A complete and formal theoretical derivation of the equations of motion and the stress–strain relations in Carcione–Leclaire three-phase theory is presented. The Lagrangian formulation is obtained on the basis of the potential and kinetic energies. We provided a new and more clear method to express the kinetic and potential energy density in an elastic solid. In particular, the deduction of the kinetic energy density in a fluid-saturated porous medium is established from a physical point of view, for which Biot’s theory did not give a detailed description. Moreover, to make the establishment of the equations more clearly expressed, the potential, kinetic and dissipation energy densities are described in great detail to obtain the equations of motion by using the Lagrangian formulation. In order to show the self-consistency of the three-phase model, the deduction of the degradation of a three-phase medium into a simplified form of a two-phase medium is proved in two physically distinct situations. One case is that the medium degenerates into a saturated porous medium. The second case is that the medium degenerates into a porous medium composed of two solids. The results show that, if the pore solid is replaced by the pore fluid, Carcione–Leclaire three-phase theory is completely consistent with Biot’s theory. Besides, if the pore fluid is substituted by the pore solid, two compressional waves and two shear waves can be generated when the reference value of the friction coefficient between two solids b130 is zero. When b130 is not neglected, there are only one compressional wave and one shear wave, which is equivalent to the single-phase elastic medium. Finally, we compute the phase velocities and attenuations versus frequency with three pore solid saturations to analyze the characteristics of the five modes.

中文翻译：

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Carcione-Leclaire三相理论的确认与完善

提出了 Carcione-Leclaire 三相理论中运动方程和应力-应变关系的完整和正式的理论推导。拉格朗日公式是基于势能和动能获得的。我们提供了一种新的更清晰的方法来表达弹性固体中的动能和势能密度。尤其是流体饱和多孔介质中动能密度的推导是从物理角度建立的，Biot的理论并未对此进行详细描述。此外，为了更清楚地表达方程的建立，对势能、动能和耗散能密度进行了详细描述，以利用拉格朗日公式获得运动方程。为了显示三相模型的自洽性，在两种物理上不同的情况下证明了将三相介质的退化推导为两相介质的简化形式。一种情况是介质退化为饱和多孔介质。第二种情况是介质退化为由两种固体组成的多孔介质。结果表明，如果用孔隙流体代替孔隙固体，Carcione-Leclaire三相理论与Biot理论完全一致。此外，如果将孔隙流体换成孔隙固体，当两个固体之间的摩擦系数参考值时，可以产生两个纵波和两个剪切波。一种情况是介质退化为饱和多孔介质。第二种情况是介质退化为由两种固体组成的多孔介质。结果表明，如果用孔隙流体代替孔隙固体，Carcione-Leclaire三相理论与Biot理论完全一致。此外，如果将孔隙流体换成孔隙固体，当两个固体之间的摩擦系数参考值时，可以产生两个纵波和两个剪切波。一种情况是介质退化为饱和多孔介质。第二种情况是介质退化为由两种固体组成的多孔介质。结果表明，如果用孔隙流体代替孔隙固体，Carcione-Leclaire三相理论与Biot理论完全一致。此外，如果将孔隙流体换成孔隙固体，当两个固体之间的摩擦系数参考值时，可以产生两个纵波和两个剪切波。b130为零。什么时候b130不忽略，只有一种纵波和一种横波，相当于单相弹性介质。最后，我们计算了三种孔隙固体饱和度下的相速度和衰减与频率的关系，以分析五种模式的特征。