We revisit the classic upscaling approach for predicting Darcy-scale colloid retention based on pore-scale processes, and explore the implicit assumption that retention is a Markov process. Whereas this assumption holds under favorable attachment conditions, it cannot be assumed to hold under unfavorable conditions due to accumulation of colloids in the near-surface fluid domain. We develop a novel link between two-layer mass transfer parameters and the topologies of the pore and collector domains, starting with an elegant outcome of classic colloid filtration theory: that the likelihood of mass transfer from the bulk- to the near-surface fluid domain depends strongly on colloid proximity to the forward flow stagnation axis of each collector (grain). Applying this concept to colloid mass transfer from the near-surface to the bulk fluid domain yields the conclusion that such mass transfer predominantly occurs at rear-flow stagnation zones on collector surfaces. We support this concept with experimental proof that the alignment of rear and forward flow stagnation zones influences colloid mass transfer to surfaces. Colloid accumulation in the near-surface fluid domain under unfavorable conditions may produce extended tailing of colloids during elution in Darcy-scale studies by: (1) long residence times of colloids in the near-surface fluid domain; (2) direct propagation of near-surface colloids from upstream to downstream collectors. The latter generates correlated motion that violates the assumed independence of colloid retention on history of transport. We suggest an approach to upscaling that accounts for the above-described influences of colloid-surface interactions and pore/collector domain topology.
 Among disease outbreaks associated with contaminated groundwater, design considerations for riverbank filtration, and delivery of bacteria (and nanoparticle aggregates) for subsurface cleanup, there exists a compelling need for a functioning theory to predict colloid transport under environmental conditions. The basis for such a theory is well established for colloid transport and removal in uniform granular porous media lacking colloid-grain (collector) repulsion, and is known as colloid filtration theory (CFT) originating from Yao et al. . CFT accounts for irreversible attachment of colloids to surfaces via processes distinct from straining (entrapment in pore throats too small to pass), which in most engineering applications leads to early filter clogging. Despite reliance on filtration for its role in water quality control for many decades [e.g., Iwasaki, 1937], our understanding of filtration processes in the context of natural porous media remains incomplete. In particular, much of current colloid transport research concerns how colloid-collector repulsion mediates colloid retention under environmental conditions. Such conditions are often characterized by a relatively small likelihood of attachment upon nearing the surface, and are often referred to as unfavorable attachment conditions.
 The theoretical prediction of colloid retention under unfavorable conditions is challenging, as evidenced by the fact that CFT-predicted filter coefficients may differ from measured ones by many orders of magnitude [e.g., Elimelech and O'Melia, 1990]. In order to address such discrepancies, mechanistic predictions of colloid retention in granular media under environmental conditions based on a comprehensive force/torque balance are currently sought in quantitative contexts by several research groups [e.g., Bradford et al., 2011; Ma et al., 2011; Nelson and Ginn, 2011; Skouras et al., 2011]. Collective analyses of experimental data at the pore scale [e.g., Johnson et al., 2010; Waske et al., 2012; Qiu et al., 2012; Zevi et al., 2012], combined with quantitative tools for testing hypotheses, are leading to rapid growth in our understanding of the interplay and order of physical and chemical processes controlling colloid fate and transport at the pore scale.
 Whereas both solutes and colloids may be subject to mass transfer between mobile and relatively immobile zones generated by connectivity variations in the pore domain, colloids are additionally, and always, subject to relatively limited diffusion and longer-range interaction with surfaces. Specifically, solutes, being two or more orders of magnitude smaller than colloids, have much greater diffusion constants than colloids. Therefore, the likelihood that a solute particle will reach the porous medium surface (a prerequisite for attachment/adsorption) is not defined by deterministic pore-scale trajectories but rather by randomly oriented diffusion. In contrast, the weaker diffusive translation of colloids yields relatively well-defined trajectories along fluid streamlines that (modified by diffusion and settling) determine the likelihood of colloids reaching porous media surfaces.
 Once colloids reach (intercept) a surface, their attachment is governed by colloid-surface interaction forces emanating from electric double layer and van der Waals interactions (in addition to Lewis acid-base, steric, and other forces). While at least some of these forces operate at the molecular scale, their scaling with size is such that some of these forces are much longer range for colloids than for solutes [Israelachvili, 2011]. Below we demonstrate that colloid-surface interaction forces give rise to behavior that must be considered in scale up of colloid transport from the pore to the Darcy scale. Hence, while dual porosity media are relevant to colloid transport in many contexts, we demonstrate below that, even for uniformly advective pore domains, colloid-surface interaction forces that predominate in the near-surface fluid domain must be considered during scale up of colloid transport and retention.
 Upon achieving good pore-scale prediction of colloid retention under environmental conditions, the subsequent goal becomes to upscale the pore-scale information to the Darcy scale, where the net result of the action of pore-scale processes acting over a multiplicity of pores of varying sizes, structure, etc., lead to, for example, colloid straining [Bradford et al., 2003] and transport in preferential flow paths [Pazmino et al., 2011]. However, even in uniform media lacking a distribution in pore domain sizes, scale-up under environmental conditions has not proven to be straightforward, and in fact remains a standing challenge. The purpose of this paper is to establish the following critical points related to upscaling of colloid retention:
 1. Upscaling colloid retention under favorable conditions involves an implicit assumption of independence of colloid retention on the history of transport in preceding collectors in series. This assumption works because the surface acts as a perfect sink boundary and because the colloid concentration in the near-surface fluid domain is replenished at each grain due to mixing between grains/pores, thereby avoiding depletion of the near-surface domain.
 2. Accumulation of colloids in the near-surface fluid domain occurs under unfavorable conditions in response to van der Waals attraction outboard of the energy barrier. This local excess of colloids yields an increased likelihood of colloid attachment relative to that expected from the bulk colloid concentration, and if it were to occur continuously, it would violate the independence assumption.
 3. Whereas current two-layer models for Darcy-scale colloid transport assume continuous mass transfer between the bulk and near-surface fluid domains, mass transfer between bulk and near-surface domains is not continuous, but rather occurs predominantly in discrete zones; i.e., in forward and rear flow stagnation zones on the open surfaces of the collectors. Upscaling should adequately account for these highly localized processes when deriving/formulating continuum-scale equations.
 4. Propagation of near-surface colloids directly from upstream to downstream collectors via alignment of upstream rear flow stagnation zones with downstream forward flow stagnation zones may yield a subset of colloids that remain in the near surface while propagating across multiple collectors, thereby producing correlated motion effects that violate the independence assumption.
 5. Long residence times of colloids in the near-surface fluid domain, and correlated motion due to direct transfer of near-surface colloids from upstream to downstream collectors, may both contribute to the observed Darcy-scale extended tailing of colloid concentrations following breakthrough-elution of a colloid pulse.
 We provide experimental verification of the effect of collector alignment on the mass transfer of colloids between bulk and near-surface fluid domains. Our overall goal is to propose a conceptual framework within which colloid transport research can be allied for the purpose of successful prediction of colloid transport and retention in porous media under unfavorable conditions, and to identify promising approaches to this long-sought objective.
2. Upscaling Under Favorable Conditions
 Mathematical particle trajectory models based on representative unit collectors predict well the retention of colloids under conditions absent colloid-collector repulsion (so-called “favorable” conditions), and where the colloids and collectors can be idealized as spherical and each of uniform size. This type of approach stems from the historical reliance of the CFT development on one such idealized unit cell, the Happel sphere-in-cell model [Happel, 1958]. The predicted number of colloids that intercept the collector (relative to the number that approach the collector) is the so-called “collector efficiency” denoted by η. The methods of calculating η are the subject of CFT applied to hydrosols in granular porous media, and this history is described, for example, in Tien and Ramarao .
 Predicted collector efficiencies for unit cells have been traditionally upscaled to the Darcy scale by considering the porous medium as a series of collectors, such that the fraction of colloids that pass one collector, (1 − η), is multiplied in a series of Nc collectors (Figure 1) to yield the fraction passing through larger scales: C/Co = (1 − η)Nc, where C is the colloid concentration after the series of collectors, and Co is the influent colloid concentration.
 Implicit in this scale-up approach is the assumption of the independence of colloid retention on the history of transport in preceding collectors. The independence assumption falls more generally within the so-called “iid assumptions” (independent and identically distributed variables) [Metzler and Klafter, 2000]. That the theory correctly predicts retention for a range of colloid sizes in uniform media at larger scales under favorable conditions [e.g., Yao et al., 1971; Rajagopalan and Tien, 1976; Tong and Johnson, 2006] is a strong indication that this assumed independence truly exists under favorable conditions. In addition, numerical simulations of filtration in Darcy-scale domains that resolve colloid transport at the pore scale suggest that unit-cell upscaling by this method yields reasonable predictions under favorable conditions [Long and Hilpert, 2009; Long et al., 2010]. The modest (factor-of-two) discrepancies that are observed between unit-cell predictions and Darcy-scale observations under favorable conditions for microsphere retention in uniform media likely result from unit-cell correlations that do not account for the topology of the pore spaces of filter media.
3. Existing Upscaling Under Unfavorable Conditions: Two-Region Conceptual Models
 Notably, whereas η represents the fraction of introduced colloids that reach the collector surface under favorable conditions, it also represents kinetically limited transport of colloids from the bulk fluid to the near-surface domain under unfavorable conditions. We formalize this relationship further below. Under favorable conditions, arrival to the near-surface domain coincides with attachment, since no energy barrier (no repulsion) exists to prevent attachment. In contrast, under unfavorable conditions, the presence of an energy barrier (repulsion) imposes a kinetic constraint on attachment. Hence, colloid deposition under unfavorable conditions has been approached (in the most general treatments) by constructing two-region models at the conceptual [e.g., Hahn and O'Melia, 2004] or explicitly mathematical [Bradford et al., 2011] levels, that subdivide the pore space occupied by the aqueous phase into two different spatial domains in which colloids are subject to different forces:
 1. In the region far away from collector surfaces, i.e., in the bulk fluid, the fate of the colloid is determined solely by the processes of advection, diffusion, and sedimentation. These processes may bring the colloid into close proximity to the surface (the near surface), where colloid-surface interaction forces become significant relative to fluid drag, diffusion, and gravity. The fraction of bulk fluid colloids that reach the near surface is assumed to be a first-order process governed by a forward rate constant; e.g., kf (Figure 2).
 2. In the near-surface region, the fate of the colloid is largely governed by colloid-surface interaction (DLVO) forces. Specifically, the near-surface fluid domain is defined by the secondary energy minimum by which colloids are attracted to the surface. In this domain, colloids remain mobile unless they become immobilized (attached to the surface). There are different conceptualizations of the process leading to attachment, which may either assume: (a) that a fraction of the near-surface colloid population attaches instantaneously, employing a deposition or attachment efficiency, α; or (b) that the attachment step itself is kinetically limited, involving a first-order rate constant, e.g., katt, applied to the concentration or number density of near-surface colloids (Figure 2). In many models, there is also a fraction of near-surface colloids that return to the bulk fluid, and this is typically governed by a reentrainment rate constant, e.g. kr (Figure 2). In this review, we neglect explicit consideration of models concerning detachment from the surface to the near surface; however, we do not mean to imply that detachment is negligible under all conditions.
 Two-region models are useful because they accommodate the occurrence of (at least) two different scales, which govern colloid transport under unfavorable conditions. While hydrodynamic dispersion in the bulk fluid can be resolved at the pore scale, the motion of a colloid that approaches a collector in the near-surface region must be resolved at the nm scale due to short-range electric double layer forces. By design, two-region models neglect certain forces and physical phenomena in either region, and depending on the model, different forces are neglected. Below, we briefly review strategies to provide a mechanistic basis for the various constants (kf, α, η, katt, kr) used in two-layer models.
3.2. Forward Rate Constant: η and kf
 In two-layer models, the forward rate constant governing mass transfer from the bulk fluid to the near-surface under unfavorable conditions is typically based on a probability of bulk fluid colloids reaching the near surface or surface (η). Under favorable conditions, there is no energy barrier to attachment, and η directly reflects the likelihood of attachment. Under unfavorable conditions, repulsion may prevent immediate near-surface colloid attachment, and so η reflects the likelihood of colloid entry into the near-surface domain.
 Translation of η to a rate coefficient (kf) can be performed by equating the C/Co − η relationship to the C/Co − kf relationship, as shown in Figure 1. The resulting η to kf relationship involves the average pore water velocity (ν) and the ratio of the number of collectors, Nc, corresponding to a given distance (L), both of which depend on the geometry of the unit cell:
 The ratio Nc/L depends on unit collector porosity (ɛ) and collector size (dc). Whereas the exact value for Nc/L requires specifying whether kf is being determined for attachment to the surface (kfs) versus entry to the near-surface domain (kfns) (see supporting information), the small thickness of the near-surface domain relative to the bulk domain renders kfs practically equivalent to kfns; hence, they are not explicitly distinguished here. The resulting relationship for the Happel sphere-in-cell geometry is (see supporting information):
and for the hemisphere-in-cell geometry [Ma et al., 2009]:
 While many publications simplify equations (2) and (3) above by substituting η ≈ −ln(1 − η), we avoid that substitution in order to retain clarity regarding the relationship between η and C/Co and kf, as shown in Figure 1. It is important to keep in mind that we distinguish transport to the near-surface domain from attachment. We now take up the subject of traditional and recent modes of accounting for attachment below.
 Classically, the steps of colloid arrival to the near-surface domain and attachment were treated as simultaneous, such that attachment rates were set equal to the product of the collision efficiency (η), that is, the fraction of colloids that approach the surface, and the deposition (or attachment) efficiency (α), which is the fraction of those colloids approaching the surface that stick to it. The resulting equation for the attachment rate coefficient in, for example, the Happel sphere is therefore almost identical to equation (2), except that η is multiplied by α. We now turn to the topic of prediction of α.
3.3.1. Mechanistic Prediction of α
 The assumption of attachment immediately upon entry into the near-surface domain (via kf) allows the attachment step to be represented by a ratio of colloids that attach to the surface relative to colloids that enter the near-surface domain. Mechanistic prediction of attachment efficiency α (primary energy minimum association with the collector surface) via numerical simulation is described in a large body of literature, but is well summarized by results presented in Elimelech and O'Melia , which demonstrate that colloid kinetic energies in the near-surface domain are sufficient to allow colloids to overcome only small (e.g., <10 kT) energy barriers (e.g., those encountered at high ionic strengths), which are far smaller than energy barriers calculated via DLVO theory for many environmental conditions [e.g., Johnson and Tong, 2006].
 With recognition that overcoming the DLVO energy barrier to attachment does not occur except for unrealistically small energy barriers, an alternative approach to predicting α considers the likelihood of near-surface colloids remaining at least loosely associated with the surface via secondary energy minimum interaction. The mechanistic basis for this prediction is provided by Maxwell theory [e.g., Hahn and O'Melia, 2004] under the assumption that the velocity distribution for gas molecules (Maxwell velocity distribution) is representative of that of colloids in aqueous suspension. The concept here is that colloids entering the near-surface domain, as a result of all operating forces including secondary energy minimum attraction, have kinetic energies that may exceed that necessary to escape the secondary energy minimum. The Maxwell frequency distribution is assumed to give the actual histogram of kinetic energies for colloids in the near-surface domain. Therefore, from the Maxwell distribution, one can determine the fraction of “hot” colloids, which are those with sufficient kinetic energy to escape from the secondary energy minimum into the bulk fluid domain. This fraction is defined as the integral of the Maxwell velocity distribution from this energy up to infinity. Likewise, the fraction of “cold” colloids unable to escape the secondary energy minimum is the complement of the “hot” colloids.
 The assumption underlying the Maxwell-based approach (velocity distribution for gas molecules is representative of colloids in aqueous suspension) is reasonable on the basis that the kinetic energy of particles (and molecules) is the same (3/2 kT or 3/2 mv2) regardless of the phase in which a particle resides (vapor or liquid). That is, one can expect velocities to obey this distribution regardless of phase (vapor or liquid); whereas, the displacements should be reduced in liquid relative to vapor due to the relatively small mean free path in the former relative to the latter [Israelachvili, 2011]. Our numerical simulations [Ma et al., 2011] indicate that for representative conditions involving a significant secondary energy minimum, the difference between the Maxwell-predicted and our simulated velocity distribution is relatively small, e.g., a factor of two to three (results not shown).
 A drawback of the Maxwell-based approach, or at least its present implementation, is that the fates of “cold” and “hot” colloids have been variously interpreted in the literature, within a range spanning from irreversible to reversible association with the surface or likewise for the near-surface domain. For example, “cold” colloids have been assumed to be irreversibly associated with the surface, that is, the “cold” fraction is considered equal to α corresponding to genuine attachment to the surface [Hahn and O'Melia, 2004]. But in fact, “cold” colloids are not immobilized by secondary energy minimum interaction, except under the circumstance where one extends surface friction to the secondary energy minimum and where the resulting torque balance supports immobilization [Johnson et al., 2009]. Shen et al.  suggest that a fraction of the “hot” colloids (1 − α) may “jump” the energy barrier; however, this is possible only for small energy barriers [e.g., Elimelech and O'Melia, 1990]. Whereas the Maxwell-based approach allows determination of the likelihood that secondary energy minimum-associated colloids will remain so on the basis of diffusion and noncontact interaction forces, one must keep in mind that this association is indefinite. These colloids may return to the bulk fluid if, for example, fluid drag forces shift the force/torque balance toward this outcome.
3.3.2. Empirical Prediction of α
 Alternative to mechanistic prediction of α under unfavorable conditions is the development of semiempirical correlation equations to predict α [Elimelech, 1992; Bai and Tien, 1999; Tien and Ramarao, 2007; Chang and Chan, 2008; Chang et al., 2009; Park et al., 2012]. However, the accuracy of such semiempirical correlations is undetermined when they are applied outside the range of experimental conditions that were used to develop the regression. Lacking explicit mechanisms, the regressions provide limited insights into the underlying mechanistic processes. Park et al.  provide correlation equations to predict α for a range of experiments involving oocysts where the collision efficiency η is given by one of several models, but the reliance on the calculated η values leads to values of α that are greater than unity (a common outcome in existing studies), thereby violating the definition of α.
3.3.3. Mechanistic Prediction of katt
 More recent models recognize that colloid entry to the near-surface domain and attachment is not necessarily (and even not typically) simultaneous; but rather involves a separate kinetically controlled attachment step after colloid entry into the near-surface domain. One model that builds on this conceptual framework [Bradford et al., 2011] quantifies katt using a torque balance that allows near-surface colloids (colloids associated with the surface via secondary energy minima) to become immobilized in zones of low fluid drag. This torque balance has been criticized [Johnson et al., 2009] because secondary energy minimum interaction has traditionally been considered “non-contact,” whereas the torque balance uses a surface friction parameter to determine whether colloids that are already assumed to be attached onto (in contact with) surfaces will begin to roll in response to increased fluid drag and shear [Hubbe, 1984, 1985; Bergendahl and Grasso, 2000; Li et al., 2005]. Notably, the secondary energy minimum torque balance by Bradford et al.  predicts colloid attachment in areas of low fluid drag (e.g., leeward areas); whereas direct observations in micromodels show attachment predominantly in forward flow stagnation zones [e.g., Johnson et al., 2010] where there is higher fluid drag, but also a higher rate of colloid delivery.
 Inability of the above mechanistic approaches to predict genuine colloid attachment (capture in primary energy minima) under unfavorable conditions arises from the fact that they rely on traditional “mean-field” DLVO theory, which treats surfaces as being perfectly smooth and having uniform charge, such that repulsion is the same everywhere across the surface. In reality, we expect nanoscale zones of attraction on surfaces (arising from charge heterogeneity and/or roughness) to locally reduce or eliminate repulsion and allow colloid attachment [e.g., Bhattacharjee et al., 1998; Hoek and Agarwal, 2006; Duffadar and Davis, 2007]. Moreover, idealized two-layer models ignore effects of (for example) surface roughness, which may influence near-surface fluid velocities and colloid kinetic energy. However, because typical energy barriers calculated from surface properties are large, we argue that the primary effect of surface roughness and surface charge heterogeneity (with respect to potential colloid attachment) is reduction or even removal of the energy barrier to attachment. Hence, we herein assume that roughness-altered near-surface fluid velocities and particle kinetic energies are secondary to this effect.
 Building upon the work of Bhattacharjee et al.  and Duffadar et al. , Ma et al.  incorporated nanoscale charge heterogeneity on the collector in the hemisphere-in-cell model. By trading in mean-field DLVO theory for discrete representation of nanoscale domains of opposite charge to the colloid, all experimentally observed nonstraining modes of colloid retention under unfavorable conditions were simulated, i.e., attachment to the open surface for smaller colloids, wedging in grain-to-grain contacts for larger colloids, and retention via secondary energy minimum interactions in zones of low fluid drag. In addition to qualitative prediction of the mode of colloid retention, incorporation of discrete heterogeneity into the hemisphere-in-cell model produced trends of colloid retention as a function of solution ionic strength that were reflective of experimental trends, and that did not show the hypersensitivity to ionic strength that is characteristic of mean-field DLVO approaches [Ma et al., 2011].
 A notable feature of colloid trajectory simulations under unfavorable conditions is the characteristic “skimming” of colloids for significant distances across the collector surface outboard of the energy barrier prior to attachment (Figure 3), which is well observed in experiments under unfavorable conditions [Johnson et al., 2010]. Figure 3 shows simulated trajectories of retained or reentrained 10 μm diameter colloids simulated in the hemispheres-in-cell model [Ma et al., 2011] under a representative unfavorable condition. Colloid trajectories along the surface are relatively long under unfavorable conditions, eventually resulting in attachment or association with the surface via the mechanisms mentioned above, or exit from the collector.
 The review above demonstrates that research regarding prediction of colloid attachment under unfavorable conditions is rapidly advancing; and that regardless of the model considered, colloid accumulation and pore-scale transport in the near-surface domain is a hallmark of unfavorable conditions. It is this behavior that we contend must be accounted for in upscaling of pore-scale predictions of colloid transport (and retention) to the Darcy scale, and so we now focus our attention on this subject, starting first with closely related experimental observations.
3.4. Colloid Remobilization and Its First-Order Rate Constant kr
 An ubiquitous attribute of colloid breakthrough-elution in column and field experiments under unfavorable conditions is the slow tailing of effluent concentrations long after the main colloid pulse has exited the system [e.g., Dabros and Van de Ven, 1982; Johnson et al., 1995; Zhang et al., 1999; Li et al., 2005]. Broadly speaking, extended tailing (for solutes and colloids) is driven by retention in regions with much lower velocity than the mean. In simulations, this is represented by a reverse rate constant for colloid return to the bulk fluid (kr), or a distribution of such rates of return, in order to accommodate the delayed colloid arrivals in effluent. Until recently, many researchers (present authors included) imagined tailing to represent detachment of colloids from the collector surface, such that the filtration process involves reversible kinetically controlled attachment. More recently, experimental evidence suggests that truly attached colloids remain so even with significant perturbation of solution chemistry and fluid drag [e.g., Johnson et al., 2010], at least for simple colloids. Accordingly, we herein adopt the term “reentrainment” to describe colloid remobilization (as evidenced by tailing) from either the near-surface fluid or the surface (detachment).
 At scales smaller than a collector grain, the probability of escape of colloids from the near-surface domain to the bulk fluid falls directly from the Maxwell-based approach for α described above, where the “hot” colloids (1 − α) may be expected to gain exit from the near-surface domain to bulk fluid. This probability (1 − α) can be directly implemented in numerical simulation of a near-surface colloid (e.g., in random walk models) by comparison to a randomly drawn number in the range between zero and one during each time step, Δt [e.g., Li et al., 2004], or alternatively, it can be converted to a reentrainment rate coefficient (kr) and implemented via the following approaches.
3.4.1. Power Law Tailing
 So-called power law tailing corresponds to effluent concentrations versus time that yield a linear trend in log-log space [Haggerty et al., 2000]. Power law tailing is not captured by conventional first-order reversible kinetics, which gives rise to linearity of log concentration versus time [Haggerty et al., 2000], since first-order kinetics are based on (Markovian) Poisson process models of the attachment and detachment frequencies. Capturing power law tailing requires more complex process models, e.g., non-Markovian [Ginn, 2000a] or which incorporate multiple Poisson processes per each of multiple sites or domains [Haggerty et al., 2000], as described below.
 One approach to accommodate tailing behavior, in particular power law tailing for biotic colloids, was introduced in Johnson et al.  and formalized in Ginn [2000a, 2000b] wherein a residence time-dependent detachment rate was invented. The need for such an approach was outlined earlier in Dabros and Van de Ven , who described factors contributing to long residence times for colloids at or near surfaces, including entrapment in secondary energy minima. Those authors anticipated that “one can expect a large influence of the contact time of the particles on their ability to escape from the collector surface.” As described above, our contemporary interpretation of tailing modifies such statements by replacing “surface” with “near surface” and “detachment” with “reentrainment.”
Cortis et al.  applied the continuous-time random walk (CTRW) empirical approach [Margolin et al., 2003] to describe tailing. In the original form of CTRW, a colloid undergoes a random walk where motion is specified to result from a sequence of jumps dx taking durations dt. The frequency distribution of jumps and durations is specified as ψ(dx,dt). Typically, the jumps are then presumed independent of the durations, and this joint distribution is factored into the product ψx(dx) ψt(dt). Cortis et al.  assigned models for these two factors, in terms of unknown parameters, and then fitted the parameters so that the resulting model matched colloid effluent data. Specifically, the model included both irreversible attachment and delays due to both reversible attachment and anomalous transport (e.g., other unspecified delays). The reversible attachment was treated as a compound Poisson process, and the anomalous transport was treated as giving rise to a truncated power law distribution of immobilization times. Both delay mechanisms were represented in a single convolution kernel function.
 The result of this factored CTRW is a model that allows particles to become reentrained at a multiplicity of rates, giving rise to long tails in the effluent concentrations. In fact, this factored form of the CTRW is mathematically equivalent to the well-known multirate mass transfer modeling (MRMT) approach of Haggerty et al. , as shown in Dentz and Berkowitz , who point out that both models can be cast in the same integrodifferential equation form. Furthermore, Ginn  showed a modified form of the MRMT can be cast in the form of the exposure time model of Ginn [2000a, 2000b]. In summary, all three of these approaches are different ways of accommodating a multiplicity of reentrainment rates of colloids.
3.5. Implications of Accumulation in Near-Surface Region to Upscaling
 If one considers the Darcy scale to be equivalent to a series of unit cells (as traditionally assumed for colloid retention under favorable conditions), then, under the independence assumption, one would expect some form of convolution of the unit cell colloid breakthrough-elution curve to generate breakthrough curves at the Darcy scale, as follows. The independence assumption says that all colloids entering one unit cell have equal access to the residence times in that unit cell's residence time distribution regardless of their history or position exiting the prior unit cell (e.g., complete mixing between unit cells is assumed). In this (linear, Markovian) case, a direct upscaling is achieved by treating each unit cell as a mixing reactor with a particular residence time distribution, and the eventual breakthrough curve is obtained by repeated convolution of the residence time distribution per each unit cell in the series. If the residence time distribution within the unit cell has a finite moment, this uncorrelated convolution rapidly yields a tailless Gaussian-shaped breakthrough-elution curve, as expected from the central limit theorem. The inability of uncorrelated convolution to predict extended tailing indicates either: (1) that the underlying independence assumption is incorrect for unfavorable conditions; or (2) the residence times of colloids within the collector are indefinite (moment of the distribution is not finite).
 Recall that scale-up under favorable conditions (section 2) assumes porous media to be represented as a series of collectors, with implicit assumption of independence of colloid retention in each collector on the history of transport in preceding collectors. Notably, the demonstrated absence of extended tailing under favorable conditions [Li et al., 2004] is consistent with uncorrelated convolution and the independence assumption. The success of this implicit assumption is likely driven by the fact that under favorable conditions two influences occur: (1) a “perfect sink” boundary condition applies on the collector surface such that colloids in the near-surface region attach to the collector surface, and therefore do not accumulate in the near-surface domain; and (2) the colloid concentration in the near-surface fluid domain is replenished by mixing between grains/pores such that monotonic depletion of the near-surface colloid concentration does not occur as the fluid parcel moves through larger scales.
 The latter point is demonstrated by colloid trajectory simulations in a radial stagnation point flow system where a laminar jet of colloid-laden solution is directed normal to a flat surface in a closed chamber with outlets distal to the impinging jet axis. Away from the impinging jet axis, the flow is increasingly radial (Figure 4). There is an uninterrupted near-surface fluid domain under laminar flow conditions, lacking mixing with the bulk flow field. Under favorable conditions, because colloids are lost from the near-surface domain to the surface (via attachment), colloid concentrations become increasingly depleted in the near-surface layer with increasing transport downgradient. In contrast, in porous media, colloids in the near-surface fluid can be expected to mix with those in the bulk fluid between each collector in the series, thereby homogenizing the near surface and bulk fluid colloid concentrations, and allowing η to remain constant with scale (Figure 1).
 In contrast to favorable conditions, under unfavorable conditions, colloids accumulate in the near-surface domain due to secondary energy minimum attraction outboard of the energy barrier, as demonstrated by colloid trajectory simulations in the radial stagnation point flow system (Figure 4). While the accumulation of colloids in the near-surface domain under unfavorable conditions is clear in unmixed systems, colloid accumulation in the near-surface layer in porous media will occur to an extent that is mediated by mixing between collectors, as described below. This accumulation is likely the source of observed extended tailing following elution, and must therefore be accounted for in upscaling strategies under unfavorable conditions.
4. Novel Upscaling Approaches Under Unfavorable Conditions
 Under unfavorable conditions, colloids that gain residence in the near-surface domain skim along the surface until they either attach to the surface; return to the larger pore-space pathline, or exit to the bulk fluid domain [Johnson et al., 2007, 2010; Ma et al., 2011]. According to the discussion above, the significance of colloid accumulation in the near-surface domain to colloid breakthrough-elution at larger scales depends on the distribution of colloid residence times in the near-surface domain of a single collector, as well as the extent to which near-surface colloids maintain this state between upstream and downstream collectors. The extent to which near-surface colloids remain such over scales larger than a single grain or pore depends on the degree of fluid mixing between grains or pores.
4.2. Relating Colloid Mass Transfer to Flow Field Geometry
 An elegant aspect of CFT is the fact that η represents not only the fraction of introduced colloids that reach the near surface, but also represents the flow field by identifying the fluid trajectories surrounding the collector surface that may actually bring the colloid into close proximity to the surface, if transport is advection dominated. In this case, colloids located in trajectories beyond some limiting trajectory, away from the forward flow stagnation axis, have little possibility of reaching the surface. Hence, the forward stagnation axes in porous media are the centers of regions of colloid entry from the bulk fluid to the near-surface domain (Figure 5a). Building on this framework, we propose that the major points of exit from the near-surface domain to the bulk fluid are rear flow stagnation zones (Figure 5a), where near-surface fluid mixes with bulk fluid. Of course colloid diffusion and sedimentation broaden these entry and exit “windows”; however, it is reasonable to consider these locations to be the major loci of mass transfer between the bulk fluid and near-surface domains.
 For example, particle trajectories simulated under unfavorable conditions can provide such information directly, as demonstrated visually by a 2-D projection onto an x-y plane (normal to flow axis) of the exit locations (Figure 6) of 10 μm colloids from the hemisphere-in-cell flow field [Ma et al., 2011]. In this plot, the exit locations of the particles having the longest residence times in the unit cell cluster around the rear flow stagnation band running between the rear flow stagnation points of the two grains. This pattern matches observations in micromodels, where colloids translating along grain surfaces under unfavorable conditions often reentrain into bulk fluid upon reaching the rear stagnation zone (see supporting information in Ma et al. ). The deficit of symbols in the flow stagnation band represents depletion of particles in the near-surface domain due to retention on the upstream collector surface [Ma et al., 2011]. The significance of the “exit trajectories” in formulating kinetic representation of colloid transfer from the near surface back to the bulk fluid has not been previously explored, yet it is clear that flow stagnation zones are important mediators of colloid mass transfer in porous media. In contrast to diffusion-driven mass transfer from the near surface to the bulk fluid, fluid mixing at rear flow stagnation zones on open grain surfaces returns the entire near-surface colloid population (at that location) to the bulk fluid. At the assemblage scale, this mixing process is likely much more significant than diffusion-driven colloid transfer in returning colloids from the near-surface to the bulk fluid domain. In contrast to the bulk fluid domain, where the trajectories proximal to the forward flow stagnation zones deliver colloids to the near-surface domain, the near-surface domain conveys colloids along the collector surface, and colloids may or may not escape by diffusion before they are delivered back to the bulk fluid, or to a downstream grain, at the rear flow stagnation zone.
 The above-described relationship between rate coefficients and fluid trajectories makes clear that one cannot expect equality between kf and kr, as previously assumed by Bradford et al. . The parameter kf is a kinetic representation of the fraction of the bulk flow field that will (largely, excepting diffusion and settling) deliver colloids to the forward stagnation zone (to the near-surface layer). In contrast, kr governs mass transfer from the near-surface fluid flow field back to the bulk fluid. The latter process occurs predominantly at rear flow stagnation zones (our simulations show little diffusive escape from the near-surface layer) thereby transferring the entire near-surface population to the bulk fluid. The flow fields, and resident colloid population transferred, differ greatly between the near-surface and bulk fluid domains; hence no equality can be expected between kf and kr.
 An important insight from consideration of trajectories is that if the porous media system were purely composed of forward and rear flow stagnation zones located on the open surfaces of grains, near-surface colloids would be mixed into the bulk fluid between each pore (collector) due to release of these colloids to the bulk fluid at the rear flow stagnation zones. Under this condition, one would expect no net accumulation of near-surface colloids from upstream to downstream collectors, and the attachment collector efficiency would presumably upscale as described previously for favorable conditions (Figure 1). In contrast, the occurrence of locales where rear flow stagnation zones are aligned with downstream forward stagnation zone, or rear/forward stagnation zones coincide with grain-to-grain contact (flow-aligned contact, as illustrated in Figure 5b) may allow direct propagation of near-surface colloids directly from upstream to downstream grains, thereby allowing colloids to remain in the near-surface domain for scales larger than single grains or pores (correlated motion), potentially yielding the observed extended tailing under unfavorable conditions.
 The frequency distribution of colloid residence times from particle trajectory simulations in the hemisphere-in-cell model for representative unfavorable conditions is shown in Figure 7. The slope of the frequency of colloids versus residence time in the hemisphere-in-cell model is linear on a log-log plot, indicating power law tailing was produced in this single collector as a result of colloid transport in the near-surface domain. The simulations show that all colloids with residence times >30 s attained separation distances ≤200 nm at some point during transport, whereas all colloids with residence times ≤12 s maintained separation distances >200 nm during transport. The power law trend lines decay much more slowly for the case with near-surface colloids (blue) versus the case subtracting near-surface colloids (green), demonstrating the importance of near-surface colloids in generating long residence times.
 Extended tailing at the assemblage scale may reflect long residence times in association with single collectors or correlated motion (non-Markovian mass transfer) via propagation of near-surface colloids from upstream to downstream grains via flow-aligned contact. The frequencies of the various fates of near-surface colloids; (1) escape to bulk fluid via diffusion or mixing at rear flow stagnation points; (2) attachment; or (3) continued residence in the near-surface fluid domain, depend on the flow field and the grain connectivity at the assemblage scale (Figure 5).
 Our proposed relationship between flow-field geometry/collector arrangement and mass transfer kinetics provides a framework for upscaling colloid transport and retention under unfavorable conditions. It should be noted that regardless of whether existing upscaling approaches have a physicochemical basis, all previous approaches assume continuous transfer of colloids between the near surface and bulk fluid domains. In contrast, our concept involves localized colloid mass transfer that occurs predominantly at both forward and rear flow stagnation zones. Furthermore, none of the existing upscaling approaches attempt to relate upscaled colloid mass transfer kinetics to flow field geometry and collector arrangement; whereas ours explicitly does so, and we hope that this provides a useful platform for continued research.
5. Preliminary Experimental Verification of Collector Alignment Influences
 Below, we provide preliminary experimental verification of our hypothesis that alignment of forward and rear flow stagnation zones influences colloid mass transfer between the bulk and near-surface fluid domains. The concept driving the experiment is that collectors in-line with flow (and each other) are effectively limited to few forward flow stagnation zones at the bounding upstream collector layer, within which colloids are delivered to the near surface from the bulk liquid. In contrast, randomly packed collectors have a multitude of forward flow stagnation points distributed through the entire column. Hence, we hypothesize that under equivalent conditions, a packed column with collectors (2 mm quartz beads from same source) strung in-line with flow (Figure 8a) should have significantly reduced colloid retention relative to a randomly packed column having the same porosity (Figure 8b).
 The simultaneous column transport experiments (18.7 cm length, 3.73 cm radius, 101.5 ± 1.5 mL pore volume, 0.54 ± 0.01 porosity) utilized a single reservoir and syringe pump (two syringes) to deliver (2 m day−1 average pore water velocity) three pore volumes of solution (Ionic strength = 0.02 M NaCl, pH 6.73 MOPS buffer) having 2.0 μm carboxylate-modified polystyrene latex microspheres (Co = 4.0E6 mL−1, PolySciences, Inc.), followed by seven pore volumes of microsphere-free solution. Recoveries relative to total microspheres injected, combined from effluent (integrated area below breakthrough-elution curve) and recovered from packing following experiment (in Milli-Q solution with sonication for 2 min), were 95 and 93% for the Strung and Random columns, respectively. Aqueous microsphere concentrations were determined using flow cytometry as described in Pazmino et al. .
 The breakthrough-elution results provided in Figure 9 support our hypothesis. The Strung column yields lesser colloid retention relative to the Random column. The effect was reproducible (Strung C/Co = 0.670 ± 0.008; Random C/Co = 0.537 ± 0.007) and significant, with two-tailed t tests (assuming unequal variances and utilizing range among replicates rather than true standard deviation) demonstrating a C/Co difference of 0.11 (22%) at the 95% confidence level despite the limited retention. Greater retention (and potentially greater difference between the two packing arrangements) would be achieved with smaller bead size and longer columns, both of which pose a daunting challenge, as demonstrated in the description of column construction (supporting information).
 The significant reduction of colloid retention in the Strung relative to the Random column under equivalent conditions is convincing evidence that colloid mass transfer from the bulk to the near-surface fluid was greatly reduced under conditions where collectors were aligned with flow. This finding supports our hypothesis that colloid mass transfer between bulk and near-surface fluid domains is not continuous (e.g., predominantly driven by diffusion), but rather is localized according to the occurrence of forward and rear flow stagnation zones. The results support our hypothesis that linkage of colloid mass transfer parameterization to these zones, which are defined by topologies of the pore and collector domains, is necessary for prediction of Darcy-scale colloid retention and transport based on pore-scale physicochemical processes.
 Extended tailing from the two columns was statistically equivalent despite lesser colloid delivery to surfaces in the strung column. Predicting an expected effect of grain alignment on tailing is complicated by the fact that while direct propagation of near-surface colloids from upstream to downstream grains should predominate in the Strung column, greater mass transfer of colloids between the bulk fluid and the near-surface fluid should occur in the Random column. It is not clear which effect should produce greater extended tailing. More specific experiments will be required to explore collector alignment influences on extended tailing. The experiments shown here clearly demonstrate the influence of collector alignment on colloid retention, and thereby support the concepts we have introduced above. Many additional experiments to explore these concepts can be envisioned, including investigation of the significance of flow-aligned contact in random-packed media via periodic interruption of structure using mesh screens, among many others.
6. Conclusions and Suggestions for Further Research
 We have introduced five novel concepts which relate colloid transport and retention to upscaling phenomena and which are anchored in proof-of-concept experiments:
 1. Scale-up of colloid retention under favorable conditions using η implicitly assumes independence of colloid retention on the history of transport in preceding collectors.
 2. Accumulation of colloids in the near-surface fluid domain in response to van der Waals attraction outboard of the energy barrier under unfavorable conditions, if it occurred continuously, would yield an increased likelihood of colloid attachment despite negligible change in the bulk colloid concentration, and so would violate the independence assumption.
 3. Colloid mass transfer from bulk to near-surface fluid domains, and back, is not continuous. Rather it is localized, occurring predominantly at forward and rear flow stagnation zones. Darcy-scale two-layer models should be formulated that account adequately for the spatially focused mass transfer processes at forward and rear flow stagnation zones and grain-to-grain contacts.
 4. The propagation of near-surface colloids directly from upstream to downstream collectors will produce correlated motion that also violates the independence assumption.
 5. Long residence times of colloids in the near-surface fluid domain, and correlated motion due to direct transfer of near-surface colloids from upstream to downstream collectors, may each contribute to the observed Darcy-scale extended tailing of colloid concentrations following elution of a colloid pulse.
 Based on the framework we have presented, and for which we have provided preliminary experimental verification, we advocate investigation of colloid transport at multiple scales to produce: (1) Information at the pore scale, which includes simulation of colloid transport under unfavorable conditions to understand surfaces in terms of the heterogeneity that drives colloid retention; and (2) Information at the assemblage scale, which includes topological characterization of packed collectors to parameterize zones of colloid mass transfer between near surface and bulk fluid domains to explore the influence of fluid mixing as colloids move from upstream to downstream grains or pores.
 With respect to pore-scale simulations, the discrete heterogeneity approach will yield an easily implemented expression for prediction of katt if we can interpret surfaces in terms of discrete heterogeneity, which is challenging due to a lack of reliable methods to directly characterize nanoscale heterogeneity. The strategy therefore involves performing arrays of colloid transport experiments on a given collector surface (or porous media) with varying colloid size, solution chemistry, and fluid velocity to allow backing out the surface characteristics (heterodomain size and spatial frequency) that are consistent with the array of experimental results. Clearly, this approach will take time even if pursued by multiple researchers, since it needs to be understood for various conditions of discrete heterogeneity, colloid size, solution chemistry, fluid velocity, etc. Increasing recognition of the role of surface heterogeneity on colloid retention, and the development of new strategies to account for it [e.g., Duffadar et al., 2009; Bendersky and Davis, 2011] will undoubtedly engender incorporation of surface heterogeneity into kinetic models, and these models should be evaluated in terms of their ability to explain arrays of data spanning parameters such as those listed above.
 The potential direct transfer of near-surface colloids from upstream to downstream collectors at flow-aligned contact points, strongly suggests correlation between residence time of the colloid in the near-surface domain and its likelihood of return to the larger pore-space pathline (reentrainment as described above in Section 3.4). This correlated behavior may persist over multiple, and perhaps many, collectors, resulting in an upscaled behavior driven by non-Markovian kinetics, or “memory”-dependent kinetics of transfer between the near surface and bulk fluid. Such effects are non-Markovian at the unit cell (single collector) scale, and we propose to incorporate such non-Markovian effects through study of correlated random walk occurrences at the sub-REV (representative elementary volume) scale, and exposure-time modeling at the super-REV scale as done in Ginn [2000b]. The development of these Eulerian transport equations (i.e., advective-dispersive-filtration transport equations) is best done in parallel with upscaling of Lagrangian (particle tracking) equations, so that colloidal particle trajectory data and simulations [e.g., Ma et al., 2009, 2011; Nelson and Ginn, 2011] can be used to compare Lagrangian and Eulerian models at the larger scale. This requires incorporation of correlations into Lagrangian models, such as random walk models. Correlated random walks are more challenging to study because the usual central-limit theorem approaches to develop governing equations are made more difficult by the correlations [e.g., O'Malley and Cushman, 2012]. Recently however, such tools have been developed as extensions of the continuous-time random walk, and are termed correlated continuous-time random walk (CCTRW) models [Le Borgne et al., 2008; Kang et al., 2011]. Le Borgne et al.  have applied this in the context transport of sinusoidal channels. These developments provide promising tools for upscaling the particle forces we study at the colloid-scale, and finally a link of the particle mechanics as specified in the force balance, with upscaled models of correlated transport, both in the Lagrangian and Eulerian contexts.
 At the assemblage scale, our proposed framework significantly alters strategies relative to existing network modeling approaches for colloid transport [Tien and Payatakes, 1979; Cushing and Lawler, 1998; Burganos et al., 2001; Chang and Chan, 2006, 2008], because existing approaches account only for transport in the pore network. More general network models are needed that also describe transport along grain surfaces and across grain-to-grain contacts. We believe that network models based on mathematical duality [Glantz and Hilpert, 2006, 2008] are particularly well-suited to approach this problem and we are working to implement such a model. Simulations in such networks are based on simple algebraic rules rather than numerical solution of differential equations (the algebraic rules are based on more complex simulations) and would therefore allow one to simulate transport under unfavorable conditions at the assemblage or column scale. The hoped-for overall result is to combine the assemblage scale topological information with pore scale colloid transport simulations to develop theoretical formulations that yield good prediction of colloid transport and retention at assemblage scales under unfavorable conditions. Accounting for the topology of the grain matrix can be expected to yield predictions of Darcy-scale behavior that are real and that cannot be explained by models/theories, which assume grains to be unconnected (e.g., Happel's model). In a way, this is similar to the case of solute transport where accounting for the topology of the pore space gives rise to Darcy-scale hydrodynamic dispersion.
 Strategies such as pore network modeling and pore/grain domain characterization via X-ray microtomography [e.g., Long et al., 2010; Pazmino et al., 2011] enable development of simulated media for testing of hypotheses via particle trajectory simulations, e.g., that a portion of near-surface colloids are transmitted directly from upstream to downstream grains. Related questions concern the degree of connectivity of these “flow-aligned contact” zones and alignment of rear/forward flow stagnation zones for grains not in contact, which determines the significance of this expected process over larger scales, and its significance to observed extended tailing under unfavorable conditions. Experimental approaches might, for example, involve comparison of colloid retention and tailing in systems where collectors are aligned versus randomly packed in order to understand the relative influences of nonaligned forward and rear flow stagnation zones versus flow-aligned contact zones on colloid transport and retention.
 We owe many thanks to Tim Ginn for contributing the text regarding CTRW and related modeling approaches and for many helpful discussions during the development of this paper. Thank you as well to Tamir Kamai for his suggestion to develop a figure to demonstrate the power law distribution of colloid residence times under unfavorable conditions. This research was supported by the U.S. National Science Foundation Hydrologic Sciences Program (award 1215726). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We are also grateful for the technical and facility support provided at the Center for High Performance Computing at the University of Utah. WPJ thanks Eddy Pazmino and Brittany Dame for their assistance in conducting the column transport experiments.