非饱和弹性多孔介质中体波与表面波的传播特性研究
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摘要
弹性多孔介质材料的波动问题研究涉及到数学、热力学、弹性力学等诸多学科,它不仅是对弹性介质动力学理论的扩充和发展,更对实际工程问题,如：岩土工程、石油工程、地球物理学、生物力学等领域有着广泛的指导价值。以往的研究多是基于两相多孔介质材料,即多孔材料的孔隙中充满液体时的完全饱和状态。然而,实际上许多多孔介质材料是处于非饱和状态的,即介质孔隙中存在气体,例如地球表面附近的岩土材料、油气田中含有天然气的土层。因此对非饱和弹性多孔介质材料的波动问题的研究显得尤为重要。本文基于三相多孔介质理论建立其动力控制方程,并在此基础上对非饱和多孔介质中的体波与表面波的传播问题进行了较为深入和系统的研究。
首先本文在广义混合物理论的框架内,以多相孔隙介质弹性理论为基础,根据三相介质组分的运动学方程,结合质量守恒方程、动量守恒方程、熵不等式及相关的本构关系式,推导得到了线弹性变形条件下的非饱和弹性多孔介质波动控制方程。所建立的三相介质波动方程可以退化到经典的Biot波动理论方程的简化形式,这验证了该方程的合理性。运用Helmholtz分解的计算方法对波动方程进行求解,计算并分析了无限空间内非饱和弹性多孔介质中体波的传播问题。通过求解波动方程,得到了体波传播的特征方程。计算结果显示：在非饱和弹性多孔介质中存在有三种压缩波(P1波、P2波和P3波)和一种剪切波(S波),这与之前许多学者的研究结果一致。通过数值算例,系统地研究了非饱和弹性多孔介质中这四种体波的传播特性(如传播速度与衰减系数)受到介质饱和度和激发频率的影响情况。计算结果显示,由于孔隙中气体的出现,非饱和弹性多孔介质中不同体波的传播特性发生了显著的变化。
接着本文在非饱和弹性多孔介质线弹性理论的基础之上,分别研究了体波在不同边界条件时的反射问题和透射问题：(i)不同体波(P1波和S波)在非饱和多孔介质自由表面上的反射问题；(ii)不同体波(P1波和S波)在非饱和多孔介质和弹性介质分界面上的反射与透射问题；(iii)不同体波(P1波和S波)在具有不同饱和度的多孔介质之间分界面上的反射与透射问题。由于在四种不同体波中P1波和S波传播速度较快且衰减速度较慢,因此本文主要考虑这两种体波的反射与透射特性。数值算例从振动幅值和入射波的能量分配等方面分析了在边界处的频率、入射角度、多孔介质饱和度的影响作用。计算结果均显示,在不同条件的分界平面处,不同体波的反射与透射特性受到多孔介质饱和多变化的影响是不可忽视的。
然后本文通过三相弹性多孔介质波动理论研究分析了非饱和半空间自由边界处的Rayleigh波的传播问题。根据半空间自由边界处的边界条件推导出Rayleigh波的弥散方程。由弥散方程可以看出,在非饱和多孔介质半空间自由边界处存在三种波型的Rayleigh波(R1波、R2波、R3波),其中波速：R1波>R2波>R3波。通过数值计算分别讨论了这三种Rayleigh波在半空间自由边界附近的传播特性(如传播速度与衰减系数)受到介质饱和度和频率变化的影响情况。计算结果显示,不同模态的Rayleigh波的传播特性受到频率和饱和度变化的影响较为明显。
最后本文基于非饱和弹性多孔介质波动理论方程,运用解析法的方法,通过分界平面处的边界条件建立和解析了非饱和多孔介质边界处中Love波的弥散特征方程。分别讨论了两种情况：(1)当非饱和多孔介质半空间上覆盖有弹性介质层的情况；(2)当弹性介质半空间上覆盖有非饱和多孔介质层的情况。通过迭代法对弥散方程进行了数值计算,讨论了不同模态下的Love传播速度及衰减系数受到饱和度变化的影响情况。此外,本文还讨论了高模态Love波的截止频率受到介质饱和度的影响情况。
本文通过进行一系列相关数值模拟计算,得到了非饱和多孔介质材料中体波和表面波传播过程中的一般规律和结论。这些规律和结论在理论上揭示在实际工程应用领域,如土动力学或者地震工程学等研究领域中,人们应更加重视饱和度变化的影响。
The problem of wave motion in poroelastic materials involves many academic subjects such as mathematics, thermodynamics and elastic mechanics. It is not only the expansion and development for the elastic medium dynamic theory, but also has extensive guidance significance for the practical engineering problems such as geotechnical engineering, petroleum engineering, geophysics, biomechanics and other fields. The previous research was based on the two-phase porous medium materials, namely the fully saturated state when the porosity of the porous materials is filled with liquid. In practice, however, most of the porous materials are in the unsaturated state, namely the gas exist in the medium pore, such as the geo-materials near the earth's surface and the soil layer containing gas in the oil and gas field. Hence it is very important to study the wave motion problem of the unsaturated poroelastic materials. The dynamic governing equations are established basing on the triphase poroelastic medium theory in this dissertation and then the propagation problems of body waves and surface waves in these materials is analyzed deeply and systematically.
In this dissertation, first of all, within the framework of generalized mixture theory, on the base of multiphase porous medium elasticity theory, according to the kinematics equations of three-phase equation, combining with mass conservation equations, momentum conservation equations, entropy inequality and relevant constitutive equations, the wave controlling equations of the unsaturated porous elastic medium in the scope of linear elastic deformation is obtained. The established three-phase wave equations can be reduced to the simplified form of Biot wave theory equations, which verifies the rationality of our equations. Then the Helmholtz decomposition method is used to solve the equations and the propagation problem of body waves in infinite space of unsaturated porous elastic medium is calculated and analyzed. By means of solving the wave equations, the characteristc equations of wave propagation are derived. The calculation results show that, three compressional waves (P1wave, P2wave, P3wave) and one shear wave (S wave) exist in the unsaturated poroelastic medium, which is consistant with the results of previous research. The influences of saturation degree and excitation frequency on the propagation characteristics such as wave velocity and attenuation coefficient are studied systematically through numerical examples. It is shown that, due to the presence of gas in the porosity, the propagation characteristics of body wave in unsaturated poroelastic medium have changed significantly.
Then, on the base of linear elastic theory of unsaturated porous elastic medium, the reflection and transmission problems of body waves at various boundary conditions are respectively studied:(ⅰ) reflection problems of different body waves (P1wave and S wave) on the free surface of unsaturated porous medium,(ⅱ) reflection and transmission problems of different body waves (P1wave and S wave) on the interface between unsaturated porous medium and elastic medium,(ⅲ) reflection and transmission problems of different body waves (P1wave and S wave) on the interface between unsaturated porous medium layers with different saturation degrees. Because P1wave and S wave are the body waves with high wave velocity and low attenuation, so this dissertation mainly consider the reflection and transmission characteristics of these two body waves. In the numerical examples, the influences of frequency, incident angle and porous media saturation on the aspects such as vibration amplitude and incident wave energy distribution are discussed. The calculation results shows that, at different interface conditions, the impact of saturation change on the reflection and transmission characteristics should not be overlooked.
After that, based on the wave theory of three-phase porous elastic medium, the Rayleigh wave propagation problems at the free boundary of unsaturated half-space are analyzed. According to the free boundary conditions on the half-space, the dispersion equation of Rayleigh wave is derived. It can be seen from the dispersion equation that, there are three Rayleigh wave modes, namely R1wave, R2wave and R3wave, with wave velocity R1wave> R2wave> R3wave. Through the numerical examples, the influences of porous media saturation on the propagation characteristics such as propagation velocity and attenuation coefficient are discussed. The calculation results show that the influences of frequency and saturation degree on the propagation characteristics of the different modes of Rayleigh waves are obvious.
Finally, based on the unsaturated porous elastic medium wave theory and the boundary condition of the plane interface, this dissertation use the analytical method to establish and solve the Love wave dispersion characteristic equation near the boundary of unsaturated porous media. And two different cases are discussed:(1) unsaturated porous media half space is covered with a layer of elastic medium;(2) the elastic medium half-space covered with a layer of unsaturated porous media. The numerical analysis is carried out through an iterative process of dispersion equation and the influences of saturation change on the propagation velocity and attenuation coefficient are calculated and discussed. At the same time, this dissertation also has discussed the impact of media saturation degree on the cutoff frequency of high mode Love waves.
In this dissertation, through a series of corresponding numerical simulation calculations, some general laws and conclusions of body waves and surface waves propagating in the unsaturated porous materials are obtained. On a theoretical level, these laws and the conclusions show that the impact of the saturation change should attract more attention in the practical engineering application fields such as soil dynamics and earthquake engineering.
首先本文在广义混合物理论的框架内,以多相孔隙介质弹性理论为基础,根据三相介质组分的运动学方程,结合质量守恒方程、动量守恒方程、熵不等式及相关的本构关系式,推导得到了线弹性变形条件下的非饱和弹性多孔介质波动控制方程。所建立的三相介质波动方程可以退化到经典的Biot波动理论方程的简化形式,这验证了该方程的合理性。运用Helmholtz分解的计算方法对波动方程进行求解,计算并分析了无限空间内非饱和弹性多孔介质中体波的传播问题。通过求解波动方程,得到了体波传播的特征方程。计算结果显示：在非饱和弹性多孔介质中存在有三种压缩波(P1波、P2波和P3波)和一种剪切波(S波),这与之前许多学者的研究结果一致。通过数值算例,系统地研究了非饱和弹性多孔介质中这四种体波的传播特性(如传播速度与衰减系数)受到介质饱和度和激发频率的影响情况。计算结果显示,由于孔隙中气体的出现,非饱和弹性多孔介质中不同体波的传播特性发生了显著的变化。
接着本文在非饱和弹性多孔介质线弹性理论的基础之上,分别研究了体波在不同边界条件时的反射问题和透射问题：(i)不同体波(P1波和S波)在非饱和多孔介质自由表面上的反射问题；(ii)不同体波(P1波和S波)在非饱和多孔介质和弹性介质分界面上的反射与透射问题；(iii)不同体波(P1波和S波)在具有不同饱和度的多孔介质之间分界面上的反射与透射问题。由于在四种不同体波中P1波和S波传播速度较快且衰减速度较慢,因此本文主要考虑这两种体波的反射与透射特性。数值算例从振动幅值和入射波的能量分配等方面分析了在边界处的频率、入射角度、多孔介质饱和度的影响作用。计算结果均显示,在不同条件的分界平面处,不同体波的反射与透射特性受到多孔介质饱和多变化的影响是不可忽视的。
然后本文通过三相弹性多孔介质波动理论研究分析了非饱和半空间自由边界处的Rayleigh波的传播问题。根据半空间自由边界处的边界条件推导出Rayleigh波的弥散方程。由弥散方程可以看出,在非饱和多孔介质半空间自由边界处存在三种波型的Rayleigh波(R1波、R2波、R3波),其中波速：R1波>R2波>R3波。通过数值计算分别讨论了这三种Rayleigh波在半空间自由边界附近的传播特性(如传播速度与衰减系数)受到介质饱和度和频率变化的影响情况。计算结果显示,不同模态的Rayleigh波的传播特性受到频率和饱和度变化的影响较为明显。
最后本文基于非饱和弹性多孔介质波动理论方程,运用解析法的方法,通过分界平面处的边界条件建立和解析了非饱和多孔介质边界处中Love波的弥散特征方程。分别讨论了两种情况：(1)当非饱和多孔介质半空间上覆盖有弹性介质层的情况；(2)当弹性介质半空间上覆盖有非饱和多孔介质层的情况。通过迭代法对弥散方程进行了数值计算,讨论了不同模态下的Love传播速度及衰减系数受到饱和度变化的影响情况。此外,本文还讨论了高模态Love波的截止频率受到介质饱和度的影响情况。
本文通过进行一系列相关数值模拟计算,得到了非饱和多孔介质材料中体波和表面波传播过程中的一般规律和结论。这些规律和结论在理论上揭示在实际工程应用领域,如土动力学或者地震工程学等研究领域中,人们应更加重视饱和度变化的影响。
The problem of wave motion in poroelastic materials involves many academic subjects such as mathematics, thermodynamics and elastic mechanics. It is not only the expansion and development for the elastic medium dynamic theory, but also has extensive guidance significance for the practical engineering problems such as geotechnical engineering, petroleum engineering, geophysics, biomechanics and other fields. The previous research was based on the two-phase porous medium materials, namely the fully saturated state when the porosity of the porous materials is filled with liquid. In practice, however, most of the porous materials are in the unsaturated state, namely the gas exist in the medium pore, such as the geo-materials near the earth's surface and the soil layer containing gas in the oil and gas field. Hence it is very important to study the wave motion problem of the unsaturated poroelastic materials. The dynamic governing equations are established basing on the triphase poroelastic medium theory in this dissertation and then the propagation problems of body waves and surface waves in these materials is analyzed deeply and systematically.
In this dissertation, first of all, within the framework of generalized mixture theory, on the base of multiphase porous medium elasticity theory, according to the kinematics equations of three-phase equation, combining with mass conservation equations, momentum conservation equations, entropy inequality and relevant constitutive equations, the wave controlling equations of the unsaturated porous elastic medium in the scope of linear elastic deformation is obtained. The established three-phase wave equations can be reduced to the simplified form of Biot wave theory equations, which verifies the rationality of our equations. Then the Helmholtz decomposition method is used to solve the equations and the propagation problem of body waves in infinite space of unsaturated porous elastic medium is calculated and analyzed. By means of solving the wave equations, the characteristc equations of wave propagation are derived. The calculation results show that, three compressional waves (P1wave, P2wave, P3wave) and one shear wave (S wave) exist in the unsaturated poroelastic medium, which is consistant with the results of previous research. The influences of saturation degree and excitation frequency on the propagation characteristics such as wave velocity and attenuation coefficient are studied systematically through numerical examples. It is shown that, due to the presence of gas in the porosity, the propagation characteristics of body wave in unsaturated poroelastic medium have changed significantly.
Then, on the base of linear elastic theory of unsaturated porous elastic medium, the reflection and transmission problems of body waves at various boundary conditions are respectively studied:(ⅰ) reflection problems of different body waves (P1wave and S wave) on the free surface of unsaturated porous medium,(ⅱ) reflection and transmission problems of different body waves (P1wave and S wave) on the interface between unsaturated porous medium and elastic medium,(ⅲ) reflection and transmission problems of different body waves (P1wave and S wave) on the interface between unsaturated porous medium layers with different saturation degrees. Because P1wave and S wave are the body waves with high wave velocity and low attenuation, so this dissertation mainly consider the reflection and transmission characteristics of these two body waves. In the numerical examples, the influences of frequency, incident angle and porous media saturation on the aspects such as vibration amplitude and incident wave energy distribution are discussed. The calculation results shows that, at different interface conditions, the impact of saturation change on the reflection and transmission characteristics should not be overlooked.
After that, based on the wave theory of three-phase porous elastic medium, the Rayleigh wave propagation problems at the free boundary of unsaturated half-space are analyzed. According to the free boundary conditions on the half-space, the dispersion equation of Rayleigh wave is derived. It can be seen from the dispersion equation that, there are three Rayleigh wave modes, namely R1wave, R2wave and R3wave, with wave velocity R1wave> R2wave> R3wave. Through the numerical examples, the influences of porous media saturation on the propagation characteristics such as propagation velocity and attenuation coefficient are discussed. The calculation results show that the influences of frequency and saturation degree on the propagation characteristics of the different modes of Rayleigh waves are obvious.
Finally, based on the unsaturated porous elastic medium wave theory and the boundary condition of the plane interface, this dissertation use the analytical method to establish and solve the Love wave dispersion characteristic equation near the boundary of unsaturated porous media. And two different cases are discussed:(1) unsaturated porous media half space is covered with a layer of elastic medium;(2) the elastic medium half-space covered with a layer of unsaturated porous media. The numerical analysis is carried out through an iterative process of dispersion equation and the influences of saturation change on the propagation velocity and attenuation coefficient are calculated and discussed. At the same time, this dissertation also has discussed the impact of media saturation degree on the cutoff frequency of high mode Love waves.
In this dissertation, through a series of corresponding numerical simulation calculations, some general laws and conclusions of body waves and surface waves propagating in the unsaturated porous materials are obtained. On a theoretical level, these laws and the conclusions show that the impact of the saturation change should attract more attention in the practical engineering application fields such as soil dynamics and earthquake engineering.
引文
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