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4 0


7 0


8 0


9 0


1 1 0


1 7 0 head wave Shear


Compressional head wave


> The  rst few m om ents of sim pli ed w avefront propagation from a m onopole transm itter in a  uid- lled b orehole ( b lue) and a fast form ation ( tan) . Both


m edia are assum ed hom ogeneous and isotropic. Tool effects are neglected. Tim e progression is to the right. Num b ers in the upper left corner correspond to tim e in µ s after the source has  red. Wavefronts in the m ud are b lack , com pressional w avefronts in the form ation are b lue, and shear w avefronts in the form ation are red. The com pressional head w ave can b e seen at 9 0 µ s, and the shear head w ave can b e seen at 17 0 µ s.


arrival at the receivers is recorded as the P arrival. The P-wave takes longer to arrive at receivers that are farther from the source. The time difference between P arrivals divided by the distance traveled is known as ∆t, or slowness, and is the reciprocal of speed. This is the most basic sonic-logging measurement.3 The P-wave that continues into the formation is known as a body wave, and travels on deeper into the formation unless a reflector sends it back toward the borehole, at which time it is called a reflected P-wave. Standard sonic logging ignores reflected P-waves, but special applica- tions, such as those described near the end of this article, take advantage of the extra information contained in reflected P-waves. The behavior of refracted S-waves is similar to that of refracted P-waves. When the refracted S-wave becomes parallel to the borehole wall, it propagates along the borehole-formation interface as a shear disturbance at speed Vs and generates another head wave in the borehole fluid. Its arrival at the receivers is recorded as the S-wave. In this way, shear slowness of a fast formation can be measured by a tool surrounded by borehole fluid, even though S-waves cannot propagate through the fluid. In cases when the shear-wave speed is less than the mud-wave speed— a situation known as a slow formation— the shear wavefront in the formation never forms a right angle with the borehole. No shear head wave develops in the fluid. In both fast and slow formations, an S body wave continues into the formation.


Another way of visualizing how P and S head waves and body waves travel near the borehole is through ray tracing. Strictly speaking, ray tracing is valid only when the wavelength is much smaller than the diameter of the borehole, or when the wavefronts can be approximated as planes rather than spheres or cones. Most borehole acoustic modes, especially those at low frequencies, do not meet these conditions, but ray tracing can still be useful for visualization. A ray is simply a line perpendicular to a wavefront, showing the direction of travel. A raypath between two points indicates the fastest travel path. Changes in raypath occur at interfaces and follow Snell’s law, an equation that relates the angles at which rays travel on either side of an interface to their wave speeds (right). Among other things, Snell’s law explains the conditions under which head waves form and why none form in slow formations.


Ray tracing is useful for understanding where waves travel and for modeling basics of sonic-tool design, such as determining the transmitter- receiver (TR) spacing that is required to ensure that the formation path is faster than the direct mud path for typical borehole sizes and formation P and S velocities. This ensures that the tool will measure formation properties rather than borehole-mud properties. Ray tracing also helps describe the relationship between TR


B ore h ol e


F orm a tion R efracted P R efracted S R eflected P θ 2 θ 1 θ 1 I ncident P -wave Source M ud velocity, V m


Sin θ 1 V m


=


Sin θ 2 = V p


P velocity, V p > V m S velocity, V s


Sin θ s V s


>Wavefront re ection and refraction at interfaces, and Snell’ s law . θ 1 is the angle of incident and reect ed P-w aves. θ 2 is the angle of refracted P-w aves. θ s is the angle of refracted S-w aves. Vm is m ud-w ave velocity . Vpis P-w ave velocity in the form ation, and Vsis S-w ave velocity in the form ation. When the angle of refraction eq uals 9 0° , a head w ave is created.


1. A hom ogeneous form ation is one w ith uniform velocity . In other w ords, the velocity is independent of location. An isotropic form ation is one w ith velocity independent of direction of propagation.


2. The head w ave has a conical w avefront in 3 D. 3 . Slow ness ty pically has units of µ s/ ft.


θ s


Spring 2006


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