Liquid inclusion: $K_f \gt\gt \pi\alpha\gamma_m$

In this limit, the polarization factor for shear is given by

Q^*f 1c + 4 G^*15 K_f,   where

1c 15[3 + 8(1-_m)(2-_m)].   In this case, we have approximated $\gamma^* \simeq G^*/2(1-\nu_m)$ in order to decouple the G^* equation from the one for bulk modulus K^*. Note that as $y \to 1$, we anticipate $\nu^* \to 0.5$, so for $\nu^*$ in the usual range from to 0.5 the factor $(1-\nu)$varies at most by a factor of 2. Therefore, the condition on K_f is not affected. The parameter c depends on a factor $(1-\nu)/(2-\nu)$ which changes at most by a factor of 3/2. Thus, we expect some small deviations between the analytical formula and the full DEM for G^* in the liquid saturated case.

Also note that we could argue, in this limit, that the first term on the right hand side of (DEMQr1) is dominant and therefore the second term should be neglected. However, for purposes of comparison with Mavko and Jizba (1991), it will prove helpful to retain the second term.

Integrating the DEM equation, we have

1G^* + 4c15K_f = (1G_m + 4c15K_f)(1-)^-1c.   In the limit of small c (i.e., small $\alpha$)and $\phi \to 0$, we have

1G^* - 1G_m = (415K_f + 1cG_m)+ ...,   which should be contrasted with the result of Mavko and Jizba [1991] for the same problem

1G^* - 1G_dry = 415(1K^* - 1K_dry).   Because we need some other results to permit the analysis to proceed, a thorough comparison of the present results with the Mavko and Jizba formula will be postponed to the section on the ratio of compliance differences.