Abstract
Aiming at the propagation characteristics of acoustic waves in a porous medium containing a solid in pores, the equations of motion and constitutive relation are deducted in the case of two-solid porous media. The frequency dispersion and attenuation characteristics of wave modes are analyzed by a plane wave analysis. In addition, based on the first-order velocity-stress equations, the time-splitting high-order staggered-grid finite-difference algorithm is proposed and constructed for understanding wave propagation mechanisms in such a medium, where the time-splitting method is used to solve the stiffness problem in the first-order velocity-stress equations. The generation mechanisms and energy distributions of different kinds of waves are investigated in detail. In particular, the influences of the friction coefficient between solid grains and pore solid as well as frequency on wave propagation are analyzed. It can be known from the results of plane wave analysis that there are two compression waves (P1 and P2) and two shear waves (S1 and S2) in a porous medium containing a solid in pores. The attenuations of P2 wave and S2 wave are much larger than those of P1 wave and S1 wave. This is due to the friction between the solid grains and the pore solid. The results show that our proposed numerical simulation algorithm can effectively solve the problem of stiffness in the velocity-stress equations, with high accuracy. The excitation mechanisms of the four wave modes are clearly revealed by the simulation results. The P1 wave and S1 wave propagate primarily in the solid grain frame, while P2 wave and S2 wave are concentrated mainly in the pore solid, which are caused by the relative motion between the solid grains and the pore solid. Besides, it should be pointed out that the wave diffusions of the P2 wave and S2 wave are influenced by the friction coefficient between solid grains and pore solid. The existence of friction coefficient between two solids makes P2 wave and S2 wave attenuate to a certain extent at high frequency, but the attenuation is much smaller than that at low frequency. This is the reason why it is difficult to observe the slow waves in practice. However, because the slow waves also carry some energy, it may not be ignored in the studying of the energy attenuation of acoustic waves in porous media.
Highlights
Biot 研究了连通固体中含连通孔隙且孔隙被液体充满的介质。本文研究 了连通孔隙空间被有别于骨架固体的另一种固相介质充填而形成的双组 分连通固体孔隙介质,简称固–固孔隙介质。本文推导出了固–固孔隙介 质的声波动力学方程和本构关系,利用平面波分析的方法分析了各种波 的频散和衰减特性。在此基础上,基于一阶速度–应力方程提出了基于该 方程的时间分裂的高阶交错网格有限差分算法,对其中的声场演化特点 进行了模拟计算,研究了各种波的产生机制和能量分布,并详细讨论了 两种固体间的摩擦系数和声源频率对各种波传播特性的影响。数值模拟 表明:固–固孔隙介质中存在两种纵波(P1 和 P2)和两种横波(S1 和 S2), 其中 P1 和 S1 波能量主要在骨架固体中传播;P2 和 S2 波是骨架固体和 孔隙固体之间相对运动产生的慢波,能量主要在孔隙固体中传播。固体 骨架和孔隙固体之间的摩擦主要影响慢波(P2 和 S2)的衰减,且低频时衰 减大于高频。.
Biot 首次提出流体饱和孔隙介质的弹性波传播理论[1,2,3,4],奠定了孔隙介质声
当流体饱和孔隙介质中的流体相被另一种固体替代时,就形成了由两种固体 组成的固–固孔隙介质,且固体骨架和孔隙固体之间存在相对运动。假设孔隙均 匀分布,其尺度远小于声波波长。根据 Biot 理论[1,2,3,4],孔隙介质的一阶速度–应力 方程可根据其动能密度、势能密度以及耗散能密度结合 Lagrange 定理得到。与 Biot 理论描述的双相介质中的动能密度[1,2,3,4]类似,孔隙中填充一种固体的固–固孔 隙介质的动能密度可以表示为
Summary
Biot 研究了连通固体中含连通孔隙且孔隙被液体充满的介质。本文研究 了连通孔隙空间被有别于骨架固体的另一种固相介质充填而形成的双组 分连通固体孔隙介质,简称固–固孔隙介质。本文推导出了固–固孔隙介 质的声波动力学方程和本构关系,利用平面波分析的方法分析了各种波 的频散和衰减特性。在此基础上,基于一阶速度–应力方程提出了基于该 方程的时间分裂的高阶交错网格有限差分算法,对其中的声场演化特点 进行了模拟计算,研究了各种波的产生机制和能量分布,并详细讨论了 两种固体间的摩擦系数和声源频率对各种波传播特性的影响。数值模拟 表明:固–固孔隙介质中存在两种纵波(P1 和 P2)和两种横波(S1 和 S2), 其中 P1 和 S1 波能量主要在骨架固体中传播;P2 和 S2 波是骨架固体和 孔隙固体之间相对运动产生的慢波,能量主要在孔隙固体中传播。固体 骨架和孔隙固体之间的摩擦主要影响慢波(P2 和 S2)的衰减,且低频时衰 减大于高频。. Biot 首次提出流体饱和孔隙介质的弹性波传播理论[1,2,3,4],奠定了孔隙介质声 当流体饱和孔隙介质中的流体相被另一种固体替代时,就形成了由两种固体 组成的固–固孔隙介质,且固体骨架和孔隙固体之间存在相对运动。假设孔隙均 匀分布,其尺度远小于声波波长。根据 Biot 理论[1,2,3,4],孔隙介质的一阶速度–应力 方程可根据其动能密度、势能密度以及耗散能密度结合 Lagrange 定理得到。与 Biot 理论描述的双相介质中的动能密度[1,2,3,4]类似,孔隙中填充一种固体的固–固孔 隙介质的动能密度可以表示为 介质中的耗散势 D 可以用 Biot 理论中对饱和流体孔隙介质中耗散函数的描述来 表示[1,2,3,4] 体和流体之间摩擦系数[1]的值,在存在摩擦时我们选取和 Guerin 和 Goldberg 研究
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