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im
neka,*
a Dep. of Environmental Sciences, Univ. of California Riverside, CA 92521
b Institute of Environmental Science & Research, Christchurch, NZ
c George E. Brown Jr. Salinity Laboratory, USDA, ARS, Riverside, CA 92521
* Corresponding author (Jiri.Simunek{at}ucr.edu)
Received 22 December 2005.
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ABSTRACT |
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 TOPEXECUTIVE SUMMARY ABSTRACT INTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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INTRODUCTION |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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The transport behavior of dissolved contaminant species has been studied for many years. By comparison, colloid transport and the mutual interactions among contaminants, colloids, and porous media are less well understood. While colloids are subject to similar subsurface fate and transport processes as chemical compounds, they are also subject to their own unique complexities (van Genuchten and imnek, 2004). Since many colloids and microbes are negatively charged, they are electrostatically repelled by negatively-charged solid surfaces. This will lead to an anion exclusion process that can cause slightly enhanced transport relative to fluid flow. The advective transport of colloids may similarly be enhanced by size exclusion, which limits their presence to the larger pores (Bradford et al., 2003, 2006). In addition to being subject to adsorption–desorption process at solid surfaces, colloids are also affected by straining in the porous matrix (Bradford et al., 2003, 2006) and may accumulate at air–water interfaces (Wan and Wilson, 1994; Thompson and Yates, 1999; Wan and Tokunaga, 2002). All of these additional complexities require colloid transport models to be more flexible than regular solute transport models.
Models that consider colloid-facilitated solute transport are based on mass balance equations for all colloid and contaminant species. The various colloid-facilitated solute transport models that have appeared in the literature differ primarily in the manner which colloid transport and contaminant interactions are handled. For example, Mills et al. (1991) and Dunnivant et al. (1992) assume that colloids are nonreactive with the solid phase, Corapcioglu and Jiang (1993) and Jiang and Corapcioglu (1993) consider a first-order kinetic attachment of colloids, Saiers and Hornberger (1996) consider an irreversible nonlinear kinetic attachment of colloids, and van de Weerd and Leijnse (1997) describe colloid attachment kinetics using the Langmuir equation. All colloid-facilitated transport models account for interactions between the contaminants and colloids. Various equilibrium and kinetic models have been used for this purpose.
Although already relatively complex, no existing model for colloid-facilitated solute transport to our knowledge includes all of the major processes contributing to colloid and colloid-facilitated solute transport. For example, the majority of models for colloid-facilitated solute transport consider flow and transport only in fully saturated groundwater systems, usually for steady-state flow, and thus do not account for colloid interactions with the air–water interface. Also, no colloid-facilitated transport model has considered straining and size exclusion as mechanisms of colloid retention and transport, respectively. These two processes, and especially straining, have recently received much attention since classical colloid transport models are often unable to describe simultaneously both breakthrough curves vs. time and concentration profiles vs. depth (Bradford et al., 2003, 2006; Li et al., 2004; Tufenkji and Elimelech, 2005).
Straining involves the entrapment of colloids in down-gradient pores that are too small to allow particle passage. The critical pore size for straining will depend on the size of the colloid and the pore-size distribution of the medium (McDowell-Boyer et al., 1986; Bradford et al., 2002, 2003). Straining may have significant implications for colloid-facilitated solute transport, as illustrated by Bradford et al. (2006). Considering average capillary pressure-saturation curves for the 12 major soil textural groups given by Carsel and Parrish (1988), they calculated that a 2 µm colloid, which is the size of a clay particle, will be excluded or strained in 10 to 86% of the soil pore space for the various soil textures (Bradford et al., 2006). These percentages should significantly increase if the soils become unsaturated.
Size exclusion, a process closely related to straining, affects the mobility of colloids by constraining them to flow domains and pore networks that are physically accessible (Ryan and Elimelech, 1996; Ginn, 2002). Electrostatic forces also play an important role in the distribution (and mobility) of colloids. Anionic colloids will be excluded from locations adjacent to negatively charged solid surfaces; similar to the much reported anion exclusion process for anionic solutes (Krupp, 1972; Gvirtzman and Gorelick, 1991; Ginn, 1995). In case of size or anion exclusion, colloids will tend to reside in larger pores and in more conductive parts of the flow domain. As a result, colloids will be transported faster than a conservative solute tracer (Reimus, 1995; Cumbie and McKay, 1999; Harter et al., 2000; Bradford et al., 2004). Differences in the dispersive flux for colloids and a conservative solute tracer are also anticipated as a result of exclusion (Scheibe and Wood, 2003). Bradford et al. (2002) observed that the dispersivity of 3.2 µm carboxyl latex colloids was up to seven times greater than bromide in saturated aquifer sand. Conversely, Sinton et al. (2000) found in a field microbial transport experiment that the apparent colloid dispersivity decreased with increasing particle size.
Colloid and colloid-facilitated contaminant transport in partially saturated porous media is even more complex than in water-saturated systems. In addition to all of the processes and difficulties discussed above, colloid transport in partially saturated porous media is further complicated by the presence of an air phase and thin water films, in addition to the solid and water phases present in saturated media. Wan and Wilson (1994) observed that colloidal particles deposit preferentially on the air–water interface via a capillary force acting on the particles, and that particle transport was tremendously retarded since the air–water interface acted as a strong sorption phase. Another physical restriction on colloid transport in unsaturated systems is imposed by thin water films; this process is often referred to as film straining (Wan and Tokunaga, 1997; Saiers and Lenhart, 2003). Wan and Tokunaga (1997) proposed that colloid transport in unsaturated systems depends on the ratio of the colloid size to water film thickness. Corapcioglu and Choi (1996) developed a mathematical model describing colloid transport in unsaturated porous media, and also studied the effect of colloids on volatile contaminant transport and air–water partitioning in unsaturated porous media.
Most current models for colloid-facilitated solute transport assume that the number of colloids with respect to contaminant is large and that kinetic reactions coefficients are not dependent on the number of colloids in the system. Although this may be true for some systems, the number of colloids (or concentrations) is often highly variable, with colloids being mobilized (or immobilized) due to changing chemical or hydrological conditions. Thus the reaction coefficients need to be adjusted to the number of colloids in the system in different phases (i.e., mobile, immobile, attached to the air–water interface). This adjustment needs to be performed also for numerical stability reasons. For example, if the number of colloids in the system decreases dramatically and the sorption constants for solute to colloids are assumed to be constant, this may lead to large sorbed concentrations, and hence numerical instabilities.
In this study we developed a one-dimensional numerical model based on the HYDRUS-1D software package that incorporates processes associated with colloid and colloid-facilitated solute transport in variably saturated porous media. The model accounts for both colloid and solute movement due to advection, diffusion, and dispersion in variably saturated soils, as well as for solute movement facilitated by colloid transport. The colloid transport module additionally considers the processes of attachment/detachment to/from the solid phase and/or the air–water interface, straining, and/or size exclusion. The model allows for different pore water velocities and dispersivities for colloids and solute. In this paper we first present the model itself, provide information about the numerical techniques used to solve the governing equations, and provide a simple application of the model to laboratory data. We also present a brief sensitivity analysis of the model to selected transport and reaction parameters.
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CONCEPTUAL MODEL |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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represent instantaneous and kinetic solute sorption to the solid phase, respectively, while the factors 
account for nonlinearity of the process.
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MATHEMATICAL MODEL |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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 | [1] |
is the volumetric water content [L3 L–3], t is time [T], x is the spatial coordinate [L] (positive upward), S is the sink term [L3 L–3 T–1], 
is the angle between the flow direction and the vertical axis, and K(h) is the unsaturated hydraulic conductivity function [L T–1]. The soil hydraulic properties, that is, the retention curve (h) and the unsaturated hydraulic conductivity function K(h) can be evaluated using, for example, the analytical model of van Genuchten (1980).
Colloid Transport
Colloid fate and transport models are commonly based on some form of the advection–dispersion equation, but modified to account for colloid filtration (Harvey and Garabedian, 1991; Hornberger et al., 1992; Corapcioglu and Choi, 1996; Bradford et al., 2003). A more comprehensive equation for colloid transport describing both colloid/matrix and colloid/air–water interface mass partitioning in one-dimensional form is given by
 | [2] |
c are colloid concentrations adsorbed to the solid phase [nm–1] and air–water interface [n L–2], respectively; w is the volumetric water content accessible to colloids [L3 L–3] (due to ion or size exclusion, w may be smaller than the total volumetric water content ), Dc is the dispersion coefficient for colloids [L2 T–1], 
is the bulk density [M L–3], Aaw is the air-water interfacial area per unit volume [L2 L–3], qc is the volumetric water flux density for colloids [L T–1], while Rc represents various chemical and biological reactions [n L–3 T–1]. The second and third terms on the left side of Eq. [2] represent colloid mass-transfer terms between the aqueous phase and the solid phase or the air–water interface [n L–3 T–1], respectively. The first two terms on the right side of Eq. [2] represent the dispersive and advective colloid fluxes, respectively. The value of w is defined as
 | [3] |
is porosity (equal to the saturated water content, s) [L3 L–3] and 
is the water saturation that is not accessible to mobile colloids [-] (Bradford et al., 2006). The volumetric water flux density for colloids, qc, moving only in pores from which they are not excluded, can be calculated from the ratio of the relative hydraulic conductivity of the entire pore space (Krw) to the relative hydraulic conductivity of the colloid-accessible pores (Krc)
 | [4] |
 | [5] |
Colloid mass transfer between the aqueous and solid phases is traditionally described using attachment–detachment models of the form:
 | [6] |
s is a dimensionless colloid retention function [-], and fs is the fraction of the solid surface area that is available for attachment. The attachment coefficient is generally calculated using filtration theory (Logan et al., 1995), a quasi-empirical formulation in terms of the median grain diameter of the porous medium (often termed the collector), the pore-water velocity, and collector and collision (or sticking) efficiencies accounting for colloid removal due to diffusion, interception and gravitational sedimentation (Rajagopalan and Tien, 1976; Logan et al., 1995; Tufenkji and Elimelech, 2004). The first-order detachment coefficient in Eq. [6] accounts for colloid mobilization, presumably as affected by changes in pore-water chemistry (ionic strength, ionic composition, and pH) and physical perturbations in flow, including changes in the flow rate and the water content.
To simulate reductions in the attachment coefficient due to filling of favorable sorption sites, s is sometimes assumed to decrease with increasing colloid mass retention. Random sequential adsorption (Johnson and Elimelech, 1995) and Langmuirian dynamics (Adamczyk et al., 1994) equations have been proposed for s to describe this blocking phenomenon, with the latter equation given by:
 | [7] |
s that increases with increasing mass of retained colloids (Tien, 1989; Tien and Chiang, 1985; Deshpande and Shonnard, 1999) We refer to several recent studies (Ginn, 2002; DeNovio et al., 2004; Rockhold et al., 2004) for more detailed discussions of the attachment and detachment coefficients in Eq. [6].
The first term on the right side of Eq. [6] is multiplied by the fraction of the solid surface area, fs, that is available for attachment. Since mobile colloids are due to size exclusion transported in only a fraction of the pore space (Sw – ), only a portion of the solid surface area is accessible for attachment. The fraction of the solid surface area that is available for attachment can be estimated as (Bradford etal., 2006):
 | [8] |
 | [9] |
Application of the first-order attachment–detachment model given by Eq. [6] typically leads to exponential colloid distributions vs. depth. Bradford et al. (2003) showed that such exponential distributions are often inconsistent with experimental data. They obtained much better results using a depth-dependent straining coefficient sstr in Eq. [9] of the form
 | [10] |
Finally, a model similar to Eq. [6] may be used to describe the partitioning of colloids to the air–water interface
 | [11] |
c is the colloid concentration adsorbed to the air–water interface [n L–2], Raca represents various chemical and biological reactions of attached colloids to the air–water interface [n L–3 T–1], Aaw is the air–water interfacial area per unit volume [L2 L–3], aca is a dimensionless colloid retention function for the air–water interface [-] similarly as used in Eq. [6], and fa is the fraction of the air–water interfacial area that is available for attachment [-], and kaca and kdca are the first-order colloid attachment and detachment coefficients to/from the air–water interface [T–1], respectively. The interfacial area model of Bradford and Leij (1997) can be used to estimate the air–water interfacial area Aaw as:
 | [12] |
is the porosity [L3 L–3], Paw and 
aw are the air-water capillary pressure [M L–1 T–2] and surface tension [M T–2], respectively; h is the pressure head [L], w is the density of water [M L–3], and g is the gravitational acceleration [L T–2].
Colloid-Facilitated Solute Transport
Colloid-facilitated solute transport model requires knowledge of colloid transport, dissolved contaminant transport, and colloid-facilitated contaminant transport, and of various interactions between colloids, solute, soil, and air phase. Transport and/or mass-balance equations must therefore be formulated for the total contaminant, for contaminant sorbed kinetically or instantaneously to the solid phase, and for contaminant sorbed to mobile colloids, to colloids attached to the soil solid phase, and to colloids accumulated at the air–water interface, that is, for contaminant in all different phases or pools.
Our colloid-facilitated transport model closely follows development of many similar models (e.g., Corapcioglu and Jiang, 1993; Jiang and Corapcioglu, 1993; Saiers and Hornberger, 1996; van de Weerd and Leijnse, 1997; Bekhit and Hassan, 2005). Major differences in our model include consideration of water contents, water fluxes, and air–water interfacial areas that may change in time and space (i.e., transient variably-saturated water flow), the presence of colloids on the air–water interface, and adjustment of all kinetic rates to the number of colloids present in the system.
Mass-Balance Equation for the Total Contaminant
The combined dissolved and colloid-facilitated contaminant transport equation (in one dimension) is given by:
 | [13] |
Mass-Balance Equation for Contaminant Sorbed to the Solid Phase
Eq. [13] invokes the concept of two-site sorption for modeling non-equilibrium adsorption–desorption reactions (e.g., van Genuchten and Wagenet, 1989). The two-site sorption concept assumes that total sorption, S, can be divided into two fractions:
 | [14] |
 | [15] |
is the first-order rate constant [T–1], f is the fraction of exchange sites assumed to be in equilibrium with the solution phase [-], 
(C) is the adsorption isotherm [M M–1] that can be expressed using Freundlich, Langmuir, Freundlich-Langmuir, or linear adsorption models (imnek et al., 1998), and Rsk represents various chemical and biological reactions of the kinetically sorbed contaminant [M L–3 T–1].
Mass-Balance Equation for Contaminant Sorbed to Mobile Colloids
The mass-balance equation for contaminant sorbed to mobile colloids can be written as
 | [16] |
m adjusts the sorption rate to the number of mobile colloids present, that is,
 | [17] |
Mass-Balance Equation for Contaminant Sorbed to Immobile Colloids
The mass-balance equation for contaminant sorbed to immobile colloids can be written as follows
 | [18] |
i adjusts the sorption rate to the number of immobile colloids present:
 | [19] |
Mass-Balance Equation for Contaminant Sorbed to Colloids Attached to the Air–Water Interface
The mass-balance equation for contaminant sorbed to colloids attached to the air–water interface may be written as
 | [20] |
g adjusts the sorption rate to the number of colloids at the air–water interface:
 | [21] |
cref is the reference concentration of immobile colloids for which sorption rate kaac is valid [nL–2]. In Eq. [20] the first two terms on the right-hand side represent the sorption and desorption, respectively, of contaminant to/from colloids at the air–water interface; the third and fourth terms account for the attachment and detachment, respectively, of colloids with sorbed contaminant to/from the air–water interface; whereas the fifth term represents degradation and other reactions of contaminant sorbed to colloids accumulated at the air–water interface.
Reaction Term
The reaction term R in Eq. [13] may be used to account for a variety of chemical and biological reactions and transformations, including degradation and production, not already explicitly incorporated in the main total contaminant mass transport equation. In the newly developed colloid-facilitated solute transport module we do not fully support current capabilities of the HYDRUS software package to simulate sequential first-order decay chains and can simulate transport of only one single compound. The reaction term R may thus include provisions for only one first-order degradation reaction that may have different rate coefficients in each phase (i.e., in the liquid, sorbed, and colloid phase). The reaction term R in Eq. [13] for colloid-facilitated transport scenarios is now given by:
 | [22] |
The terms Rsk, Rmc, Ric, and Rac for reactions in the kinetically sorbed phase, on mobile colloids and on colloids associated either with the solid phase or the air–water interface, respectively, are as follows:
 | [23] |
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SOLUTION OF THE GOVERNING EQUATIONS |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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c, C, Sk, Smc, Sic, and Sac). Four partial-differential equations pertain to four forms of colloids, that is, mobile colloids (Cc), strained colloids (Scstr), attached colloids to the solid phase (Scatt), and colloids attached to the air–water interface (c). Additional five partial-differential equations pertain to five forms of contaminants, that is, dissolved contaminant (C), contaminant kinetically sorbed to the solid phase (Sk), contaminant sorbed to mobile colloids (Smc), contaminant sorbed to colloids associated with the solid phase (Sic), and contaminant sorbed to colloids attached to the air–water interface (Sac). Note that the instantaneously sorbed contaminant does not need a special partial differential equation, since it is described by the algebraic equation (adsorption isotherm) and treated simultaneously with the dissolved contaminant.
To find a solution for the flow Eq. [1] and the nine transport equations (Eq. [2], [6], [9], [11], [13], [15], [16], [18], and [20]) analytically is difficult, if not impossible. Numerical methods hence must be used to solve the flow and transport equations. Our numerical solution was implemented into the HYDRUS-1D software package (imnek et al., 1998). The numerical solution is performed at each time step in three sequential steps.
First, because of the assumption that colloids do not affect the flow and transport properties of the porous medium, the variably-saturated flow Eq. [1] can be solved independently of the colloid and solute transport equations. Had we considered the effect of attached and strained colloids on the hydraulic conductivity (i.e., clogging), the soil hydraulic properties would have had to be updated at each time step depending on the number of strained and attached colloids. HYDRUS-1D uses a mass-lumped linear finite elements scheme for the spatial discretization and a fully implicit finite difference scheme for the temporal discretization (imnek et al., 1998). The mixed form of the nonlinear Richards Eq. [1] was solved with a Picard iterative solution scheme. After achieving convergence, the nodal values of the fluid flux are determined from nodal values of the pressure head by applying Darcy's law, and together with nodal values of the water content subsequently used as input to the numerical solution of the colloid transport and colloid-facilitated solute transport equations.
Second, because of the assumption that the transport of colloids is not affected by sorption of contaminants on colloids, the four partial differential equations describing colloid transport can be solved independently of the five solute transport equations. The four governing equations (2, 6, 9, and 11) form a linear system when the dimensionless colloid retention functions s, sstr, and aca are equal to one (i.e., when blocking and depth-dependent phenomena are not considered). In that case, Eq. [6], [9], and [11], none of which contains spatial derivatives, are discretized using finite differences, similarly as for physical nonequilibrium solute transport in imnek et al. (1998), and incorporated directly into the discretized form of Eq. [2]. The Galerkin finite element method (spatial discretization) coupled with the Crank–Nicholson finite difference method (temporal discretization) was used to solve the colloid transport Eq. [2] subject to appropriate initial and boundary conditions. When the dimensionless colloid retention functions s, sstr, and aca are not equal to one (i.e., when blocking or depth-dependent processes are considered), then the system of equations is nonlinear. For those cases we used a Picard iterative solution scheme to solve the system of nonlinear equations. Again, after the solution of the colloid transport equations is obtained, nodal values of colloid concentrations in the different states (e.g., mobile, strained, and/or attached) are used as input to the numerical solution of the colloid-facilitated solute transport equations.
Finally, a Picard iterative solution scheme was used to solve the system of five equations describing colloid-facilitated solute transport (13, 15, 16, 18, and 20). We used again the finite element method for spatial discretization (of 13 and 15) and finite difference method for temporal discretization (of all five equations). After achieving convergence, and hence the solute concentrations of the different phases, the numerical solution can proceed to the next time step, starting again with the Richards equation.
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MODEL VERIFICATION |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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Figure 2 shows the observed breakthrough curves (BTCs) of Br, B. subtilis spores and Cd from the column experiment. Although Br and Cd solutes were injected simultaneously in the beginning of the experiment, the BTC of Cd showed a slow increase of concentrations compared to the BTC of Br, which suggests a higher adsorption capability of Cd onto the solid matrix. After injection of the spores, a simultaneous steep rise and flat plateau of both B. subtilis spores and Cd occurred. This observation implies that Cd was co-transported with the spores.
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imnek, 2006). First, the Br breakthrough curve was used to optimize the pore-water velocity and longitudinal dispersivity. Next, the B. subtilis breakthrough curve was used to optimize the first-order attachment and detachment coefficients, while the pore-water velocity and longitudinal dispersivity were fixed at the values determined for Br. Finally, the Cd breakthrough curve was analyzed to obtain the distribution coefficient, the first-order rate constant for exchanging between two sorption sites, and the adsorption and desorption rate coefficients to/from immobile bacteria. Pang and imnek (2006) independently determined adsorption and desorption rate coefficients to/from mobile bacteria from a batch kinetic study in the absence of aquifer media. Results of preliminary simulations suggest that almost all Cd sorption sites of the solid phase were kinetic. These rate coefficients, together with parameters for the bacteria and flow, were fixed during simulations of the Cd data. Parameters were optimized using the internal HYDRUS optimization routine (Marquardt, 1963) for the Br and bacteria data and the PEST software (Doherty, 1994) for the Cd data. The optimized parameters are given in Table 3, while a comparison of model-predicted results and observed experimental data is presented in Fig. 3
. Notice an excellent correspondence between optimized and measured breakthrough curves for all three compounds. Since the column experiment was conducted under saturated condition, the simulation did not consider the attachment-detachment of colloids to/from the air–water interface. As we assumed the same pore-water velocity and dispersion for Br and the bacteria, size exclusion was also not considered here. Pang and imnek (2006) provide more details about the experimental set up, batch studies, as well as more examples of model experimental validation involving Cd transport with either B. subtilis spores or Escherichia coli.
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SENSITIVITY ANALYSIS |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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imnek and Valocchi, 2002). In the sensitivity analysis here we assume that all physical properties, such as porosity, bulk density, water flux, dispersivity, and pore-water velocity are the same as for the column experiment discussed above for B. subtilis spores and Cd. Since the sensitivity of colloid BTCs to various parameters and processes, such as straining and size exclusion, was presented elsewhere (e.g., Bradford et al., 2003), we evaluate here only the sensitivity of model output to the various solute transport parameters.
In this section we evaluate the effects of the solute sorption coefficients on the solute BTC. The analysis considers the same properties for colloids as before (Table 4). We assume that the column is fully saturated and that all pores are accessible to colloids. We further assume that solute sorption to almost all sorption sites of the solid phase is kinetic (f = 0.01) and that the distribution coefficient Kd is equal to 2.5 cm3g–1 (and 0, 1, and 2.5 in the sensitivity analysis). The adsorption rate of contaminant to mobile colloids is considered to be equal to the adsorption rate to immobile colloids (i.e., kamc = kaic) and, similarly, the desorption rate from mobile colloids is assumed to be equal to the desorption rate from immobile colloids (i.e., kdmc = kdic). While colloids were applied during the time interval between 1 and 1.9 h, solute was applied between 0 and 1.9 h.
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The colloid BTCs were the same in all three runs (Fig.4 ). Figure 4 (top) shows additionally solute BTCs for three different values of Kd. Recall that almost all sorption sites are kinetic and thus that the distribution coefficient does not lead to a retardation of the solute front. Kinetic adsorption to the solid phase leads to a gradual increase in solute concentration after arrival of the solute front, and an additional gradual increase in concentration after arrival of the colloid front. For the case of no adsorption to the solid phase (i.e., Kd = 0), solute concentrations quickly reached unit relative concentration after arrival of the solute front, and stayed at this level until the end of the solute pulse. Colloids did not have any effect on the solute breakthrough for this set of solute transport parameters since solute was not retarded and thus its transport cannot be accelerated by colloids unless they are excluded from some part of the pore space and travel faster than unretarded solute. The effects of colloids on solute transport become apparent when solute is kinetically sorbed. Increasing the kinetic sorption coefficient of the solute produced a more pronounced effect of colloids on solute transport (Fig. 4, top).
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Similar results were obtained when varying the desorption rates from the mobile and the immobile colloids (kdmc = kdic = 0.1, 1, or 10 h–1). We obtained again the same initial phase of the solute concentration BTC for all three runs (Fig. 4, bottom). Then due to relatively high sorption to both mobile and immobile colloids, concentrations relatively quickly increased with arriving colloids. Due to different desorption rates from the colloids, the plateaus of the solute concentration BTCs were at different levels.
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SUMMARY |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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The solute transport module uses the concept of two-site sorption to describe nonequilibrium adsorption–desorption reactions to the solid phase. The module further assumes that contaminants can be sorbed onto surfaces of both deposited and mobile colloids, thus fully accounting for the dynamics of colloid transfer between different phases. While a majority of models for colloid-facilitated solute transport assume that the number of colloids with respect to the contaminant is large and that kinetic reaction coefficients are not dependent on the number of colloids in the system, our newly developed model accounts for a variable number of colloids in the system; this since colloids may be mobilized (or immobilized) due to changing chemical or hydrological conditions. Our model can also account for different transport velocities of colloids and solute due to the size exclusion of colloids from small pores.
The use of the model was demonstrated using experimental data from a saturated column experiment, conducted to investigate the transport of Cd in the presence of Bacillus subtilis spores. We also presented a brief sensitivity analysis of the model to selected reaction parameters characterizing various sorption and desorption coefficients.
Our future work will involve further verification of the code using experimental data derived for more complex conditions, such as transient water flow with mobilization and immobilization of colloids, with colloids interacting with the air–water interface and/or being excluded from part of the pore space.
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APPENDIX: LIST OF VARIABLES |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONCONCEPTUAL MODELMATHEMATICAL MODELSOLUTION OF THE GOVERNING...MODEL VERIFICATIONSENSITIVITY ANALYSISSUMMARYAPPENDIX: LIST OF VARIABLESREFERENCES |
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