On the Normal-Incidence Reflection Coefficient in Porous Media

We compare the exact normal-incidence PP reflection coefficient [Geertsma-Smit expression] to approximations reported by several authors, based on open-pore boundary conditions at a plane interface between two porous media. The approximations correspond to low frequencies. Two of them are derived from the low-frequency Biot theory below the Biot characteristic frequency, but the results show significant differences much below the Biot frequency. Then, we extend the Geertsma-Smit equations by including the high-frequency viscodynamic operator (i.e., the full-frequency range Biot theory), showing that there are additional substantial differences at the high-frequency range. Use of this latter expression is required to honor the physics in the whole frequency range. We further generalize the Geertsma-Smit equations to the case of general boundary conditions other than the open-pore interface. At the seismic band, it is shown that the lossless (elastic) expression based on the Gassmann P-wave impedance is the reflection coefficient to use for practical applications. It is inferred that interpretations based on the frequency dependency of these approximations can be misleading, since this dependency does not provide a suitable description of the physics.

Automated summary

The study compares the exact normal-incidence PP reflection coefficient with approximations reported by several authors based on open-pore boundary conditions between two porous media. It is found that there are significant differences at low frequencies, and further differences are shown at high frequencies when extending the Geertsma-Smit equations to include the high-frequency viscodynamic operator. The use of the correct expression is crucial to accurately describe the physics across the frequency range.