INTRODUCTION

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 $\lambda$ and $\mu$ of elasticity theory, and the overall density $\rho$. These relationships are well-known (Ewing et al., 1957; Aki and Richards, 1980), but the parameter $\lambda$is seldom used to analyze seismic data. Our first new way of displaying seismic data is to plot data points in the ($\rho/\mu$, $\lambda/\mu$)-plane -- instead of (for example) the (v_p, v_s)-plane. (Note that $\rho/\mu = 1/v_s^2$.) 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 ($\rho/\lambda$, $\mu/\lambda$)-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.