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 .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) = Km, i.e., the bulk modulus of the pure solid. Then, integrating (DEMK1) from y = 0 to ( 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 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 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 gives (DEMKr12) to first order in . Note however that Gassmann's full equation (Gassmann) has the further advantage that it is valid for all values of Kf (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.