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Description
The present work focuses on the impact of gas bubble dynamics on effective acoustic properties in partially saturated poroelastic media. It's key objective is the bubbles’ interface mechanics on various length scales. The analysis starts from a single air bubble embedded in surrounding water. The classical Minnaert solution [Minnaert 1933] is covered by first taking into account the bulk compressibility. Turning the view towards smaller scales, we extend the description by surface tension and capillary effects [Leighton 1994, de Gennes 2003]. This does not only modify the bubbles’ stiffness, it also introduces new oscillation modes. The length-scale analysis of the interface mechanics is eventually completed by incorporation of higher-order curvature effects [Helfrich 1986] such as the Tolman-length [Tolman 1949].
A macroscopic acoustic model is then derived for multiple gas bubbles in saturated porous media. The system thus contains a poroelastic frame (e.g., rock) [Biot 1956a,b], a continuous wetting phase (e.g., water) and a discontinuous non-wetting phase (e.g., air bubbles). Homogenization of the gas bubble ensemble yields a set of continuum equations that account for the individual bubbles’ resonance frequencies and damping [Frehner 2010, Steeb 2012, Kurzeja 2014a,b]. Like in Biot's theory, the model accounts for two propagating P-waves and one shear wave, respectively. In addition to classical poroelasticity, the discrete (discontinuous) bubble oscillation contribute to the dispersive behaviour of the system.
The evolution of wave speed and intrinsic attenuation of waves is illustrated numerically for water-saturated rock with air bubbles of various size and distribution. Respective assumptions and limitations are summarized in this process to provide a quick reference for choosing the best compromise between a model’s complexity and applicability. Opportunities and limitations of the predictions are discussed with respect to characterization. Open questions in terms of material parameters and simplifying assumptions will conclude the discussion and shall motivate subsequent investigations across disciplines.
References
[Biot 1956a] M. A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. Journal of the Acoustical Society of America, 28: 168–178, 1956.
[Biot 1956b] M. A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher-frequency range. Journal of the Acoustical Society of America, 28: 179–191, 1956.
[de Gennes 2003] P.-G. de Gennes, F. Brochard-Wyart, and D. Quéré. Capillarity and wetting phenomena. Springer, 2003.
[Frehner 2010] M. Frehner, H. Steeb, and S. M. Schmalholz. Wave velocity dispersion and attenuation in media exhibiting internal oscillations. In A. Petrin, editor, Wave Propagation in Materials for Modern Applications, 455–476. In-Tech Education and Publishing, Vukovar, Croatia, 2010.
[Helfrich 1986] W. Helfrich. Size distributions of vesicles: the role of the effective rigidity of membranes. Journal de Physique, 47(2): 321-329, 1986.
[Kurzeja 2014a] P. S. Kurzeja, and H. Steeb. Variational formulation of oscillating fluid clusters and oscillator-like classification. I. Theory. Physics of Fluids, 26(4): 042106, 2014.
[Kurzeja 2014b] P. S. Kurzeja, and H. Steeb. Variational formulation of oscillating fluid clusters and oscillator-like classification. II. Numerical Study of Pinned Liquid Clusters. Physics of Fluids, 26(4): 042107, 2014.
[Leighton 1994] T. G. Leighton. Acoustic bubble detection - I. The detection of stable gas bodies. Environmental Engineering, 7: 9-16, 1994.
[Minnaert 1933] M. Minnaert. XVI. On musical air-bubbles and the sounds of running water. Philosophical Magazine Series 7, 16(104): 235–248, 1933.
[Steeb 2012] H. Steeb, P. S. Kurzeja, M. Frehner, and S. M. Schmalholz. Phase velocity dispersion and attenuation of seismic waves due to trapped fluids in residual saturated porous media. Vadose Zone Journal, 11, 2012.
[Tolman 1949] R. C. Tolman. The effect of droplet size on surface tension. Journal of Chemical Physics, 17(3): 333–337, 1949.
| Participation | Online |
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| Country | Germany |
| MDPI Energies Student Poster Award | No, do not submit my presenation for the student posters award. |
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