We quantitatively account for the measured concentration profile from a laboratory experiment of bimolecular (A + B → C) reactive transport in a porous medium with a particle tracking (PT) model. The PT results are in contrast to the analytical solution of the continuum scale advection-dispersion-reaction equation, which results in an excess quantity of reaction product (C). The approaches differ in the treatment of the mixing zone, the fluctuations due to the low reactant concentrations, and the localized nature of the reaction. The PT can accommodate a range of transport modes with different temporal distributions. An exponential temporal distribution is equivalent to Fickian transport, which we use for the comparison to the laboratory data; a truncated power-law (TPL) temporal distribution yields a non-Fickian transport characteristic of heterogeneous media. We study the influence of disorder on the mixing zone and the product concentration profiles via these contrasting transport modes.
 Considerable effort has been invested over the last three decades in conceptualizing and modeling reactive transport in porous media. The problem is of key importance for understanding the evolution of different chemical contaminants in groundwater. In porous media, the reactions are localized to pore-scale sites and heterogeneity can affect the nature of the transport of reactants. The high degree of fluctuations in pore-scale mixing, especially for low concentrations, can impact the larger-scale reactive transport behavior. The actual amount of mixing between reactants (and hence the amount of product) may be less than would be otherwise predicted by continuum models, due to averaging the concentrations and the mixing zone [e.g., Rashidi et al., 1996; Cao and Kitanidis, 1998; Raje and Kapoor, 2000]. Palanichamy et al.  discussed the inappropriateness of assumptions made in a continuum picture of low concentrations of reactants undergoing random collisions. Methods to account for non-uniform mixing and random reaction of the reactants, in both space and time, therefore appear necessary. The objective of this study is to develop such a method, and to compare the results to available data from a laboratory experiment of bimolecular reactive transport in a porous medium and to solutions of continuum models.
 We have used a PT approach previously to demonstrate non-Fickian transport in a random fracture network, modeling it within the continuous time random walk (CTRW) framework [Berkowitz and Scher, 1998]. The results of direct CTRW-PT simulations are in excellent agreement with analytical and numerical solutions of the CTRW equations [Dentz et al., 2004]. The CTRW-PT approach allows considerable flexibility in the choice of probability density functions (pdf's) governing the spatial and temporal aspects of the particle motion [Dentz et al., 2008]. The limiting choice of an exponential temporal pdf with a spatial pdf having finite moments (e.g., an exponential pdf) conveniently reproduces the solutions of the advective-dispersion equation (ADE). Changing the temporal pdf to a truncated power law (TPL) allows treatment of a full range of non-Fickian behaviors [Berkowitz et al., 2006; Dentz et al., 2008].
 Here we focus on the bimolecular chemical reaction A + B → C, introducing into the CTRW-PT framework interactions among migrating particles. We maintain the same degree of flexibility for reaction (i.e., local equilibrium or kinetic controlled) as for transport. For our present purposes we simplify a general picture of the reacting species; we specify that the reaction time is fast compared to the transport time and focus on the type of transport and on the spatial feature of the bimolecular process, which we use to capture the local nature of the interaction in a porous medium. The radius of interaction R is a key parameter in defining the mixing zone in this picture; it is the upper limit of the volume within which A and B react. The variation of R from local interaction to uniform mixing presents an opportunity to compare the PT results to the spatial distribution of the reactant product (i.e., C) in laboratory experiments and to the analytical solution of the advection-dispersion-reaction equation (ADRE) discussed below. Few detailed comparisons of PT for reactive transport, laboratory experiments and the ADRE have been undertaken to date. The ADRE is limited to Fickian transport. Depending on the degree of porous medium heterogeneity, we consider the influence of both Fickian and non-Fickian transport as well as the stochastic nature of the bimolecular chemical reaction on the effective mixing zone. We stress that consideration of PT simulations in two and three dimensions (2d, 3d) is critical. In contrast to real systems, random walks in 1d guarantee that all A and B particles will interact, generally over short length scales, giving rise to a distorted mixing zone.
 A prime example of the continuum approach mentioned above is the ADRE [e.g., Raje and Kapoor, 2000; Gramling et al., 2002], which for the bimolecular reaction and in one dimension is written as ∂ci/∂t + v∂ci/∂x −∂(D∂ci/∂x)/∂x = r(cA, cB) where ci = ci(x, t) (i = A, B) is the concentration, v is the velocity, D is the dispersion coefficient and ∣r(cA, cB)∣ is the total rate of product creation via reaction. A common form is r(cA, cB) = −ΓcAcB; this coupled system involves an equation for each reactant and one for the product (cC) with the source ΓcAcB. Many studies are based on analytical and numerical solutions of the ADRE [e.g., Najafi and Hajinezhad, 2008; Rubio et al., 2008]. The ADRE assumes that dispersion is Fickian and that the reactants undergo continuous mixing along the flow path. As such, the reactants are assumed to be well-mixed so that deterministic rate laws based on local concentrations of reacting species are employed. While Eulerian and Lagrangian descriptions of (reactive) transport should in principle yield identical results [e.g., Gardiner, 2004], the ADRE description may in the case of reactive transport require high spatial resolution to fully capture, if at all, the local-scale heterogeneous (non-uniform) reactant and product concentration distributions. An informative critique of this approach is given by Palanichamy et al. . We stress that in contrast to the continuum (ADRE or other) framework, the PT approach monitors particle positions in both space and time, and accounts naturally for both particle spreading (characterized by variance) and mixing (i.e., relative positions between particles, characterized by covariance).
 As an example of bimolecular reactive transport in porous media, we consider the laboratory column experiment described by Gramling et al. . Colorimetric reactions between CuSO4 and EDTA4− were measured in a 1d flow field, in a cell (36 cm (length) × 5.5 cm (width) × 1.8 cm (depth)) packed with cryolite. The flow cell was saturated with EDTA4− (denoted as B) and CuSO4 (denoted as A) was subsequently injected as a step input. The mixing and reaction zone of the product CuEDTA2− was measured and compared to an analytical solution of the ADRE. An earlier, similar experimental study was reported by Raje and Kapoor .
 We consider a general CTRW-PT model that incorporates bimolecular reactions. The model involves numerical simulation via application of the equations of motion [Dentz et al., 2004, 2008], s(N+1) = s(N) + ς(N), t(N+1) = t(N)+τ(N) where (s(N), t(N)) denotes the location of a particle in space-time after N steps. The spatial and temporal random increments ς(N) and τ(N) are distributed according to the joint transition displacement and time probability density ψ(s, t). These equations comprise a full range of CTRWs, including, e.g., general coupled space-time behavior and correlation with previous steps. A particular feature is that this model accounts for the time difference between the particle “step” movements and their typical reaction times.
 Here, we use a decoupled form of the random walk, ψ(s, t) = F(s)ψ(t), wherein the spatial s, angle ϕ and time steps are sampled from different, independent distributions, p(s), Ω(ϕ) and ψ(t), respectively (see Dentz et al.  for detailed discussions of coupled and decoupled formulations and the angular dependence Ω(ϕ)). To obtain an ADE from these distributions, we assign decaying exponential distributions, exp(−λst) and exp(−λtt), to p(s) and ψ(t), respectively. We derive λs and λt using the definitions (for 2d) vψ = and Dψ = , where vψ and Dψ are the generalized particle velocity and dispersion, respectively [Berkowitz et al., 2006]. In simulations of non-Fickian transport, we choose a truncated power law (TPL) for the time distribution and an exponential distribution for the spatial transition. The TPL is characterized by three parameters: the power law exponent β, the characteristic transition time t1, and the cut-off time to Fickian transport, t2 (see, e.g., Berkowitz et al.  for details).
 To simulate bimolecular reactions, two types of particles, marked “A” and “B” are introduced into the domain. We record the position of each particle at specified times, as it migrates through the domain. A sampling time step, Δt, is specified, at which we calculate the distance between a given A particle and a B particle. If this distance is smaller than a prescribed effective reaction radius, R, then A and B are replaced by a “C” particle. The C particles are allowed to migrate through space by the same rules as the A and B particles, although if desired, they can be held in place (“precipitated”), or the reaction can be treated as reversible. A further modification can be introduced to effect different reaction rates, i.e., a probability to produce a “C” particle even if the A, B separation is ≤R. A simple way to effect this stochastic variation in the rate is to replace the fixed R by a normal distribution of R, N(Rm, σ), where Rm is the mean radius and σ is the standard deviation. The reaction rate features of the model are quite flexible and other investigations are in progress (e.g., by relating R directly to specific reaction rate coefficients). Clearly, the choice of R will affect the degree of particle interaction and production of C particles; comparing simulations to experiments can in fact be used to estimate the effective reaction radius in a physical system (see discussion below).
 The simulation procedure is numerically intensive, but not impractical; calculations with a C compiler, using for example 50,000 particles each of A and B, required only one hour on a Pentium 4, 2.4 GHz processor with 1 GB of RAM. With 50,000 particles, we find that discrete artifacts in the spatial distribution are virtually eliminated; the difference in peak value of the spatial distribution of C particles between 50,000 particles and 100,000 particles is less than 1%.
 We focus on an experiment of Gramling et al. . To compare the CTRW-PT model against this experiment and the ADRE solution, the initial conditions of the experiment require uniformly distributed B particles in the bounded flow domain. From this spatial distribution, the concentration of particles per unit length (c0 = number of particles/length of flow cell (36 cm)) is used to define the influx of A particles distributed uniformly along the inlet face. Specifically, the inflow of A particles per time step Δt is set equal to the number of particles in a length of flow cell Δx/v. Thus, the inflow spatial concentration of A particles equals the outflow spatial concentration of B particles, with C particles being formed in the mixing zone.
 We set vψ = 0.67 cm/s and Dψ = 0.175 cm2/s, to match the fluid velocity and dispersion values of a specific experiment of Gramling et al. , so that λs = 5.74 cm−1 and λt = 1.92 s−1 for the decaying exponential forms of p(s) and ψ(t), respectively. The angle ϕ representing the direction of each spatial transition was chosen from a normal N(0, 1) distribution between (, ); each particle that “hit” the boundary of the domain was advanced at the same angle of transition. With this angular range, the PT simulations using the prescribed values of λs and λt yielded concentration profiles with approximately the same values of v and D chosen above. Employing the ADE form of our PT model with only the uniformly distributed A particles, we confirmed that the calculated x position of the particle center of mass at different times matched the center of mass velocity. The dispersion coefficient, D, was estimated by fitting the solution cA(x, t)/c0 = 0.5 erfc [(x−vt)/(2)] to the PT simulations, which was run for a constant influx of A particles. This is an approximate solution valid for D ≪ vx in a flow domain with infinite boundaries. Note that because in general vψ ≠ v and Dψ ≠ D, small variations between the results can arise. These values of vψ and Dψ were then used in the analytical solution of the ADRE for the distribution of C particles [Gramling et al., 2002]: cC(x, t)/c0 = 0.5 erfc [(vt−x)/(2)], for x < vt; 0.5 erfc [(x−vt)/(2)], for x > vt.
3. Results and Discussion
 Results are shown for particle profiles c/c0, where c denotes the PT simulation of cA,B,C, the analytical solution of the ADRE and the measured profile [Gramling et al., 2002]. A feature of the experiment is the direct measurement of a profile rather than only breakthrough curves. There are three basic comparisons among these various c: the PT cA,B,C in the Fickian (ADE) mode (Figure 1), with the experimental data and the analytical ADRE cC (Figure 2) and with the PT-TPL (non-Fickian) mode (Figure 3).
Figure 1 shows the spatial distribution of particles A, B, and C, with R = 0.5 cm, using the ADE form. The A particles show the step inflow (cA/c0 = 1) and then a sharp decline and forward tail. The B particles, originally distributed uniformly, leave a backward tail as they advance with the bulk fluid flow. Between these two profiles, C particles are produced in the mixing zone; cC is distributed approximately symmetrically between the forward and backward tails of the A and B particles. Increasing R leads to an increased number of C particles. The time step Δt was determined to be the value where further decrease caused no change in the production of C particles.
Figure 2 shows PT-ADE mode simulations, compared to the ADRE analytical solution and to the experimental measurements (points) of Gramling et al. . The simulations are given for two cases, with a fixed R = 0.5 cm, and with normally distributed N(0.5, 1) values of R sampled at each time step. Gramling et al.  showed that the analytical solution of the ADRE significantly over-predicts the experimentally measured reaction product by up to 40% of the peak value, as seen in Figure 2. The experiments of Raje and Kapoor  also found that while the analytical solution of the ADE without a reactive component quantifies transport of each reactant separately, the analytical solution of the ADRE over-predicts the amount of reaction product by this amount. Both studies suggested that this error is due to the high degree of mixing at the local scale assumed in the ADRE formulation.
 On the other hand the resemblance of the C particle concentration profiles between the PT simulations (Figure 2) and the measurements of Gramling et al.  is striking. The value R = 0.5 cm can be considered representative of the mixing volume in which the reaction takes place; this value seems reasonable given the average grain size of 0.13 cm, and the relatively high fluid velocity of 0.67 cm/s. Comparing the PT simulations for the fixed R and a variable R (N(0.5,1) distributed) at each time step, it is seen that the latter condition leads to a slightly broader concentration profile. Moreover, an R can be defined above which the particles appear well mixed, i.e., the peak of the C profile increases, so that the ADRE can be recovered. The inset in Figure 2 shows a comparison of the fit of the PT-ADE against the measurements, allowing for a small degree of experimental error. Clearly, even a 1% deviation in the estimated inlet fluid velocity or initial time at which A particles enter the system – not unreasonable given the practical limitations of controlling the change in fluid inflows, and the resolution of images used to estimate cC – modifies the location of the predicted peak concentration and of the relative fit to the tails of the concentration profile. These results clearly contrast the ADRE assumption of uniform mixing and the particle tracking accounting of local-scale heterogeneity and non-uniform mixing, and support the arguments against the use of the ADRE given in the Introduction. In other words, the ADRE picture of mixing does not hold because the degree of reaction is strongly dependent on the local-scale spatial distribution of the reactants.
 While the column experiment of Gramling et al.  is well-matched by the ADE form of the CTRW-PT model, asymmetric spatial particle plumes that are distinct from Fickian patterns (whether in the forward and/or backward directions) are often present in both “homogeneous” and heterogeneous porous media [Berkowitz et al., 2006]. In such cases, transport is often better described within the CTRW framework [Berkowitz et al., 2006] using a temporal distribution with a significant power-law region. Particle tracking simulations with the TPL form were therefore considered, assuming a moderate degree of non-Fickian transport (β = 1.6); this value is typical of transport behavior in many columns [e.g., Berkowitz et al., 2006], and leads to spatial concentration profiles with heavier backward tails.
Figure 3 shows the spatial distribution of particles A, B, and C, with normally distributed N(0.5, 1) values of R sampled at each time step, using the TPL form (β = 1.6, t1 = 2L/v, where L = 0.13 cm (the grain size) and v = 0.67 cm/s, t2 = 105 s); for comparison, the corresponding C profile using the PT-ADE form is also shown. With the TPL, the backward tail of B particles is heavier than the forward tail of A particles, and as a result, the concentration profile of reaction product C particles is highly asymmetrical. In general, the spatial distribution of C particles is broader than that for the ADE form. Note that the relative concentration of A reaches a value greater than 1, which is related to the integral effect of the inflows for a TPL pdf (which will be investigated in a further study).
4. Concluding Remarks
 The reactive transport problem considered here involves a separation between two different length scales. One is the length scale of the flow cell, at which on average, A and B particles may appear well mixed. The second length scale is of the pores where the mixing particles react. As a consequence, there is less reaction than might be expected at the large scale, such as from the analytical solution of the ADRE which assumes a greater degree of mixing.
 The CTRW-PT model (i.e., ADE-PT, TPL-PT) enables a general representation of reactive transport, which is not bound by the assumptions of ADRE-like models that require the concentrations to be continuous and deterministic, and the transport to be Fickian. Though the CTRW-PT model does not account explicitly for physical pores in the medium, it includes the effects of local-scale dynamics via the pdf's applied to describe the particle transport. The effective distance over which particles mix and react in any specific experiment is given by the effective reaction distance R; increasing R has the effect of increasing the mixing between A and B particles, and thus of increasing the amount of product C.
 The analysis here leads to two principal conclusions. First, the use of partial differential equation approaches to quantify reactive transport must be called into question, because of the difficulty in capturing local scale mixing and reaction. Particle tracking approaches thus appear more attractive, in spite of their potential computational limitations for large-scale domains. Second, and related to the use of a PT approach, reactions require that reactant particles be found in close proximity in both space and time.
 We thank the Israel Ministry of Science and Technology for financial support.