| In petroleum exploration and development, inclined and fractural formations are often encountered. It is necessary to solve in situ how to detect inclined formation, fractures, and their density and orientation. Thus, the study of the acoustic field of the layer and fractures in and outside of a borehole is the theoretical base for sonic logs.This paper focuses on the theoretical analysis and numerical modeling of acoustic field in and outside of the borehole. This paper can be grouped into two parts.In part one, the 2-D finite-difference formula are set up. Based on the previous work, the equations of motion and elastodynamics are changed. A high-order staggered velocity-stress finite-difference formula of the acoustic field in borehole are set up in cylindrical coordinate, and the corresponding code is accomplished. The stability of stagger-grid , boundary conditions and the source function implementation are discussed.In part two, the acoustic field in and outside of a borehole is analyzed theoretically and numerically for thedifferent physical model of the borehole. The characteristic of excitation of the acoustic field is investigated forsources with various frequencies. The effects of an isotropic medium, inclined formation of different azimuth,horizontal or vertical fracture and its width or density on the propagation of the acoustic field are investigated.The fluid in borehole is also investigated. The results show that, when the depth of the interface is the same asthat of the source, the reflected wave is observed in the waveform apparently, and the location of the interfaceis also observed in the snapshots. The interface can be identified with these characteristics. But when the depthof the interface is not the same as that of the source, the character of the interface is not obvious in thewaveform. When the vertical formation lies outside of the borehole, the effect of the borehole on the acousticfield decreases with the increase of the distance between the formation and the axis of the borehole; when thesource locates on the interface, with the increase of the azimuth of the dipping interface, the transmitted energyof the horizontal component gradually increases, but that of the vertical component decreases. When thesource locates under the interface, the inversed result is obtained. When the acoustic impedance of the fluid inborehole decreases, and other conditions keep unchanged, the acoustic energy in the borehole increasesrelatively. When the acoustic impedance contrast between the thin dipping interface and its adjacent formationsexits, their interfaces can be observed from the snapshots, but the acoustic field is rather complex. There aretwo lines for the first arrivals in the waveform clearly. With the increase of the thickness of the dippingformation, the two lines are clearer and clearer. The amplitude of the waveform in a borehole with a verticalfracture gradually decreases with the increase of the source distance, but the decrease velocity is varied withthe location of the fracture. The decrease velocity is relative low when the distance between the fracture andthe axis of the borehole is small, and the successive reflected wave back in the borehole becomes strong. Whenthe depth of the horizontal fracture is not same as that of the source, the fracture can be clearly observed in thesnapshots, and a reflected wave appears in the waveforms. But when their depths are same, there is noapparent signal character of the fracture. When the source frequency increases, the dispersion and theattenuation of signals become great, but the resolution is better. When the source frequency is low (lower than2kHz) and the widths of fractures are approximate, the difference of the snapshots and waveforms betweendifferent fracture models is not obvious. When tiie density of fractures is great, the effect of multi-fractures is similar to the single fracture with the thicknesses equating to the total thickness of multi-fractu...
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