Introduction

Surface-engineered silica nanoparticles (nSiO2) have been produced with food-grade quality and used in biotechnology, food-processing, and medical research (Lundqvist et al. 2004; Roy et al. 2005). Organically modified nSiO2 particles have overcome limitations of their unmodified silica counterparts (Roy et al. 2005) granting them robust amphiphilic properties (Wang et al. 2012) that yield high colloidal stability in suspensions (Espinoza et al. 2010) able to endure elevated temperatures (Aroonsri et al. 2013) and high ionic strength levels (Kim et al. 2015). Hence and more recently, polymer-coated nSiO2 has been extensively researched for utility in subsurface engineering and enhanced oil recovery (Aroonsri et al. 2013; Worthen et al. 2013; Miranda et al. 2012). This includes improving the volumetric sweep efficiency of injected fluids in CO2-flooding (Yu et al. 2012; Mo et al. 2012) and water-flooding (Ponnapati et al. 2011) applications, mobility control through the spontaneous formation of nanoparticle-stabilized emulsions and foams (DiCarlo et al. 2011; Zhang et al. 2010), wettability alteration (Roustaei et al. 2013; Ju et al. 2002), and interfacial tension reduction (Roustaei et al. 2013).

Effective deployment and delivery of nSiO2 to the target zone within the subsurface will be essential to the successful field-scale implementation of this class of nanoparticle-based technologies. This necessitates an accurate understanding of the mobility behavior of nSiO2 in porous media for development of particle transport models as effective predictive and design tools. Equilibrium adsorption isotherms have served as conventional fate descriptors for solute transport; however, their application to nanoparticles lacks scientific justification (Praetorius et al. 2014; Cornelis 2015). Nanoparticle suspensions are thermodynamically unstable, and their mobility behavior is controlled by kinetic attachment and detachment processes (Petosa et al. 2010) that are described by Derjaguin-Landeau-Verwey-Overbeek (DLVO) theory of colloidal stability (Derjaguin and Landau 1941; Verwey and Overbeek 1948). At high ionic strength levels with suppression of electrostatic repulsion, non-DLVO forces such as steric, hydrophobic, and hydration forces can control particle mobility (Petosa et al. 2010). Macroscopic scale mass transfer of nanoparticles from the fluid phase to solid surfaces is commonly described by advection-dispersion equation (Praetorius et al. 2014; Elimelech et al. 2013), incorporating an additional term to account for surface accumulation of particles on solid phase where the clean-bed filtration theory (CFT) (Yao et al. 1971) or a modified version of it which considers a maximum attainable surface coverage or retention capacity (MRC) (Ko and Elimelech 2000; Li et al. 2008) is commonly adopted. Few recent modeling studies have outlined a need for more complex multi-attachment-site (Zhang et al. 2015, 2016; Park et al. 2016; Sasidharan et al. 2014) or multi-porosity region (Suddaby et al. 2014; Simunek et al. 2013) filtration models for proper core-scale description of the transport and deposition of nanocolloids in porous media. These alternative calls of model approaches include the two site filtration approaches that allows for (i) fast (equilibrium) and slow (kinetic) attachment (Martin and Olusegun 2014; Wei et al. 2017) or (ii) reversible and irreversible attachment to fixed capacity MRC sites (Zhang et al. 2015, 2016) or mixed CFT and MRC types (Fang et al. 2013).

Kim et al. (2015) investigated the effects of flow velocity, particle size, and concentrations on the mobility of nSiO2 aggregates in sand columns saturated with API brine (8% wt. NaCl and 2% wt. CaCl2). In their study, a gradual increase in the efflux of particles with time was observed in all experiments that could not be explained by CFT. Although an MRC approach with rate-limited particle detachment captured the experimental particle breakthrough and elution, the scaling of the fitted attachment efficiency and site blocking parameters with variations in flow velocity, mean particle diameter, and effective viscosity were not consistent with the underpinning colloid filtration theories. This implicated the necessity of further refinement of existing mathematical modeling frameworks for particle filtration in order to have an accurate description of nSiO2 mobility under high salinity conditions. This is especially important in the upscaling process to make large-scale predictions of particle mobility. For instance, foams have been used as an enhanced oil recovery technique with application as hydrocarbon mobilizing agents and/or flood-conformance enhancers (DiCarlo et al. 2011; Baran and Cabrera 2006). In the latter case and specifically for CO2 in brine foams stabilized by surface-engineered nSiO2, the foam-induced flow redistribution in stratified formations has been demonstrated to be highly sensitive to slight variations in core-scale nSiO2 mobility parameters (Worthen et al. 2015).

In this mathematical modeling study, in order to provide an improved mechanistic description of Kim et al. (2015) observations of aggregated nSiO2 mobility under API brine salinity, we develop a class of two-site attachment filtration models (2S) and evaluate its applicability to simulating pre-aggregated nSiO2 elution data. Inter-model and intra-class comparisons are made between CFT model, MFT model, and four subclasses of 2S model, respectively. Model performances are gauged and analyzed in terms of (i) the goodness of fit criteria and (ii) consistency of model fits with underlying colloid filtration theories by introducing two theoretical compliance indices to quantify the consistency of the velocity, viscosity, and size dependence of the fitted particle mobility parameters with colloid filtration and shadow zone theories.

Methods

Experiments

Mobility tests, fully described in Kim et al. (2015), were carried out by injecting pre-aggregated nSiO2 dispersions ranging in mean particle size between 5 nm and 2 μm through a 30-cm-long Ottawa sand column (d50 = 350 μm). In each experiment prior to injection of nanoaggregates, the sand column was preconditioned with API brine (8% wt. NaCl and 2% wt. CaCl2). The nanoparticle suspension was pumped from a 400-mL floating-piston accumulator to the column at a controlled variable rate of 2.5, 5, or 25 mL/min corresponding to a hydraulic residence time ranging from 2 to 24 min. An approximate 4.5 pore volume wide (PV) pulse of nSiO2 suspension was injected and was followed by another 4.5 PV of particle-free background brine. The influent and effluent samples were analyzed for silica concentration using inductively coupled plasma-optical emission spectrometry (ICP-OES, Varian). The schematic test setup is shown in Fig. 1, and the experimental conditions are summarized in Table 1.

Fig. 1
figure 1

Schematic view of nSiO2 transport test setup (Kim et al. 2015)

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Table 1 Summary of experimental conditions of nSiO2 mobility tests
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Viscosity measurement (Kim et al. 2015) for nSiO2 suspensions varying in concentration between 0.5 and 2% wt. was taken in steady rate sweep mode between 10 and 1000 s−1 shear rate using a rheometer (ARES-LS1, TA Instruments) with the double-wall couette geometry. The temperature was controlled by a circulator at 25 °C. Samples of different aggregate sizes were prepared by keeping nanoparticle dispersions exposed to API brine for extended periods of time (at least 48 h). The measured viscosity at 100 s−1 of shear rate increased from 1 cP to above 20 cP as nSiO2 concentration increased from 0.5 to 2% wt. in aqueous suspension. The viscosity of nSiO2 suspension was dependent on the nSiO2 concentration under the shear rates tested. The viscosity value corresponding to the shear rate of each mobility test was selected for the simulation.

Mathematical model

Mass balance for particle retention and transport was expressed using 1-D advection-dispersion equation with an additional term for surface mass accumulation on the solid phase to describe nSiO2 mobility in sand columns (Wang et al. 2008):

$$ \frac{\partial }{\partial t}\left(\phi C+{\rho}_bS\right)+\frac{\partial }{\partial x}\left[\phi \left({v}_wC-{D}_w^h\frac{\partial C}{\partial x}\right)\right]=0 $$
(1)

Here x [L] and t [T] denote space and time, C [ML−3] and S [M/M] are the suspended and retained nSiO2 concentrations, ρb [ML−3] is the bulk density of the porous medium, vw [LT−1] and \( {D}_w^h\ \left[{L}^2{T}^{-1}\right] \) denote pore velocity and hydrodynamic dispersion coefficient, and ϕ [−] is the effective porosity expressed as a function of retained concentration:

$$ \phi ={\phi}_0-\frac{\rho_b}{\rho_p}\bullet S $$
(2)

where ϕ0 [−] is clean-bed porosity, and ρp [ML−3] is density of particles (for details of the derivation of Eq. 2 see Supporting Information (SI)—Appendix 0). Changes in effective porosity due to particle retention are generally neglected in dilute particle suspension systems, i.e., milligrams per liter concentrations that was exceeded several orders of magnitude by the range of nSiO2 concentrations used in Kim et al. (2015) (i.e., 20–40 g/L). For this reason, changes in local effective porosity and subsequent effect on spatial distribution of flow velocity and particle attachment kinetics were included in this study.

The second term on the time derivative on the left hand side of Eq. 1 accounts for NP solid-phase interactions (i.e., nSiO2 attachment and detachment). A multi-site attachment model, with similarities to the approach presented in Zhang et al. (2016), was employed that allows for particle deposition in two types of capture sites:

$$ \frac{\rho_b}{\phi}\frac{\partial {S}_{CFT}}{\partial t}=\frac{3}{2}\frac{\left(1-\phi \right){v}_w}{d_c}{\alpha}_{CFT}{\boldsymbol{\eta}}_0C-\frac{\rho_b}{\phi }{k}_{d, CFT}{S}_{CFT} $$
(3a)
$$ \frac{\rho_b}{\phi}\frac{\partial {S}_{MRC}}{\partial t}=\frac{3}{2}\frac{\left(1-\phi \right){v}_w}{d_c}{\psi}_{MRC}{\alpha}_{MRC}{\boldsymbol{\eta}}_0C-\frac{\rho_b}{\phi }{k}_{d, MRC}{S}_{MRC} $$
(3b)
$$ S={S}_{CFT}+{S}_{MRC} $$
(3c)

where SCFT and SMRC [M/M] respectively denote the nSiO2 mass concentration attached to (i) CFT sites following clean-bed filtration (Yao et al. 1971) with attachment efficiency αCFT [−] and (ii) MRC sites with attachment efficiency αMRC [−]. η0 [–], the single collector contact efficiency, is a measure of the frequency of NP-collector surface collisions and was determined using the semi-empirical correlation by Tufenkji and Elimelech (2004) as a function of physicochemical properties of particles, collector surface and flow phase including factors such as particle and collector sizes and fluid velocity and viscosity. The last term (i.e., effective viscosity) was selected based on Kim et al. viscosity measurements and the shear rate corresponding to constant flow velocity of each experiment. The deposition rate of particles in CFT sites (i.e., the attachment rate constant in clean-bed filtration model \( {k}_{att}=\frac{3}{2}\frac{3}{2}{\alpha}_{CFT}{\boldsymbol{\eta}}_{\mathbf{0}} \) (Yao et al. 1971)) is independent of deposition history (i.e., no blocking of CFT capture sites); however, the MRC deposition rate approaches zero with full saturation of MRC sites (Li et al. 2008):

$$ {\psi}_{MRC}\left({S}_{MRC},x\right)=\left(1-\frac{S_{MRC}}{S_{max}}\right){\left(\frac{d_c+x}{d_c}\right)}^{-\beta } $$
(4)

where ψMRC[−] is the site-blocking function, and Smax [M/M] is the maximum retention capacity parameter. The second term on the right-hand side of Eq. 4 accounts for physical straining effects where a micro-scale blocking of pores may occur as a result of deposition of larger aggregates (Bradford et al. 2004). A β value of 0.432 has been used in several existing reports for modeling straining effects for colloidal particles ranging in a hydrodynamic size in the submicron to micrometer scale. (Bradford et al. 2002, 2004). A β value of zero eliminates the straining effects, reducing the site blocking function to the mathematical form proposed in Ko and Elimelech (2000). The last term on the right-hand side of Eq. 3a and Eq. 3b is a kinetic model for particle reentrainment with a first-order dependence on the concentration of deposited NPs and deposition rate constant kd[T−1].

Numerical implementation

The transport equation (i.e., Eq. 1) was discretized using a Crank-Nicolson finite difference scheme implementing a sequential time-lag approach to solve Eq. 3a,b,c. Three models were fit to nSiO2 mobility data: the two-parameter CFT model (αCFT, kdet) (M1), three-parameter MRC model (αMRC, Smax, and kdet) (M2), and the 2S model class with a maximum of four model parameters (αCFT, αMRC, Smax, and kdet) where four subclasses were considered (i) the unbounded 2S (M3), (ii) 2S with favorable irreversible attachment to MRC sites (i.e., αMRC = 1 and kd, MRC = 0) (M4), (iii) 2S with physical straining-based irreversible attachment to MRC sites (i.e., β = 0.432 and kd, MRC = 0) (M5), and (iv) 2S with irreversible physical straining under favorable attachment conditions (M6). Note that the latter three subclasses of 2S allow for particle detachment only from the CFT sites consistent with the expected irreversibility of particle deposition in deep primary energy minima under favorable attachment conditions (Petosa et al. 2010) and previous reports on the irreversibility of physical straining (Bradford et al. 2004, 2007).

The nonlinear least squares method was used in the inverse analyses. Parameter estimation follows a two-stage iterative procedure: (stage 1) the injected pulse portion of effluent data was used to estimate all model parameters except for kdet which had a fixed value (starting with an initial guess), and (stage 2) kdet parameter was then fitted incorporating tailing part of nSiO2 effluent concentrations (corresponding to chase water phase of mobility experiments) with all other mobility parameters fixed at values estimated in stage 1. Stage 1 was repeated using the updated fixed for kdet, and iterations continue until a set tolerance of 1% on kdet between two successive iterations was satisfied.

Results and discussion

Table 2 provides the fitted particle mobility parameters for experiments 1–4 based on models M1-M6. The nSiO2 in the non-aggregated state (experiment 1), with a mean particle diameter of 48 nm, was highly mobile behaving almost like a conservative tracer (with a fitted αCFT = 3.7 × 10−4) at a pore velocity of 7.1 m/day (the lowest in this study). The pre-aggregated nSiO2 with a mean aggregate diameter ranging between 1.5 and 2 μm, on the other hand, showed remarkably low mobility, far less than what can be attributed to the effect of gravitational sedimentation alone (Fig. 2). Even though the dramatic increase in the mean particle size was associated with a three orders-of-magnitude magnification of the gravity component of contact efficiency (ηG; Tufenkji and Elimelech 2004) from 1.66 × 10−5 (experiment 1) to 0.0162 (experiment 2), the overall single collector contact efficiency (η0) decreased due to the dampening of Brownian motions of particles with the size increase (0.0592 (experiment 1) versus 0.0214 (experiment 2)). This corresponded to a significant increase of the CFT fitted particle-collector attachment efficiency (αPC; Tufenkji and Elimelech 2004) from 3.7 × 10−4 (experiment 1) to 0.569 (experiment 2). MRC model fitted αPC increased 13-fold between experiments 1 and 2, suggestive of a lower magnitude of the same trend captured by CFT model. According to classical colloid filtration theory, a change in physical system properties should merely impact η0-efficiency while αPC is anticipated to be unaffected (Li et al. 2008; Tufenkji and Elimelech 2004). This notable increase of αPC, hence, suggests that the change in particle sizes during the aggregation process reflects not only a change in the physical characteristics of nSiO2 suspension but also a shift in particle surface chemistry. This hypothesis is further corroborated by the presence of persistent tailings of nSiO2 in effluent samples during the post flush with particle-free brine in all experiments with pre-aggregated nSiO2 indicative of reversible attachment as opposed to irreversible attachment observed for non-aggregated nSiO2 in experiment 1 (no tailing on the effluent breakthrough curve was detected during the post flush period in this experiment). Another notable difference in model predictions stemming from the aggregation state of the particles was the computed reduction of the effective porosity of soil (see SI—Appendix B, Table B1), while negligible for non-aggregated nSiO2 (limited to about 0.1%), the effective porosity decreased moderately for pre-aggregated particles (by roughly 0.5–2.5% inversely correlated with flow velocity).

Table 2 List of the fitted model parameters for filtration models considered herein
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Fig. 2
figure 2

The experimental nSiO2 effluent concentration and respective best fit by clean-bed filtration (CFT) model

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For more in-depth comparison of the implemented modeling approaches, the numerical performance of the CFT and MRC models and four subclasses of 2S model were evaluated through the analysis of (i) modeling residuals (i.e., the difference between modeled and measured effluent concentrations) and (ii) the theoretical conformance of model-specific fitted parameters. The two aspects of model performance assessment are discussed separately hereinafter.

Analysis of model residuals

Comparative analysis of the numerical performance of models M1-M6 was done based on two criteria on the (i) goodness of fit and (ii) distribution of modeling residuals.

Criterion I. Normalized Sum of Squared Residuals (NSSR), as a measure of the goodness of fit, was defined as follows:

$$ \mathrm{NSS}{\mathrm{R}}_{i,m}=\frac{\sum_{k=1}^{N_e}{\left({C}_{m,i}^k-{C}_i^k\right)}^2}{C_{0,i}^2\bullet P{W}_i\bullet \left({N}_i-{N}_m-1\right)\ } $$
(5)

where NSSRi, m [–] denotes the NSSR for experiment i computed based on model m, Ci [ML−3] is a 1-D matrix that stores experimental effluent concentrations, Cm, i [ML−3] contains the respective modeled effluent concentrations for Ci given by model m, C0, i [ML−3] and PWi [pore volumes] are experiment-specific influent concentration and injected pulse width, and lastly, Ni and Nm denote the numbers of experimental effluent concentration data points and model fitting parameters, respectively. The lower the NSSR value of a model, the better the numerical fit provided by that model. Models with greater number of fitting parameters were penalized following a method similar to calculating adjusted R2 in multivariate regression (Helland 1987).

Computed NSSRs are shown in Fig. 4. A general improvement in the goodness of fit for MRC and 2S models was observed over the CFT model fits. The unbounded 2S NSSRs were 20–90% lower than CFT-NSSRs in all experiments but remained within a 10% range of MRC-NSSRs with only one exception (experiment 3), implying that the additional intricacies of the 2S class did not help improve model performance with respect to NSSR metric compared to less sophisticated MRC model. The 2S subclass with favorable straining (M6) representation of nSiO2 mobility was notably similar to CFT model in terms of the shape of fitted breakthrough curves (Fig. 3b–d) and calculated NSSR (gray rods vs. hollow blue cones in Fig. 4).

Fig. 3
figure 3

Best fits provided by CFT, MRC, and 2S class models to nSiO2 effluent data from a control non-aggregated (exp no. 1), and experiments conducted using pre-aggregated nSiO2 at fixed interstitial velocity of b 7.1 (exp no. 2), c 14.2 (exp no. 3), and d 71 m/day (exp no. 4)

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Fig. 4
figure 4

Normalized squared sum of residuals computed for experiments 1–4 based on different models

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Criterion II. Normality of Model Residuals was characterized for experiments 2–4 by plotting the computed residuals of each model class (i.e., CFT, MRC, and 2S) versus the Gaussian quantiles (referred to as a normal quantile-quantile or Q-Q plot). Models were then compared based on the degree of linearity (R2 value) of the alignment of model residuals on Q-Q plots (Fig. 5). For a clear visual representation, only the subclass of 2S with favorable attachment to MRC sites (M4) was included in Fig. 5. The MRC (M2) and 2S (M4) models (both implementing three fitting parameters) exhibited generally higher R2 values than the two-parameter CFT model. All models performed poorly on experiment 4; however, 2S (M4) resulted in a remarkably higher R2 value compared to MRC model. No consistently discernible improvement to the normality of model residuals was observed for 2S class over MRC, albeit, both model classes showed clear improvement over CFT.

Fig. 5
figure 5

CFT, MRC, and 2S-αMRC = 1 model residual quantiles plotted versus standard normal quantiles for experiments on pre-aggregated nSiO2 at fixed interstitial velocity of a 7.1 (experiment 2), b 14.2 (experiment 3), and c 71 m/day (experiment 4)

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Theoretical conformance of implemented modeling approaches

Comparison of the consistency of the tested filtration models with the underlying theories of colloid deposition in porous media, is outlined in this section. CFT, MRC, and 2S model classes incorporate either one (CFT and MRC) or two (2S) attachment efficiency term(s) (αPC) with the latter two models also implementing a maximum retention capacity parameter (Smax). Here we define two indices to quantify the consistency of the scaling of αPC and Smax parameters across experiments 2–4.

A quantitative analysis of the fitted detachment rates was not undertaken. To the best of our knowledge, colloid reentrainment kinetics in the absence of double-layer interactions has not been mechanistically characterized at macro-scale, and the existing literature only qualitatively explains the observed trends in particle detachment by balancing microscopic adhesive and hydrodynamic drag torques acting on deposited particles (Raychoudhury et al. 2012; Bergendahl and Grasso 2000). An increase in pore water velocity is expected to increase the drag torque on particles submerged in a Newtonian fluid. However, the higher shear rate associated with a higher velocity can reduce the drag torque due to a reduction of the effective viscosity caused by shear thinning behavior of nSiO2 suspension in pre-aggregated state. Increasing the interstitial velocity from 7.1 (experiment 2) to 71 m/day (experiment 4) resulted in a 19–30-fold increase of fitted kd by all models with the exception of the unbounded 2S (M3) with a 2 orders of magnitude increase of kd. The CFT- and unbounded 2S-fitted detachment rate constant changed very little (< 10%) when pore velocity increased to 14.2 m/day (experiment 3). In comparison, kd fitted values of MRC and 2S with straining (M5) decreased more than 2-fold. M4 subclass of 2S was the only model that indicated a kd increase of 1.5-fold. Note that this subclass considered favorable irreversible attachment to MRC sites.

Attachment efficiency scaling

Filtration theory predicts that the attachment efficiency, the fraction of particle-collector collisions resulting in sticking, depends solely on the solution and surface chemistry (Tufenkji and Elimelech 2004). This implies an expected insensitivity of the fitted αPC to variations of physical factors such as particle size and flow velocity. That is, the ratio of attachment efficiencies for two mobility experiments conducted under similar chemistry is expected to be close to unity. Therefore, a deviance from unity of this ratio obtained from model m fits can be regarded as a measure of inadequacy of that filtration model for capturing the physics of particle mobility behavior under the respective experimental conditions. On this basis, a theoretical deviance index was defined as follows:

$$ TD{I}_m=\frac{\alpha_{PC}^{m,0}}{\sqrt[N-1]{\prod_{i=1}^{N-1}{\alpha}_{PC}^{m,i}}} $$
(6)

where \( {\alpha}_{PC}^{m,0} \) and \( {\alpha}_{PC}^{m,i} \) are model m fitted attachment efficiencies for experiment i, with superscript “0” denoting the reference experiment with the highest αPC (experiment 2 in this study). Because of similarity in chemical conditions, data from experiments 2–4 were used to determine TDI values. For the 2S class, the geometric mean of \( {\alpha}_{PC}^{CFT} \) and \( {\alpha}_{PC}^{MRC} \) values was used. Figure 6a shows the computed TDIs. CFT model was found most deviant with a TDI value of 4.8 whereas the lowest TDI value of 2.2 was calculated for a subclasses of 2S that considered irreversible favorable attachment to MRC sites (M4 and M6). These subclasses are three-parameter versions of 2S with the same number of fitting parameters as the MRC (M2) model that yielded a notably higher TDI of 3.4. The unbounded 2S (M3) and staining-based 2S (M5) TDIs of 3.1 and 3.2 also exhibited a higher degree of deviance compared to M4 and M6. Care must be taken in the interpretation of staining-based 2S model results (M5 and M6). In the case of unbounded straining (M5), the fitted \( {\alpha}_{PC}^{MRC} \) ranged between 1.7 and 11.5 where an αPC value in excess of one is not theoretically meaningful (Yao et al. 1971; Elimelech and O’Melia 1990). Nevertheless, instances of attachment efficiencies greater than one are not uncommon in colloid filtration literature (Kuhnen et al. 2000; Litton and Olson 1993). Even though the straining model with favorable attachment (M6 with \( {\alpha}_{PC}^{MRC}=1 \)) yielded one of the lowest TDI values, it should be noted that M6 performed poorly relative to other 2S subclasses with respect to model residuals.

Fig. 6
figure 6

Model-specific a theoretical deviance index (TDI) and b theoretical conformity index (TCI) for implemented model classes

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Retention capacity scaling

The scaling of fitted Smax parameter with changes in aggregate size, flow rate, and measured effective viscosity across experiments 2–4 is compared with the predicted Smax scaling according to the shadow zone theory (Ko and Elimelech 2000). A shadow zone is thought to be created on the collector surface down gradient of deposited particles, where the probability of subsequent attachment is greatly reduced due to the combined effects of the tangential component of flow around collector grains and double-layer repulsion. Particle size has been shown to be an important factor in the dynamics of particle deposition and blocking in stagnation point flow cell studies (Böhmer et al. 1998). In theory, an increase in the size of particles, fluid viscosity, and/or approach velocity can lead to an increase in the excluded surface area for particle-collector collisions (reduction in maximum attainable surface coverage), which attributed to an enhancement of the shadow effect. Based on this theory, Li et al. (2008) suggested Smax = 19.6 Λ−1.2 with Λ [–] as a normalized mass flux term approximated by Peclet number as Λ ≈ Pe1/3 ∙ (dc/dM) in which dc [L] and dM [L] denote the median and a medium sand grain diameter (Li et al. 2008). Peclet number is defined as Pe = (vw ∙ dc)/Dp with Dp [L2T−1] as the particle diffusivity coefficient calculate via Einstein-Stokes equation as follows:

$$ {D}_p=\frac{k_B\bullet T}{3\pi \bullet {\mu}_w\bullet {d}_p} $$
(7)

where kB [J K−1] and T [K] are Boltzmann constant and temperature, respectively. The following theoretical expression can be derived to explain the scaling of Smax:

$$ {S}_{max}\propto \frac{1}{\Lambda^{1.2}}\propto \frac{1}{{\left(P{e}^{1/3}\right)}^{1.2}}\propto \frac{1}{{\left({\left({v}_p/{D}_p\right)}^{1/3}\right)}^{1.2}}\propto \frac{1}{{\left({\left({v}_p/\left(\frac{1}{\mu_w}\bullet \frac{1}{d_p}\right)\right)}^{1/3}\right)}^{1.2}} $$
(8)

In other words, if for a given set of \( {\mu}_w^0 \), \( {d}_p^0 \), and \( {v}_p^0 \) at a reference state, \( {S}_{max}^0 \) is measured, then, it is expected that Smax for μw, dp, and vp will theoretically follow:

$$ {S}_{max}^{theory}\left({\mu}_w,{d}_p,{v}_p\right)={\left(\frac{\mu_w^0\bullet {d}_p^0\bullet {v}_p^0}{\mu_w\bullet {d}_p\bullet {v}_p}\right)}^{0.4}\bullet {S}_{max}^0 $$
(9)

Using experiment 2 as the reference experiment and the respective model-specific Smax as \( {S}_{max}^0 \), the theoretical \( {S}_{max}^{theory} \) values were derived for experiment 3 and 4 from Eq. 9. The overall consistency of each model with the shadow zone theory predictions was then determined through the computation of a theoretical conformity index (TCI) defined as follows:

$$ TC{I}_m=\sqrt[N-1]{\prod \limits_{i=1}^{N-1}\frac{S_{max}^{i,m}}{S_{max}^{theory}}} $$
(10)

where \( {S}_{max}^{i,m} \) denotes the fitted Smax for experiment i by model m, and N − 1 = 2 (in this study) is the number of experiments with the exclusion of the reference experiment.

Figure 6b shows the TCIs for MRC and 2S class. The calculated TCIs reveal overall superiority of 2S class compared to MRC class in terms of conformance with shadow zone theory. A TCI value of 0.21 was calculated for MRC (M2) model which was less than respective value for all 2S subclasses (M3-M5). A special one-parameter case of MRC that considered favorable attachment conditions (i.e., \( {\alpha}_{PC}^{MRC}=1 \)) produced the lowest TCI (0.02—fits not shown) among all implemented models. Within the 2S class, subclasses with favorable attachment to MRC sites (i.e., M4 and M6) were most conforming with shadow zone theory returning the highest TCIs among the tested models. M4 and M6 returned TCI values of 0.58 and 0.85, respectively, where in comparison, the unbounded 2S (M3) and unbounded 2S with physical straining (M5) yielded a TCI of 0.3 and 0.33.

Conclusions

The transport and retention of pre-aggregated nSiO2 under API brine-saturated conditions were simulated using three classes of colloid filtration models: a clean-bed filtration (CFT) model, a maximum retention capacity (MRC) model, and a two-site mixed type (2S) model. Despite penalizing models for using additional fitting parameters, a general improvement in the goodness of fit (NSSR) was observed for MRC and 2S models over the CFT model. Our findings did not substantiate the adequacy of either of the single-site approaches (i.e., CFT or MRC) to modeling nSiO2 aggregates mobility at high salinity. CFT approach performed poorly with respect to both the goodness of fit and theoretical conformity criteria. This poor performance stems from the inherent incapability of CFT model to explain the gradual increase in particle effluent concentration observed in all pre-aggregated mobility experiments. The reason lies in the assumption of an infinite capacity for attachment to the CFT sites that is independent from the history of colloid retention. Therefore, CFT model cannot explain the observed reduction in the retention capacity of sand columns. CFT model may explain incremental change in particle breakthrough concentrations if the kinetic rate of detachment is comparable to the rate of attachment (same order of magnitude). This explanation, however, is not supported by the low tailing concentrations measured during the post-particle injection chase water flushing period in all experiments. The MRC approach, on the other hand, provided general good fits to mobility data, but the scaling of MRC fitted parameters with variations in physical properties (e.g., velocity and effective viscosity) provided poor theoretical compliance with the physics of colloid filtration and site-blocking.

On the contrary, the 2S class exhibits a superior numerical performance demonstrated by a higher degree of normality of the distribution of model residuals for 2S class compared to CFT model. This superiority was further corroborated by the greater consistency of the 2S class with the underlying principles of colloid filtration in terms of (i) the relative insensitivity of fitted attachment efficiency to changes in system physical properties and (ii) the scaling of fitted retention capacity in accordance with shadow zone theory predictions across a range of experimental conditions. The highest level of conformity with the shadow zone theory and the lowest deviance from filtration theory was yielded by two three-parameter subclasses of 2S (i.e., M4 and M6) that considered irreversible favorable attachment to MRC sites (fixed limited retention capacity) and reversible attachment to CFT sites (infinite capacity) under unfavorable conditions. The 2S subclass with favorable irreversible attachment to MRC subclass (M4) provided (a) one of the closest numerical fits to mobility tests results and (b) the second highest overall theoretical conformity among the tested models. The 2S subclass with favorable straining (M6) yielded the highest overall theoretical conformance among all tested models suggesting the importance of physical straining as a secondary particle removal mechanism at high salinity, which would be in concert with the larger than colloid straining literature specified critical ratio of 0.002 (Bradford et al. 2004) between mean particle diameter and grain size ratio in pre-aggregated nSiO2 experiments. Both models suggest the existence of favorable attachment sites. A possible explanation is that the same aging or particle surface passivation process that decayed the steric/hydrophobic forces between the interacting particles thereby causing the formation of nSiO2 aggregates might have also created a patch-wise heterogeneity in the distribution of steric forces on the surface of particles colliding collector surfaces. In the absence of electrostatic double layer interactions (due to high salinity), this might have warranted favorable attachment conditions for a fraction of particle-collector collisions. Care must be taken, however, in the interpretation of the results of 2S subclass with favorable straining because of the respective poor fits relative to other 2S subclasses. It must also be noted that numerical fits of the unbounded straining model (M5) were on a par with other subclasses of 2S; albeit, a clear departure from the theoretical principles of filtration was associated with M5 subclass: (i) the significantly larger than unity fitted attachment efficiencies were theoretically inexplicable and (ii) the TDI and TCI indices indicated primitive performance compared to M6 (favorable straining).

Lastly, we should stress the importance of model selection on the upscaling of core-scale particle transport to field-scale mobility predictions, which is culminated in the sensitivity of the estimated effective travel distance to the choice of particle transport model. For instance, for constant velocity fields, CFT model yields the most conservative estimate of particle transport with a distance to three-log removal (a measure of effective travel length (Wang et al. 2008) of \( 3.07\times \frac{d_c}{\left(1-\phi \right)\bullet {\alpha}_{PC}\bullet {\eta}_0} \), an estimation that is independent of the injected mass of particles. On the contrary, the MRC model considers a finite capacity of attachment where distance to three-log removal is a strong function of injected mass of particles. Here upon complete blocking of MRC sites, particles can continue to migrate indefinitely in the subsurface given a sufficient amount of sacrificial particles is injected. 2S model class incorporates a mix of both types, and thus, the upscaled predictions based on 2S class are expected to lie somewhere between the two theoretical extremes depending on the physicochemical characteristics of nanoparticle suspension and site-specific properties of flow and porous media.