Hydrology and Land Surface Studies
A colloid-facilitated transport model with variable colloid transport properties
 Anomalous contaminant transport velocities in groundwater for species generally considered to be immobile are often attributed to the mechanism of colloid-facilitated transport. In some of the field observations attributed to colloid facilitation, an extremely small fraction of the total contaminant mass introduced to the groundwater is detected downstream. In this study, a new model of colloid-facilitated contaminant transport is proposed that explains this phenomenon as the variability of mobility of individual colloids in the population. The process of retardation via attachment and detachment of colloids on immobile surfaces is often modeled with time and space invariant parameters; here it is modeled assuming a diverse population of transport properties that account for the inherent variability of colloid size, surface charge and chemical properties, mineralogy, and the concomitant impact on colloid mobility. When the contaminant is assumed to irreversibly attach to or form colloids, the migration of the contaminant plume exhibits extremely non-Fickian behavior. The plume's center of mass travels with a velocity governed by the groundwater velocity divided by the mean colloid retardation factor. However, small quantities of contaminant attached to a few highly mobile colloids travel at velocities up to the groundwater velocity, far exceeding the velocity of the centroid of the plume. This paper introduces the colloid diversity model, presents some sensitivity calculations for an idealized case, and discusses the implications of such a model on data needs, simulation of field observations, and model scaling.
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 Colloids at well-known sites are implicated in facilitating contaminant transport over anomalously large distances [McCarthy and Zachara, 1989; Penrose et al., 1990; Kersting et al., 1999]. For example, when sorption coefficients are large, aqueous transport of contaminants should be slow, however, the field-study fluid samples have indicated that most of a strongly sorbing contaminant was bound to mobile colloidal material and not in the aqueous phase. This is not surprising because the same attractive forces that immobilize a contaminant on rock surfaces might also be expected to occur on mobile colloid particles. In specific cases counterarguments pointing to other transport pathways have been proposed [e.g., Marty et al., 1997], but the case for colloids potentially playing a role in contaminant migration in groundwater studies in general remains strong.
 Simply applying the conventional, aqueous-phase transport models, even when applying a low retardation factor to the contaminant, does not adequately reconcile observations in the cited field investigations. For example, at the Nevada Test Site, the concentration of colloidal plutonium (∼10−14 M) represents a miniscule portion of the total plutonium associated with the underground nuclear test and it has changed very little over nearly a decade of sampling [Rose et al., 2006]. Likewise, measured colloidal 239,240Pu and 241Am concentrations in Mortandad Canyon, Los Alamos, New Mexico by Penrose et al.  are approximately 10−13 M and 10−14 M, respectively. By contrast, effluent concentrations in the source water were orders of magnitude higher for 239,240Pu and 241Am. This severe attenuation is not due to hydrologic mixing because data presented by Marty et al.  for tritium in this same system suggest a downgradient dilution of only about one order of magnitude. Therefore, despite the observed largescale transport, it is inconsistent to treat colloid-facilitated contaminants at these field sites as conservative. Rather, a more plausible explanation of these observations is that a small fraction of the total contaminant mass is mobile, and the rest is immobile on the time scales of the observations.
 Early models of colloid-facilitated contaminant transport in fractures considered equilibrium partitioning of contaminants between the solution phase, colloid surfaces, and media surfaces [c.f. Smith and Degueldre, 1993; Grindrod, 1993], allowing analytical or semi-analytical solutions to be implemented. Corapcioglu and Jiang  introduced a numerical model with first-order reversible rate expressions to describe contaminant sorption onto both colloid and media surfaces as well as first-order expressions describing colloid attachment and detachment onto media surfaces. Ibaraki and Sudicky  were the first to implement kinetic expressions in a model to explicitly describe colloid-facilitated contaminant transport in discrete fractures and fracture networks. Saiers  introduced a colloid-facilitated transport model that accommodates distributions of both colloid transport parameters and solute sorption parameters to account for heterogeneities in both colloid and media properties. Given the affinity of immobile minerals for strongly sorbing species, colloid-facilitated migration over substantial distances in the field may only occur for those contaminants that are essentially irreversibly sorbed to colloid particles or that are embedded within the colloid structure. Reimus  examined colloid-facilitated plutonium transport in sub-meter length fractures with multiple colloid types. Substantial desorption occurred with certain colloids types but not with others, indicating colloid-specific potential for large-scale migration.
 Another research line has focused on better understanding of mechanistic dependencies of colloid attachment and detachment on media surfaces and migration rates considering colloid size and density [James and Chrysikopoulos, 2003], colloid and media surface properties, solution ionic strength, solution pH, flow rates, and media geometry as reviewed by Ryan and Elimelech  and Sen and Khilar . Coupled experimental and simulation studies have sought to develop model parameters for attachment and detachment rates for a variety of colloids with varying flow rates and different rock types [e.g., Reimus, 2003]. Simunek et al.  combined elements of mechanistic models and colloid transport models by linking attachment/detachment parameters to the number of colloids in the system and Sun et al.  and Chen et al.  include the effects of media heterogeneity on colloid transport.
 In the present study, we develop the “colloid diversity model” that allows each colloid in the system to be assigned a different set of transport parameters to explain the large-scale migration of a small fraction of an otherwise immobile contaminant. The critical concept is the assumption that the transport properties of colloids in nature are heterogeneous. Therefore, a single set of colloid transport parameters is insufficient to capture the composite transport behavior of a contaminant that is sorbing or is intrinsically bound to colloidal material. The model is formulated to account for these heterogeneities, which could be due to a variety of factors, including colloid size, attachment/detachment behavior, surface charge, and capacity for sorbing contaminants. A heterogeneous distribution is built into the model, and sensitivity analyses are performed to illustrate model behavior, after which the implications for contaminant transport predictions and data needs are discussed.
2. Model Development
2.1. Colloid-Facilitated Transport: Homogeneous Transport Properties
 As a prelude to development of the colloid diversity model, we begin by considering colloid-facilitated transport of a contaminant through a porous medium. Assuming that the colloids transport in a manner similar to that of a dissolved species, we may write the advection-dispersion equation (ADE) for transport of contaminants attached to colloids as:
where Ccoll is the concentration of contaminant attached to colloids, t is time, z is distance in the direction of flow, ν is the pore-water velocity, Rcoll is the retardation factor of the transporting colloids, and Deff is the effective dispersion coefficient for colloids. Aqueous contaminants also migrate via the ADE with Rf as the sorption retardation factor. Therefore, if they sorb onto colloids via equilibrium reactions where C is the aqueous concentration of the contaminant, a sorption parameter for contaminant onto colloids Kc = Ccoll/C enables equation 1 to be rewritten as:
2.2. Heterogeneous Property Extension to the Colloid-Facilitated Transport Model
 As a practical matter, there are numerous physical and chemical properties that could be important in the determination of the effective retardation factor Reff. Natural colloids have significant variability in size, mineral content, surface area and charge, and capacity to sorb contaminants. This variability in turn gives rise to variability in the parameters related to colloid mobility and contaminant carrying capacity. Therefore, we should not expect a simplified expression such as equation 3 to capture the composite behavior of a contaminant undergoing colloid-facilitated transport.
 Unfortunately, there is a lack of fundamental measurements and theory related to colloid mobility in natural systems. Therefore, a reasonable first step is to develop a simplified model based on the hypothesis that there is a heterogeneous distribution of colloid transport properties, and to explore the implications of such a model on data needs and the types of extrapolation that might be appropriate in practical applications. To this end, we propose the colloid diversity model, in which the colloidal particles possess a distribution of colloid retardation factors Rcoll. In addition, to further simplify the model, we assume that contaminants are irreversibly bound to the colloids, so that transport heterogeneities are due only to variability in colloid mobility parameters. This is done by taking the limit of equation 3 as Kc → ∞, which yields Reff = Rcoll, and is justified by some observations by of Reimus . Although these are significant simplifications, complexity is introduced into this model by assuming a distribution for Rcoll. In contrast to the situation of spatial heterogeneity of a transport property, this heterogeneity is due to a population of colloids possessing a distribution of transport properties. Because it is the contaminant that is of interest rather than the colloids themselves, it is more convenient to define this heterogeneity in terms of the contaminant concentrations rather than a colloid distribution. Therefore, for the case of a chemical species irreversibly sorbed to colloids of various Rcoll, we define the probability density function η(Rcoll) as the fraction of the total contaminant mass with retardation factors between Rcoll and Rcoll + dRcoll, from which it follows that
 The lower bound of the integral reflects an unretarded colloid. Ginn  developed a model in which transport velocities of some colloids are greater than the average groundwater flow velocity, implying that Rcoll < 1. If data suggest that this phenomenon occurs for a particular application, this could easily be accommodated in the present model by reducing the lower bound of the integral. For a population of colloids and associated contaminant, we write the transport equation for the portion of the contaminant bound to colloids of retardation factor Rcoll:
where the contaminant concentration associated with colloids of retardation factor between Rcoll and Rcoll + dRcoll is denoted by C(Rcoll). Because measurements of contaminant migration involve only bulk measurements of total concentration rather than more detailed descriptions such as C(Rcoll), it is most convenient to write an expression for the total contaminant concentration associated with colloids:
 In summary, for the model introduced here, the total concentration is found by computing the individual concentration distributions given by solving equation 5, and superimposing the solutions to these individual C(Rcoll) values using equation 6. In this manner, colloid-facilitated transport of a contaminant with a distribution of colloid transport properties can be simulated easily. In the remainder of this paper, we explore the implications of the colloid diversity model, compared to more traditional approaches that assume that the colloid population can be characterized with a single set of transport parameters.
2.3. Plug Flow Case for Assumed Distribution Functions
 Although the model described above can be solved numerically for a given distribution function Rcoll, further simplifications can be made to obtain an analytical solution to examine the sensitivity of the model to the colloid transport input parameters. The first of these simplifications is the assumption of no longitudinal dispersion due to flow heterogeneities, that is, Deff = 0. For this case, equation 5 reduces to
 For a Dirac pulse of contaminant introduced at the inlet, solving equation 7 gives the following relationship for the location of the contaminant of given Rcoll in time and space:
where v is the pore-water velocity. Contaminants will separate along the flow path chromatographically subject to equation 8 for different values of Rcoll. Likewise, when C(Rcoll) in equation 6 is a Dirac pulse for a specific Rcoll, equation 8 gives the location at any given time for that value of Rcoll.
 The final step in this derivation is to define a distribution function for η(Rcoll) and derive the corresponding relation for the mass distribution of contaminant along the flow path. In this study, both normal and lognormal distributions are examined:Normal:
 In these equations, coll is the mean and σ is the standard deviation of the normal distribution, and 〈ln(Rcol)〉 and σl are the corresponding terms for the lognormal distribution. To obtain an expression for the contaminant mass density as a function of time and location along the flow path in response to a pulse injected at the input at time 0, we substitute equation 8 into either equation 9 or 10 to eliminate Rcoll:Normal:
3. Model Results
 We now illustrate the impact of heterogeneity in the colloid-transport properties, as reflected in the distribution Rcoll, on the predicted distribution of contaminant mass along a one-dimensional flow path. Beginning with the normal distribution and assuming coll = 10, we use equation 11 with different σ values, to compute the dimensionless concentration profiles for these Rcoll distributions. Figure 1a shows concentration for three σ values at a time corresponding to the transport of an unretarded species to the end point of the path, at dimensionless z of unity. These profiles, shown in Figure 1, illustrate the extreme non-Fickian transport behavior that arises for the colloid diversity model, despite the relatively narrow distributions used. The peak and center of mass of the contaminant occurs at roughly the position that would be predicted based on the mean of the retardation factor distribution (coll), but the model also predicts small quantities traveling far in advance of the center of mass. Greater travel distances and larger concentrations of the rapidly moving mass are predicted for larger σ.
 In field-scale observations of contaminant migration in which colloids are implicated, the focus is usually on the observation that some mass has moved much farther than expected based only on contaminant retardation properties. The unique feature of the colloid diversity model is that it predicts a small quantity of the total plume mass at great distances, while the majority of the mass remains relatively close to the source. Models that assume a Fickian dispersion mechanism and homogeneous colloid properties do not yield this behavior. For example, Figure 1b shows the predicted concentration profile for a one-dimensional transport model with dispersion (equation 5) but without the distribution of colloid retardation factors. The plumes are symmetrical, and concentrations a significant distance from the center of mass are essentially zero for reasonable values of the dispersion coefficient. By contrast, the model in the present study (e.g., Figure 1a) yields small but non-zero concentrations a great distance from the source and the center of mass. Similar model calculations for coll = 100 are shown in Figure 1c, with values of σ scaled so that the normalized standard deviations are the same as for the coll = 10 case. The center of mass for coll = 100 is even closer to the inlet due to the larger retardation factors, but the same transport phenomenon for the fastest moving portion of the contaminant mass remains.
 In real groundwater systems, longitudinal dispersion due to small-scale velocity variations is a process that spreads a solute along the flow path. For clarity, we call this process hydrodynamic dispersion to distinguish it from the spreading mechanism resulting from heterogeneous colloid transport properties. Recall that in simplifying the colloid transport model, we neglected hydrodynamic dispersion of colloids of a given value of Rcoll. If it had been included, we expect that hydrodynamic dispersion would have a minimal impact on the concentration profile. In general, the distribution of Rcoll would need to be quite narrow for the competing hydrodynamic dispersion mechanism to be important. Therefore, the analytical expressions developed in the present study may be useful in a large number of practical applications.
 For the lognormal distributions, the model yields qualitatively similar results to the normal distribution scenarios as shown in Figure 1d. The parameter 〈ln(Rcoll)〉 is 2.303 for these simulations, yielding distributions that peak at Rcoll = 10 so that results can be compared easily to the normal distribution cases. Slightly more mass is held up near the inlet for the lognormal distribution profiles because of the skewing of the distributions toward large values of Rcoll as compared with the normal distribution scenarios. At large distances from the source, small quantities of contaminant outpace the bulk of the mass. The tails of the concentration profile are different between the two cases due to differences in the shapes of the distributions for small values of Rcoll . The important result, that small amounts of mass travel significant distances relative to the plume's center of mass, is yielded regardless of the exact nature of the distribution. For the colloid diversity model, concentrations of contaminant downstream from the center of mass are controlled by the proportion of the colloid population that is relatively mobile, that is, those that possess low values of Rcoll . For narrower distributions, the quantity of mobile colloids is smaller, resulting in lower concentrations downstream of the source.
4. Discussion and Conclusions
 Several observations of large-scale transport of contaminants typically thought to be immobile have cast doubt into the validity of transport models that consider only aqueous species, and have spawned numerous investigations on the role of colloids in facilitating contaminant transport in groundwater. Although the large transport distances are well publicized, it is perhaps less well known that in the cases cited here (Nevada Test Site and Mortandad Canyon), the concentrations of contaminants found to have traveled a great distance are exceedingly small, measurable only because of the highly sensitive techniques available for detecting radionuclides. Plumes of these species are not found to behave like conservative solutes such as tritium. Rather, the center of mass of the plume remains very close to the source due to aqueous solute retardation although a small fraction of the mass is able to migrate a significant distance.
 We believe that the colloid diversity model developed here is a reasonable alternative to colloid transport models available in the literature. Conceptually, the model ascribes the rapid transport to the mobility of a small fraction of the total colloid mass, to colloids carrying irreversibly sorbed contaminants. Variability in transport characteristics of the colloids gives rise to a plume-spreading phenomenon that can be highly non-Fickian.
 According to the colloid diversity model, it is critical to measure the transport characteristics of the population of colloids present in the hydrogeochemical setting of interest. Mechanisms controlling the migration rates of contaminant-carrying colloids of different size and charge characteristics are critical. Therefore, experiments using particles of uniform size and charge characteristics will not shed light on the colloid variability controlling the behavior predicted by the model, which we believe approximates the behavior of actual colloid-facilitated contaminant plumes. If average plume behavior is of interest, perhaps mean retardation factors and mean transport velocities of typical colloids are sufficient. However, to understand the reasons for rapid transport of trace quantities of a contaminant, there is a need for extensive characterization focusing on the highly mobile portion of the colloid population. This would require procedures for identifying and separating those colloids from the rest of the colloid population, followed by detailed characterization of their physicochemical properties. In addition, the partition coefficients of contaminants onto various types of colloids and onto competing immobile minerals are required to determine the source concentration Ccoll.
 Several assumptions in the model must be examined critically. First, irreversible attachment of contaminant to colloid is an approximation and that may be erroneous for some contaminant-colloid systems. However, it is easy to show that unless the contaminant is irreversibly sorbed to or intrinsically a part of the colloid, strongly sorbing contaminants remain relatively immobile because on a unit mass basis, they sorb equally to the stationary porous medium and the colloids. Therefore, assuming completely irreversible attachment to colloids seems a reasonable approximation for those sites where contaminants have migrated unexpectedly large distances. Kinetics of contaminant-colloid interactions could be added to the model, albeit at the expense of increased complexity. Such enhancements could potentially explain differences in mobility of different contaminants. In such a case, we still believe that variability of the mobility of colloids should be included, given the inherent variability of colloids in nature.
 The reduction of the complex process of colloid transport to a retardation factor is another simplification. Colloid attachment and de-attachment, filtration, and stability are complex processes that depend on groundwater chemistry, particle surface charge, particle size, and particle concentration [e.g., McCarthy and Zachara, 1989; Roy and Dzombak, 1997; Kaplan et al., 1997]. The complexity of these processes underscores the necessity of assuming that a population of colloids possesses a range of transport properties. Selection of a retardation factor model is for convenience in developing an approximate mathematical model: more complex versions of the model (kinetically controlled colloid-contaminant interactions, multidimensional dispersed flow) are possible and are the subject of current research. Nevertheless, we believe the colloid diversity model is a useful engineering approach for capturing the key transport features in field settings where colloids are suspected to be important but detailed colloid characterization is lacking.
 We are grateful to Sharad Kelkar and two anonymous reviewers of this manuscript. This study was performed under the auspices of the Department of Energy, Office of Civilian Radioactive Waste Management, for the Yucca Mountain Project.