There are various ways of trying to resolve the apparent paradox found
in our attempt to invert ultrasonic data to find Biot-Gassmann parameters.
Space limitations prevent us from elaborating on these possibilities
at length in the present paper, but we can give a brief discussion of
the main points. The three most likely sources of the apparent discrepancies
are these: (1) The assumption of negligible velocity dispersion may be invalid.
(2) The ``effective porosity'' at high frequencies may be smaller than the
measured porosity. (3) The appropriate value of the drained modulus *K*_{d}
may not be that of the air-dry porous material, but another value that must
also be determined from the data.

Since the theory does very well at predicting velocities in glass bead samples at ultrasonic frequencies (see Figure 1), it seems unlikely that any small effect of velocity dispersion we have neglected is likely to explain the observed large anomaly.

For analysis of fluid flow through porous media, it is known that the effective
porosity is less than true porosity. When a porous sample has been drained of
water, it is possible to measure the *irreducible water saturation*
*S*_{wi} (Vernik and Nur, 1992). Then, the effective porosity is given by
. However, the change in effective porosity
obtained has generally been shown (Vernik and Nur, 1992) to be substantially
less than what would be needed to explain the anomalous results obtained by our
inversion method. The irreducible water saturation *S*_{wi} thus provides
a measure of the *static* effective porosity. The *dynamic* effective
porosity we seek presumably depends on the magnitude of the fluid permeability
and is also presumably less than the static effective porosity,
but it is difficult to estimate at present how much less. So, while
we believe that it must be the case that
the effective porosity is less than that measured, we think it is unlikely that
the entire observed anomaly is due to a reduction in effective porosity.

Of the three possibilities we are considering,
the one remaining is that the value of *K*_{d} found by both draining
and subsequently drying the samples is not the appropriate value to be
used in Gassmann's equation for some types of porous materials. It has often
been noted that the presence of water in a rock containing clay minerals
can have a strong effect on the mechanical properties of the rock.
The effect can either strengthen or weaken the rock depending on
details of the microstructure and the actual fluid saturant. Clay tends
to soften and expand in the presence of a polar fluid like water.
Thus, the shear modulus might be expected to decrease in the absence of
confining pressure; however, in the presence of confining pressure, the
fact that the clay expands may dominate the softening effect and tend to
strengthen the sample overall (in the same way that an increase in pore
pressure tends to strengthen a rock). Gregory (1976) and Mavko and
Jizba (1991) also show examples
of anomalous behavior in low porosity rocks and attribute the anomalies to
the presence of microcracks that are open at low confining pressure, but
closed at high confining pressure. While the microcracks remain open, they
introduce nonlinear behavior into the rock (at low confining stress)
since the crack closure process dominates the mechanical response
until a critical pressure is achieved above which essentially all of
the cracks remain closed. However, for fixed confining pressure
(as is normally provided in ultrasonic experiments), the main contribution of
the microcracks should appear as a small reduction in the effective
porosity from the value measured in the unconfined state.
We conclude that there can be dramatic differences between the measured
*K*_{d} for air-dry samples and the effective *K*_{d} for a saturated sample.
We consider this effect to be the most likely explanation of the observed
anomaly.

Space limitations prevent us from exploring these ideas in greater depth here.
However, we have shown elsewhere that it is possible to obtain sensible values
of Biot-Gassmann parameters by treating the drained modulus *K*_{d} as an
independent parameter, computing the slope *b* and intercept *a* for each
assumed value of *K*_{d}, and then solving for and *K*_{s}
assuming that the unjacketed pore modulus satisfies .We find that both and *K*_{s} are then monotonically increasing
functions of *K*_{d}. Other physical constraints [such as the previously
measured values of , upper bounds on the possible range of *K*_{s}
from the known mineralogy of the rock, and the fact that the intercept
satisfies (Berryman and Milton, 1991)] provide sufficient
information that we can find the range of values that the various physical
constants can take and still be consistent with the ultrasonic data. We will
present a more thorough discussion of these issues elsewhere.

11/17/1997