## 1. Introduction

[2] 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.* [2007] 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.

[3] We use a particle tracking (PT) technique to simulate the subtle features of reactions and different transport modes in porous media. The PT approach has a long history, having been first described by Smoluchowski about a century ago to treat Brownian motion. The extension of PT to account for reactions, by specifying probabilistic rules of particle interaction, has been discussed extensively [e.g., *Gillespie*, 1976, 1977; *Fabriol et al.*, 1993; *Sun*, 1999; *Cao et al.*, 2005; *Lindenberg and Romero*, 2007; *Palanichamy et al.*, 2007; *Srinivasan et al.*, 2007; *Yuste et al.*, 2008].

[4] 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].

[5] 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.

[6] 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 ∂*c*_{i}/∂*t* + *v*∂*c*_{i}/∂*x* −∂(*D*∂*c*_{i}/∂*x*)/∂*x* = *r*(*c*_{A}, *c*_{B}) where *c*_{i} = *c*_{i}(*x*, *t*) (*i* = *A*, *B*) is the concentration, *v* is the velocity, *D* is the dispersion coefficient and ∣*r*(*c*_{A}, *c*_{B})∣ is the total rate of product creation via reaction. A common form is *r*(*c*_{A}, *c*_{B}) = −Γ*c*_{A}*c*_{B}; this coupled system involves an equation for each reactant and one for the product (*c*_{C}) with the source Γ*c*_{A}*c*_{B}. 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.* [2007]. 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).

[7] As an example of bimolecular reactive transport in porous media, we consider the laboratory column experiment described by *Gramling et al.* [2002]. Colorimetric reactions between CuSO_{4} and EDTA^{4−} 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 EDTA^{4−} (denoted as *B*) and CuSO_{4} (denoted as *A*) was subsequently injected as a step input. The mixing and reaction zone of the product CuEDTA^{2−} was measured and compared to an analytical solution of the ADRE. An earlier, similar experimental study was reported by *Raje and Kapoor* [2000].