Wave propagation characteristics in porous medium containing a solid in pores

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. Thefrequency dispersion and attenuation characteristics of wave modes are analyzed by a plane wave analysis. Inaddition, 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 amedium, where the time-splitting method is used to solve the stiffness problem in the first-order velocity-stressequations. The generation mechanisms and energy distributions of different kinds of waves are investigated indetail. In particular, the influences of the friction coefficient between solid grains and pore solid as well asfrequency on wave propagation are analyzed. It can be known from the results of plane wave analysis that thereare two compression waves (P1 and P2) and two shear waves (S1 and S2) in a porous medium containing asolid 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 proposednumerical 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 simulationresults. The P1 wave and S1 wave propagate primarily in the solid grain frame, while P2 wave and S2 wave areconcentrated mainly in the pore solid, which are caused by the relative motion between the solid grains and thepore solid. Besides, it should be pointed out that the wave diffusions of the P2 wave and S2 wave are influencedby the friction coefficient between solid grains and pore solid. The existence of friction coefficient between twosolids makes P2 wave and S2 wave attenuate to a certain extent at high frequency, but the attenuation is muchsmaller 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 energyattenuation of acoustic waves in porous media