Analysis of the solid phase stress tensor in multiphase porous media

William G. Gray, Bernhard A. Schrefler

Research output: Contribution to journalArticlepeer-review

127 Scopus citations


Conservation equations for mass, momentum, energy, and entropy are formulated for the phases and interfaces of a three-phase system consisting of a solid and two immiscible fluids. The microscale equations are averaged to the macroscale by integration over a representative elementary volume. Thermodynamic statements for each of the phases and interface entities are also formulated at the microscale and then averaged to the macroscale. This departure from most uses of thermodynamics in macroscale analysis ensures consistency between models and parameters at the two scales. The expressions for the macroscale rates of change of internal energy are obtained by differentiating the derived forms for energy and making use of averaging theorems. These thermodynamic expressions, along with the conservation equations, serve as constraints on the entropy inequality. A linearization of the resulting equations is employed to investigate the theoretical origins of the Biot coefficient that relates the hydrostatic part of the total stress tensor to the normal force applied at the solid surface by the pore fluids. The results here are placed in the context of other formulations and expressions that appear in the literature.

Original languageEnglish (US)
Pages (from-to)541-581
Number of pages41
JournalInternational Journal for Numerical and Analytical Methods in Geomechanics
Issue number4
StatePublished - Apr 10 2007


  • Biot coefficient
  • Bishop parameters
  • Elasticity
  • Porous media
  • Skempton stress

ASJC Scopus subject areas

  • Geotechnical Engineering and Engineering Geology
  • Materials Science(all)
  • Mechanics of Materials
  • Computational Mechanics


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