The Linear Theory of Anisotropic Poroelastic Solids

  • Florian K. Lehner
Part of the CISM Courses and Lectures book series (CISM, volume 533)


This chapter offers a comprehensive derivation of the constitutive equations of linear poroelasticity. A main purpose of this survey is to assist with the formulation of experimental strategies for the measurement of poroelastic constants. The complete set of linearized constitutive relations is phrased alternatively in terms of undrained bulk parameters or drained skeleton parameters. Displayed in the form of a mnemonic diagram, this can provide a rapid overview of the theory. The principal relationships between alternative sets of material parameters are tabulated, among them the well-known Gassmann or Brown-Korringa fluid substitution relations that are rederived here without any pore-scale considerations. The possibility of an isotropic unjacketed response is pointed out, which — if verified experimentally — will make for an interesting and practically useful special case of anisotropic poroelasticity.


Pore Pressure Bulk Modulus Linear Theory Solid Skeleton Pore Volume Fraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berge, P.A., and J.G. Berryman (1995). Realizability of negative pore compressibility in poroelastic composites. J. Appl. Mech. 62, 1053–1062.CrossRefGoogle Scholar
  2. Berryman, J.G. (1995). Mixture theories for rock properties. In: Rock Physics and Phase Relations. A Handbook of Physical Constants, edited by T.J. Ahrens, Am. Geophys. Union, Washington D.C., pp. 205–228.CrossRefGoogle Scholar
  3. Biot, M.A. (1941). General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164.zbMATHCrossRefGoogle Scholar
  4. Biot, M.A. (1955). Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Biot, M.A. (1956a). Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Biot, M.A. (1972). Theory of finite deformation of porous solids. Indiana Univ. Math. J. 21(7), 597–620.MathSciNetCrossRefGoogle Scholar
  7. Biot, M.A. (1973). Nonlinear and semilinear rheology of porous solids. J. Geophys. Res. 78, 4924–4937.CrossRefGoogle Scholar
  8. Biot, M.A., and D.G. Willis (1957). The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601.MathSciNetGoogle Scholar
  9. Brown, R.J.S., and J. Korringa (1975). On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid. Geophysics 40, 608–616.CrossRefGoogle Scholar
  10. Callen, H.B. (1960). Thermodynamics, John Wiley, New York.zbMATHGoogle Scholar
  11. Carcione, J.M., and U. Tinivella (2001). The seismic response to overpressure: a modelling study based on laboratory, well and seismic data. Geophysical Prospecting 49, 523–539.CrossRefGoogle Scholar
  12. Carroll, M.M. (1979). An effective stress law for anisotropic elastic deformation. J. Geophys. Res. 84, 7510–7512.CrossRefGoogle Scholar
  13. Cheng, A.H.D. (1997). Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci. 34, 199–205.CrossRefGoogle Scholar
  14. Coussy, O. (2004). Poromechanics, John Wiley & Sons Ltd., Chichester UK.Google Scholar
  15. Detournay, E., and A.H.D. Cheng (1993). Fundamentals of poroelasticity. In: Comprehensive Rock Engineering Vol. 2, edited by J. A. Hudson, Pergamon Press, Oxford, Chap. 5, pp. 113–171.Google Scholar
  16. Gassmann, F. (1951). Über die Elastizität poröser Medien. Vierteljahrsschr. Naturforsch. Ges. Zürich 96, 1–23.MathSciNetGoogle Scholar
  17. Geertsma, J. (1957a). A remark on the analogy between thermoelasticity and the elasticity of saturated porous media. J. Mech. Phys. Solids 6, 13–16.zbMATHCrossRefGoogle Scholar
  18. Geertsma, J. (1957b). The effect of fluid pressure decline on volumetric changes in porous rocks. Trans. AIME 210, 331–340.Google Scholar
  19. Geertsma, J. (1966). Problems of rock mechanics in petroleum production engineering. Proc. 1st Congr. Int. Society of Rock Mechanics, 585–594.Google Scholar
  20. Green, D.H., and H.F. Wang (1986). Fluid pressure response to undrained compression in saturated sedimentary rocks. Geophysics 51, 948–956.CrossRefGoogle Scholar
  21. Green, D.H., and H.F. Wang (1990). Specific storage as a poroelastic coefficient. Water Resources Res. 26, 1631–37.CrossRefGoogle Scholar
  22. Guéguen, Y., L. Dormieux, and M. Boutéca (2004). Fundamentals of Poromechanics. In: Mechanics of Fluid-Saturated Rocks, edited by Y. Guéguen and M. Boutéca, Elsevier Academic Press, Burlington MA, pp. 1–54.Google Scholar
  23. Jaeger, J.C., N.G.W. Cook, and R.W. Zimmerman (2007). Fundamentals of Rock Mechanics (4th edition), Blackwell Publishing Ltd, Oxford, etc.Google Scholar
  24. Kümpel, H-J. (1991). Poroelasticity: parameters reviewed. Geophys. J. Int. 105, 783–799.CrossRefGoogle Scholar
  25. Mavko, G., T. Mukerji, and J. Dvorkin (1998). The Rock Physics Handbook. Tools for Seismic Analysis in Porous Media. Cambridge U. Press, Cambridge UK.Google Scholar
  26. McTigue, D.F. (1986). Thermoelastic response of fluid-saturated porous rock. J. Geophys. Res. 91, 9533–9542.CrossRefGoogle Scholar
  27. Nur, A., and J.D. Byerlee (1971). An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res. 76, 6414–6419.CrossRefGoogle Scholar
  28. Nye, J.F. (1957). Physical Properties of Crystals. Oxford University Press, Oxford etc., (reprinted in 1979).zbMATHGoogle Scholar
  29. Rice, J.R., and M.P. Cleary (1976). Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys. 14, 227–241.CrossRefGoogle Scholar
  30. Rudnicki, J.W. (1985). Effect of pore fluid diffusion on deformation and failure of rock. In: Mechanics of Geomaterials, edited by E. Bažant, John Wiley & Sons Ltd., New York, Chap. 15, pp. 315–347.Google Scholar
  31. Rudnicki, J.W. (2001). Coupled deformation-diffusion effects in the mechanics of faulting and failure of geomaterials. Appl. Mech. Rev. 54, 1–20.CrossRefGoogle Scholar
  32. Terzaghi, K., and O.K. Fröhlich (1936). Theorie der Setzung von Tonschichten, F. Deutike, Wien.zbMATHGoogle Scholar
  33. Thompson, M., and J.R. Willis (1991). A reformulation of the equations of anisotropic poroelasticity. J. Appl. Mech. 58, 612–616.zbMATHCrossRefGoogle Scholar
  34. Wang, H.F. (2000). Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton and Oxford.Google Scholar
  35. Weiner, J.H. (1983). Statistical Mechanics of Elasticity, John Wiley & Sons, New York.zbMATHGoogle Scholar
  36. Zimmerman, R.W., W.H. Somerton, and M.S. King (1986). Compressibility of porous rocks. J. Geophys. Res. 91,12,765–12,777.Google Scholar

Copyright information

© CISM, Udine 2011

Authors and Affiliations

  • Florian K. Lehner
    • 1
  1. 1.Department of Geography and GeologyUniversity of SalzburgAustria

Personalised recommendations