Spherical-wave seismic inversion for petrophysical properties in fluid-saturated porous media

Summary Analytical expressions for the reflection and transmission coefficients of a spherical fast compressional wave (fast P-wave or P1-wave) incident on an interface within fluid-saturated porous media are derived. These coefficients exhibit explicit dependence on frequency, propagation depth, incidence angle, and the material properties of porous rocks. A comparison of the amplitudes and phases of the spherical reflected P1-waves at oil-water and gas-oil interfaces reveals significant differences in wave characteristics between these two interfacial environments. Moreover, notable distinctions are observed between the reflection behaviors of spherical and plane P1-waves. We further derive an approximate spherical P1-wave reflection coefficient ($R_{{\rm{p}}1}^{{\rm{Sph}}}$) and the corresponding spherical P1-wave elastic impedance ($Z_{{\rm{p}}1}^{{\rm{Sph}}}$), both expressed as functions of the bulk modulus of the solid grain (Ks), bulk modulus of the pore fluid (Kfl), density of the solid grain (ρs), density of the pore fluid (ρfl), shear modulus of the solid grain (μs), and porosity ($\phi $). Using the approximate $R_{{\rm{p}}1}^{{\rm{Sph}}}$ and $Z_{{\rm{p}}1}^{{\rm{Sph}}}$, a spherical-wave inversion approach in fluid-saturated porous media is established to estimate Ks, Kfl, ρs, ρfl, μs, and $\phi $ from seismic data with different incidence angles and frequency components. The feasibility and robustness of the proposed inversion approach are verified using synthetic data with added Gaussian random noise. Finally, the approach is applied to field data acquired over a hydrocarbon reservoir. The inversion results show good agreement with filtered well-log data.