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

In this limit, it is somewhat more convenient to rewrite the DEM equations in terms of compliances, rather than stiffnesses, so we have

(1-y)ddy(1K^*(y)) = [K^*(y)-K_f]P^*f/(K^*)^2 = (1K_f - 1K^*).   The only terms that couple the equations for bulk to shear have been readily neglected in this case, since $P^{*i} \simeq K^*/K_f$.Thus, we expect little if any deviation between the analytical results and the full DEM for the liquid saturated case. We are treating here the granite-like case such that the limit of zero inclusion volume fraction corresponds to K^*(0) = K_m, i.e., the bulk modulus of the pure solid. Then, integrating (DEMK1) from y = 0 to $y=\phi$ ($\phi$ is the resulting porosity in the composite medium) gives directly

(1K_f- 1K^*) = (1K_f- 1K_m)(1-).   which may be rearranged as

(1K^*- 1K_m) = (1K_f- 1K_m).  

Eqn.(DEMKr12) can also be obtained as the small $\phi$ limit of Gassmann's equation when the saturating fluid is a liquid. Gassmann's result for the bulk modulus (Gassmann, 1951) is expressible as

1K^* = 1-BK_dry,   where $\theta = 1 - K_{dry}/K_m$ is the Biot-Willis parameter (Biot and Willis, 1957), and B is Skempton's coefficient (Skempton, 1954)

B = /K_dry/K_dry + (1/K_f - 1/K_m).   Expanding (Gassmann) for small $\phi$ gives (DEMKr12) to first order in $\phi$. Note however that Gassmann's full equation (Gassmann) has the further advantage that it is valid for all values of K_f (right down to zero), not just for values in the liquid range.

Eqn.(DEMKr12) is also the result of Mavko and Jizba (1991) for a granite-like material under high confining pressure so that the crack-like pores are closed. Their result is stated for a sandstone-like material including both crack-like pores and other pores. But since we have not considered the presence of any other pores except the crack-like pores in this argument, the correct comparison material is just the mineral matrix.

Appendix A shows how to obtain the result of Mavko and Jizba (1991) from a modified DEM scheme.