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SPECIAL SECTION: VADOSE ZONE MODELING

Modeling Colloid-Facilitated Contaminant Transport in the Vadose Zone

Dep. of Crop and Soil Sciences, Center for Multiphase Environmental Research, Washington State Univ., Pullman, WA 99164
^* Corresponding author (flury{at}mail.wsu.edu).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Received 2 April 2007.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 
Subsurface colloids can enhance the movement of strongly sorbing contaminants, a phenomenon called colloid-facilitated contaminant transport. In the presence of mobile subsurface colloids, contaminants may move faster and farther than in the absence of colloids, thereby bypassing the filter and buffer capacity of soils and sediments. Fate and transport models neglecting colloid-facilitated transport therefore often underpredict contaminant movement. Long-term predictions of contaminant fate and transport as well as risk assessment rely on an accurate representation of subsurface processes, and in the case of strongly sorbing contaminants, need to consider mobile colloids as potential contaminant carriers. The purpose of this review is to discuss the current knowledge and recent developments of modeling colloid-facilitated contaminant transport in the vadose zone. The main part of this review is devoted to the discussion of conceptual models used to describe colloid-facilitated contaminant transport in the vadose zone and their mathematical implementation. Modeling of colloid-facilitated contaminant transport involves various interactions, including colloid attachment to and detachment from the solid matrix and the air–water interface, contaminant adsorption to and desorption from colloids and transport with mobile colloids, and contaminant adsorption to and desorption from the solid matrix. Most of these processes in colloid-facilitated contaminant transport models have been described by first- or second-order kinetics. The unique feature of the vadose zone is the presence of an air phase, which affects colloid and contaminant transport in several ways. Colloids can be trapped in immobile water, strained in thin water films and in the smallest regions of the pore space, or attached to the air–water interface itself. The modeling of colloid-facilitated contaminant transport in the vadose zone has mostly been theoretical, and tested only with column experiments; field applications are still lacking.

Abbreviations: RSA, random sequential adsorption

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 
IT HAS LONG BEEN ECOGNIZED that certain contaminants move considerable distances through the subsurface only if they can sorb or attach to mobile colloidal particles. The colloids act as carriers for these contaminants. This phenomenon is called colloid-facilitated contaminant transport. The prerequisites for colloid-facilitated contaminant transport are: (i) colloids need to be present and mobile in the subsurface, and (ii) contaminants need to sorb or attach strongly to the colloids (Honeyman,1999; Kretzschmar et al., 1999). Certain contaminants can also form colloids on their own by precipitation or polymerization from the solution phase to a colloidal solid phase. These type of colloids are called "intrinsic colloids," "true colloids," "real colloids," or "Eigencolloids" (Kim, 1986; Lieser et al., 1986; Choppin and Morgenstern, 2001). Intrinsic colloids are particularly important for actinides, such as Pu and Am. If contaminants associate with organic or inorganic colloids present in the subsurface, the contaminant–colloid associations are often called "pseudocolloids" (Kim, 1986; Honeyman and Ranville, 2002).

The relevance of colloid-facilitated contaminant transport is that strongly sorbing contaminants can move much farther than in the absence of a colloidal carrier. Such enhanced movement can have positive or negative consequences. On the positive side, if strongly sorbing contaminants need to be removed by leaching or washing, colloid-facilitated transport may offer a viable remediation strategy. More often, however, the consequences of colloid-facilitated transport are negative. Enhanced mobility of contaminants may cause contamination of groundwater. A prominent example of undesired colloid-facilitated transport is the migration of radionuclides from underground testing operations or storage facilities. Such enhanced migration for Pu in groundwater has been reported from the Nevada Test Site in the United States (Kersting et al., 1999) and the Mayak Production Association site in Russia (Novikov et al., 2006). Other examples of colloid-facilitated transport have been reported for pesticides (de Jonge et al., 1998; Sprague et al., 2000), heavy metals (Karathanasis, 1999; Sen et al., 2002), and P (Heckrath et al., 1995; de Jonge et al., 2004). Several reviews on colloid-facilitated contaminant transport have been published in the past 20 yr, most of them focused on the saturated zone (McCarthy and Zachara, 1989; Mills et al., 1991; Ouyang et al., 1996; McCarthy, 1998; Kretzschmar et al., 1999; McGechan and Lewis, 2002; Honeyman and Ranville, 2002; Sen and Khilar, 2006).

There is only scarce field evidence for facilitated transport of contaminants by colloids in the vadose zone, and much of this evidence is indirect, because either the direct association of contaminants with colloids has not been proven or, in cases where the association with colloids was demonstrated, there was no direct proof that the contaminant-bearing colloid really had moved through the subsurface. For instance, based on the temporal correlation of particle concentrations and Pu activities measured in zero-tension lysimeters, it was inferred that Pu was probably transported through a soil profile by mobile particles (Ryan et al., 1998). Fractionation of interstitial soil water showed that Pu and Am were associated with suspended particles larger than 0.45 µm (Litaor et al., 1998). Such findings are strong evidence of colloid-facilitated contaminant transport, although the process itself has not been shown in field studies. One big limitation in field studies is that there is no good method to sample vadose zone pore water for colloids without disturbing the hydraulic regime.

From laboratory transport experiments, we know that colloids can facilitate contaminant movement under unsaturated flow, but the amounts of contaminants transported are usually less than under saturated flow. As the amounts of colloids transported usually decrease with decreasing water saturation, so do the amounts of colloid-associated contaminants (Chen et al., 2005). On the other hand, transient flow, which is common at least in the near-surface vadose zone, has been found to enhance colloid mobilization (Saiers and Lenhart, 2003; Levin et al., 2006; Zhuang et al., 2007) and thus could potentially be an important contributor to colloid-facilitated contaminant transport.

Modeling is particularly important for the study of colloid-facilitated contaminant transport, because time scales of interest are often large. For instance, compliance periods for nuclear waste storage can be 10,000 yr or longer (Contardi et al., 2001), requiring mathematical models to make long-term predictions. The modeling of colloid-facilitated contaminant transport in the vadose zone builds on the modeling of water flow and solute transport, in which colloid transport and colloid–contaminant interactions are incorporated.

The purpose of this review is to discuss current approaches for modeling colloid-facilitated contaminant transport with emphasis on the vadose zone. We first briefly discuss some principles of colloid transport in the vadose zone, as the transport of colloids is the prerequisite for colloid-facilitated contaminant transport. We then present a conceptual model for colloid-facilitated transport in the vadose zone. Three specific examples of conceptual models are discussed for the transport of the radionuclide Pu, for the interaction of Am with humic colloids, and for organic colloids. Next, we discuss mathematical models for colloid-facilitated contaminant transport, i.e., how the individual reactions of the conceptual models are formulated mathematically and implemented in transport codes.

	Colloid Transport in the Vadose Zone

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 
Several review articles have summarized colloid transport in the saturated zone (Swanton, 1995; Ryan and Elimelech, 1996; Kretzschmar et al., 1999; Sen and Khilar, 2006; Tufenkji, 2007). Recent reviews by DeNovio et al. (2004) and McCarthy and McKay (2004) specifically focused on colloid movement in the unsaturated zone. We refer to these reviews for a comprehensive treatment of colloid movement in the subsurface. Here, we provide only a brief summary relevant for the vadose zone and describe some of the new developments in recent years.

Colloid transport in the vadose zone differs from transport in the saturated zone mainly by the following aspects (see also DeNovio et al., 2004; McCarthy and McKay, 2004).

The Presence of an Air–Water Interface
Colloidal particles may attach to the air–water interface (Wan and Wilson, 1994; Schäfer et al., 1998; Sirivithayapakorn and Keller, 2003; Lazouskaya et al., 2006); thus the air–water interface provides an additional site for colloid attachment and immobilization. Experiments under steady-state flow have shown that colloid transport usually decreases if the water saturation of the porous medium decreases (Wan et al., 1994; Lenhart and Saiers, 2002; Cherrey et al., 2003). The air–water interface can also cause colloids to become immobilized when the water films on the solid surfaces become thinner than colloidal diameters, a mechanism called film straining. In this case, strong capillary forces pin colloids to the solid surfaces (Zimon, 1969; Wan and Tokunaga, 1997; Veerapaneni et al., 2000) or colloids get trapped in pendular ring regions separated from the remaining fluids by thin water films (Wan and Tokunaga, 1997). If the water films expand, these trapped colloids can be remobilized (Gao et al., 2006). It was also suggested that colloids can be immobilized at or near the air–water–solid interface (Crist et al., 2004, 2005; Chen and Flury, 2005; Zevi et al., 2005). This latter mechanism is different from air–water interface attachment, because colloids do not have to penetrate the air–water interface to become immobilized.

Colloid attachment to the air–water interface usually leads to strong bonding due to capillary forces (Huh and Mason, 1974; Scheludko and Nikolov, 1975; Pitois and Chateau, 2002). Moving air–water interfaces are therefore likely to carry attached colloids along (Saiers and Lenhart, 2003; Zhuang et al., 2007) and can even scour colloids from solid–water interfaces (Gomez-Suarez et al., 1999a,b; Saiers et al., 2003). Such movement of air–water interfaces frequently occurs in the vadose zone during infiltration and drainage.

As shown, the role of the air–water interface is manifold: the air–water interface may cause both enhanced colloid retention and enhanced colloid mobilization and transport in the vadose zone. There is still much debate on how exactly the air–water interface affects colloid transport in porous media, but it is well established that it plays an important role.

The Presence of Nonuniform Flow
Flow in the vadose zone is often nonuniform. This nonuniformity of flow is manifested in various forms of preferential flow: macropore flow, unstable flow, fracture flow, funnel flow, or mobile–immobile water (Beven and Germann, 1982; Hendrickx and Flury, 2001). The nonuniform flow can promote colloid mobilization and transport in macropores (Ryan et al., 1998; Schelde et al., 2002). Moving air–water interfaces and increased water content in preferential flow channels can enhance colloid migration (DiCarlo et al., 2006). On the other hand, colloids can get trapped in immobile water regions, resulting in decreased movement of colloids.

The Occurrence of Transient Flow
Water flow in the vadose zone is not only spatially nonuniform, but flow rates and water saturations change with time. Transient flow is indeed the standard flow type in the near-surface vadose zone, i.e., the region close to the soil surface. Rainfall, snowmelt, irrigation, and evapotranspiration lead to transient flow, which is associated with expanding and shrinking water films, changing and moving air–water interfaces, and changing mobile–immobile water regions (El-Farhan et al., 2000). All of these processes affect colloid transport.

Pronounced Gradients in Pore Water Chemistry
Infiltrating water from precipitation, either rain or snow, usually has a lower ionic strength than the pore water of the soil. Displacement of higher ionic strength with lower ionic strength water can mobilize colloids, as observed in the case of precipitation and groundwater recharge (Nightingale and Bianchi, 1977; Kaplan et al., 1993; Ryan et al., 1998). Such displacement-associated colloid mobilization can also occur at waste disposal sites (Grolimund et al., 1996; Flury et al., 2002). The near-surface vadose zone, i.e., soil, is characterized by pronounced vertical stratifications in solid substrate and solution chemistry. For instance, in humid climates, solution pH usually increases with depth. Such changes in surface and solution chemistry can cause colloid mobilization or immobilization. Various soil-forming processes, where organic colloids or colloidal clay are translocated along the soil profile, are testimony of colloid migration and deposition as affected by soil stratifications (Brady and Weil, 2002).

Enhanced Weathering and Chemical Reactions
In the near-surface vadose zone, chemical and physical weathering processes are intense, causing minerals to break apart, transform, dissolve, and precipitate. Organic acids exuded from plants and other soil organisms enhance weathering. Colloidal particles in soils can form in situ, a process known in soil science as clay formation (Brady and Weil, 2002). Organic matter, carbonates, and sesquioxides can bind colloidal particles together, thereby preventing mobilization. Upon degradation of organic matter or dissolution of carbonates and sesquioxides, colloids can again be mobilized.

All these factors make the vadose zone unique in terms of colloid fate and transport. Many of these factors are not well understood quantitatively in terms of colloid fate and transport, making it difficult to develop mathematical models to predict these processes (McCarthy and McKay, 2004).

	Conceptual Models for Colloid-Facilitated Contaminant Transport in the Vadose Zone

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 
General Conceptual Model
A conceptual model for colloid-facilitated contaminant transport in the vadose zone builds on the colloid transport models developed for the saturated zone (Corapcioglu and Jiang, 1993; Kretzschmar et al., 1999; Honeyman and Ranville, 2002), but in addition the air phase and its interfaces with the solid and aqueous phases have to be included. The main components of such a conceptual vadose zone model have been described by Corapcioglu and Choi (1996), Simunek et al. (2006), and Massoudieh and Ginn (2007).

A general model for colloid-facilitated contaminant transport in the vadose zone consists of four distinct phases: (i) the stationary solid phase, (ii) the aqueous phase, (iii) the colloidal phase (carrier), and (iv) the air phase (Fig. 1 ). These different phases are separated by interfaces, which can act as reaction sites for contaminants and colloids. Colloids can be either inorganic or organic materials reacting with contaminants (pseudocolloids) or colloids formed by the contaminants themselves (intrinsic colloids). The contaminants, either as aqueous species or intrinsic colloids, can interact with all interfaces in the system, as indicated by the arrows in Fig. 1. These arrows represent different types of reactions: (i) sorption or precipitation of aqueous contaminants to stationary sediments; (ii) sorption of aqueous contaminants to mobile (suspended) inorganic or organic colloids; (iii) sorption of aqueous contaminants to colloids attached at the solid phase or the air–water interface; (iv) sorption of aqueous contaminants to the air–water interface; (v) precipitation or polymerization of aqueous contaminants to form intrinsic colloids; (vi) attachment, detachment, or straining of colloids at the solid–water interface; (vii) attachment, detachment, or straining of colloids at the air–water interface; and (viii) attachment, detachment, straining, or trapping of colloids through a combined effect of the solid–water and air–water interfaces. If contaminants are volatile, an additional mass transfer from the aqueous phases into the bulk gas phase has to be considered.

View larger version (51K): [in this window] [in a new window]   FIG. 1. Conceptual model of colloid-facilitated contaminant transport in the vadose zone.

Example 1: General Conceptual Model for Plutonium
The radionuclide Pu is an ideal candidate for colloid-facilitated transport. Plutonium strongly sorbs to natural subsurface materials and has a low aqueous solubility (Honeyman, 1999). Consequently, Pu will not move far through the subsurface if transported only as the pure soluble species. Attached to colloidal particles, however, transport velocities can be as fast as the average speed of water, and Pu can move unretarded through the subsurface. Enhanced mobility of Pu in the subsurface has indeed been linked to colloidal transport (Kersting et al., 1999; Novikov et al., 2006).

A specific conceptual model for Pu reaction and transport processes in the vadose zone is depicted in Fig. 2 . The model contains the major geochemical reactions that Pu can undergo, including hydrolysis, complexation, polymerization, precipitation, sorption, and colloid attachment or detachment to interfaces. Plutonium has four oxidation states and can occur in all four oxidation states simultaneously, yet one oxidation state will usually dominate the aqueous system depending on pH and Eh conditions. Plutonium sorbs strongly to inorganic mineral colloids, and this association has been found to be responsible for large-scale field transport of Pu (Kersting et al., 1999; Novikov et al., 2006). Plutonium can also form aqueous complexes with dissolved organic matter, humic colloids, or anions such as CO₃^2– (Cleveland and Rees, 1981; Nelson et al., 1985; Kim et al., 1989). Furthermore, Pu can form intrinsic colloids by polymerization, where the particles are initially very small and grow with time. For Pu, particle sizes of intrinsic colloids vary from <1 nm to >15 µm (Ichikawa and Sato, 1984; Kim et al., 1985). Finally, Pu can also associate with biocolloids such as bacteria (Mahara and Kudo, 1998; Mahara and Kudo, 2001). The model depicted in Fig. 2 shows the complexity involved when dealing with an element like Pu.

View larger version (52K): [in this window] [in a new window]   FIG. 2. Conceptual model of Pu reaction processes in an unsaturated subsurface system. Arrows indicate direction of reactions.

Example 2: Conceptual Model for Americium in the Presence of Humic Acids
Americium has a general conceptual model very similar to that of Pu except that Am does not undergo redox reactions. In this example, we focus on just one of the many reactions that can take place in the subsurface, namely the association of Am with dissolved organic matter. The interactions of Am with humic acid were found to be kinetically controlled and, based on experimental evidence, a three-compartment model has been developed (Artinger et al., 1998; Schüssler et al., 2000). The model describes sorption of Am to both stationary sediments and mobile humic acids, where the humic acid sorption is considered a sequential two-step process, including fast and slow sorption reactions (Fig. 3 ).

View larger version (11K): [in this window] [in a new window]   FIG. 3. Conceptual model of Am reaction with sediments and humic acid. Arrows indicate direction of reactions (after Schüssler et al., 2000).

View larger version (44K): [in this window] [in a new window]   FIG. 4. Conceptual model of organic sorbent affected contaminant transport. Arrows indicate direction of reactions (after Totsche and Kögel-Knabner, 2004).

	Mathematical Models for Colloid-Facilitated Contaminant Transport in the Vadose Zone

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

The General Advection–Dispersion Equation
For the mathematical description of fate and transport, we closely follow the notation used in Simunek et al. (2006) (all symbols are defined in the Appendix). The general mass balance equation for the transport of colloids in one dimension can be written as (Corapcioglu and Choi, 1996; Simunek et al., 2006)

[1]

where _w is the volumetric water content accessible to colloids [L³ L^–3], C_c, S_c, and _c are the colloid concentrations suspended in the aqueous phase [M L^–3], attached to the solid phase [M M^–1], and attached to the air–water interface [M L^–2], respectively; is the bulk density of the solid phase [M L^–3], A_aw is the area of the air–water interface per unit volume [L² L^–3], D_c is the hydrodynamic dispersion coefficient for colloids [L² T^–1], q_c is the volumetric water flux for colloids [L T^–1], R_c represents various chemical and biological reactions that generate or destroy colloids, e.g., formation and dissolution of intrinsic colloids [M L^–3 T^–1], t is time [T], and x is space [L]. The subscript c denotes that the variable applies to the colloidal phase. The terms on the left side of Eq. [1] represent the rate of mass change of colloids in the aqueous and solid phases as well as the air–water interface. The right-hand side of Eq. [1] denotes the dispersive and advective colloid fluxes, respectively, and the colloid mass change due to colloid formation or dissolution. The volumetric water content accessible to colloids, _w, can be smaller than the actual volumetric water content, , because of size exclusion or regions that are inaccessible to colloids (Lichtner et al., 2002; Bradford et al., 2006). The actual velocity, v_c [L T^–1], of the colloids is given as

[2]

The colloid concentrations can be denoted as mass or number concentrations. The choice of which concentration is used depends on what type of colloids are investigated (e.g., monodisperse or polydisperse), how the colloids are measured (e.g., on a mass or number basis), or the purpose of the study (e.g., relevance of total mass or individual particles). For the remainder of this review, we use the mass concentration for colloids (e.g., M L^–3), but one can replace this by number concentrations (e.g., number L^–3). In the latter case, all related concentrations (e.g., contaminant concentrations on colloids) would have to be adapted accordingly.

The mass balance equation for the contaminant can be written as (Corapcioglu and Choi, 1996; Saiers and Hornberger, 1996a; van de Weerd et al., 1998; Simunek et al., 2006)

[3]

where C, S, and are the contaminant concentrations in the aqueous phase [M L^–3], adsorbed to the solid phase [M M^–1], and adsorbed to the air–water interface [M L^–2], respectively; S_mc, S_ic, and S_ac are contaminant concentrations sorbed to mobile colloids in the aqueous phase, to the immobile colloids at the solid phase, and to the immobile colloids at the air–water interface [M M^–1], respectively; D is the hydrodynamic dispersion coefficient for contaminants [L² T^–1], q is the volumetric water flux for the contaminant [L T^–1], and R represents various chemical and biological reactions [M L^–3 T^–1]. The first three terms on the left-hand side of Eq. [3] represent the rates of mass changes of the contaminant in the aqueous phase, solid phase, and air–water interface. The third term, representing contaminants sorbed to the air–water interface, is relevant for any surface-active contaminant, such as organic chemicals or surfactants. The fourth, fifth, and sixth terms describe mass changes of contaminants associated with colloids in suspension, attached to the solid phase, and attached to the air–water interface. The right-hand side of Eq. [3] represents mass transfer due to dispersive and advective transport of the dissolved contaminant (first and second term), and colloid-associated contaminants (third and fourth term), and the last term denotes sink–source reactions of contaminants.

In the case of a volatile contaminant, and assuming that the air phase is stagnant, an additional term for the contaminant in the air phase would have to be included on the left-hand side of Eq. [3] (Choi and Corapcioglu, 1997a):

[4]

where a is the air content [L³ L^–3], C_a is the contaminant concentration in the air phase [M L^–3], k_aw is the rate coefficient for contaminant mass transfer across the air–water interface [T^–1], and H is Henry's constant (dimensionless).

Various modifications and simplifications of Eq. [3] have been made or used by researchers. Some researchers assume that the colloids have the same dispersion coefficient as the contaminants, others assume that the colloids move with the same velocity as the water (Corapcioglu and Jiang, 1993). Equations [1] and [3] are also based on the assumption that the air–water interface is stationary and therefore does not contribute to colloid and contaminant movement through the porous medium. During infiltration and drainage, however, air–water interfaces are known to move (Saiers and Lenhart, 2003; Zevi et al., 2005; Gao et al., 2006; Zhuang et al., 2007), and this movement (which probably includes both advective and dispersive components) would have to be included on the right-hand side of Eq. [1] and [3].

Colloid Interactions with the Solid Phase
The colloid attachment to the stationary solid phase can be described as a kinetic process. Both irreversible attachment and reversible attachment–detachment kinetics have been used (Corapcioglu and Jiang, 1993; Saiers and Hornberger, 1996b; Choi and Corapcioglu, 1997b; Compere et al., 2001). For organic colloids, such as dissolved organic matter, equilibrium and nonequilibrium sorption have been used (Corapcioglu and Jiang, 1993; Knabner et al., 1996). It is reasonable to assume that the attachment sites for colloids on the solid phase are limited, particularly when colloid concentrations are high. A blocking function and a second-order kinetics for colloid attachment are well accepted to represent the limited attachment capacity of the solid phase (Saiers et al., 1994; van de Weerd et al., 1998; Turner et al., 2006; Simunek et al., 2006). With the blocking function, the mass balance for colloids attached to the solid phase can be written as (Simunek et al., 2006)

[5]

where k_ac is the colloid attachment coefficient and k_dc the detachment coefficient [T^–1], _s is a blocking function (dimensionless), and f_s is the available fraction of the solid surface area for attachment (dimensionless).

It is often found that colloid attachment is irreversible, so that Eq. [5] reduces to

[6]

A special case of Eq. [5] is the equilibrium condition when (S_c)/t = 0. Although this latter condition has been used for colloid modeling, it is usually considered inappropriate for colloids (Tufenkji, 2007).

Some researchers also considered that there are two sources of colloids, namely, colloids initially released from the stationary solid phase and colloids captured and released again later (Sen et al., 2004; Massoudieh and Ginn, 2007). Thus, colloid release and capture can occur at different sites. In this case, the release term in Eq. [5] can explicitly be written as (Sen et al., 2004 Massoudieh and Ginn, 2007)

[7]

where k_dc1 and k_dc2 are detachment coefficients [T^–1] for Site 1 and 2, respectively, and S_c1 and S_c2 are the colloid concentrations at the two sites [M M^–1].

For porous media with spherical grains, the colloid attachment coefficient k_ac can be expressed by (Harvey and Garabedian, 1991; Logan et al., 1995)

[8]

where is the porosity of the porous medium [L³ L^–3], d_g is the grain diameter [L], is the collision efficiency (dimensionless), is the single-collector efficiency (dimensionless), and v_c is the velocity of the colloids [L T^–1]. The collision efficiency describes the fraction of collisions that lead to attachment (Stumm and Morgan, 1996):

[9]

The solid surface has a limited capacity for colloid deposition, which is represented by a blocking function, _s. The simplest form of such a function is of Langmuir type (e.g., Saiers et al., 1994; van de Weerd et al., 1998):

[10]

where S_max is the colloid deposition capacity of the solid phase [M M^–1]. Another blocking function, random sequential adsorption (RSA), which is based on the irreversible attachment of spherical particles, is given as (Schaaf and Talbot, 1989)

[11]

For colloid-facilitated contaminant transport, it is often assumed that the contaminants do not compete with colloids for attachment sites, otherwise the blocking functions would have to be modified to include this process. The function _s can also be used to represent increased colloid deposition with increasing amount of colloids attached, e.g., ripening processes. Another assumption often made implicitly is that when contaminants sorb to colloids, the physicochemical characteristics of the colloids do not change, i.e., their transport and attachment behavior remains the same.

Filtration theory describes colloid attachment under favorable conditions, i.e., in the absence of an energy barrier. For subsurface colloids, however, there often exists an energy barrier for attachment because colloids and sediment surfaces are often similarly charged (Honeyman and Ranville, 2002). Under these conditions, filtration theory would predict no attachment or retention of colloids during transport. Experiments have repeatedly shown, however, that even in presence of a repulsive barrier, colloids are retained in porous media, and several modifications to filtration theory have been proposed to explain these observations (see Johnson et al. [2005] for a detailed discussion). Processes not described by filtration theory include physical straining of colloids in pore constrictions (Herzig et al., 1970; Bradford et al., 2002, 2006), deposition in secondary energy minima (Hahn and O'Melia, 2004; Tong and Johnson, 2006), and wedging of colloids between sediment grains and retention in flow-stagnation regions (Herzig et al., 1970; Johnson et al., 2007).

Physical straining of colloids due to pore constrictions becomes important when the ratio of the colloid to grain size, d_c/d_g, exceeds a certain threshold value. A threshold value of d_c/d_g = 0.08 was suggested by Herzig et al. (1970), but recently much lower ratios of d_c/d_g = 0.005 (Bradford et al., 2002) and d_c/d_g = 0.008 (Xu et al., 2006a) have been proposed. Based on experimental observations, Bradford et al. (2003) proposed the following equations for colloid straining:

[12]

and

[13]

where S_c^str is the concentration of strained colloids [M M^–1], _str is a straining function (dimensionless), k_str is a first-order straining coefficient [T^–1], d_g is the median grain size of the porous medium [L], x is the distance from the inlet [L], and β is an empirical parameter (dimensionless). Combining physicochemical filtration (Eq. [5]) and straining (Eq. [12]) then leads to

[14]

An alternative straining formulation was proposed by Xu et al. (2006a). In their formulation, it is assumed that as colloids are strained out, the fluid streamlines are diverted away from the strained areas because of the increased flow resistance. The streamlines are then concentrated in larger pores and straining consequently decreases. The following mathematical representation of this process was proposed (Xu et al., 2006a):

[15]

where k_str,0 is an initial straining rate coefficient [T^–1] and is an empirical parameter describing the decline of straining [M M^–1].

Compared with deposition, colloid release is much less understood from a modeling perspective. Colloid release has been modeled by first-order kinetics (Dahneke, 1975; Ruckenstein and Prieve, 1976; Roy and Dzombak, 1996) or multiple first-order kinetics (Grolimund and Borkovec, 1999; Grolimund et al., 2001a). For ideal geometries, colloid release can be predicted as a function of flow rates and shear forces (Sharma et al., 1992). Under typical subsurface conditions, colloid concentrations in pore water are small (McCarthy and McKay, 2004), and colloid deposition and release will not change the porosity and permeability of the porous medium. Under extreme geochemical conditions, however, as for instance found under waste storage facilities (Wan et al.,2004; Mashal et al., 2004), colloid deposition or release may affect porosity and permeability. A thorough mathematical treatment of clogging in porous media as applied to deep bed filtration was given by Herzig et al. (1970). Biological colloids, such as bacteria, can cause changes in the physical properties of the porous media by growth and biofilm production (e.g., Taylor et al., 1990). The change in porosity can be formulated as (Sen et al., 2002)

[16]

where is the porosity [L³ L^–3], r_r is the rate of release or deposition of the colloids [M L^–3 T^–1], and _c is the specific density of the colloids [M L^–3].

Colloid Interactions with the Air–Water Interface
First-order kinetics is often used to describe colloid interactions with the air–water interface (Corapcioglu and Choi, 1996; Chu et al., 2001; Lenhart and Saiers, 2002; Simunek et al., 2006). The interaction of colloids with the air–water interface can be because of direct attachment to the interface itself or because of straining. We discuss these two mechanisms in the following.

Different mathematical formulations for colloid attachment to the air–water interface have been proposed in the literature. Some researchers define the colloid concentrations at the air–water interface as mass per unit volume of air (Corapcioglu and Choi, 1996; Chu et al., 2001; Massoudieh and Ginn, 2007), while others express it as mass per unit interfacial surface area (Simunek et al., 2006). Differences also exist in the way the attachment rates are formulated and how the equilibrium partition coefficient is expressed. Here, we define the colloid concentration attached to the air–water interface in terms of mass per unit interfacial surface area, and we use a kinetic formulation similar to the one proposed by Massoudieh and Ginn (2007):

[17]

where _aca is a blocking function (dimensionless), f_a is the fraction of the air–water interface available for colloid attachment (dimensionless), and k_aca [L T^–1] and k_dca [T^–1] are the first-order attachment and detachment rate coefficients, respectively, for the interactions of colloids with the air–water interface. The first term on the right-hand side of Eq. [17] describes the colloid attachment to the air–water interface. Note that the attachment coefficient has different dimensions than the detachment coefficient.

Similar to the solid–water interface attachment, a blocking function _aca can account for decreasing available interfacial area as more and more colloids attach to the interface. It is probable that such blocking functions assume similar mathematical forms as in the case of the solid–water interface. A Langmuir-type blocking function has been suggested (Corapcioglu and Choi, 1996; Chu et al., 2001):

[18]

where _max is the maximum attainable colloid concentration at the air–water interface [M L^–2]. The RSA-type blocking function can be written as (Schaaf and Talbot, 1989)

[19]

Neglecting blocking of attachment sites (_aca = f_a = 1), Eq. [17] reduces under equilibrium conditions to

[20]

where k_aca/k_dca is the air–water partition coefficient for colloids (Wan and Tokunaga, 2002; Massoudieh and Ginn, 2007).

Different models have been proposed to estimate the area of the air–water interface, A_aw. Most of these approaches are based on the capillary-pressure relationship _aw(S_w) (Skopp, 1985; Cary, 1994; Bradford and Leij, 1997; Or and Tuller, 1999). As an example, we show the following relationship (Bradford and Leij, 1997):

[21]

where _aw is the air–water interfacial tension [M T^–2], S_w is the effective water saturation (dimensionless), and _aw is the capillary pressure [M L^–1 T^–1].

When colloids are attached to the air–water interface, strong capillary forces will prevent the colloids from detaching (Scheludko and Nikolov, 1975; Pitois and Chateau, 2002). The detachment is also thermodynamically unfavorable based on interfacial energy considerations (Israelachvili, 1992). As a result, the interaction of a colloid with the air–water interface can often be considered to be irreversible, and the second term on the right-hand side of Eq. [17] can be omitted.

Another mechanism for colloid retention occurs when the water saturation decreases such that the water films become smaller than the colloid diameters. Then colloids will be trapped either in pendular rings or in the water films themselves. This water film straining can be expressed as a first-order kinetics (Wan and Tokunaga, 1997):

[22]

with

[23]

where _c^str is the colloid concentration strained by the air–water interface [M L^–2], k_aca,str is the rate coefficient for film straining [L T^–1], P() is the probability of pendular ring discontinuity (dimensionless) as a function of capillary pressure, w is the water film thickness [L], and , N, and m are empirical parameters (dimensionless). If the probability function P() in Eq. [23] is expressed as a function of water content instead of capillary pressure, then (Lenhart and Saiers, 2002; Saiers and Lenhart, 2003)

[24]

where _r is the residual water content [L³ L^–3] and is an empirical parameter (dimensionless). Combining attachment (Eq. [17]) and straining (Eq. [22]) leads to

[25]

Contaminant Interactions with Colloids
Colloid-facilitated transport will only be an effective means for contaminant movement if the contaminants sorb strongly to the colloids. Due to their small size, subsurface colloids have a large surface area and indeed can adsorb certain contaminants strongly. The specific interactions between contaminants and colloids are key to colloid-facilitated contaminant transport and its modeling. The simplest model for sorption of contaminants to colloids is equilibrium sorption (Corapcioglu and Jiang, 1993; Contardi et al., 2001). Sorption of contaminants to colloid particles is also often found to be kinetically controlled (e.g., Artinger et al., 1998, 2002). Various sorption models have been developed and used in colloid-facilitated contaminant transport modeling. Schüssler et al. (2000) introduced a sequential reaction model to characterize the association of Am with humic colloids. The sorption and desorption were described by first-order kinetics. In this model, Am first reacts fast with humic acid, and further sorbs more strongly through a slow reaction (Fig. 3). Smith and Degueldre (1993) assumed two types of sorption sites on the colloids: sorption on Type 1 sites is irreversible, sorption on Type 2 sites is reversible and kinetically controlled.

A general mass balance equation for contaminants associated with mobile colloids with a first-order kinetic sorption model was developed by Corapcioglu and coworkers (Corapcioglu and Jiang, 1993; Choi and Corapcioglu, 1997b), and further expanded to include limited sorption capacity by Simunek et al. (2006). We assume here that sorption rates of contaminants to mobile and immobile colloids are identical. The overall mass balance for contaminants sorbed onto colloids can be written as

[26]

where k_acoll [L³ M^–1 T^–1] and k_dcoll [T^–1] are rates of contaminant attachment to and detachment from colloids, and _m, _i, and _g are parameters accounting for available sorption space. The terms on the right-hand side of Eq. [26] represent: (i) contaminant transport due to dispersive and advective fluxes of mobile colloids, (ii) contaminant adsorption to and desorption from mobile colloids, (iii) contaminant adsorption to and desorption from immobile colloids at the solid phase, and (iv) contaminant adsorption to and desorption from immobile colloids at the air–water interface. Note that the rate expressions in Eq. [26] are different than those used by Simunek et al. (2006). In our formulation, the sorption rates are independent of bulk density or air–water interfacial areas.

Contaminant Interactions with Solid Phase
Contaminant interactions with the solid phase are well described in many textbooks and review articles (e.g., Helfferich, 1962; Sposito, 1989). We list here only those approaches that are commonly used in colloid-facilitated transport models. These include linear and nonlinear, equilibrium and nonequilibrium sorption models of Freundlich or Langmuir type (Smith and Degueldre, 1993; Corapcioglu and Choi, 1996; Saiers and Hornberger, 1996a; Corapcioglu et al., 1999; Sen et al., 2004). The sorption kinetics can be written as

[27]

where k_a is the first-order adsorption rate [L³ M^–1 T^–1], represents a blocking of sorption sites (dimensionless), and k_d is the first-order desorption rate [T^–1]. Equation [27] is often also expressed as

[28]

where is the sorption rate coefficient [T^–1] and K = k_a/k_d is the equilibrium sorption coefficient [L³ M^–1].

For a Langmuir-type sorption isotherm, the blocking function is given as

[29]

where S_max is the maximum amount of contaminant that can be sorbed to the solid phase [M M^–1]. For equilibrium conditions, Eq. [27] can be reduced to Freundlich or Langmuir isotherms:

[30]

or

[31]

It is sometimes assumed that the sorption sites for the contaminants consist of two types, an equilibrium and a nonequilibrium type, also commonly denoted as two-site models (Noell et al., 1998; Compere et al., 2001; Simunek et al., 2006). This two-site model can be written as (Simunek et al., 2006)

[32]

where S_e and S_k are contaminant concentrations sorbed at equilibrium and nonequilibrium sites [M M^–1], respectively.

The sorption process in Eq. [27–32] is treated as an empirical reaction, and the effect of the specific chemical conditions on sorption cannot be considered explicitly. Mechanistic models for ion exchange, mineral precipitation, and surface complexation are available and can be used to replace the empirical sorption models in transport codes (Lichtner, 1996; Steefel et al., 2003; Lichtner et al., 2004). Reactive transport codes, such as FLOTRAN (Lichtner, 2003), CRUNCH (Steefel and Lasaga, 1994; Maher et al., 2006), or TOUGHREACT (Xu et al., 2006b) provide such mechanistic sorption models.

Special Considerations
Colloid Aggregation
Colloids in suspension are thermodynamically unstable, i.e., colloids will ultimately aggregate and settle out of suspension (Stumm and Morgan, 1996). Whether colloids can be transported considerable distances in the subsurface therefore depends on the rate of colloid aggregation in relation to the rate of transport (Czigany et al., 2005). The stability of colloidal suspension can be described by the stability ratio W (dimensionless), which is defined as the ratio of fast to slow aggregation rates (Holthoff et al., 1996):

[33]

where k_agg,fast and k_agg,slow are the fast and slow aggregation rates [T^–1], respectively. The inverse of the stability ratio is equivalent to the collision efficiency in colloid deposition, i.e., 1/W = (Stumm and Morgan, 1996; Grolimund et al., 2001b) and can be calculated as (Holthoff et al., 1996; Behrens et al., 2000)

[34]

where n_fast and n_slow are the initial particle number concentrations in the fast and slow aggregation regimes [number L^–3], respectively, and R_h is the hydrodynamic radius of the aggregate [L]. The particle aggregation process is often also modeled as a second-order kinetics (Stumm and Morgan, 1996):

[35]

where n is the colloid number concentration [number L^–3] and k_agg is the aggregation rate coefficient [T^–1], which, for monodisperse suspensions and aggregation due to Brownian motion, is related to the collision efficiency by (Stumm and Morgan, 1996)

[36]

where k_B is the Boltzmann constant [M L² T^–2 K^–1], T is absolute temperature [K], and _d is the dynamic viscosity [M L^–1 T^–1].

Linear Equilibrium Sorption of Contaminants to Colloids
A simple approach to estimate colloid-facilitated transport of contaminants has been proposed by Vilks et al. (1998) and has been used by several researchers (Contardi et al., 2001; Lichtner et al., 2002; Honeyman and Ranville, 2002) to illustrate the magnitude of colloid-facilitated transport. In this approach, it is assumed that contaminants sorb to the solid phase and colloids (pseudocolloids) with linear equilibrium sorption, and that the colloids are stable in suspension and do not attach to the solid phase or the air–water interface. If the sorption coefficient of contaminants to colloids, K_c [L³ M^–1], is larger by a factor F (dimensionless) than the sorption coefficient to the solid phase, K, then we can write (Vilks et al., 1998)

[37]

and the effective sorption coefficient K_eff [L³ M^–1] is given as

[38]

The retardation of the contaminants in the presence of colloids compared with a nonreactive chemical can then be expressed by (Vilks et al., 1998)

[39]

where R_eff is an effective retardation coefficient (dimensionless) and is the bulk density of the porous medium [M L^–3]. In the case where the colloids are made of the same material as the solid phase, denoted as a symmetrical system by Honeyman and Ranville (2002), Eq. [37] can be expressed as (Contardi et al., 2001)

[40]

where A_c and A are the specific surface areas of the colloids and the solid phase [L² M^–1], respectively. As pointed out by Honeyman and Ranville (2002), this model predicts only limited colloid-facilitated contaminant transport because the solid phase will compete effectively for contaminant sorption as colloids and contaminants move through the subsurface.

Numerical Implementation of Colloid-Facilitated Contaminant Transport Models
The equations shown above have been implemented to various degrees into analytical and numerical codes. Due to the numerous processes involved in modeling colloid-facilitated contaminant transport in the vadose zone, the coupled differential equations generally have to be solved numerically. Only under simplified assumptions are analytical solutions available and feasible. Table 1 lists models used for colloid and colloid-facilitated contaminant transport in the vadose zone. While there are many models simulating colloid and colloid-facilitated contaminant transport under water-saturated conditions, we list only those models that are specific to the vadose zone, i.e., models that include the air phase.

Stochastic Models
Deviations between experimental observations and modeling results have also led to the development of a stochastic representation of colloid mass transfer processes in porous media. Heterogeneity of porous media and colloid surfaces has been accounted for by assuming that the colloid attachment coefficient (Eq. [8]) is not constant, but rather a random variable with a specific probability distribution (Tufenkji et al., 2003; Li et al., 2004). Bradford and Toride (2007) considered both the attachment coefficient and the detachment coefficient to be random variables and used a joint probability density function to describe the correlation between the two variables. Such representations have usually led to better description of experimental observations, especially under conditions unfavorable for colloid attachment to the solid surfaces, although not all aspects of the observations can be resolved.

	Case Studies of Colloid-Facilitated Contaminant Transport in the Vadose Zone

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 
Colloid-Facilitated Transport of Cesium
Turner et al. (2006) studied colloid-facilitated transport of Cs and Sr in 15-cm-long quartz sand columns under water-saturated and steady-state flow conditions. Illite with a particle size of 467 ± 190 nm and a concentration of 100 mg L^–1 was used as the colloid. The column tests were run under two ionic strengths of 1 and 2 mmol L^–1 background solution adjusted with Na₂CO₃ and NaCl. They analyzed the experimental data using the advection–dispersion equation coupled with the following processes: (i) kinetic colloid attachment to and detachment from the quartz sand (Eq. [5] and [10]); (ii) two-site kinetic contaminant adsorption to and desorption from colloids (similar to Eq. [26]), where the two sites are characterized by fast and slow kinetics, respectively; and (iii) kinetic contaminant adsorption to and desorption from the quartz sand (similar to Eq. [27]).

The results of experiments and modeling for Cs transport under a solution ionic strength of 2 mmol L^–1 are shown in Fig. 5 . Without colloids, Cs breakthrough occurred 17 h (14 pore volumes) after injection (Fig. 5a); however, in the presence of colloids, Cs breakthrough occurred after only 9 h (7.5 pore volumes; Fig. 5b). Cesium was transported as aqueous species as well as attached on mobile colloids. The model could reproduce the experimental data after rate coefficients were adjusted by fitting. Modeling and experimental data point to the importance of sorption kinetics, i.e., the kinetics of contaminant adsorption to and desorption from mobile colloids. Overall, the model fitted colloid and contaminant phases well, indicating that the model described the processes adequately.

View larger version (31K): [in this window] [in a new window]   FIG. 5. Cesium transport through quartz sand in (a) the absence and (b) the presence of 100 mg/L illite colloids. Cesium concentration in the inflow was 7.5 x 10–4 mmol/L and solution ionic strength was 2 mmol/L (taken from Fig. 2 and 4 of Turner et al., 2006; reproduced/modified by permission of the American Geophysical Union.)

The effect of the air–water interface and its colloid-attachment capacity on column breakthrough curves of colloids and contaminants is illustrated in Fig. 6 . A larger sorption capacity leads to retarded colloid transport, but as all attachment sites are being filled up, the final colloid breakthrough concentration reaches a constant value, given by the colloid attachment to the solid phase (Fig. 6a). Contaminants in the presence of colloids are transported in the water phase and attached to mobile colloids, and contaminants break through the column earlier than in the absence of colloids (Fig. 6b). Complete contaminant breakthrough is delayed, however, because colloids attach to the air–water interface, and as colloid breakthrough is delayed, so is the breakthrough of the colloid-bound contaminants to a later time.

View larger version (24K): [in this window] [in a new window]   FIG. 6. Effect of the air–water interface on (a) colloid transport and (b) total contaminant transport, i.e., the sum of aqueous-phase and colloid-bound contaminant. 'max denotes a dimensionless colloid attachment capacity to the air–water interface (taken from Fig. 4 of Choi and Corapcioglu, 1997b; reprinted with permission from Elsevier).

	Conclusions

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

Several mathematical models have been developed for colloid-facilitated contaminant transport, but most of them only consider the saturated zone. Only a few models consider the air phase and its interactions with colloids, and are applicable to the vadose zone. Models vary in complexity and are mostly based on steady or non-steady water flow and advective–dispersive contaminant and colloid transport. Various colloid transport and colloid–contaminant interactions are used in these models. As the importance of the vadose zone for contaminant transport in general, and colloid-facilitated contaminant transport in particular, is being increasingly recognized, colloid-facilitated contaminant transport mechanisms are being incorporated into comprehensive flow and transport codes.

	Appendix

TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 

A specific surface area of solid phase [L2 M–1] Aaw area of air–water interface per volume of porous medium [L2 L–3] Ac specific surface area of colloids [L2 M–1] a air content of unsaturated porous medium [L3 L–3] C contaminant concentration in the aqueous phase [M L–3] Ca contaminant concentration in the air phase [M L–3] Cc colloid concentration in the aqueous phase [M L–3] D dispersion coefficient for contaminants [L2 T–1] Dc dispersion coefficient for colloids [L2 T–1] dc diameter of colloids [L] dg median grain size of porous medium [L] F empirical factor (dimensionless) fs fraction of solid surface area available for colloid attachment (dimensionless) fa fraction of air–water interface area available for colloid attachment (dimensionless) H Henry's constant (dimensionless) K sorption coefficient for contaminant sorption [L3 M–1] Keff effective sorption coefficient [L3 M–1] ka first-order adsorption rate for contaminant sorption on solid phase [L3 M–1 T–1] kac attachment coefficient of colloids to solid phase [T–1] kaca first-order attachment rate coefficient for colloid/air-water interfacial interactions [L T–1] kaca,str rate coefficient for film straining [L T–1] kacoll contaminant attachment rate to colloids [L3 M–1 T–1] kagg aggregation rate coefficient [T–1] kagg,fast fast aggregation rate coefficient [T–1] kagg,slow slow aggregation rate coefficient [T–1] kaw rate coefficient for contaminant mass transfer across the air–water interface [T–1] kB Boltzmann constant [M L2 T–2 K–1] Kc sorption coefficient of contaminants to colloids [L3 M–1] kd first-order desorption rate coefficient [T–1] kdc detachment rate coefficient of colloids from solid phase [T–1] kdc1,2 detachment rate coefficient of colloids from solid phase at sites 1 or 2 [T–1] kdca first-order detachment rate coefficient for colloid/air-water interfacial interactions [T–1] kdcoll contaminant detachment rate coefficient from colloids [T–1] Keff effective sorption coefficient [L3 M–1] kstr first-order straining rate coefficient of colloids [T–1] kstr,0 initial straining rate coefficient [T–1] m empirical parameter (dimensionless) N empirical parameter (dimensionless) n colloid number concentration [number L–3] nfast initial particle number concentration in the fast aggregation regimes [number L–3] nslow initial particle number concentration in the slow aggregation regimes [number L–3] P() probability of pendular ring discontinuity as a function of capillary pressure (dimensionless) q volumetric water flux for contaminants [L T–1] qc volumetric water flux for colloids [L T–1] R reaction term for degradation, production, etc. [M L–3 T–1] Rc reaction term for formation and dissolution of intrinsic colloids [M L–3 T–1] Reff effective retardation coefficient (dimensionless) Rh hydrodynamic radius of the aggregate [L] rr rate of release or deposition of colloids leading to porosity change [M L–3 T–1] S contaminant concentration sorbed to solid phase [M M–1] Sac contaminant concentration sorbed to colloids that are attached to air–water interface [M M–1] Sc colloid concentration attached to solid phase [M M–1] Sc1,2 colloid concentration attached to solid phase at sites 1 or 2 [M M–1] Se contaminant concentration sorbed at equilibrium sites [M M–1] Sk contaminant concentration sorbed at non-equilibrium sites [M M–1] Sic contaminant concentration sorbed to colloids that are attached to solid phase [M M–1] Smc contaminant concentration sorbed to mobile colloids [M M–1] Smax colloid deposition capacity of solid phase [M M–1] Scstr concentration of colloids strained by pore constrictions [M M–1] Sw effective water saturation (dimensionless) T absolute temperature [K] t time [T] v pore water velocity [L T–1] vc actual velocity of colloids [L T–1] W stability ratio (dimensionless) w water film thickness [L] x space [L] collision efficiency of colloids (dimensionless) β fitting parameter (dimensionless) contaminant concentration sorbed to air–water interface [M L–2] c colloid concentration attached to air–water interface [M L–2] cstr colloid concentration strained by the air–water interface [M L–2] max maximum attainable colloid concentration at air–water interface [M L–2] empirical parameter (dimensionless) porosity of porous medium [L3 L–3] single-collector efficiency of colloids (dimensionless) d dynamic viscosity [M L–1 T–1] empirical parameter (dimensionless) sorption rate coefficient [T–1] volumetric water content [L3 L–3] r residual water content [L3 L–3] w volumetric water content accessible to colloids [L3 L–3] blocking factor of sorption sites (dimensionless) empirical parameter describing the decline of straining [M M–1] bulk density of solid phase [M L–3] c specific density of colloids [M L–3] aw air–water interfacial tension [M T–2] aw capillary pressure of the unsaturated porous media [M L–1 T–1] aca blocking function for colloids on air–water interface (dimensionless) g factor accounting for available sorption sites for contaminant on immobile colloids at the air–water interface (dimensionless) i factor accounting for available sorption sites for contaminant on immobile colloids at the solid phase (dimensionless) m factor accounting for available sorption sites for contaminant on mobile colloids (dimensionless) str straining function for colloids due to pore constrictions (dimensionless) s blocking function for colloid attachment to solid phase (dimensionless) integration variable (dimensionless)

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TOP ABSTRACT INTRODUCTION Colloid Transport in the... Conceptual Models for Colloid... Mathematical Models for Colloid... Case Studies of Colloid... Conclusions Appendix REFERENCES

 

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