Skip to main content
SpringerLink
Log in

Porothermoelasticity

  • Chapter
  • First Online:
Poroelasticity

Part of the book series: Theory and Applications of Transport in Porous Media ((TATP,volume 27))

Abstract

The poroelasticity theory presented so far assumes an isothermal condition, that is, temperature remains unchanged during the deformation and diffusion process. In practice, however, temperature of a porous medium can change. Not only it can change if it is in contact with a body of different temperature, and heat is transferred by conduction, but also the deformation itself can generate internal heat. In addition, heat can be transported in and out of the porous medium by a fluid flow.

If two corpuscles of a body lie infinitely close and have different temperature, the warmer corpuscle transmits a certain amount of its heat to the other one; and this heat—given from the warmer corpuscle to the colder one at a given time and during a given moment—is proportional to the temperature difference, if the difference has a small value.

—Joseph Fourier (1822)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    McTigue [47] used the notations \((\alpha '_{s},\,\alpha ''_{s},\,\alpha _{f})\) for coefficient of thermal expansion. The correspondences with the current notations are α′ s  = β d and ϕ(α f −α″ s ) = β v . In the table, McTigue further assumed α′ s  = α″ s  = α s , which corresponds to the ideal porous medium model of K ψ  = β ψ  = 0. These assumptions lead to the consistent result of β d  = β s and β v  = ϕ(β f −β s ) in (11.54) and (11.60) in the current model. Also, the value of α e ( = b′ of McTigue) for rock salt has been corrected in the table.

References

  1. Abousleiman Y, Ekbote S (2005) Solutions for the inclined borehole in a porothermoelastic transversely isotropic medium. J Appl Mech ASME 72(1):102–114

    Article  MATH  Google Scholar 

  2. Bear J, Corapcioglu MY (1981) A mathematical-model for consolidation in a thermoelastic aquifer due to hot water injection or pumping. Water Resour Res 17(3):723–736

    Article  Google Scholar 

  3. Belotserkovets A, Prevost JH (2011) Thermoporoelastic response of a fluid-saturated porous sphere: an analytical solution. Int J Eng Sci 49(12):1415–1423

    Article  MathSciNet  Google Scholar 

  4. Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):240–253

    Article  MathSciNet  MATH  Google Scholar 

  5. Biot MA (1977) Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int J Solids Struct 13(6):579–597

    Article  MathSciNet  MATH  Google Scholar 

  6. Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New York, 586pp

    MATH  Google Scholar 

  7. Booker JR, Savvidou C (1984) Consolidation around a spherical heat source. Int J Solids Struct 20(11–12):1079–1090

    Article  Google Scholar 

  8. Booker JR, Savvidou C (1985) Consolidation around a point heat source. Int J Numer Anal Methods Geomech 9(2):173–184

    Article  Google Scholar 

  9. Carnahan CL (1983) Thermodynamic coupling of heat and matter flows in near-field regions of nuclear waste repositories. MRS Proc 26:1023–1030

    Article  Google Scholar 

  10. Carson JK, Lovatt SJ, Tanner DJ, Cleland AC (2005) Thermal conductivity bounds for isotropic, porous materials. Int J Heat Mass Transf 48(11):2150–2158

    Article  MATH  Google Scholar 

  11. Chen GZ, Ewy RT (2005) Thermoporoelastic effect on wellbore stability. SPE J 10(2):121–129

    Article  Google Scholar 

  12. Cheng AHD (2000) Multilayered aquifer systems—fundamentals and applications. Marcel Dekker, New York, Basel, 384pp

    Google Scholar 

  13. Cheng AHD, Morohunfola OK (1993) Multilayered leaky aquifer systems: I. Pumping well solution. Water Resour Res 29(8):2787–2800

    Article  Google Scholar 

  14. Cheng AHD, Morohunfola OK (1993) Multilayered leaky aquifer systems: II. Boundary element solution. Water Resour Res 29(8):2801–2811

    Article  Google Scholar 

  15. Coussy O (2004) Poromechanics. Wiley, Chichester, 298pp

    MATH  Google Scholar 

  16. Coussy O (2005) Poromechanics of freezing materials. J Mech Phys Solids 53(8):1689–1718

    Article  MATH  Google Scholar 

  17. Coussy O, Monteiro PJM (2008) Poroelastic model for concrete exposed to freezing temperatures. Cem Concr Res 38(1):40–48

    Article  Google Scholar 

  18. Cussler EL (2009) Diffusion: mass transfer in fluid systems, 3rd edn. Cambridge University Press, Cambridge/New York, 647pp

    Book  Google Scholar 

  19. de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, New York, 526pp

    MATH  Google Scholar 

  20. de Marsily G (1986) Quantitative hydrogeology: groundwater hydrology for engineers. Academic, Orlando, 440pp

    Google Scholar 

  21. Demirel Y (2014) Nonequilibrium thermodynamics: transport and rate processes in physical, chemical and biological systems, 3rd edn. Elsevier, Amsterdam, 758pp

    MATH  Google Scholar 

  22. Derjaguin BV, Churaev NV, Muller VM (1987) Surface Forces. Plenum, New York, 440pp

    Book  Google Scholar 

  23. Derski W, Kowalski S (1979) Equations of linear thermoconsolidation. Arch Mech 31(3):303–316

    MATH  Google Scholar 

  24. Dirksen C (1969) Thermo-osmosis through compacted saturated clay membranes. Soil Sci Soc Am J 33(6):821–826

    Article  Google Scholar 

  25. Fourier JBJ (1822) Théorie Analytique de la Chaleur (The analytical theory of heat). Chez Firmin Didot, Père et Fils, Paris

    Google Scholar 

  26. Ghabezloo S, Sulem J, Saint-Marc J (2009) The effect of undrained heating on a fluid-saturated hardened cement paste. Cem Concr Res 39(1):54–64

    Article  Google Scholar 

  27. Ghassemi A, Diek A (2002) Porothermoelasticity for swelling shales. J Pet Sci Eng 34(1–4):123–135

    Article  Google Scholar 

  28. Ghassemi A, Zhou X (2011) A three-dimensional thermo-poroelastic model for fracture response to injection/extraction in enhanced geothermal systems. Geothermics 40(1):39–49

    Article  Google Scholar 

  29. Ghassemi A, Nygren A, Cheng AHD (2008) Effects of heat extraction on fracture aperture: a poro-thermoelastic analysis. Geothermics 37(5):525–539

    Article  Google Scholar 

  30. 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):57–64

    Article  Google Scholar 

  31. Gonçalvès J, Trémosa J (2010) Estimating thermo-osmotic coefficients in clay-rocks: I. Theoretical insights. J Colloid Interface Sci 342(1):166–174

    Article  Google Scholar 

  32. Gray DH, Mitchell JK (1967) Fundamental aspects of electro-osmosis in soils. J Soil Mech Found Div ASCE 93(SM6):209–236

    Google Scholar 

  33. Hashin Z, Shtrikman S (1962) A variational approach to theory of effective magnetic permeability of multiphase materials. J Appl Phys 33(10):3125–3131

    Article  MATH  Google Scholar 

  34. Hollister CD, Anderson DR, Heath GR (1981) Subseabed disposal of nuclear wastes. Science 213(4514):1321–1326

    Article  Google Scholar 

  35. Hörmander L (1969) Linear partial differential operators. Springer, Berlin/Heidelberg/New York, 285pp

    Book  MATH  Google Scholar 

  36. Kanj M, Abousleiman Y (2005) Porothermoelastic analyses of anisotropic hollow cylinders with applications. Int J Numer Anal Methods Geomech 29(2):103–126

    Article  MATH  Google Scholar 

  37. Katchalsky A, Curran PF (1967) Nonequilibrium thermodynamics in biophysics. Harvard University Press, Cambridge, 248pp

    Google Scholar 

  38. Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45(4):574–588

    Article  Google Scholar 

  39. Kurashige M (1989) A thermoelastic theory of fluid-filled porous materials. Int J Solids Struct 25(9):1039–1052

    Article  Google Scholar 

  40. Lachenbruch AH (1980) Frictional heating, fluid pressure, and the resistance to fault motion. J Geophys Res Solid Earth 85(B11):6097–6112

    Article  Google Scholar 

  41. Landauer R (1952) The electrical resistance of binary metallic mixtures. J Appl Phys 23(7):779–784

    Article  Google Scholar 

  42. Letey J, Kemper WD (1969) Movement of water and salt through a clay-water system: experimental verification of Onsager reciprocal relation. Soil Sci Soc Am J 33(1):25–29

    Article  Google Scholar 

  43. Mase CW, Smith L (1985) Pore-fluid pressures and frictional heating on a fault surface. Pure Appl Geophys 122(2–4):583–607

    Article  Google Scholar 

  44. Mase CW, Smith L (1987) Effects of frictional heating on the thermal, hydrologic, and mechanical response of a fault. J Geophys Res Solid Earth 92(B7):6249–6272

    Article  Google Scholar 

  45. Mason EA, Wendt RP, Bresler EH (1972) Test of Onsager relation for ideal gas transport in membranes. J Chem Soc Faraday Trans II 68(11):1938–1959

    Article  Google Scholar 

  46. Maxwell JC (1954) A treatise on electricity and magnetism, 3rd edn. Dover, New York, 560pp

    MATH  Google Scholar 

  47. McTigue DF (1986) Thermoelastic response of fluid-saturated porous rock. J Geophys Res Solid Earth 91(B9):9533–9542

    Article  Google Scholar 

  48. McTigue DF (1990) Flow to a heated borehole in porous, thermoelastic rock: analysis. Water Resour Res 26(8):1763–1774

    Article  Google Scholar 

  49. Miller DG (1959) Ternary isothermal diffusion and the validity of the Onsager reciprocity relations. J Phys Chem 63(4):570–578

    Article  Google Scholar 

  50. Miller DG (1960) Thermodynamics of irreversible processes—the experimental verification of the Onsager reciprocal relations. Chem Rev 60(1):15–37

    Article  Google Scholar 

  51. Msaad Y (2007) Comparison between hydraulic and thermal spalling in heated concrete based on numerical modeling. J Eng Mech ASCE 133(6):608–615

    Article  Google Scholar 

  52. Neaupane KM, Yamabe T, Yoshinaka R (1999) Simulation of a fully coupled thermo-hydro-mechanical system in freezing and thawing rock. Int J Rock Mech Min Sci 36(5):563–580

    Article  Google Scholar 

  53. Nowacki W (1986) Thermoelasticity, 2nd edn. Pergamon, Oxford/New York, 566pp

    MATH  Google Scholar 

  54. Onsager L (1931) Reciprocal relations in irreversible process I. Phys Rev 37(4):405–426

    Article  MATH  Google Scholar 

  55. Onsager L (1931) Reciprocal relations in irreversible process II. Phys Rev 38(12):2265–2279

    Article  MATH  Google Scholar 

  56. Palciauskas VV, Domenico PA (1982) Characterization of drained and undrained response of thermally loaded repository rocks. Water Resour Res 18(2):281–290

    Article  Google Scholar 

  57. Powers DW, Lambert SJ, Shaffer SE, Hill LR, Weart WD (eds) (1978) Geological characterization report, Waste Isolation Pilot Plant (WIPP) site, Southeastern New Mexico, Technical report SAND78-1596, Sandia National Laboratories, Albuquerque

    Google Scholar 

  58. Rahman S, Grasley Z (2014) A poromechanical model of freezing concrete to elucidate damage mechanisms associated with substandard aggregates. Cem Concr Res 55:88–101

    Article  Google Scholar 

  59. Rastogi RP, Singh K (1966) Cross-phenomenological coefficients. Part 5. Thermo-osmosis of liquids through cellophane membrane. Trans Faraday Soc 62:1754–1761

    Article  Google Scholar 

  60. Rastogi RP, Agarwal RK, Blokhra RL (1964) Cross-phenomenological coefficients. Part 1. Studies on thermo-osmosis. Trans Faraday Soc 60:1386–1390

    Article  Google Scholar 

  61. Schiffman RL (1971) A thermoelastic theory of consolidation. In: Cremers CJ, Kreith F, Clark JA (eds) Environmental and geophysical heat transfer. ASME, New York, pp 78–84

    Google Scholar 

  62. Selvadurai APS, Suvorov AP (2012) Boundary heating of poro-elastic and poro-elasto-plastic spheres. Proc R Soc A Math Phys Eng Sci 468(2145):2779–2806

    Article  MathSciNet  Google Scholar 

  63. Selvadurai APS, Suvorov AP (2014) Thermo-poromechanics of a fluid-filled cavity in a fluid-saturated geomaterial. Proc R Soc A Math Phys Eng Sci 470(2163):20130634

    Article  MathSciNet  Google Scholar 

  64. Skinner BJ (1966) Thermal expansion. In: Clark SP (ed) Handbook of physical constants. Memoirs (Geological Society of America), vol 97. Geological Society of America, New York, pp 75–96

    Chapter  Google Scholar 

  65. Smith DW, Booker JR (1993) Green’s functions for a fully coupled thermoporoelastic material. Int J Numer Anal Methods Geomech 17(3):139–163

    Article  MATH  Google Scholar 

  66. Smith DW, Booker JR (1996) Boundary element analysis of linear thermoelastic consolidation. Int J Numer Anal Methods Geomech 20(7):457–488

    Article  MATH  Google Scholar 

  67. Soler JM (2001) The effect of coupled transport phenomena in the Opalinus Clay and implications for radionuclide transport. J Contam Hydrol 53(1–2):63–84

    Article  Google Scholar 

  68. Somerton WH (1992) Thermal properties and temperature-related behaviour of rock/fluid systems. Elsevier, Amsterdam, 256pp

    Google Scholar 

  69. Srivastava RC, Avasthi PK (1975) Nonequilibrium thermodynamics of thermo-osmosis of water through kaolinite. J Hydrol 24(1–2):111–120

    Article  Google Scholar 

  70. Sugawara A, Yoshizawa Y (1961) An investigation on thermal conductivity of porous materials and its application to porous rock. Aust J Phys 14(4):468–469

    Article  Google Scholar 

  71. Suzuki T, Yamashita T (2006) Nonlinear thermoporoelastic effects on dynamic earthquake ruptur. J Geophys Res Solid Earth 111(B3):B03307

    Article  Google Scholar 

  72. Tao Q, Ghassemi A (2010) Poro-thermoelastic borehole stress analysis for determination of the in situ stress and rock strength. Geothermics 39(3):250–259

    Article  Google Scholar 

  73. Tavman IH (1996) Effective thermal conductivity of granular porous materials. Int Commun Heat Mass Transf 23(2):169–176

    Article  Google Scholar 

  74. Trémosa J, Gonçalvès J, Matray JM, Violette S (2010) Estimating thermo-osmotic coefficients in clay-rocks: II. In situ experimental approach. J Colloid Interface Sci 342(1):175–184

    Article  Google Scholar 

  75. Vardoulakis I (2000) Catastrophic landslides due to frictional heating of the failure plane. Mech Cohesive-Frict Mater 5(6):443–467

    Article  Google Scholar 

  76. Vardoulakis I (2002) Dynamic thermo-poro-mechanical analysis of catastrophic landslides. Géotechnique 52(3):157–171

    Article  Google Scholar 

  77. Wang Y, Papamichos E (1994) Conductive heat flow and thermally induced fluid flow around a well bore in a poroelastic medium. Water Resour Res 30(12):3375–3384

    Article  Google Scholar 

  78. Wiener O (1912) Die theorie des mischkörpers für das feld der stationären strömung. Erste abhandlung: Die mittelswertsätze für kraft, polarisation und energie (The theory of mixtures for the stationary flow field. First treatise: The averaging theorem for power, polarization and energy), Abhandlungen der Mathematisch-Physischen Klasse der Königlich-Sächsischen Gesellschaft der Wissenschaften 32:509–604

    Google Scholar 

  79. Woodside W, Messmer JH (1961) Thermal conductivity of porous media. I. Unconsolidated sands. J Appl Phys 32(9):1688–1699

    Google Scholar 

  80. Woodside W, Messmer JH (1961) Thermal conductivity of porous media. II. Consolidated rocks. J Appl Phys 32(9):1699–1706

    Google Scholar 

  81. Yeung AT, Mitchell JK (1993) Coupled fluid, electrical and chemical flows in soil. Géotechnique 43(1):121–134

    Article  Google Scholar 

  82. Zimmerman RW (2000) Coupling in poroelasticity and thermoelasticity. Int J Rock Mech Min Sci 37(1–2):79–87

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Mississippi, Oxford, Mississippi, USA

    Alexander H.-D. Cheng

Authors
  1. Alexander H.-D. Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Cheng, A.HD. (2016). Porothermoelasticity. In: Poroelasticity. Theory and Applications of Transport in Porous Media, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-25202-5_11

Download citation

Publish with us

Policies and ethics

Search

Navigation