Confirmation and Perfection of Carcione–Leclaire Three-Phase Theory

Abstract

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 [Formula: see text] is zero. When [Formula: see text] 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.