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6.4.2 Attenuation of the Pseudo-Rayleigh Wave (MODE M2) The dependence of Q-value of mode M2 on the Q-value of the S wave in the formation is shown in Figure 6.4 by the partition coefficient ( v 2 / )( / v 2 ) [see equation (6.14)]. It can be seen that this coefficient is sensitive to Qs at the cutoff frequency. This provides an opportunity to calculate Qs from Q2 using equation (6.14). Similarly, we can calculate the partition coefficients for the formation P wave and the fluid P wave. The partition coefficient for the formation P wave is negligible at all frequencies, which means that Q2 is almost independent of Qp. Therefore, it is impossible to estimate Qp from Q2. This independence of Qp, however, makes the estimation of Qs from equation (6.14) more accurate, due to ( v 2 / )( / v 2 ) 0 . The next test shows how to estimate Qs from Q2 near the cutoff frequency for the simple borehole. P a r t it io n C o e f f ic ie n t 0.8 0.6 0.4 0.2 1 0 6 8 10 12 14 16 18 20 Frequency (KHz) - 145 - Invaded Hole Simple Hole Cased Hole Figure 6.4 Attenuation partition coefficients for the three borehole models.
6.4.3 Attenuation Estimation from the Pseudo-Rayleigh Wave A linear sweep with the frequency band of 0.1-29 KHz is used as the source to calculate a synthetic excitation log (Medlin & Schmitt, 1992) for the simple borehole model given in Table 6.1. Figure 6.5 shows the excitation log which is the borehole response to a source with a flat spectrum. Around f=9.5 KHz there is a peak on the excitation log. This frequency is close to the cutoff frequency of the fundamental pseudo-Rayleigh wave. Then, I run a short sweep of 1.2 ms long with a frequency band of 9-9.5 KHz. This gives an enhanced pseudo-Rayleigh wave shown in Figure 6.6. Using the amplitude ratio method [equation (5.10) with p=1], I find that the estimated QPR =130. The given S wave Qs is 80, and Q-values of other waves are infinite. Figure 6.4 shows that the partition coefficient is about 0.45 for this case. Then, I get the estimated Qs=60. The pseudo-Rayleigh wave is dominant in Figure 6.6. But other waves, e.g., the Stoneley wave, direct P and S waves are also in it, though they are weak. This is one of the possible reasons why the estimated Qs is smaller than the given value. Figure 6.5 Excitation log for the simple borehole. - 146 -
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SEISMOGRAM SYNTHESIS IN COMPLEX BOR
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I certify that I have read this dis
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developed in this thesis do not hav
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Finally, I would like to thank my w
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2.4.2 Cased borehole 2.4.3 Borehole
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6.4.3 Attenuation estimation from t
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3.4b Seismograms calculated using t
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List of Tables 2.1 Model parameters
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Seismogram synthesis can help us un
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calculate the wave fields (e.g., St
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modeling techniques to crosswell se
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oth synthetic data and field data.
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simultaneously determine both phase
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to measure formation properties fro
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x ( r , z ) are used. Since there
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normalized Hankel functions of n th
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(1 ) ?u r ( r ) (1 ) ? ( r )
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Figure 2.2 Pictorial explanation of
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Figure 2.3 Pictorial explanation of
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(1 ) (1 ) ( 1) (1 ) (1 ) ? ( r )
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imaginary angular frequency I also
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Figure 2.4 (a) Seismogram calculate
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Figure 2.5 (a) Seismograms calculat
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Table 2.3a Model parameters of a si
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profiling. Chen et al. (1994) gave
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Figure 2.7 (a) Configuration of the
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Figure 2.8a An invaded zone model u
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Two special examples for single bor
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measurement technique: seismic atte
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Let us start with the elastodynamic
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Substituting equation (3.2) into eq
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where { E pq with ( j) ( r ; l , n
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different dimensions. Moreover, R
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3.4 GENERALIZED REFLECTION AND TRAN
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3.6 DETERMINATION OF EIGEN FUNCTION
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Eigenvalues ( n ( j ) ) 2 and ( n
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peak frequency 1KHz. Figures 3.4b g
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Figure 3.4b A common source gather
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3.5b). Figure 3.5a shows the synthe
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(a) Synthetic full waveform sonic l
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(a) Seismograms. The peak frequency
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3.9 CONCLUSIONS A semi-analytical a
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Seismic wave attenuation includes i
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Under this model, the high frequenc
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the amplitude. In this study, I con
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and the variance to be 2 S 0 (
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o dl 18 ( f S f R ) / B 2 - 92 -
Page 95 and 96: Spectrum Shape Figure 4.3a A boxcar
Page 97 and 98: decreases from 370 Hz to 280 Hz ove
Page 99 and 100: Here, the index i represents the it
Page 101 and 102: comparison. We can see that the tom
Page 103 and 104: Figure 4.7 Synthetic test on 2-D at
Page 105 and 106: Figure 4.8 The data picked from the
Page 107 and 108: profile curves within the tomograms
Page 109 and 110: Figure 4.10 The attenuation and vel
Page 111 and 112: Chapter 5 Acoustic Attenuation Logg
Page 113 and 114: decreases with the increasing offse
Page 115 and 116: Assume that the intrinsic attenuati
Page 117 and 118: 2 i 0 ( f f ) i 2 R ( f ) df i
Page 119 and 120: Figure 5.3b Micro-seismograms calcu
Page 121 and 122: This simulation shows the central f
Page 123 and 124: Figure 5.5b Micro-seismograms recor
Page 125 and 126: Figure 5.8 Centroid frequency picks
Page 127 and 128: p 1 log [ z i 1 z i log [ ] R ( f
Page 129 and 130: spreading. To understand how the in
Page 131 and 132: 5.9b, in fact, shows the velocity s
Page 133 and 134: The generalized reflection and tran
Page 135 and 136: eflection and transmission (R/T) co
Page 137 and 138: where, signs "+" and "-" refer to o
Page 139 and 140: Therefore, equation (6.5) is a disp
Page 141 and 142: [ ?v n ( ), ?v p , ?v s ] [ v n
Page 143 and 144: tested for three typical radially l
Page 145: Figure 6.2 A plot of the function d
Page 149 and 150: References Aki, K, and Richards, P.
Page 151 and 152: Parker, K., Lerner, R. & Waag, R.,
Page 153 and 154: A-2 ELASTODYNAMIC EQUATION IN TERMS
Page 155 and 156: ? (1 ) (1 ) ( 1) (1 ) ( r , k , )
Page 157 and 158: where, ( j ) k 2 ( j) e 32 ( j )
Page 159 and 160: ( j ) ( j ) ( j ) ( j) ( j ) ( j) E
Page 161 and 162: - 160 -
Page 163 and 164: ( j) E 31 ( j) E 32 ( j) E 33 ( j)
Page 165 and 166: or rewrite it as ( j ) D 11 D 21 (
Page 167 and 168: Appendix C Relationships between th
Page 169: a dl 12 ( f f ) / B o S R ray 2 C
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