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系統識別號 U0026-1008201023513300
論文名稱(中文) 波在半無限含兩非混合相流體孔彈性介質中之傳遞與反射特性
論文名稱(英文) WAVE PROPAGATION AND REFLECTION THROUGH A POROELASTIC HALF-SPACE SATURATED BY TWO IMMISCIBLE FLUIDS
校院名稱 成功大學
系所名稱(中) 水利及海洋工程學系碩博士班
系所名稱(英) Department of Hydraulics & Ocean Engineering
學年度 98
學期 2
出版年 99
研究生(中文) 陳由聖
研究生(英文) Yu-Sheng Chen
學號 N8893102
學位類別 博士
語文別 英文
論文頁數 125頁
口試委員 召集委員-陳主惠
口試委員-譚義績
口試委員-李振誥
口試委員-王裕民
口試委員-簡仲和
共同指導教授-羅偉誠
指導教授-呂珍謀
中文關鍵字 反射  孔彈性  雷利表面波  飽和度 
英文關鍵字 Poroelasticity  Rayleigh wave  Reflection  Saturation 
學科別分類
中文摘要 本文旨在探討彈性波(elastic waves)在半無限(seim-infinite)含兩非混合相(immiscible)流體孔彈性(poroelastic)介質中的傳遞及反射行為,以Lo等人在2005年推導之含兩非混合相、黏稠性流體之孔彈性理論為基礎,且在孔彈性介質表面不受應力之邊界條件下,分別進行兩個模式(雷利表面波及彈性波反射)控制方程式之推導及解析解之推求。
第一個發展之表面波傳遞模式可用來預測雷利表面波(Rayleigh surface waves)在沿半無限不透水薄層下非飽和孔隙介質中之傳遞與衰減特性。此理論可證明在非飽和(unsaturated)孔隙介質中因三個膨脹波(dilatational waves)(依速度大小表示為P1波、P2波及P3波)之存在而分別與剪力波(shear wave)(S波)合成產生三個雷利表面波之模式,依其相速度(phase speed)值之大小可標示為R1波、R2波及R3波。當震盪頻率(excitation frequency)及水飽和度(water saturation)已知之情況下,此三次冪次解析擴散(dispersive)方程式經過程式運算可以依速度值大小順序求出R1波、R2波及R3波之相速度及衰減係數(attenuation coefficient)。為了方便與前人研究結果比較,在此以未飽和(水飽和度0.01~0.99)哥倫比亞細質砂壤土(Columbia fine sandy loam)及震盪頻率50~200Hz為例,計算結果顯示:R1波之相速度與頻率不相關,其值分別約為剪力波相速度值之93~95%及P1波相速度值的28~49%。然而,R2波及R3波之相速度卻與頻率有關,且其值分別約為P2波相速度值及P3波相速度值之56~90%。另一方面,與Lo(2008)發表半無限透水邊界下之雷利表面波模式進行比較後發現,R1波相速度在不透水表面下約是透水表面時之1.01~1.37倍,尤其在孔隙水含量高時,R1波之相速度仍可維持與S波相速度值間保持一定之比例(93~95%)關係,此係因表層不透水邊界侷限孔隙流體於孔彈性介質內所造成之影響。
前述模式除應用在非飽和案例外,可衍生擴大應用至含兩非混合相黏稠性流體之情況,進而探討孔隙中兩黏稠性流體組合對雷利表面波傳遞及衰減之影響。在此以三種含兩非混合相流體(空氣-水、空氣-油及油-水)組合之林肯砂土(Lincoln sand)為例進行計算,模擬雷利表面波在含潤濕流體飽和度(wetting fluid saturation)1~99%且受到震波頻率1~100Hz時之三個波的相速度及衰減係數。計算結果顯示:R1波在空氣-水及空氣-油流體組合中,其相速度約為剪力波相速度之93~95%,但R1波在油-水流體組合中之相速度約為剪力波相速度之95~96%。R1波衰減係數在空氣-水及空氣-油兩個案例中皆與孔隙中兩流體密度的差異及流體與固體間相對運動性有關。但是R1波衰減係數在油-水系統中卻與有效運動黏滯性(effective kinematic viscosity)相關。其次,R2波與R3波之相速度與Lo等人在2005年預測之P2波及P3波之趨勢相近,意指R2波如同P2波受到流體固體間反向(out-of-phase)運動有關,且R3波如同P3波受毛細壓力(capillary pressure)影響。雖然R3波之衰減係數在油-水系統中最大,但在三種流體組合中R3波之衰減係數都非常接近,彼此間差異性都不大。
第二個開發之模式係用來模擬半無限非飽孔彈性介質透水表面附近之波反射問題。當P1波自介質中入射不受應力表面之邊界後,會產生四個反射波(P1波、P2波、P3波及S波)。此模式可用在探討四個反射波與原入射P1波之振幅比,並可剖析在此情況下固體相之表面位移量及表面水平應力的變化。在此同樣以非飽和林肯砂土受到震波震盪頻率1~100Hz時為例進行運算,計算結果顯示:反射波之振幅、表面位移量及表面承受水平應力量皆為水飽和度及波入射角度(incident angle)之函數。在P1波正向入射(normal incidence)或平行入射(grazing incidence)表面時,原預期在任意角度入射之P1波在碰觸邊界反射後會產生四個反射波,但在此特例中僅會產生一個反射波(P1波),至於其它三個反射波(P2波、P3波及S波)在此情況下並不存在。另外,反射之P1波與S波之振幅比與震盪頻率無關,但反射之P2波與P3波之振幅比卻與震盪頻率相關,此與一般瞭解之P1波、P2波、P3波及S波相速度的特性相同。
英文摘要 This study presents two analytical models for describing the propagation and reflection of elastic waves through a porous half-space permeated by two immiscible viscous fluids. These models are based on the poroelastic equations developed by Lo et al. (2005) for a porous medium containing two immiscible, viscous, and compressible fluids.
The first developed model depicts the propagation and attenuation of Rayleigh waves along the impermeable surface of an unsaturated poroelastic half-space. This demonstrates the existence of three modes of Rayleigh waves. These three Rayleigh waves, which are induced by the three dilatational waves (P1, P2, and P3 waves) and one shear wave (S wave) in a two-fluid saturated medium, can be expressed as the R1, R2, and R3 waves in a descending order of magnitude of phase speed. As the excitation frequency and water saturation are given, the dispersion equation of a cubic polynomial can be solved mathematically to obtain the phase speeds and attenuation coefficients of the R1, R2, and R3 waves.
Computational results demonstrate that the phase speed of the R1 wave is frequency-independent (non-dispersive) in Columbia fine sandy loam. Its value is approximately 93-95% of the shear wave speed, and nearly 28% to 49% of the P1 wave speed at frequencies of 50-200 Hz and relative water saturation ranging from 0.01 to 0.99. However, phase speeds and attenuation coefficients of the R2 and R3 waves are dispersive at the frequencies examined. The P2 and P3 waves phase speeds range 56-90% of the R2 and R3 wave speeds. The R1 wave attenuates the least while the R3 wave has the highest attenuation. Furthermore, the phase speed of the R1 wave under an impermeable surface is 1.01-1.37 times of that under a permeable boundary. Surface impermeability causes the R1 wave phase speed to match the S wave phase speed closely, compared to permeable surfaces which exhibits reduced speed at high water saturation.
In addition to the previous unsaturated case, the first model presented is extended to include a two-fluid system to investigate the impact of viscous pore-fluid mixtures on Rayleigh wave propagation and attenuation. As the seismic frequency (1-100 Hz) is stipulated, the dispersion equation is mathematically solved to determine the phase speeds and attenuation coefficients of Rayleigh waves in Lincoln sand respectively permeated by three different fluid mixtures (air-water, air-oil and oil-water). Lincoln sand is filled by two-fluid mixtures at wetting fluid saturation ranging from 1% to 99%. Computational results show that the phase speed of the R1 wave is approximately equal to 93-95% of the S wave speed in the air-water and air-oil mixtures, but the oil-water mixture is 95-96% of the S wave speed. The attenuation coefficients of the R1 wave in both air-water and air-oil systems are dependent on the difference in two fluid densities and the relative motion between the solid and fluid phases. However, the attenuation coefficient of the R1 wave in the oil-water system depends on the effective kinematic viscosity. The phase speeds of the R2 and R3 waves possess similar trends to those of the P2 and P3 waves found in Lo et al. (2005). This implies the out-of-phase motion among these three phases (solid, non-wetting, and wetting) influence the R2 wave phase speed while capillary pressure affects the R3 wave phase speed. The attenuation coefficient of the R2 wave is shown to be positively correlated to the effective dynamic viscosity similar to that of the P2 wave. The attenuation coefficient of the R3 wave in an oil-water system is highest among the three fluid mixtures, but the differences in the three two-fluid mixtures are not obvious.
The second model derived describes an incident P1 wave traveling through a traction-free porous unsaturated half-space, then four reflection waves (P1, P2, P3, S) yield due to the incident P1 wave. The amplitude ratios of the P1, P2, P3, and S waves to the incident P1 wave are derived. This model also portrays the surface displacement and stress of the solid phase during the wave reflection process. Computational results show that the reflection amplitude, surface displacement, and surface stress are functions of water saturation and angles of incidence at seismic frequency ranging from 1 Hz to 100 Hz in Lincoln sand. At normal and grazing incidence, the reflected P1 wave exists, but the reflected P2, P3, and S waves vanish during the reflection process. The reflected amplitude ratios of the P1 and S waves to the incident P1 wave are frequency-independent, while those of the P2 and P3 waves depend on excitation frequency.
論文目次 摘 要 I
Abstract III
Acknowledgement V
List of Tables IX
List of Figures X
Notations XV

Chapter 1 Introduction and Literature Review 1
1.1 Statement of the problem 1
1.2 Literature Review 3
1.3 Structure of the dissertation 8
Chapter 2 Review of the Theory of Wave Propagating in Elastic Porous Media 12
2.1 Introduction 12
2.2 Classification of typical wave motion 12
2.3 Model equations for a two-fluid system 15
2.4 Linear stress-strain relations 20
2.5 Dispersion relations for free dilatational waves 22
2.6 Dispersion relations for a free shear wave 25
2.7 Computational simulation 27
2.7.1 Viscous and inertial coupling coefficients 27
2.7.2 Relationships among capillary pressure, fluid saturation and relative permeability, and fluid saturation 29
2.7.3 Computational results 30
2.7.4 Viscous coupling effect on the body wave motions 32
Chapter 3 Rayleigh Waves Propagating along the impermeable surface of Semi-infinite Unsaturated Poroelastic Media 43
3.1 Introduction 43
3.2. Relations between body and Rayleigh waves 43
3.3 Characteristic model of Rayleigh-type waves 48
3.4 Computational results and discussion 51
3.4.1 Case for a sealed-pore boundary 52
3.4.2 Comparison of the sealed-pore case with an open-pore case 54
Chapter 4 Effect of Pore Fluid Mixtures on the Propagation of Rayleigh Waves through Sands 62
4.1 Introduction 62
4.2 Pore-fluid contact angle and wettability 62
4.3 Soil and fluid parameters 63
4.4 Computational results and discussion 65
4.4.1 Sealed-pore (impermeable) case 65
4.4.2 Comparison with the open-pore (permeable) case 75
Chapter 5 Reflection of an Incident Dilatational Wave in Semi-infinite Unsaturated Poroelastic Media 83
5.1 Introduction 83
5.2 Reflection wave model 83
5.3 Computational results 95
5.3.1 Amplitude ratio coefficients 95
5.3.2 Surface displacements of the solid phase 102
5.3.3 Surface stress of the solid phase 103
Chapter 6 Conclusions 108
6.1 Summary 108
6.2 Areas for further research 112
References 114
Appendix A Phase speeds of dilatational and shear waves in unsaturated Lincoln sand 122
Appendix B Ratio of phase speed in the case of non-viscous coupling terms to that of the viscous case in unsaturated Lincoln sand 124
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