spherical diffusion and are calculated using the diffusion rate constant (α = ).
 It can be observed that the error in general is very small and that with significantly less than 30 terms no further improvement is visible. This is not surprising as the capacity ratios of the multirate series approach rapidly very small values whereas the mass transfer rates for the different immobile compartments approximate large DaI,j values. These compartments can therefore be lumped together and modeled with a single equilibrium mass transfer relationship (according to Haggerty and Gorelick ). Furthermore, when using particle tracking, an increase in the number of terms used for the multirate series does not necessarily result in a higher precision of the outcome as there are two restrictive constraints to the time step size Δt: the rate of mass transfer and the approximation of the exponential matrix using a third-order Taylor series.
4. Effect of a Heterogeneous Intraparticle Pore Diffusion Distribution: An Example
 One of the main advantages of the method presented here is that it does not impose any spatial restrictions on the different types of mass transfer while preserving computational efficiency. To illustrate this a synthetic example of the effect of heterogeneous distribution of intraparticle pore diffusion rates is presented in this section. Some of the parameters of this synthetic example, i.e., spatial correlation, pore-scale dispersivities, mobile/immobile porosities, and diffusion rates, are representative of the Borden aquifer. However, the objective of this example is not to reproduce solute transport at the Borden site, but to illustrate the application of the presented random walk approach in a realistic setting.
 For this purpose one realization of a sequential gaussian simulation [Gómez-Hernández and Journel, 1993] was chosen. The following standardized exponential semivariogram was applied for the simulation of the hydraulic conductivity field
where λx,y,z [L] are the directional correlation length scales, hx,y,z [L] are the directional lag spaces, and σ2 is the variance of the natural logarithm of the hydraulic conductivity ln K [L/T]. According to Woodbury and Sudicky , a correlation length of λx = λy = 5.1 m (horizontal) and λz = 0.21 m (vertical) was selected. It should be noted that the variance of ln K was increased with respect to the Borden aquifer in this example to a value of σ2ln K = 2.5 and the average hydraulic gradient to a value of 0.043. The computational domain is parallelepipedic with dimensions of x = 80 m, y = 15 m, and z = 4 m and a discretization of Δx = Δy = 0.5 m, and Δz = 0.04 m was chosen, resulting in a total of 480,000 grid cells. The aquifer was assumed to be confined and with constant head boundaries at x = 0 m and x = 80 m and with no-flow boundaries at the remaining model faces. A total of 20,000 particles randomly distributed in a plane shaped, rectangular area of 10 m width and 3 m height located orthogonal to the principal flow direction at a distance of x = 5 m were released at t = 0. All particles were initially released in the mobile domain. Mass arrival was measured at a control plane located at x = 78 m. Pore-scale longitudinal and transverse dispersivities were assumed to be αL = 0.0005 m and αT = 0.00005 m [Brusseau and Srivastava, 1997], and mobile/immobile domain porosities were selected to be θm = 0.293 and θim = 0.037, respectively [Brusseau and Srivastava, 1997].
 We considered here the following four different models: Solute transport in model A was purely influenced by advection and dispersion, whereas in models B, C, and D different types of mass transfer were added to the advection-dispersion equation. Model B utilizes a spatially heterogeneous intraparticle pore diffusion. Model C has a uniform coefficient for intraparticle pore diffusion. Finally, model D employs a uniform first-order mass transfer coefficient. Intraparticle diffusion was modeled employing diffusion into a spherical geometry.
 In order to obtain the field of intraparticle pore diffusion rates for model B the Kozeny-Carmen relationship [Bear, 1972] was used to calculate a representative grain size diameter for each cell:
where ρw[M/L3] is the fluid density, g[L/T2] is the gravitational constant, μ [M/(LT)] is the fluid viscosity, d50 [L] is the grain size diameter, and θ is the total porosity, which corresponds to the sum of mobile and immobile porosity. The grain sizes were then employed to assign the corresponding diffusion rate coefficients to each grid cell pursuant to Ball and Roberts , which estimated diffusive PCE uptake into different size fractions of a Borden sample (see Table 5). The reader should keep in mind that we are not suggesting that d50 is necessarily the representative length scale for intraparticle diffusion. In fact, intraparticle diffusion might even be better represented with several grain sizes [e.g., Haggerty and Gorelick, 1995]. However, in this example we consider the assumptions taken as valid.
Table 5. Grain Sizes and Diffusion Rate Coefficients for a Borden Sand Sample Measured by Ball and Roberts  for PCE Desorption
Size Range (d50 = 2a)
0.85 – 1.7 mm
3.1 × 10−8 s−1
0.42 – 0.85 mm
9.2 × 10−8 s−1
0.25 – 0.42 mm
2.3 × 10−7 s−1
0.18 – 0.25 mm
2.7 × 10−7 s−1
0.125 – 0.18 mm
9.4 × 10−7 s−1
0.075 – 0.125 mm
1.7 × 10−6 s−1
1.4 × 10−6 s−1
 The uniform coefficient for model C was calculated by applying (44) to the geometric mean of the conductivity field K = 6.182 m/d [Burr and Sudicky, 1994] resulting in a grain size diameter of d50 = 0.128 mm and thus into a diffusion rate of Da/a2 = 9.4 × 10−7s−1 (see Table 5). Model D in turn uses the following relationship to calculate the first-order mass transfer rate
which has shown to be the best effective rate coefficient used in “equivalent” first-order models of mass transfer [e.g., Young and Ball, 1995; Haggerty et al., 2000]. To calculate α in this example the uniform diffusion rate coefficient of model C is used resulting in a first-order mass transfer coefficient of α = 1.41 × 10−5s−1.
 Concerning the numerical implementation of the presented approach the following issues had to be addressed: (1) the negative correlation between hydraulic conductivity and mass transfer rates in model B lead to high Damköhler numbers in low-velocity zones; (2) due to the constant displacement scheme employed Δt was considerably large in high hydraulic conductivity areas, thus exceeding the criteria for the matrix norm for models B and C in some grid cells established in section 3.3. Therefore the Courant number was adapted for each grid cell before starting the solute transport based on the cell specific velocity field and mass transfer rate, in order to avoid defining one maximum time step size for the entire model domain and therewith decreasing computational efficiency.
 The breakthrough curves and the relative mass fraction remaining in the aquifer of the four models are presented in Figures 7 and 8. It can be seen clearly that the diffusion process acts as a retardation factor on solute transport. However, it can also be observed that models B, C, and D produce similar results for this example, indicating that the heterogeneous representation of mass transfer rates does not influence mean transport behavior significantly for the given input parameters. Instead, it even seems that using an “equivalent” first-order model, is able to reproduce correctly the main features of solute transport. Nevertheless, model D appears to underestimate the low-concentration tailing in comparison to model B, whereas model C slightly overestimates the tailing. This indicates that the choice of mass transfer type as well as the spatial distribution of mass transfer rates can potentially have a significant effect on low-concentration tailing. Similar observations on the effect of heterogeneous rate-limited mass transfer were also made by Li and Brusseau  and Cunningham and Roberts .
 We have developed a new numerical method to solve dual-domain multirate mass transfer coupled with advective-dispersive transport using the random walk particle tracking method. Phase transition probabilities which are calculated based on the zeroth spatial moments of the multirate mass transfer equations are used to simulate the particle distribution between the mobile and various immobile domains. The two major advantages of this approach are the flexibility in the sense that it does not impose any restrictive assumptions on the spatial variability of advection, dispersion, and mass transfer and its low computational cost even for highly discretized models having a spatially heterogeneous mass transfer rate. The flexibility of the multirate model to describe a variety of different mass transfer processes is preserved using this approach as well as the advantages of the random walk method: the nonexistence of numerical dispersion even for highly advection-dominated solute transport and the local as well as global mass conservation.
 However, there are also two disadvantages when using this approach: First, high mass transfer rates require an increasingly smaller time step size. Secondly, using a third-order Taylor series to approximate the matrix exponential can possibly result in an incorrect calculation of the phase transition probabilities when either a large number of immobile domains to simulate diffusion into various geometries is used or the time step is not sufficiently small. Nevertheless, introducing a restrictive criteria for the time step size, which can be adjusted for each grid cell separately instead of defining one maximum time step for the whole model domain, a correct simulation of mass transfer processes can be assured. Hence the herein presented approach constitutes a valuable tool for the evaluation of the effects of a variety of mass transfer processes on solute transport especially in highly heterogeneous three-dimensional systems.
 Financial support from the Spanish Ministry of Education and Science (projects REN2002-02428 and CGL2004-02008) and the European Union (FUN-MIG, contract FP6-516514) is gratefully acknowledged.