Results for constituents A and B

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 ${\bf u}_A$ 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 $\partial I_f$ with the fluid (or pore space) and one $\partial I_B$ with the other solid (B).

Since

<u_A> = 1V _E_A B<>n_Au_A dS,   and a similar expression for $\nabla\cdot{\bf u}_B$, we find easily from the identity

1V _E_s B<>n_su_s dS = 1V _E_A B<>n_Au_A dS + 1V _E_B B<>n_Bu_B dS   that

<u_s> = <u_A> + <u_B>.  

In order to determine the physical significance of $\nabla\cdot\left<{\bf u}_A\right\gt$, 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 $V_A(1-\phi_A)$ 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 $1-\phi_A$ of the total interface area and A-pore the remaining fraction $\phi_A$ 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: $\eta_{AB} = \eta_{BA}$ is the fraction of the interface on which solid A touches solid B, $\eta_{Af}$ is the fraction of the interface on which solid A touches the fluid in B, and similarly $\eta_{Bf}$ 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 $\eta_{AB} + \eta_{Af} = 1 -\phi_A$ and $\eta_{BA} + \eta_{Bf} = 1 - \phi_B$.One immediate general result is that the difference $\eta_{Af}-\eta_{Bf} = \phi_B - \phi_A$.The solid/solid contact area should be proportional to $\eta_{AB}$, which may be very small or it can be as large as the minimum of the two solid fractions $(1-\phi_A)$, $(1-\phi_B)$.For uncorrelated surfaces, we expect $\eta_{AB} = (1-\phi_A)(1-\phi_B)$,$\eta_{Af} = (1-\phi_A)\phi_B$, and $\eta_{Bf} = (1-\phi_B)\phi_A$.These identities are easily shown to satisfy the statistical sum rules for these coefficients. For correlated interfaces, we may view $\eta_{AB}$ 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 $V_A\phi_A$, 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 ${\bf u}_A$), so the correct expression for this change in the absence of other solids is clearly $V_A\delta\phi_A$. 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_su_s dS = _I_f B<>n_Au_A dS + _I_f B<>n_Bu_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 $\delta\phi= v_A \delta\phi_A + v_B \delta\phi_B + (\phi_A -\phi_B)\delta v_A$.

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 $\delta v_A$ 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 $\delta v_A$.Since the AB interface occupies only a fraction $\eta_{AB}$ 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_Au_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 $(1-\phi)\delta V$ 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 $\left<{\bf u}_A\right\gt$ 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 $\tilde{\bf u}_A$ satisfying

B<>u_A V_AV_A,   which is related to $\bar{\bf u}_A$ by

B<>u_A = B<>u_A - v_Av_A,   where the change in volume fraction is itself related to $\tilde{\bf u}_A$ and the corresponding expression of B by $\delta v_A = v_Av_B(\nabla\cdot\tilde{\bf u}_A - \nabla\cdot\tilde{\bf u}_B)$.These definitions and interrelations will be important for our analysis of wave propagation issues for multicomponent rocks.