Differential effective medium (DEM) 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
effective medium 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 (Cleary *et al.*, 1980;
Norris, 1985; Avellaneda, 1987) -- and one version of the
self-consistent approach (Korringa *et al.*, 1979;
Berryman, 1980a,b; Milton, 1985) is the other. This fact,
together with the associated analytical capabilities (including ease of
computation and flexibility of application),
provides strong motivation to study the predictions of both of these
schemes and the differential scheme in particular.
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 the inclusions
are sufficiently sparse that they do not form a single connected
network throughout the composite,
it is most appropriate to use the Differential Effective Medium (DEM)
to model their elastic behavior (Berge *et al.*, 1993).
Assume that the host material has moduli *K*_{m} and ,while the inclusion material has moduli *K*_{i} and . Then,
the effective bulk and shear moduli (indicated as such by the asterisks)
of the composite are parametrized by *K ^{*}*(

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

(1-y)d^*(y)dy = [_i-^*(y)]Q^*i,
where the scalar factors, *P*^{*i} and *Q*^{*i}, will be explained in the
following paragraph,
*y* is porosity which equals inclusion volume fraction here, and
the subscript *i* again stands for inclusion phase. We assume that
the reader is somewhat familiar with this approach, and will therefore
not dwell on its derivation, which is easily found in many places
including, for example, Berryman (1992).
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 (DEMmu) are the so-called
polarization factors for bulk and shear modulus (Eshelby, 1957; Wu, 1966).
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 usually have 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) and Mavko *et al.*
(1998).

Because it is relevant both to low porosity granites and to sandstones
having equant (*i.e.*, close to spherical) porosity as well as flat cracks,
the case we consider here is that of penny-shaped cracks, where

P^*i = K^* + 43_iK_i+43_i+^* and

Q^*i = 15[1 + 8^*4_i+ (^*+2^*) + 2K_i + 23(_i+^*) K_i+43_i+^*], with ()being the crack (oblate spheroidal) aspect ratio,

^* ^*[(3K^*+^*)/(3K^* +4^*)],
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 and allows *K*_{i}/*K*_{m} << 1,
with these approximations being made before any 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
quite inappropriate for the bulk modulus when the inclusion phase
is a gas such as air (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 cracked rocks.

In general the DEM equations (DEMK) and (DEMmu) 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. Later in the paper, we will show results obtained from integrating the DEM equations numerically.

11/11/2002