Study on the Numerical Simulation of Array Sonic Logging Responses and Their Velocity Dispersion Characteristics in Fractured Formation

To understand the effects of fractured formation on the responses of array sonic logging, the wave propagation of array acoustic logging was simulated based on the staggered grid finite difference method and poroelastic theory. Methods of velocity dispersion analysis and attenuation estimation were carried out on simulated sonic waveforms, and the differences of velocity dispersion and wave amplitude attenuation among these wave modes were summarized. First, the velocity-stress equation in cylindrical coordinates was deduced for the Biot poroelastic model. Next, the velocity-stress wave equation was discretized by using the staggered grid finite difference method. Furthermore, the higher order finite difference format and nearly perfect match layer (NPML) were adopted to improve the efficiency and accuracy of the finite difference method. Finally, simulated sonic waveforms in formation models with a tilted fracture zone and collapsed interval were applied to the velocity dispersion analysis by using forward and back amplitudes and the phase estimation method. The results show that the time difference of the P and S wave field in the low-velocity fracture zone increases and that the attenuation amplitude changes with the dip angle of the fracture zone. The attenuation of Stoneley waves and pseudo-Rayleigh waves is enhanced when the shaft collapse expands in the radial and axial directions. The time difference of acoustic waves, velocity dispersion, and amplitude attenuation can reflect the characteristics of the fracture zone, which is of great significance to the evaluation and development of fractured reservoirs.