Experimental and finite difference modelling of borehole Mach waves¹

A series of model experiments was performed in an ultrasonic laboratory to study the radiation of downhole sources in a variety of formations. Three models were used in the experiments. They were a Lucite model, a Lucite model with a free glass pipe in the centre, and a glass‐cased soil model. In addition, a finite‐difference modelling technique was used to simulate the wave propagation in these models and the results of the laboratory and numerical experiments are compared. In the Lucite borehole model the waveforms recorded in the experiment agree very well with the finite‐difference synthetics. The snapshots of the wavefield from the finite‐difference simulation show the radiation pattern of the P‐ and S‐waves in the Lucite formation. These patterns are consistent with the theoretical calculations. In the Lucite model with the free glass pipe, the finite‐difference synthetics are also in good agreement with the experimental observations, especially for the conical P‐wave arrival. The angle between the wavefront of the conical P‐wave and the borehole axis, observed from the snapshot, agrees with the theory. In the cased soil model, the arrival time of the finite‐difference synthetics is in good agreement with the laboratory measurements. The relative amplitudes of the P‐wave and the Mach wave are not correctly modelled because intrinsic attenuation is not included in the finite‐difference calculation. The Mach cone angle from the snapshot agrees with the theoretical prediction. Finally, a finite‐difference method was used to simulate Mach‐wave propagation in a formation with two horizontal layers. In the case of two slow layers, the Mach‐wave generated in the first layer is reflected back from and transmitted through the boundary and another Mach wave is generated at the second layer when the Stoneley wave travels into the second layer. In the case of a formation having one slow and one fast layer, the Mach wave generated in the slow layer is reflected back at the boundary and leaked into the fast layer. There is no Mach wave in the fast layer.