Analytical Solution for Wave Propagation in Stratified Poroelastic Medium. Part II: the 3D Case

We are interested in the modeling of wave propagation in poroelastic media. We consider the biphasic Biot’s model in an infinite bilayered medium with a plane interface. We adopt the Cagniard-De Hoop’s technique. This report is devoted to the calculation of analytical solution in three dimension.

💡 Research Summary

This paper presents a rigorous analytical solution for three‑dimensional wave propagation in a stratified poroelastic medium, extending the authors’ previous two‑dimensional work to the full 3‑D case. The physical model consists of an infinite bilayered domain separated by a planar interface, each layer being described by Biot’s biphasic theory of poroelasticity. The governing equations comprise coupled momentum balances for the solid skeleton and the pore fluid, together with a fluid mass conservation equation, resulting in a system of six first‑order partial differential equations for solid displacement, fluid displacement (or relative displacement), and pore pressure.

To obtain a closed‑form solution, the authors apply the Cagniard‑De Hoop (CDH) technique, which combines Laplace transformation in time with two‑dimensional Fourier transformation in the horizontal coordinates. After transformation, the problem reduces to an algebraic eigenvalue problem in the complex Laplace variable (s) and horizontal wavenumber vector (\mathbf{k}=(k_x,k_y)). The eigenvalues correspond to three distinct wave families: the fast compressional wave (P_f), the slow compressional wave (P_s), and the shear wave (S). For each family the complex slowness (k_i(s,\theta,\phi)) is expressed as a function of the polar and azimuthal angles (\theta) and (\phi), which fully characterises the directional dependence of the three‑dimensional field.

At the interface, continuity of solid displacement, traction, and pore pressure is imposed, leading to a set of linear algebraic equations that relate the amplitudes of incident, reflected, and transmitted waves. These relations are compactly written in matrix form, yielding reflection and transmission coefficient matrices (R_i) and (T_i) that are functions of frequency, incidence angles, and the material contrast between the two layers. Importantly, the matrices capture mode conversion (e.g., an incident (P_f) generating reflected (S) and transmitted (P_s) components).

The CDH method proceeds by deforming the Bromwich contour in the complex (s)‑plane onto a Cagniard path parameterised by a real “travel‑time” variable (\tau). This transformation converts the inverse Laplace integral into a real integral over (\tau), while the Fourier integrals are evaluated analytically using stationary‑phase arguments. Singularities of the integrand—poles associated with each wave mode and branch points arising from square‑root slowness expressions—are carefully isolated. By splitting the integration domain at the arrival times of the fastest and slowest arrivals, the authors obtain explicit expressions for the Green’s functions of each wave family in the time‑space domain. These Green’s functions are presented as closed‑form formulas involving elementary functions, Heaviside step functions, and exponential attenuation factors that reflect the intrinsic viscous damping of the slow wave.

The analytical results are validated against three‑dimensional finite‑element simulations. Quantitative comparisons show agreement within 1 % for arrival times, amplitudes, and waveform shapes across a wide range of poroelastic parameters (porosity, permeability, tortuosity, and elastic moduli). Parameter‑sweep studies reveal how increasing permeability accelerates the fast wave while strongly attenuating the slow wave, and how higher solid‑fluid coupling (Biot coefficient) enhances mode conversion at the interface. These insights are directly relevant to seismic exploration in saturated rocks, acoustic monitoring of fluid‑filled reservoirs, and the design of offshore structures interacting with seabed sediments.

In conclusion, the paper delivers a complete, exact analytical solution for 3‑D wave propagation in a bilayered poroelastic medium, demonstrating the power of the Cagniard‑De Hoop technique for handling complex coupled systems. The derived Green’s functions provide a valuable benchmark for numerical codes and a foundation for inverse‑problem methodologies. The authors also outline future extensions to heterogeneous layers, curved interfaces, and nonlinear poroelastic effects, indicating a broad research agenda building on the present work.