My PhD Thesis, PDF 3MB - Stanford University

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Triangular 0.0008 0.000765 266.3 800 2 239.1 800 2 4.3 VALIDATION TEST Before using equation (4.11), let us first examine its validity by considering an ideal synthetic example in the zero offset VSP geometry. For this geometry both input and output centroid frequencies fs and fR are measurable since signals in two successive receivers can be viewed as the incident and transmitted spectra. In this case, we write equation (4.11) as oi 1 2 i f i z i , (4.14) where fi = fi – fi+1 is the centroid frequency difference between two successive depth levels, zi is the distance between these two receivers, oi is the average attenuation coefficient between the two levels, and i 2 the variance at i th receiver. The generalized R/T coefficients method for vertically layered media (Apsel, 1979) is used to model the seismic response. Attenuation in this synthetic model is introduced through the complex velocity [see equation (2.26)]. The full wavefield is calculated at 220 receivers. Figure 4.4a shows the model. The frequency band of the source is 10 Hz – 2010 Hz. This frequency band is much broader than the real VSP, but it is used here to illustrate the range of typical crosswell data for which this algorithm was initially developed. I select a time window that isolates the first arrival. Figure 4.4d is a plot of the centroid frequency picked from the data. The centroid frequency - 95 -
decreases from 370 Hz to 280 Hz over a range of nearly 1000 feet. After smoothing the data and computing equation (4.14), I obtain an estimate of the attenuation coefficient o. Then using the definition Q = / (ov), we get the estimated Q-values shown in Figure 4.4c. The reconstructed Q-value (dotted line) fits the original synthetic model (solid line) very well. The deviations of the inversion result near interfaces are caused by the interference of reflections in the estimation of the frequency shift near interfaces. Therefore, the thicker the layer, the better the inversion result. However, even for a relatively thin layer, the layer around the depth of 1770 feet, I obtain a satisfactory result. 4.4 SEISMIC ATTENUATION TOMOGRAPHY The ideal VSP test done in previous section shows the frequency shift can be used for seismic attenuation estimation. In this section I apply this method to attenuation tomography. I first solve some practical problems related to attenuation tomography. Then, I present four examples for synthetic data and field data. - 96 -
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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
Page 45 and 46: Figure 2.5 (a) Seismograms calculat
Page 47 and 48: Table 2.3a Model parameters of a si
Page 49 and 50: profiling. Chen et al. (1994) gave
Page 51 and 52: Figure 2.7 (a) Configuration of the
Page 53 and 54: Figure 2.8a An invaded zone model u
Page 55 and 56: Two special examples for single bor
Page 57 and 58: measurement technique: seismic atte
Page 59 and 60: Let us start with the elastodynamic
Page 61 and 62: Substituting equation (3.2) into eq
Page 63 and 64: where { E pq with ( j) ( r ; l , n
Page 65 and 66: different dimensions. Moreover, R
Page 67 and 68: 3.4 GENERALIZED REFLECTION AND TRAN
Page 69 and 70: 3.6 DETERMINATION OF EIGEN FUNCTION
Page 71 and 72: Eigenvalues ( n ( j ) ) 2 and ( n
Page 73 and 74: peak frequency 1KHz. Figures 3.4b g
Page 75 and 76: Figure 3.4b A common source gather
Page 77 and 78: 3.5b). Figure 3.5a shows the synthe
Page 79 and 80: (a) Synthetic full waveform sonic l
Page 81 and 82: (a) Seismograms. The peak frequency
Page 83 and 84: 3.9 CONCLUSIONS A semi-analytical a
Page 85 and 86: Seismic wave attenuation includes i
Page 87 and 88: Under this model, the high frequenc
Page 89 and 90: the amplitude. In this study, I con
Page 91 and 92: and the variance to be 2 S 0 (
Page 93 and 94: o dl 18 ( f S f R ) / B 2 - 92 -
Page 95: Spectrum Shape Figure 4.3a A boxcar
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 and 146: Figure 6.2 A plot of the function d
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6.4.3 Attenuation Estimation from t
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References Aki, K, and Richards, P.
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Parker, K., Lerner, R. & Waag, R.,
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A-2 ELASTODYNAMIC EQUATION IN TERMS
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? (1 ) (1 ) ( 1) (1 ) ( r , k , )
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where, ( j ) k 2 ( j) e 32 ( j )
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( j ) ( j ) ( j ) ( j) ( j ) ( j) E
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- 160 -
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( j) E 31 ( j) E 32 ( j) E 33 ( j)
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or rewrite it as ( j ) D 11 D 21 (
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Appendix C Relationships between th
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a dl 12 ( f f ) / B o S R ray 2 C
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