Effect of linear bubble vibration on wave propagation in unsaturated porous medium containing air bubbles

Abstract

Biot model has widely been used in geophysics, petroleum engineering, civil engineering, and ocean engineering since it was presented, and thus the research on the wave propagation in saturated porous medium has made much progress. However, fully saturated porous medium is rarely found in nature. Almost all the rocks or soils contain two kinds of fluids, such as gas and petroleum. Many researches have been done on the wave propagation in unsaturated porous medium. As is well known, a small volume of gas bubbles existing in a liquid can greatly change the velocity and attenuation of acoustic wave in the liquid. Evidences are beginning to be accumulated that the velocity and attenuation of acoustic wave in a saturated marine sediment can be affected by the gas bubbles existing in the saturated liquid. To investigate the sound propagation in a porous medium when the pore water contains a small number of air bubbles, in this paper we integrate the volume vibrations of bubbles in pore water into the continuity equation of pore-fluid filtration in porous medium based on Biot theory, so as to obtain the continuity equation of pore-fluid filtration with bubble volume vibration. On this basis, according to the relationship between the instantaneous radius of bubble and the background pressure of the medium under the linear vibration of bubble, as well as the equations of motion of the fluid medium and porous medium, a new displacement vector wave equation of porous medium under the influence of bubble is derived, which establishes the model for the sound velocity dispersion and attenuation prediction under the unsaturated porous medium. The presence of air bubbles increases the compressibility of pore fluid, which leads to the decrease in the sound velocity of the bubbly saturated porous medium. When the wave frequency equals the resonance frequency of the bubbles, the bubbles in pore water will produce resonance; the medium will present high dispersion and the velocity can greatly exceed the gas-free velocity. However, these have not been measured in field data. The absorption cross section of the air bubble can reach a maximum value, which leads to the maximum attenuation of the porous medium. It should be noted that the attenuation coefficient calculated with this model is related to the damping of the bubble motion due to radiation, thermal and internal friction, and the dissipation of the relative motion between the pore water and porous solid frame. The obtained numerical analysis is consistent with the above conclusion, which indicates that the volume concentration, the bubble size and the excitation frequency of the sound field are important parameters affecting the sound wave propagation in the saturated porous medium containing few bubbles.