Colloid Transport in Geochemically Heterogeneous Porous Media

Environ. Sci. Technol. 1996, 30, 3284-3293

Colloid Transport in Geochemically Heterogeneous Porous Media: Modeling and Measurements PHILIP R. JOHNSON,† NING SUN, AND MENACHEM ELIMELECH* Department of Civil and Environmental Engineering, University of California, Los Angeles, California 90095-1593

The transport of colloids in geochemically heterogeneous porous media is investigated. A model describing the transport and deposition of colloids onto heterogeneously charged mineral grains is developed and applied to column experiments. The model characterizes mineral grain surfaces according to a patchwise charge distribution, with individual patches being either favorable or unfavorable for deposition depending on their electrostatic charge. Separate rate expressions are used in the model to depict favorable and unfavorable deposition kinetics. Declining deposition kinetics that are produced when previously retained particles block subsequent attachment of colloids are quantified in the model by dynamic blocking functions. Column experiments involving colloid transport in geochemically heterogeneous porous media were performed using silica colloids and quartz sand. Surface charge heterogeneity was introduced into the porous medium by coating a fraction of the quartz sand with iron oxyhydroxide. Theoretical breakthrough curves generated by the model using experimentally determined parameters compared quite well to the experimental results, demonstrating the importance of geochemically heterogeneous surfaces in determining the transport behavior of colloidal particles in heterogeneous aquatic environments.

Introduction Recent investigations pertaining to the subsurface aquatic environment have provided compelling evidence that colloid-mediated contaminant transport may be a significant process governing the mobility of contaminants in groundwater (1-3). The realization that colloids facilitate the transport of low-solubility compounds, metals, and polar organics in aqueous settings has provided much of * Author to whom correspondence should be addressed; telephone: (310) 825-1774; fax: (310) 206-2222; e-mail address: elim@ seas.ucla.edu. † Present address: Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556-0767.

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the impetus behind the current research aimed at delineating colloidal transport mechanisms in porous media (e.g., refs 4 and 5). A thorough characterization of colloidal transport processes in porous media is not only required for assessing the migration of dissolved contaminants that sorb to colloidal mineral surfaces (6) and detrital organic matter (7) but is also necessary when predicting the mobility of colloidal contaminants such as viruses (8), pathogenic bacteria (9), and radioactive particulate matter (10). Experimental investigations involving the transport of colloidal particles in aqueous porous media have been conducted under controlled laboratory conditions using granular media packed in vertical columns (4, 5, 11-14) as well as under field conditions using both natural- and forced-gradient groundwater flow (15-17). The results of these investigations underscore the significance of advection, dispersion, deposition, and mobilization (release) as the primary processes controlling particle transport in porous media. Several models for colloid transport in porous media have been proposed as a means of characterizing the laboratory and field experiments (18-23). In these models, an analogy is implicitly drawn between the transport behavior of colloidal particles and molecular solutes, as each model invokes the advection-dispersion equation as originally derived for solute transport in porous media (24), with slight modifications to account for the retention and release of colloids by the immobile solid matrix. In most aquatic environments, colloid deposition is inhibited by repulsive electrostatic forces between suspended colloids and stationary mineral grain surfaces. These repulsive forces are a result of the preponderance of negatively charged surfaces found in aquatic environments (25), which produce unfavorable deposition conditions. In spite of the unfavorable conditions, colloid deposition rates have been experimentally observed to be several orders of magnitude greater than theoretical predictions based on the classic theory of Derjaguin-Landau-Verwey-Overbeek (i.e., DLVO theory) (5, 26-29). Various attempts have been made to explain the anomalously high deposition rates observed under unfavorable conditions, including attachment in secondary energy minima (25, 30), interfacial dynamics of double-layer interaction (29, 31), and surface nonidealities (29, 32). Of these explanations, surface nonidealities such as cracks, dislocations, and geochemical impurities are considered to be the most probable cause of the anomalous colloid deposition rates observed in porous media (2, 32). These nonidealities give rise to nonuniform charge distribution and are known as surface charge heterogeneities (2, 29, 32, 33). Surface charge heterogeneities are ubiquitous in natural and engineered aquatic environments, being a product of the geochemical, biological, and structural variability commonly associated with mineral surfaces. Variable surface charge may result when diverse functional groups are exposed on adjacent facets of mineral surfaces (32) or from minerals that contain bulk or surface-bound chemical impurities (25, 34). The oxides of iron, aluminum, and manganese represent the most common source of surface charge heterogeneity in subsurface aqueous environments, since these compounds carry a positive surface charge at

S0013-936X(96)00053-3 CCC: $12.00

 1996 American Chemical Society

neutral pH and are generally present as anisopachous coatings on silicate mineral grains (2, 32-35) or as separate, accessory minerals (25, 36). In this paper, we examine the impact of geochemically heterogeneous surfaces on colloid transport in porous media. First, a mathematical model for colloid transport in geochemically heterogeneous porous media is presented. Nonuniform charge distribution resulting from geochemically heterogeneous surfaces is represented by a patchwise charge heterogeneity model, with separate kinetic expressions that describe particle deposition onto favorable and unfavorable patches. A dynamic blocking function is incorporated into the kinetic rate expressions as a means of characterizing declining deposition rates that are caused by the blocking effects of previously retained particles. The model is compared to the results of colloid transport experiments conducted in geochemically heterogeneous porous media. The experiments consisted of colloidal silica suspensions flowing through columns packed with quartz granules. Geochemical variability was introduced into the porous medium by coating a fraction of the quartz granules with iron oxyhydroxide. The experimental results are compared to model predictions that are based on experimentally determined model parameters.

Model Development Transport Equation. For saturated porous media, the onedimensional transport equation may be used to describe the advection, dispersion, and deposition of colloids (3739)

∂n ∂n ∂2n f ) -νp + Dh 2 ∂t ∂x ∂x πa

∂θ 2 ∂t

particle to the interstitial fluid velocity and aspect ratio between colloid radius and pore radius. Their expression, adapted for spherical particles flowing in a cylindrical tube, is

νp )

[ ( )]

ap U 2- 1 r0

p

where the independent variables x and t represent the spatial and temporal dimensions, while n(x,t) and θ(x,t) are dependent variables corresponding to the number concentration of suspended particles and the local fraction of the immobile matrix that is covered by deposited particles, respectively. Model parameters contained in the transport equation (eq 1) include the interstitial particle velocity νp, the hydrodynamic dispersion coefficient Dh, the particle radius ap, and the specific surface area f. The specific surface area for uniform, spherical collector grains may be determined as a function of grain size and bed porosity from an analytical expression that is given elsewhere (37, 40). Previous colloid transport models typically quantify the deposition process according to the mass of particles retained on collector grains (e.g., refs 21 and 23). In contrast to previous models, the colloid transport model as represented by eq 1 quantifies deposition according to the fractional surface coverage of collector grains (θ) rather than mass of retained particles. The surficial coverage approach to deposition is essential when quantifying changes in colloid deposition kinetics that are produced by the blocking effects of previously retained particles (37). Particle Advection. Colloidal particles suspended within an aqueous porous medium will generally travel at slightly higher velocities than the advecting fluid medium. This enhancement of particle velocity with respect to fluid velocity is a result of a particle’s finite dimensions, which prohibit it from contacting the slower moving fluid near the pore boundaries. DiMarzio and Guttman (41) derived an expression that relates the velocity of a suspended

(2)

where U is the superficial fluid (Darcy) velocity, is the porosity, and r0 is the pore radius. Values for the pore radius (r0) may be experimentally obtained using the water retention characteristics of the porous medium (42). When the immobile matrix is composed of well-sorted spherical granules, the mean pore radius may be geometrically determined. Johnson and Elimelech (37) presented an interpolative method for determining the average pore radius of a granular medium that is based on the spatial packing arrangement of collector grains. Their expression is limited to porosity values between 25.96 and 47.60%, the values corresponding to rhombohedral and cubic close-packing arrangements of spherical grains, respectively (43). Hydrodynamic Particle Dispersion. Most existing onedimensional dispersion models for colloidal particles utilize an expression that is based on longitudinal dispersivity (19, 20, 22, 23). Longitudinal dispersivity for one-dimensional flow in porous media exhibits a linear dependence on particle velocity, viz.

Dh ) (1)

2

D∞ + RLνp τ

(3)

where D∞ is the bulk particle diffusion coefficient, τ is the porous medium tortuosity, and RL is the dispersivity parameter. Values for the Brownian diffusion coefficient are readily determined from the Stokes-Einstein equation. The dispersivity parameter must be experimentally determined, however, since there are no fundamental expressions available for obtaining RL. Geochemically Heterogeneous Surfaces. All natural and engineered solids develop surface charge when immersed in water (25). Surface charge may result from ionic substitution within the crystalline framework of minerals, complexation or ionization of surface functional groups, or specific adsorption of ions to solid surfaces (25, 29). Surfaces are inherently heterogeneous due to physical and chemical imperfections. These imperfections include cracks, edges, lattice defects, and chemical impurities that produce variations in the density and sign of electrostatic charge on solid surfaces (2, 44). Such surficial variations in electrostatic charge are known as surface charge heterogeneities (32, 33, 39). Surfaces containing patchwise charge heterogeneities are ubiquitous in the groundwater environment, due to geochemical variability inherent in mineral grains. The oxides of Fe, Al, and Mn are the main source of geochemical heterogeneity in aquatic environments. These mineral phases are positively charged at near-neutral pH, whereas the primary immobile matrix minerals such as quartz and feldspar carry a negative surface charge. In most nearsurface geologic settings, the oxides of Fe and Mn constitute a relatively minor fraction of the total mass of the soil framework, typically in the range of 0.5-2% (35, 45, 46). They may occur as individual minerals, as microaggregates

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(47), or as discontinuous, anisopachous coatings on clay minerals and other primary silicate minerals (2, 35, 36). Because of the patchy, discontinuous nature of surface charge commonly encountered in natural porous media, collector grains will be characterized as patchwise heterogeneous surfaces in the transport model. A two-patch charge heterogeneity model will be utilized where the total surface area of the collector grains is divided into favorable and unfavorable fractions. In most instances, the immobile matrix will have surface characteristics unfavorable for particle deposition, so that favorable patches will generally constitute only a minor fraction of the total surface area. With the immobile solid matrix being subdivided into favorable and unfavorable patches, the fractional surface coverage θ becomes a linear combination of the coverage corresponding to favorable and unfavorable patches

θ ) λθf + (1 - λ)θu

(4)

where λ is the heterogeneity parameter, and the subscripts f and u refer to patches having favorable and unfavorable charge characteristics, respectively. The heterogeneity parameter (λ) represents the fraction of spatially available surface area that is favorable for colloid deposition. Kinetics and Dynamics of Particle Deposition. Unlike solutes, which typically adsorb to surfaces in a reversible fashion, most colloids remain attached after depositing on a surface (12, 29, 33, 48). As a consequence, colloid deposition may be modeled as a kinetic process (37, 48)

∂θ ) πap2kn ∂t

(5)

where k is a kinetic rate constant known as the particle transfer coefficient (37). The particle transfer coefficient describes the initial kinetics of deposition when collector surfaces are devoid of particles. Because deposition is being modeled from the perspective of surface coverage rather than retained particle mass, the deposition rate is equivalent to the change in surface coverage, ∂θ/∂t. An examination of the kinetic rate equation (eq 5) reveals that the coverage rate ∂θ/∂t is proportional to the projected area of a colloidal particle (πap2) as well as the kinetic rate constant (k) and the number concentration of suspended particles (n). Under irreversible conditions, the kinetic rate equation is only valid during the initial stages of deposition while collector surfaces are essentially devoid of retained colloids. For most cases involving deposition of colloids in porous media, the rate of deposition declines as retained particles block subsequent attachment (33). Because of blocking effects, the kinetic rate equation must be modified to consider the dynamic or transient aspects of colloid deposition. Deposition dynamics that are produced by blocking effects may be quantified by appending a dynamic blocking function to the kinetic rate equation (37, 48, 49)

∂θ ) πap2knB(θ) ∂t

(6)

The dynamic blocking function B(θ) describes the probability of a particle contacting a portion of collector surface that is unoccupied by previously deposited particles. It provides a correction for the blocking effects of deposited particles on the initial deposition rate. Two types of dynamic blocking functions are generally recognized. The

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simplest dynamic blocking function has been derived from the molecular adsorption model of Langmuir (50) and, as a consequence, is generally referred to as the Langmuirian blocking function (37, 48, 49)

B(θ) ) 1 - βθ

(7)

The parameter β included in the Langmuirian blocking function is the excluded area parameter (37, 38, 48). The excluded area parameter is the normalized area of collector surface that is blocked by a deposited particle, taken as the ratio of the average excluded area to the projected particle area (πap2). Because repulsive electrostatic forces between colloids limit the maximum surface density of attached particles to a single monolayer, the reciprocal value of the excluded area parameter corresponds to the maximum attainable surface coverage θmax or jamming limit. For deposition under irreversible conditions, θmax cannot exceed a value of 0.546. This unique value for the jamming limit corresponds to uncharged or “hard” particles and is called the hard sphere jamming limit, symbolized by θ∞ (51). For charged particles depositing onto spherical surfaces, the actual jamming limit value will usually be considerably less than the hard sphere jamming limit value because of the combined effects of repulsive electrostatic forces and hydrodynamic interactions between particles (33, 37, 48). Previous colloid transport models have relied exclusively on Langmuirian blocking functions as a means of characterizing deposition dynamics (21-23). Although the Langmuirian blocking function may be sufficient when describing the excluded area effects of point-size entities such as solute molecules, it has been shown to underestimate blocking associated with the areal dimensions of colloidal particles (37, 49). Schaaf and Talbot (49) introduced a dynamic blocking function based on random sequential adsorption (RSA) mechanics that considers the excluded area effects of particles having areal dimensions. The so-called RSA dynamic blocking function was originally intended for describing deposition dynamics of uncharged particles onto flat surfaces. Johnson and Elimelech (37) modified the original RSA dynamic blocking function to accommodate deposition of charged particles onto spherical surfaces. The modified RSA blocking function assumes spherical geometry for both colloids and collector grains:

B(θ) ) 1 - 4θ∞βθ +

(

)

40 6x3 176 (θ∞βθ)3 (θ∞βθ)2 + π x3π 3π2 (8)

This RSA-based blocking function is adequate when quantifying excluded area effects for low to moderate surface coverages (θ < 0.8θmax). Other dynamic blocking functions must be used for higher surface coverages (37, 49). When modeling deposition in geochemically heterogeneous porous media, separate rates of deposition are expected for the favorable and unfavorable fractions. These rate equations for favorable and unfavorable surfaces are directly obtainable from eq 6:

∂θf ) πap2kfnB(θf) ∂t

(9)

∂θu ) πap2kunB(θu) ∂t

(10)

The subscripts f and u refer to the appropriate parameters

for favorable and unfavorable surfaces. The overall rate of surface coverage in geochemically heterogeneous porous media is expressed as a linear combination of the individual rate expressions for favorable and unfavorable surfaces, i.e.

∂θ ) πap2n[λkfB(θf) + (1 - λ)kuB(θu)] ∂t

(11)

The dynamic rate equations for geochemically heterogeneous surfaces (eqs 9-11) along with the transport equation (eq 1) comprise the governing equations of the transport model for colloids in geochemically heterogeneous porous media. Solution of the Colloid Transport Model. The governing equations (eqs 1 and 9-11) apply to one-dimensional, laminar flow of colloidal suspensions through a packed column. The initial and boundary conditions for a square pulse input of colloids into a semi-infinite packed column with initially “clean” collector grains are

n)0

te0

(12a)

θ)0

te0

(12b)

x ) 0, 0 e t e t0

(12c)

n)0

x ) 0, t > t0

(12d)

∂n )0 ∂x

x ) L, t > 0

(12e)

n ) n0

where t0 is the time duration of a square pulse of colloidal suspension having an initial concentration n0. A Crank-Nicolson finite difference numerical scheme was used to solve the coupled governing equations subject to the conditions as stated in eq 12. An upstream weighting factor (53, 54) was applied to the first-order spatial derivative terms of the finite difference analog when mesh Peclet numbers were large in order to eliminate the overshoot commonly associated with the near-hyperbolic problem (54). The amount of numerical dispersion produced by the upstream weighting factor was determined to be negligible when a sensitivity analysis was performed on the model. The accuracy of the numerical scheme was tested and verified against an analytical solution for advection and dispersion (54, 55). A sensitivity analysis was performed on the numerical model for each of the parameters contained in the model. The analysis revealed a high degree of model sensitivity to the parameters for heterogeneity (λ), fluid approach velocity (U), and excluded area (β). The model sensitivity to these parameters indicates the importance of considering geochemical properties of the fluid and porous medium when modeling colloid transport. Because the rate of colloid deposition onto favorably charged mineral grain surfaces is typically several orders of magnitude larger than that for unfavorable surfaces, mineral grains having only minor amounts of charge heterogeneity will exert a major influence on colloid transport and mobility. Variations in solution chemistry can also create significant changes in colloid mobility, due to the sensitivity of the excluded area parameter to solution ionic strength. Additional discussion

of model sensitivity to other model parameters, including advection and dispersion, are presented elsewhere (56).

Materials and Methods Colloidal Suspension. Silica (SiO2) microspheres (PST-3, Nissan Chemicals, Tarrytown, NY) were selected as the colloidal phase. The microspheres are reported by the manufacturer to be monodisperse, with a mean diameter of 0.30 µm. The nominal size of the silica was verified in our laboratory using dynamic light scattering measurements (Nicomp Model 370, Nicomp Particle Sizing Systems, Santa Barbara, CA). The size and sphericity of the silica colloids were also examined by a scanning electron microscope (SEM). Gravimetric measurements were used to determine a particle density of 2.28 g/cm3 for the colloidal silica. The particle density measurement was used in conjunction with the nominal particle diameter to calculate the number concentration of particles in suspension. An indifferent, monovalent electrolyte (reagent-grade KCl) was used to control the ionic strength of colloidal suspensions. Porous Media. Quartz sand (SiO2) was used as the column packing material. Uniform quartz grains were selected, having a reported sieve size of 50-70 mesh (Aldrich Chemical, Milwaukee, WI). The sand was thoroughly cleaned prior to use. The cleaning procedure consisted of ultrasonication for 30 min in deionized water, followed by immersion in sodium dithionite solution (0.1 M Na2S2O4) for 2 h to remove surficial metallic compounds such as iron oxide and manganese oxide. Organic impurities were removed by soaking the sand in hydrogen peroxide (5% H2O2) for 3 h, followed by a wash with deionized water and subsequent overnight soaking in hydrochloric acid (12 N HCl). The cleansed grains were treated to a final rinse in deionized water, followed by additional ultrasonication for 30 min. Visual examination of the quartz sand was performed at various magnifications using both petrographic microscopy and scanning electron microscopy. The petrographic examinations revealed the sand grains to be prolate spheroidal in shape (57), with surfaces having a subangular to subrounded appearance (58). The quartz grains are well sorted, with a nominal grain diameter of 0.32 mm. The nominal grain diameter was determined by averaging the major and minor axes of 35 separate quartz granules. Highmagnification SEM imagery of individual quartz grain surfaces revealed a substantial amount of surface roughness. The surfaces appear highly irregular, having sharp, angular terminations, jagged protrusions, and deep crevices. The prominence of the surface irregularities are of the same magnitude as the colloidal silica, having an amplitude of ca. 1-2 µm. In order to produce geochemically heterogeneous porous media surfaces that reflect conditions encountered in the subsurface environment, a portion of the cleaned quartz grains was coated with iron oxyhydroxide. A procedure similar to that reported by Penners and Koopal (59) for preparing hematite colloids by the forced hydrolysis method was used to coat the grains. In this procedure, 25 mL of a solution having 0.72 M FeCl3 and 3.75 × 10-3 M HCl is mixed with 975 mL of preheated 3.75 × 10-3 M HCl solution. The quartz sand is then added to the resulting solution, and the mixture is heated for 36 h at 100 °C and allowed to cool back to room temperature. The coated grains are then flushed with deionized water until a clear


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supernatant is obtained, followed by oven drying at 65 °C for 24 h. The procedure was repeated in order to ensure complete coating of the sand surfaces. An analysis of the coated sand by X-ray diffractometry indicated that the coating material is composed primarily of hematite (R-Fe2O3), with other iron-bearing oxide phases being detected in small amounts. Examination of individual grains via SEM imagery indicated a relatively uniform layer of iron oxyhydroxide present over the entire quartz grain surface. Although surfaces displayed uniform coatings, the coating process did little to mask the geometric irregularities, since coated grains exhibit the same degree of surface roughness as clean grains. Electrokinetic Measurements. Electrophoretic mobility measurements were used to determine the surface potential of colloidal silica and porous media surfaces. The electrophoretic mobilities were measured using Lazer Zee Model 501 apparatus (Pen Kem Inc., Bedford Hills, NY). Particle suspensions having solution chemistry similar to the column experiments were used in the electrokinetic measurements. Silica colloid suspensions were prepared by diluting stock suspension in KCl solution. Colloidal quartz sand suspensions were obtained by ultrasonication of clean sand grains in distilled water for 10 min. Colloidal iron oxyhydroxide was obtained from the supernatant of a solution similar to that used in the sand coating procedure described in the previous section. Aliquots of colloidal suspensions were adjusted to the appropriate pH using reagent-grade HCl and KOH solutions. ζ-potentials were calculated from the measured electrophoretic mobilities using the tabulated values of Ottewill and Shaw (60), which have been corrected for relaxation and retardation effects. Column Experiments. Colloid transport experiments were conducted in a glass column packed with various ratios of clean and coated sand grains. Coated sand was ultrasonicated for 30 min prior to use in order to remove loosely attached coating fragments that might detach during experimentation. A constant flow of solution was delivered to the packed column by a series of peristaltic pumps (Masterflex, Cole-Palmer, Vernon Hills, IL). An adjustableheight glass column was used (Rainin Instruments, Woburn, MA). The column was “wet-packed” in a solution having the same chemical characteristics as the solution used during the subsequent experiment. Standard gravimetric methods were used to determine the column packing density. Based on a grain density of 2.59 g/cm3, the column porosity for each experiment was 0.392. A typical column experiment involved pumping an aqueous particle suspension through the packed column at constant flow rate. A fluid approach velocity of 1.02 × 10-4 m/s was maintained. The particle suspension was preceded and followed by a particle-free solution having the same chemical characteristics as the colloidal suspension. The appropriate dosage of KCl solution was applied in-line to the flowing suspension ahead of the column by a low-flow rate peristaltic pump. Particle concentrations in the column effluent were monitored at 10-s intervals using optical density measurements obtained with a UV/ Vis spectrophotometer (Hewlett-Packard Model 8452A) and a 1-cm flow-through cell. Room temperature was maintained during experimentation (about 22 °C). Solution pH of influent and effluent showed little or no variation during the experiments, remaining between 5.6 and 5.8. Two sets of column experiments were conducted. The first set of experiments provided the information necessary

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FIGURE 1. Electrokinetic (ζ) potentials of colloids and collector grains. Potentials were determined from electrophoretic mobility measurements. Electrophoretic mobilities were measured as a function of pH for silica colloids, quartz sand fragments, and iron oxyhydroxide coating material at the ionic strengths shown in the legend.

for determining the excluded area parameter of colloidal silica over a wide range of ionic strengths. Iron oxyhydroxide-coated sand was used as the packing material, and the columns were packed to a depth of 2 cm. The initial colloid concentration used was 72.4 mg/L, corresponding to a number concentration of 2.246 × 109 cm-3. Experiments were performed at various ionic strengths corresponding to the decade intervals spanning the range 10-5 to 10-2 M. A square pulse of suspended colloids was applied to the packed column during each experiment for a duration of 1.5-6.3 h, depending on the ionic strength. Following the particle pulse, columns were flushed with colloid-free solution having the same chemistry as that used during the particle pulse. The second set of experiments was designed to investigate the effect of changing surface charge heterogeneity on particle transport behavior. Various mixtures of coated and clean sand were used to represent changes in geochemical heterogeneity. Columns were packed to a depth of 10.1 cm, and a square particle pulse having a concentration of 18 mg/L (5.584 × 108 cm-3) was delivered to the column at a constant fluid approach velocity of 1.02 × 10-4 m/s. Pulses had a duration of 1.5 h, followed by a flush with particle-free solution. Solution ionic strength was maintained at 10-3 M using KCl as the electrolyte. The surface charge heterogeneity was varied between 0 and 16% in this set of experiments by packing the column with the appropriate mixture of clean and coated sand. Care was taken to minimize abrasion between clean and coated sand during the mixing and packing procedure. All column apparatus and accessories were carefully cleaned between experimental runs with strong acid (HCl), EDTA solution, and a final rinse with deionized water.

Results and Discussion Surface Potentials of Colloids and Collector Grains. The electrokinetic potentials of surfaces used in the colloid transport experiments were determined by electrophoretic mobility. The calculated electrokinetic (ζ) potentials are shown in Figure 1 for the silica colloids, quartz sand grains, and iron oxyhydroxide coatings. Trends in ζ-potential for

FIGURE 2. Hydrodynamic dispersion of silica colloids in porous media. Tracer (NO3-) breakthrough is shown in the insets. Clean sand was used as the column packing. Column length, 10.1 cm; grain diameter, 0.32 mm; bed porosity, 0.39; approach velocity, 1.02 × 10-4 m/s; particle concentration, 18 mg/L; colloid diameter, 0.30 µm; pulse duration, 14.03 pore volumes. A value of 0.692 mm was calculated for the dispersivity parameter (rL). Solid line represents model predictions based on the experimental parameters.

both colloidal silica and quartz sand suggest an isoelectric point of 2, whereas the isoelectric point of the iron oxyhydroxide coating occurs at about pH 6. These isoelectric points are in general agreement with published values for silica, quartz, and iron oxyhydroxide (25, 61, 62). The surface potentials indicate that silica colloids and clean quartz granules are negatively charged under experimental conditions used in the column experiments (i.e., pH ≈ 5.7), while the iron oxyhydroxide surfaces have a slight positive charge. As such, electrostatic interactions between colloids and collector grains are expected to be favorable for deposition onto iron oxyhydroxide-coated collector surfaces and unfavorable for deposition onto bare collectors. Particle Advection and Hydrodynamic Dispersion. In order to isolate the effects of advection and dispersion on the transport of colloidal particles in porous media, a column experiment was conducted under unfavorable conditions using clean quartz collector grains as the packing material. This experimental setup minimizes particle deposition onto collector surfaces, a process that interferes with the determination of advective velocity and particle dispersivity. The results of the column experiment conducted under unfavorable conditions are shown in Figure 2 along with model predictions. Normalized effluent particle concentrations (n/n0) are portrayed in the figure, where n represents the number concentration of particles in the column effluent, and n0 refers to the concentration of particles at the column inlet. Time units have been converted to pore volumes, where one time unit is equivalent to the ratio between bed pore volume and interstitial fluid velocity. A square pulse of colloidal suspension having a concentration of 18 mg/L was intro-

duced into a packed column having a length of 10.1 cm and a porosity of 0.392. The particle pulse had a duration of 14.03 pore volumes and was followed with a particle-free solution having a chemical makeup identical to that of the particle suspension (10-3 M ionic strength and pH 5.8). The conservative molecular tracer (NO3-) breakthrough data are displayed in the insets of Figure 2. These insets display an enhanced view of the ascending and descending portions of the square pulse for both tracer and colloids. The conservative tracer may be used to compare the interstitial particle velocity to the advective velocity of pore fluids. Note the earlier arrival of colloids with respect to the tracer as shown in the insets. This suggests that the advective velocity of the colloidal particles within the porous medium is greater than the fluid velocity, if one assumes that the tracer and fluid medium maintain identical velocities. This velocity enhancement experienced by particles is a result of the pore exclusion phenomenon known as hydrodynamic chromatography (41). A comparison of the tracer and particle breakthrough curves indicates that the colloids are traveling 1.01 times faster than the conservative tracer. This value agrees quite well with the theoretical value of 1.007 obtained from the particle advection equation (eq 2). The experimental particle breakthrough curve shown in Figure 2 was used to obtain a value of 0.692 mm for the dispersivity parameter (RL). The dispersivity may be determined from the variance of the ascending arm of the particle breakthrough curve, according to the method provided in Bear (ref 24, p 608). Particle Deposition Parameters. A series of column experiments were conducted under favorable conditions in order to determine the other deposition parameters required by the theoretical model, namely, the excluded


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FIGURE 3. Particle breakthrough curves and deposition dynamics under favorable chemical conditions. Iron oxyhydroxide-coated sand was used as column packing material. Colloid concentration, 72.4 mg/L; column length, 2.0 cm; other parameters are similar to those listed in the caption of Figure 2. The experimental breakthrough curve data (open circles) were used to obtain the kinetic parameters shown in Table 1 as well as the excluded area parameter (β) values shown in the figure. Theoretical predictions shown in the figure have been generated with the experimentally determined kinetic parameters listed in Table 1. Predictions are based on the random sequential adsorption (RSA) blocking function (solid line) and the Langmuirian blocking function (broken line). TABLE 1

Experimentally Determined Deposition Parameters ionic strength (M)

kinetic parameter kf (m/s)

excluded area parameter, β

10-5 10-4 10-3 10-2

6.43 × 10-7 6.89 × 10-7 7.32 × 10-7 7.57 × 10-7

14.03 7.63 4.44 1.92

area parameter and the particle deposition rate onto favorably charged surfaces. Because electrolyte concentration is known to have a pronounced effect on the kinetic and dynamic aspects of colloid deposition (33, 37), experiments were conducted under favorable conditions using a wide range of ionic strengths. Favorable deposition conditions were promoted by using iron oxyhydroxide-coated collector grains as column packing material. The collective results of the favorable deposition experiments are contained in Figure 3. These results have been used to obtain the experimental values for deposition kinetics and excluded area effects that are found in Table 1. The methods for obtaining these values from column breakthrough data have been previously presented elsewhere (33, 37). The experimental values for the kinetic deposition rate (kf) shown in Table 1 may be compared to deposition rates that have been theoretically determined from rigorous solution of the convective diffusion equation for favorable surfaces (52). The ratio of experimental and theoretical deposition rates gives a value for the collision efficiency, R. For Table 1 deposition rates, the R values are 1 ( 10%, indicating that the experimental deposition rate is close to

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theoretical predictions. This comparison provides evidence that the iron oxyhydroxide coating has reversed the surface charge on the quartz grains, making them favorable for deposition. The effects of blocking are manifested in Figure 3 as rising effluent concentration. These rising concentrations are a direct result of the excluded area effects of previously retained particles. As deposition proceeds, particles begin to accumulate on collector surfaces. This accumulation of colloids produces a decline in the rate of deposition as retained particles impede the deposition process by blocking subsequent particle attachment (33, 37). Blocking is most pronounced at low ionic strengths, as evidenced by the sharper rise in effluent concentration (with respect to time) and the larger values calculated for the excluded area parameter (β). (Note that each plot in Figure 3 has a different time period.) This sensitivity to electrolyte concentration has been attributed to electrical double-layer interactions between colloidal particles (33, 37). As ionic strength is reduced, the range of repulsive forces increases due to expansion of the diffuse ionic cloud surrounding colloids (33). The experimental deposition parameters contained in Table 1 are used to generate the theoretical model output shown in Figure 3. The model results were generated using dynamic blocking functions based on Langmuirian mechanics and the random sequential adsorption (RSA) mechanism. These dynamic blocking functions provide a separate means of quantifying deposition dynamics associated with excluded area effects of retained particles. Although the fit is not consistently superior, the model

FIGURE 4. Particle transport in geochemically heterogeneous porous media. Experimental particle breakthrough curves correspond to columns packed with various fractions of iron oxyhydroxide-coated sand shown in the figure. Experimental parameters are listed in Table 2.

magnitude of the excluded area as well as the coefficients used in the RSA dynamic blocking function (see eq 8).

TABLE 2

Heterogeneous Column Parameter Values parameter

value

approach velocity, U column length, L bed porosity, collector grain diameter, dc colloid diameter, dp influent particle mass concn colloidal particle density influent particle number concn, n0 ionic strength solution pH heterogeneity parameter, λ dispersivity,a RL favorable single collector efficiency,b ηf unfavorable single collector efficiency,c ηu excluded area parameter, β

1.02 × 10-4 m/s 10.1 cm 0.392 0.32 mm 0.30 µm 18 mg/L 2.28 g/cm3 5.58 × 108 cm-3 10-3 M 5.4-5.8 0-0.16 0.692 mm 2.87 × 10-2 4.20 × 10-5 4.44

a Determined from particle breakthrough curve shown in Figure 2. Determined from the 10-3 M particle breakthrough curve in Figure 3 and eq 14. c Determined from the λ ) 0 particle breakthrough curve in Figure 4 and eq 14. b

predictions based on RSA mechanics provide the best overall match to the experimental data for the range of ionic strengths that were investigated. In a previous study involving spherical collector grains and monodisperse colloids, the RSA blocking function was found to be superior to the Langmuirian blocking function over a wide range ionic strengths (37). In this investigation, the observed inability of the RSA mechanism to provide a consistently better fit than the Langmuirian blocking function is attributed to the nonideal nature of mineral grains used in the experiments. Since the RSA mechanism was developed for smooth collector surfaces, deviations from sphericity and prominent surface roughness that were noted for the quartz mineral grains may result in nonideal deposition dynamics. The geometrical deviations from ideality exhibited by the collector grains may have affected the

Colloid Transport in Geochemically Heterogeneous Porous Media. In addition to examining the behavior of colloids under totally favorable or unfavorable conditions, experiments were also conducted under a range of geochemically heterogeneous conditions by packing columns with various ratios of clean and coated sand grains. The results of the heterogeneous column experiments are shown in Figure 4. These experiments were performed under the set of conditions summarized in Table 2. The Figure 4 experimental results clearly indicate the importance of surface charge heterogeneities in controlling colloid mobility in porous media. These experiments show a direct relationship between colloid deposition rate and the fraction of favorably charged collector grains. When there are no favorable surfaces available for deposition (λ ) 0), almost total particle breakthrough is attained (i.e., n/n0 f 1). The introduction of geochemical heterogeneities has a profound impact on colloidal transport, with relatively minor fractions of favorably charged surfaces causing significant departure from total breakthrough. This suggests that the degree of geochemical variability and the availability of favorably charged surfaces may play a major role in determining the mobility of colloids in subsurface aquatic environments. In groundwater environments having a preponderance of iron oxyhydroxides as mineral coatings and low in organic carbon, colloid mobility is likely to be severely curtailed due to the favorable conditions for the attachment of colloids to surfaces. A comparison of the experimental data and model predictions is presented in Figure 5. Since colloid mobility is sensitive to values of the heterogeneity parameter (λ) and the deposition rate for favorable surfaces (kf), two separate sets of model predictions are contained in the figure. The first set of model output (panel a) was generated by allowing the deposition kinetic parameter for favorable


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bed) removal efficiency, with ne and n0 being the column effluent and influent particle concentrations, respectively. This expression can be used to obtain single collector efficiency values for favorable and unfavorable surfaces as well as overall values for heterogeneous porous media. The respective values of the single collector efficiency are then used to determine the kinetic rate constants for deposition onto favorable and unfavorable surfaces according to the following relationship (64):

k)

FIGURE 5. Comparison of model predictions to particle transport in geochemically heterogeneous porous media. In panel a, the single collector efficiency for favorable surfaces (ηf) was allowed to vary in the model. Panel a single collector efficiencies: 1%, ηf ) 0.0287; 2%, ηf ) 0.0354; 4%, ηf ) 0.0367; 8%, ηf ) 0.0424; 16%, ηf ) 0.0443. ηu ) 4.2 × 10-5 in all cases. In panel b, the heterogeneity parameter (λ) was allowed to vary. Panel b heterogeneity values: 0%, λ ) 0; 1%, λ ) 0.0114; 2%, λ ) 0.0235; 4%, λ ) 0.0512; 8%, λ ) 0.1182; 16%, λ ) 0.2473. Other model parameters are listed in Table 2.

surfaces (kf) to vary while fixing the value of the heterogeneity parameter (λ). In the second set (panel b), the heterogeneity values were allowed to vary while the kinetic parameter was held constant. The relation between kinetic parameters and surface charge heterogeneity is described by the single collector efficiencies corresponding to favorable and unfavorable surfaces (32):

η ) ληf + (1 - λ)ηu

(13)

The single collector efficiency η is a dimensionless parameter representing the ratio of the rate of particles striking a collector to the rate of particles approaching the collector (63). Here, the overall single collector efficiency for the packed column η is represented as a linear combination of the respective single collector efficiencies for favorable (ηf) and unfavorable (ηu) fractions. The fraction of collector surface area having favorable charge characteristics is represented by the heterogeneity parameter (λ). Values for the overall single collector efficiency can be obtained directly from experimental data by identifying the clean bed removal efficiency of the particle breakthrough curve (29, 63)

η)-

2dc 3L(1 - )

ln (ne/n0)

(14)

where dc is the collector grain diameter, L is the column length, is the bed porosity, and ne/n0 is the initial (clean

3292

9


ηU 4

(15)

where k is the particle transfer rate and U is the fluid approach velocity. The model results displayed in Figure 5a were obtained by determining separate values for the favorable single collector efficiency for each set of experimental data shown in Figure 4. This was accomplished by first determining the overall single collector efficiency from the clean bed removal efficiency for each data set, according to eq 14. Next, the unfavorable single collector efficiency was determined (ηu ) 4.2 × 10-5) using the experimental data collected under unfavorable conditions (i.e., Figure 4 for λ ) 0 or Figure 2). Finally, the values for ηf were found for each value of λ using eq 13. Two sets of model predictions were generated for each set of experimental data based on the experimental kinetic parameters and Table 2 model parameters. One set was generated with a blocking function based on RSA mechanics, while the other set is based on Langmuirian blocking. The model results shown in Figure 5a provide an adequate portrayal of the experimental data over the range of geochemical heterogeneity that was investigated. The overall results indicate that RSA mechanics is more effective than Langmuirian mechanics when describing blocking in heterogeneously charged porous media. The deviations between the blocking models become more pronounced as deposition proceeds, with RSA mechanics showing an overprediction of blocking effects while the Langmuirian model underpredicts blocking. The deviation between model output and experimental results is attributable to geometrical nonidealities inherent in the column experiments that have not been accounted for in the theoretical model. These nonidealities include serious deviations from sphericity in the sand grains and prominent surface roughness. The experimental results were also modeled by allowing the value of the heterogeneity parameter to vary while fixing the value of the deposition parameters (Figure 5b). Instead of assuming that the favorable surface fraction was identical to the mixing ratio of coated and clean collector grains used to pack the column, we set the favorable single collector efficiency to the value obtained from Figure 3 results (i.e., ηf ) 0.0287). This value was then used along with ηu (Table 2) and the overall single collector efficiency values for each breakthrough curve to calculate the nominal values for λ from eq 13. This procedure produced nominal λ values that are greater than the mixing ratios used in the experiments (see Figure 5 caption for nominal λ values). We attribute this discrepancy to abrasion of iron oxyhydroxide onto clean sand grains during the mixing and packing process. This transfer of iron oxyhydroxide onto unfavorable surfaces would cause an increase in the fraction of favorable surface area (λ) that is present on porous medium grains.

Environmental Implications. The experimental results illustrate the importance of surface charge heterogeneities in controlling the transport and mobility of colloids in granular porous media. Because subsurface aquatic environments are generally unfavorable settings for particle deposition due to the charge similarity between colloids and mineral grain surfaces, small amounts of favorably charged patches can have a profound effect on colloidal mobility. The results demonstrate that the rate of deposition is controlled by favorably charged surfaces, with the overall deposition rate being directly related to the favorable surface fraction. As colloidal particles accumulate on the favorable surfaces, the deposition rate drops as previously deposited colloids block subsequent particle attachment to collector surfaces. The presence of dissolved organic matter and microorganisms on mineral surfaces complicates the characterization of colloid mobility in subsurface aquatic environments. When adsorbed on mineral surfaces, exogenic polysaccharoidal substances, humic material, and other ionogenic organic compounds may mask underlying geochemical heterogeneities, thereby altering the surface charge characteristics and deposition kinetics. Although the effects of organic matter are not included in the transport model, they should be considered in future investigations pertaining to colloidal transport in geochemically heterogeneous porous media.

Acknowledgments The authors acknowledge the support of the National Science Foundation under Research Grants BCS-9308118 and EAR-9418172.

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Received for review January 19, 1996. Revised manuscript received July 10, 1996. Accepted July 12, 1996.X ES960053+ X

Abstract published in Advance ACS Abstracts, October 1, 1996.


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Colloid Transport in Geochemically Heterogeneous Porous Media

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