On the normal-incidence reflection coefficient in porous media

We compare the exact normal-incidence PP reflection coefficient [Geertsma–Smit expres- sion] to approximations reported by several authors, based on open-pore boundary condi- tions 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-fre- quency 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 gen- eralize 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) expres- sion based on the Gassmann P-wave impedance is the reflection coefficient to use for prac- tical 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.