Resolution of various practical and scientific issues in the earth sciences depends on knowledge of fluid properties underground. In environmental cleanup applications, the contaminant to be removed from the earth is often a liquid such as gasoline or oil, or ground water contaminated with traces of harmful chemicals. In commercial oil and gas exploration, the fluids of interest are hydrocarbons in liquid or gaseous form. In analysis of the earth structure, partially melted rock is key to determining temperature and local changes of structure in the earth's mantle. In all cases the tool commonly used to analyze the fluid content is measurements of seismic (compressional and shear) wave velocities in the earth. Depending on the application, the sources of these waves may be naturally occurring such as earthquakes, or man-made such as reflection seismic surveys at the surface of the earth, vertical seismic profiling, or still more direct measurements using logging tools in either shallow or deep boreholes.

Underground fluids occupy voids between and among the solid earth grains.
When liquid or gas completely fills interconnnecting voids, a well-known result
due to Gassmann (1951) predicts how the composite elastic constants that
determine seismic velocities should depend on
the fluid and drained rock or soil elastic constants and densities
[also see tutorial by Berryman (1999)].
The formulas due to Gassmann are low frequency (seismic) results and both
laboratory and well-log measurements of wave velocities have been
observed to deviate markedly from Gassmann's predictions at higher
(sonic and ultrasonic) frequencies. This is especially so for partial
saturation conditions
(*i.e.*, when the fluid in each pore is a mixture of gas and
liquid). In some cases these deviations can be attributed
(Berryman *et al.*, 1988; Endres and Knight, 1989;
Mavko and Nolen-Hoeksema, 1994; Dvorkin and Nur, 1998) to ``patchy
saturation,'' meaning that some void regions are fully saturated with
liquid and others are filled with gas. When the concept of patchy saturation
is applicable, Gassmann's formulas
apply locally (but not globally) and must be averaged over the volume
to obtain the overall seismic velocity of the system. In other cases,
neither Gassmann's formulas nor
the ``patchy saturation'' model seem to apply to seismic data. In these
cases a variety of possible reasons for the observed velocity discrepancies
have been put forward, including viscoelastic effects (velocity
decrement due to frequency-dependent attenuation), fluid-enhanced
softening of intragranular cementing materials, chemical changes in
wet clays that alter mechanical properties, etc.

The objective of the present study therefore has been to find a method of
using seismic data to estimate porosity and saturation, regardless of
whether the rock or soil fits the Gassmann, the patchy saturation, or
some other model.
Seismic data typically provide two measured parameters,
*v*_{p} and *v*_{s} (compressional and shear wave velocities,
respectively).
Simple algebraic expressions relate *v*_{p} and *v*_{s}
to the Lamé parameters and of elasticity theory,
and the overall density .
These relationships are well-known
(Ewing *et al.*, 1957; Aki and Richards, 1980), but the parameter is seldom used to analyze seismic data.
Our first
new way of displaying seismic data is to plot data points in the
(, )-plane -- instead of (for example) the
(*v*_{p}, *v*_{s})-plane.
(Note that .)
The advantage of this plot is that, when
the liquid and gas are either mixed homogeneously throughout
(Gassmann's assumption) or are fully segregated throughout
(patchy saturation),
most of the data will fall along one or the other of two straight lines.
Significant deviations from these two expected behaviors then provide
a clear indication that the data violate some of the assumptions in
Gassmann's simple model, and furthermore provide clues to help
determine which assumptions are being violated.
Our second innovation in displaying seismic data is to plot the data points in
the (, )-plane. This second approach
involves the use of an easily understood mathematical trick
that leads naturally to universal and easily interpreted behavior;
virtually all laboratory data on partial saturation for similar rocks
that we have analyzed plot with minimal scatter along straight lines
in this plane. The length and slope of these lines have quantitative
predictive capabilities for measurements of both partial saturation
and porosity. We have used sonic and ultrasonic laboratory data
in the present study, but the results provide very strong indications
that equally useful relationships among seismic parameters, porosity,
and saturation will be obtained from seismic data collected at lower
frequencies in the field.

10/25/1999