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Appendix A: Effective Stress Coefficient and Partial Saturation

Although Eq. (20) for the overall volume effective stress coefficient $\alpha^*$ is known to be true for homogeneous pore saturation, we also need to have a corresponding result here for patchy saturation. It turns out that the same formula applies for arbitrarily patchy saturated media, as long as there are only two types of solid components. To show this, consider  
{c} \delta e^{(n)} \\ -\delta\zeta^{(n)}...
{c} -\delta p_c \\  -\delta p_f^{(n)} \end{array}\right),
 \end{displaymath} (26)
where $\delta e^{(n)}$ and $\delta\zeta^{(n)}$ are the change in overall strain and the increment of fluid content in component n, where n = 1,2. [There are also similar formulas for all the overall properties with (n) replaced by * for the corresponding effective properties. See Berryman and Wang (1995) for more discussion.] Similarly, the change in overall confining (external) pressure is $\delta p_c$, and the pore pressure change of component n is $\delta p_f^{(n)}$.The porous material coefficients are defined as in the main text, Kd(n) is the drained bulk modulus, $\alpha^{(n)}$ is the volume effective stress coefficient, and B(n) is Skempton's coefficient for the n-th constituent.

Now the rest of the argument follows that given in Berryman and Milton (1991) exactly, since it is not important what fluid is in the pores when trying to determine the overall effective stress coefficient at long times (when fluid pressure in the system has equilibrated). We simply postulate the existence of any fixed ratio $r = \delta p_f^{(1)}/\delta p_c = \delta p_f^{(2)}/\delta p_c$such that $\delta e^{(1)} = \delta e^{(2)}$. If there is such a ratio (valid at appropriately long times), then $\delta e^* = \delta e^{(1)} = \delta e^{(2)}$also follows immediately and we have the condition that must be satisfied:  
\frac{\delta p_c}{K_d^{(1)}}\left[1-\alpha^{(1)}r\right] =
\frac{\delta p_c}{K_d^{(2)}}\left[1-\alpha^{(2)}r\right],
 \end{displaymath} (27)
which is just a linear relation for ratio r. The result shows that the postulated value of r does exist unless the denominator of the following expression vanishes:  
r = 
 \end{displaymath} (28)
If the numerator of (28) vanishes, the results are trivial because Gassmann's microhomogeneity condition is then satisfied. Once the value of r is known, it is easy to see that $\delta e^* = \delta e^{(1)} = \delta e^{(2)}$ implies  
\frac{\delta p_c}{K_d^*}\left[1-\alpha^*r\right] =
\frac{\delta p_c}{K_d^{(1)}}\left[1-\alpha^{(1)}r\right].
 \end{displaymath} (29)
This equation can be rearranged into the form (20), as has been shown previously by Berryman and Milton (1991).

Arguments similar to the one just given have also been used, for example, in the context of thermal expansion by Benveniste and Dvorak (1990) and Dvorak and Benveniste (1997), who call this approach ``the theory of uniform fields.'' It turns out this method is not restricted to isotropic constituents as one might infer from the arguments presented here and also in (Berryman and Milton, 1991).

A somewhat more difficult task than the one just accomplished involves deducing the overall effective pore bulk modulus $K_\phi^*$ as was also done previously by Berryman and Milton (1991). However, this coefficient does not play any direct role in our present analysis, so we will leave this exercise to the interested reader.

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