Characterization of Bimolecular Reactive Transport in Heterogeneous Porous Media

Abstract

We characterize the role of preferential pathways in controlling the dynamics of bimolecular reactive transport in a representative model of a heterogeneous porous medium. We examine a suite of numerical simulations that quantifies the irreversible bimolecular reaction $A+B\rightarrow C$, in a two-dimensional heterogeneous domain (with log-conductivity, Y), wherein solute A is injected along an inlet boundary to displace the resident solute B under uniform (in the mean) flow conditions. We explore the feedback between the reactive process and (a) the degree of system heterogeneity, as quantified by the unconditional variance of Y, $1 \le \sigma _Y^2\le 7$, representing moderately to strongly heterogeneous media, and (b) the relative strengths of advective and diffusive mechanisms, as quantified by a grid Péclet number, $\textit{Pe}_ {\Delta }$. Our analysis is based on the identification of particle preferential pathways, focusing on particle residence time within cells employed to discretize the flow domain. These preferential pathways are formed mainly by high conductivity cells and generally contain an important component of (sometimes isolated and a relatively small number of) lower conductivity values. A key finding of our analysis is that while the former dominate the behavior, the latter are shown to provide a non-negligible contribution to the global number of reactions taking place in the domain for strongly heterogeneous media, i.e., for the largest investigated values of $\sigma _Y^2$. Reactions are detected across the complete simulation time window (of about 5.5 pore volumes) for the strongly advective case. When diffusion plays an important role, the reactive process essentially stops after the injection of a limited amount ($\sim$2.5) of pore volumes.

Appendix: Dependence on Number of Realizations

We illustrate here selected results of a convergence study that examined the dependence of our results on the number of realizations, $n_Y$. All results presented here are for the case with $\sigma _Y^2 =7$. Figure 11 shows how the total number of reactions, $N_R$, displayed in Fig. 2, depends on $n_Y$ for various pore volumes eluted from the system. Here, $N_R$ is normalized with respect to the results shown in Fig. 2, obtained for $n_Y=20$. Note that oscillations of limited amplitude are observed for $n_Y>$ 15 at all simulation times. Figure 12a depicts the frequency distribution $F_{\textit{YR}}$ shown in Fig. 4a for $n_Y =5$, 10, 15, 20. These results are complemented by Fig. 12b which shows the convergence of the quantities $M_{\textit{YR}}$ (with $M =\mu$, $\sigma ^{2}$, or $\gamma )$ that are considered in Fig. 4b. All three statistical moments display only small variations with respect to the result obtained with 20 realizations for $n_Y>15$. Results in Figs. 11 and 12 suggest that the number of realizations employed in this study is sufficient to characterize the reactive process in the entire domain and the distribution of reactions conditional to conductivity values.

Fig. 11

Total number of reactions, $N_R$, versus $n_Y$ for various pore volumes eluted from the system and for $\sigma _Y^2 =7$. Here, $N_R$ is normalized with respect to $N_R$(20), i.e., to the results shown in Fig. 2 and obtained for $n_Y=20$. Thin colored curves correspond to $N_R$ evaluated at a set of selected pore volumes (${\textit{PV}}<5.5$). The black bold curve corresponds to $N_R$ at the final simulation time (${\textit{PV}} = 5.5$)

Fig. 12

Impact of the number of realizations $n_Y$ on the results shown in Fig. 4; a same results as in Fig. 4a for $\sigma _Y^2 =7$ computed on the basis of $n_Y = 5$ (green), 10 (red), 15 (blue) and 20 (bold black) realizations; b related statistical moments of the distributions (results are normalized with respect to the results shown in Fig. 4b–d and obtained for $n_Y=20$ and $\sigma _Y^2 =7$