My PhD Thesis, PDF 3MB - Stanford University

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$Drilling-induced tensile wall-fractures - Stanford University$

where Q is the quality factor for either P waves or S waves, and V is either the P wave velocity or S wave velocity. The imaginary term in equation (2.26) is determined from the desired Q and added to the elastic velocity. It is this imaginary term that causes an exponential decay into the waveform as the wave propagates. The decay or attenuation is proportional to the frequency. To retain causality, the real part of the velocity is modified by introducing velocity dispersion. The second term in equation (2.26) gives this velocity dispersion. Table 2.1 Model parameters used for Figures 4a and 4b. Layer r (cm) (km/s) (km/s) (g/cm 3 ) Qp Qs Fluid 4.7 1.68 – 1.2 20 – Casing 5.72 6.1 3.35 7.5 1000 1000 Cement 10.2 2.82 1.73 1.92 40 30 Formation 4.88 2.6 2.16 60 60 - 41 -
Figure 2.4 (a) Seismogram calculated by the generalized R/T coefficient method for a four-layer cased borehole model; (b) Seismogram taken from Page 1055 in Tubman (1984) for comparison with figure 2a. Figure 2.4a is the seismogram calculated by the generalized R/T coefficient method, and Figure 2.4b is the result of Tubman et al. (1984). It can be seen that the travel times of all phases are the same. The waveforms are the similar too, except that the waveform around 1.8 ms is slightly different. The comparison shows an overall good agreement. I will give more comparisons in the next section. 2.4.3 Boreholes with invaded zones I choose two invaded zone models from Baker (1984) for comparison. There are two typical invaded zones: the damaged zone and the flushed zone. In Baker's terminology, a damaged zone has a velocity slower than the unperturbed formation velocity, and a flushed zone has a velocity faster than the unperturbed formation velocity. The source function is a Ricker wavelet whose spectrum is - 42 -
Page 1 and 2: SEISMOGRAM SYNTHESIS IN COMPLEX BOR
Page 3 and 4: I certify that I have read this dis
Page 5 and 6: developed in this thesis do not hav
Page 7 and 8: Finally, I would like to thank my w
Page 9 and 10: 2.4.2 Cased borehole 2.4.3 Borehole
Page 11 and 12: 6.4.3 Attenuation estimation from t
Page 13 and 14: 3.4b Seismograms calculated using t
Page 15 and 16: List of Tables 2.1 Model parameters
Page 17 and 18: Seismogram synthesis can help us un
Page 19 and 20: calculate the wave fields (e.g., St
Page 21 and 22: modeling techniques to crosswell se
Page 23 and 24: oth synthetic data and field data.
Page 25 and 26: simultaneously determine both phase
Page 27 and 28: to measure formation properties fro
Page 29 and 30: x ( r , z ) are used. Since there
Page 31 and 32: normalized Hankel functions of n th
Page 33 and 34: (1 ) ?u r ( r ) (1 ) ? ( r )
Page 35 and 36: Figure 2.2 Pictorial explanation of
Page 37 and 38: Figure 2.3 Pictorial explanation of
Page 39 and 40: (1 ) (1 ) ( 1) (1 ) (1 ) ? ( r )
Page 41: imaginary angular frequency I also
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 and 96:
Spectrum Shape Figure 4.3a A boxcar
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decreases from 370 Hz to 280 Hz ove
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Here, the index i represents the it
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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
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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
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p 1 log [ z i 1 z i log [ ] R ( f
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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
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tested for three typical radially l
Page 145 and 146:
Figure 6.2 A plot of the function d
Page 147 and 148:
6.4.3 Attenuation Estimation from t
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)
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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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My PhD Thesis, PDF 3MB - Stanford University

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