Skip to main content
SpringerLink
Log in

A porothermoelasticity theory for anisotropic medium

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The concept of porothermoelasticity has been developed to investigate the thermal and mechanical behavior of elastic materials with porosity. This approach combines the theory of heat conduction with poroelastic constitutive equations and by incorporating the coupling of temperature field with the stresses and pore pressure. The mathematical modeling of the porothermoelastic material to various practical problems has attracted researchers in recent years due to its wide applications to geomechanics, where the coupled thermal and poro-mechanical processes play an essential role. The porothermoelasticity theory with relaxation parameter and predicting the finite speed for the isotropic medium has been established by Youssef (Int J Rock Mech Min 44(2):222–227, 2007) and by Sherief and Hussein (Transp Porous Med 91(1):199–223, 2012). However, the governing equations of the porothermoelasticity theory including the effects of temperature as well as temperature-rate terms for an anisotropic medium are yet to be derived. Hence, in view of the useful applications of the subject, the present work is motivated to derive the set of governing equations for the theory of temperature-rate-dependent fluid-saturated anisotropic poroelastic medium from the fundamental laws of thermodynamics. Further, we establish a uniqueness theorem for the general porothermoelastic problem of an anisotropic medium based on the present theory. Lastly, we solve a half-space problem of porothermoelasticity for isotropic medium to show the implementation of the present theory and investigate the effects of the relaxation times and porosity parameter over the distributions of various field variables by considering kerosene-saturated sandstone medium.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27(3), 240–253 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  3. Treitel, S.: On the attenuation of small-amplitude plane stress waves in a thermoelastic solid. J. Geophys. Res. 64(6), 661–665 (1959)

    Article  ADS  Google Scholar 

  4. Armstrong, B.H.: Models for thermoelastic in heterogeneous solids attenuation of waves. Geophysics 49(7), 1032–1040 (1984)

    Article  ADS  Google Scholar 

  5. Jacquey, A.B., Cacace, M., Blöcher, G., Scheck-Wenderoth, M.: Numerical investigation of thermoelastic effects on fault slip tendency during injection and production of geothermal fluids. Energy Proc. 76, 311–320 (2015)

    Article  Google Scholar 

  6. Fu, L.Y.: Evaluation of sweet spot and geopressure in Xihu. Sag. Technical report, CCL2012-SHPS-0018ADM. Key Laboratory of Petroleum Resource Research, Institute of Geology and Geophysics, Chinese Academy of Sciences (2012)

  7. Bear, J., Sorek, S., Ben-Dor, G., Mazor, G.: Displacement waves in saturated thermoelastic porous media. I. Basic equations. Fluid Dyn. Res. 9(4), 155 (1992)

    Article  ADS  Google Scholar 

  8. Levy, A., Sorek, S., Ben-Dor, G., Bear, J.: Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature. Transp. Porous Med. 21(3), 241–268 (1995)

    Article  Google Scholar 

  9. Biot, M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4), 1482–1498 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Biot, M.A.: Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34(9A), 1254–1264 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  11. Biot, M.A.: Theory of buckling of a porous slab and its thermoelastic analogy. J. Appl. Mech. ASME 31(2), 194–198 (1964). https://doi.org/10.1115/1.3629586

    Article  ADS  MathSciNet  Google Scholar 

  12. Biot, M.A., Temple, G.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21(7), 597–620 (1972)

    Article  MATH  Google Scholar 

  13. Rice, J.R., Cleary, M.P.: Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. 14(2), 227–241 (1976)

    Article  ADS  Google Scholar 

  14. Pecker, C., Deresiewicz, H.: Thermal effects on wave propagation in liquid-filled porous media. Acta Mech. 16(1–2), 45–64 (1973)

    Article  Google Scholar 

  15. Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13(2), 125–147 (1983)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Fourie, J.G., Du Plessis, J.P.: A two-equation model for heat conduction in porous media (I: theory). Transp. Porous Med. 53(2), 145–161 (2003)

    Article  MathSciNet  Google Scholar 

  20. Wang, H.F.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, vol. 2. Princeton University Press, Princeton (2000)

    Google Scholar 

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

    Article  Google Scholar 

  22. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)

    Article  ADS  MATH  Google Scholar 

  23. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)

    Article  MATH  Google Scholar 

  24. Youssef, H.M.: Theory of generalized porothermoelasticity. Int. J. Rock Mech. Min. 44(2), 222–227 (2007)

    Article  Google Scholar 

  25. Sherief, H.H., Hussein, E.M.: A mathematical model for short-time filtration in poroelastic media with thermal relaxation and two temperatures. Transp. Porous Med. 91(1), 199–223 (2012)

    Article  MathSciNet  Google Scholar 

  26. Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72(2), 175–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with a double porosity structure. J. Therm. Stress. 37(9), 1017–1036 (2014)

    Article  Google Scholar 

  28. Emin, A.N., Florea, O.A., Crăciun, E.M.: Some uniqueness results for thermoelastic materials with double porosity structure. Contin. Mech. Thermodyn. (2020). https://doi.org/10.1007/s00161-020-00952-7

    Article  Google Scholar 

  29. Rohan, E., Naili, S., Lemaire, T.: Double porosity in fluid-saturated elastic media: deriving effective parameters by hierarchical homogenization of static problem. Contin. Mech. Thermodyn. 28(5), 1263–1293 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Ezzat, M., Ezzat, S.: Fractional thermoelasticity applications for porous asphaltic materials. Pet. Sci. 13(3), 550–560 (2016)

    Article  Google Scholar 

  31. Roubíček, T.: Geophysical models of heat and fluid flow in damageable poro-elastic continua. Contin. Mech. Thermodyn. 29(2), 625–646 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Miller, L., Penta, R.: Effective balance equations for poroelastic composites. Contin. Mech. Thermodyn. 32, 1–25 (2020)

    Article  MathSciNet  Google Scholar 

  33. Iovane, G., Passarella, F.: Saint-Venant’s principle in dynamic porous thermoelastic media with memory for heat flux. J. Therm. Stress. 27(11), 983–999 (2004)

    Article  MathSciNet  Google Scholar 

  34. Marin, M., Nicaise, S.: Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin. Mech. Thermodyn. 28(6), 1645–1657 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Marin, M.S., Vlase, I.S., Paun, M.: Considerations on double porosity structure for micropolar bodies. AIP Adv. 5(3), 037113 (2015)

  36. Marin, M., Öchsner, A., Craciun, E.M.: A generalization of the Gurtin’s variational principle in thermoelasticity without energy dissipation of dipolar bodies. Contin. Mech. Thermodyn. 32, 1685–1694 (2020). https://doi.org/10.1007/s00161-020-00873-5

    Article  ADS  MathSciNet  Google Scholar 

  37. Marin, M., Öchsner, A., Craciun, E.M.: A generalization of the Saint-Venant’s principle for an elastic body with dipolar structure. Contin. Mech. Thermodyn. 32(1), 269–278 (2020). https://doi.org/10.1007/s001161-019-00827-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Liu, W., Chen, M.: Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback. Contin. Mech. Thermodyn. 29(3), 731–746 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Zampoli, V., Amendola, A.: Uniqueness, continuous dependence, and spatial behavior of the solution in linear porous thermoelasticity with two relaxation times. J. Therm. Stress. 42(12), 1582–1602 (2019)

    Article  Google Scholar 

  40. Marin, M., Othman, M.I., Vlase, S., Codarcea-Munteanu, L.: Thermoelasticity of initially stressed bodies with voids: a domain of influence. Symmetry 11(4), 573 (2019)

    Article  MATH  Google Scholar 

  41. Wei, J., Fu, L.Y.: The fundamental solution of porothermoelastic theory. In: 2nd SEG Rock Physics Workshop: Challenges in Deep and Unconventional Oil/Gas Exploration, p. 52. Society of Exploration Geophysicists (2020)

  42. Marin, M., Öchsner, A., Taus, D.: On structural stability for an elastic body with voids having dipolar structure. Contin. Mech. Thermodyn. 32(1), 147–160 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Othman, M.I.A., Marin, M.: Effect of thermal loading due to laser pulse on thermoelastic porous medium under GN theory. Results Phys. 7, 3863–3872 (2017)

    Article  ADS  Google Scholar 

  44. Sharma, M.D.: Wave propagation in thermoelastic saturated porous medium. J. Earth Syst. Sci. 117(6), 951 (2008)

    Article  ADS  Google Scholar 

  45. Carcione, J.M., Cavallini, F., Wang, E., Ba, J., Fu, L.Y.: Physics and simulation of wave propagation in linear thermoporoelastic media. J. Geophys. Res. Solid Earth 124(8), 8147–8166 (2019)

    Article  ADS  Google Scholar 

  46. Sur, A.: Wave propagation analysis of porous asphalts on account of memory responses. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1712553

  47. Alzahrani, F., Abbas, I.A.: Generalized thermoelastic interactions in a poroelastic material without energy dissipations. Int. J. Thermophys. 41, 1–13 (2020). https://doi.org/10.1007/s10765-020-02673-0

  48. Saeed, T., Abbas, I., Marin, M.: A GL model on thermo-elastic interaction in a poroelastic material using finite element method. Symmetry 12(3), 488 (2020)

    Article  Google Scholar 

  49. Alzahrani, F.S., Abbas, I.A.: Fractional order GL model on thermoelastic interaction in porous media due to pulse heat flux. Geomech. Eng. 23(3), 217–225 (2020)

    Google Scholar 

  50. Guo, Y., Xiong, C., Zhu, H.: Dynamic response of coupled thermo-hydro-elastodynamic problem for saturated foundation under GL generalized thermoelasticity. J. Porous Med. 22(13) (2019). https://doi.org/10.1615/JPorMedia.2019025579

  51. Green, A.E., Laws, N.: On the entropy production inequality. Arch. Ration. Mech. Anal. 45(1), 47–53 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  52. Stehfest, H.: Algorithm 368: numerical inversion of Laplace transforms [D5]. Commun. ACM 13(1), 47–49 (1970)

    Article  Google Scholar 

  53. Sherief, H.H., Dhaliwal, R.S.: Generalized one-dimensional thermal-shock problem for small times. J. Therm. Stress. 4(3–4), 407–420 (1981)

    Article  Google Scholar 

Download references

Acknowledgements

One of the authors, Om Namha Shivay, thankfully acknowledges the full financial assistance of the SRF Fellowship (Roll Number 433492, reference number 21/06/2015 (i) EU-V) by the University Grant Commission (UGC), India, to carry out the present work.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Om Namha Shivay & Santwana Mukhopadhyay

Authors
  1. Om Namha Shivay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Santwana Mukhopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Om Namha Shivay.

Additional information

Communicated by Andreas Ochsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shivay, O.N., Mukhopadhyay, S. A porothermoelasticity theory for anisotropic medium. Continuum Mech. Thermodyn. 33, 2515–2532 (2021). https://doi.org/10.1007/s00161-021-01030-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-021-01030-2

Keywords

Search

Navigation