Biot介质中倾斜入射平面P波与井孔耦合的理论研究和数值模拟
摘要
声波与井孔的耦合是典型的弹性波散射问题,其理论结果对解释井间地震勘探资料具有指导意义。
本文研究了Biot介质(孔隙介质)中平面P波倾斜入射时与井孔的耦合,推导了全频域的井孔内外声波场的理论公式;并在此基础上,进行了数值分析,讨论了入射角度不同时,井内流体声压(振幅比)随频率的变化;对入射角为90°时,井内流体声压及相位差随频率的变化进行分析,并与已有文献作了比较,讨论了井壁上不同方位的声压(振幅比)分布;并且考察了渗透率和孔隙度对井孔内外声压场的影响;从而更深刻的了解平面P波倾斜入射时波与井孔耦合的规律。
对Biot介质中倾斜入射平面P波与井的耦合进行理论与数值计算,不仅有利于掌握波在孔隙介质中的传播规律特性,而且对孔隙介质中波识别与分离具有一定的实际意义。本文的研究是至今无人做过的工作,具有一定的创新性。
Acoustic logging is a sort of very developed logging method which holds a widespread applied foreground in physical geography filed. A Acoustic logging has the same physics base with the seismic prospecting, it is a tie of combining the geophysical log with the seismic prospecting data. In twenty or thirty years, acoustic logging develops quickly, no matter what theory investigation and open country field working measure or the well logging data treatment application all have great development. However, recognizing of the well sound field is not enough, people can not use the already possess of acoustic wave data sufficiently. Wherefore, it necessitates further investigations of acoustic wave propagation rule in the hole, inspecting characteristic outside the hole and formation bounding surface influence on the full wave sound field and the partial wave sound field, building more perfect logging interpretation model . These are physics base of acoustic logging, these are also the research subject which geological prospecting scholar and petroleum developing scientific worker are interested in.
As a matter of fact, ideal elastic formation not exists. Formation frequently belongs to multiphase mixture-small opening medium, belonging to mixture is on account for solid part (then rock frame) often being made up particles, which has different chemical characteristic and crystal characteristic, this kind of solid part always combining gas phase and liquid phase of holding loophole space among particles, so it belongs multiphase. Microinhomogeneities of the porous medium cause a complex macrophysics shape, the shape treats comparing sensitization of minimal change group of fluid content or solid constitution, porous medium acoustic of the porous medium combines strictness of mechanics rule with natural random of porous medium, besides attempting to describe its state characteristic.
The theoretical study of cross well have two ways: one is base on the method of geometry optics or radial theory, the other one is base on the method of wave-equation. Radial theory can be applied when seismic wavelength great for the scale of anuniformity body. But radial theory can not applied when seismic wavelength close to the scale of anunfoumity body. Because diffraction and scattering rise leading action in this time. We must apply the method of wave-equation. We must establish physics model base on the fact circs on cross well seismic. Resolving and computing acoustic field radiation, propagation and effect between boreholes from strict wave-equation are the foundation of comprehend and explain for the method of measure.
This article consider a comparison simple model, incident wave outside the hole is plane P-wave case. An exact parsed expression of the coupling of an external oblique plane P-wave to a fluid-filled in Biot saturation porous solid is present. Discussing angle of incidence is 90 degree, sound field of the pressure in a hole is analyzed theoretically and numerically, the theory is compared with literature to testify the correctness of the theory. The mechanism of the resonance response of the pressure in a hole is analyzed numerically in different angles of incidence. In different angles of incidence, resonance hump of the pressure in a hole is not linearity variation, resonance hump of the pressure in a hole appear minimal value in angles of incidence between 30 and 40 degree. The changes of the pressure for off-axis are investigated in different angles of incidence. The frequency of signal is under 500Hz, the difference that the pressure in a hole ( ) and the sidewall ( ) of different points following frequency variation curve may neglect, the frequency of signal is over 1 kHz, they divided obviously. The frequency of signal is about 3 kHz, two pressure curve of polar angle and appear concurrent hump, but angles of incidence is less than 60 degree the concurrent hump vanish. The effect of the difference of permeability and porosity for the pressure in a hole are investigated emphatically. In low frequency region magnification of r=0r =bθ=00θ=1800 permeability leads to rise of the pressure in a hole, the frequency of signal is over 3 kHz that high frequency lead to falling of the pressure in a hole. In higher frequency of signal the difference of permeability only homologize identical pressure in a hole. The angle of incidence is 90 degrees, no matter what is low and high frequency, the pressure in a hole even drives to stable value in high or low permeability, the limit of low permeability is corresponded to the case of elastic solid. At the variation angles of incidence, the width of different permeability curve also generates variation. In low frequency region magnification of porosity leads to falling of the pressure in a hole, the frequency of signal is over 3 kHz that high frequency lead to rise of the pressure in a hole. In higher frequency of signal the difference of porosity only homologize identical pressure in a hole. No matter what is low and high frequency, the pressure in a hole even not drive to stable value. The angle of incidence is 90 degree, at the magnification of the pressure in a hole reduce in size. The angle of incidence is 60 and 15 degree, in low frequency region magnification of porosity leads to falling of the pressure in a hole, in high frequency region magnification of porosity leads to magnification of the pressure in a hole. The magnification of porosity leads to falling of the pressure outside a hole. The angle of incidence is only 60 degree, the magnification of porosity leads to falling of the pressure outside a hole and magnification again. The results show that the well hole has a significant effect on the pressure measured in the hole over the complete frequency range. The different angles of incidence have effect on resonance hump of the pressure in a hole. When the frequency of the seismic signal measured is over 500Hz, the effect of the off-axis measurement and the difference azimuth must be considered. The effect of permeability and porosity dimension must be considered obviously. It made us recognized more deep recognition of porous solid formation.
Total teachings, through assuming the different angles of incidence carry on calculation and analogue. Body work is research work that nobody do hitherto, have definite innovation. Its result have definite reference value on identify and separation of the wave field in Biot solid and explanation of seismic exploration data between two holes.
本文研究了Biot介质(孔隙介质)中平面P波倾斜入射时与井孔的耦合,推导了全频域的井孔内外声波场的理论公式;并在此基础上,进行了数值分析,讨论了入射角度不同时,井内流体声压(振幅比)随频率的变化;对入射角为90°时,井内流体声压及相位差随频率的变化进行分析,并与已有文献作了比较,讨论了井壁上不同方位的声压(振幅比)分布;并且考察了渗透率和孔隙度对井孔内外声压场的影响;从而更深刻的了解平面P波倾斜入射时波与井孔耦合的规律。
对Biot介质中倾斜入射平面P波与井的耦合进行理论与数值计算,不仅有利于掌握波在孔隙介质中的传播规律特性,而且对孔隙介质中波识别与分离具有一定的实际意义。本文的研究是至今无人做过的工作,具有一定的创新性。
Acoustic logging is a sort of very developed logging method which holds a widespread applied foreground in physical geography filed. A Acoustic logging has the same physics base with the seismic prospecting, it is a tie of combining the geophysical log with the seismic prospecting data. In twenty or thirty years, acoustic logging develops quickly, no matter what theory investigation and open country field working measure or the well logging data treatment application all have great development. However, recognizing of the well sound field is not enough, people can not use the already possess of acoustic wave data sufficiently. Wherefore, it necessitates further investigations of acoustic wave propagation rule in the hole, inspecting characteristic outside the hole and formation bounding surface influence on the full wave sound field and the partial wave sound field, building more perfect logging interpretation model . These are physics base of acoustic logging, these are also the research subject which geological prospecting scholar and petroleum developing scientific worker are interested in.
As a matter of fact, ideal elastic formation not exists. Formation frequently belongs to multiphase mixture-small opening medium, belonging to mixture is on account for solid part (then rock frame) often being made up particles, which has different chemical characteristic and crystal characteristic, this kind of solid part always combining gas phase and liquid phase of holding loophole space among particles, so it belongs multiphase. Microinhomogeneities of the porous medium cause a complex macrophysics shape, the shape treats comparing sensitization of minimal change group of fluid content or solid constitution, porous medium acoustic of the porous medium combines strictness of mechanics rule with natural random of porous medium, besides attempting to describe its state characteristic.
The theoretical study of cross well have two ways: one is base on the method of geometry optics or radial theory, the other one is base on the method of wave-equation. Radial theory can be applied when seismic wavelength great for the scale of anuniformity body. But radial theory can not applied when seismic wavelength close to the scale of anunfoumity body. Because diffraction and scattering rise leading action in this time. We must apply the method of wave-equation. We must establish physics model base on the fact circs on cross well seismic. Resolving and computing acoustic field radiation, propagation and effect between boreholes from strict wave-equation are the foundation of comprehend and explain for the method of measure.
This article consider a comparison simple model, incident wave outside the hole is plane P-wave case. An exact parsed expression of the coupling of an external oblique plane P-wave to a fluid-filled in Biot saturation porous solid is present. Discussing angle of incidence is 90 degree, sound field of the pressure in a hole is analyzed theoretically and numerically, the theory is compared with literature to testify the correctness of the theory. The mechanism of the resonance response of the pressure in a hole is analyzed numerically in different angles of incidence. In different angles of incidence, resonance hump of the pressure in a hole is not linearity variation, resonance hump of the pressure in a hole appear minimal value in angles of incidence between 30 and 40 degree. The changes of the pressure for off-axis are investigated in different angles of incidence. The frequency of signal is under 500Hz, the difference that the pressure in a hole ( ) and the sidewall ( ) of different points following frequency variation curve may neglect, the frequency of signal is over 1 kHz, they divided obviously. The frequency of signal is about 3 kHz, two pressure curve of polar angle and appear concurrent hump, but angles of incidence is less than 60 degree the concurrent hump vanish. The effect of the difference of permeability and porosity for the pressure in a hole are investigated emphatically. In low frequency region magnification of r=0r =bθ=00θ=1800 permeability leads to rise of the pressure in a hole, the frequency of signal is over 3 kHz that high frequency lead to falling of the pressure in a hole. In higher frequency of signal the difference of permeability only homologize identical pressure in a hole. The angle of incidence is 90 degrees, no matter what is low and high frequency, the pressure in a hole even drives to stable value in high or low permeability, the limit of low permeability is corresponded to the case of elastic solid. At the variation angles of incidence, the width of different permeability curve also generates variation. In low frequency region magnification of porosity leads to falling of the pressure in a hole, the frequency of signal is over 3 kHz that high frequency lead to rise of the pressure in a hole. In higher frequency of signal the difference of porosity only homologize identical pressure in a hole. No matter what is low and high frequency, the pressure in a hole even not drive to stable value. The angle of incidence is 90 degree, at the magnification of the pressure in a hole reduce in size. The angle of incidence is 60 and 15 degree, in low frequency region magnification of porosity leads to falling of the pressure in a hole, in high frequency region magnification of porosity leads to magnification of the pressure in a hole. The magnification of porosity leads to falling of the pressure outside a hole. The angle of incidence is only 60 degree, the magnification of porosity leads to falling of the pressure outside a hole and magnification again. The results show that the well hole has a significant effect on the pressure measured in the hole over the complete frequency range. The different angles of incidence have effect on resonance hump of the pressure in a hole. When the frequency of the seismic signal measured is over 500Hz, the effect of the off-axis measurement and the difference azimuth must be considered. The effect of permeability and porosity dimension must be considered obviously. It made us recognized more deep recognition of porous solid formation.
Total teachings, through assuming the different angles of incidence carry on calculation and analogue. Body work is research work that nobody do hitherto, have definite innovation. Its result have definite reference value on identify and separation of the wave field in Biot solid and explanation of seismic exploration data between two holes.
引文
1. 张海澜,王秀明,张碧星等,《井孔的声场和波》,科学出版社,2004,第一版.
2. 吕伟国,用孔隙柱管模拟测井声系的数值分析与交叉偶极声测井资料反演的应用研究,吉林大学,硕士学位论文,2004.
3. 丛健生,利用有限差分法模拟计算具有分层和裂缝地层井内外声场,大庆石油学院,硕士学位论文,2004.
4. Biot, M.A., Theory of propagation of elastic waves in a fluid-filled saturated porous solid, I: Low-frequency range,J.Acoust.Soc.Am.,1956a,28:168-191.
5.顾金海,叶学千,《水声学基础》,国防工业出版社,1981.
6. 应崇福,张守玉,沈建中,《超声在固体中的散射》,国防工业出版社,1994.
7. 钟伟芳,聂国华,《弹性波的散射理论》,华中理工大学出版社,1997.
8.Sezawa K, Scattering of Elastic Waves and Some Allied Problem, Bull Earthquake Res Inst Tokyo Imperial Univ,1927,3:19.
9.Wolf A, Motion of a Ridid sphere in an Acoustic Wave Field,Geophsics,1945,10:91.
10.Nagase M, Diffraction of Elastic Wave by a spherical Cross section, ASME J APP Mech,1982,49(1):157-164.
11.Knopoffl, Scattering of Compressional Waves by spherical Obstacle, Geophysics,1959,24(1):30.
12.Ying C F, Trull R, Scattering of a Plane Longitudinal Wave by a spherical Obstacle in an Isotropic Elastic solid, J Appl phys,1956,27(9):1086-1097.
13.Einspruch N G, Witterholt E J, Truell R, Scattering of a Plane Transverse Wave by a Spherical Obstacle in an Elastic Medium, J Appl phys,1960,31(5):806-818.
14.Gautesen A K, Achenbach J D, Mcmaken H, Surface Wave Rays in Elastodynamic Diffraction by Cracks, J of Acoust Soc of Am,1978,63:1828-1831.
15.黎在良,刘殿魁,各向异性介质中圆柱对 SH 波散射的射线理论,地震工作与工程振动,1987,7(1):1-8.
16.Barakat R, Diffraction of Plane Waves by an Elliptic Cylinder, J Acoust Soc Am,1963,35:1990.
17.Simon M M, Elastic Wave Scattering from Elliptical shells, J Acoust Soc Am,1982,17(2):273-281.
18.王克协,J.E.White,伍先运, Biot 介质中 P 波与井孔的耦合,地球物理学报,1996,39(1):103-112。
19. [苏]E.I.加尔比林著,朱光明,肖慈珣,余惠宁等译,《垂直地震剖面》,石油工业出版社,1983.
20. 董世泰,井间地震应用技术,中国石油勘探开发研究院,博士学位论文,2004.
21.周来江,井外声源激发井孔声场及井间地震的理论研究与数值模拟,吉林大学,硕士学位论文,2006.
22. 吴律,《层析基础及其在井间地震中的应用》,石油出版社,1997.
23. 刘合,王玉普,隋军,薛家峰,刘兵编写,《国外井间地震技术》,石油工业出版社,1998.
24. 黎在良,刘殿魁,《固体中的波》,科学出版社,1995.
25. [法]T.布尔贝,O.库索,B.甄斯纳,《孔隙介质声学》,石油工业出版社,1994.
26. Biot, M.A., Theory of propagation of elastic waves in a fluid-filled saturated porous solid, I: High-frequency range,J.Acoust.Soc.Am.,1956b,28:179-191.
27.Bensoussan A, Lions.J.L, Papanicolaou.S, Asymptotic analysis for periodic structures, North Holland, Amsterdam,1978.
28.Levy.Th, Sanchez-Palencia.E, Equations and interface condition for acoustic phenomena in porous media, J. of Mathematical Analysis and Applications,1977,61:813-834.
29.Levy.Th, Propagation of waves in a fluid-saturated porous elastic solid, Int.J.Eng.Sci,1979,17:1005-1014.
30.Suquet.P, Plasticite a homogeneisation, Thesis Universite P. et M. Curie, Paris,1982.
31. Biot, M.A., Mechanics of deformation and acoustic propagation in porous dedia, J.Appl.Phys,1962,23:1482-1498.
32.Mandel, J., Essai sur mecanique des pserdosolides, Annales des Ponts. Et Chaussees,1950.
33.Biot, M.A., Variational iagrangian-thermody namics of nonisoternal finite strain mechanics of porous solids and thermonuclear diffusion, Int.J.Solids structures,1977,13:579-597.
34.Michael Schoenberg, fluid and solid motion in the neighborhood of a fluid-filled borehole due to the passage of a low-frequency elastic plane wave, Geophysics,
35. J.R.Lovell and B.E.Hornby, Borehole coupling at sonic frequencies,Geophysics,1990,55:806-814.
36. 胡嗣柱,倪光炯,《数学物理方法》,复旦大学出版社,1989.
2. 吕伟国,用孔隙柱管模拟测井声系的数值分析与交叉偶极声测井资料反演的应用研究,吉林大学,硕士学位论文,2004.
3. 丛健生,利用有限差分法模拟计算具有分层和裂缝地层井内外声场,大庆石油学院,硕士学位论文,2004.
4. Biot, M.A., Theory of propagation of elastic waves in a fluid-filled saturated porous solid, I: Low-frequency range,J.Acoust.Soc.Am.,1956a,28:168-191.
5.顾金海,叶学千,《水声学基础》,国防工业出版社,1981.
6. 应崇福,张守玉,沈建中,《超声在固体中的散射》,国防工业出版社,1994.
7. 钟伟芳,聂国华,《弹性波的散射理论》,华中理工大学出版社,1997.
8.Sezawa K, Scattering of Elastic Waves and Some Allied Problem, Bull Earthquake Res Inst Tokyo Imperial Univ,1927,3:19.
9.Wolf A, Motion of a Ridid sphere in an Acoustic Wave Field,Geophsics,1945,10:91.
10.Nagase M, Diffraction of Elastic Wave by a spherical Cross section, ASME J APP Mech,1982,49(1):157-164.
11.Knopoffl, Scattering of Compressional Waves by spherical Obstacle, Geophysics,1959,24(1):30.
12.Ying C F, Trull R, Scattering of a Plane Longitudinal Wave by a spherical Obstacle in an Isotropic Elastic solid, J Appl phys,1956,27(9):1086-1097.
13.Einspruch N G, Witterholt E J, Truell R, Scattering of a Plane Transverse Wave by a Spherical Obstacle in an Elastic Medium, J Appl phys,1960,31(5):806-818.
14.Gautesen A K, Achenbach J D, Mcmaken H, Surface Wave Rays in Elastodynamic Diffraction by Cracks, J of Acoust Soc of Am,1978,63:1828-1831.
15.黎在良,刘殿魁,各向异性介质中圆柱对 SH 波散射的射线理论,地震工作与工程振动,1987,7(1):1-8.
16.Barakat R, Diffraction of Plane Waves by an Elliptic Cylinder, J Acoust Soc Am,1963,35:1990.
17.Simon M M, Elastic Wave Scattering from Elliptical shells, J Acoust Soc Am,1982,17(2):273-281.
18.王克协,J.E.White,伍先运, Biot 介质中 P 波与井孔的耦合,地球物理学报,1996,39(1):103-112。
19. [苏]E.I.加尔比林著,朱光明,肖慈珣,余惠宁等译,《垂直地震剖面》,石油工业出版社,1983.
20. 董世泰,井间地震应用技术,中国石油勘探开发研究院,博士学位论文,2004.
21.周来江,井外声源激发井孔声场及井间地震的理论研究与数值模拟,吉林大学,硕士学位论文,2006.
22. 吴律,《层析基础及其在井间地震中的应用》,石油出版社,1997.
23. 刘合,王玉普,隋军,薛家峰,刘兵编写,《国外井间地震技术》,石油工业出版社,1998.
24. 黎在良,刘殿魁,《固体中的波》,科学出版社,1995.
25. [法]T.布尔贝,O.库索,B.甄斯纳,《孔隙介质声学》,石油工业出版社,1994.
26. Biot, M.A., Theory of propagation of elastic waves in a fluid-filled saturated porous solid, I: High-frequency range,J.Acoust.Soc.Am.,1956b,28:179-191.
27.Bensoussan A, Lions.J.L, Papanicolaou.S, Asymptotic analysis for periodic structures, North Holland, Amsterdam,1978.
28.Levy.Th, Sanchez-Palencia.E, Equations and interface condition for acoustic phenomena in porous media, J. of Mathematical Analysis and Applications,1977,61:813-834.
29.Levy.Th, Propagation of waves in a fluid-saturated porous elastic solid, Int.J.Eng.Sci,1979,17:1005-1014.
30.Suquet.P, Plasticite a homogeneisation, Thesis Universite P. et M. Curie, Paris,1982.
31. Biot, M.A., Mechanics of deformation and acoustic propagation in porous dedia, J.Appl.Phys,1962,23:1482-1498.
32.Mandel, J., Essai sur mecanique des pserdosolides, Annales des Ponts. Et Chaussees,1950.
33.Biot, M.A., Variational iagrangian-thermody namics of nonisoternal finite strain mechanics of porous solids and thermonuclear diffusion, Int.J.Solids structures,1977,13:579-597.
34.Michael Schoenberg, fluid and solid motion in the neighborhood of a fluid-filled borehole due to the passage of a low-frequency elastic plane wave, Geophysics,
35. J.R.Lovell and B.E.Hornby, Borehole coupling at sonic frequencies,Geophysics,1990,55:806-814.
36. 胡嗣柱,倪光炯,《数学物理方法》,复旦大学出版社,1989.