My PhD Thesis, PDF 3MB - Stanford University

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

approach. We do not need to discretize the model into grids. Therefore, there is not grid dispersion involved, and the calculation is faster and more accurate. I will give two synthetic data examples: one is for 1-D geological structures and another is for 2-D geological structures. 4.2 ATTENUATION COEFFICIENT AND CENTROID FREQUENCY SHIFT Since we want a quantitative estimate of the attenuation from the centroid frequency shift phenomenon, we need an equation which links the frequency shift to the attenuation parameters describing the medium. To do this, we need a model that describes the attenuation feature of a materiel, and a parameter that defines the frequency shift of a signal. 4.2.1 Attenuation model For the purpose of estimating attenuation, I assume that the process of wave propagation is described by the linear system theory. If the amplitude spectrum of an incident wave is S(f) and the medium/instrument response is G(f)H(f), then the received amplitude spectrum R(f) may be, in general, expressed as (see Figure 4.1) R(f) = G(f)H(f)S(f) (4.1) where the factor G(f) includes geometrical spreading, instrument response, source/receiver coupling, radiation patterns, reflection and transmission coefficients, and the phase accumulation due to propagation; and H(f) describes the attenuation effect on - 87 -
the amplitude. In this study, I concentrate on the absorption property of the medium, and call Figure 4.1 Linear system model for attenuation. H(f) the attenuation filter. Experiments indicate the attenuation is usually proportional to frequency, that is, we may write H ( f ) exp ( f o dl ) , (4.2) ray where the integral is taken along the ray path, and o is attenuation coefficient defined by o Qv - 88 - , (4.3) i.e., attenuation is linearly proportional to frequency, where Q is medium's quality factor, and v is wave propagation velocity. Note that the attenuation factor o is different from the usually defined attenuation coefficient o f . This linear frequency model is useful in demonstrating the frequency shift method. More complex models, for example with higher power dependence on frequency, e.g.,
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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
Page 37 and 38: Figure 2.3 Pictorial explanation of
Page 39 and 40: (1 ) (1 ) ( 1) (1 ) (1 ) ? ( r )
Page 41 and 42: imaginary angular frequency I also
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: Under this model, the high frequenc
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
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
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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
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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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