The inverse problem for inhomogeneous frames in the Brown and Korringa
picture is complicated by the presence of three bulk moduli characterizing
the porous frame -- all three of which must be determined from the same data.
Measurements of the three ultrasonic velocities of poroelasticity
(slow compressional, shear, and fast compressional) may be used
to determine pore-space tortuosity, shear modulus, and saturated bulk modulus,
respectively, for a single fluid saturant. In principle, three more constants
(the drained frame modulus and the two additional bulk moduli of Brown and
Korringa) can be determined by repeating the ultrasonic measurements on the
same porous sample when it is saturated with different fluids
(*e.g.*, air and hydrocarbons). The theory suggests therefore that,
in principle, the inversion problem can be solved. However, available real
ultrasonic data on rocks (Wang, 1988) shows paradoxically (but conclusively)
either
that the effective porosity at ultrasonic frequencies is substantially smaller
than the measured porosity or that the effective drained bulk modulus is larger
than the measured air-dry modulus. Thus, either the effective porosity or the
effective drained modulus must also be determined from the
data, giving us one too many unknowns. The effective porosity might
be interpreted as resulting from a type of high-frequency dispersion caused
by the inability of the entire mass of pore fluid to respond instantaneously
to acoustically induced pressure changes. Such an interpretation would allow
us to estimate an effective porosity as a function of frequency and thereby
resolve ambiguities in the inverse problem. However, to establish this
mechanism as the correct one, we had to establish first that there were
no static effects that could be the true cause. It is at least
equally likely that the effective drained modulus of a rock differs
from the air-dry modulus when the rock has complex mineralogy. These
considerations show that the process of inverting ultrasonic data
for Biot-Gassmann parameters must incorporate other physical constraints
and then produces a range of possible values for each coefficient.

11/17/1997