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 Kd 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 Swi (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 Swi 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 Kd 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 Kd for air-dry samples and the effective Kd 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 Kd as an independent parameter, computing the slope b and intercept a for each assumed value of Kd, and then solving for and Ks assuming that the unjacketed pore modulus satisfies .We find that both and Ks are then monotonically increasing functions of Kd. Other physical constraints [such as the previously measured values of , upper bounds on the possible range of Ks 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.