Poroelasticity pp 703-773 | Cite as


  • Alexander H.-D. Cheng
Part of the Theory and Applications of Transport in Porous Media book series (TATP, volume 27)


The poroelasticity theory presented in Chaps.  2 through 10 models the force and energy interaction of mechanical origin between two material phases, a solid and a fluid. The thermal energy and force field are introduced in Chap.  11 In the physical world, there exist other types of energy and forces, such as those of electrical, magnetic, and chemical origins, and their coupling, such as the electromechanical (piezoelectric), electromagnetic, electrochemical, and magnetohydrodynamic forces. The simultaneous presence of these multiple physical phenomena, particularly their interactions, is known as multiphysics.Depending on the engineering applications on hand and practical considerations, not all forces are present or significant, and need to be modeled. In geoscience, geophysical, and geomechanical applications, it is generally recognized that four processes, thermal (T), hydraulic (H), mechanical (M), and chemical (C), known as THMC processes (Taron J, Elsworth D, Thermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal reservoirs. Int J Rock Mech Min Sci 46(5):855–864, 2009; Taron J, Elsworth D, Min K-B (2009) Numerical simulation of thermal-hydrologic-mechanical-chemical processes in deformable, fractured porous media. Int J Rock Mech Min Sci 46(5):842–854; Tsang C-F (2009) Introductory editorial to the special issue on the DECOVALEX-THMC project. Environ Geol 57(6):1217–1219), are more important. The poroelasticity theory considers the hydraulic and mechanical coupling (HM). The porothermoelasticity theory introduces the additional thermal coupling, and is a THM theory. In this chapter, we shall examine the THMC processes by including the chemical energy and force field and introduce the porothermochemoelasticity theory; though, for simplicity, we shall refer to it as the porochemoelasticity theory in this book.Many geotechnical, biological, and synthetic porous media are chemically active, and exhibit swelling or shrinking behaviors when brought in contact with aqueous solutions. This phenomenon, observed in clays (Bennethum et al., Transp Porous Media 39(2):187–225, 2000; Low, Langmuir 3(1):18–25, 1987; Sposito et al., Proc Natl Acad Sci 96(7):3358–3364, 1999; van Olphen, An introduction to clay colloid chemistry, 2nd edn. Wiley, New York, 318pp, 1977), shales (Ghassemi, Diek, J Pet Sci Eng 38(3–4):199–212, 2003; Nguyen, Abousleiman, Anais Da Academia Brasileira De Ciencias 82(1):195–222, 2010; Sherwood, Proc R Soc A–Math Phys Eng Sci 440(1909):365–377, 1993; Sherwood, Langmuir 10(7):2480–2486, 1994), cartilage (Gu et al., J Biomech Eng ASME 120(2):169–180, 1998; Lai et al. J Biomech Eng ASME 113(3):245–258, 1991; Wilson et al., J Biomech Eng 127(1):158–165, 2005) and gels (Hong et al., Int J Solids Struct 46(17):3282–3289, 2009; Marcombe et al., Soft Matter 6(4):784–793, 2010), is caused by electric charges fixed to the solid, counteracted by corresponding charges in the fluid. These charges result in a variety of features, including swelling, chemico-osmosis, electro-osmosis, streaming potentials, streaming currents, and electrophoresis (Mitchell, Soga, Fundamentals of soil behavior, 3rd edn. Wiley, New York, 592pp, 2005; Sachs, Grodzinsky, Physicochem Hydrodyn 11(4):585–614, 1989).In biological and medical applications, swelling behavior is observed in cartilage, glycogalyx, cell, and skin (van Meerveld et al., Transp Porous Media 50(1–2):111–126, 2003). Swelling of shales is of major problem for petroleum engineering. According to Steiger and Leung (SPE Drill Eng 7(3):181–185, 1992), shales make up more than 75 % of drilled formations and cause at least 90 % of wellbore-stability problems. If shale surrounding the wellbore swells, the wellbore diameter will be reduced, and the drill string and drill bit can become trapped, costing both time and money. Alternatively, if swollen shale disintegrates, motion of the drill string will be hindered by the soft, roughened walls of the well (Mody, Hale, J Pet Technol 45(11):1093–1101, 1993; Sherwood, Bailey, Proc R Soc Lond Ser A–Math Phys Eng Sci 444(1920):161–184, 1994). Although the use of oil based drilling mud can reduce the chemical effect, its use is much restricted due to the environmental concern of its disposal (van Oort et al., SPE Drill Complet 11(3):137–146, 1996). Clay is used as liner and buffer for containment of landfill leachate and underground burial of nuclear or hazardous wastes in environmental geotechnology. The integrity of the barrier can be much affected by the swelling or shrinkage of the material.Shales are fine-grained sedimentary rocks consisting of clay, silt, and mud. Clay minerals are basically crystalline and their properties are determined by the atomic structure of their crystals. When these minerals are exposed to a fluid with different physic-chemical properties, the microscopic changes take place, which manifest in macroscopic scale as swelling (or shrinkage) and an apparent pressure as osmotic pressure. Swelling mostly occurs in smectites and particularly montmorillonite, due to its expanding lattice and finely laminated structure, subject to cation exchange and water content.In the following, we shall examine the modeling of the chemical effects through the concept of a chemical potential. The constitutive relations and transport laws are constructed to form governing equations that allow the mathematical solution of the various applications.


  1. 1.
    Achanta S, Cushman JH, Okos MR (1994) On multicomponent, multiphase thermomechanics with interfaces. Int J Eng Sci 32(11):1717–1738CrossRefzbMATHGoogle Scholar
  2. 2.
    Alonso EE, Vaunat J, Gens A (1999) Modelling the mechanical behaviour of expansive clays. Eng Geol 54(1–2):173–183CrossRefGoogle Scholar
  3. 3.
    Baierlein R (2001) The elusive chemical potential. Am J Phys 69(4):423–434CrossRefGoogle Scholar
  4. 4.
    Bear J, Cheng AHD (2010) Modeling groundwater flow and contaminant transport. Springer, Dordrecht/London, 834ppCrossRefzbMATHGoogle Scholar
  5. 5.
    Bennethum L, Murad M, Cushman J (2000) Macroscale thermodynamics and the chemical potential for swelling porous media. Transp Porous Media 39(2):187–225MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bronick CJ, Lal R (2005) Soil structure and management: a review. Geoderma 124(1–2):3–22CrossRefGoogle Scholar
  7. 7.
    Das BM (1997) Advanced soil mechanics, 2nd edn. Taylor & Francis, London/New York, 457ppGoogle Scholar
  8. 8.
    Dormieux L, Lemarchand E, Coussy O (2003) Macroscopic and micromechanical approaches to the modelling of the osmotic swelling in clays. Transp Porous Media 50(1–2):75–91CrossRefGoogle Scholar
  9. 9.
    Ekbote S, Abousleiman Y (2006) Porochemoelastic solution for an inclined borehole in a transversely isotropic formation. J Eng Mech ASCE 132(7):754–763CrossRefGoogle Scholar
  10. 10.
    Epstein N (1989) On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem Eng Sci 44(3):777–779CrossRefGoogle Scholar
  11. 11.
    Fritz SJ (1986) Ideality of clay membranes in osmotic processes–a review. Clays Clay Miner 34(2):214–223CrossRefGoogle Scholar
  12. 12.
    Gajo A, Loret B, Hueckel T (2002) Electro-chemo-mechanical couplings in saturated porous media: elastic-plastic behaviour of heteroionic expansive clays. Int J Solids Struct 39(16):4327–4362CrossRefzbMATHGoogle Scholar
  13. 13.
    Ghassemi A, Diek A (2002) Porothermoelasticity for swelling shales. J Pet Sci Eng 34(1–4):123–135CrossRefGoogle Scholar
  14. 14.
    Ghassemi A, Diek A (2003) Linear chemo-poroelasticity for swelling shales: theory and application. J Pet Sci Eng 38(3–4):199–212CrossRefGoogle Scholar
  15. 15.
    Ghassemi A, Tao Q, Diek A (2009) Influence of coupled chemo-poro-thermoelastic processes on pore pressure and stress distributions around a wellbore in swelling shale. J Pet Sci Eng 67(1–2):57–64CrossRefGoogle Scholar
  16. 16.
    Gibbs JW (1873) Graphical methods in the thermodynamics of fluids. Trans Conn Acad Arts Sci 2:309–342zbMATHGoogle Scholar
  17. 17.
    Gibbs JW (1873) A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans Conn Acad Arts Sci 2:382–404zbMATHGoogle Scholar
  18. 18.
    Gibbs JW (1876/1878) On the equilibrium of heterogeneous substances. Trans Conn Acad Arts Sci 3:108–248, 343–524Google Scholar
  19. 19.
    Gu WY, Lai WM, Mow VC (1998) A mixture theory for charged hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. J Biomech Eng ASME 120(2):169–180CrossRefGoogle Scholar
  20. 20.
    Heidug WK, Wong SW (1996) Hydration swelling of water-absorbing rocks: a constitutive model. Int J Numer Anal Methods Geomech 20(6):403–430CrossRefzbMATHGoogle Scholar
  21. 21.
    Hensen EJM, Smit B (2002) Why clays swell. J Phys Chem B 106(49):12664–12667CrossRefGoogle Scholar
  22. 22.
    Hong W, Liu Z, Suo Z (2009) Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int J Solids Struct 46(17):3282–3289CrossRefzbMATHGoogle Scholar
  23. 23.
    Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35(8):793–802CrossRefzbMATHGoogle Scholar
  24. 24.
    Karaborni S, Smit B, Heidug W, Urai J, van Oort E (1996) The swelling of clays: molecular simulations of the hydration of montmorillonite. Science 271(5252):1102–1104CrossRefGoogle Scholar
  25. 25.
    Katchalsky A, Curran PF (1967) Nonequilibrium thermodynamics in biophysics. Harvard University Press, Cambridge, 248ppGoogle Scholar
  26. 26.
    Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng ASME 113(3):245–258CrossRefGoogle Scholar
  27. 27.
    Loret B, Hueckel T, Gajo A (2002) Chemo-mechanical coupling in saturated porous media: elastic-plastic behaviour of homoionic expansive clays. Int J Solids Struct 39(10):2773–2806CrossRefzbMATHGoogle Scholar
  28. 28.
    Low PF (1980) The swelling of clay. 2. Montmorillonites. Soil Sci Soc Am J 44(4):667–676CrossRefGoogle Scholar
  29. 29.
    Low PF (1987) Structural component of the swelling pressure of clays. Langmuir 3(1):18–25CrossRefGoogle Scholar
  30. 30.
    Low PF, Margheim JF (1979) Swelling of clay. 1. Basic concepts and empirical equations. Soil Sci Soc Am J 43(3):473–481CrossRefGoogle Scholar
  31. 31.
    Madsen FT, Müller-Vonmoos M (1989) The swelling behaviour of clays. Appl Clay Sci 4(2):143–156CrossRefGoogle Scholar
  32. 32.
    Marcombe R, Cai S, Hong W, Zhao X, Lapusta Y, Suo Z (2010) A theory of constrained swelling of a pH-sensitive hydrogel. Soft Matter 6(4):784–793CrossRefGoogle Scholar
  33. 33.
    McBride MB (1997) A critique of diffuse double layer models applied to colloid and surface chemistry. Clays Clay Miner 45(4):598–608CrossRefGoogle Scholar
  34. 34.
    Mitchell JK, Soga K (2005) Fundamentals of soil behavior, 3rd edn. Wiley, New York, 592ppGoogle Scholar
  35. 35.
    Mody FK, Hale AH (1993) Borehole-stability model to couple the mechanics and chemistry of drilling-fluid/shale interactions. J Pet Technol 45(11):1093–1101CrossRefGoogle Scholar
  36. 36.
    Moyne C, Murad MA (2002) Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure. Int J Solids Struct 39(25):6159–6190CrossRefzbMATHGoogle Scholar
  37. 37.
    Murad MA, Cushman JH (2000) Thermomechanical theories for swelling porous media with microstructure. Int J Eng Sci 38(5):517–564MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nguyen V, Abousleiman Y (2005) Analysis of Biot’s poroelastic parameters in chemically active porous media. In: Abousleiman YN, Cheng AHD, Ulm F-J (eds) Poromechanics III—Biot Centennial (1905–2005), Proceedings of the 3rd Biot conference on poromechanics. Balkema, Leiden, 695–703Google Scholar
  39. 39.
    Nguyen VX, Abousleiman YN (2010) Incorporating electrokinetic effects in the porochemoelastic inclined wellbore formulation and solution. Anais Da Academia Brasileira De Ciencias 82(1):195–222CrossRefGoogle Scholar
  40. 40.
    Rogers DW (1962) Deriving the Gibbs-Duhem equation. J Chem Educ 39(10):527–528CrossRefGoogle Scholar
  41. 41.
    Roshan H, Oeser M (2012) A non-isothermal constitutive model for chemically active elastoplastic rocks. Rock Mech Rock Eng 45(3):361–374CrossRefGoogle Scholar
  42. 42.
    Roshan H, Rahman SS (2011) A fully coupled chemo-poroelastic analysis of pore pressure and stress distribution around a wellbore in water active rocks. Rock Mech Rock Eng 44(2):199–210CrossRefGoogle Scholar
  43. 43.
    Sachs JR, Grodzinsky AJ (1989) An electromechanically coupled poroelastic medium driven by an applied electric-current—surface detection of bulk material properties. Physicochem Hydrodyn 11(4):585–614Google Scholar
  44. 44.
    Sarout J, Detournay E (2011) Chemoporoelastic analysis and experimental validation of the pore pressure transmission test for reactive shales. Int J Rock Mech Min Sci 48:759–772CrossRefGoogle Scholar
  45. 45.
    Sherwood JD (1993) Biot poroelasticity of a chemically active shale. Proc R Soc A–Math Phys Eng Sci 440(1909):365–377CrossRefzbMATHGoogle Scholar
  46. 46.
    Sherwood JD (1994) A model for the flow of water and ions into swelling shale. Langmuir 10(7):2480–2486CrossRefGoogle Scholar
  47. 47.
    Sherwood JD, Bailey L (1994) Swelling of shale around a cylindrical wellbore. Proc R Soc Lond Ser A–Math Phys Eng Sci 444(1920):161–184CrossRefGoogle Scholar
  48. 48.
    Sherwood JD, Craster B (2000) Transport of water and ions through a clay membrane. J Colloid Interface Sci 230(2):349–358CrossRefGoogle Scholar
  49. 49.
    Sposito G, Skipper NT, Sutton R, Park SH, Soper AK, Greathouse JA (1999) Surface geochemistry of the clay minerals. Proc Natl Acad Sci 96(7):3358–3364CrossRefGoogle Scholar
  50. 50.
    Steiger RP, Leung PK (1992) Quantitative determination of the mechanical properties of shales. SPE Drill Eng 7(3):181–185CrossRefGoogle Scholar
  51. 51.
    Taron J, Elsworth D (2009) Thermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal reservoirs. Int J Rock Mech Min Sci 46(5):855–864CrossRefGoogle Scholar
  52. 52.
    Taron J, Elsworth D, Min K-B (2009) Numerical simulation of thermal-hydrologic-mechanical-chemical processes in deformable, fractured porous media. Int J Rock Mech Min Sci 46(5):842–854CrossRefGoogle Scholar
  53. 53.
    Tsang C-F (2009) Introductory editorial to the special issue on the DECOVALEX-THMC project. Environ Geol 57(6):1217–1219CrossRefGoogle Scholar
  54. 54.
    van Meerveld J, Molenaar MM, Huyghe JM, Baaijens FPT (2003) Analytical solution of compression, free swelling and electrical loading of saturated charged porous media. Transp Porous Media 50(1–2):111–126MathSciNetCrossRefGoogle Scholar
  55. 55.
    van Olphen H (1977) An introduction to clay colloid chemistry, 2nd edn. Wiley, New York, 318ppGoogle Scholar
  56. 56.
    van Oort E (2003) On the physical and chemical stability of shales. J Pet Sci Eng 38(3–4):213–235CrossRefGoogle Scholar
  57. 57.
    van Oort E, Hale AH, Mody FK, Roy S (1996) Transport in shales and the design of improved water-based shale drilling fluids. SPE Drill Complet 11(3):137–146CrossRefGoogle Scholar
  58. 58.
    van’t Hoff JH (1901) Osmotic pressure and chemical equilibrium, Nobel Prize LectureGoogle Scholar
  59. 59.
    Velde B (1992) Introduction to clay minerals: chemistry, origins, uses and environmental significance. Springer, Dordrecht, 198ppCrossRefGoogle Scholar
  60. 60.
    Viani BE, Low PF, Roth CB (1983) Direct measurement of the relation between interlayer force and interlayer distance in the swelling of montmorillonite. J Colloid Interface Sci 96(1):229–244CrossRefGoogle Scholar
  61. 61.
    Wheeler SJ, Sivakumar V (1995) An elasto-plastic critical state framework for unsaturated soil. Géotechnique 45(1):35–53CrossRefGoogle Scholar
  62. 62.
    Wilson W, van Donkelaar CC, Huyghe JM (2005) A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues. J Biomech Eng 127(1):158–165CrossRefGoogle Scholar
  63. 63.
    Yu M, Chenevert ME, Sharma MM (2003) Chemical-mechanical wellbore instability model for shales: accounting for solute diffusion. J Pet Sci Eng 38(3–4):131–143CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Alexander H.-D. Cheng
    • 1
  1. 1.University of MississippiOxfordUSA

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