Notes on Colloid transport and filtration in saturated porous media

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## Notes on Colloid transport and filtration in saturated porous media

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**Notes on Colloid transport and filtration in saturated**porous media Tim Ginn, Patricia Culligan, Kirk Nelson Purdue Summerschool in Geophysics 2006**But first, we start with**• Brief review of general reactive transport formalism**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach • Blocking • Issues • Return to macroscale: multisite/population**Gone to mathcad**• Some analytical solutions - hope it runs • Just transport • Irreversible filtration no dispersion • Reversible filtration no dispersion • (Dispersion included by superposition.)**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview of colloids in hydrogeology • Processes catwalk • Classical approach • Blocking • Issues • Return to macroscale: multisite/population**1. Introduction - Background**Particle Sizes 10-5 10-3 10-2 10-10 10-9 10-8 10-7 10-6 10-4 (diameter, m) 1 nm 1 Å 1 cm 1 mm 1 mm Soils Sand Gravel Clay Silt Microorganisms Bacteria Viruses Protozoa Red blood cell Blood cells White blood cell Atoms, molecules Atoms Molecules Macromolecules Colloids Suspended particles Depth-filtration range Electron microscope Human eye Light microscope**Problems Involving Particle Transport through Porous Media**in Environmental and Health Systems • Water treatment system • Deep Bed Filtration (DBF) • Membrane-based filtration • Transport of pollutants in aquifers • Colloidal particle transport1 • Colloid-facilitated contaminant transport2 • Transport of microorganisms • Pathogen transport in groundwater • Bioremediation of aquifers • … • Ryan, J.N., and M. Elimelech. 1996. Colloids Surf. A, 107:1–56. • de Jonge, Kjaergaard, Moldrup. 2004. Vadose Zone Journal, 3:321–325**…and some more**• In situ bioremediation • transport of bacteria to contaminants1 • excessive attachment to aquifer grains – biofouling • Bacteria-facilitated contaminant transport (e.g.,DDT2) • Clinical settings • Blood cell filtration • Bacteria and viruses filtration • Ginn et al., Advances in Water Resources, 2002,25, 1017-1042. • Lindqvist & Enfield. 1992. Appl. Environ. Microbiol, 58: 2211-2218.**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach • Blocking • Issues • Return to macroscale: multisite/population**Processes in colloid-surface interaction**• Actual colloid, • Inertia in (arbitrary) velocity field • Torque, drag due to nonuniform flow • Diffusion, • hydrodynamic retardation/lubrication • Effective increase in viscosity near surface • Electrostatic (dynamic) interaction • DLVO (=LvdW + doublelayer model electrostatics) • Buoyancy/gravitational force**Overview**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach – “Colloid filtration theory” and some Details • Blocking • Issues • Return to macroscale: multisite/population**Classical take on Processes in colloid-surface interaction**• Inert, Spherical colloid to Sphere (flat) • Inertia in (Stokes) velocity field • Torque, drag due to nonuniform flow • approximated • Diffusion (superposed) • hydrodynamic retardation/lubrication • Electrostatic (dynamic) interaction • DLVO (=LvdW + doublelayer model electrostatics ) • Buoyancy/gravitational force added • So flow must be downward**Forces And Torques – RT model**Trajectory Analysis Smoluchowski-Levich Solution (particle has finite diameter) (particle diameter = 0) TD TD FG FI FvdW FI FBR FD FD h = + FI = inertial force due to Stokes flow FD = drag force due to Stokes flow TD = drag torque due to Stokes flow FBR = random Brownian force FB FI = inertial force due to Stokes flow* FD = drag force due to Stokes flow* TD = drag torque due to Stokes flow* FG = gravitational force FB = buoyancy force FvdW = van der Waals force *with corrections near surface SURFACE**Single collector efficiency**Filtration coefficient First-order deposition rate Classical CFT :Happel sphere-in-cell • Clean-bed “Filtration Theory” • Single “collector” represents a solid phase grain. A fraction h of the particles are brought to surface of the collector by the mechanisms of Brownian diffusion, Interception and/or Gravitational sedimentation. • A fraction a of the particles that reach the collector surface attach to the surface (electrostatic and ionic strength) • The single collector efficiency is then “scaled up” to a macroscopic filtration coefficient, which can be related to first-order attachment rate of the particles to the solid phase of the medium.**Bulk “kf” by classical filtration theory**First-order removal Rate = filter coefficient * porewater velocity => two-step process • n porosity • C aqueous phase concentration of colloid suspension • fc flux of C • U groundwater (Darcy) specific flux • a fraction of colloids encountering solid surface that stick (empirical2,3) • fraction of aqueous colloids that encounter solid surface (modeled1,3-6) 1. Rajagoplan & Tien. 1976. AIChE J. 22: 523-533. 2. Harvey & Garabedian. 1991. ES&T 25: 178-185. 3. Logan et al. 1995. J. Environ. Eng. 121: 869-873. 3. Nelson & Ginn. 2001 Langmuir 17: 5636-5645 4. Tufenkji & Elimelech. 2004 ES&T 38: 529-536. 5. Nelson & Ginn. 2005Langmuir21: 2173-2184**Details1:Happel sphere-in-cell model2**A1 A2 • Happel sphere-in-cell is porous medium • Stokes’ flow field • h calculated via trajectory analysis1 • Additive decomposition • h=hI+hG+hD • Initial point of limiting trajectory • h = A1/A2 = sin2qs • Rajagoplan & Tien. 1976. AIChE J. 22: 523-533. • Happel. 1958. AIChE J. 4: 197-201.**Detail: Basic solution (analytical) due to Rajagopalan &**Tien (1976) • Hydrodynamic retardation effect = the increased drag force a particle experiences as it approaches a surface. • a deviation from Stokes’ law • Hydrodynamic correction factors • Particle velocity expressions gives: • where frt, frm, s1, s2, and s3 are the drag correction factors. Interception by boundary condition Sedimentation group London van der Waals group**Detail: h vs. a**• irreversible adsorption constant, kirr = f(a,h) • h = fraction of colloids contacting solid phase, calculated a priori from RT model • a = fraction of colloids contacting solid phase that stick, treated as a calibration parameter accounting for all forces and mechanisms not considered in calculation of h Role of electrostatic forces : aside**Detail: Surface Forces in CFT – DLVO**• RT model uses DLVO theory for surface interaction forces: potential = van der Waals + double layer • Theory predicts negligible collection when repulsive surface interaction exists RT model neglects double layer force. attractive repulsive for like charges**Detail: Surface Forces in CFT – DLVO**• RT model uses DLVO theory for surface interaction forces: potential = van der Waals + double layer • Theory predicts negligible collection when repulsive surface interaction exists RT model neglects double layer force. • Thus, double layer force implicit in a. attractive repulsive for like charges**Highlights of Formulae for h**• Yao (1971) • hydrodynamic retardation and van der Waals force not included • Rajagopalan and Tien (1976) • deterministic trajectory analysis • torque correction factors • Brownian h added on separately from Eulerian analysis • Tufenkji and Elimelech (2004) • convective-diffusion equation solution • influence of van der Waals force and hydrodynamic retardation on diffusion • Diffusion, interception, & sedimentation considered additive • Nelson and Ginn (2005) • Particle tracking in Happel cell – all forces together**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach – “Colloid filtration theory” and some Details • Blocking • Issues • Return to macroscale: multisite/population**Dynamic surface blocking (ME)**• initial deposition rate (kinetics) • later, when deposition rate drops due to surface coverage (dynamics) • retained particles block sites, B is the dynamic blocking function (misnomer).**B's**• B = fraction of particle-surface collisions that involve open seats (cake walk). • Random Sequential Adsorption • Power series in S, for spherical geometry • Langmuirian Dynamic Blocking**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach – “Colloid filtration theory” and some Details • Blocking • Issues • Return to macroscale: multisite/population**Issues**• CFT coarse idealized model • Chem/env. Engineering, not natural p.m. • Biofilms, organic matter, asperities, heterogeneity (gsd, psd, surface area, electrostatic (dynamic), transience, flow reversal, temperature, etc. • Reversibility ??? • CFT good for trend prediction • Attachment goes up with colloid size, gw velocity, ionic strength, etc. • Ultimately need equs for bulk media • Lab • field**Outline**• General reactive transport intro • Multicomponent/two-phase/multireaction • colloid filtration “Miller lite” • Stop and smell the characteristic plane - mcad • Colloid Filtration “Guiness” • Overview • Processes catwalk • Classical approach – “Colloid filtration theory” and some Details • Blocking • Issues • Return to macroscale: See the data !**Field/Lab observations**• Microbes 1,2,3 and viruses 4,5 first showed apparent multipopulation rates due to decreased attachment with scale • Sticky bugs leave early • Readily explained by subpopulations • Some suggest geochemical “heterogeneity” • Recent surprize is that inert monotype, monosize and polysize colloids exhibit same6 • Albinger et al., FEMS Microbio Ltr., 124:321 (1994) • Ginn et al., Advances in Water Resources, 25:1017 (2002). • DeFlaun et al., FEMS Microbio Ltr., 20:473 (1997) • Redman et al., EST 35:1798 (2001); Schijven et al., WRR 35:1101 (1999) • Bales et al., WRR 33:639 (1997) • Li et al., EST 38:5616 (2004); Tufenkji and Elim. Langmuir 21:841 (2005)Yoon et al., WRR June 2006**Ability-based modeling (because we can)**• BTCs (first) exhibit long flat tails • Two-site, multisite model1 (google “patchwise”) • Two-population, multipop’n model2 (UAz, Arnold/Baygents) • Can’t tell the difference • Profiles (recently) are steeper than expected • Multipopulation works, not multisite (Li et al in 2), 3 • This is the location of the front in practice • Upscaling • Alternative explanations • E.g., Sun et al., WRR 37:209 (2001); “patchwise heterogeneity”, CXTFIT ease of use (sorta) • E.g., Redman et al., EST 35:1798 (2001); Li et al. EST 38:5616 (2004) • Johnson and Li, Langmuir 21:10895 (2005); Comment/Reply**Research Needs (at least)**• Formal upscaling • Forces complex but well understood • Approximations tested • Analytical results (Smoluchowski-Levitch1) • Alternative explanations • C<-> S -> S’ surface transformations 2 • Mainly bacteria; need RTD for attachment events • Physical straining of larger sizes (a pop’n model)3 • Reentrainment4 • Contact (CFT) and surface (multipopn) filtration5 • For CFT/Happel cell without interception or sfc forces (LvdW =-hyd. Retardation) • Davros & van de Ven JCIS 93:576 (1983); Meinders et al. JCIS 152:265 (1992); Johnson et al. WRR 31: 2649 (1995); Ginn WRR 36:2895 (2000) • Bradford et al WRR 38:1327 (2002); Bradford et al. EST 37:2242 (2003) • Grolimund et al WRR 37:571 (2001) 5. Yoon et al. WRR June 2006**Appendix: DNS Approach**• Langevin equation of motion • Happel sphere-in-cell • Contemporaneous accounting of all forces • Solution per colloid • Calculating h • Monte carlo colloidal release per qs => • P(qs) frequency of attachment per qs • h as an expectation over P(qs)**Langevin Equation**• Deterministic and Brownian displacements are combined per time step: • mp is the particle mass, u is the particle velocity vector, Fh is the hydrodynamic force vector, Fe is the external force vector, and Fb is the random Brownian force vector. • All three components of random displacement must be modeled in the axisymmetric (3D 2D) flow field.**Solution**• R = 3D displacement, • udet = deterministic velocity vector • n =3 N(0,1), • sR = standard deviations of Brownian displacements. • negligible particle inertia assumed • Dt >> tB (Kanaoka et al., 1983) • tB particle’s momentum relaxation time (=mp/6pmap). • Thus, tB << Dt <tu • tu is the time increment at which udet is considered constant.**Highlights of numerical solution**• Stokes’ flow in two-dimensions • R&T (1976) hydrodynamic drag correction factors1 • Brownian diffusion algorithm of Kanaoka et al. (1983)2 for diffusive aerosols • Coordinate transformation to 2D model • Brenner, H., Chem. Eng. Sci. 1961, 16, 242-251; Dahneke, B.E., J. Colloid Interface Sci., 1974, 48, 520-522. • Kanaoka, C.; Emi, H.; Tanthapanichakoon, W., AIChE J., 1983, 29, 895-902.**Coordinates for diffusion**• The Happel model: 3-D -> 2-D polar coordinates • convert 3-D Brownian Cartesian displacement to spherical, to polar • y,z, contribute to angular displacements • And thus to r**Calculating h**• qS starting angle of a colloid • Pc(qS) frequency of contact with the collector. • reduces to classical equation when deterministic (e.g., when Pc(qS) equals one for all qS<qLT and zero for all qS > qLT). • task of stochastic trajectory analysis for h is to find Pc(qS).**Colloid transport and Colloid Filtration Theory**• Classical approach • Issues • Direct numerical simulation: • Approach • Examples, Convergence, Testing • Results • Blocking - pages from Elimelech's site • Conclusions**Convergence to deterministic trajectory analysis of**Rajagopalan and Tien (when diffusion is neglected), Parameters: e = 0.2, as = 50 mm, ap = 0.1 mm, and U = 3.4375 * 10-4 m/s. The approximate analytical solution is h = 1.5 NR2g2AS (Rajagopalan and Tien, 1976).**Convergence of stochastic simulations for**Smoluchowski-Levich approximation. Parameters: ap = 0.1 mm, as = 163.5 mm, e = 0.372, U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec, T = 298 K.**Colloid transport and Colloid Filtration Theory**• Classical approach • Issues • Direct numerical simulation: • Approach • Convergence • Results • Smoluchowski-Levitch approximation • General case • Blocking - pages from Elimelech's site • Conclusions**Testing comparison to the Smoluchowski-Levich**approximation (external forces, interception neglected). h m Parameters: as = 163.5 mm, e = 0.372, U = 3.4375*10-4 m/sec, m = 8.9*10-4 kg*m/sec, T = 298 K, Dt = 1 ms, N = 6000.**Comparison of h calculations**R&T (1976) X N&G - - - T&E (2004) o N&G Additive R&T (1976) deterministic N&G deterministic h**Conclusions**• Lagrangean analysis is viable tool with modern computers • Stochastic trajectory analysis suggests diffusion and sedimentation may not be additive • More realistic “unit cell” models could be used • Lagrangean approach allows for arbitrary interaction potentials • Chemical (mineralogical, patchwise) heterogeneity • Exocellular polymeric substances in bacteria • Polymer bridging, hysteretic force potentials**Parameter**Value Collector radius, as 163.5 mm Porosity, e 0.372 Approach velocity, U 3.4375 * 10-4 sec Fluid viscosity, m 8.9 * 10-4 kg·m / sec Hamaker constant, H 10-20 J Bacterial density, rp 1070 kg / m3 Fluid density, rf 997 kg / m3 Absolute temperature, T 298 K Time step, Dt 1 ms Number of realizations, N 6000 Parameters used in stochastic trajectory simulations.**Modification of CFT to Account for EPS**• Distribution of polymer lengths on the cell surface • Repulsion modeled by steric force, Fst(h)1,2 depends on polymer density and brush length • If sufficient polymers contact collector, cell attaches depends on polymer density, length, and adhesion forces Hypothetical cell (drawn to scale) C O L L E C T O R KT2442 h 0.695 mm mean polymer length = 160 nm 1. de Gennes. 1987. Adv. Colloid Interface Sci. 27: 189-209. 2. Camesano & Logan. 2000. Environ. Sci. Technol. 34: 3354-3362.**Steric repulsive force**Polymer bridging Interception Sedimentation Brownian motion London van der Waals attractive force Hydrodynamic retardation effect Theoretical Sticking EfficiencyNumerical Calculation of Trajectories Incorporation of Brownian motion and polymer interactions into trajectory analysis allows for computation of a theoretical sticking efficiency.