Analytical Solution for Wave Propagation in Stratified Poroelastic Medium. Part I: the 2D 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 solutions in two dimensions. The solutions we present here have been used to validate numerical codes.

💡 Research Summary

The paper presents a complete analytical solution for two‑dimensional wave propagation in a stratified poroelastic medium composed of two infinite homogeneous layers separated by a planar interface. The governing equations are the Biot biphasic model, which couples the solid displacement Uₛ, the relative fluid displacement W, and the fluid pressure P. Material parameters such as overall density ρ, porosity φ, shear modulus µ, Lamé constant λ, bulk moduli, tortuosity a, and permeability κ appear in the system. The authors consider a point source acting on the solid, the relative displacement, and the pressure, with the source terms written as spatial derivatives of Dirac deltas multiplied by a temporal function f(t).

The transmission conditions at the interface enforce continuity of solid displacement, normal component of relative displacement, fluid pressure, and normal stress. To solve the problem, the authors first rewrite the second‑order Biot equations in a compact matrix form and then decompose the displacement fields into irrotational (scalar potentials Θ) and solenoidal (vector potentials Ψ) parts. This decomposition separates compressional (P‑type) and shear (S‑type) motions. By introducing the matrices A and B that contain the physical coefficients, they diagonalize the system (A⁻¹B = PDP⁻¹) to obtain uncoupled wave equations for three independent modes: fast P‑wave (V₊Pf), slow P‑wave (V₊Ps), and shear wave (V₊S). The eigenvalues of the diagonal matrix D are the squared phase velocities of these modes.

The Green’s function for each mode is constructed using the Cagniard‑De Hoop technique. The method transforms the time‑domain integral into a complex‑plane integral over the Laplace variable q, introducing the complex square‑root function κᵢ(q)=√(1/Vᵢ²+q²) with a branch cut along the negative real axis. The authors define reflection and transmission coefficients Rᵢⱼ(q) and Tᵢⱼ(q) by solving a linear system A(q)·