Acoustic Wave Propagation in a Borehole Penetrating an Inclined Layered Formation

Abstract

Abstract Understanding acoustic wave propagation in an inclined fluid–filled borehole embedded in 3‐D media is essential for acoustic wave logging data processing and interpretation. A 3‐D staggered finite‐difference (FD) method is used to simulate borehole sonic waves in isotropic inclined layered formations. First, the FD results are compared with those obtained by a Real Axis Integration method for a monopole source in an open borehole vertically imbedded in homogeneous formation. A good agreement has been made for the two methods. Consequently, the FD solutions are confirmed by the analytical solutions. Then, the waveforms for a borehole with various deviations between the borehole axis and formation interface are calculated. Those for dipole sources are also simulated at a low source center frequency. The numerical results show that, when the source is under the inclined interface, equivalently going up in a fast (lower) formation, and also as the interface inclined angle increases, the determined slowness decreases from the slowness approximately correspondent to that of the slow (upper) formation, to the one in downward formation, and never reaches the real compressional slowness in the upper formation. The correspondent slowness curve, with respect to record points, and above the formation interface, becomes flatter with increasing distance from the source; while the short spacing detection makes the curve sharper. For all spacing with receivers in the upper formation, and the source under the interface in the lower formation, the apparent slowness becomes smaller than the real wave slowness in the upper formation. The larger the inclined angle is, the flatter are the curves away from the interface range for the long spacing. The shear wave slowness shows the similar features but more sensitive to the inclined angle than the compressional slowness. Also, this effect is controlled by the velocity contrast between the upper and lower formations. The above phenomenon is shown clearly in both snapshot visualization and slowness calculation with synthetic waveforms, and is well explained with the theory of ray acoustics.