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 *G*_{eff} 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.

7/8/2003