In a variety of applied problems, it is important to determine the state of saturation of a porous medium from acoustic or seismic measurements. In the oil and gas industry, it is common to use amplitude-versus-offset (AVO) processing of seismic reflection data to reach conclusions about the presence of gas, oil, and their relative abundances on the opposite sides of a reflecting interface underground [e.g., Castagna and Backus (1993)]. For environmental applications, we can expect to be working in the near surface where sensor geometries other than surface reflection surveys become practical. For example, when boreholes are present, it is possible to do crosswell seismic tomography, or borehole sonic logging to determine velocities [e.g., Harris et al. (1995)]. For AVO processing the data obtained are the seismic impedances and (where is the density, and vp, vs are the seismic compressional and shear wave velocities, respectively), which arise naturally in reflectance measurements. (In this paper, we will use the term ``velocities'' to refer to measured velocities at seismic, sonic, or ultrasonic frequencies, unless otherwise specified.) However, for crosswell applications, we are more likely to have simply velocity data, i.e., vp and vs themselves without density information. For well-logging applications, separate measurements of the velocities as well as density are possible. Although a great deal of effort has been expended on AVO analysis, relatively little has been done to invert the simple velocity data for porosity and saturation. It is our purpose to present one method that shows promise for using velocity data to obtain porosity and saturation estimates. The key physical idea used here is the fact that the Lamé parameter and the density are the two parameters containing information about saturation, while both of these together with shear modulus contain information about porosity ( and are defined in the next section). These facts are well-known from earlier work of Gassmann (1951), Domenico (1974), and many others. (It is well-established that even though the Gassmann-Domenico relations are derived for the static case, they have been found to describe behavior measured in the field at sonic and seismic frequencies, and, in some cases, even in laboratory ultrasonic experiments.) The same facts are used explicitly in AVO analysis (Castagna and Backus, 1993; Ostrander, 1984; Castagna et al., 1985; Foster et al., 1997), but in ways that are significantly different from those to be described here. A major point of departure is that the present work allows direct information to be obtained about, not only the level of the saturation, but also concerning the state of saturation, i.e., whether the liquid and gas present are mixed homogeneously, or are instead physically separated and therefore in a state of patchy saturation (Berryman et al., 1988; Endres and Knight, 1989; Knight and Nolen-Hoeksema, 1990; Mavko and Nolen-Hoeksema, 1994; Dvorkin and Nur, 1998; Cadoret et al., 1998). Another advantage is that this method uses velocity rather than amplitude information, and therefore may have less uncertainty and may also require less data processing for some types of field experiments.
One of the main points of the analysis to be presented is the purposeful avoidance of the well-known complications that arise at high frequencies, due in large part to velocity dispersion and attenuation (Biot, 1956a,b; Biot, 1962; O'Connell and Budiansky, 1977; Mavko and Nur, 1978; Berryman, 1981; McCann and McCann, 1985; Johnson et al., 1987; Norris, 1993; Best and McCann, 1995). Our point of view is that seismic data (as well as most sonic, and some ultrasonic data) do not suffer from contamination by the frequency-dependent effects to the same degree typically seen for high frequency laboratory measurements. By restricting our range of frequencies to those most useful in the field, we anticipate a significant simplification of the analysis and therefore an improvement in our ability to provide both simple and robust interpretations of field data. In the Discussion section, we also provide a means of identifying data in need of correction for dispersion effects.
We introduce the basic physical ideas in the next section. Then we present two new methods of displaying the velocity data. One method is used to sort data points into sets that have similar physical attributes, such as porosity. Then, the second method is used to identify both the level of saturation and the type of saturation, whether homogeneous, patchy, or a combination of the two. We show a subset of the large set of data we have examined that confirms these conclusions empirically. We then provide some discussion of the results and what we foresee as possible future applications of the ideas. Finally, we summarize the accomplishments of the paper in the concluding section.