My subject is the treatment of rocks -- and, especially, fluid-saturated and partially saturated rocks -- as composite poroelastic media. By this I mean to study and partially answer the question of how the elastic/poroelastic constants of the rock can be estimated from a knowledge of the constituents of the rock, their volume fractions, and possibly the geometry of individual grains and/or pores -- when that information is also available. The main new results I obtain concern means of estimating the effective-stress coefficient known as the Biot-Willis parameter (Biot and Willis, 1957) and the fluid storage coefficient known as Skempton's coefficient (Skempton, 1954). These two are the key new parameters arising in generalizing from elasticity to poroelasticity, and the ones that are not accounted for in previous theories of elastic/poroelastic composites like rocks.
Having an identity analogous to Eshelby's classic result (Eshelby, 1957) -- for the response of a single ellipsoidal elastic inclusion in an elastic whole space to a strain imposed at infinity -- available in more complex problems in composites analysis (such as poroelastic or thermoelastic composites) is of great practical interest. In Biot-Gassmann poroelasticity (Biot, 1941; Gassmann, 1951; Biot, 1962), 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 an externally applied stress or strain. With a rigorous generalization of Eshelby's formula valid for poroelasticity, the hard part of Eshelby's work (in computing the elliptic integrals needed to evaluate the fourth-rank tensors for inclusions shaped like spheres, oblate and prolate spheroids, needles, disks and/or penny-shaped cracks) can be carried over to these new results with only trivial modifications. Then, effective medium theories for poroelastic composites like rocks can be formulated easily by analogy to well-established theories for elastic composites (Korringa et al., 1979; Berryman, 1980; 1995).
The author (Berryman, 1997) has discovered a simple mathematical trick, applicable to media having isotropic constituents and based on a linear combination of results from two thought experiments, that makes the derivation of such a generalization of Eshelby's formula to poroelasticity an elementary task. In earlier work (Berryman, 1985; 1992), the problem of acoustical scattering by a spherical inhomogeneity of one poroelastic material imbedded in another was solved and the results then used to construct various single-scattering-based effective medium theories. The Eshelby generalization now permits incorporation of Eshelby's results for arbitrary ellipsoidal-shaped inclusions into both quasistatic formulations of effective medium theory and/or into scattering formulas. The resulting improved estimates of poroelastic material properties has important applications in geothermal and oil reservoir modeling, nuclear waste disposal, and hydrology, among others.
Generalization of almost all effective medium theories [see Berryman and Berge (1996) for a discussion] now can proceed more easily into the complex realm of poroelastic composites by making use of this generalization of Eshelby's results.