Evaluation of the Effects of Porous Media Structure on Mixing

The objectives of this work were to determine if a pore-scale model could accurately capture the physical and chemical processes that control transver...
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Environ. Sci. Technol. 2008, 42, 3185–3193

Evaluation of the Effects of Porous Media Structure on Mixing-Controlled Reactions Using Pore-Scale Modeling and Micromodel Experiments THOMAS W. WILLINGHAM,† CHARLES J. WERTH,* AND ALBERT J. VALOCCHI Department of Civil and Environmental Engineering, University of Illinois Urbana–Champaign, Urbana Illinois

Received September 18, 2007. Accepted February 07, 2008.. Revised manuscript received February 04, 2008

The objectives of this work were to determine if a porescale model could accurately capture the physical and chemical processes that control transverse mixing and reaction in microfluidic pore structures (i.e., micromodels), and to directly evaluate the effects of porous media geometry on a transverse mixing-limited chemical reaction. We directly compare porescale numerical simulations using a lattice-Boltzmann finite volume model (LB-FVM) with micromodel experiments using identical pore structures and flow rates, and we examine the effects of grain size, grain orientation, and intraparticle porosity upon the extent of a fast bimolecular reaction. For both the micromodel experiments and LB-FVM simulations, two reactive substrates are introduced into a network of pores via two separate and parallel fluid streams. The substrates mix within the porous media transverse to flow and undergo instantaneous reaction. Results indicate that (i) the LB-FVM simulationsaccuratelycapturedthephysicalandchemicalprocess in the micromodel experiments, (ii) grain size alone is not sufficient to quantify mixing at the pore scale, (iii) interfacial contact area between reactive species plumes is a controlling factor for mixing and extent of chemical reaction, (iv) at steady state, mixing and chemical reaction can occur within aggregates due to interconnected intra-aggregate porosity, (v) grain orientation significantly affects mixing and extent of reaction, and (vi) flow focusing enhances transverse mixing by bringing stream lines which were initially distal into close proximity thereby enhancing transverse concentration gradients. This study suggests that subcontinuum effects can play an important role in the overall extent of mixing and reaction in groundwater, and hence may need to be considered when evaluating reactive transport.

1.0 Introduction Degradation of contaminants through chemical and biological processes has proven to be one of the most effective mechanisms for decreasing chemical contamination in groundwater. For contaminants from a persistent source, * Corresponding author e-mail: [email protected]; fax: 217-3336968. † Currently at ExxonMobil Upstream Research Corporation, Houston, Texas. 10.1021/es7022835 CCC: $40.75

Published on Web 04/05/2008

 2008 American Chemical Society

degradation primarily occurs in narrow regions along plume fringes where limiting substrates from the surrounding groundwater mix with the contaminant plume transverse to the direction of flow (1–6). Most of the work in the literature has focused on measuring dispersion as a surrogate to quantify chemical mixing; however, several recent studies have indicated that dispersion may not accurately quantify mixing (e.g., refs 7–9). Despite the extensive research on dispersion, it is not clear what effect pore geometry will have on a transverse mixing-limited chemical reaction. One approach to evaluate transverse mixing-limited degradation is to use pore-scale modeling (1). By evaluating transport at the pore scale, it is possible to directly include, and hence evaluate, the physical mechanisms controlling transverse mixing and chemical reaction within porous media. In a network-model study, Sahimi et al. (10) found that transverse dispersion was related to the pore radius distribution, pore coordination (number of pores at a junction), pore blockage, and orientation of pores. In another study, Eidsath et al. (11) evaluated the effect of grain size distribution on dispersion by using grains with two different radii for a porous media that consisted of a regular array of staggered cylinders. They found that when the ratio of cylinder radii was decreased from 5 to 2, transverse dispersion increased by approximately a factor of 1.5. Grain size can also affect transverse dispersion. De Josselin de Jong (12) found that transverse dispersion increases with mean grain size, whereas results from Buyuktas and Wallender (13) indicated that transverse dispersion decreased in disordered packing structures with increasing grain size. Several authors have also conducted pore-scale studies to evaluate the effect of packing structure on transverse mixing. Edwards et al. (14) found that for an inline 2D array of cylinders, transverse dispersion increased with bed porosity, while for a 2D hexagonal array of cylinders, transverse dispersion initially increased and then decreased with increasing bed porosity. The same authors also found that transverse dispersion increased with bed tortuosity. In addition to porosity and tortuosity, packing structure also affects pore body and throat sizes. Cao and Kitanidis (15) found that transverse dispersion was enhanced in narrow pore throats due to the focusing of streamlines. These results all indicate that grain and pore size geometry affect transverse mixing. Immobile fluid zones in aggregates have also been found to affect solute dispersion in porous media. Zhang and Kang (16) found that transverse dispersion initially increased then approached a constant value within a short time scale for a conservative solute injected into the fracture of a dual permeability porous media. While most of the work reported in the literature has focused on dispersion of conservative solutes, one study evaluated the effect of pore geometry on chemical degradation. Dykaar and Kitanidis (17) evaluated chemical degradation in a stagnant biofilm which uniformly coated an undulating pore. Biomass located near pore throats more effectively degraded contaminants compared to biomass within the pore body due to enhanced mixing within pore throats. Despite the numerous pore-scale studies evaluating dispersion available in the literature, in general, pore-scale models of mixing and reactive transport have not been directly tested against experimental data due to measurement limitations. The main objectives of this work were first to determine if a pore-scale model could accurately capture the physical and chemical processes that control transverse mixing and reaction in microfluidic pore structures (i.e., micromodels), VOL. 42, NO. 9, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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and second to directly evaluate the effects of porous media geometry on transverse mixing and chemical reaction. Five different pore structures are evaluated to test the following hypotheses: (1) mixing and product formation are controlled by the contact area between plumes of reactive substrates, and hence can not be predicted based on grain size alone, (2) mixing and product formation decrease with increased bypassing of intra-aggregate porosity, and (3) mixing and reaction are affected by pore geometry. In both the experiments and simulations, we control the exact porous-media structure and therefore are able to directly evaluate how changes in porous media structure affect mixing and product formation. For all experiments and simulations, we evaluate flow and transport at steady state. This replicates field conditions where contaminant degradation is controlled by transverse mixing of solutes along plume fringes. We outline the methods in Section 2, discuss the five model porous media structures in Section 3, directly compare nonreactive and reactive LB-FVM simulations with micromodel experiments in Section 4.1, and present LB-FVM simulations evaluating the effect of pore geometry on mixing and product formation in Section 4.2. Discussion and conclusions are presented in Section 5.

2.0 Materials and Methods 2.1. Micromodels. The microfluidic devices used in this study were fabricated at the Micro-Nano-Mechanical Systems Cleanroom Laboratory at the University of Illinois Urbana–Champaign. Standard photolithography techniques were used (18). Silicon wafers were coated with a photoresist (PR) polymer, and selectively exposed to UV light by placing a mask directly above each wafer. The areas of PR which were exposed to UV light were weakened, and were removed utilizing a developer. Exposed areas of silicon (e.g., areas not covered by PR) were etched using plasma generated from an inductively coupled plasma-deep reactive ion etching (ICPDRIE) system. Flow channels were sealed by anodically bonding thin Pyrex glass to the top of each silicon wafer. Each micromodel contains a porous media section 1 cm wide by 2 cm long. Solutes were injected through two separate inlet ports, and hence were unable to mix until they entered the porous media. We recognize that 2D pore structures do not capture all of the physics present in real 3D porous media (e.g., twisting of streamlines); however, the diffusive and mechanical dispersion processes that control mixing in 2D also affect mixing in 3D, and this work allows us to isolate and quantify these effects. 2.1.1. Micromodel Visualization. In all experiments, the low-affinity Ca fluorophore Oregon Green 488 Bapta-5N (OG5N) in the hexapotassium salt form was used (Invitrogen Inc.). OG5N reacts with Ca2+ in a 1:1 stoichiometric ratio to form a fluorescent product. While no studies to our knowledge have measured the reaction rate of OG5N with Ca, the reaction rate between Ca and a similar low-affinity Ca fluorophore, Ca-orange-5N (Invitrogen Inc.), has been reported to have a reaction rate of 3 × 108 M-1 s-1 (19). In our work, the reactive time scale is much faster than the advective and diffusive time scales, and hence is treated as instantaneous (4). A Nikon Epiphot 200 epi-fluorescent microscope with a 5× objective, Prior Scientific Instrument motorized stage, and a RT Spot CCD digital camera were used to capture images of fluorescent intensity within the pore space for each micromodel experiment. A total of 27 separate images were taken at steady state for each experiment. Image correction was systematically applied to each image to correct for nonuniform illumination prior to montaging the 27 separate images to form a single image which captured the entire mixing zone region. 3186

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2.2. Lattice-Boltzmann Finite Volume Method. The lattice-Boltzmann (LB) method was used to solve the steady Stokes equation for interstitial pore velocities within the model pore structures. We used a standard single relaxation time Bhatnagar-Gross-Krook (BGK) (20) implementation (e.g., refs 21–23). No-slip boundary conditions were imposed at all soil grain surfaces via a bounce-back boundary condition (22, 23). Node spacing was set to 5 µm for all simulations and was tested using a sensitivity analysis to ensure that grid spacing did not affect system results. Reactive transport was simulated using the finite-volume method (FVM) to solve the steady advection-diffusion-reaction equation. Since there are no adjustable parameters, it is possible to rigorously evaluate the ability of our pore-scale numerical model to capture the physical and chemical process controlling transversemixingandreactioninourmicromodelexperiments. 2.2.1. Reactive Transport. We represent the reaction between OG5N and Ca using the elementary reaction A+BfC

(1)

where the rate of chemical transformation for each species is defined by d[C] d[A] d[B] ) )) -k[A][B] dt dt dt

(2)

Transport and reaction of solutes A, B, and C are described by the steady-state advection-diffusion-reaction equation bA) - kAB 0 ) DA∇2A - ∇ (u

(3)

0 ) DB∇2B - ∇ (u bB) - kAB

(4)

0 ) DC∇2C - ∇ (u bC) + kAB

(5)

where k is the intrinsic reaction rate, DA, DB, and DC are the molecular diffusion coefficients of solutes A, B, and C, respectively, and b u is the steady-state velocity field which is determined from the LB solution. Inlet, outlet, and grain boundary conditions are defined to represent the conditions of the micromodel experiments. Dirichlet boundary conditions are used along the inlet boundary where cin ) Bo for the top half and cin ) Ao for the bottom half. A zero gradient condition is used at the downstream outlet (dc/dx ) 0), a free exit boundary condition (24) is used along the top and bottom boundaries, and a zero flux boundary condition is imposed at grain-fluid boundaries. Standard second-order approximations are used for the advective and dispersive flux (e.g., ref 25). The system of equations is linearized and solved using an iterative Newton–Raphson approach. For all LB-FVM simulations, k was set to an arbitrarily large value to mimic an instantaneous reaction. Solute A was modeled as calcium, solute B was modeled as OG5N, and solute C was modeled as OG5N+Ca. The molecular diffusion coefficient for Ca in water at infinite dilution is 7.9 × 10-6 cm2 /s (26), while the molecular diffusion coefficient for OG5N+Ca is 3.6 × 10-6 cm2 /s (27). Willingham (27) measured the molecular diffusion coefficient for 0G5N+Ca by measuring the transverse spreading of OG5N+Ca in a microfluidic channel that did not contain any grains; hence, the only component that contributed to transverse spreading was molecular diffusion. To evaluate the molecular diffusion coefficient for OG5N, we used the Hayduk and Laudie (28) method to estimate the approximate difference in molecular diffusion coefficients for OG5N and OG5N+Ca. Due to the large molecular weight (and hence molar volume) of OG5N compared with Ca, the estimated difference in molecular diffusion coefficients was less than 1.4%; hence, we used the same molecular diffusion coefficient for OG5N+Ca and OG5N (i.e., 3.6 × 10-6 cm2 /s). Additional information, including an example calibration curve and details regarding the conversion of LB-FVM

FIGURE 1. SEM images of porous media structures evaluated: (A) periodic array of small cylinders, (B) periodic array of large cylinders, (C) aggregate porous media, (D) horizontal ellipses, and (E) vertical ellipses.

TABLE 1. Micromodel Experimental Conditionsa structure

flow rate (µL/hr)

depth (µm)

porosity

Darcy velocity (m/s)

small cylinders large cylinders aggregates vertical ellipses horizontal ellipses

400 1000 400 200 400

28.0 33.8 41.7 42.4 25.9

0.450 0.378 0.373 0.397 0.330

3.97 × 10-4 8.22 × 10-4 2.66 × 10-4 1.31 × 10-4 4.29 × 10-4

a Experimental conditions for micromodel flow experiments presented in section 4.1; flow rate is the total flow rate for the two pumps.

simulations to intensities, is included in the Supporting Information.

3.0 Porous Media Structure The five different pore structures evaluated in this work are shown in Figure 1. The base (small cylinder) case consists of a periodic staggered array of cylinders, 300 µm in diameter with 40 µm pore throats. The second pore structure has the same periodic array, but larger grain (600 µm) and pore throat (80 µm) sizes (large cylinder). The third consists of aggregates, each comprising 16 primary cylinders, 300 µm in diameter, with 9 smaller secondary cylinders, 100 µm in diameter, placed inside the pore bodies of the primary cylinders. The intra-aggregate porosity for this case accounts for 43% of the total porosity, which is in good agreement with the 48% intraaggregate porosity for the aggregated soil used by Brusseau (29). The fourth and fifth pore structures are periodic staggered arrays of horizontal and vertical ellipses, respectively. Each ellipse has the same area as a small (300 µm) cylinder. Experimental conditions for all five porous media structures are detailed in Table 1.

4.0 Results 4.1. Comparison of LB-FVM with Micromodel Experiments. A direct comparison between nonreactive micromodel experimental results and LB-FVM simulations is shown in Figure 2 for the small cylinder and aggregate pore structures. Experimental and simulation results for other pore structures are not presented but show similar agreement.

Recall that the only input parameters for the LB-FVM are inflow rate, pore geometry, reaction rate, and species molecular diffusion coefficients. Qualitatively, the intensity distribution in the experiments and simulations agree. Experimental and simulation intensity profiles were closely compared at multiple cross sections, but results from only one downgradient cross section are shown here for brevity. Intensity profiles at other locations are provided in the Supporting Information. For the small cylinder pore structure, point measurements from the micromodel experiment and LB-FVM simulation at the downgradient cross section (i.e., at 16 530 µm) are shown in Figure 3A. In general there is good agreement. However, there is some discrepancy in the magnitude of intensity due to the center line in the micromodel experiment being shifted toward the upper section. This discrepancy may be due to a small difference in flow rate between the two ISCO syringe pumps used in the micromodel experiments. For the aggregate pore structure, point measurements from the micromodel experiment and LB-FVM simulation at the downgradient cross section are shown in Figure 3B. As with the small cylinder pore structure, there is generally good agreement. However, for transverse distances greater than zero, experimental intensities are less than LB-FVM predictions. This discrepancy can be explained by noting that in Figure 2, the mixing interface between OG5N+Ca and MilliQ water is shifted slightly toward the OG5N+Ca side at the downgradient location. As before, this discrepancy is most likely due to small variations in the flow rate between the two syringe pumps. Direct comparisons between reactive micromodel experiments and LB-FVM simulations for the small cylinder and aggregate pore structures are shown in Figure 4. In the unbound state, the fluorophore OG5N has some background fluorescence, which is noticeable in the top half of the micromodel images. Qualitatively, the intensity distribution in the experimental and model systems agree. As with the nonreactive experiments, experimental and model intensity profiles were closely compared at multiple cross sections, but results from only one downgradient cross section are shown here for brevity. For the small cylinder pore structures, point measurements from the micromodel experiment and LB-FVM simulation at the downgradient cross section are shown in VOL. 42, NO. 9, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 2. Direct comparison of nonreactive (A and C) fluorescence intensity measurements in the micromodel obtained from digital images of the pore structure taken through the objective of an epi-fluorescent microscope and (B and D) LB-FVM simulations in a numerical 2D pore network converted to fluorescence intensity. Fluorescence intensity is linearly related to individual solute concentrations (see Supporting Information). Micromodels and numerical pore networks contain (A and B) a periodic array of cylinders and 2D circles with a Darcy flow rate of 0.0397 cm/s, or (C and D) a periodic array of aggregates with a Darcy flow rate of 0.0267 cm/s.

FIGURE 3. Evaluation of intensities under nonreactive conditions for the small cylinder (A) and aggregate eighth unit cell (B). The intensity cross section for the small cylinders was taken at a longitudinal distance of 16 530 µm while the intensity cross section for the aggregate was taken at the centerline of the eighth aggregate unit cell, 14 587 µm. Figure 5A. Agreement is generally good. As before, the micromodel intensity profile is shifted toward the OG5N (i.e., left) section. Again, this may be due to slight differences in the flow rate between the two ISCO syringe pumps. For the aggregate pore structure, point measurements from the micromodel experiment and LB-FVM simulation are shown in Figure 5B. Overall, there is very good agreement between the micromodel and LB-FVM simulation except for the peak at a transverse distance of –1000 µm. To compare the overall ability of the LB-FVM to capture the extent of reaction and product formation for the micromodel experiments, we directly compare transverseintegrated fluorescent intensity as a function of distance. The results are plotted in Figure 6. It is important to note 3188

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that the flow rate (Table 1) and imaging conditions are slightly different for each experiment; hence, it is not possible to directly compare product intensities between the different experiments. For the vertical ellipse case, Figure 6 indicates a dip in total intensity around a longitudinal distance of 0.5 cm for the micromodel experiment. This dip occurs in an area where physical discoloration of the micromodel occurred during ICP-DRIE etching. This discoloration led to a decrease in fluorescence in the affected area. Downgradient, intensity values approach LB-FVM predictions indicating that the decreased fluorescence was most likely due to increased absorbance of emitted photons on the micromodel surface rather than a chemical effect, which would have been evident downgradient also. For the other porous media structures,

FIGURE 4. Direct comparison of reactive (A and C) light intensity in the micromodel and (B and D) light intensity in a numerical 2D pore network obtained from LB-FVM simulations. Micromodels and numerical pore networks contain (A and B) a periodic array of cylinders and 2D circles with a Darcy flow rate of 0.0397 cm/s, or (C and D) a periodic array of aggregates with a Darcy flow rate of 0.0266 cm/s.

FIGURE 5. Evaluation of intensities under reactive conditions for the small cylinder (A) and the aggregate eighth unit cell (B). The intensity cross section for the small cylinders was taken at a longitudinal distance of 16 530 µm while the intensity cross section for the aggregate was taken at the centerline of the eighth aggregate unit cell, 14 587 µm. modeled and measured total intensity values match closely over the entire longitudinal distance. It is important to note that all LB-FVM simulations are true predictions since all input parameters were determined independently. The good agreement between the cross section profiles for the nonreactive and reactive LB-FVM simulations with the micromodel experiments, and the strong agreement between LBFVM predictions of total intensity with experimental measurements, indicate that we adequately captured the physical and chemical processes controlling the nonreactive and reactive micromodel experiments using our pore-scale LB-FVM.

4.2. Effects of Porous Media Structure on Mixing and Chemical Transformation. In this section we use the LBFVM to directly evaluate the effect of porous media structure on mixing and product formation. To achieve this objective, each porous media structure was designed to have an equivalent porosity (Table 2), and each simulation was run at a Darcy flow rate of 0.0111 cm/s. As a result, the residence time for each scenario is identical. Since numerical simulations are used exclusively in this section, all results are presented in terms of either concentration or product mass. Product formation results for all porous media configurations are shown in Figure 7. We continue to use the molecular VOL. 42, NO. 9, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 6. Comparison of total intensity measured from LB-FVM simulations and micromodel experiments for all porous media structures. Dashed lines indicate LB-FVM simulations and solid lines indicate micromodel experiments.

TABLE 2. Flow and Porous Media Parameters for LB-FVM Simulations structure small cylinders large cylinders aggregate vertical ellipse horizontal ellipse

porosity Darcy velocity (m/s) 0.40 0.39 0.39 0.41 0.39

1.11 × 10-4 1.11 × 10-4 1.11 × 10-4 1.11 × 10-4 1.11 × 10-4

Vx (m/s)

Pea

2.84 × 10-4 12.2 2.85 × 10-4 24.2 2.66 × 10-4 752.1 2.66 × 10-4 221.6 2.88 × 10-4 0.6

a Pe ) (Vx * λT)/D where λT is the transverse length scale of the unit cell.

diffusion coefficients for Ca, OG5N, and OG5N+Ca for solutes A, B, and C, respectively, in this section. As reported in Section 2.2.1, the molecular diffusion coefficient for Ca is ∼2.2 times the molecular diffusion coefficient for OG5N; hence, product formation for each case is shifted slightly toward the OG5N side (i.e., upward) due to Ca having a higher molecular diffusion coefficient (Figure 7). Product mass as a function of distance is plotted for each porous media configuration in Figure 8. Product mass was calculated by integrating product concentrations over the width for each porous media configuration. Results are smoothed by averaging product mass profiles over a unit cell in the longitudinal direction using a moving average. For both small and large cylinders, product formation, and hence extent of mixing, is nearly identical. This can be explained by noting that although the grain size is doubled in the large periodic cylinder scenario, the porosity is kept the same. Hence, pore throat diameter and grain spacing are doubled also. This results in equivalent contact time and interfacial area between solute species for the two scenarios. In agreement with our first hypothesis, this indicates that grain size alone cannot account for variations in transverse mixing. To evaluate the effect of intra-aggregate porosity on transverse mixing, we compare product formation results from the aggregate scenario with the two periodic cylinder scenarios. For steady-state conditions, product formation rates are nearly identical to those for the large and small 3190

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cylinder scenarios (Figure 8). In contrast to our second hypothesis, this indicates that transverse mixing is not significantly affected by intra-aggregate porosity under steady-state conditions. This can be explained by noting that the aggregate structure has the same total porosity as the periodic cylinder scenarios; hence, this scenario has approximately the same contact area and contact time between reactive solute species as in the periodic cylinder scenarios. We also evaluated two ellipse structures, one oriented parallel and the other transverse to the flow direction. The horizontally aligned ellipse structure (Figure 8) has the lowest product formation rates of the porous media structures evaluated in this study. This finding can be explained by three mechanisms: first, the horizontally aligned ellipse structure limits contact between stream lines above and below grains to the narrow horizontal pore throats between grains. Second, this scenario does not create significant transverse convective fluxes which were shown by Salles et al. (30) and Buyuktas and Wallender (13) to be important for transverse dispersion. Third, this scenario does not significantly compress stream lines which would promote mixing by bringing reactants into closer contact (15). On the other hand, the vertical ellipse structure demonstrates the highest amount of mixing and product formation of all the cases evaluated in this study (Figure 8). For this scenario, the three mechanisms listed for the horizontal ellipse structure work to enhance transverse mixing. Specifically, the vertical ellipse structure has the highest contact area between stream lines immediately above and below grains. Second, the vertical ellipse structure creates significant transverse convective fluxes. Third, flow experiences significant focusing, and hence enhanced mixing, at the pore throats between the vertically aligned ellipses. Conceptually, the focusing of stream lines between vertically aligned ellipses enhance the transverse concentration gradient by bringing solutes which are initially distal into close proximity. These results, when combined with those from the periodic cylinder cases, indicate the important impact grain orientation can have on mixing and product formation, as hypothesized.

FIGURE 7. LB-FVM results showing product concentrations for all porous media configurations. The bright region along the centerline is the product from the reaction of solutes A and B. Images correspond to (A) small cylinders, (B) large cylinders, (C) aggregate, (D) vertical ellipses and (E) horizontal ellipses.

FIGURE 8. Evaluation of product as function of distance for all porous media structures using LB-FVM simulations. All simulations were evaluated using equivalent conditions at a Darcy flow rate of 0.0111 cm/s.

5.0 Discussion Results from this study indicate that the LB-FVM simulations were able to adequately capture the physical processes that control mixing and mixing limited chemical reactions within the micromodel experiments. This study is one of the first to directly compare pore-scale numerical simulations with micromodel experimental data. The good agreement obtained between the simulations and the experimental results is direct validation that numerical simulations based upon the Stokes and advection-diffusion-reaction equation can accurately describe the physics controlling mixing and reaction in porous media. Results from the two periodic cylinder cases indicate that the commonly used approximation that transverse dispersivity, and hence mixing, is directly related to mean grain size (12, 31) does not completely account for mixing at the

pore scale. Instead, our work indicates that contact area between interfaces of reacting solute species is more important than grain size in determining the amount of mixing and extent of chemical reaction. The apparent difference between our results and the theoretical predictions by de Josselin de Jong (12) and Saffman (31) can be explained by noting than in the works by de Josselin de Jong (12) and Saffman (31), both authors assumed complete mixing at pore junctions. As a result, when solutes reached the end of a tube, they were assumed to mix completely at the junction. Hence, the probability of selecting the next tube was based purely on the relative flow rates of the tubes connected at the junction, and does not depend on the location of where the solute enters the junction. In contrast, in our simulations solutes are transported around grains following the path of a stream line. As a result, when a solute enters a junction, VOL. 42, NO. 9, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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a pore throat in our work, the transverse displacement is dependent on the stream lines followed and the molecular diffusion coefficient of the solute. In the work by Buyuktas and Wallender (13), transverse dispersion was actually observed to decrease with increasing grain size. In their work however, they evaluated disordered porous media which also had slight variations in porosity. Considering that they only evaluated one porous media realization for each grain size, and had slight variations in porosity between the different realizations, no absolute conclusions on the effect of grain size on dispersion can be drawn from their work. In our work however, we specifically evaluated the effect of grain size on dispersion by using ordered porous media of two different grain sizes with equivalent porosity. For the aggregate scenario, results indicate that at steady state, the effective porosity can be assumed equal to the total porosity. Here, overall product formation rates were similar to the periodic cylinder structures. This result is initially surprising considering the strong effect low permeability intraparticle water has on longitudinal dispersion (16, 29); however, this effect can be explained by considering that the contact area between stream lines above and below grains is similar to the periodic cylinder cases, and that for the aggregate scenario, there is no significant focusing of stream lines compared to the periodic cylinder scenarios. This finding is supported by the modeling work by Zhang and Kang (16) where the transverse dispersion coefficient of a conservative tracer injected into a fractured matrix approached a constant within a short time scale, and remained small and on the order of molecular diffusion. These results, however, may need to be revaluated for highly cemented aggregates where intra-aggregate connectivity may become discontinuous (e.g., dead end pore channels), potentially decreasing mixing and chemical transformation. Comparison of the periodic cylinder cases and the two ellipse scenarios indicates that grain orientation also plays a significant role in overall extent of mixing and product formation. This finding is supported by the network-model results by Sahimi et al. (10) who found that dispersion was related to the relationship between pore orientation and flow direction. The importance of grain orientation on mixing is also supported by the theoretical results from Souto and Moyne (32) who studied the effect of flow angle on dispersion using 2D periodic arrays of square columns. Due to the square columns, it is possible to show that grain orientation (e.g., squares compared with diamonds) was responsible for enhanced dispersion in the staggered array compared to the appropriate rotated inline array due to the narrowing (e.g., focusing) of flow channels between grains for the staggered array compared with uniform flow channels for the inline structure. In our case, the difference between the horizontal and vertical aligned ellipse structures is attributed to the combined effects of enhanced mixing in the pore throats, increased transverse convective fluxes, and increased contact time between steam tubes above and below grains. Our finding that mixing, and hence chemical reaction, can be enhanced due to flow focusing in pore throats agrees with the pore-scale findings by Dykaar and Kitanidis (17) and Cao and Kitanidis (15). Similarly, flow focusing at the macro scale has also been found to enhance mixing and chemical reaction (2, 33). In conclusion, our results indicate that (1) microfluidicbased pore networks are a promising experimental tool to study mixing and reaction, (2) the LB-FVM is able to capture the governing physics controlling mixing and mixing controlled reactions in micromodel experiments, (3) grain size alone is not sufficient to explain mixing at the pore scale, (4) interfacial contact area between solute species is important in determining solute mixing and product formation, (5) under steady-state conditions, mixing and chemical reaction 3192

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can occur within aggregates due to interconnected intraaggregate porosity, (6) grain orientation can have a significant effect on mixing and product formation, and (7) flow focusing can significantly enhance the effect of molecular diffusion on mixing by enhancing the transverse concentration gradient. Additional studies evaluating different flow velocities and packing structures are needed in order to extend the results of this study to a wider range of experimental conditions. Based on our findings, pore-scale structure can have a significant effect on mixing and product formation and hence may need to be taken into consideration when evaluating mixing controlled reactions.

Acknowledgments This work was supported in part by the National Science Foundation through grant BES-496714 and through an IPENG fellowship for T.W. A portion of the research was performed in the Environmental Molecular Sciences Laboratory sponsored by the Department of Energy’s Office of Biological and Environmental Research located at the Pacific Northwest National Laboratory.

Supporting Information Available Additional information on the calibration process, as well as comparisons taken from additional cross sections closer to the inlet mixing zone region. This material is available free of charge via the Internet at http://pubs.acs.org.

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