Transport in Porous Media

, Volume 54, Issue 1, pp 1–34 | Cite as

Three Pressures in Porous Media

  • Lynn S. Bennethum
  • Tessa Weinstein

Abstract

In a thermodynamic setting for a single phase (usually fluid), the thermodynamically defined pressure, involving the change in energy with respect to volume, is often assumed to be equal to the physically measurable pressure, related to the trace of the stress tensor. This assumption holds under certain conditions such as a small rate of deformation tensor for a fluid. For a two-phase porous medium, an additional thermodynamic pressure has been previously defined for each phase, relating the change in energy with respect to volume fraction. Within the framework of Hybrid Mixture Theory and hence the Coleman and Noll technique of exploiting the entropy inequality, we show how these three macroscopic pressures (the two thermodynamically defined pressures and the pressure relating to the trace of the stress tensor) are related and discuss the physical interpretation of each of them. In the process, we show how one can convert directly between different combinations of independent variables without re-exploiting the entropy inequality. The physical interpretation of these three pressures is investigated by examining four media: a single solid phase, a porous solid saturated with a fluid which has negligible physico-chemical interaction with the solid phase, a swelling porous medium with a non-interacting solid phase, such as well-layered clay, and a swelling porous medium with an interacting solid phase such as swelling polymers.

pressure porous media mixture theory swelling constitutive equations clay polymers 

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References

  1. Achanta, S. and Cushman, J. H.: 1994, Non-equilibrium swelling and capillary pressure relations for colloidal systems, J. Colloid Interf. Sci. 168, 266-268.Google Scholar
  2. Achanta, S., Cushman, J. H. and Okos, M. R.: 1994, On multicomponent, multiphase thermomechanics with interfaces, Int. J. Eng. Sci. 32(11), 1717-1738.Google Scholar
  3. Bear, J.: 1972, Dynamics of Fluids in Porous Media, Dover, New York.Google Scholar
  4. Bennethum, L. S.: 1994, Multiscale, hybrid mixture theory for swelling systems with interfaces, PhD Thesis, Purdue University, West Lafayette, Indiana, 47907.Google Scholar
  5. Bennethum, L. S. and Cushman, J. H.: 1996a, Multiscale, hybrid mixture theory for swelling systems. I. Balance laws, Int. J. Eng. Sci. 34(2), 125-145.Google Scholar
  6. Bennethum, L. S. and Cushman, J. H.: 1996b, Multiscale, hybrid mixture theory for swelling systems. II. Constitutive theory, Int. J. Eng. Sci. 34(2), 147-169.Google Scholar
  7. Bennethum, L. S. and Cushman, J. H.: 1999, Coupled solvent and heat transport of a mixture of swelling porous particles and fluids: single time-scale problem, Transport Porous Med. 36(2), 211-244.Google Scholar
  8. Bennethum, L. S. and Cushman, J. H.: 2002a, Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics. I. Macroscale field equations, Transport Porous Med. 47(3), 309-336.Google Scholar
  9. Bennethum, L. S. and Cushman, J. H.: 2002b, Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics. I. Constitutive theory, Transport Porous Med. 47(3), 337-362.Google Scholar
  10. Bennethum, L. S. and Giorgi, T.: 1997, Generalized forchheimer law for two-phase flow based on hybrid mixture theory, Transport Porous Med. 26(3), 261-275.Google Scholar
  11. Bennethum, L. S., Murad, M. A. and Cushman, J. H.: 1996, Clarifying mixture theory and the macroscale chemical potential for porous media, Int. J. Eng. Sci. 34(14), 1611-1621.Google Scholar
  12. Bennethum, L. S., Murad, M. A. and Cushman, J. H.: 1997, Modified Darcy's law, terzaghi's effective stress principle and fick's law for swelling clay soils, Comput. Geotech. 20(3/4), 245-266.Google Scholar
  13. Bennethum, L. S., Murad, M. A. and Cushman, J. H.: 2000, Macroscale thermodynamics and the chemical potential for swelling porous media, Transport Porous Med. 39(2), 187-225.Google Scholar
  14. Bowen, R. M.: 1976, Theory of mixtures, in: A. C. Eringen (ed.), Continuum Physics, Academic Press, New York.Google Scholar
  15. Bowen, R. M.: 1982, Compressible porous media models by use of the theory of mixtures, Int. J. Eng. Sci. 20, 697-735.Google Scholar
  16. Callen, H. B.: 1985, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York.Google Scholar
  17. Coleman, B. D. and Noll, W.: 1963, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. An. 13, 167-178.Google Scholar
  18. Darcy, H.: 1856, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris.Google Scholar
  19. de la Cruz, V., Sahay, P. N. and Spanos, T. J. T.: 1993, Thermodynamics of porous media, P. Roy. Soc. Lond. A. Mat. 443, 247-255.Google Scholar
  20. Eringen, A. C.: 1967, Mechanics of Continua, Wiley, New York.Google Scholar
  21. Gray, W. G.: 1999, Thermodynamics and constitutive theory for multiphase porous-media flow considering internal geometric constraints, Adv. Water Resour. 22(5), 521-547.Google Scholar
  22. Gray, W. G. and Hassanizadeh, S. M.: 1998, Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines, and common points, Adv. Water Resour. 21(4), 261-281.Google Scholar
  23. Hassanizadeh, S.M.: 1986a, Derivation of basic equations of mass transport in porous media. Part 1. Macroscopic balance laws, Adv. Water Resour., 9, 196-206.Google Scholar
  24. Hassanizadeh, S. M.: 1986b, Derivation of basic equations of mass transport in porous media. Part 2. Generalized Darcy's and Fick's laws, Adv. Water Resour. 9, 207-222.Google Scholar
  25. Hassanizadeh, S.M. and Gray,W. G.: 1979a, General conservation equations for multiphase systems. 1. Averaging procedure, Adv. Water Resour. 2, 131-144.Google Scholar
  26. Hassanizadeh, S.M. and Gray,W. G.: 1979b, General conservation equations for multiphase systems. 2. Mass, momenta, energy, and entropy equations, Adv. Water Resour. 2, 191-208.Google Scholar
  27. Hassanizadeh, S.M. and Gray, W. G.: 1980, General conservation equations for multiphase systems. 3. Constitutive theory for porous media, Adv. Water Resour. 3, 25-40.Google Scholar
  28. Hassanizadeh, S. M. and Gray, W. G.: 1987, High velocity flow in porous media, Transport Porous Med. 2, 521-531.Google Scholar
  29. Hassanizadeh, S. M. and Gray, W. G.: 1990, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resour. 13, 169-186.Google Scholar
  30. Hassanizadeh, S. M. and Gray, W. G.: 1993a, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res. 29(10), 3389-3405.Google Scholar
  31. Hassanizadeh, S. M. and Gray, W. G.: 1993b, Toward an improved description of the physics of two-phase flow, Adv. Water Resour. 16, 53-67.Google Scholar
  32. Low, P. F.: 1980, The swelling of clay. II. Montmorillonites-water systems, Soil Sci. Soc. Am. J. 44, 667-676.Google Scholar
  33. Low, P. F.: 1987, Structural component of the swelling pressure of clays, Langmuir 3, 18-25.Google Scholar
  34. Low, P. F.: 1994, The clay/water interface and its role in the environment, in: Progress in Colloid and Polymer Science, Vol. 40, pp. 500-505.Google Scholar
  35. Malvern, L. E.: 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  36. Murad, M. A., Bennethum, L. S. and Cushman, J. H.: 1995, A multi-scale theory of swelling porous media. I. Application to one-dimensional consolidation, Transport Porous Med. 19, 93-122.Google Scholar
  37. Nemat-Nasser, S.: 2002, Micromechanics of actuation of ionic polymer-metal composites, J. Appl. Phys. 92(5), 2899-2915.Google Scholar
  38. Pride, S. R., Gangi, A. F. and Morgan, F. D.: 1992, Deriving the equations of motion for porous isotropic media, J. Acoust. Soc. Am. 92(6), 3278-3290.Google Scholar
  39. Spanos, T. J. T.: 2001, The Thermodynamics of Porous Media, Chapman & Hall/CRC Press, New York.Google Scholar
  40. Terzaghi, K.: 1943, Theoretical Soil Mechanics, Wiley, New York.Google Scholar
  41. Thomas, N. L. and Windle, A. H.: 1982, A theory of case II diffusion, Polymer 23, 529-542.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Lynn S. Bennethum
    • 1
  • Tessa Weinstein
    • 1
  1. 1.Center for Computational MathematicsUniversity of Colorado at DenverDenverUSA

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