Propagation Of Acoustic Waves In A Saturated Porous Medium Containing A Small Amount Of Bubbly Fluid

Biot established the acoustics theory of fluid saturated porous media,which has been widely used.However,the actual porous formation contains not only single phase medium of rock particles but also contains two phase of liquid and gas or multiphase.In this paper,the sound propagation in a porous medium model containing fluid and a small amount of gas is studied.And it is applied to the study of plane interface reflection,transmission and acoustic logging response simulation.First,considering a porous medium model that contains fluid and a small volume of bubbles,and the bubbles do the linear vibration under the action of sound waves.This paper studied the propagation characteristics of acoustic waves transmitting in this porous medium.Firstly,we derived the seepage continuity equation according to the fluid mass conservation equation and the relationship between porosity differentiation and fluid pressure differentiation.In order to deal with the derivative of the gas volume fraction to the time in the seepage continuity equation,we have adopted the relationship between the instantaneous radius of bubble and the background pressure under the bubble linear vibration given by Commander.Then we got the seepage continuity equation of modified Biot model.Last,the Biot displacement field equation with bubbles is obtained by combining the Biot dynamic equations.So we can obtain not only the acoustic properties of the two kinds of longitudinal waves,but also the acoustic properties of the transverse waves.Through the numerical calculation of dispersion,attenuation and other acoustic characteristics of the two longitudinal waves and the transverse waves in the super soft formation,soft and hard formation respectively,it is found that the existence of a small volume of bubbles has a great influence on the acoustic properties of the fast compressional waves and the slow compressional wave,and the influence on the super soft formation is more obvious than that of soft and hard strata.But the presence of a small number of bubbles has little effect on the shear waves.Second,based on the acoustic theory of fluid saturated porous media with less gas,the reflection,transmission coefficient and the dispersion,attenuation characteristics of the Scholte waves on the interface of the fluid and fluid saturated porous media with less gas are studied.By using the Helmholtz decomposition and introducing the displacement potential function,we get the expression of the basic acoustic field of the upper layer fluid and the lower porous layer with the displacement potential function.Then combining with the boundary conditions,we obtained the reflection coefficient,transmission coefficient and the characteristic equation of Scholte wave.So the relationship between the reflection,transmission coefficient and the incident angle,frequency is calculated under different gas volume fraction.The relationship between Scholte wave velocity,attenuation and frequency,gas volume fraction is also researched.Finally,the acoustic field excited by the point source in the saturated porous medium with less gas fluid is simulated by using the acoustic model established in this paper.In the cylindrical coordinate system,using the Helmholtz decomposition,introducing the displacement potential function to get the basic sound field expression.Then combining with the boundary conditions,the expression of fluid sound pressure in the well is obtained.The sound potential at the same source distance but different gas volume fraction,and at the same gas volume fraction but different source spacing are examined in the soft and hard stratum.It is found that with the increase of the gas volume fraction,the arrival of the longitudinal wave is delayed,indicating that the existence of the bubble has an influence on the velocity of the longitudinal wave.The longitudinal wave velocity calculated from the array waveform is consistent with the influence of the volume fraction of gas on longitudinal wave velocity.