Kirk E. Nelson

Processes in microbial transport in the natural subsurface

This is a review of physical, chemical, and biological processes governing microbial transport in the saturated subsurface. We begin with the conceptual models of the biophase that underlie mathematical descriptions of these processes and the physical processes that provide the framework for recent focus on less understood processes. Novel conceptual models of the interactions between cell surface structures and other surfaces are introduced, that are more realistic than the oft-relied upon DLVO theory of colloid stability. Biological processes reviewed include active adhesion/detachment (cell partitioning between aqueous and solid phase initiated by cell metabolism) and chemotaxis (motility in response to chemical gradients). We also discuss mathematical issues involved in upscaling results from the cell scale to the Darcy and field scales. Finally, recent studies at the Oyster, Virginia field site are discussed in terms of relating laboratory results to field scale problems of bioremediation and pathogen transport in the natural subsurface.

Comment on "extending applicability of correlation equations to predict colloidal retention in porous media at low fluid velocity".

Predict Colloidal Retention in Porous Media at Low Fluid Velocity” R Ma et al. sought “to extend the correction offered by Nelson and Ginn” for the nonphysical collector efficiency (η) values given by correlation equations (including theirs, the MPFJ equation) of the colloid filtration theory (CFT) at low fluid velocities. Ma et al. demonstrate that the NG equation still yields η > 1 when small values of fluid velocity (∼10−7 m/s), porosity (∼0.25), and colloid diameter (dp ≤ 100 nm) exist concurrently, and address this by adopting the NG equation with a modified diffusion term (referred to herein as the MHJ equation) but stating it is “based on regression to mechanistic simulation results”. In this comment, we discuss important issues regarding equation comparisons, address the claim that low fluid velocity applications of CFT are “largely hypothetical”, and present a mathematical constraint that preserves the physics lost by the MHJ equation. The MHJ equation is presented as “an improvement for predicting η under a wider variety of fluid velocities” than prior correlation equations including the NG equation. However, Ma et al. recognize that their approach results in an incorrect dependence on velocity as η approaches unity and suggest correcting this via “asymptotes at the diffusion limit”; the transition point is never defined, though, leaving the suggested approach prone to ambiguous interpretation. Moreover, since the asymptote is η = 1 and the error of the MHJ equation is that it misrepresents the inverse relationship of η with velocity for η < 1, use of the asymptote will yield even larger errors. Not acknowledged by the authors is that the MHJ equation can also fail to capture the inverse relationship with colloid size (Figure 1a). Thus, the strategy employed to constrain the MHJ equation sacrifices key aspects of the physics of colloid deposition at low fluid velocities. The benefit of never exceeding unity must be weighed against these deficiencies to evaluate the claim that this is an “improvement”. Regarding inapplicability of the NG equation to “certain parametric conditions”, we note that η equations are commonly used to elucidate trends with respect to input parameters. Before we addressed the application of CFT at low fluid velocities, prior equations employed a power law dependence on the Peclet number (NPE) without any limiting factors. This results in high sensitivity with respect to NPE that gives extremely large errors for nanoparticles at low fluid velocities, that is, prior equations yield η values dramatically above unity with errors increasing as NPE decreases. In contrast, when the NG equation exceeds unity, the errors are relatively minor as the limiting factor moderates the error. Even when unity is exceeded slightly, the physical trends are maintained and, thus, the equation is qualitatively correct (whereas the MHJ equation is qualitatively incorrect in its reversing the trends with respect to velocity and colloid size). The benefits of constraining the MHJ equation to stay below unity are not worth the cost of losing salient aspects of the physics. Moreover, this trade-off is unnecessary as mathematical means exist to achieve the desired constraint without compromising the physics (as shown below). Regarding Ma et al.’s statements implying that all available η equations agree well with each other and with available data, this is demonstrated to be false in our prior comparison of available equations with 112 experiments, and disparities are much greater with respect to the new MHJ equation (Figure 1c). The MPFJ equation exceeded the factor-of-two level agreement 39 out of 112 times (versus 10 for the LH equation, 17 for both NG and TE, and 34 for RT). The MHJ equation exceeds this threshold 43 times. The MHJ equation performs particularly poorly for nanoparticles (dp ≤ 100 nm). The factorof-two threshold is exceeded for all nanoparticles in the data set with an average difference of a factor of 4.3 (compared to 1.6 for NG) greater than the experimental value and a maximum of 6.7 (compared to 2.3 for NG). For other submicrometer particles (100 nm < dp ≤ 1 μm), the MHJ equation also fares worse than all prior equations (average Figure 1. (a) NG equation, NG constrained equation, MHJ equation, and MHJ asymptote (dashed red line) applied to data of Nagasaki et al.(MPFJ equation values range from 1.9 to 9.4 and are not viewable). Note that the MHJ equation reverses the dependence on colloid size and use of the MHJ asymptote increases error magnitudes of subunity η values (b) Comparison of NG equation, NG constrained equation, MPFJ equation, and MHJ equation for low Darcy velocity (U = 0.04 m/d) and low porosity (0.25) (c) NG equation, NG constrained equation, MPFJ equation, and MHJ equation compared to data set of 112 experiments. The first-order deposition rate coefficient (kf) is computed based on each equation’s η value. Experiments 1−15 are nanoparticles (dp ≤ 100 nm); experiments 16−48 are larger submicrometer particles (100 nm < dp < 1 μm); experiments 49− 112 are large colloids (1 μm < dp ≤ 10.1 μm). The factor-of-two level agreement is bracketed by the dashed black lines at kf model/kf experiment = 0.5 and kf model/kf experiment = 2; perfect agreement (ratio = 1) is denoted by the solid black line. Correspondence/Rebuttal

Kinetics of conjugative gene transfer on surfaces in granular porous media.

The transfer of genetic material among bacteria in the environment can occur both in the planktonic and attached state. Given the propensity of organisms to exist in sessile microbial communities in oligotrophic subsurface conditions, and that such conditions typify the subsurface, this study focuses on exploratory modeling of horizontal gene transfer among surface-associated Escherichiacoli in the subsurface. The mathematics so far used to describe the kinetics of conjugation in biofilms are developed largely from experimental observations of planktonic gene transfer, and are absent of lags or plasmid stability that appear experimentally. We develop a model and experimental system to quantify bacterial filtration and gene transfer in the attached state, on granular porous media. We include attachment kinetics described in Nelson et al. (2007) using the filtration theory approach of Nelson and Ginn (2001, 2005) with motility of E. coli described according to Biondi et al. (1998).

Colloid filtration prediction by mapping the correlation-equation parameters from transport experiments in porous media

Colloid filtration theory (CFT) is a conceptual construct for predicting the characteristic rate of colloid-surface collisions during transport in granular porous media. A central product of this theory is the correlation equation for predicting collection-efficiency (η), based exclusively on theoretical model development. Specifically, the η-equation has terms combining dimensionless groups (of physicochemical properties) with unknown parameters that are usually fitted so that the predicted η matches that determined by colloid-surface collisions simulated in idealized pore-scale models. In this study, we replace the simulated colloid-surface collisions in idealized models with experimental column-scale data on apparent colloid-surface collisions. A new correlation equation is obtained by minimizing the difference between η determined by the correlation equation and that determined experimentally, using data from a collection of experiments for favorable conditions for colloid filtration. In this way we parameterize a mechanistically-based η-equation with empirical evidence. The impact of different properties of colloids and porous media are studied by comparing experimental properties with different terms of the correlation equation. This comparison enables insight about the different processes that occur during colloid transport and retention in porous media, such as diffusion and interception, and provides directions for future CFT developments that will need to account for these processes differently than the current theory does. Additionally, we find that while most of the parameters of the presented η equation are only slightly different than those proposed in previous theoretical studies, the match between theory and observation is significantly improved. For the available experimental data, which provides a reasonable representation of property ranges for many applications of CFT, the proposed equation provides a closer match to the experimentally measured collection efficiencies compared to available theories to date. This article is protected by copyright. All rights reserved.

Comment on Particle Tracking Model for Colloid Transport near Planar Surfaces Covered with Spherical Asperities

The approach presented by Kemps and Bhattacharjee (2009) for considering the deposition of colloidal particles onto spherical asperities covering a planar surface in the presence of shear flow indicates some important aspects about how surface roughness may impact particle hydrodynamics and deposition. However, clarification is required regarding their claim that omitting the effect of hydrodynamic retardation on the diffusion coefficient would serve to decrease deposition. This is incorrect. A positive correlation between diffusion and deposition is well established. Our points of clarification below stem from both a priori theoretical considerations and recent work on our particle tracking model that shares some similarities with that of Kemps and Bhattacharjee. In the Validation of Approach section, the authors present results obtained from their method applied to the determination of the single-collector collection efficiency (η) of colloid filtration theory, with comparison to the simulation results of Nelson and Ginn, as well as the correlation equations of Rajagopalan and Tien and Tufenkji and Elimelech. Figure 3 in Kemps and Bhattacharjee shows markedly lower collection efficiencies for Brownian particles computed in ref 1 than those calculated by all others. Kemps and Bhattacharjee provide the explanation that the reason for the difference between the results in ref 1 and their own results is that the effects of hydrodynamic retardation on particle diffusivity are not considered in ref 1. The authors suggest that the hydrodynamic retardation of diffusivity would lead to more frequent collisions and larger η by stating the following complement: “This absence of hydrodynamic interactions would make it more difficult for each particle to contact the collector’s surface since the particle’s Brownian displacement at each time step would not be reduced as the particle approaches the collector surface.” However, this explanation is contrary to the actual effects of diffusion on particle collision with surfaces in both physical and mathematical contexts. Physically, a larger diffusivity leads to a larger effective mean free path and on average to more, not less, frequent collisions with a fixed boundary. Mathematically, it is established that greater diffusivity results in greater deposition as evidenced by the negative exponent on the Peclet number in all common correlations (and analytical solutions) for the collector efficiency, including those in refs 2 and 3. The incorporation of hydrodynamic retardation in the diffusion coefficient would then be expected to decrease diffusivity and thus deposition. There are a number of other reasons that could explain the discrepancy shown in Kemps and Bhattacharjee’s Figure 3. First, the computational domain of Kemps and Bhattacharjee is the unit cell surrounding a spherical asperity protruding from a planar surface, which is significantly different from the Happel model computational domain (used in all of the computational results used for comparison) in which the spherical collector is surrounded by a concentric liquid envelope. Did the authors remove the planar surface for this set of simulations and revise the particle tracking for strictly Happel sphere velocity conditions? If not, it would seem that the presence of the planar surface, through its effect of reducing the velocities from the original Stokes’ flow around the Happel sphere, would serve to increase depositions and, thus, the difference in results with those in ref 1. Also, if the authors’ planar surface remains, it is unclear how uniform particle start locations are established over the whole spherical collector. Second, we note that the authors cite a definition of the collector efficiency as the ratio of the flux of particles onto the collector to the flux of particles in the projected area of the collector upstream; this can be referred to as the isolated sphere definition. This is only one of two possible definitions, and in ref 1 it is noted that for appropriate comparison to their results it is necessary to use the definition that is correct for theHappelmodel (i.e., the ratio of flux onto the collector to flux in theprojected area of the entire Happel sphere (collector plus liquid envelope)). By using the first definition for the results of Kemps and Bhattacharjee, as well as those of refs 2 and 3, the difference with the results of ref 1 are exaggerated. It is also worthwhile to note that theHappelmodel definition ofη is used in the original paper developing the Rajagopalan and Tien equation for η. However, this equation is commonly applied in the form based on the incorrect isolated sphere definition; see also refs 4 and 5.Using the isolated sphere definition for deposition calculations based on the Happel flow field results in a nonphysical definition that overestimates η, including values greater than unity under highdeposition conditions. Thus, use of the isolated sphere definition inKemps and Bhattacharjee’s Figure 3 for all results except those in ref 1 clearly accounts for some of the observed discrepancies. Third, we have recently updated our own Lagrangianmethod and in the process have discovered that our approximations for hydrodynamic retardation overestimated the effect at large separation distances. After correcting this, the discrepancy with refs 2 and 3 for Brownian particles diminished. Additionally, we have incorporated the effect of hydrodynamic retardation into the diffusion coefficient, and as expected, this results in lowered deposition and smaller values of η. Figure 1 shows the results of our simulations with the corrected hydrodynamic retardation factors on both the deterministic motion alone and deterministic plus Brownian motion, along with the results in refs 1-3 (all plotted using the Happel definition of η). This clearly shows that some of the observed discrepancies in Kemps and Bhattacharjee’s Figure 3 are due to the errors in the hydrodynamic retardation factors (used on the deterministic component of motion) in ref 1. We also address Kemps and Bhattacharjee’s neglect of gravity, which was noted to be a possible means of increasing deposition when the gravitational force acts in the opposite direction to forces leading to deposition. For the particular case of downward flow that is adopted in all of the studies used by Kemps and Bhattacharjee for comparison, the inclusion of gravity will result in more deposition on the upper half of the collector and less