It has been shown recently (Berryman and Wang, 2001) that local anisotropy in heterogeneous porous media must play a significant role in deviations from the well-known fluid substitution formulas of Gassmann (Gassmann, 1951; Berryman, 1999). In particular, it is not easy to see in an explicit way how fluid dependence of the effective shear modulus Geff of such systems can arise within a Gassmann-like derivation. Even though such effects have been observed in experimental data (Berryman et al., 2002a,b), locally isotropic materials have been shown to be incapable of producing such results. So a simple theoretical means of introducing anisotropy into such a poroelastic system is desirable in order to aid our physical intuition about these problems.
When viewed from a point close to the surface of the Earth, the structure of the Earth is often idealized as being that of a layered (or laminated) medium with essentially homogeneous physical properties within each layer. Such an idealization has a long history and is well represented by famous textbooks such as Ewing et al. (1957), Brekhovskikh (1980), and White (1983). The importance of anisotropy due to fine layering (i.e., layer thicknesses small compared to the wavelength of the seismic or other waves used to probe the Earth) has been realized more recently, but efforts in this area are also well represented in the literature by the work of Postma (1955), Backus (1962), Berryman (1979), Schoenberg and Muir (1987), Anderson (1989), Katsube and Wu (1998), and many others.
In a completely different context, because of the relative ease with which their effective properties may be computed, finely layered composite laminates have been used for theoretical purposes to construct idealized but, in principle, realizable materials to test the optimality of various rigorous bounds on the effective properties of general composites. This line of research includes the work of Tartar (1976), Schulgasser (1977), Tartar (1985), Francfort and Murat (1986), Kohn and Milton (1986), Lurie and Cherkaev (1986), Milton (1986), Avellaneda (1987), Milton (1990), deBotton and Castañeda (1992), and Zhikov et al. (1994), among others. Recent books on composites by Cherkaev (2000), Milton (2002), and Torquato (2002) also make frequent use of these ideas.
In this work, we will study some simple means of estimating the effects of fluids on elastic and poroelastic constants and, in particular, we will derive formulas for anisotropic poroelastic media using a straightforward generalization of the method of Backus (1962) originally formulated for determining the effective constants of a laminated elastic material. There has been some prior work in this area for poroelastic problems by Norris (1993), Gurevich and Lopatnikov (1995), Gelinsky et al. (1998), and others. However, our focus is specific to the issue of shear modulus dependence on pore fluids [see Mavko and Jizba (1991) and Berryman et al. (2002b)]. We initially review facts about elastic layered systems and then show that, of all the possible candidates for an effective shear modulus exhibiting mechanical dependence on pore fluids, the evidence shows that one choice is unambiguously preferred. We then use Backus averaging to obtain an explicit formula for this shear modulus in terms of layer elastic parameters. The results agree with prior physical arguments indicating that fluid presence stiffens the medium in shear, but the layered material needs substantial inhomogeneity in its shear properties for the effect to be observed. An Appendix provides a simple derivation of some useful product formulas that arise in the analysis.