Petroleum Engineers

PETROLEUM ENGINEERING HANDBOOKPseudo-Rayleightraveling with a velocity, vf. in the mud. When thesewaves reach the borehole face, they are both reflectedand refracted. For angles of incidence less than the Pmwave critical angle tI1,,CompressionalIIMt+Fig. 51.23-AcousticiIAiry Phasewaveform.part of the energy is transmitted into the formation in theform of compressional wave and another part as a shearwave, and the remainder is reflected back into the mudas a compressional wave, all according to Snell’s law.At or near the P-wave critical angle, a shear wave isstill transmitted into the formation and P-wave reflectedback into the mud, but a P-wave is critically refractedand travels with the v,’ in the formation, close andparallel to the borehole wall, while continuouslyradiating P-wave energy back into the mud at the sameP-wave critical angle (Fig. 5 I .22).At the S-wave critical angle (o,,).Acoustic Wave Propagation in aFluid-Filled BoreholeThe propagation of elastic waves in a borehole filledwith liquid has been studied extensively.60-70 Only aqualitative description of the phenomenon will be givenhere for identifying the components of an acoustic pulsereoorded in a borehole.The general geometry for the transmission method isillustrated in Fig. 51.22, which shows a single receiverlogging sonde. Two pressure transducers are spaced onan acoustically insulated body, the upper one to generatecompressional waves in the borehole fluid and the lowerone to detect compressional waves reaching it. Thereceiver converts these waves to electrical signals. Theseare transmitted to the surface and displayed on anoscilloscope as a record of received-signal amplitude vs.time and recorded either in analog form on film ordigitally on magnetic tape.This received signal, which is referred to as theacoustic waveform, represents several acoustic wavesand is illustrated by the synthetic waveform trace shownin Fig. 51.23. For the usual case of a liquid-filledborehole in a formation with both the compressional- andshear-wave velocities higher than borehole fluid velocity,two body (or head) waves and two guided waves arepropagated. These waves are shown in Fig. 5 1.23 in theorder of their arrival time at the receiver: (I) compressionalwave, (2) shear wave. (3) pseudo-Rayleighwaves, and (4) Stoneley waves.Compressional and shear waves, which are also calledP. primary, and S, secondary waves, respectively, arehead or body waves because they travel in the body ofthe formation. Pseudo-Raylcigh and Stoneley waves,which also are called reflected conical (or normal mode)and tube wave (or water arrival). respectively, arc guidedwaves because they require the presence of theborehole for their existence.A description of the various ray paths of these wavesmay help further in understanding elastic wave propagationin and around the borehole. The acoustic transmittershown in Fig. 51.22 generates compressional wavesO,y=sin-’ “f ,( v 5 >the S-wave is critically refracted and travels with the \J,,in the formation along a path similar to that of therefracted P-wave. It also continuously radiates P-waveenergy back into the mud at the S-wave critical angle(Fig. 51.22). Beyond the S-wave critical angle, all theincident energy is reflected back into the mud to form theguided pseudo-Rayleigh waves (Fig. 5 1.24).To summarize, the compressional wave travels as a P-wave between the transmitter and the formation, in theformation, and also between the formation and thereceiver (PPP); the shear wave travels as a P-wave betweenthe transmitter and the formation, an S-wave inthe formation, and again as a P-wave between the formationand the receiver (PSP). If the formation shear-wavevelocity is slower than borehole fluid velocity, shearwaves cannot be refracted along the borehole wall;therefore, no shear head wave is generated.As described earlier, compressional and shear wavestravel at velocities determined by the elastic moduli andthe density of the formation:and( . . . ..(7)p,, is the bulk density of formation, and I,, and I, arecompressional- and shear-wave transit times.The body waves travel at all frequencies at speedsgiven by Eqs. 8 and 9. They are nondispersive (variationof velocity with frequency is negligible), and undergo attenuationand geometric spreading. Attenuation, 01, ofthe body waves is proportional to the logarithmic ratio ofthe amplitudes, A 1 and A?, at distances s t and s? fromthe source 15,t6: