We investigate the potential effects of inorganic colloids on radionuclide transport in groundwater using generic sensitivity studies and an example based on the alluvial aquifer near Yucca Mountain, Nevada. Our emphasis is on kinetically controlled sorption of radionuclides on mobile and immobile colloids. Three kinetic sorption models are considered for the sensitivity analysis: bilinear, Langmuir, and linear. Plutonium is assumed to be injected into the Yucca Mountain alluvial aquifer at a constant rate and follows a random stream tube to a monitoring boundary. The linear sorption model provides a reasonable upper bound on colloid-facilitated plutonium transport for the site-specific conditions. In the absence of colloid filtration and retardation, colloids enhance the plutonium discharge by a large factor over the situation without colloids. Exchange of plutonium between solution and reversibly attached colloids makes colloid retardation relatively ineffective at reducing colloid-facilitated transport except when the retardation factor is large. Irreversible removal of colloids (filtration) is more effective than retardation at reducing colloid-facilitated transport. For fixed filtration rate the degree of attenuation depends sensitively and nonmonotonically on the rate of plutonium desorption from colloids. These results emphasize the need for accurate measurements of rates of desorption from colloids as well as in situ studies of filtration of naturally occurring colloids.
 Colloids have been suggested as a potentially critical factor in enhancing radionuclide migration in the subsurface [Buddemeier and Hunt, 1988; McCarthy and Zachara, 1989]. Because radionuclides bound to mobile colloids have reduced interaction with the geologic media and can move essentially unretarded with the groundwater flow, colloidal transport of radionuclides is potentially an important factor in the long-term risks associated with potential geological repositories for high-level nuclear waste.
 If sorption on colloids is reversible and relatively rapid, then the effect of colloid-facilitated radionuclide transport (CFRT) is likely to be minor, given the relatively low colloidal concentrations found in most aquifers. Only if binding of radionuclides on colloids is irreversible (or slowly reversible) relative to the timescale of the transport problem are colloids likely to play a potentially significant role in subsurface contaminant transport.
 Irreversible (or slowly reversible) sorption of plutonium on inorganic colloids has been suspected in relation to a few field observations [Penrose et al., 1990]. Of particular interest is the fact that plutonium associated with natural colloids detected at the Nevada Test Site appeared to be irreversibly attached to colloids [Kersting et al., 1999]. Despite these field-scale indications, verifying irreversible or slowly reversible sorption under laboratory conditions is difficult since the timescales required for observation may be prohibitively long. Using a two-site sorption model, Painter et al.  reevaluated data of Lu et al. [1998, 2000] for plutonium sorption on four minerals: hematite, montmorillonite, silica, and smectite. Indication of slow kinetics was found, and a range of kinetic rates was estimated.
 Various models for the solution-colloid-porous matrix exchange which account for sorption kinetics have been considered in the literature. Saiers and Hornberger  proposed a linear first-order sorption-desorption model for the solution-colloid exchange and a nonlinear (Langmuir) kinetic model for the solution-porous matrix exchange; the model was used for interpreting results of cesium transport by kaolinite colloids in a sand column. Van de Weerd and Leijnse  used a nonlinear (Langmuir) kinetic model for all the exchanges between the solution, colloids (mobile as well as immobile), and the porous matrix; their model was used for interpreting americium breakthrough curves in columns with aquifer material and groundwater which contained humic colloids. A Langmuir kinetic sorption model was also used by Lührmann and Noseck  for interpreting breakthrough of europium from porous media columns with humic-rich groundwater. Noell et al.  interpreted transport of cesium through glass bead columns with silica colloids using both linear and nonlinear kinetic sorption models for the solution-colloid-porous matrix exchange; a bilinear kinetic sorption model was used as the nonlinear model, similar to the model of Corapcioglu and Jiang .
 Although the aforementioned studies indicate that CFRT is a potentially significant mechanism for transport of radionuclides in columns, CFRT for controlled conditions on the field scale has yet to be demonstrated. Field-scale applications will generally imply large temporal and spatial scales, with hydrogeological variability being an important factor. Most of the current modeling approaches are based on the advection-dispersion equation and were designed for laboratory conditions; their applicability to heterogeneous aquifers and nonergodic transport (i.e., for sources significantly smaller than the heterogeneity integral scale [e.g., Dagan, 1989]), where quantifying prediction uncertainty is essential, is not obvious. Moreover, the most significant parameters controlling solution-colloid-matrix exchanges and the exchange model (linear, bilinear, Langmuir) most appropriate for field-scale applications remains to be determined. These issues clearly cannot be resolved without combining comprehensive field and laboratory experiments. Nevertheless, modeling CFRT for field-scale conditions using site-specific parameters combined with relevant laboratory data can provide important clues on controlling mechanisms and relevant parameters.
 In this work we build on the Lagrangian modeling framework proposed by Cvetkovic  for field-scale CFRT. By considering a continuous radionuclide release into a heterogeneous aquifer, we focus our study on the steady state radionuclide discharge across a monitoring boundary downstream from the injection plane. We first compare three kinetic sorption models and their impact on colloid facilitated transport: the linear sorption model, the bilinear sorption model, and the Langmuir sorption model. The goal is to identify the key controlling parameters and the parameter ranges over which the three models provide comparable results. Using sorption rates estimated [Painter et al., 2002] from previous laboratory experiments, and site-specific estimates of the maximum radionuclide concentration on colloids (Appendix D), we investigate the potential effect of colloids on radionuclide transport in the alluvial aquifer near Yucca Mountain, Nevada. In this example we address the uncertainty in the total plutonium normalized discharge at a monitoring plane due to spatial variability in the hydraulic conductivity of the alluvial aquifer. Our particular interest is to set bounds on controlling parameters for mechanisms which mitigate CFRT, such as colloidal retardation (reversible attachment-detachment) and filtration (irreversible attachment).
2. Transport Formulation and Assumptions
 Our basic transport configuration is given in Figure 1, similar to the one shown by Cvetkovic . We consider a heterogeneous aquifer with meandering stream tubes extending from a specified injection plane to a monitoring boundary (control plane) downstream. Steady state groundwater flow is assumed with a mean velocity U [L/T] that is set parallel to the x1-axis. A radionuclide is continuously released (as a step function) at the injection plane over a small surface A0 [L2]; this surface defines a stream tube with a (constant) volumetric flow rate q [L3/T].
 Natural colloids are assumed to be present in the groundwater. They can be generated along the stream tube and can be subject to linear equilibrium retardation due to reversible attachment-detachment, and/or to permanent removal (filtration). If retardation is neglected, we assume colloids to move with groundwater velocity, which implies that velocity enhancement due to exclusion is neglected.
 Quantifying the dynamics of colloidal movement, generation, and filtration on the field scale is difficult, since available data on the controlling parameters are typically very limited. The colloidal concentration will in general vary in space and time, with possible trends combined with random fluctuations.
where Cc [M/L3] is the colloidal concentration defined per bulk volume, ε [1/T] is the rate of irreversible colloidal removal, χ [M/L3T] is the generation rate, and τ [T] is the groundwater residence time from the injection point to the monitoring plane (Figure 1). In reality, colloidal concentration for a given aquifer will generally vary both in space and in time. In equation (1), Cc should be considered as a mean value for a given aquifer, formulated to capture the spatial trend.
 We shall simplify our following analysis by assuming that in the mean, the irreversible removal of colloids is compensated by colloidal generation, i.e., dCc/dτ = 0, where χ = εCc. Thus there is no spatial trend and Cc is approximately uniform as well as being at steady state. We assume that attachment and detachment of colloids (reversible or irreversible) does not affect the aquifer porosity. Using Cc, the concentration Sc [M/L3] of colloids that are reversibly attached to the porous matrix is Sc = KcCc, where Kc [ ] is a dimensionless distribution coefficient for the colloids.
 Let the boundary condition for transport be defined at the injection plane as J0H(t), where J0 [M/T] is the radionuclide injection rate in solution and H(t) is the Heaviside step function. With this input, radionuclide discharge at the monitoring plane increases monotonically until a steady state is reached. For carrying out the sensitivity analysis and providing site-specific estimates on the impact of CFRT, we simplify the transport problem by considering the steady state limit, as t → ∞. Thus our comparative transport measure is the upper bound of the radionuclide discharge at the monitoring plane, irrespective of when in time this value is reached.
 Let C [M/L3] denote the radionuclide concentration in solution, S [M/L3] be the radionuclide concentration attached to mobile colloids, C* [M/L3] be the radionuclide concentration immobilized on the porous matrix from solution, S* [M/L3] be the radionuclide concentration on reversibly immobilized colloids, and S** [M/L3] be the radionuclide concentration on irreversibly immobilized colloids; C, S, C*, S*, and S** are all defined on a bulk aquifer volume basis. Principal components of radionuclide partitioning C, S, C*, S*, and S** and their exchange connections that will be considered in this work are illustrated in Figure 2.
 Focusing on field-scale transport in the presence of significant uncertainty about groundwater travel time, we neglect pore-scale dispersion and consider advection only. Using the methodology of Cvetkovic and Dagan , we can transform the Eulerian mass balance equations onto one-dimensional stream tubes (trajectories) that are set in a three-dimensional advective flow field (Figure 1). Eulerian and Lagrangian transport equations are given in Appendix A. Note that the Lagrangian transport equations are mathematically similar to those of plug flow. However, the groundwater velocity does not appear explicitly in the Lagrangian transport equations. The spatially variable velocity is replaced by an uncertain limit of integration of the differential equations. This is an advantageous trade-off because in the Lagrangian framework, the transport equations only need to be solved once to obtain the solution for all values of the uncertain travel time.
2.3. Linear Sorption Model
 Colloidal concentrations, in particular of inorganic colloids in deep aquifers, are generally low. However, the radionuclide concentrations relevant for field-scale applications are also typically very low. Hence linear sorption models may be applicable in many cases, in addition to being attractive for field-scale modeling due to their simplicity [Cvetkovic, 2000].
 Eliminating X*, Y*, and Y** from equation (A12) or (A13) yields two ordinary differential equations (ODEs) in a compact form:
where X ≡ Cq/nJ0 and Y ≡ Sq/nJ0 are dimensionless radionuclide mass discharge in solution and on mobile colloids, respectively; q [L3/T] is the volumetric flow rate in a stream tube; and n is the aquifer porosity (assumed uniform). Since coefficients aij do not depend on τ, an analytical solution of equation (2) can be obtained for X(0) = 1 and Y(0) = 0 (Appendix C).
 The elements of aij (i, j = 1, 2) for the full kinetic model (A12) are
 A few special cases are of interest. If the sorption of the dissolved radionuclides on the porous matrix is relatively rapid and attachment-detachment of colloids on the porous matrix is also rapid (Appendix A), then the elements of the matrix aij (i, j = 1, 2) become
where R = 1 + Kd′ and Rc = 1 + Kc are retardation coefficients. A further simplification is if we neglect exchanges between the aqueous phase and immobilized colloids (see Figure 2). The elements of aij then reflect “mobile exchanges” only (i.e., exchanges between the solution and mobile colloids, Figure 2) and become
In the special case where the reversible rate α r is negligible, we obtain the irreversible case with αr → 0; analytical transient solutions for this case have been derived by Cvetkovic .
2.4. Nonlinear Sorption Models
 We consider two nonlinear sorption models: Langmuir and bilinear. Both have been used previously in interpreting laboratory experiments Lührmann and Noseck, 1998; van de Weerd and Leijnse, 1997; Noell et al., 1998. In both cases we simplify the transport problem by neglecting colloid retardation and filtration. In addition, we assume that sorption of the radionuclide on the porous matrix is rapid (at equilibrium). Thus the focus of CFRT sensitivity in this case is on the capacity of mobile colloids to sorb radionuclides.
 Neglecting colloid retardation and filtration, the mass balance equations using the bilinear sorption model are written as
The rate αf0 is defined per unit concentration, and Sm′ is the sorption capacity, the maximum concentration of the radionuclide on the colloids in a unit fluid volume. Note that Sm′ corresponds to CcQ0 using the notation of Noell et al. .
 Transformation to a normalized mass discharge is obtained by following steps. Multiplication of the transport equation with q/(n J0) yields for the nonlinear term of (6)
where αf = αf0Sm′ [1/T], Ym = Sm′ q/J0, and S′/Sm′ = Y/Ym. The Lagrangian steady state mass balance equations for the bilinear model then become
 For the Langmuir sorption model the mass balance equations are written in terms of normalized radionuclide discharge as
where β = b nJ0/q, and a [ ] and b [L3/M] are model parameters. The Langmuir model can also be written as
where αf ≡ αra; thus a is a dimensionless ratio between the forward and reverse sorption rate, corresponding to a distribution coefficient for the linear model once equilibrium is reached. If β is sufficiently small such that βX ≪ 1, then the Langmuir sorption model converges to the linear model.
 Both the bilinear and Langmuir sorption models are three-parameter models: Ym, αf, and αr for the bilinear, and β, αf, and αr for the Langmuir model. Since the forward and reverse rates αf and αr appear in both models, the question is how to relate, for comparative purposes, Ym with β. To this end we use the equilibrium limit for the bilinear and Langmuir batch sorption models, to obtain β = a/Ym (Appendix B), which we use for model comparison.
Equations (8) and (10) are two pairs of coupled ordinary differential equations. We solve these equations numerically with X(0) = 1 and Y(0) = 0. The solution for the total normalized radionuclide discharge Z(τ) = X(τ) + Y(τ) is a random variable since τ is random.
3. Model Sensitivities
 We consider first some generic sensitivities to the sorption models and parameters. For simplicity, we reduce the number of sensitivity parameters by neglecting retardation and filtration of colloids, i.e., assuming Rc = 1 and ε = 0. We also normalize τ and all the rates by λR. This normalization eliminates λ from the equations. The retardation factor is fixed at R = 500. All other parameters are treated as sensitivity parameters.
 The dimensionless total discharges Z = X + Y as obtained from the three sorption models are compared in Figures 3 and 4. The horizontal axis is normalized travel time τ′ = τλ R. Also included for comparison is the case without colloids, for which Y = 0, Z = X, and X(τ′) = exp(−τ′). Without colloids, the total radionuclide discharge decreases rapidly for τ′ > 1 and drops below 10−12 for τ′ >20. When colloids are present, Z can, depending on the sorption parameters, exhibit a plateau value that can be considerably larger than 10−12. For sufficiently long travel times (τ′ > R, corresponding to τ > 1/λ), even those radionuclides moving unretarded attached to colloids have time to decay and the discharge again decreases rapidly with increasing travel time. In all cases, the linear model predicts the greatest effect of colloids and the Langmuir model predicts the smallest, although the differences may be small depending on the combination of parameters.
 Consider first the case of small sorption capacity Ym = 10−5 (Figure 3). If the forward sorption rate αf is low (Figures 3a and 3c), then the linear model is essentially independent of the reversibility ratio a ≡ αf/αr. By contrast, the Langmuir model is sensitive to both the forward rate and the reversibility ratio (compare Figures 3a and 3c). For strong reversibility and relatively large forward sorption rate (Figure 3d), Z from the two nonlinear models coincides over the entire τ′ range.
 If Ym is small, the most significant impact of colloids is predicted by the linear model with fast forward rate and weak reversibility (Figure 3b); the impact of colloids as predicted by the nonlinear models is comparatively small in this case.
 Consider next the situation of large sorption capacity Ym = 1 (Figure 4). For larger sorption capacity the nonlinear models predict a greater effect of colloids, and the differences between the linear and nonlinear models are much smaller. For the weakly reversible cases (Figures 4c and 4d) the three models coincide over the entire τ′ range. The linear and bilinear models coincide over the entire τ′ range for all four combinations of forward rate αf′ and reversibility ratio a. The normalized discharge Z predicted by the Langmuir model is sensitive to both the reversibility ratio a and the forward rate αf′, similar to the situation of low sorption capacity (Figure 3).
 The most significant impact of colloids is predicted by the linear model (and by the bilinear model with large Ym) when the reversibility is weak and the forward rate is relatively large (Figure 4b). In these situations, Z is in the range between 0.1 and 1. The Langmuir model predicts significantly smaller effect of colloids (Z ≈ 10−4 for large travel times).
4. Field-Scale Example
 As a site-specific example, we consider colloid-facilitated transport of plutonium that is hypothetically released into the alluvial aquifer located south of the proposed repository at Yucca Mountain, Nevada. We emphasize that this study is designed to investigate the potential sensitivities with respect to the effect of colloids on transport in the alluvial aquifer. It is not intended to be an assessment of radionuclide transport in repository performance. Such an assessment would require absolute (not relative) values for the radionuclide release into the alluvial aquifer, and thus the alluvial aquifer would need to be considered in the context of the entire multiple-barrier system envisioned for the proposed Yucca Mountain repository, which is well outside the scope of this work.
 Our basic concept for transport is a single flow path (stream tube) which extends through the alluvial aquifer to a monitoring plane located downstream. This stream tube approach with an uncertain groundwater travel time in the stream tube is consistent with the stochastic Lagrangian framework [Cvetkovic and Dagan, 1994] and similar to models used in performance assessment studies of the high-level nuclear waste repository proposed for Yucca Mountain, Nevada [Mohanty and McCartin, 2001]. The groundwater flow is at steady state. Plutonium is injected in solution and is transported by groundwater as a dissolved phase and bound to colloids. In this section we consider the bilinear and linear sorption models.
4.1. Advective Transport
 The parameters describing advective transport in the alluvial aquifer are selected to be roughly consistent with those used in previous studies [e.g., Painter et al., 2001; Mohanty and McCartin, 2001]. Groundwater travel time τL ≡ τ(L) through the aquifer is a random variable with mean 〈τL〉 = L/U = 500 years, where L = 5 km is the travel distance through the alluvial aquifer and U = 0.01 km/yr is the mean groundwater velocity. In addition, we assume that the τL distribution is lognormal with variance [Shapiro and Cvetkovic, 1988]
where σlnK2 = 1.56 and IlnK = 2 km are the log conductivity variance and integral scale, respectively. The grain density of the aquifer material is ρ = 2500 kg m−3, from which the bulk density is obtained as ρ b = (1-n) ρ = 2125 kg m−3, with n = 0.15 being the effective porosity.
 The volumetric flow rate q for a single flow path is assumed perfectly correlated to the groundwater residence time and computed as q = A0Ln/τL, where A0 is the area of the hypothetical release. Although the perfect correlation is a simplifying assumption, it is supported by the relatively large integral scale of ln K, compared to the transport distance (2 km versus 5 km). A more rigorous approach would be to relate q to the Eulerian velocity at the injection point and then use a joint probability density function for this velocity and the groundwater residence time. Such an approach would require additional parameters and/or assumptions and is not considered warranted for our current purpose.
4.2. Radionuclide for Site-Specific Example
 Assumptions for the radionuclide are summarized as follows.
 1. We consider Pu-239 in Pu(V) form as a representative radionuclide. The half-life of Pu-239 is 2.4 × 104 years, which implies a mass decay rate of λ = 2.88 × 10−5 yr−1.
 2. We assume a hypothetical release of plutonium into the alluvium. Because we will present results in normalized form, the magnitude of the release is immaterial when considering the linear model. However, the bilinear model depends indirectly on the magnitude of the input flux even when the equations are written in normalized form, because of the need to normalize the sorption capacity by the inlet flux. Thus an estimate of the inlet flux is needed for purposes of assessing the bilinear model. The flux of plutonium per unit area is taken as J0/A0 = 10−4 g/m2 yr−1, which is consistent with the flux at the water table when averaged over many realizations in probabilistic performance assessment calculations [Mohanty and McCartin, 2001]. This value is overestimating the inlet flux in the alluvium for a variety of reasons but is adequate for evaluating the effect of limited sorption capacity on colloid facilitated transport.
 3. Radionuclide sorbs onto the solid (immobile) phase at equilibrium with a sorption coefficient Kd [L3/M]. The sorption coefficient is fixed at a representative value of Kd = 0.24 m3 kg−1 [Contardi et al., 2001; Turner et al., 2002], which yields a dimensionless sorption coefficient of ρb × 0.24/n ≈ 3400.
 4. Maximum plutonium loading on colloids is estimated based on colloid surface area, a sorption site density of 2.3 sites/nm2 [Davis and Kent, 1990], and an assumption of monodentate sorption in a monolayer.
4.3. Colloids for Site-Specific Example
 Colloid concentration data for the alluvial aquifer are available from a well at Lathrop Wells [Kingston and Whitbeck, 1991] as Cc/n = 0.54 mg/L; this value is assumed to be representative for the alluvial aquifer.
Kingston and Whitbeck  report the particle size distribution for Lathrop Wells. If spherical particles are assumed, the surface area per unit groundwater volume is about 0.23 m2/L. For mineral sorption studies Davis and Kent  recommend using a uniform number of sites per unit area of 2.3 sites/nm2. Using this value, the total number of available sites per unit volume is estimated as 0.88 × 10−6 mol site/L (Appendix D). Assuming that sorption of plutonium on colloids is monolayer, mononuclear (1 atom Pu per sorbed complex), and monodentate (1 complex sorbed per site), the maximum sorbed Pu on the available colloid phase, Sm′ (defined per unit fluid volume), is estimated as Sm′ = 2.1 × 10−4 g Pu/L.
 We provide an alternative estimate of the maximum sorbed Pu concentration which is based on estimated Kd for Pu. Contardi et al.  calculated Kd for Pu on colloids in the alluvial aquifer based on water chemistry, colloid concentration, colloid size distribution, and surface complexation sorption modeling. Extending the continuum idea of sorption/precipitation, we assume that the maximum amount of Pu irreversibly sorbed is limited by the amount sorbed as represented by Kd. The resulting maximum irreversibly sorbed concentration is considerably lower, Sm′ = 7.5 × 10−7 g Pu/L (Appendix D).
 On the basis of experimental data of Lu et al. [1998, 2000], the forward sorption rate αf for Pu was estimated for hematite, montmorillonite, smectite, and silicate. Assuming a linear relationship as αf = α0Cc/n, where α0 [L3/TM] is an intrinsic sorption rate per unit colloidal concentration (fluid volume basis), we obtain an estimate of the field-scale forward sorption rate αf as 0.1 yr−1 (Appendix E), which we consider fixed in all our calculations. Similarly, we assume that αf* = α0Sc/n, where Sc is the (reversibly) immobilized colloidal concentration. For equilibrium attachment-detachment we have Sc = KcCc, and hence αf* = α fKc = 0.1Kc yr−1.
 The degree of reversibility for radionuclide sorption on mobile colloids is quantified by the ratio a ≡ αf/αr; a = ∞ indicates the irreversible case (αr = 0). We currently have no site-specific data on the reverse rate αr (or a) and use it here as a sensitivity parameter. For simplicity, all reverse rates are assumed identical, i.e., αr = αr* = αr**.
 Site-specific data on Rc and ε for natural inorganic colloids are currently unavailable. A distribution of possible Rc values was estimated using filtration theory and an assumed detachment rate, with the median of approximately Rc = 20 [Wolfsberg and Reimus, 2000, Figure 6]. In the following calculations we use Rc as a sensitivity parameter in a relatively wide range, but we also use the value Rc = 20 for assessing the potential impact of colloid retardation on colloid-facilitated Pu transport.
 The irreversible removal (filtration) rate ε is also treated as a sensitivity parameter. The values of ε used in the calculations are compared with estimates obtained by using classical filtration theory (Appendix F).
4.4. Transport Models
 For the site-specific case we consider the bilinear model, for which Sm′ has been estimated (Appendix D). The bilinear model is given in equation (8), where the normalized maximum discharge Ym is a function of τL because q is assumed to be a function of τL, whereas αf is assumed fixed (i.e., independent of τL). The site-specific transport equations for the bilinear model (8) are then formulated as
where values for all parameters have been specified in the preceding sections. Equations (12) are solved numerically over the domain 0 ≤ τ ≤ τL, and the normalized total plutonium discharge is computed as Z(τL) = X(τL) + Y(τL).
 The linear sorption transport model is given in equation (C2) with aij defined either in equation (4) or (5). We use the model aij(5) to compare the linear model (C2) and bilinear model (12). Alternatively, we use equation (C2) with the more general expression aij(4) when only the linear model is considered.
5. Site-Specific Results
 Because the total travel time τL is an uncertain parameter, the normalized discharge Z is a random variable. Thus results need to be expressed in terms of summary statistics. In the following, we use the exceedance probability (complementary cumulative distribution) of the total radionuclide normalized discharge, Z,
where Z(τL) is obtained either from equation (12), or from equation (C2) with (5) or (4). In a limited number of cases we present the full CCDF curve, but in most cases we use the 99-percentile, median, and 1-percentile of Z to summarize results. These three percentiles correspond to τL = 40, 334 and 2702 years, respectively, as determined from the lognormal model in section 4.1.
5.1. Colloid-Facilitated Transport in the Absence of Colloid Immobilization Processes
 Consider first the case of negligible retardation and filtration of colloids (Rc = 1 and ε = 0). The complementary cumulative distribution of the total normalized Pu discharge is illustrated in Figure 5 for the linear model. Results are shown for different reversibility ratios a.
 Without colloids, the median discharge Z = X is negligible, and the 99-percentile is 0.02. By comparing this value with Figure 5, it is clear that in the absence of colloid filtration and retardation, the effect of colloids is to enhance the normalized discharge by orders of magnitude. The enhancement increases with increasing values of the reversibility ratio but is large even in the moderately reversible situation a = 10. The difference in the discharge distribution for a = ∞ and a = 100 is relatively small, implying negligible sensitivity to the reversibility ratio for a > 100. The extreme tail of the distribution, corresponding to long travel times, is insensitive to the degree of irreversibility.
 The effect of travel time uncertainty is quantified by spreading in the distribution in Figure 5. The spreading is strongly dependent on a and decreases with increasing a. In the irreversible case, spreading is particularly low, except in the extreme tail of the distribution.
5.2. Evaluation of the Linear Model
 As was noted in section 2, the nonlinear models are equivalent to the linear model if the sorption capacity is large. Linear models are more convenient to use than the nonlinear models, are conservative in the sense that they overestimate the discharge, and are often used in practice. An important modeling issue is to understand the degree to which linear models are overestimating the discharge for various values of the sorption capacity Sm′. This issue is explored in Figure 6, where the discharge predicted by the bilinear model is shown versus the sorption capacity. In Figure 6a, Z50 is plotted versus Sm′ for three values of reversibility ratio a. In Figure 6b, Z01, Z50, and Z99 are plotted versus Sm′ for a = 100. The horizontal lines in each case show the results of the linear model. Recall that the relevant range of values for Sm′ is 7.5 × 10−7 to 2 × 10−4 g Pu/L.
 It can be seen from Figure 6 that the linear model provides a good approximation to the bilinear model over the relevant ranges. For this reason we use the linear model to study the effect of exchanges between the aqueous phase and immobile colloids.
5.3. Effect of Colloid Retardation and Filtration
 The transport model described by equation (2) with (3) or (4) includes two processes (reversible attachment of colloids to the host rock and physical filtration of colloids) that delay the downstream movement of colloids. These two processes could potentially mitigate colloid-facilitated transport. The effects of these retardation and filtration processes, however, are not straightforward because radionuclides that are bound to immobilized colloids may desorb, return to solution, and attach to mobile colloids. Sorting out the relative contributions of these competing processes is important for understanding the potential role of colloids in facilitating radionuclide transport.
 The 99-percentile, median, and 1-percentile of normalized discharge are plotted in Figure 7 as a function of the colloid retardation factor Rc. The reversibility ratio is 1000 in this example, and there is no filtration (ε = 0). The three curves in Figure 7 all show nonmonotonic behavior, first increasing and then decreasing with increasing Rc. The initial increase in the discharge with increasing Rc is somewhat counterintuitive but can be understood by considering the transfer of radionuclides from solution to the reversibly attached colloids (C → S* link in Figure 2). The rate of this transfer is proportional to the density of colloids reversibly attached to the host rock. Increasing Rc with Cc fixed increases the number of these sorbed colloids and increases the transfer of Pu along the C → S* link. The sorbed colloids, though, are free to detach and carry any attached radionuclide with them. Thus increasing Rc has the net effect of increasing the transfer to mobile colloids. Or more simply, increasing Rc increases the total number of colloids in the system and allows a greater fraction of Pu to be transported by colloids. Of course, the effect of this net increase in colloid-bound Pu is countered by the slower migration of the colloids, thus giving the colloid-bound Pu longer to decay. For smaller Rc the increased net transfer to colloids is the larger of the competing effects, while the opposite is true for large Rc, leading to the nonmonotonic curves in Figure 7.
 The nonmonotonic shape of the Z versus Rc curve is more pronounced when filtration is present (not shown) because the discharge at Rc = 1 starts at a lower value. For example, for ε = 0.01 yr−1 and a = 100, Z50 increases from 0.08 at Rc = 1 to 0.45 at Rc = 10.
 The main point to be made by Figure 7 is that relatively large values of Rc are required to significantly reduce colloid-facilitated transport. For example, Rc values of a few hundred are needed to reduce the median discharge by two factors of 10. For Z99, Rc values of a few thousand are required. For comparison, Rc = 20 is approximately the median of Rc as provided by the statistical analysis of Wolfsberg and Reimus  for the Yucca Mountain region.
 Permanent removal of colloids (filtration) has a more significant effect on reducing the Pu discharge. In Figure 8 we fix Rc at 20 and plot Z50 versus the filtration rate ε for three values of the reversibility ratio a. The median discharge Z50 is clearly sensitive to the filtration rate once the filtration rate has reached a critical value that depends on the reversibility ratio. For the moderately reversible situation (a = 10), filtration rates ε > 1 yr−1 are sufficient to reduce the discharge to low levels. For the irreversible situation (a = ∞), much smaller values of ε are required to reduce the discharge to low levels.
 One important feature in Figure 8 is that the discharge in the situation of irreversible binding to colloids is much smaller than the discharge for the weakly or moderately reversible situations. The reason for this is that plutonium is permanently immobilized if it is irreversibly bound to a colloid that is permanently immobilized, whereas plutonium reversibly attached to a permanently immobilized colloid can desorb and become mobile again. Ignoring desorption from colloids does not provide a conservative upper bound on the radionuclide discharge if significant filtration is also included.
6. Discussion and Conclusions
 Naturally occurring inorganic colloids have the potential to significantly enhance field-scale transport of radionuclides. The degree and significance of colloid-facilitated transport at any given site depends on the radionuclide release scenario, time period of interest, and other site-specific details that are beyond the scope of this preliminary sensitivity study. Moreover, field information on some of the controlling parameters is incomplete. We can, however, make some general statements about which model parameters and modeling assumptions are most important. Such sensitivity information can guide future modeling studies and field investigations aimed at assessing colloid-facilitated transport for particular sites.
 Of the three kinetic sorption models considered, the Langmuir model predicts the smallest effect of colloids and the linear model predicts the largest. The bilinear model is intermediate between the two but is generally quite close to the linear model. In the absence of site-specific information pointing strongly to one of the sorption models, the linear model provides the most conservative predictions of radionuclide discharge. Moreover, it is close to the bilinear model for a wide range of conditions. However, the applicability of the linear sorption model does depend on the initial concentration of the radionuclide compared with sorption capacity and should be reevaluated in applications once a release scenario has been specified.
 For our site-specific example based on the Yucca Mountain alluvial aquifer, there are three parameters that are currently poorly constrained by field-specific data: the rate of desorption from colloids, the colloid retardation factor, and the rate for irreversible removal (filtration) of colloids. The results depend strongly on the reversibility ratio and filtration rate and to a lesser degree on the colloid retardation.
 That colloid retardation is relatively ineffective at reducing the discharge is caused by transfer of radionuclides between the solution and reversibly attached colloids, which has the net effect of increasing the transfer to mobile colloids. For relatively small values of the retardation factor, this increased transfer more than compensates for the increased colloid travel times. For many cases, the discharge in our steady state analysis peaks at retardation factors around 10–100. Thus setting the retardation factor to unity, which might intuitively seem to provide an upper bound on the effect of colloids, may actually be underestimating the effect. This finding underscores the difficulty in selecting “conservative” values for uncertain parameters when multiple competing processes are present.
 Colloid filtration is more effective than colloid retardation at reducing facilitated transport. Filtration rates of the order of ε >1 yr−1 are sufficient to reduce the discharge to small values for our specific example. These and larger rates are consistent with “clean-bed” filtration theory. Clean-bed filtration theory may be relevant for introduced colloids with characteristics different from the host rock (biocolloids, colloids formed from the degradation of waste forms, or engineered structures), but applicability of clean-bed filtration theory to natural inorganic colloids in deep undisturbed aquifers is not clear. The site-specific analysis of Wolfsberg and Reimus  neglects irreversible removal altogether, although the forward (attachment) rate is estimated using the filtration theory Wolfsberg and Reimus . Without reliable in situ estimates of the filtration rate and retardation factor for natural inorganic colloids, estimates of colloid facilitated transport will remain highly uncertain.
 The calculated discharge is also sensitive to the rate of desorption from colloids. This rate can be estimated from laboratory experiments, but the required durations of the experiments need to be relatively large. In the absence of constraining data on desorption rates, the assumption of irreversible sorption provides an upper bound, but only if filtration is neglected. If filtration is included, then irreversible sorption on colloids is not a conservative (bounding) assumption.
 These sensitivities were obtained in the context of a constant source of radionuclides after the discharge has reached a steady state, irrespective of when that steady state level is reached. Although useful for the purposes of this broad-scope sensitivity study, specific applications involving a finite time period of interest and/or time-dependent radionuclide source may result in different sensitivities. Assessments of colloid-facilitated radionuclide transport within a finite time window and for a time-dependent radionuclide source will require transient calculation.
 Current analysis was based on several simplifying assumptions, and a few particular issues require further scrutiny. The colloidal concentration (in an average sense) was assumed constant in space and time, based on a few data values available. Once a more comprehensive field database for Cc would be available, possible trends and fluctuations in Cc could be accounted for in modeling. Furthermore, as shown by recent laboratory studies, suspensions are generally heterogeneous, consisting of colloid minerals with varying properties [Saiers, 2002]. Thus, rather than a single population, colloids could be grouped into several populations with different transport characteristics. Although in terms of computations such an analysis would be a straightforward extension of the current model, it would be considerably more demanding in terms of field data requirements. Finally, in our field-scale applications, the rate α f was assumed constant, estimated from laboratory data [Lu et al., 2000; Painter et al., 2002] and dependent on Cc. A more accurate and direct estimate of αf would require site-specific field and/or laboratory experiments for estimating an intrinsic sorption rate α0 (Appendix E), and also accounting for possible trends in the colloidal concentration Cc.
Appendix A: Governing Equations
 The mass balance equations for radionuclide concentration in solution (C [M/L3]) and on colloids (S [M/L3]) (both defined per unit bulk/porous media volume) are written in the general form
where C* [M/L3] is radionuclide concentration immobilized on the porous matrix from solution, S* [M/L3] is the radionuclide concentration on reversibly immobilized colloids, S** [M/L3] is the radionuclide concentration on irreversibly immobilized colloids, V is spatially variable Darcy velocity, and D is a dispersion tensor; C*, S*, and S** are all defined per unit porous media (bulk) volume. The exchange rates ψ take different forms, depending on the sorption model assumed.
 For the case of linear sorption we assume ψc and ψs in the following form:
where subscript f denotes the forward sorption rates and subscript r denotes the reversible sorption rates, all given in [1/T], and λ [1/T] is the rate of radioactive decay; the exchange links consistent with equation (A2) are given in Figure 2. In equation (A2), kf, kr are exchange rates of the radionuclide between the solution and the porous matrix, αf, αr are exchange rates of the radionuclide between the solution and mobile colloids, αf*, αr* are exchange rates of radionuclide between the solution and reversibly immobilized colloids, αr** is the release rate of the radionuclide from the irreversibly immobilized (filtrated) colloids into the solution, κf, κr are exchange rates of the colloids with the porous matrix, and ε is the filtration rate.
Equations (A1) and (A2) (with αr** = 0) are identical to the transport equations for the linear sorption model presented by Noell et al. . In particular, the mobile concentrations C and S correspond to the ϕCD and ϕσcmCc, respectively (in the notation of Noell et al. ) provided that the presence of colloids does not change the (uniform) porosity. Similarly, the immobilized concentrations C* and S* in equation (A2) correspond to ρb σD and ρc σcc σc, respectively, as given by Noell et al. .
 The Lagrangian form of the equations is obtained by first neglecting the dispersive term in equation (A1) and then transforming V · ∇C and V · ∇S onto a stream tube (trajectory) as presented by Cvetkovic and Dagan ; we get
where τ is the groundwater residence time from the injection point to a control plane located an arbitrary distance x1 from the injection point,
where V(x)[V1, V2, V3] is the groundwater velocity vector and X(x1)[x1, X2(x1), X3(x1)] is the usual Lagrangian displacement vector [Dagan, 1984], here parameterized with respect to the Cartesian length along the mean flow direction, x1. Equations (A3) are mathematically equivalent to plug-flow transport if τ is interpreted as the distance along a column scaled with a constant uniform velocity. In the present analysis, however, the groundwater velocity V is a random space function due to the spatial variability of the hydraulic conductivity; hence τ (equation (A4)) is a random variable. The first two moments of τ can be related analytically to statistical parameters of the hydraulic conductivity using first-order analysis Shapiro and Cvetkovic . Distribution of the concentrations in equation (A3) can be obtained from the distribution of τ; that is, randomness of the concentrations in equation (A3) is due to uncertainty in groundwater residence time τ.
 Steady state equations are obtained by setting the time derivatives to zero as
which yields a coupled system of two ODEs and three algebraic equations.
 It can be noted that the steady state assumption does not necessarily imply equilibrium conditions for the kinetic exchanges. To illustrate this point, consider the simplest case of transport in the absence of colloids. Then we have for C the Lagrangian (transient) transport equations
The steady state form is
wherefrom we get the steady state concentration at the control plane with groundwater residence time τ, as
with the injection given in constant concentration C0. With the additional assumption of equilibrium exchange, we have
Equations (A3)–(A5) are written in terms of radionuclide concentration. We wish to write the Lagrangian balance equations in terms of radionuclide discharge, i.e., mass per unit time as a function of τ [Cvetkovic, 2000]. If q is the volumetric flow rate for a stream tube, we compute the radionuclide mass discharge at the control plane from a stream tube as Jc = C q/n [M/T] and Js = S q/n [M/T]. If J0 [M/T] is the mass discharge at the injection point x1 = 0, then we define normalized discharge as X ≡ Jc/J0 and Y ≡ Js/J0. Inserting equation (A2) into (A1) and multiplying with q/(n J0) [L3/TM], we obtain a Lagrangian equation system (i.e., following a stream tube, or trajectory) for the dimensionless discharge as
where X/C = Y/S = X*/C* = Y*/S* = Y**/S** = q/(n J0) = const. Note that X*, Y*, Y** do not have a direct physical interpretation (as “immobilized radionuclide discharge”) and should be considered as auxiliary quantities.
 If the sorption of the dissolved radionuclides on the porous matrix is relatively rapid (kr → ∞, kf/kr = Kd′, where Kd′ ≡ (1-n)ρKd/n is the dimensionless sorption coefficient, n [ ] is the porosity, and ρ [M/L3] is the density of the porous matrix) and attachment-detachment of colloids on the porous matrix is also rapid (κ r → ∞, κf/κ r = Kc), then (A10) becomes
 The steady state system of coupled equations is obtained from equation (A10) as
 Inversion of equation (C1) yields the solution for X and Y as
Appendix D: Estimates of Sm′ From Kd
 Information on colloid concentrations and water chemistry reported by Kingston and Whitbeck  includes a sampling point located at Lathrop Wells in the alluvial aquifer south of Yucca Mountain. Two methods were used to calculate Sm′ for the Lathrop Wells sample: The first method is a simple geometric argument that incorporates assumptions about sorption site density from Davis and Kent . The second method requires an estimate of a Kd for plutonium on the colloidal phase.
 Using either method for estimation of Sm′ requires an estimate of colloidal surface area. For the Lathrop Wells sample, surface area was estimated by assuming spherical colloidal particles with a radius at the midpoint of a given filter size fraction by Kingston and Whitbeck  (four sizes from 30 nm to >1 μm). Colloid concentration for each size fraction (in mg/L) was converted to surface area by assuming that colloid density is related to the lithology of the aquifer such that ρ = 2500 kg/m3. For the Lathrop Wells sample, the total colloid concentration is 0.54 mg/L and the estimated surface area is 0.23 m2/L.
 The pronounced effects of aqueous chemistry on actinide sorption behavior suggest that sorption modeling should account for changing physicochemical conditions. In particular, the two most important chemical parameters that typically control actinide sorption behavior are pH and carbonate concentration Turner et al. . A number of different modeling approaches of varying complexity can be used to incorporate the effects of chemistry on radionuclide sorption. A class of models that has been used with success in modeling pH-dependent sorption for actinides and other metals is the electrostatic surface complexation model. These models are equilibrium representations of sorption at the mineral-water interface and are discussed in detail elsewhere [Davis and Kent, 1990].
 As described by Contardi et al. , a diffuse-layer (DLM) surface complexation modeling approach was used in combination with the geochemical speciation code MINTEQA2, Version 3.11, to calculate chemistry-dependent sorption for Pu(V) under the chemical conditions specific to the sample from Lathrop Wells presented by Kingston and Whitbeck . For the Lathrop Wells sample, the field pH of 8.13 was used in preference to laboratory pH. Measured water temperatures were used to model aqueous speciation, but because of uncertainties in temperature effects on surface complexation, DLM parameters for Pu(V) sorption at T = 25°C were taken from Turner et al. . On the basis of modeling assumptions, the DLM was used to estimate a value for Kd = 11.6 m3/kg. To calculate Sm′, the Kd value is multiplied by the colloid concentration (0.54 mg/L from Kingston and Whitbeck ) and the estimated solubility limit for Pu of 1.2 × 10−4 g/L [Mohanty and McCartin, 2001]. It is important to note that because of limits on available sorption data for plutonium, this study considers colloidal transport of Pu(V) rather than the more strongly sorbing Pu(IV).
Appendix E: Estimate of Field-Scale Forward Sorption Rate αf
 On the basis of experimental data of Lu et al. [1998, 2000], the forward sorption rates αf for Pu on different inorganic colloids (hematite, monmorillonite, smectite, and silicate) were estimated by Painter et al.  and are summarized in Table E1. The experimental data were available for a limited range of relatively high colloidal concentrations, and our data evaluation could not establish a dependence of the sorption rate αf on the colloidal concentration. However, it seems reasonable to assume for low colloidal concentration a linear relationship as αf = α0Cc/n, where α0 [L/yr g] is an “intrinsic” sorption rate per unit colloidal concentration, similar to what was done for interpreting laboratory experiments by Saiers and Hornberger . We compute α0 for the considered minerals in Table E1. The arithmetic mean intrinsic rate α0 is 228 L/g yr. With the site-specific value Cc/n = 0.54 mg/L, the linear relationship αf = α0Cc/n yields for the forward sorption rate in the alluvial aquifer αf = 0.123 yr−1. Taking into account the reliability of the estimates of αf for the considered minerals (because a different number of data points were used in the estimates of Painter et al. ), we shall consider a somewhat lower value of αf = 0.1 yr−1 as the site-specific forward sorption rate.
 If the alluvial aquifer is regarded as a deep-bed filter for the inorganic colloids, then the filtration (or irreversible removal) rate ε can be predicted using the classical filtration theory [Yao et al., 1971]. The expression for ε is
where U is the flow velocity, n is the porosity, d is a characteristic grain diameter, η is the single-collector efficiency, and α is the collision efficiency factor. Neglecting the colloid collector collision caused by interception and settling, the single-collector efficiency is estimated as
where k = 1.38 × 10−23 kg m2/s2 K is the Boltzmann constant, T = 288 K is the estimated temperature, μ = 0.0014 kg/s m is water viscosity, and dc [m] is the colloid diameter.
 Inserting η into equation (F1), we get the expression for ε [yr−1]:
We see from equation (F2) that the dependence of ε on U is relatively weak. With U = 10 m/yr, the final expression to be used for estimating ε [yr−1] is
where d, dc are both given in meters.
 For α we use the lower bound α = 0.005 calibrated by Harvey and Garabedian  for microorganisms in the Cape Cod aquifer; this value was also used as a lower bound in earlier site-specific calculations [Wolfsberg and Reimus, 2000]. Colloid sizes for the alluvial aquifer have been determined for Lathrop Wells (Table F1). There are currently no data on aquifer grain size; for illustration purposes, we shall consider a sensitivity range of d = 0.2–2 mm. Using equation (F3), we estimate ε as summarized in Table F1.
Table F1. Estimation of Irreversible Removal Rate for Colloids ε (Equation (F3)) Using Classical Filtration Theorya
 This report was prepared to document work performed by the Center for Nuclear Waste Regulatory Analyses (CNWRA) for the U.S. Nuclear Regulatory Commission (NRC) under contract NRC-02-02-012. The activities reported here were performed on behalf of the NRC Office of Nuclear Material Safety and Safeguards, Division of Waste Management. The report is an independent product of the CNWRA and does not necessarily reflect the views or regulatory position of the NRC.