Abstract
We study the anelastic properties (attenuation and velocity dispersion) of surface waves at an interface between a finite water layer and a porous medium described by Biot theory including the frequency-dependent effects due to mesoscopic flow. A closed-form dispersion equation is derived, based on potential functions and open and sealed boundary conditions (BC) at the interface. The analysis indicates the existence of high-order surface modes for both BCs and a slow true surface mode only for sealed BC. The formulation reduces to two particular cases in the absence of water and with infinite-thickness water layer, with the presence of pseudo-versions of Rayleigh and Stoneley waves. The mesoscopic flow affects the propagation of all the pseudo-surface waves, causing significant velocity dispersion and attenuation, whereas the effect of the BC is mainly evident at high frequencies, due to the presence of the slow Biot wave. The mesoscopic-flow peak moves to low frequencies as the thickness of the water layer increases. In all cases, the true surface wave resembles the slow P2 wave, and is hardly affected by the flow.
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Acknowledgements
We appreciate the comments from the Editor Michael J. Rycroft and three anonymous reviewers, which significantly improve this manuscript. This work has been supported by the “National Nature Science Foundation of China (42174148, 41804095),” the “China Postdoctoral Science Foundation (2020M682242),” and the Qingdao Postdoctoral Applied Research Project.
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Appendices
Appendix A: Complex Coefficients of the Effective Biot Medium
From Pride et al. (2004) and Liu et al. (2009), the complex coefficients are
where \(a_{ij}\) are the real double-porosity constants corresponding to the high-frequency response for which no internal fluid pressure relaxation can take place. These real coefficients are (Pride et al. 2004)
where subscript \(i=\) 1 or 2 denotes the phase 1 or 2, respectively, \(v_i\) is the volume fraction, \(K_i^d\) is the bulk modulus of the dry-rock frame, \(K_\textrm{d}\) is the dry-rock bulk modulus of the composite,
\(B_i\) is the Skempton coefficient,
where \(K_\textrm{s}\) and \(K_\textrm{f}\) are the bulk moduli of the grains and fluid, respectively, and \(\phi _i\) is the porosity. Moreover, \(\alpha _i\) is the Biot-Willis coefficient of phase i, given by
where \(K_i^u\) is the Gassmann wet-rock bulk modulus (confining pressure change divided by dilatation for a sealed sample), and is given by
Substituting equation (A6) into (A5), we obtain
The frequency-dependent internal transport coefficient \(\gamma (\omega )\) is derived by Pride et al. (2004) as
where \(\gamma _\textrm{m}\) and \(\omega _\textrm{m}\) are parameters dependent on the constituent properties and the mesoscopic geometry. When the embedded phase 2 is very permeable, \(\gamma _\textrm{m}\) can be expressed by
where \(\eta\) is the fluid viscosity, \(\kappa _i\) is the permeability of phase i, \(B_0\) is the static Skempton coefficient of the composite, given by
\(R_1\) is the ratio of the average confining pressure in phase 1 to the pressure applied to the external surface of the double-porosity composite, given by
and \(L_1\) represents the characteristic length of the fluid pressure gradient.
In terms of \(\gamma _\textrm{m}\), \(\omega _\textrm{m}\) is
with V/S representing the volume-to-surface ratio, where S is the surface area of the interface between the two phases in each volume V of composite.
In a concentric sphere geometry (a composite (phase 1) of radius R contains a small sphere of radius a of phase 2), satisfying \(\kappa _1/\kappa _2 \ll 1\),
In this case, the volume fractions \(v_1\) and \(v_2\) are defined as
and the total porosity is
Appendix B: Components of Matrix \({\textbf{M}}\)
The elements of matrix \({\textbf{M}}\) are
Next, we compare the above equations with those of Feng and Johnson (1983a). By letting \(H =+\infty\) and \(P=A+2N\), and defining
in the same manner as Feng and Johnson (1983a), the first equation in (30) becomes
By substituting \(\xi _j\) in equation (25) into (B6), we derive equation (C2) of Feng and Johnson (1983a).
The second equation in (30) becomes
which is Eq. (C1) of Feng and Johnson (1983a).
The third equation in (30) becomes
which is equation (C3) of Feng and Johnson (1983a), where they instead use \(\alpha =\tau\).
The last equation in (30) becomes
By multiplying \(-\phi c^2\) on both sides, we derive equation (C4) of Feng and Johnson (1983a).
It is evident that the equations of Feng and Johnson (1983a) are special cases of our equations when \(H=+\infty\). Moreover, our equations use frequency-dependent elastic coefficients associated with the mesoscopic fluid flow, and hence are more realistic for low-frequency surface-wave propagation.
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Wang, E., Yan, J., He, B. et al. Surface-Wave Anelasticity in Porous Media: Effects of Wave-Induced Mesoscopic Flow. Surv Geophys 44, 1953–1983 (2023). https://doi.org/10.1007/s10712-023-09780-1
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DOI: https://doi.org/10.1007/s10712-023-09780-1