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为了研究孔隙水含少量气泡时多孔介质中波的传播，本文在Biot模型的基础上，将孔隙水中气泡的体积振动融合到多孔介质的孔隙流体渗流连续性方程中，从而得到了考虑气泡体积振动的孔隙流体渗流连续性方程.在此基础上，根据气泡线性振动下气泡瞬时半径和介质背景压力的关系，以及多孔介质运动方程和流体介质运动方程，导出了受气泡影响下多孔介质位移矢量波动方程，建立了非水饱和多孔介质声速频散和衰减预报模型.气泡的存在增大了孔隙水的压缩率，导致含气泡水饱和多孔介质声速的降低.当声波频率等于气泡的共振频率时，在声波激励下，介质呈现高频散，且孔隙水中的气泡产生共振，吸收截面达到最大，使得多孔介质的声衰减也达到最大.文中数值分析验证了上述结论，表明了气泡含量、大小和驱动声场频率是影响声波在含少量气泡的水饱和多孔介质中传播的主要因素.Biot model has widely been used in geophysics, petroleum engineering, civil engineering, and ocean engineering since it was presented, and thus the research on the wave propagation in saturated porous medium has made much progress. However, fully saturated porous medium is rarely found in nature. Almost all the rocks or soils contain two kinds of fluids, such as gas and petroleum. Many researches have been done on the wave propagation in unsaturated porous medium. As is well known, a small volume of gas bubbles existing in a liquid can greatly change the velocity and attenuation of acoustic wave in the liquid. Evidences are beginning to be accumulated that the velocity and attenuation of acoustic wave in a saturated marine sediment can be affected by the gas bubbles existing in the saturated liquid. To investigate the sound propagation in a porous medium when the pore water contains a small number of air bubbles, in this paper we integrate the volume vibrations of bubbles in pore water into the continuity equation of pore-fluid filtration in porous medium based on Biot theory, so as to obtain the continuity equation of pore-fluid filtration with bubble volume vibration. On this basis, according to the relationship between the instantaneous radius of bubble and the background pressure of the medium under the linear vibration of bubble, as well as the equations of motion of the fluid medium and porous medium, a new displacement vector wave equation of porous medium under the influence of bubble is derived, which establishes the model for the sound velocity dispersion and attenuation prediction under the unsaturated porous medium. The presence of air bubbles increases the compressibility of pore fluid, which leads to the decrease in the sound velocity of the bubbly saturated porous medium. When the wave frequency equals the resonance frequency of the bubbles, the bubbles in pore water will produce resonance; the medium will present high dispersion and the velocity can greatly exceed the gas-free velocity. However, these have not been measured in field data. The absorption cross section of the air bubble can reach a maximum value, which leads to the maximum attenuation of the porous medium. It should be noted that the attenuation coefficient calculated with this model is related to the damping of the bubble motion due to radiation, thermal and internal friction, and the dissipation of the relative motion between the pore water and porous solid frame. The obtained numerical analysis is consistent with the above conclusion, which indicates that the volume concentration, the bubble size and the excitation frequency of the sound field are important parameters affecting the sound wave propagation in the saturated porous medium containing few bubbles.
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Keywords:
- linear vibration of bubbles /
- sound velocity dispersion /
- attenuation /
- Biot theory
[1] Biot M A 1956 J. Acoust. Soc. Am. 28 168
[2] Biot M A 1956 J. Acoust. Soc. Am. 28 179
[3] Domenico S N 1974 Geophysics 39 759
[4] Domenico S N 1976 Geophysics 41 882
[5] Domenico S N 1977 Geophysics 42 1339
[6] Gassmann F 1951 Geophysics 16 673
[7] Geertsma J 1961 Geophysics 26 169
[8] Santos J E, Corbero J M, Jim D J 1990 J. Acoust. Soc. Am. 87 1428
[9] Santos J E, Jim D J, Corbero J M, Lovera O M 1990 J. Acoust. Soc. Am. 87 1439
[10] Ravazzoli C L, Santos J E, Carcione J M 2003 J. Acoust. Soc. Am. 113 1801
[11] Carcione J M, Cavallini F, Santos J E, et al. 2004 Wave Motion 39 227
[12] Li B Z 2007 Ph. D. Dissertation (Hangzhou:Zhejiang University) (in Chinese)[李保忠2007博士学位论文(杭州:浙江大学)]
[13] Commander K W, Prosperetti A 1989 J. Acoust. Soc. Am. 85 732
[14] Prosperetti A, Crum L A, Commander K W 1988 J. Acoust. Soc. Am. 83 502
[15] Wang Y, Lin S Y, Zhang X L 2013 Acta Phys. Sin. 62 064304 (in Chinese)[王勇, 林书玉, 张小丽2013 62 064304]
[16] Wang Y, Lin S Y, Zhang X L 2014 Acta Phys. Sin. 63 034301 (in Chinese)[王勇, 林书玉, 张小丽2014 63 034301]
[17] Bedford A, Stem M 1983 J. Acoust. Soc. Am. 73 409
[18] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1865
[19] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1890
[20] Stoll R D 1974 Acoustic Waves in Saturated Sediments (New York:Plenum Press) pp19-39
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[1] Biot M A 1956 J. Acoust. Soc. Am. 28 168
[2] Biot M A 1956 J. Acoust. Soc. Am. 28 179
[3] Domenico S N 1974 Geophysics 39 759
[4] Domenico S N 1976 Geophysics 41 882
[5] Domenico S N 1977 Geophysics 42 1339
[6] Gassmann F 1951 Geophysics 16 673
[7] Geertsma J 1961 Geophysics 26 169
[8] Santos J E, Corbero J M, Jim D J 1990 J. Acoust. Soc. Am. 87 1428
[9] Santos J E, Jim D J, Corbero J M, Lovera O M 1990 J. Acoust. Soc. Am. 87 1439
[10] Ravazzoli C L, Santos J E, Carcione J M 2003 J. Acoust. Soc. Am. 113 1801
[11] Carcione J M, Cavallini F, Santos J E, et al. 2004 Wave Motion 39 227
[12] Li B Z 2007 Ph. D. Dissertation (Hangzhou:Zhejiang University) (in Chinese)[李保忠2007博士学位论文(杭州:浙江大学)]
[13] Commander K W, Prosperetti A 1989 J. Acoust. Soc. Am. 85 732
[14] Prosperetti A, Crum L A, Commander K W 1988 J. Acoust. Soc. Am. 83 502
[15] Wang Y, Lin S Y, Zhang X L 2013 Acta Phys. Sin. 62 064304 (in Chinese)[王勇, 林书玉, 张小丽2013 62 064304]
[16] Wang Y, Lin S Y, Zhang X L 2014 Acta Phys. Sin. 63 034301 (in Chinese)[王勇, 林书玉, 张小丽2014 63 034301]
[17] Bedford A, Stem M 1983 J. Acoust. Soc. Am. 73 409
[18] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1865
[19] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1890
[20] Stoll R D 1974 Acoustic Waves in Saturated Sediments (New York:Plenum Press) pp19-39
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