Now, when we want to distinguish the properties of the solids *A* and *B*, we can break up the averaging volume into two
pieces such that

<**u**_s> = <**u**_A> +
<**u**_B>,
which follows from the fact that material *A* and *B* occupy disjoint parts of the
averaging volume.
The averaging theorem for the divergence of alone then states that

<**u**_A> = <**u**_A> +
1V_I_f B<>n_A **u**_A dS
+ 1V_I_B B<>n_A **u**_A dS,
where we have explicitly noted that the interior interface has two parts:
one boundary with the fluid (or pore space) and one with the other solid (*B*).

Since

<**u**_A> = 1V _E_A B<>n_A**u**_A dS,
and a similar expression for , we find easily from
the identity

1V _E_s B<>n_s**u**_s dS =
1V _E_A B<>n_A**u**_A dS +
1V _E_B B<>n_B**u**_B dS
that

In order to determine the physical significance of , we
need to repeat the analysis for all solids, taking into account the fact that when
there are two or more solids there must also be additional interior interfaces between
these various constituents.
Of the four terms in (averaginguA), each requires some interpretion.
First, the left hand side has an interpretation similar to that of the left hand side
of (divergencetheorem).
Thus, we have the volume average of the dilatation of *A* material must be

<**u**_A> = [V_A(1-_A)]V,
where *V*_{A} is the total porous volume of *A* material and is the total solid volume
of *A* material.

The two integrals on the right hand side of (averaginguA) are more difficult to interpret because they involve the
contact region of two porous materials having possibly different porosities.
Statistically the *A* material should have solid material at this interface
occupying the fraction of the total interface area and *A*-pore the remaining fraction of the total.
The *B* material has corresponding proportions. Now these continuous surfaces may be
statistically correlated or uncorrelated. If uncorrelated, we can easily compute
the coefficients we will need. But if they are correlated, we must introduce
some new constants with the following properties: is the
fraction of the interface on which solid *A* touches solid *B*, is
the fraction of the interface on which solid *A* touches the fluid in *B*,
and similarly is the fraction on which the solid *B* touches the fluid in
*A*. Within our general assumption of statistical homogeneity, these constants should
obey the relations and
.One immediate general result is that the difference .The solid/solid contact area should be proportional to
, which may be very small or it can be as large as the minimum of the two
solid fractions , .For uncorrelated surfaces, we expect ,, and .These identities are easily shown to satisfy the statistical sum rules for these
coefficients.
For correlated interfaces,
we may view as a new microstructural parameter that characterizes the
internal (to the averaging volume) solid/solid interface.

The first integral on the right hand side of (averaginguA) is the surface integral of displacement along the
fluid boundary. This term has the same significance as the corresponding one for the
whole solid; it is the change in porosity associated with *A* material.
The total pore volume associated with *A*
is , so the change in pore volume must be a change in this quantity. However,
the surface integral is strictly over the original boundary of the *A* material (prior to
the displacements ), so the correct expression for this change in the absence
of other solids is clearly . But, in the presence of other solids, we
must account for the possibility of changes in overall porosity due to changes in volume fraction.
Thus, the full contribution of this term is

_I_f B<>n_A **u**_A dS
= - V_A_A + V_Afv_A,
using one of the constants introduced in the preceding paragraph.
When the volume fraction does not change, as in the case when the averaging volume happens to
contain only *A* material, we see that this expression reduces correctly to
(porositychange2).
When we write the corresponding relation for the *B* phase and then consider that
it must be true that

_I_f B<>n_s**u**_s dS =
_I_f B<>n_A**u**_A dS +
_I_f B<>n_B**u**_B dS,
then we see that the extra terms proportional to change in volume fraction are
exactly what were needed to guarantee that
(porositytotals) is equivalent to
.

The second integral on the right hand side of (averaginguA) is the surface integral over the *AB* solid/solid
interface.
This term also has the important characteristic that it must be exactly the
negative of the corresponding term for the *B* material. So however we interpret it,
the expression should be easily identified by the fact that interchanging *A* and *B*
should change the sign of the term.
Making the identification that

1V_I_B B<>n_A **u**_A dS
_AB v_A,
where is the change in volume fraction of porous constituent
*A*, we find that this interpretation is reasonable. If an imaginary continuous surface
between the porous constituents is drawn and the corresponding surface integral taken,
then the result would be exactly .Since the *AB* interface occupies only a fraction of this total interface area,
we see that (tirdterm) follows.

The remaining term to be interpreted in (averaginguA) is proportional to the surface integral of the *A* component
displacement. Combining the previous results, this average must be given by

_E_A B<>n_A**u**_A dS =
(1-_A)[V_A - Vv_A]
= (1-_A)v_AV,
with a matching expression for the *B* phase. That these two integrals must satisfy the
sum rule in (sumofsurface) together with (divergenceofaverage2) implies that their sum
must be equal to which is easily seen to be true.

Comparing all these expressions, we finally obtain the result

<**u**_A> = v_A(1-_A)V_AV_A - (1-_A)v_A,
which is the desired expression for divergence of the average displacement of *A*.

Although the divergence of is given rigorously by (gendivuA), the quantities that actually appear in the quasistatic equations of motion are simply the dilatations of the constituents, so we will define a new quantity satisfying

B<>u_A V_AV_A, which is related to by

B<>u_A = B<>u_A -
v_Av_A,
where the change in volume fraction is itself related to and the corresponding
expression of *B* by .These definitions and interrelations will be important for our analysis of wave propagation issues
for multicomponent rocks.

11/12/1997