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Rocks as poroelastic composites

James G. Berryman


In Biot's theory of poroelasticity, elastic materials contain connected voids or pores and these pores may be filled with fluids under pressure. The fluid pressure then couples to the mechanical effects of stress or strain applied externally to the solid matrix. Eshelby's formula (for the response of a single ellipsoidal elastic inclusion in an elastic whole space to a strain imposed at infinity) is a very well-known and important result in elasticity. The hardest technical part of Eshelby's work was in computing the elliptic integrals needed to evaluate the fourth-rank tensors for inclusions shaped like spheres, oblate and prolate spheroids, needles and disks. Having a rigorous generalization of Eshelby's results valid for poroelasticity means that the hard part of Eshelby's work can be carried over from elasticity to poroelasticity - and also thermoelasticity - with only trivial modifications. Effective medium theories for poroelastic composites such as rocks can then be formulated easily by analogy to well-established methods used for elastic composites. An identity analogous to Eshelby's classic result has been previously derived by the author for use in these more complex and more realistic problems in rock mechanics analysis. Using these results as the starting point for new methods of estimation, I apply these techniques to the Biot-Willis parameter, which is the technical name for the effective-stress coefficient for total volume strain. The results show that poroelastic parameters can now be estimated as easily as elastic parameters for arbitrary ellipsoidal inclusions using any of the standard effective medium theories.

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