INTRODUCTION

One of the perennial problems in rock physics has been the difficulty of understanding how seismic wave speeds in fluid saturated and partially fluid saturated rocks depend on the wave frequency. Field methods for exciting seismic waves are usually in the 1-100 Hz band, while well-logging tools might be in the 1-50 kHz band. However, the careful controlled experiments needed to verify the predictions of the theory can normally only be done at still higher frequencies, typically in the 200-1000 kHz band. This segmentation of the frequency band into its distinct regions of application has caused and continues to cause much confusion about what is known and unknown about wave propagation and attenuation in rocks.

Theoretical analyses of Gassmann (1951) and Biot (1956a,b) provide low frequency results. Gassmann's results in particular are very low frequency, really applicable to the quasi-static domain, and therefore strictly apply only to the very lowest seismic frequencies. But a skeptical scientist wants proof of these theories, and it is sometimes hard to find convincing verifications among experiments done on rocks. Plona (1980) provided one very nice series of ultrasound experiments ($\simeq$ 1 MHz) using water-saturated sintered glass-bead samples (instead of rocks) showing (Chin et al., 1983) that the Biot-Gassmann theory is in fact correct even in this high frequency regime, at least for such simple porous materials. The likely reason for this success is the high permeability ($1\sim 10$D) and lack of microcracks in the porous glass-bead samples.

Difficulties still exist in explaining some high frequency laboratory data, especially in situations having low fluid permeability (and therefore making it unlikely that Gassmann's quasi-static conditions are close to being satisfied) and also having partial saturation conditions (mixtures of gas and liquid are present in the pores). There has been extensive work on partial saturation by Nur and Simmons (1969), Domenico (1974), Walls (1982), Murphy (1984), Berryman et al. (1988), Endres and Knight (1989), Knight and Nolen-Hoeksema (1990), Dvorkin and Nur (1998), Dvorkin et al. (1999a,b), Johnson (2001), and Berryman et al. (2002a), among others. In particular, the work of Knight and Nolen-Hoeksema (1990) makes it particularly apparent that great care must be taken in modeling these types of materials because details clearly matter. Whether the gas and liquid components are homogeneously mixed or are distributed in a patchy manner (i.e., gas here, liquid there) makes a significant difference in the measured wave speeds. Even for the shear waves, where according to Gassmann's low frequency calculations we might conclude that there should be no difference at all (Berryman, 1999), we find clearly observable differences at higher frequencies.

Through a series of recent publications (Berryman et al., 2000; 2002a), it has become clear that the most appropriate of the simple effective medium models for partial saturation conditions at high frequencies is the differential effective medium (DEM) theory (Berryman et al., 2002b). The present paper will show specifically how to use this theory to fit data on partial and patchy saturation in a low-porosity, low-permeability granite and two tight sandstones.