Theory of compressional and transverse wave propagation in consolidated porous media

A theory of compressional and shear wave propagation in consolidated porous media (rocks) is developed by extending ideas already introduced in connection with unconsolidated marine sediments. The consolidated material is treated as an elastic medium which exhibits a specific form of stress relaxation associated with grain boundaries and microcracks. The stress relaxation, which is linear in the sense that it obeys superposition, shows hysteresis, as characterized by a material response function. Two linear wave equations are derived, one for compressional and the second for shear waves, from which expressions for the wave speeds and attenuations are established. In both cases, the attenuation is found to scale with the first power of frequency, consistent with many observations of attenuation in sandstones, limestones, and shales; the wave speeds show weak logarithmic dispersion. These expressions for the wave speeds and attenuations satisfy the Kronig–Kramers dispersion relationships, as they must if the response of the medium to disturbances is to be causal. Some comments are offered on the nature of the material response, notably that it appears to be primarily associated with grain-boundary interactions occurring at a molecular level, rather than being related to the macroscopic properties of the material, such as density or porosity.