How does the constant *c _{1}* 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
(

Gassmann's equation for fluid substitution is often written
to emphasize the change in saturated bulk modulus *K*_{sat} from that
of the drained bulk modulus *K*_{dr}. The well-known result is

K_sat = K_dr + ^2(-)/K_g + /K_f,
where is the fluid-saturated porosity, *K*_{g} is the solid or grain
material bulk modulus, *K*_{f} 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 *K*_{f} deviates from
the value of the solid bulk modulus *K*_{g}. The result is

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 *K*_{f} = *K*_{g} 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.

10/25/1999