The first scheme I consider is sometimes called the
Coherent Potential Approximation (CPA)
(Gubernatis and Krumhansl, 1975; Berryman, 1992; Berryman and Berge, 1996)
or the Self-Consistent Scheme (Korringa *et al.*, 1979; Berryman, 1980).

When there is no pore fluid present (*i.e.*, drained frame
conditions), the equations of poroelasticity reduce to those of elasticity
(general) for the
porous frame material. Within CPA, the idea is to treat all constituents on
an equal footing, so no single material serves as host medium for the
others. For this reason, the CPA is sometimes known as a symmetrical
self-consistent scheme. To find the formulas for the CPA, we
take the reference material to be the composite itself, so *r*=*.
The formula (general) reduces to

v_i (^(i)-^*_CPA)^*i = 0, where I have now approximated the unknown linear coefficient by the Eshelby-Wu tensor corresponding to inclusions of stiffness in host material of stiffness .

To make use of the generalization of Eshelby's formula for poroelasticity in the case when pore fluid and pore pressure are significant factors, I note that each inclusion is effectively imbedded in the composite material *, so it makes sense to consider the formula for inclusion strain

^(i) = e^*i(p_f) + ^*i[^* - e^*i(p_f)],
where the strain corresponding to equal expansion or contraction of
both materials *i* and * is given by

e^*i_pq =
(^*-^(i)K^*-K^(i))p_f3_pq.
If the mixture were composed only of the two materials *i* and *, then
the uniform expansion result would apply exactly. In the composite
poroelastic material, (CPAEshelby) should be viewed as an
estimate of the true strain of the *i*th constituent. This estimate
is conceptually on the same footing as that traditionally used when
saying that is a reasonable
approximation of the strain in the *i*th constituent of an elastic
composite, even though there may be many other types of materials
present.

To derive a formula within CPA for the Biot-Willis constant ,I want to make use of (CPAEshelby). For elasticity, the
average confining stress equals the total confining stress, so
. This fact was actually used to derive
(general). However, for poroelasticity with finite pore
pressure *p*_{f}, it is important to distinguish confining stress
from the stress in the solid components and so
it is no longer true that the average confining stress is equal
to the total confining stress, *i.e.*, . The correct relation for the pertinent stress is more complicated than
this. (We could learn some important things about our problem by studying
this issue, but the analysis becomes rather more technical than what
follows and it seems preferable to avoid this discussion here.)
However, it is still necessarily true that the average strain is
equal to the total strain, *i.e.*,

v_i ^(i) = ^*. Furthermore, this relation is just what is needed to make use of (CPAEshelby). Substituting (CPAEshelby) into (averagestrain) and then rearranging terms, I find that

v_i (- ^*i) e^*i(p_f) = v_i (- ^*i)^*, where is the identity matrix and I used the fact that .Equation (CPAalpha) is almost what I want, but the right hand side seems to be a problem, because it depends explicitly on , which is arbitrary. It is known however that (Hill, 1963; Berryman and Berge, 1996), and, since is my approximation to , it is clear that the right hand side of (CPAalpha) should be set identically to zero. Thus, after making use of (CPAstrain) in (CPAalpha), the CPA for is

v_i (1 - P^*i)^*_CPA-^(i)K^*_CPA - K^(i)
= 0,
where *P*^{*i} is the coefficient for the compressional
modulus, and the corresponding coefficient for the
shear modulus is *Q*^{*i} (see Table 1).
Some care should be taken however to check the degree of satisfaction of the
subsidiary condition to make
sure that it is at least approximately satisfied by the estimate obtained
for . It turns out that this condition is satisfied
exactly for spherical inclusions. Furthermore, in the case of spheres
I have ,and it is easy to show that (CPAalpha) reduces to

v_i (^(i)-^*_CPA)P^*i = 0, which should be compared to

v_i (K^(i)-K_CPA^*)P^*i = 0, which follows from (CPA).

TABLE 1. Four examples of coefficients *P* and *Q* for
spherical and nonspherical scatterers. The superscripts *h* and *i* refer to
host and inclusion phases, respectively.
Special characters are defined by ,, and .The expression for spheres, needles, and disks were derived by Wu (1966)
and Walpole (1969). The expressions for penny-shaped cracks were
derived by Walsh (1969) and assume *K*^{(i)}/*K*^{(h)} << 1 and
.The aspect ratio of the cracks is *a*.

7/5/1998