Applicability of colloid filtration theory in size-distributed, reduced porosity, granular media in the absence of energy barriers

The vast majority of colloid transport experiments have been conducted in granular porous media with narrow size distribution, which allows a single collector size and narrow values of porosity to be used in colloid filtration theory to predict deposition rates under favorable conditions (absence of energy barriers). In this work, deposition of colloids (ranging from 0.21 mm to 9.1 mm) in packed columns was examined with three different borosilicate glass bead porous media: uniform, monomodal nonuniform and bimodal nonuniform. The corresponding porosities to these media were 0.38, 0.34 and 0.28. The effect of reduced porosity on the flow field was studied using high resolution computerized X-ray micro tomography (HRXMT), Lattice-Boltzmann (LBM) flow simulation methods, and direct observations in packed flow cells. This allowed discerning the influence of preferential flow paths on fluid velocities, and it demonstrated that straining in pore throats too small to pass was not a significant contributor to colloid retention even up to particle to grain size ratios of 0.05. Mechanistic simulations successfully predicted the experimentally-observed sensitivity to porosity and absolute values of the deposition rate coefficients when a number-based mean grain size was used to represent the size-distributed porous media.

APPLICABILITY OF COLLOID FILTRATION THEORY IN SIZE-DISTRIBUTED, REDUCED POROSITY, GRANULAR MEDIA IN THE ABSENCE OF ENERGY BARRIERS by Eddy Fernando Pazmiño A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science in Geological Engineering Department of Geology and Geophysics The University of Utah May 2011 Copyright © Eddy Fernando Pazmiño 2011 All Rights Reserved Th e Uni v e r s i t y o f Ut a h Gr a dua t e S cho o l STATEMENT OF THESIS APPROVAL The thesis of Eddy Fernando Pazmiño has been approved by the following supervisory committee members: William P. Johnson , Chair 3/10/2011 Date Approved Aurelian Trandafir , Member 3/10/2011 Date Approved Diego Fernandez , Member 3/10/2011 Date Approved and by D. Kip Solomon , Chair of the Department of Geology and Geophysics and by Charles A. Wight, Dean of The Graduate School. ABSTRACT The vast majority of colloid transport experiments have been conducted in granular porous media with narrow size distribution, which allows a single collector size and narrow values of porosity to be used in colloid filtration theory to predict deposition rates under favorable conditions (absence of energy barriers). In this work, deposition of colloids (ranging from 0.21 mm to 9.1 mm) in packed columns was examined with three different borosilicate glass bead porous media: uniform, monomodal nonuniform and bimodal nonuniform. The corresponding porosities to these media were 0.38, 0.34 and 0.28. The effect of reduced porosity on the flow field was studied using high resolution computerized X-ray micro tomography (HRXMT), Lattice-Boltzmann (LBM) flow simulation methods, and direct observations in packed flow cells. This allowed discerning the influence of preferential flow paths on fluid velocities, and it demonstrated that straining in pore throats too small to pass was not a significant contributor to colloid retention even up to particle to grain size ratios of 0.05. Mechanistic simulations successfully predicted the experimentally-observed sensitivity to porosity and absolute values of the deposition rate coefficients when a number-based mean grain size was used to represent the size-distributed porous media. A mis padres con mucho cariño y gratitud CONTENTS ABSTRACT .................................................................................................................................... iii LIST OF FIGURES .......................................................................................................................... vi LIST OF TABLES .......................................................................................................................... vii ACKNOWLEDGEMENTS ........................................................................................................... viii 1. INTRODUCTION ..........................................................................................................................1 2. MATERIALS AND METHODS ...................................................................................................4 2.1 Materials ..................................................................................................................................4 2.2 Column Experiments ................................................................................................................5 2.3 Flow Cell Experiments .............................................................................................................7 2.4 Data Analysis ............................................................................................................................7 3. RESULTS ...................................................................................................................................10 4. DISCUSSION ..............................................................................................................................14 APPENDIX .....................................................................................................................................22 REFERENCES ................................................................................................................................29 LIST OF FIGURES 1 Experimental kf values for the three different porous media studied. ...........................................11 2 Experimental kf value results and predictions for the uniform medium .......................................12 3 HRXMT images and breaktrough curves of the three porous media studied ...............................15 4 Three-dimensional sections of the flow field in each porous media .............................................16 5 Corrected kf values for the bimodal nonuniform medium . ...........................................................18 6 Comparison of predicted kf values usign diferrent collector sizes ...............................................19 7 Weight-based size distribution of both nonuniform porous media ...............................................26 8 Number-based size distribution of both nonuniform porous media ..............................................27 9 Image of 9.1 mm colloids attached in the bimodal nonuniform medium ......................................28 LIST OF TABLES 1 Characteristic sizes of the three porous media studied ...................................................................5 2 Summary of mass balances corresponding to each experimental kf . ...........................................23 3 Two-region model parameter fitting for the bimodal nonuniform media .....................................24 4 Comparison between predicted kf values using different characteristic collector sizes and composite number and weight based fractions with observed kf values...........................25 ACKNOWLEDGEMENTS I am really grateful to Bill Jonson for his guide, help, and support during all the stages of this work. Thanks to Huilian Ma for sharing her ideas and expertise in this field of study. I want to acknowledge Professors Jan Miller and C. L. Lin at the Department of Metallurgical Engineering of the University of Utah for their assistance in generating HRXMT reconstructions of the porous media, and LBM simulation of fluid flow fields. Thanks a lot to my family, friends from Ecuador, and new friends I've made in SLC, who supported me all the time. This study is based upon work supported by the National Science Foundation Chemical, Biological, and Environmental Transport, and Hydrologic Science Programs (0822102). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. CHAPTER 1 INTRODUCTION Over the past several decades scores of experiments have been performed to examine colloid filtration in granular porous media. Surprisingly, all of these experiments fall within the porosity range from 0.33 to 0.38, at least to the author's reasonably extensive knowledge. This limited porosity range likely results from the fact that most experiments involve media of a relatively narrow grain size distribution. Additionally, significantly decreased porosity requires not only a wide grain size distribution, but also one characterized by particle size extremes, e.g., bimodal. Many engineered, and especially natural, granular media are distributed with respect to grain size. However, existing colloid filtration theory idealizes porous media using a single grain (collector) size. Hence, prediction of colloid transport via existing theory in size-distributed porous medium requires determining a collector size to represent the medium. To date, two studies have examined this issue Martin et al. (1) and Porubcan and Xu (2). Martin et al. (1) and Porubcan and Xu (2) concluded that the volume-based d10, and number-based mean, respectively, best represented their distributed porous media for the purpose of predicting colloid retention. The two studies employed two different strategies. The strategy used by Martin et al. (1) was to identify the representative grain size that collapsed deposition efficiency versus collision number onto a single trend, where collision number equals kfvL, kf is the deposition rate 2 coefficient, v is the average pore water velocity, and L is the column length. This strategy assumed that deposition efficiency should be independent of the different fluid velocity fields expected for the different porous media size distributions examined. However, subsequent research shows that deposition efficiency varies with fluid velocity (3), indicating that the strategy may be compromised. Additionally, Martin et al. (1) examined only a single colloid size, limiting the generality of their observations. Porubcan and Xu (2) examined multiple colloid sizes (0.46, 2.9, 5.1, 6.1 mm) under unfavorable deposition conditions. Their strategy was to identify the representative grain size that collapsed kf versus ratio of particle to grain diameter (dp/dg) onto a single trend under the assumption that straining was the mechanism of deposition. However, this assumption neglects the possibility that deposition occurred in response to surface roughness or heterogeneity. Furthermore, a directly-observed lack of straining in pore throats too small to pass in our experiments (described below) (maximum dp/dg ratio = 0.05) seems to negate straining as a prevalent mechanism in the experiments of Porubcan and Xu (2) (maximum dp/dg ratio = 0.03). Porubcan and Xu (2) attributed the deposition to straining on the basis that the trend for kf as a function of colloid size was inconsistent with filtration theory. Specifically, lesser (negligible) deposition was observed for the 0.46 mm colloid relative to the 2.9 mm colloid. We would note, however, that the influent concentration was 3 orders of magnitude greater for the 0.46 mm colloid relative to the 2.9 mm colloid, possibly yielding large differences in ability to detect deposition. Consistent with the existing literature, Martin et al. (1) and Porubcan and Xu (2) examined a narrow porosity range (0.32 - 0.38). The lack of experimental results over a larger porosity range stands in contrast with existing filtration theory, which explicitly includes porosity as a parameter that describes the flow field that delivers colloids to the collector surfaces (4). Practically speaking, 3 filtration theory has never been compared to experimental results conducted in granular media over a significant porosity range. In this paper we conduct colloid filtration experiments in porous media spanning a larger porosity range than previously examined (0.28 to 0.38). The primary goal is to discern the influence of porosity on colloid retention in granular media. A secondary goal is determine a representative collector size that reconciles experimental results to theoretical predictions. Since mechanistic theory is presently available only for favorable deposition conditions (absent of a colloid-surface energy barrier), the experiments were conducted under these conditions. Additional issues that arise under conditions of decreased porosity are the development of preferential flow paths and straining of colloids in pore throats too small to pass. We show that colloid deposition in the reduced porosity granular media was strongly influenced by preferential flow paths. We also observed that straining in pore throats too small to pass was not a significant influence on colloid deposition in the reduced porosity media, suggesting that it is not a major influence in granular media under unfavorable conditions. CHAPTER 2 MATERIALS AND METHODS Materials Spherical mono-dispersed fluorescent carboxylate-modified polystyrene latex microspheres were used in the experiments. Six sizes were used: 0.21, 0.5, 1.1, 2.0 mm (505 nm excitation, 515 nm emission wavelengths, Molecular Probes, Inc., Eugene, OR), 4.4 and 9.1 mm (441 nm excitation, 486 nm emission wavelengths, Polysciences, Inc., Warrington, PA). The solids concentration of the stock solutions was in the range of 2-3%. The solutions used for the experiments were prepared by diluting the stocks using a buffered solution at the desired ionic strength and pH. Colloid injection concentration varied among the different sizes due to different light intensities among the different microsphere stocks, i.e., 1E6, 1E6, 1E6, 5E5, 1E5/ml for the 0.21, 1.1, 2.0, 4.4, 9.1 mm sizes, respectively. Borosilicate glass beads (Cataphote Inc., Jackson, MS) were used as the porous media in the experiments. The glass beads were sieved to produce discrete size fractions ranging from 850 to 74 mm. The size fractions were combined to produce three porous media of different porosities (Table 1), including: 1) A uniform mono-disperse porous medium (porosity 0.38) consisting solely of 510-mm beads; 2) A bimodal nonuniform porous medium (porosity 0.28), consisting of predominant weight fractions of 850-600 mm and 250-147 mm beads; and 3) A monomodal (normally distributed by 5 Table 1 Characteristic sizes for the three porous media studied Sediment Porosity Uniform 0.38 da dgm d10 da dgm d10 Monomodal non-uniform 0.34 414 372 222 225 210 183 Bimodal non-uniform 0.28 507 425 214 169 147 115 weight-based number-based Grain size (mm) 510 weight) nonuniform porous medium (porosity 0.34), consisting of beads ranging from 800 to 74 mm. The parameters da and dgm correspond to the arithmetic and geometric means, the parameter d10 corresponds to the size below which 10% of the overall collector population is smaller; weight based d10 is equivalent to volume based since density is the same for all size fractions. The glass beads and the flow cell were cleaned using the SC-1 cleaning procedure as described in Johnson et al. (5).A portion of the glass beads was coated with iron oxide by titrating a suspension of glass beads in Fe(NO3)3 (0.001 M) with NaOH (0.1 M) according to Johnson and Logan (6), yielding a iron oxide content of 0.14 mg per gram of sediment. Column Experiments All experiments were conducted under favorable attachment conditions using two strategies: 1) Using the cleaned glass beads and setting the solution ionic strength to 50 mM and pH to 2.0 (7); 2) Using iron oxide-coated glass beads and setting the solution ionic strength to 10 mM and pH to 6.72. Favorable conditions were replicated by using both strategies in the bimodal nonuniform 6 medium for experiments involving 0.5 2.0 and 9.1 mm colloids. Deposition was equivalent for both strategies; hence, these strategies are not distinguished in the results below. Solution pH was maintained using 2.2 mM MOPS buffer (3-(N-Morpholino) propanesulfonic acid, 4-Morpholinepropanesulfonic acid; Sigma-Aldrich Corp.) Cylindrical Plexiglas columns (20 cm long, 3.81 cm i.d.) were dry-packed with glass beads flushed with CO2, and equilibrated with microsphere-free solution. The procedure of packing and preequilibration is described in previous publications (7, 8). After preequilibration, a solution with microspheres was injected (three pore volumes). This was followed by elution with microsphere-free solution (one pore volume). The suspensions and solutions were injected in up-flow mode, and in down-flow mode for selected experiments (as described in Results), using a syringe pump (Harvard Apparatus, Inc, Holliston, MA). During injection, aggregation and settling of microspheres were minimized by sonicating for 1 minute, and vertically rotating the syringe pump, every 15 minutes. Pore water velocities of 4.0 m/day, 3.3 m/day and 3.0 m/day were examined in the bimodal nonuniform, monomodal nonuniform and uniform media, respectively. Column effluent samples were collected in 5 ml polystyrene vials every 15 minutes. Colloids were recovered from sediment following each column experiment by dissecting the porous media into ten 2-cm-long segments, placing into specified volumes of Milli-Q water, sonicating for 1 minute, followed by manual vigorous shaking for 0.5 minutes. Aqueous effluent samples, and supernatant samples from recovery of retained microspheres, were analyzed using flow cytometry (BD FACScan, Becton Dickinson & Co., Franklin Lakes, NJ); details are provided in Li et al. (8). 7 Flow Cell Experiments Direct observation of colloid retention was performed in a 1 cm2 x 4 cm length flow-through quartz cell (46-Q-10, Starna Cells, Inc., Atascadero, CA) oriented horizontally over an inverted fluorescence microscope. The microsphere suspension was introduced by a syringe pump (Harvard Apparatus, Holliston, MA), with an average pore water velocity a factor of 10 greater than the pore water velocity of the column experiments. This was done for each porous medium to prevent settling due to the low pumping rate required for this small volume cell. The cell was preconditioned for 60 minutes (pH = 2.0, IS = 50 mM NaCl) with microsphere-free solution. Bulk epifluorescence microscopy (Eclipse TE2000-S inverted microscope; Nikon, Japan) was used for direct observation in the flow cell. A wide field fluorescence lamp (X-Cite 120 PC; Lumen Dynamics Group Inc., Canada) was used to illuminate the fluorescent microspheres. Images were acquired using a high-speed camera (Cool Snap HQ, Photometrics Inc., Tucson, AZ) every 0.15 seconds to capture attachment events. In order to ensure generality, observations were made in at least 10 locations across the porous medium. Data Analysis The overall recovery (mass balance) of colloids was determined by summing the numbers of retained colloids and colloids that exited the column. The total number of retained colloids was obtained by summing the colloids recovered from all column segments. The total number of colloids that exited the column was obtained by integrating the area under the breakthrough-elution curve. The colloid deposition rate coefficient was determined from the experimental effluent breakthrough and the profile of retained colloids using equations 1 and 2, respectively: 8 (1) where C is the microsphere concentration in the aqueous phase, Co is the concentration of microspheres at the source, x is the travel distance in the column and, q is the porosity. (2) where S is the concentration of microspheres in the sediment phase, rb is the bulk density of the porous medium and to is the duration of injection. Tracer breakthrough curves were simulated using the two region model in CXTFIT (9, 10), according to the following equation: ! " " # $% & ' (3) where qm is the mobile porosity Cm the aqueous concentration in the mobile region, Cim the aqueous concentration in the immobile region. vm is the average pore water velocity in the mobile pore space, and a is a first order coefficient describing colloid transfer from mobile to immobile regions. Three-dimensional reconstructions of each porous media were obtained using X-ray high resolution micro computerized tomography, HRXMT (MicroXCT-200, Xradia Pleasanton, CA), as described in Miller and Lin (11). Each porous medium was packed in micro columns (4.5 mm x 10 mm) to reconstruct a cylindrical domain (4.46 mm diameter x 5.07 mm height). The parameters of 9 operation were: 60 KV, 4X lens, 10 seconds exposure time, 150 mm glass filter. Reconstruction resolution: 5.02 mm. A subset (2.75 mm or 550 voxel cube) of the XMT reconstruction was used to solve the fluid flow field using Lattice Boltzmann methods (LBM) (12). In the simulation the fluid is body forced-induced (gravity field) to flow in a defined direction. A periodic boundary condition is used streaming the velocity field at the exit plane as an input for the fluid plane buffering the cube entry plane for subsequent iteration until a steady state flow field was obtained. The permeability of the domain was obtained from the LBM-computed flow field. CHAPTER 3 RESULTS Both of the nonuniform media demonstrate the expected minimum value of kf corresponding to colloid sizes in the range 2 to 4 mm as was observed for the uniform media (Figure 1). The open symbols correspond to column experiments conducted in upflow, and closed symbols experiments conducted in downflow. Data for uniform media and colloids < 2 mm correspond to references (7) and (8). Trend lines connect data points for ease of presentation. Error bars denote maximum and minimum values. The trends with colloid size for the three media indicate that kf generally increases with decreasing porosity (in qualitative agreement with classic filtration theory as discussed below). However, the trends differed for the bimodal medium relative to the monomodal and uniform media in two important respects: 1) Disproportionately low values were observed for the bimodal nonuniform media for the < 2.0 mm colloid size range. 2) Kf values were approximately the same among the two nonuniform media for the 9.1-mm colloid size. These exceptions will be discussed further below. 11 Figure 1 Experimental kf values for the three different porous media studied. 1) Uniform (q= 0.38); 2) Monomodal nonuniform (q= 0.34); and 3) Bimodal nonuniform (q= 0.28). The values of kf observed in the uniform media show reasonable agreement with predicted values from the, LH, MPFJ, TE and NG correlation equations (13-17); i.e., experimental data and predictions agree within a factor of two to three for all colloids sizes (Figure 2). Predictions from the original Happel-based model (RT model) (4) are not shown due to their similarity to the TE model predictions. For colloid sizes of 2 mm and larger, the predicted values of kf using the TE, LH, NG and to a lesser extent MPFJ, correlation equations are higher and show a steeper slope relative to the experimental results. 1.0 10.0 100.0 0 2 4 6 8 10 kf (hr-1) size (mm) BIMODAL down-flow BIMODAL MONOMODAL down-flow MONOMODAL UNIFORM down-flow UNIFORM 12 Figure 2 Experimental kf results (open circles) for uniform porous media, collector size 510 mm, and predictions from four correlations: LH (13), MPFJ (14, 15), TE (16), and NG (17). Error bars denote maximum and minimum values. The lower kf values predicted for the larger colloids by the MPFJ model (relative to the other models) result from the grain-to-grain contact influence on the flow field, as described in (18). The experimental results were largely run in upflow mode, where the flow direction was opposite to gravity. In contrast, the models are based on force balance simulations involving downflow mode. The effect of gravity may be significant for microspheres larger than approximately 2 mm; hence, additional experiments were run in downflow mode for the 9.1-mm microspheres (Figure 1). Values of kf were increased in downflow mode relative to upflow mode by a factor of two or less, which is similar to the error of the experiment, but which likely reflects the effect of gravity since this effect was observed for all three porous media (Figure 1). 1 10 100 0 1 2 3 4 5 6 7 8 9 10 kf (hr-1) size (mm) UNIFORM TE LH MPFJ NG 13 The downflow results for the monomodal nonuniform medium more closely match theory (Figure 2), as expected since the underlying numerical models were run in downflow mode (13-17). The average mass recovery from all experiments was 87%, reflecting good experimental recovery. In the few individual cases where recovery was low, the general trend of the retention profile was used to guide a model fit that accounted for accounted for all injected colloids. All experimental retention profiles were log linear, corroborating favorable conditions of attachment. CHAPTER 4 DISCUSSION The relatively low values of kf for < 2 mm and the 9.1 mm size colloids observed for the bimodal medium likely reflect influences of the porous media structure not accounted for in colloid filtration theory. X-ray micro tomography (XMT) images of the three porous media reveal increasingly nonuniform structure in the order from uniform, monomodal nonuniform and bimodal nonuniform media (Figure 3 top). Breakthrough curves from tracer tests (Figure 3 bottom) show increasing dispersion in the order from uniform, monomodal nonuniform, and bimodal nonuniform media. The two-region model fits (CXTFIT) indicate that practically all pore space was mobile for the uniform and monomodal nonuniform media (Figure 3 bottom), as shown by unitary ratios of mobile to total porosity backed out from the simulations. In contrast, the model fits to the bimodal medium indicated that a fraction of the pore space was immobile. The fraction of mobile pore space relative to total pore space (qm/q) for which a good fit to the tracer breakthrough was obtained ranged from 10% to 90%, with the mass rate transfer coefficient (a) correspondingly ranging from 12 to 0.20 hr- 1. The values of qm/q and a backed out from non-linear least squares analyses were 48% and 4 hr-1, respectively. 15 Figure 3 Top: HRXMT images of the three porous media and porosity values. Bottom: corresponding tracer breakthrough curves for each media and ratios of mobile to total porosity shown. For a given discharge, the pore water velocity in mobile pore space must increase proportionally to the fraction of immobile pore space. Because the deposition rate coefficient varies with fluid velocity, the relatively low values of kf observed for the < 2.0 mm and 9.1 mm colloids in the bimodal nonuniform medium (Figure 2) may reflect the effect of higher fluid velocity in the preferential flow paths in this medium, as discussed further below. To investigate whether the XMT-rendered pore domains yield flow fields consistent with the tracer tests, Lattice-Boltzmann (LB) simulations were run in the XMT-domains. The resulting pore water velocity distributions in each porous media are shown for the three media (Figure 4). 0 1 2 Pore volume 0 0.2 0.4 0.6 0.8 1 0 1 2 C/Co Pore volume 0 1 2 Pore volume q = 0.34 Uniform Monomodal nonuniform Bimodal nonuniform q = 0.38 q = 0.28 qm/q = 1 qm/q = 1 qm/q = 0.48 Figure 4 Three-dimensional sections of the flow field in each porous media obtained from Lattice Boltzmann simulations: uniform (left), monomodal (right). Each section is 500 mm thick, main flow direction: from left to lower 30% of the velocity histogram distribution are shown in blue. Fluid velocities in the upper 30% of the velocity histogram distribution are shown in green. The LB flow field for the uniform porous medium shows relat domains evenly distributed through the medium. nonuniform porous medium shows relatively small advective pore domains that also evenly distributed through the medium. In further contrast, the LB flow field for the bimodal media shows relatively small advective pore domains that are not evenly distrib media, indicating a larger contrast between mobile and immobile pore space relative to the other two media. Volume-averaged permeabilities derived from the LB flow fields were 0.19 0.21x10-6 cm2 for the bimodal and monomodal or silt), respectively, which is a factor of five lower than the uniform media ( Although the two nonuniform distinguished by its uneven spatial distribution of advective pore domains. The combined XMT, Lattice of the pore domain in the bimodal nonuniform (center) and, bimodal right. Fluid velocities in the relatively large advective pore In contrast, the LB flow field for the monomodal dal nonuniform media (corresponding to very fine sand media show similar permeabilities, the bimodal medium is Lattice-Boltzmann, and tracer test results indicate that only a fraction nonuniform media was significantly advective. 16 Lattice- nonuniform ively nonuniform distributed through the in the bimodal medium 0.19x10-6 and 1.10x10-6 cm2). , The value of 17 48% mobile porosity backed out from non-linear least squares optimization was herein used to represent the fraction of advective pore domain in the bimodal medium in order to demonstrate the effect of the preferential flow paths in colloid transport, yielding an average pore water velocity in the advective pore space of the medium two times greater than the whole-domain-averaged pore water velocity. Since the bulk of colloid flux was carried in the advective pore domain, a correction for the corresponding fluid velocity can be performed to yield a corrected value for kf from Equations (1) and (2). The resulting trends in kf as a function of colloid size are shown in Figure 5, where it is seen that, with the velocity correction for the bimodal media, kf increases with decreasing porosity across the entire porosity range examined. It should be noted that the velocity correction was equivalent for all colloid sizes for lack of a known method to account for possible differential effects of preferential flow paths among the colloid sizes (due to differing diffusion coefficients). This potential differential effect may contribute to different trend in kf versus colloid size for the bimodal relative to the other media. The preponderance of the 147-250 mm collector size fraction of the bimodal medium yields a ratio of particle to grain diameter (dp/dg) ranging from 0.062 to 0.036 for the 9.1 mm microspheres relative to these collectors. Given that previous studies infer straining to be the mechanism of deposition under much lower dp/dg ratios, e.g. 0.008, 0.003 (2, 19), the potential role of straining in our results warrants further examination. Direct observation experiments conducted for the 9.1-mm colloids in all three media showed that attachment occurred predominantly on the open surface of the glass beads. There was no evidence of straining in pore throats, which were much larger than even the largest colloid size studied. Images documenting lack of straining in pore throats are provided in the Appendix. 18 Figure 5 Corrected kf values for the bimodal nonuniform medium after preferential paths are considered. The observed lack of straining for up to dp/dg ratios up to 0.05 runs counter to previous studies that have inferred straining in pore throats too small to pass as the mechanism of retention under lower dp/dg ratios (2, 19) . The clear inverse relationship between porosity and kf observed in Figure 5 is not driven by straining, and is therefore expected to reflect the influence of mass transfer processes that are enveloped in classic filtration theory. The trends in kf as a function of colloid size obtained under favorable deposition conditions (Figure 5) provide a platform for comparison to mechanistic theoretical predictions (Figure 6). 1 10 100 0 2 4 6 8 10 kf (hr-1) size (mm) BIMODAL MONOMODAL UNIFORM 19 Figure 6 Comparison between experimental data (open symbols) of uniform (blue) , monomodal nonuniform (red) and bimodal nonuniform (green) porous media with kf predictions using the MPFJ correlation (lines) for different collector sizes: (a) weight based mean, (b) number based mean, (c) composite weight-based kf ; and, (d) composite number- based kf . The theoretical unit cell predictions are based on a representative collector size, for which there are multiple options to choose from, including arithmetic versus geometric means, as well as number-based versus weight-based size fractions). The number-based mean grain size (based on numbers of collectors within each size fraction) is lower than the weight-based average grain size (based on the weight of collectors within each size fraction) (Table 1). The number-based arithmetic means are 169 mm and 224 mm for the bimodal and monomodal media, respectively; whereas, the weight-based arithmetic means are 507 mm and 414 mm for the bimodal and 20 monomodal media, respectively Figure 6 shows that the weight-based average grain size does not reflect the sensitivity to porosity that was observed in the experiments (Figure 6a), and this is also true if one develops a composite kf by determining kf for each size fraction and then averaging according to the weight-based contribution to the total size distribution (Figure 6c). In contrast, the number-based average grain size does capture the experimentally-observed sensitivity of kf to porosity (Figure 6b), although the absolute values of kf predicted by the number-based average grain size are higher than the experimental values (Figure 6d). Notably, this over-prediction is largely eliminated by use of a composite number-based kf (Figure 6d). The number-based geometric mean, and the weight-based d10, are similar to the number-based arithmetic mean (Table 1). Therefore, predictions using the number-based geometric mean and weight-based d10, will yield similar values as the ones shown in (Figure 6b). By comparing experimental observations and theoretical predictions for deposition in grain size-distributed media, we herein provide a theory-based approach for determining the appropriate grain size with which to represent a porous medium for prediction of colloid deposition. Our finding, that the number-based mean captures the sensitivity of kf to porosity, is consistent with the conclusions of Martin et al. (1) and Porubcan and Xu (2) who concluded that the volume-based d10, and number-based mean, respectively, best represented their distributed porous media. Both of these parameters emphasize the smaller size fractions of the porous media size distribution. However, improved predictions were obtained by determining kf for each size fraction and then averaging according to the number-based contribution to the total distribution (Figure 6d). This indicates that a single representative grain size did not fully reflect the mass transfer processes for the overall porous media grain size distribution, reflecting a limitation of the unit cell approach, which idealizes the porous media using a single grain or pore-scale representative structure. The 21 observed influence of preferential flow paths on colloid deposition also highlights a limitation of single-pore unit cells, since they cannot explicitly capture this attribute of porous media. Although assemblage-scale (e.g., Figure 4) mechanistic simulations of colloid transport have been performed under favorable deposition conditions, e.g., Long and Hilpert (13), it is doubtful that accurate colloid transport simulations at this scale are tractable for unfavorable deposition conditions, suggesting the need for a hybrid approach that combines the numerical tractability of the unit cell approach with the representativeness of the fluid flow field at the assemblage scale. One of the most important findings of extending colloid filtration experiments to lower porosities is the determination that the observed values of kf, and their trends with colloid size, are described by classic filtration theory. This finding, and the direct observation that no straining occurred in pore throats too small to pass even for dp/dg ratios up to 0.05, negates the notion that physical straining in pore throats too small to pass is a predominant mechanism of retention in media above the dp/dg threshold of 0.003-0.008 reported in the literature (2, 19). Both observation and theory clearly showed that the colloid deposition observed in our experiments was physicochemical in nature. Such deposition would of course eventually lead to colloid aggregation in, and eventual clogging of, pore throats. But this should be addressed mechanistically via physicochemical processes such as colloid-colloid interaction, and are not mechanistically represented by straining in pore throats too small to pass. APPENDIX SUPPORTING INFORMATION 23 Table 2 Summary of mass balances corresponding to each experimental kf . All experimental profiles of retained colloids were used to determine kf (Equation 2 of text). Effluent breakthrough concentrations were also used to further constrain kf when breakthrough occurred. Uniform medium diameter k f IS pH Mass balance Remarks (mm) (hr-1) (mM) % 0.1* 10 50 2 101.3 0.21* 8 50 2 105.3 0.5* 6.5 50 2 81.5 0.93** 4 1 6.92 101.8 2* 4.1 50 2 102.4 2 2.2 50 2 93 4.4 2.9 50 2 106 9.1 6.5 50 2 94 9.1 11 50 2 92 9.1 15 50 2 81 downflow Monomodal medium 0.21 17 50 2 24 0.5 14 50 2 47 1.1 10 50 2 84 2 6.5 50 2 84 44 7 50 2 87 9.1 20 50 2 70 9.1 14 50 2 97 9.1 36 50 2 88 downflow Bimodal medium 0.21 15 50 2 134 0.5 10 50 2 41 1.1 4.5 50 2 136 2 11.4 50 2 105 2 12 1 6.72 67 FeOxide 4.4 12.6 50 2 65.3 9.1 15 50 2 66 9.1 20 1 6.72 87 FeOxide 9.1 14 50 2 113 9.1 23 50 2 87 downflow * Tong, M.; Johnson, W. P., Excess colloid retention in porous media as a function of colloid size, fluid velocity, and grain angularity. Environmental Science and Technology 2006, 40, (24), 7725- 7731. ** Li, X.; Scheibe, T. D.; Johnson, W. P., Apparent decreases in colloid deposition rate coefficients with distance of transport under unfavorable deposition conditions: A general phenomenon. Environmental Science and Technology 2004, 38, (21), 5616-5625. 24 Table 3 Two-region model parameter fitted for the bimodal nonuniform media Parameters fitted 1 2 3 4 5 6 7 8 qm/q 0.10 0.20 0.30 0.40 0.48 0.60 0.80 0.90 a (hr-1) 1.20E+01 9.44E+00 7.21E+00 5.28E+00 4.00E+00 2.33E+00 6.08E-01 1.96E-01 Sum of squared residuals 4.62E-02 4.56E-02 4.49E-02 4.43E-02 4.41E-02 4.53E-02 8.02E-02 2.32E-01 Coefficient of determination R2 0.99742 0.99746 0.99750 0.99753 0.99754 0.99748 0.99553 0.98709 Runs 25 Table 4 Comparison between predicted kf values using different characteristic collector sizes and composite number and weight based fractions with observed kf values. The MPFJ correlation equation was used to calculate kf. Logarithmic values of kf were used to calculate the total sum of residuals squared to prevent overestimation of residuals of the relatively higher kf values. Weight arithmetic mean Number arithmetic mean Composite kf weight based Composite kf number based MONOMODAL Experimental (Residual)2 (Residual)2 (Residual)2 (Residual)2 Colloid size (mm) log kf (hr-1) 0.21 1.230 0.027 0.359 0.002 0.009 0.5 1.146 0.002 0.152 0.063 0.013 1.1 1.000 0.022 0.085 0.123 0.043 2 0.813 0.006 0.135 0.076 0.016 4.4 0.845 0.003 0.232 0.020 0.000 9.1 1.368 0.001 0.172 0.029 0.005 BIMODAL 0.21 1.495 0.002 0.533 0.014 0.027 0.5 1.319 0.028 0.378 0.055 0.003 1.1 0.972 0.005 0.533 0.016 0.041 2 1.387 0.358 0.053 0.407 0.053 4.4 1.419 0.214 0.145 0.234 0.000 9.1 1.566 0.011 0.451 0.023 0.097 Sum of squared residuals 0.680 3.229 1.062 0.307 26 Figure 7 Monomodal nonuniform medium (top) and bimodal nonuniform medium (bottom) size distributions by weight. 0 10 20 30 40 50 60 800 - 600 600 - 500 500 - 300 300 - 212 212 - 147 147 - 74 < 74 % weight mm 0 10 20 30 40 50 60 850 - 600 600 - 500 500 - 300 300 - 250 250 - 147 147 - 74 <74 % weight mm 27 Figure 8 Monomodal nonuniform medium (top) and bimodal nonuniform medium (bottom) size distributions by weight. 0 10 20 30 40 50 60 800 - 600 600 - 500 500 - 300 300 - 212 212 - 147 147 - 74 < 74 % number mm 0 10 20 30 40 50 60 850 - 600 600 - 500 500 - 300 300 - 250 250 - 147 147 - 74 <74 % number mm 28 Figure 9 Image of 9.1 mm colloids attached in the bimodal nonuniform media after 10 pore volumes of injection. 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