Differential effective medium theory
(Bruggeman, 1935; Cleary *et al.*, 1980; Walsh, 1980;
Norris, 1985; Avellaneda, 1987)
takes the point of view
that a composite material may be constructed by making infinitesimal
changes in an already existing composite. There are only two schemes
known at present that are realizable, *i.e.*, that have a definite
microgeometry associated with the modeling scheme. The differential
scheme is one of these (Norris, 1985; Avellaneda, 1987). This fact
provides a strong motivation to study the predictions of this theory
because we can have confidence that the results will always satisfy
physical and mathematical constraints, such as the Hashin-Shtrikman
bounds (Hashin and Shtrikman, 1961; 1962).

When inclusions
are sufficiently sparse that they do not form a single connected
network throughout the composite,
it is appropriate to use the Differential Effective Medium (DEM)
to model their elastic behavior (Berge *et al.*, 1993).
If the effective bulk and
shear constants of the composite are *K ^{*}*(

(1-y)dK^*(y)dy = [K_i-K^*(y)]P^*i and

(1-y)dG^*(y)dy = [G_i-G^*(y)]Q^*i,
where *y* is also porosity in the present case and
the subscript *i* stands for inclusion phase.
These equations are typically integrated starting from porosity
*y* = 0 with values *K ^{*}*(0) =

The factors *P*^{*i} and *Q*^{*i} appearing in (DEMK)
and (DEMG) are the so-called
polarization factors for bulk and shear modulus.
These depend in general on the bulk and shear moduli of both the
inclusion, the host medium (assumed to be the existing composite
medium
* in DEM), and on the shapes of the inclusions. The polarization
factors have usually been computed from Eshelby's well-known
results (Eshelby, 1957) for ellipsoids, and Wu's work (Wu, 1966)
on identifying the isotropically averaged tensor based on
Eshelby's formulas. These results can be found
in many places including Berryman (1980b, 1995) and Mavko *et al.*
(1998).

The special case of most interest to us here is that for penny-shaped cracks, where

P^*i = K^* + 43G_iK_i+43G_i+^* and

Q^*i = 15[1 +
8G^*4G_i+ (G^*+2^*) +
2K_i + 23(G_i+G^*)
K_i+43G_i+^*],
with being the crack (oblate spheroidal) aspect ratio,
, and
where the superscript * identifies constants of the matrix material
when the inclusion volume fraction is *y*.
This formula is a special limit of Eshelby's results not included in
Wu's paper, but apparently first obtained by Walsh (1969).
Walsh's derivation assumes that *K*_{i}/*K*_{m} << 1 and *G*_{i}/*G*_{m} << 1,
and makes these approximations before making assumptions about smallness
of the aspect ratio . By taking these approximations in the opposite
order, *i.e.*, letting
aspect ratio be small first and then making assumptions about
smallness
of the inclusion constants, we would obtain instead the commonly used
approximation for disks. But this latter approximation is actually
inappropriate for the bulk modulus when the inclusion phase
is air or gas (for then the ratio *K*_{i}/*K*_{m} << 1)
or for the shear modulus when the inclusion phase is
any fluid (for then ),
as the formulas become singular in these limits.
This is why the penny-shaped crack model is commonly used instead
for rocks.

In general the DEM equations (DEMK) and (DEMG) are coupled, as both equations depend on both the bulk and shear modulus of the composite. This coupling is not a serious problem for numerical integration, and we will show results obtained from integrating the DEM equations numerically later in the paper. But, the coupling is a problem in some cases if we want analytical results to aid our intuition. We will now present several analytical results for both bulk and shear modulus, and then compare these results to the fully integrated DEM results later on.

4/29/2001