Acta Geotechnica

, 3:153

A three-scale model for pH-dependent steady flows in 1:1 clays

  • Sidarta Araújo de Lima
  • Márcio A. Murad
  • Christian Moyne
  • Didier Stemmelen
Research Paper


A new three-scale model to describe the coupling between electro-chemistry and hydrodynamics in non-swelling kaolinite clays in steady-state conditions is proposed. The medium is characterized by three separate nano-micro and macroscopic length scales. At the pore (micro)-scale the portrait of the clay consists of micro-pores saturated by an aqueous solution containing four monovalent ions (Na+, H+, Cl, OH) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst–Planck equations and the influence of the double layers upon the hydrodynamics is modeled by a slip boundary condition in the tangential velocity governed by the Stokes problem. To capture the correct form of the interface condition we invoke the nanoscopic modeling of the thin electrical double layer based on Poisson–Boltzmann problem with varying surface charge density ruled by the protonation/deprotonation reactions which occur at the surface of the particles. The two-scale nano/micro model is homogenized to the macroscale leading to a precise derivation of effective governing equations. The macroscopic model is discretized by the finite volume method and applied to numerically simulate desalination of a clay sample induced by an external electrical field generated by the placement of electrodes. Numerical results indicate strong pH-dependence of the electrokinetics.


Electrical double layer Electrokinetics Homogenization Kaolinite Nernst–Planck Poisson–Boltzmann Protonation reactions 


  1. 1.
    Acar YB, Gale RJ, Alshawabkeh AN, Marks RE, Puppala S, Bricka M, Parker R (1995) Electrokinetic remediation: basic and thechnology status. J Hazard Mater 40:117–137CrossRefGoogle Scholar
  2. 2.
    Acar YB, Alshawabkeh AN (1996) Electrokinetic remediation: I. Pilot-scale tests with lead spiked Kaolinite. J Geotech Eng 122:173–185CrossRefGoogle Scholar
  3. 3.
    Acar YB, Alshawabkeh AN (1993) Principles of electrokinetics remediation. Environ Sci Technol 27:2638–2647CrossRefGoogle Scholar
  4. 4.
    Achari G, Joshi RC, Bentley LR, Chatterji S (1999) Prediction of the hydraulic conductivity of clays using the electric double layer theory. Can Geotech J 36:783-792CrossRefGoogle Scholar
  5. 5.
    Alshawabkeh AN, Acar YB (1996) Electrokinetic remediation: theoretical model. J Geotech Eng 122:186–196CrossRefGoogle Scholar
  6. 6.
    Anandarajah A (1997) Influence of particle orientation on one-dimensional compression of montmorillonite. J Colloid Interface Sci 194:44–52CrossRefGoogle Scholar
  7. 7.
    Auriault JL (1991) Heterogeneous media: is an equivalent homogeneous description always possible?. Int J Eng Sci 29:785–795MATHCrossRefGoogle Scholar
  8. 8.
    Auriault JL, et Lewandowska J (1996) Diffusion/adsorption/advection macrotransport in soils. Eur J Mech A Solids 15:681–704MATHGoogle Scholar
  9. 9.
    Avena MJ, Mariscal MM, De Pauli CP (2003) Proton binding at clay surface in water. Appl Clay Sci 24:3–9CrossRefGoogle Scholar
  10. 10.
    Avena MJ, De Pauli CP (1996) Modeling the interfacial properties of an amorphous aluminosilicate dispersed in aqueous NaCl solution. Colloids Surface A Physicochem Eng Aspects 118:75–87CrossRefGoogle Scholar
  11. 11.
    Basu S, Shrarma MM (1997) An improved space-charge model for flow through charged microporous membranes. J Membr Sci 124:77–91CrossRefGoogle Scholar
  12. 12.
    Beddiar K, Fen-Chong T, Dupas A, Berthaud Y, Dangla P (2005) Role of pH in electro-osmosis: experimental study on NaCl–water saturated Kaolinite. Aus Transport in Porous Media 61:93–107CrossRefGoogle Scholar
  13. 13.
    Bolland MDA, Posner AM, Quirk JP (1976) Surface Charge on Kaolinites in aqueous suspension. Aus J Soil Res 14:197–216CrossRefGoogle Scholar
  14. 14.
    Brady PV, Cygan RT, Nagy KL (1996) Molecular control on Kaolinite surface charge. J Colloid Interface Sci 183:356–364CrossRefGoogle Scholar
  15. 15.
    Chi Ma, Eggleton RA (1999) Cation exchange capacity of Kaolinite. Clays Clay Miner 47(2):174–180CrossRefGoogle Scholar
  16. 16.
    Chorover J, Sposito G (1995) Surface charge characteristics of kaolinite tropical soils. J Geochim Cosmochim Acta 59:875–884CrossRefGoogle Scholar
  17. 17.
    Chung HI, Kang BH (1999) Lead removal from contaminated marine clay by electrokinetic soil decontamination. Eng Geol 53:139–150CrossRefGoogle Scholar
  18. 18.
    Dangla P, Chong TF, Gaulard F (2004) Modelling of pH-dependent electro-osmotic flows. C R Mecanique 332:915–920Google Scholar
  19. 19.
    Denaro AR (1971) Elementary electrochemistry. Butterworths, LondonGoogle Scholar
  20. 20.
    Dormieux L, Barboux P, Coussy O, Dangla P (1995) A macroscopic model of the swelling phenomenon of a saturated clay. Euro J Mech A/Solids 14(6):981–1004MATHGoogle Scholar
  21. 21.
    Edwards DA (1995) Charge transport through a spatially periodic porous medium: electrokinetic and convective dispersion phenomena. Phil Trans Roy Soc Lond A353:174–180CrossRefGoogle Scholar
  22. 22.
    Eykholt GR (1997) Development of pore pressures by nonuniform electroosmosis in clays. J Hazard Mater 55:171–186CrossRefGoogle Scholar
  23. 23.
    Fair JC, Osterlé JF (1971) Reverse electrodialysis in charged capillary membranes. J Chem Phys 54(8):3307–3316CrossRefGoogle Scholar
  24. 24.
    Gajo A, Loret B (2007) The mechanics of active clays circulated by salts, acids and bases. J Mech Phys Solids 55(8):1762–1801CrossRefGoogle Scholar
  25. 25.
    Ganor J, Cama J, Metz V (2003) Surface protonation data of Kaolinite-reevaluation based on dissolution experiments. J Colloid Interface Sci 264:67–75CrossRefGoogle Scholar
  26. 26.
    Grim RE (1968) Clay mineralogy. McGraw-Hill Book Company, New YorkGoogle Scholar
  27. 27.
    Guimaraes LD, Gens A, Olivella S (2007) Coupled thermo-hydro-mechanical and chemical analysis of expansive clay subjected to heating and hydration. Transport Porous Media 66(3):341–372CrossRefGoogle Scholar
  28. 28.
    de Groot SR, Mazur P (1969) Non-equilibrium thermodynamics. North-Holland, AmsterdamGoogle Scholar
  29. 29.
    Gross RJ, Osterlé JF (1968) Membrane transport characteristics of ultrafine capillaries. J Chem Phys 49(1):228–234CrossRefGoogle Scholar
  30. 30.
    Gupta AK, Coelho D, Adler PM (2007) Ionic transport in porous media for high zeta potentials. J Colloid Interface Sci 314(2):733–747CrossRefGoogle Scholar
  31. 31.
    Hlushkou D, Morgenstern AS, Tallarek U (2005) Numerical analysis of electroosmotic flow in dense regular and random arrays of impermeable nonconductiong spheres. Langmuir 21:6097–6112CrossRefGoogle Scholar
  32. 32.
    Hoch M, Weerasooriya M (2005) Modeling interactions at the tributyltin kaolinite interface. Chemosphere 59(5):743-752CrossRefGoogle Scholar
  33. 33.
    Huertas FJ, Chou L, Wollast R (1997) Mechanism of Kaolinite dissolution at room temperature an in pressure: Part 1. Surface speciation. Geochim Cosmochim Acta 62:417–431CrossRefGoogle Scholar
  34. 34.
    Hunter RJ (1994) Introduction to modern colloid science. Oxford University Press, OxfordGoogle Scholar
  35. 35.
    Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 25(8):793–802CrossRefGoogle Scholar
  36. 36.
    Ichikawa Y, Kawamura K, Fujii N, Kitayama K (2004) Microstructure and micro/macro-diffusion behavior of tritium in bentonite. Appl Clay Sci 26:75–90CrossRefGoogle Scholar
  37. 37.
    Ichikawa Y, Prayongphan S, Kawamura K, Kitayama K (2004) Water flow and diffusion problem in bentonite: molecular simulation and homogenization analysis . Elsevier Geo-Engineering Book Series 2:457–464Google Scholar
  38. 38.
    Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113:245–258CrossRefGoogle Scholar
  39. 39.
    Landau LD, Lifshitz EM (1960) Electrodynamics of continuous media. Pergamon Press, OxfordMATHGoogle Scholar
  40. 40.
    Lemaire T, Moyne C, Stemmelen D (2007) Modelling of electro-osmosis in clayey materials including pH effects. Phys Chem Earth 32:441–452Google Scholar
  41. 41.
    Leroy P, Revil A (2004) A triple-layer model of the surface electrochemical properties of clay minerals. J Colloid Interface Sci 270(2):371–380CrossRefGoogle Scholar
  42. 42.
    Li YL, Li RS (2000) The role of clay minerals and the effect of H+ ions on removal of heavy metal (Pb2+) from contaminated soils. Can Geotech J 37:296–307CrossRefGoogle Scholar
  43. 43.
    Looker JR, Carnie SL (2006) Homogenization of the ionic transport equations in periodic porous media. Transport Porous Media 65:107–131CrossRefMathSciNetGoogle Scholar
  44. 44.
    Loret B, Hueckel T, Gajo A (2002) Chemo-mechanical coupling in saturated porous media: elastic-plastic behavior of homoionic expansive clays. Int J Solids Struct 39:2773–2806MATHCrossRefGoogle Scholar
  45. 45.
    Lyklema J (1993) Fundamentals of colloid and interface science. Academic Press, LondonGoogle Scholar
  46. 46.
    Mitchell J (1993) Fundamentals of soil behavior. Wiley, New YorkGoogle Scholar
  47. 47.
    Moyne C, Murad M (2006) A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager’s reciprocity relations in expansive clays: I. Homogenization analysis. Transport Porous Media 62(3):333–380CrossRefMathSciNetGoogle Scholar
  48. 48.
    Moyne C, Murad M (2006) A two-scale model for coupled electro-chemo-mechanical phenomena and Onsager’s reciprocity relations in expansive clays: II. Computational validation. Transport Porous Media 63(1):13–56CrossRefGoogle Scholar
  49. 49.
    Murad M, Moyne C (2002) Micromechanical computational modeling of expansive porous media. C R Mecanique 330:865–870CrossRefMATHGoogle Scholar
  50. 50.
    Murad M, Moyne C (2008) A dual-porosity model for ionic solute transport in expansive clays. Comput Geosci. Online firstGoogle Scholar
  51. 51.
    Murray HH (1997) Applied clay mineralogy today and tomorrow. Clay Minerals 34:39–49CrossRefGoogle Scholar
  52. 52.
    Narasimhan B, Sri Ranjan R (2000) Electrokinetic barrier to prevent subsurface contaminant migration: theoretical model development and validation. J Cont Hydrol 42:1–17CrossRefGoogle Scholar
  53. 53.
    Newman JS (1973) Electrochemical systems. Prentice-Hall, Inc., Englewood CliffsGoogle Scholar
  54. 54.
    Van Olphen H (1977) An introduction to clay colloid chemistry: for clay technologists, geologists, and soil scientists. Wiley, New YorkGoogle Scholar
  55. 55.
    Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing Corporation, New YorkMATHGoogle Scholar
  56. 56.
    Revil A (1999) Ionic diffusivity, electrical conductivity, membrane and thermoelectric potentials in colloids and granular porous media: a Unified model. J Colloid Interface Sci 212:503–522CrossRefGoogle Scholar
  57. 57.
    Rosanne M, Paszkuta M, Adler PM (2006) Electrokinetic phenomena in saturated compact clays. J Colloid Interface Sci 297:353–364CrossRefGoogle Scholar
  58. 58.
    Rosanne M, Paszkuta M, Thovert JF, Adler PM (2004) Electro-osmotic coupling in compact clays. Geophys Res Lett 31(18):L18614.1–L18614.5CrossRefGoogle Scholar
  59. 59.
    Samson E, Marchand J, Robert J-L, Bournazel J-P (1999) Modelling ion diffusion mechanisms in porous media. Int J Numer Methods Eng 46:2043–2060MATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory, Lectures Notes in Physics. Springer, HeidelbergGoogle Scholar
  61. 61.
    Sasidhar V, Ruckenstein E (1981) Electrolyte osmosis through capillaries. J Colloid Interface Sci 82(2):439–457CrossRefGoogle Scholar
  62. 62.
    Schroth BK, Sposito G (1997) Surface charge properties of Kaolinite. Clays Clay Minerals 45(1):85–91CrossRefGoogle Scholar
  63. 63.
    Shang JQ, Lo KY (1997) Electrokinetic dewatering of a phosphate clay. J Hazard Mater 55:117–133CrossRefGoogle Scholar
  64. 64.
    Sherwood JD (1994) A model for the flow of water and ions into swelling shale. Langmuir 10:2480–2486CrossRefGoogle Scholar
  65. 65.
    Sposito G (1989) The chemistry of soils. Oxford University PressGoogle Scholar
  66. 66.
    Vane LM, Zang GM (1997) Effect of aqueous phase properties on clay particle zeta potential and electro-osmotic permeability: implications for electro-kinetic soil remediation process. J Hazard Mater 55:1–22CrossRefGoogle Scholar
  67. 67.
    Virkutyte J, Sillanpaa M, Latostenmaa P (2002) Electrokinetics soil remediation—critical overview. Sci Total Environ 289:97–121CrossRefGoogle Scholar
  68. 68.
    Yeung AT, Subbaraju D (1995) Fundamental formulation of electrokinetic extraction of contaminants from soil. Can Geotech J 32:569–583CrossRefGoogle Scholar
  69. 69.
    Yeung AT, Mitchell JK (1993) Coupled fluid, electrical and chemical flows in soil. Geotechnique 43(1):121–134CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Sidarta Araújo de Lima
    • 1
  • Márcio A. Murad
    • 1
  • Christian Moyne
    • 2
  • Didier Stemmelen
    • 2
  1. 1.Laboratório Nacional de ComputaçãoCientífica LNCC/MCTPetrópolisBrazil
  2. 2.LEMTA, Nancy-University, CNRSVandoeuvre les Nancy CedexFrance

Personalised recommendations