Pressure Effects on Plane Wave Reflection and Transmission in Fluid-Saturated Porous Media

Abstract

The wave reflection and transmission (R/T) coefficients in fluid-saturated porous media with the effect of effective pressure are rarely studied, despite the ubiquitous presence of in situ pressure in the subsurface Earth. To fill this knowledge gap, we derive exact R/T coefficient equations for a plane wave incident obliquely at the interface between the dissimilar pressured fluid-saturated porous half-spaces described by the theory of poro-acoustoelasticity (PAE). The central result of the classic PAE theory is first reviewed, and then a dual-porosity model is employed to generalize this theory by incorporating the impact of nonlinear crack deformation. The new velocity equations of generalized PAE theory can describe the nonlinear pressure dependence of fast P-, S- and slow P-wave velocities and have a reasonable agreement with the laboratory measurements. The general boundary conditions associated with membrane stiffness are used to yield the exact pressure-dependent wave R/T coefficient equations. We then model the impacts of effective pressure on the angle and frequency dependence of wave R/T coefficients and synthetic seismic responses in detail and compare our equations to the previously reported equations in zero-pressure case. It is inferred that the existing R/T coefficient equations for porous media may be misleading, since they lack consideration for inevitable in situ pressure effects. Modeling results also indicate that effective pressure and membrane stiffness significantly affect the amplitude variation with offset characteristics of reflected seismic signatures, which emphasizes the significance of considering the effects of both in practical applications related to the observed seismic data. By comparing the modeled R/T coefficients to the results computed with laboratory measured velocities, we preliminarily confirm the validity of our equations. Our equations and results are relevant to hydrocarbon exploration, in situ pressure detection and geofluid discrimination in high-pressure fields.

Appendices

Appendix A: Constitutive Relationship in Fluid-Saturated Porous Media Considering the Effect of Effective Pressure

In the context of acoustoelasticity theory, when the wave propagates through a rock subjected to the in situ effective pressure, the particles of this rock will undergo three different states in the entire dynamic process (Pao et al. 1984; Chen et al. 2021). The first state is usually called as “natural state” that means a particle is in the rock free of pressure and no wave disturbs. Then the in situ pressure is applied to the rock and the rock particle enters the second state termed as “initial state”. Finally, the rock particle goes to the third state, “final state,” when a wave propagates through the stressed rock. The sketch of a rock particle in three states is shown in Fig. 

Fig. 21

Sketch of a rock particle in natural, initial and final states

21. The physical quantities in the natural, initial and final states are symbolized by the superscripts 0, $s$ and $f$, respectively.

The primary focus of the early studies associated to acoustoelasticity theory is the analysis on the wave propagation in nonporous isotropic materials, for instance, metals and polymers, under the influence of initial pressure (Hughes and Kelly 1953; Thurston and Brugger 1964; Pao et al. 1984; Degtyar and Rokhlin 1998). Winkler and McGowan (2004) used this theory to estimate the 3oECs of a fluid-saturated porous rock with the laboratory measured wave velocities against pressure. There is a substantial bias between their estimated result and the real values. They attributed this bias to the disregard of wave velocity dispersion induced by wave-induced fluid flow in pores. Then several academics (e.g., Grinfeld and Norris 1996; Ba et al. 2013; Liu et al. 2021; Zong et al. 2023) successively advance and refine this theory by allowing the addition of pores and internal fluid flow.

The Helmholtz free energy of a fluid-saturated porous medium is given by (Grinfeld and Norris 1996; Liu et al. 2021)

$$\begin{aligned} H & = \frac{1}{2}\lambda_{u} E^{2} + \mu_{m} E_{ij}^{2} - \alpha ME\zeta + \frac{1}{2}M\zeta^{2} + \frac{1}{6}\nu^{\prime}_{1} E^{3} \\ & \quad + \nu^{\prime}_{2} E_{ij}^{2} E + \frac{4}{3}\nu^{\prime}_{3} E_{ik} E_{il} E_{kl} + \gamma_{1} \zeta^{3} + \gamma_{2} E^{2} \zeta + \gamma_{3} E\zeta^{2} + \frac{1}{2}\gamma \left( {E^{2} - E_{ij}^{2} } \right)\zeta , \\ \end{aligned}$$

(70)

where $E_{ij} = {{\left( {u_{i,j} + u_{j,i} { + }u_{k,i} u_{k,j} } \right)} \mathord{\left/ {\vphantom {{\left( {u_{i,j} + u_{j,i} { + }u_{k,i} u_{k,j} } \right)} 2}} \right. \kern-0pt} 2}$ is the Lagrangian strain and $E = E_{ij} \delta_{ij}$.

The stress tensor of the medium in any state can be computed with

$$T_{ij} = \frac{\partial H}{{\partial E_{ij} }},$$

(71)

and thus the stress tensors in the initial and final states can be derived from Eqs. (70) and (71), given by

$$\begin{aligned} T_{ij}^{s} & = \lambda_{u} E^{s} \delta_{ij} + 2\mu_{m} E_{ij}^{s} - \alpha M\zeta^{s} \delta_{ij} + \frac{1}{2}\nu^{\prime}_{1} \left( {E^{s} } \right)^{2} \delta_{ij} + 3\nu^{\prime}_{2} \left( {E_{ij}^{s} } \right)^{2} \delta_{ij} + 4\nu^{\prime}_{3} E_{ik}^{s} E_{jk}^{s} \\ & \quad + 2\gamma_{2} E^{s} \zeta^{s} \delta_{ij} + \gamma_{3} \left( {\zeta^{s} } \right)^{2} \delta_{ij} + \gamma \left( {E^{s} \delta_{ij} - E_{ij}^{s} } \right)\zeta^{s} , \\ \end{aligned}$$

(72)

and

$$\begin{aligned} T_{ij}^{f} & = \lambda_{u} E^{f} \delta_{ij} + 2\mu_{m} E_{ij}^{f} - \alpha M\zeta^{f} \delta_{ij} + \frac{1}{2}\nu^{\prime}_{1} \left( {E^{f} } \right)^{2} \delta_{ij} + 3\nu^{\prime}_{2} \left( {E_{ij}^{f} } \right)^{2} \delta_{ij} + 4\nu^{\prime}_{3} E_{ik}^{f} E_{jk}^{f} \\ & \quad + 2\gamma_{2} E^{f} \zeta^{f} \delta_{ij} + \gamma_{3} \left( {\zeta^{f} } \right)^{2} \delta_{ij} + \gamma \left( {E^{f} \delta_{ij} - E_{ij}^{f} } \right)\zeta^{f} , \\ \end{aligned}$$

(73)

Subtracting Eqs. (72) from (73), we obtain the stress increment induced by wave perturbation, as

$$\begin{aligned} T_{ij} & = \left( {\lambda_{u} + 2\gamma_{2} \zeta^{s} + \gamma \zeta^{s} } \right)\delta_{ij} E + \left( {2\mu_{m} - \gamma \zeta^{s} } \right)E_{ij} + \nu^{\prime}_{1} \delta_{ij} e^{s} e \\ & \quad + \left( {6\nu^{\prime}_{2} e^{s} + 8\nu^{\prime}_{3} \delta_{ik} e_{ik}^{s} } \right)e_{ij} + \left( {2\gamma_{2} \delta_{ij} E^{s} + 2\gamma_{3} \delta_{ij} \zeta^{s} + \gamma \delta_{ij} E^{s} - \gamma E_{ij}^{s} - \alpha M\delta_{ij} } \right)\zeta \\ \end{aligned}$$

(74)

Then we express Eq. (74) in terms of the displacement gradient, yields

$$T_{ij} = \Gamma_{ij} \frac{{\partial u_{i} }}{{\partial x_{i} }} + \Lambda \left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right) + \Upsilon_{ij} \frac{{\partial U_{i} }}{{\partial x_{i} }},$$

(75)

where

$$\left\{ \begin{gathered} \Gamma_{ij} = \left( {1 + e^{s} } \right)\left( {\lambda_{u} + 2\gamma_{2} \zeta^{s} + \gamma \zeta^{s} } \right)\delta_{ij} + \nu^{\prime}_{1} e^{s} \delta_{ij} , \hfill \\ \Lambda = \mu_{m} - {{\left[ {\gamma \zeta^{s} - \left( {2\mu_{m} - \gamma \zeta^{s} + 6\nu^{\prime}_{2} + 8\nu^{\prime}_{3} } \right)} \right]e^{s} } \mathord{\left/ {\vphantom {{\left[ {\gamma \zeta^{s} - \left( {2\mu_{m} - \gamma \zeta^{s} + 6\nu^{\prime}_{2} + 8\nu^{\prime}_{3} } \right)} \right]e^{s} } 2}} \right. \kern-0pt} 2}, \hfill \\ \Upsilon_{ij} = \left( {2\gamma_{2} + \gamma } \right)\delta_{ij} e^{s} - \gamma e_{ij}^{s} + 2\gamma_{3} \zeta^{s} \delta_{ij} - \alpha M\delta_{ij} . \hfill \\ \end{gathered} \right.$$

(76)

A similar operation is next used to derive the fluid pressure increment in a fluid-porous medium. The fluid pressure of the medium in any state can be computed using

$$p_{f} = \frac{\partial H}{{\partial \zeta }}.$$

(77)

According to Eqs. (70) and (77), the fluid pressure in the initial and final states can be expressed as

$$p_{f}^{s} = - \alpha ME^{s} + M\zeta^{s} + 3\gamma_{1} \left( {\zeta^{s} } \right)^{2} + 2\gamma_{3} E^{s} \zeta^{s} + \left( {\gamma_{2} + \frac{1}{2}\gamma } \right)\left( {E^{s} } \right)^{2} - \frac{1}{2}\gamma \left( {E_{ij}^{s} } \right)^{2} ,$$

(78)

and

$$p_{f}^{f} = - \alpha ME^{f} + M\zeta^{f} + 3\gamma_{1} \left( {\zeta^{f} } \right)^{2} + 2\gamma_{3} E^{f} \zeta^{f} + \left( {\gamma_{2} + \frac{1}{2}\gamma } \right)\left( {E^{f} } \right)^{2} - \frac{1}{2}\gamma \left( {E_{ij}^{f} } \right)^{2} .$$

(79)

Then the fluid pressure increment (induced by the propagating wave) from the initial state to final state is

$$p_{f} = p_{f}^{f} - p_{f}^{s} = - \alpha ME + M\zeta + 6\gamma_{1} \zeta^{s} \zeta + 2\gamma_{3} \left( {E^{s} \zeta + \zeta^{s} E} \right) + \left( {2\gamma_{2} + \gamma } \right)e^{s} e - \gamma e_{ij}^{s} e_{ij} .$$

(80)

Rewriting fluid pressure increment with the wave displacement gradients yields

$$p_{f} = \tilde{\Gamma }\frac{{\partial u_{i} }}{{\partial x_{i} }} + \tilde{\Lambda }_{ij} \frac{{\partial u_{i} }}{{\partial x_{j} }} + \tilde{\Upsilon }\frac{{\partial U_{i} }}{{\partial x_{i} }},$$

(81)

where

$$\left\{ \begin{gathered} \tilde{\Gamma } = - \alpha M + 2\gamma_{3} \zeta^{s} + \left( {2\gamma_{2} + \gamma } \right)e^{s} , \hfill \\ \tilde{\Lambda }_{ij} = - \gamma e_{ij}^{s} , \hfill \\ \tilde{\Upsilon } = M + 6\gamma_{1} \zeta^{s} + 2\gamma_{3} e^{s} . \hfill \\ \end{gathered} \right.$$

(82)

From Eqs. (75), (76), (81) and (82), we can see that if the media are free of pressure, the initial static strains of solid skeleton and fluid flow will vanish and, in this case, the constitutive equations shown in Eqs. (85) and (81) will degrade to the Biot’s ones (Biot 1956a, b).

Appendix B: Coefficients of Dynamic Equations

Under the “Small-on-Large” assumption, the high-order small terms of strains can be omitted when deriving the dynamic equations in the fluid-saturated porous media under the effect of initial pressure. In this case, Liu et al. (2021) derived the expressions for the four coefficients of dynamic Eqs. (19) and (20), given by

$$\Upsilon_{ij} = \alpha M\left( {\delta_{ij} + u_{i,j}^{s} } \right) + {{\gamma \left( {u_{i,j}^{s} + u_{j,i}^{s} } \right)} \mathord{\left/ {\vphantom {{\gamma \left( {u_{i,j}^{s} + u_{j,i}^{s} } \right)} 2}} \right. \kern-0pt} 2} - \left( {2\gamma_{2} + \gamma } \right)\delta_{ij} u_{i,i}^{s} + \left( {\alpha M + 2\gamma_{3} } \right)\delta_{ij} U_{i,i}^{s} ,$$

(83)

$$\Upsilon_{kj} = \alpha M\left( {\delta_{kj} + u_{k,j}^{s} } \right) + {{\gamma \left( {u_{k,j}^{s} + u_{j,k}^{s} } \right)} \mathord{\left/ {\vphantom {{\gamma \left( {u_{k,j}^{s} + u_{j,k}^{s} } \right)} 2}} \right. \kern-0pt} 2} - \left( {2\gamma_{2} + \gamma } \right)\delta_{kj} u_{i,i}^{s} + \left( {\alpha M + 2\gamma_{3} } \right)\delta_{kj} U_{i,i}^{s} ,$$

(84)

$$\Upsilon = M + \left( {2\gamma_{3} + \alpha M} \right)e^{s} - 3\left( {M - 2\gamma_{1} } \right)\zeta^{s} ,$$

(85)

$$\Xi_{ijkl} = \left( {c_{ijkl} + \alpha^{2} M\delta_{ij} \delta_{kl} } \right) + a_{ijklmn} u_{m,n}^{s} + a_{ijkl} U_{i,i}^{s} ,$$

(86)

where

$$c_{ijkl} = \lambda_{b} \delta_{ij} \delta_{kl} + \mu_{b} \delta_{ik} \delta_{jl} + \mu_{b} \delta_{jk} \delta_{il} ,$$

(87)

$$\begin{aligned} a_{ijklmn} & = d_{ijklmn} + \lambda_{u} \left( {\delta_{ij} \delta_{nl} \delta_{mk} + \delta_{ik} \delta_{jl} \delta_{mn} + \delta_{im} \delta_{jn} \delta_{kl} } \right) \\ & \quad + \mu_{m} \left( {\delta_{jl} \delta_{in} \delta_{mk} + \delta_{il} \delta_{jn} \delta_{mk} + \delta_{ik} \delta_{ml} \delta_{jn} + \delta_{im} \delta_{jl} \delta_{kn} + \delta_{ik} \delta_{nl} \delta_{jm} + \delta_{im} \delta_{jk} \delta_{nl} } \right), \\ \end{aligned}$$

(88)

$$a_{ijkl} = \left[ {\left( {\alpha M + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-0pt} 2}} \right)\delta_{ik} \delta_{jl} - 2\left( {\gamma_{2} + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-0pt} 2}} \right)\delta_{ij} \delta_{kl} + {{\gamma \delta_{jk} \delta_{il} } \mathord{\left/ {\vphantom {{\gamma \delta_{jk} \delta_{il} } 2}} \right. \kern-0pt} 2}} \right],$$

(89)

$$\begin{aligned} d_{ijklmn} & = \nu^{\prime}_{1} \delta_{ij} \delta_{kl} \delta_{mn} + 2\nu^{\prime}_{2} \left( {\delta_{ij} Y_{klmn} + \delta_{kl} Y_{mnij} + \delta_{mn} Y_{ijkl} } \right) \\ & \quad + 2\nu^{\prime}_{3} \left( {\delta_{ik} Y_{jlmn} + \delta_{il} Y_{jkmn} + \delta_{jk} Y_{ilmn} + \delta_{jl} Y_{ikmn} } \right), \\ \end{aligned}$$

(90)

$$Y_{ijkl} = {{\left( {\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} } \right)} 2}} \right. \kern-0pt} 2}.$$

(91)

Combining Eqs. (83)–(90) and dynamic Eqs. (19) and (20), one can conduct the wave simulation in high-pressure field. In fact, by utilizing Eqs. (83)–(90), we can easily analyze not only the wave propagation in the media under confining pressure and pore pressure (i.e., effective pressure) but also wave propagation in cases with axial stress (such as uniaxial, biaxial and triaxial, see the Appendix in Liu et al. 2021). According to the generalized Hooke’s law, the static strains of a pressured (stressed) rock change as the pressure (stress) state changes.

Appendix C: Explicit Expressions for the Wave R/T Coefficients

The linear relationship among the amplitudes of solid displacement potentials of seven waves is derived as

$$\sum\limits_{j = 1}^{6} {G_{ij} d_{j} } = b_{i} ,$$

(92)

where

$$G_{11} = f_{I} ,G_{12} = f_{I} ,G_{13} = - g_{1S} ,G_{14} = - f_{I} ,G_{15} = - f_{I} ,G_{16} = g_{2S} ,$$

(93)

$$G_{21} = g_{1P} ,G_{22} = g_{1SP} ,G_{23} = f_{I} ,G_{24} = - g_{2P} ,G_{25} = - g_{2SP} ,G_{26} = - f_{I} ,$$

(94)

$$G_{31} = 0,G_{32} = \lambda^{\left( 1 \right)} g_{1SP} ,G_{33} = 0,G_{34} = 0,G_{35} = - \lambda^{\left( 2 \right)} g_{2SP} ,G_{36} = 0,$$

(95)

$$G_{41} = \left( {H_{1}^{\left( 1 \right)} + H_{2}^{\left( 1 \right)} } \right)k_{1P}^{2} - H_{2}^{\left( 1 \right)} f_{I}^{2} ,G_{42} = \left( {H_{1}^{\left( 1 \right)} + H_{2}^{\left( 1 \right)} + \lambda^{\left( 1 \right)} H_{3}^{\left( 1 \right)} } \right)k_{1SP}^{2} - H_{2}^{\left( 1 \right)} f_{I}^{2} ,$$

(96)

$$G_{43} = H_{2}^{\left( 1 \right)} f_{I} g_{1S} ,G_{44} = - \left( {H_{1}^{\left( 2 \right)} + H_{2}^{\left( 2 \right)} } \right)k_{2P}^{2} + H_{2}^{\left( 2 \right)} f_{I}^{2} ,$$

(97)

$$G_{45} = - \left( {H_{1}^{\left( 2 \right)} + H_{2}^{\left( 2 \right)} + \lambda^{s\left( 2 \right)} H_{3}^{\left( 2 \right)} } \right)k_{2SP}^{2} + H_{2}^{\left( 2 \right)} f_{I}^{2} ,G_{46} = - H_{2}^{\left( 2 \right)} f_{I} g_{2S} ,$$

(98)

$$G_{51} = H_{2}^{\left( 1 \right)} f_{I} g_{1P} ,G_{52} = \left( {H_{2}^{\left( 1 \right)} + \lambda^{s\left( 1 \right)} H_{4}^{\left( 1 \right)} } \right)f_{I} g_{1SP} ,G_{53} = \frac{1}{2}H_{2}^{\left( 1 \right)} \left( {2f_{I}^{2} - k_{1S}^{2} } \right),$$

(99)

$$G_{54} = - H_{2}^{\left( 2 \right)} f_{I} g_{2P} ,G_{55} = - \left( {H_{2}^{\left( 2 \right)} + \lambda^{s\left( 2 \right)} H_{4}^{\left( 2 \right)} } \right)f_{I} g_{2SP} ,G_{56} = - \frac{1}{2}H_{2}^{\left( 2 \right)} \left( {2f_{I}^{2} - k_{2S}^{2} } \right),$$

(100)

$$G_{61} = \tilde{\Gamma }^{\left( 1 \right)} k_{1P}^{2} - \gamma^{\left( 1 \right)} e_{11}^{s\left( 1 \right)} f_{I}^{2} - 2\gamma^{\left( 1 \right)} e_{13}^{s\left( 1 \right)} f_{I} g_{1P} - \gamma^{\left( 1 \right)} e_{33}^{s\left( 1 \right)} g_{1P}^{2} ,$$

(101)

$$G_{62} = \left( {\tilde{\Gamma }^{\left( 1 \right)} + \lambda^{s\left( 1 \right)} \tilde{\Upsilon }^{\left( 1 \right)} } \right)k_{1SP}^{2} - \gamma^{\left( 1 \right)} e_{11}^{s\left( 1 \right)} f_{I}^{2} - 2\gamma^{\left( 1 \right)} e_{13}^{s\left( 1 \right)} f_{I} g_{1SP} - \gamma^{\left( 1 \right)} e_{33}^{s\left( 1 \right)} g_{1SP}^{2} ,$$

(102)

$$G_{63} = \gamma^{\left( 1 \right)} e_{11}^{s\left( 1 \right)} f_{I} g_{1S} - \gamma^{\left( 1 \right)} e_{13}^{s\left( 1 \right)} \left( {f_{I}^{2} - g_{1S}^{2} } \right) - \gamma^{\left( 1 \right)} e_{33}^{s\left( 1 \right)} f_{I} g_{1S} ,$$

(103)

$$G_{64} = - \tilde{\Gamma }^{\left( 2 \right)} k_{2P}^{2} + \gamma^{\left( 2 \right)} e_{11}^{s\left( 2 \right)} f_{I}^{2} + 2\gamma^{\left( 2 \right)} e_{13}^{s\left( 2 \right)} f_{I} g_{2P} + \gamma^{\left( 2 \right)} e_{33}^{s\left( 2 \right)} g_{2P}^{2} ,$$

(104)

$$G_{65} = - \left( {\tilde{\Gamma }^{\left( 2 \right)} + \lambda^{s\left( 2 \right)} \tilde{\Upsilon }^{\left( 2 \right)} } \right)k_{2SP}^{2} + \gamma^{\left( 2 \right)} e_{11}^{s\left( 2 \right)} f_{I}^{2} + 2\gamma^{\left( 2 \right)} e_{13}^{s\left( 2 \right)} f_{I} g_{2SP} + \gamma^{\left( 2 \right)} e_{33}^{s\left( 2 \right)} g_{2SP}^{2} + \lambda^{\left( 2 \right)} Wg_{2SP} ,$$

(105)

$$G_{66} = - \gamma^{\left( 2 \right)} e_{11}^{s\left( 2 \right)} f_{I} g_{2S} + \gamma^{\left( 2 \right)} e_{13}^{s\left( 2 \right)} \left( {f_{I}^{2} - g_{2S}^{2} } \right) + \gamma^{\left( 2 \right)} e_{33}^{s\left( 2 \right)} f_{I} g_{2S} ,$$

(106)

and

$$b_{1} = - \vartheta f_{I} { + }\left( {1 - \vartheta } \right)g_{I} ,b_{2} = - \vartheta g_{I} - \left( {1 - \vartheta } \right)f_{I} ,b_{3} = 0,$$

(107)

$$b_{4} = - \vartheta \left[ {H_{1}^{\left( 1 \right)} k_{I}^{2} + H_{2}^{\left( 1 \right)} g_{I}^{2} } \right] - \left( {1 - \vartheta } \right)H_{2}^{\left( 1 \right)} f_{I} g_{I} ,$$

(108)

$$b_{5} = - \vartheta H_{2}^{\left( 1 \right)} f_{I} g_{I} + \frac{1}{2}\left( {1 - \vartheta } \right)H_{2}^{\left( 1 \right)} \left( {g_{I}^{2} - f_{I}^{2} } \right),$$

(109)

$$b_{6} = - \vartheta \left( {\tilde{\Gamma }^{\left( 1 \right)} k_{I}^{2} - \gamma^{\left( 1 \right)} e_{11}^{s\left( 1 \right)} f_{I}^{2} - \gamma^{\left( 1 \right)} e_{33}^{s\left( 1 \right)} g_{I}^{2} } \right) + \left( {1 - \vartheta } \right)\left( {\gamma^{\left( 1 \right)} e_{33}^{s\left( 1 \right)} - \gamma^{\left( 1 \right)} e_{11}^{s\left( 1 \right)} } \right)f_{I} g_{I} ,$$

(110)

where

$$H_{1}^{{\left( r \right)}} = \left( {1 + e^{{s\left( r \right)}} } \right)\left( {\lambda _{u}^{{\left( r \right)}} + 2\gamma _{2}^{{\left( r \right)}} \zeta ^{{s\left( r \right)}} + \gamma ^{{\left( r \right)}} \zeta ^{{s\left( r \right)}} } \right) + \nu ^{{_{1}^{{\prime \left( r \right)}} }} e^{{s\left( r \right)}} ,$$

(111)

$$H_{2}^{{\left( r \right)}} = 2\mu _{m}^{{s\left( r \right)}} - \gamma ^{{\left( r \right)}} \zeta ^{{s\left( r \right)}} + \left( {2\mu _{m}^{{s\left( r \right)}} - \gamma ^{{\left( r \right)}} \zeta ^{{s\left( r \right)}} + 6\nu ^{{_{2}^{{\prime \left( r \right)}} }} + 8\nu ^{{_{3}^{{\prime \left( r \right)}} }} } \right) ,$$

(112)

$$H_{3}^{{\left( r \right)}} = \left( {\gamma _{2}^{{\left( r \right)}} + \gamma ^{{\left( r \right)}} } \right)e^{{s\left( r \right)}} - \gamma ^{{\left( r \right)}} e_{{33}}^{{s\left( r \right)}} + 2\gamma _{3}^{{\left( i \right)}} \zeta ^{{s\left( r \right)}} - \alpha ^{{s\left( r \right)}} M^{{s\left( r \right)}} ,$$

(113)

$$H_{4}^{\left( r \right)} = 2\gamma_{2}^{\left( r \right)} e^{s\left( r \right)} - \gamma^{\left( r \right)} e_{13}^{s\left( r \right)} ,\quad r = 1, \, 2.$$

(114)

The membrane stiffness $W$ is only related to the component $G_{65}$. Substituting Eqs. (92)–(114) in Eqs. (66) and (67) can obtain the wave R/T coefficients of an oblique-incidence plane wave on the plane interface across two dissimilar media filled with the viscous fluids.

Appendix D: Qi et al. (2021) R/T Coefficient Equations

Qi et al. (2021) considered the seismic wave with the frequency much lower than Biot’s characteristic frequency, i.e.,

$$\omega \ll \omega_{c} = \frac{\eta \phi }{{a_{t} k_{0} \rho_{f} }}.$$

(115)

In this case, the wavenumbers of fast P, slow P and S waves are given by

$$k_{P} = \frac{\omega }{{\sqrt {{H \mathord{\left/ {\vphantom {H \rho }} \right. \kern-0pt} \rho }} }},k_{SP} = \sqrt {\frac{i\omega \eta }{{k_{0} N}}} ,{\text{and}}\,k_{S} = \frac{\omega }{{\sqrt {{{\mu_{m} } \mathord{\left/ {\vphantom {{\mu_{m} } \rho }} \right. \kern-0pt} \rho }} }},$$

(116)

where $N = {{ML} \mathord{\left/ {\vphantom {{ML} H}} \right. \kern-0pt} H}$, $L = K_{m} + {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-0pt} 3}\mu_{m}$, $H = L + \alpha^{2} M$. Combining the continuous boundary conditions of wave displacement components and traction components, Qi et al. (2021) further derived the exact R/T coefficient equations for fluid-saturated porous media in the absence of in situ effective pressure. A system of linear equations in terms of the amplitudes of seven waves in Qi et al. (2021) can be expressed as

$$\sum\limits_{j = 1}^{6} {G_{ij} d_{j} } = b_{i} ,$$

(117)

where

$${\varvec{b}} = \left[ { - f_{I} , \, - g_{1P} , \, 0, \, - k_{1P}^{2} H^{\left( 1 \right)} + 2\mu_{m} f_{I}^{2} , \, - 2\mu_{m} f_{I} g_{1P} , \, - k_{1P}^{2} C^{\left( 1 \right)} } \right]^{T} ,$$

(118)

and

$$G_{11} = f_{I} ,G_{12} = f_{I} ,G_{13} = - g_{1S} ,G_{14} = - f_{I} ,G_{15} = - f_{I} ,G_{16} = g_{2S} ,$$

(119)

$$G_{21} = g_{1P} ,G_{22} = g_{1SP} ,G_{23} = f_{I} ,G_{24} = - g_{2P} ,G_{25} = - g_{2SP} ,G_{26} = - f_{I} ,$$

(120)

$$G_{31} = 0,G_{32} = \lambda^{0\left( 1 \right)} g_{1SP} ,G_{33} = 0,G_{34} = 0,G_{35} = - \lambda^{0\left( 2 \right)} g_{2SP} ,G_{36} = 0,$$

(121)

$$G_{41} = H^{\left( 1 \right)} k_{1P}^{2} - 2\mu_{m} f_{I}^{2} ,G_{42} = \left( {H^{\left( 1 \right)} + C^{\left( 1 \right)} \lambda^{0\left( 1 \right)} } \right)k_{1SP}^{2} - 2\mu_{m} f_{I}^{2} ,G_{43} = 2\mu_{m} f_{I} g_{1S} ,$$

(122)

$$G_{44} = - H^{\left( 2 \right)} k_{2P}^{2} + 2\mu_{m} f_{I}^{2} ,G_{45} = - \left( {H^{\left( 2 \right)} + \lambda^{0\left( 2 \right)} C^{\left( 2 \right)} } \right)k_{2SP}^{2} + 2\mu_{m} f_{I}^{2} ,G_{46} = - 2\mu_{m} f_{I} g_{2S} ,$$

(123)

$$G_{51} = 2\mu_{m} f_{I} g_{1P} ,G_{52} = 2\mu_{m} f_{I} g_{1SP} ,G_{53} = \mu_{m} \left( {f_{I}^{2} - g_{1S}^{2} } \right),$$

(124)

$$G_{54} = - 2\mu_{m} f_{I} g_{2P} ,G_{55} = - 2\mu_{m} f_{I} g_{2SP} ,G_{56} = - \mu_{m} \left( {f_{I}^{2} - g_{2S}^{2} } \right),$$

(125)

$$G_{61} = C^{\left( 1 \right)} k_{1P}^{2} ,G_{62} = \left( {C^{\left( 1 \right)} + \lambda^{0\left( 1 \right)} M^{\left( 1 \right)} } \right)k_{1SP}^{2} ,G_{63} = 0,$$

(126)

$$G_{64} = - C^{\left( 2 \right)} k_{2P}^{2} ,G_{65} = - \left( {C^{\left( 2 \right)} + \lambda^{0\left( 2 \right)} M^{\left( 2 \right)} } \right)k_{2SP}^{2} + \lambda^{0\left( 2 \right)} Wg_{2SP} ,G_{66} = 0,$$

(127)

where

$$\lambda^{0} = - \frac{{K_{m} + {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-0pt} 3}\mu_{m} + \alpha^{2} M}}{\alpha M}.$$

(128)

Substituting Eqs. (119)–(127) in Eqs. (66) and (67) can obtain the wave R/T coefficients on the plane interface across two dissimilar fluid-saturated media free of effective pressure.