How does the constant c1 depend on the fluid bulk modulus in a region of partial melt? The analysis usually quoted for addressing this problem in regions of partial melt have normally used some type of classical effective medium theory, which is most appropriately used for systems in which the inclusions are both disconnected and of small volume fraction . However, partial melt systems in the upper mantle are generally believed to be dominated by connected tubes of melt lying along grain edges (Waff and Bulau, 1979; Mavko, 1980; Toramaru and Fujii, 1986). When the fluid is in pressure-temperature equilibrium with its surroundings, it therefore makes sense to consider Gassmann's equations (Gassmann, 1951; Berryman, 1995) from the theory of poroelasticity for the system. This approach is particularly appealing for this problem because, except for an assumption of fluid connectedness, Gassmann's equations do not depend explicitly on the microgeometry, and this simplification should permit universal behavior to be predicted by the resulting theory.
Gassmann's equation for fluid substitution is often written to emphasize the change in saturated bulk modulus Ksat from that of the drained bulk modulus Kdr. The well-known result is
K_sat = K_dr + ^2(-)/K_g + /K_f, where is the fluid-saturated porosity, Kg is the solid or grain material bulk modulus, Kf is the bulk modulus of the saturating fluid (the melt for this application), and
= 1 - K_drK_g is the Biot-Willis (or effective stress) parameter. Formula (gassmann) can be rearranged to emphasize how the saturated bulk modulus changes as the value of Kf deviates from the value of the solid bulk modulus Kg. The result is
K_sat = K_g(1 - c_1), where
c_1 = K_g/K_f -11 + (/)(K_g/K_f -1). An important observation follows easily from (c1). If the fluid bulk modulus satisfies , then and the saturated bulk modulus is the same as that of the solid material. This is a definite prediction of Gassmann's formula. This is not a surprising result however, because it is also a quite general prediction of homogenization theory (for example, the well-known Hashin-Shtrikman bounds (Hashin and Shtrikman, 1963) also degenerate to a constant value when the constituents have the same moduli). If Kf = Kg then the bulk modulus is actually uniform throughout the medium. The point of (c1) is that it shows in addition how to compute deviations from this case when but . The result is independent of the details of the geometry of the melt system as long as the melt is connected (percolating) throughout the volume.