My PhD Thesis, PDF 3MB - Stanford University

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Normal Modes in Radially Layered Media and Application to Attenuation Estimation The P wave in acoustic full waveform logs can be used to estimate P wave attenuation (see Chapter 5). But it is difficult to estimate S wave attenuation from the S wave in acoustic logs. The S wave is immediately followed by tube waves. It is difficult or impossible to separate the S wave train from acoustic logs. An alternative way to estimate S wave attenuation is to use tube waves, or normal modes. In this chapter, the generalized reflection and transmission coefficients method introduced in Chapter 2 is further developed to systematically determine normal modes in radially multi-layered media. This approach can simultaneously determine both phase velocities and corresponding eigen functions for a borehole with an arbitrary number of radial layers. Therefore, it can simulate tube waves for the cased borehole and the invaded borehole. A formula which describes the attenuation property of each mode in absorptive media is derived. We can use this formula to estimate the S wave attenuation from tube waves, especially the pseudo-Rayleigh wave. 6.1 INTRODUCTION The dispersion relation, which gives the phase velocities of normal modes, is very important for studying guided waves in acoustic logging. It has been studied by many authors, e.g., Biot (1952), Cheng and Toks ??o z (1981). In this chapter the generalized - 133 -
eflection and transmission (R/T) coefficients method is further developed to systematically and simultaneously determine the phase velocities and eigen functions of normal modes in radially multi-layered media. These normal modes are known as the Stoneley wave and pseudo Rayleigh waves. Chen (1993) used the generalized R/T coefficients method to compute normal modes in horizontal layers. If we know the relationship between the Q-value of a normal mode and the Q- value of the formation, we may estimate the formation Q-value from that mode using this relation. Cheng et al. (1982) gave a formula describing the attenuation of tube waves for a simple open borehole. In this chapter, a more general formula to calculate the attenuation of each normal mode for a cased borehole and other complicated boreholes is derived from the new dispersion relation. Additionally, I use a numerical example demonstrating how to estimate the S wave attenuation from tube waves. 6.2 NORMAL MODES IN RADIALLY LAYERED MEDIA The normal modes are the non-trivial solutions of the source-free elastodynamic equation under given boundary conditions. Starting from this point, we can derive an effective algorithm to determine the phase velocities and the corresponding eigen functions of the normal modes in radially layered media. Figure 2.1 shows such a radially multi-layered model. In the acoustic logging problem, these normal modes are known as tube waves or guided waves. In the source-free case, the elastodynamic equation can be written as ??u (x , t ) ( 2 ) u( x , t ) u( x , t ) , (6.1) - 134 -
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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
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
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: The generalized reflection and tran
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
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)
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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