On the Normal-Incidence Reflection Coefficient in Porous Media

Normal incidence reflection coefficient approximation at an isotropic–poroelastic interface

“…Four boundary conditions need to be followed across the interface (Deresiewicz & Skalak, 1963), including (1) the solid displacements are continuous; (2) the relative fluid displacements are continuous; (3) the normal solid stresses are continuous and (4) the fluid pressures are continuous. Based on the previous boundary conditions, normal R/T coefficient matrix equation is given as follows (Carcione et al, 2021;Geertsma & Smit, 1961),…”

“…Four boundary conditions need to be followed across the interface (Deresiewicz & Skalak, 1963), including (1) the solid displacements are continuous; (2) the relative fluid displacements are continuous; (3) the normal solid stresses are continuous and (4) the fluid pressures are continuous. Based on the previous boundary conditions, normal R / T coefficient matrix equation is given as follows (Carcione et al., 2021; Geertsma & Smit, 1961), $$$$\begin{equation}\left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} 1 & \quad 1 & \quad 1 & \quad 1\\ {m_1^{\left( 1 \right)}} & \quad{m_2^{\left( 1 \right)}} & \quad{m_1^{\left( 2 \right)}} & \quad{m_2^{\left( 2 \right)}}\\ {B_1^{\left( 1 \right)}} & \quad{B_2^{\left( 1 \right)}} & \quad{ - B_1^{\left( 2 \right)}} & \quad{ - B_2^{\left( 2 \right)}}\\ {C_1^{\left( 1 \right)}} & \quad{C_2^{\left( 1 \right)}} & \quad{ - C_1^{\left( 2 \right)}} & \quad{ - C_2^{\left( 2 \right)}} \end{array} } \right]\left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} R\\ {{R}_2}\\ T\\ {{T}_2} \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} 1\\ {m_1^{\left( 1 \right)}}\\ { - B_1^{\left( 1...$$…”

“…Based on it, they gave explicit expressions of normal incidence R/T coefficients at a poroelastic/poroelastic interface in terms of first-order κ under the 𝜅 ≪ 1 assumption. Deresiewicz-Rice approximation is quoted by Bourbié et al (1987) and is also called Bourbié et al approximation in Carcione's paper (Carcione et al, 2021). Gurevich et al (2004) derived the R/T coefficient approximation from an interface between two poroelastic media or fluid/poroelastic media by ignoring the fluid-flow effects on the classical compressional wave.…”

“…(1987) and is also called Bourbié et al. approximation in Carcione's paper (Carcione et al., 2021). Gurevich et al.…”

“…Carcione et al. (2021) compared several P‐wave reflection coefficient approximate formulas for the poroelastic/poroelastic interface, including the Deresiewicz–Rice approximation, the Gurevich et al. approximation and the Silin–Goloshubin approximation, with the exact one computed by the Geertsma–Smit expression.…”