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.

我们基于两个多孔介质之间平面界面上的开孔边界条件，将精确的垂直入射PP反射系数[Geertsma–Smit表达式]与几位作者报道的近似值进行了比较。近似值对应于低频。其中两个是从比奥特特征频率以下的低频比奥特理论推导出来的，但结果表明，比奥特特征频率以下的频率存在显着差异。然后，我们通过包括高频粘动力算子（即，全频率范围比奥特理论）来扩展Geertsma-Smit方程，表明在高频范围内还存在其他实质性差异。需要使用后一种表达式来表示整个频率范围内的物理特性。我们进一步将Geertsma-Smit方程推广到除了开孔界面之外的一般边界条件的情况。在地震带上，表明基于加斯曼P波阻抗的无损（弹性）表达式是在实际应用中使用的反射系数。可以推断，基于这些近似值的频率依赖性的解释可能会产生误导，因为这种依赖性无法提供对物理学的适当描述。