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  1. Notes on Colloid transport and filtration in saturated porous media Tim Ginn, Patricia Culligan, Kirk Nelson Purdue Summerschool in Geophysics 2006

  2. But first, we start with • Brief review of general reactive transport formalism

  3. 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

  4. Gone to mathcad • Some analytical solutions - hope it runs • Just transport • Irreversible filtration no dispersion • Reversible filtration no dispersion • (Dispersion included by superposition.)

  5. 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

  6. 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

  7. 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

  8. …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.

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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.

  15. 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

  16. 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.

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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).

  24. 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

  25. 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

  26. 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

  27. 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 !

  28. 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

  29. 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

  30. 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

  31. 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)

  32. 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.

  33. 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.

  34. 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.

  35. 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

  36. 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).

  37. Colloid transport and Colloid Filtration Theory • Classical approach • Issues • Direct numerical simulation: • Approach • Examples, Convergence, Testing • Results • Blocking - pages from Elimelech's site • Conclusions

  38. Example Brownian Trajectory

  39. P(qs)

  40. Convergence of a trajectory - 50K realizations

  41. 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).

  42. 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.

  43. 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

  44. 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.

  45. 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

  46. 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

  47. 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.

  48. 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.

  49. 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.