My PhD Thesis, PDF 3MB - Stanford University

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

the discrete wavenumber method makes the calculation fast and stable (Chen, 1991, Page 89). To check the validity of the formulation, I compare the numerical results with available previous work. 2.4.1 Discrete wavenumber integration We can numerically evaluate the integral over wavenumber k in equation (2.20) by discretizing k along the real axis. To remove the singularities from the real k-axis, we can introduce a small positive imaginary frequency I, that is, let An integral as equation (2.20) can be generally written as = R + i I. (2.21) ik z i t ( r , z , t ) f ( r , k , ) e e dkd . (2.22) Using equation (2.21) and numerically integrating equation (2.22), we get ( r , z , t ) e I t e i R t M ( ) { f (r , n k , i ) exp (n kz) k }d . (2.23) R I R n M ( ) The effect of introducing the imaginary frequency I is removed by the multiplying factor e I t in the time domain result. Integration interval k is equivalent to an infinite distribution of sources separated by a distance L = 2/k from each other along the z- axis. L should be large enough (or k small enough) so that first arrivals from neighboring sources are out of the time window under consideration. The introduction of - 39 -
imaginary angular frequency I also makes it possible to choose a larger k , and therefore makes the calculation faster (Chen, 1991, Page 89). Let Tmax be the time window length, Vmax be the maximum velocity in the study area. Then we choose L = 1.5VmaxTmax, k = 2/L, so that the first arrivals from other artificial sources are beyond the time window selected. A proper choice of imaginary frequency is I = /Tmax. The smaller I is, the smaller distortion of the waveform will be. However, if I is too small, some numerical artifacts caused by discrete Fourier transform may show up at the beginning of a wave trace. To make the numerical integral convergent, the summation upper limit M() should be large enough (e.g., M ( ) B | | / V m in , B=1.5 ~ 3). 2.4.2 Cased borehole The first verification test is a four-layer cased borehole model chosen from Tubman et al. (1984), (see Figure 2.1). The model parameters are listed in Table 2.1. The spectrum of the source function is described by F ( ) 1 2 8 o ( i ) [ ( i ) 2 - 40 - 2 ] o 2 , (2.25) where o 2 13 KHz and 0 . 5 o / The source-receiver separation is 3.048 m. Attenuation is introduced through the complex velocity defined by (see, e.g., Aki and Richards, 1980) V ( ) V ( re f )[1 1 Q log ( re f ) i 2 Q ] , (2.26)
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: (1 ) (1 ) ( 1) (1 ) (1 ) ? ( r )
Page 43 and 44: 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 and 96:
Spectrum Shape Figure 4.3a A boxcar
Page 97 and 98:
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
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Figure 4.7 Synthetic test on 2-D at
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Figure 4.8 The data picked from the
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profile curves within the tomograms
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Figure 4.10 The attenuation and vel
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Chapter 5 Acoustic Attenuation Logg
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decreases with the increasing offse
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Assume that the intrinsic attenuati
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2 i 0 ( f f ) i 2 R ( f ) df i
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Figure 5.3b Micro-seismograms calcu
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This simulation shows the central f
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Figure 5.5b Micro-seismograms recor
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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
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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
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Therefore, equation (6.5) is a disp
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[ ?v n ( ), ?v p , ?v s ] [ v n
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tested for three typical radially l
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Figure 6.2 A plot of the function d
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6.4.3 Attenuation Estimation from t
Page 149 and 150:
References Aki, K, and Richards, P.
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Parker, K., Lerner, R. & Waag, R.,
Page 153 and 154:
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 )
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
show all

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