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.
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Acknowledgements
The authors are deeply grateful to the Editor in Chief, Michael J. Rycroft, reviewer Junxin Guo and two other anonymous reviewers for their invaluable suggestions, which significantly enriched this work. The authors thank Boris Gurevich for valuable discussions; and the sponsorships of National Natural Science Foundation of China (42174139, 41974119, 42030103), Shandong Province Foundation for Laoshan National Laboratory of Science and Technology Foundation (LSKJ202203400), Science Foundation from Innovation and Technology Support Program for Young Scientists in Colleges of Shandong Province and Ministry of Science and Technology of China (2019RA2136), China Scholarship Council (202206450050), and Innovation Fund Project for Graduate Students of China University of Petroleum (East China) (23CX04003A).
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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.
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)
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
and thus the stress tensors in the initial and final states can be derived from Eqs. (70) and (71), given by
and
Subtracting Eqs. (72) from (73), we obtain the stress increment induced by wave perturbation, as
Then we express Eq. (74) in terms of the displacement gradient, yields
where
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
According to Eqs. (70) and (77), the fluid pressure in the initial and final states can be expressed as
and
Then the fluid pressure increment (induced by the propagating wave) from the initial state to final state is
Rewriting fluid pressure increment with the wave displacement gradients yields
where
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
where
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
where
and
where
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.,
In this case, the wavenumbers of fast P, slow P and S waves are given by
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
where
and
where
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.
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Chen, F., Zong, Z., Rezaee, R. et al. Pressure Effects on Plane Wave Reflection and Transmission in Fluid-Saturated Porous Media. Surv Geophys (2024). https://doi.org/10.1007/s10712-024-09829-9
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DOI: https://doi.org/10.1007/s10712-024-09829-9