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It is very important to understand the acoustical properties of porous medium. To study the relationship between acoustical and other physical properties of porous medium will help us to use acoustical tools for determining the physical properties of porous medium. Many researchers have paid much attention to the properties of acoustic wave propagation in the gassy marine sediments based on the Biot model which is popularly used to predict the dispersion and attenuation of sound in saturated porous medium. The patchy model which contains gas inside the spherical water predicts that the existence of gas just has little effect on the propagation of acoustic wave in porous medium when the gas content is very small. However, the presence of a small number of bubbles in a fluid saturated sediment will lead to different acoustic responses. As is well known, the bubble vibration theory proposed by Keller and Miksis shows that a small number of bubbles existing in the liquid will have a great influence on sound velocity and attenuation. Therefore, in order to study the effect of a small amount of gas existing in fluid saturated porous medium on the property of acoustic wave propagation, we investigate a bubbly liquid saturated porous medium and consider the case of the bubbles vibrating linearly under the action of sound waves. First, we derive the continuity equation of the seepage according to the mass conservation of the pore fluid and the relationship between porosity differentiation and pore fluid pressure differentiation. Then, the bubble linear vibration theory given by Commander is used to deal with the time derivative of gas volume fraction in the continuity equation of the seepage, The bubble linear vibration theory gives the relationship between instantaneous bubble radius and background pressure of the medium. Through this relationship, we obtain the equation of time derivative of gas volume fraction and time derivative of pore fluid pressure. Then, we combine the obtained equation with the continuity equation of seepage, and obtain the modified continuity equation of seepage whose form is similar to that of Biot model. Finally, the modified Biot's equations for fluid saturated porous medium containing a small amount of bubbly fluid is obtained. As is well known, an effective density fluid model for acoustic propagation in sediments, derived from Biot theory, just can predict the acoustic properties of the fast compressional waves. However, the present model can predict the acoustic properties of fast, slow compressional waves and shear waves propagating in sediments. Through numerically calculating the dispersion, attenuation, amplitude ratios of pore fluid displacement to solid displacement for fast and slow compressional waves, it is found that the existence of a small number of bubbles has an influence on the acoustic properties of both the fast compressional waves and the slow compressional waves, especially the velocity of the fast compressional wave. In addition, the low-frequency speed approximation formula for the fast compressional wave is also presented. The approximate formula directly indicates the relationship between the velocity of fast compressional wave and the parameters of porous medium such as the gas volume fraction and the bubble radius. This study shows that the influence of a small number of bubbles in fluid saturated on acoustic wave propagation is noticeable. The modified Biot model presented in this paper provides one model to study the properties of acoustic waves in fluid saturated porous medium with a small number of bubbles.
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Keywords:
- porous medium /
- bubbly liquid /
- Biot theory /
- elastic wave
[1] Qiao W X, Wu W Q, Wang Y J 1996 Prog. Phys. 16 386(in Chinese) [乔文孝, 吴文虬, 王耀俊 1996 物理学进展 16 386]
[2] Biot M A 1956 J. Acoust. Soc. Am. 28 168
[3] Cui Z W, Wang K X, Cao Z L, Hu H S 2004 Acta Phys. Sin. 53 3083(in Chinese) [崔志文, 王克协, 曹正良, 胡恒山 2004 物理学报 53 3083]
[4] Plona T J 1980 Appl. Phys. Lett. 36 259
[5] Cui Z W, Wang K X 2003 Int. J. Eng. Sci. 41 2179
[6] Pride S R, Berryman J G 2003 Phys. Rev. E 68 036604
[7] Wang X M 2009 Appl. Acoust. 28 1(in Chinese) [王秀明 2009 应用声学 28 1]
[8] Santos J E, Corber J M, Douglas J 1990 J. Acoust. Soc. Am. 87 1428
[9] Cai Y Q, Li B Z, Xu C J 2006 Chin. J. Rock Mech. Eng. 25 2009(in Chinese) [蔡袁强, 李宝忠, 徐长节 2006 岩石力学与工程学报 25 2009]
[10] Li W H 2002 Northwest. Seismological J. 24 303(in Chinese) [李伟华 2002 西北地震学报 24 303]
[11] White J E, Mikhaylova N G, Lyakhovitskiy F M 1975 J. Acoust. Soc. Am. 57 S30
[12] Johnson D L 2001 J. Acoust. Soc. Am. 110 682
[13] Keller J B, Miksis M 1980 J. Acoust. Soc. Am. 68 628
[14] Prosperetti A, Crum L A, Commander K W 1988 J. Acoust. Soc. Am. 83 502
[15] Commander K W, Prosperetti A 1989 J. Acoust. Soc. Am. 85 732
[16] 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 物理学报 62 064304]
[17] Wang Y, Lin S Y, Mo R Y, Zhang X L 2013 Acta Phys. Sin. 62 134304(in Chinese) [王勇, 林书玉, 莫润阳, 张小丽 2013 物理学报 62 134304]
[18] 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 物理学报 63 034301]
[19] Zhu L G 2009 Ship Sci. Tech. 10 64(in Chinese) [祝令国 2009 舰船科学技术 31 64]
[20] Wang H B, Wang Z Q, Zhang H Y, Zhang W P 2005 Shipbuild. China 46 44(in Chinese) [王虹斌, 王芝秋, 张洪雨, 张文平 2005 中国造船 46 44]
[21] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1890
[22] Yang X M, Church C C 2005 J. Acoust. Soc. Am. 118 3595
[23] Mantouka A, Dogan H, White P R, Leighton T G 2016 J. Acoust. Soc. Am. 140 274
[24] Zheng G Y, Huang Y W 2016 Acta Phys. Sin. 65 234301(in Chinese) [郑广赢, 黄益旺 2016 物理学报 65 234301]
[25] Dvorkin J, Nur A 1993 Geophysics 58 523
[26] Biot M A 1941 J. Appl. Phys. 12 155
[27] Hu H S 2003 Acta Phys. Sin. 52 1954(in Chinese) [胡恒山 2003 物理学报 52 1954]
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[1] Qiao W X, Wu W Q, Wang Y J 1996 Prog. Phys. 16 386(in Chinese) [乔文孝, 吴文虬, 王耀俊 1996 物理学进展 16 386]
[2] Biot M A 1956 J. Acoust. Soc. Am. 28 168
[3] Cui Z W, Wang K X, Cao Z L, Hu H S 2004 Acta Phys. Sin. 53 3083(in Chinese) [崔志文, 王克协, 曹正良, 胡恒山 2004 物理学报 53 3083]
[4] Plona T J 1980 Appl. Phys. Lett. 36 259
[5] Cui Z W, Wang K X 2003 Int. J. Eng. Sci. 41 2179
[6] Pride S R, Berryman J G 2003 Phys. Rev. E 68 036604
[7] Wang X M 2009 Appl. Acoust. 28 1(in Chinese) [王秀明 2009 应用声学 28 1]
[8] Santos J E, Corber J M, Douglas J 1990 J. Acoust. Soc. Am. 87 1428
[9] Cai Y Q, Li B Z, Xu C J 2006 Chin. J. Rock Mech. Eng. 25 2009(in Chinese) [蔡袁强, 李宝忠, 徐长节 2006 岩石力学与工程学报 25 2009]
[10] Li W H 2002 Northwest. Seismological J. 24 303(in Chinese) [李伟华 2002 西北地震学报 24 303]
[11] White J E, Mikhaylova N G, Lyakhovitskiy F M 1975 J. Acoust. Soc. Am. 57 S30
[12] Johnson D L 2001 J. Acoust. Soc. Am. 110 682
[13] Keller J B, Miksis M 1980 J. Acoust. Soc. Am. 68 628
[14] Prosperetti A, Crum L A, Commander K W 1988 J. Acoust. Soc. Am. 83 502
[15] Commander K W, Prosperetti A 1989 J. Acoust. Soc. Am. 85 732
[16] 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 物理学报 62 064304]
[17] Wang Y, Lin S Y, Mo R Y, Zhang X L 2013 Acta Phys. Sin. 62 134304(in Chinese) [王勇, 林书玉, 莫润阳, 张小丽 2013 物理学报 62 134304]
[18] 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 物理学报 63 034301]
[19] Zhu L G 2009 Ship Sci. Tech. 10 64(in Chinese) [祝令国 2009 舰船科学技术 31 64]
[20] Wang H B, Wang Z Q, Zhang H Y, Zhang W P 2005 Shipbuild. China 46 44(in Chinese) [王虹斌, 王芝秋, 张洪雨, 张文平 2005 中国造船 46 44]
[21] Anderson A L, Hampton L D 1980 J. Acoust. Soc. Am. 67 1890
[22] Yang X M, Church C C 2005 J. Acoust. Soc. Am. 118 3595
[23] Mantouka A, Dogan H, White P R, Leighton T G 2016 J. Acoust. Soc. Am. 140 274
[24] Zheng G Y, Huang Y W 2016 Acta Phys. Sin. 65 234301(in Chinese) [郑广赢, 黄益旺 2016 物理学报 65 234301]
[25] Dvorkin J, Nur A 1993 Geophysics 58 523
[26] Biot M A 1941 J. Appl. Phys. 12 155
[27] Hu H S 2003 Acta Phys. Sin. 52 1954(in Chinese) [胡恒山 2003 物理学报 52 1954]
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