Poroelasticity pp 113-169 | Cite as

Variational Energy Formulation

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


In Chap.  2, the constitutive equations for poroelasticity were constructed using the phenomenological approach. In such approach, we attempt to model a new phenomenon, such as the deformation of a saturated porous body, by drawing an analogy with a familiar phenomenon, such as the deformation of an elastic body. In the analogy, we define stresses and strains for a porous body following the elasticity concept, even though their interpretation may not be clear. (For example, see the illustration in Fig.  2.1 for the lack of a clear definition for a continuous stress field in a porous body.) We then bring in the additional force component, namely the pore pressure, and its conjugate deformation, the fluid strain (or actually the relative fluid to solid strain), to build a linear relation that is similar to that of the elasticity theory. This type of ad hoc construction of a working theory is typically motivated first by the observation, and then supported by physical insight, without formally resorting to the laws of physics. To gain additional physical insight, in Chap.  3 we utilized the micromechanics approach to explicitly model the material phases in a porous medium, not only the solid and the fluid, but also the pore space as an additional “phase”, together with their interactions. These micromechanical constitutive laws were assembled and matched up with the bulk continuum theory to provide physical insight. However, these constitutive laws were largely constructed using the “effective stress” concept; hence their theoretical basis is still phenomenological. The material constants associated with the theory, such as K, K s , and K s , are still empirical constants, as their physical mechanisms are based on composite responses, and are not fully isolated to tie to the equation of the state of the phases. There are many attempts to build porous medium constitutive models that go beyond the phenomenological model. Among the more widely pursued approaches are the theory of mixtures (Atkin and Craine, Q J Mech Appl Math 29:209–244, 1976; Bowen, Int J Eng Sci 18(9):1129–1148, 1980; Bowen, Int J Eng Sci 20(6):697–735, 1982; Crochet and Naghdi, Int J Eng Sci 4:383–401, 1966; de Boer R, Theory of porous media, highlights in the historical development and current state. Springer, Brelin/New York, 2000, 618pp; Morland, J Geophys Res 77(5):890–900, 1972), and the homogenization theory (Auriault and Sanchez-Palencia, J de Méc 16(4):575–603, 1977; Chateau and Dormieux, Int J Numer Anal Methods Geomech 26(8):831–844, 2002; Mei and Auriault, Proc R Soc Lond Ser A Math Phys Eng Sci 426(1871):391–423, 1989; Terada et al., Comput Methods Appl Mech Eng 153(3–4):223–257, 1998). The theory of mixtures uses the mathematical assumption that the solid and fluid phases occupies the same space, whose presence and influence are weighted by a volume fraction, in order to fulfill the requirement of a continuous mathematical function. This approach is largely mathematical, and the “material coefficients” generally do not possess physical meaning until a comparison is made with a theory that is in use, such as a phenomenological theory. The homogenization approach, on the other hand, explicitly recognizes solid and fluid phases occupying different space at the microscopic level, with an assumed periodic pore geometry. The full partial differential equations, such as Navier-Stokes equation for the fluid and elasticity equation for the solid, are prescribed, together with boundary conditions. Effort is then made to simplify these equations based on the perturbation of small parameters, in order to extract mathematical terms, and physical phenomena, of the first order, second order, etc. The homogenization approach typically involves heavy mathematics. Different judgement on the mathematical treatment sometimes leads to different, and even inconsistent results. Its ultimate justification still requires the validation by physical experiments. In this chapter we take a different approach from those above. We shall build the poroelasticity theory based on the thermodynamics principles of work and energy, first based on the reversible processes for the elastic constitutive laws, and then in the later chapters on the irreversible processes for the fluid, heat flux, and chemical diffusion laws. To cope with the highly heterogeneous nature of porous materials, the volume averaging method developed in the flow through porous medium theory (Bear J, Bachmat Y, Introduction to modeling phenomena of transport in porous media. Kluwer, Dordrecht/Boston, 1990, 553pp), and the theory of heterogeneous (composite) materials (Christensen RM, Mechanics of composite materials. Wiley Interscience, New York, 1979, 384pp; Nemat-Nasser S, Hori M, Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, 1999, 810pp), are used to define the continuous and smooth functions needed in the construction of partial differential equations. The porous materials are considered as consisting of a heterogeneous solid phase and a homogeneous fluid phase (Lopatnikov and Cheng, Mech Mater 34(11):685–704, 2002). The macroscopic stresses and strains are defined as microscopically volume or surface averaged quantities depending on whether internal energy or external work is concerned. Constitutive equations are constructed using the variational energy principle stemming from the classical physics law of minimum potential energy (Landau LD, Lifshitz EM, Theory of elasticity, 3rd edn. Course of theoretical physics, vol 7. Butterworth-Heinemann, Oxford/Burlington, 1986, 195pp). The resultant model consists of a set of intrinsic material constants (Cheng and Abousleiman, Int J Numer Anal Methods Geomech 32(7):803–831, 2008), which are directly associated with the equations of state of the solid and fluid phase, and the fundamental deformation modes of the pore structure.


  1. 1.
    Atkin RJ, Craine RE (1976) Continuum theories of mixtures—basic theory and historical development. Q J Mech Appl Math 29:209–244MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auriault JL, Sanchez-Palencia E (1977) Etude du comportement d’un milieu poreux saturé déformable (Study of macroscopic behavior of a deformable saturated porous medium). J de Méc 16(4):575–603MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bear J (1972) Dynamics of fluids in porous media. American Elsevier, New York, 764pp (also published by Dover, 1988)Google Scholar
  4. 4.
    Bear J, Bachmat Y (1990) Introduction to modeling phenomena of transport in porous media. Kluwer, Dordrecht/Boston, 553ppCrossRefzbMATHGoogle Scholar
  5. 5.
    Bear J, Cheng AHD (2010) Modeling groundwater flow and contaminant transport. Springer, Dordrecht/London, 834ppCrossRefzbMATHGoogle Scholar
  6. 6.
    Berge PA, Wang HF, Bonner BP (1993) Pore pressure buildup coefficient in synthetic and natural sandstones. Int J Rock Mech Mining Sci 30(7):1135–1141CrossRefGoogle Scholar
  7. 7.
    Berryman JG (2006) Effective medium theories for multicomponent poroelastic composites. J Eng Mech ASCE 132(5):519–531CrossRefGoogle Scholar
  8. 8.
    Berryman JG, Pride SR (1998) Volume averaging, effective stress rules, and inversion for microstructural response of multicomponent porous media. Int J Solids Struct 35(34–35):4811–4843CrossRefzbMATHGoogle Scholar
  9. 9.
    Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164CrossRefzbMATHGoogle Scholar
  10. 10.
    Biot MA (1955) Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys Rev 97(6):1463–1469MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):240–253MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Biot MA (1956) Variational and Lagrangian methods in viscoelasticity. In: Grammel R (ed) Deformation and flow of solids, IUTAM colloquium Madrid 1955. Springer, Berlin, pp 251–263Google Scholar
  13. 13.
    Biot MA (1962) Generalized theory of acoustic propagation in porous dissipative media. J Acoust Soc Am 34(9):1254–1264MathSciNetCrossRefGoogle Scholar
  14. 14.
    Biot MA (1973) Nonlinear and semilinear rheology of porous solids. J Geophys Res 78(23):4924–4937CrossRefGoogle Scholar
  15. 15.
    Biot MA, Willis DG (1957) The elastic coefficients of the theory of consolidation. J Appl Mech ASME 24:594–601MathSciNetGoogle Scholar
  16. 16.
    Boresi AP, Chong KP, Lee JD (2010) Elasticity in engineering mechanics, 3rd edn. Wiley, Hoboken, 656ppCrossRefzbMATHGoogle Scholar
  17. 17.
    Borja RI (2006) On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct 43(6):1764–1786CrossRefzbMATHGoogle Scholar
  18. 18.
    Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 18(9):1129–1148CrossRefzbMATHGoogle Scholar
  19. 19.
    Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20(6):697–735CrossRefzbMATHGoogle Scholar
  20. 20.
    Brown RJS, Korringa J (1975) Dependence of elastic properties of a porous rock on compressibility of pore fluid. Geophysics 40(4):608–616CrossRefGoogle Scholar
  21. 21.
    Budiansky B, O’Connell RJ (1976) Elastic moduli of a cracked solid. Int J Solids Struct 12(2):81–97CrossRefzbMATHGoogle Scholar
  22. 22.
    Chateau X, Dormieux L (2002) Micromechanics of saturated and unsaturated porous media. Int J Numer Anal Methods Geomech 26(8):831–844CrossRefzbMATHGoogle Scholar
  23. 23.
    Cheng AHD, Abousleiman Y (2008) Intrinsic poroelasticity constants and a semilinear model. Int J Numer Anal Methods Geomech 32(7):803–831CrossRefzbMATHGoogle Scholar
  24. 24.
    Cheng AHD, Lopatnikov SL (2002) Variational formulation of fluid infiltrated porous materials and physically-based micromechanical material constants. In: Auriault JL, Geindreau C, Royer P, Bloch JF, Boutin C, Lewandowska J (eds) Poromechanics II, proceedings of the second biot conference on poromechanics. Balkema, Grenoble, pp 385–390Google Scholar
  25. 25.
    Christensen RM (1979) Mechanics of composite materials. Wiley Interscience, New York, 384ppGoogle Scholar
  26. 26.
    Cleary MP (1978) Elastic and dynamic response regimes of fluid-impregnated solids with diverse microstructures. Int J Solids Struct 14(10):795–819CrossRefzbMATHGoogle Scholar
  27. 27.
    Coussy O (1995) Mechanics of porous continua, 2nd edn. Wiley, Chichester/New York, 472ppzbMATHGoogle Scholar
  28. 28.
    Coussy O (2004) Poromechanics. Wiley, Chichester, 298ppzbMATHGoogle Scholar
  29. 29.
    Coussy O (2005) Poromechanics of freezing materials. J Mech Phys Solids 53(8):1689–1718CrossRefzbMATHGoogle Scholar
  30. 30.
    Coussy O (2006) Deformation and stress from in-pore drying-induced crystallization of salt. J Mech Phys Solids 54(8):1517–1547CrossRefzbMATHGoogle Scholar
  31. 31.
    Coussy O (2007) Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept. Int J Numer Anal Methods Geomech 31(15):1675–1694CrossRefzbMATHGoogle Scholar
  32. 32.
    Coussy O, Eymard R, Lassabatere T (1998) Constitutive modeling of unsaturated drying deformable materials. J Eng Mech ASCE 124(6):658–667CrossRefGoogle Scholar
  33. 33.
    Crochet MJ, Naghdi PM (1966) On constitutive equations for flow of fluid through an elastic solid. Int J Eng Sci 4:383–401CrossRefGoogle Scholar
  34. 34.
    Cruz-Orive LM, Myking AO (1981) Stereological estimation of volume ratios by systematic sections. J Microsc 122:143–157CrossRefGoogle Scholar
  35. 35.
    de Boer R (1998) The thermodynamic structure and constitutive equations for fluid-saturated compressible and incompressible elastic porous solids. Int J Solids Struct 35(34–35):4557–4573CrossRefzbMATHGoogle Scholar
  36. 36.
    de Boer R (2000) Theory of porous media, highlights in the historical development and current state. Springer, Brelin/New York, 618ppCrossRefzbMATHGoogle Scholar
  37. 37.
    de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, New York, 526ppzbMATHGoogle Scholar
  38. 38.
    Delesse A (1848) Procédé mécanique pour déterminer la composition des roches (Mechanical process to determine the composition of rocks). Ann des Mines Paris (Ser 4) 13:379–388Google Scholar
  39. 39.
    Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A Math Phys Sci 241(1226):376–396MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gray WG, Schrefler BA (2007) Analysis of the solid phase stress tensor in multiphase porous media. Int J Numer Anal Methods Geomech 31(4):541–581CrossRefzbMATHGoogle Scholar
  41. 41.
    Green DH, Wang HF (1986) Fluid pressure response to undrained compression in saturated sedimentary rock. Geophysics 51(4):948–956CrossRefGoogle Scholar
  42. 42.
    Hart DJ, Wang HF (1995) Laboratory measurements of a complete set of poroelastic moduli for Berea sandstone and Indiana limestone. J Geophys Res Solid Earth 100(B9):17741–17751CrossRefGoogle Scholar
  43. 43.
    Hart DJ, Wang HF (2010) Variation of unjacketed pore compressibility using Gassmann’s equation and an overdetermined set of volumetric poroelastic measurements. Geophysics 75(1):N9–N18CrossRefGoogle Scholar
  44. 44.
    Hashin Z (1964) Theory of mechanical behaviour of heterogeneous media. Appl Mech Rev 17:1–9Google Scholar
  45. 45.
    Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372CrossRefzbMATHGoogle Scholar
  47. 47.
    Hill R (1967) The essential structure of constitutive laws for metal composites and polycrystals. J Mech Phys Solids 15(2):79–95CrossRefGoogle Scholar
  48. 48.
    Kachanov M (1987) Elastic solids with many cracks—a simple method of analysis. Int J Solids Struct 23(1):23–43CrossRefzbMATHGoogle Scholar
  49. 49.
    Kachanov M, Sevostianov I (2005) On quantitative characterization of microstructures and effective properties. Int J Solids Struct 42(2):309–336CrossRefzbMATHGoogle Scholar
  50. 50.
    Katchalsky A, Curran PF (1967) Nonequilibrium thermodynamics in biophysics. Harvard University Press, Cambridge, 248ppGoogle Scholar
  51. 51.
    Kim YK, Kingsbury HB (1979) Dynamic characterization of poroelastic materials. Exp Mech 19(7):252–258CrossRefGoogle Scholar
  52. 52.
    Landau LD, Lifshitz EM (1986) Theory of elasticity, 3rd edn. Course of theoretical physics, vol 7. Butterworth-Heinemann, Oxford/Burlington, 195ppGoogle Scholar
  53. 53.
    Laurent J, Boutéca MJ, Sarda JP, Bary D (1993) Pore-pressure influence in the poroelastic behavior of rocks—experimental studies and results. SPE Form Eval 8(2):117–122CrossRefGoogle Scholar
  54. 54.
    Lopatnikov SL, Cheng AHD (2002) Variational formulation of fluid infiltrated porous material in thermal and mechanical equilibrium. Mech Mater 34(11):685–704CrossRefGoogle Scholar
  55. 55.
    Mackenzie JK (1950) The elastic constants of a solid containing spherical holes. Proc Phys Soc Lond Sect B 63(361):2–11CrossRefzbMATHGoogle Scholar
  56. 56.
    Mei CC, Auriault JL (1989) Mechanics of heterogeneous porous media with several spatial scales. Proc R Soc Lond Ser A Math Phys Eng Sci 426(1871):391–423CrossRefGoogle Scholar
  57. 57.
    Morland LW (1972) A simple constitutive theory for a fluid-saturate porous solid. J Geophys Res 77(5):890–900CrossRefGoogle Scholar
  58. 58.
    Mura T (1987) Micromechanics of defects in solids, 2nd edn. Springer, Dordrecht, 588ppCrossRefzbMATHGoogle Scholar
  59. 59.
    Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, Amsterdam/London, 810ppzbMATHGoogle Scholar
  60. 60.
    Nowacki W (1986) Thermoelasticity, 2nd edn. Pergamon, Oxford/New York, 566ppzbMATHGoogle Scholar
  61. 61.
    Nur A, Byerlee JD (1971) Exact effective stress law for elastic deformation of rock with fluids. J Geophys Res 76(26):6414–6419CrossRefGoogle Scholar
  62. 62.
    Sokolnikoff IS (1956) Mathematical theory of elasticity, 2nd edn. McGraw-Hill, New York, 476ppzbMATHGoogle Scholar
  63. 63.
    Terada K, Ito T, Kikuchi N (1998) Characterization of the mechanical behaviors of solid-fluid mixture by the homogenization method. Comput Methods Appl Mech Eng 153(3–4):223–257MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Thomas SD (1989) A finite element model for the analysis of wave induced stresses, displacements and pore pressures in an unsaturated seabed I: theory. Comput Geotech 8(1):1–38CrossRefGoogle Scholar
  65. 65.
    Thomson (Lord Kelvin) W (1852) On the universal tendency in nature to the dissipation of mechanical energy. Philos Mag Ser 4 4(25):304–306Google Scholar
  66. 66.
    Vajdova V, Baud P, Wong TF (2004) Compaction, dilatancy, and failure in porous carbonate rocks. J Geophys Res Solid Earth 109(B5):16CrossRefGoogle Scholar
  67. 67.
    van der Knaap W (1959) Nonlinear behavior of elastic porous media. Trans Am Inst Mining Metall Eng 216:179–186Google Scholar
  68. 68.
    Verruijt A (1969) Elastic storage of aquifers. In: DeWiest RJM (ed) Flow through porous media. Academic, New York, pp 331–376Google Scholar
  69. 69.
    Wei C, Muraleetharan KK (2002) A continuum theory of porous media saturated by multiple immiscible fluids: I. Linear poroelasticity. Int J Eng Sci 40(16):1807–1833MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Wei C, Muraleetharan KK (2002) A continuum theory of porous media saturated by multiple immiscible fluids: II. Lagrangian description and variational structure. Int J Eng Sci 40(16):1835–1854MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Wong TF, Baud P (1999) Mechanical compaction of porous sandstone. Oil Gas Sci Technol Revue de l’Inst Fr du Pet 54(6):715–727CrossRefGoogle Scholar
  72. 72.
    Wong TF, David C, Zhu WL (1997) The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. J Geophys Res Solid Earth 102(B2):3009–3025CrossRefGoogle Scholar
  73. 73.
    Wood AB (1941) Textbook of sound. Being an account of the physics of vibrations with special reference to recent theoretical and technical developments, 2nd edn. Bell & Sons, London, 578ppGoogle Scholar
  74. 74.
    Wu XY, Baud P, Wong TF (2000) Micromechanics of compressive failure and spatial evolution of anisotropic damage in Darley Dale sandstone. Int J Rock Mech Mining Sci 37(1–2):143–160CrossRefGoogle Scholar
  75. 75.
    Zhang JX, Wong TF, Davis DM (1990) Micromechanics of pressure-induced grain crushing in porous rocks. J Geophys Res Solid Earth Planets 95(B1):341–352CrossRefGoogle Scholar
  76. 76.
    Zimmerman RW, Somerton WH, King MS (1986) Compressibility of porous rocks. J Geophys Res Solid Earth Planets 91(B12):2765–2777CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

Personalised recommendations