My PhD Thesis, PDF 3MB - Stanford University

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

(1 ) (1 ) (1 ) e k 21 2 ( 2 ) (1 ) H ( r )e 0 i (1 ) ( 1) ( r r ) , (1 ) (1 ) (1 ) e k 22 2 ( 1) (1 ) H ( r )e 0 i (1 ) r . A-5 DERIVATION OF THE MODIFIED R/T MATRIX Let us define and ( j ) E ( r ) 11 ( j ) E ( r ) 12 ( j ) E ( r ) 21 ( j ) E ( r ) 22 ( j ) c ( j ) c p ( j ) c s ( j ) e11 ( j ) e 21 ( j ) e13 ( j ) e 23 ( j ) e 31 ( j ) e 41 ( j ) e 33 ( j ) e 43 . ( j ) ( r ) e ( r ) 12 , ( j ) ( r ) e ( r ) 22 ( j ) ( r ) e 14 ( r ) , ( j ) ( r ) e ( r ) 24 ( j ) ( r ) e ( r ) 32 , ( j ) ( r ) e ( r ) 42 ( j ) ( r ) e ( r ) 34 . ( j ) ( r ) e ( r ) 44 Then, the boundary conditions, equation (2.11), can be rewritten as - 157 -
( j ) ( j ) ( j ) ( j) ( j ) ( j) E ( r ) c + E ( r ) c 11 12 ( j ) ( j ) ( j ) ( j) ( j ) ( j) E ( r ) c + E ( r ) c 21 22 ( j 1 ) ( j ) ( j 1 ) ( j 1) ( j ) ( j 1 ) E ( r ) c + E ( r ) c 11 12 ( j 1 ) ( j ) ( j 1 ) ( j 1) ( j ) ( j 1) E ( r ) c + E ( r ) c 21 22 Rearranging the left side and right side of equation (A-5.1), we have ( j) E 11 E 21 ( j) or rewrite it as ( j ) c c ( j 1) ( j ) ( j 1 ) ( j) ( r ) - E ( r 12 ( j ) ( j 1 ) ( j) ( r ) - E ( r 22 ( j ) ) c ) c ( j 1) ( j ) ( j ) ( j 1 ) ( j ) E ( r ) - E ( r ) 11 12 ( j ) ( j ) ( j 1 ) ( j ) E ( r ) - E ( r ) 21 22 - E 12 - E 22 1 ( j 1) ( j 1) - E 12 - E 22 - 158 - ( j ) ( j ) ( j ) ( r ) E ( r ) 11 ( j ) ( j ) ( j ) ( r ) E ( r ) 21 ( j 1 ) ( j 1 ) ( r ( r ( j ) ( j ) ( j ) ) E ( r 11 ( j ) ( j ) ( j ) ) E ( r 21 ( j ) c c ( j 1 ) ( j ) ) c ) c ( j 1) . (A-5.1) Comparing (A-5.3) with equation (2.12a) (the definition of modified R/T matrix): we obtain ( j ) R T T ( j ) ( j ) ( j ) R ( j ) c c ( j ) E 11 E 21 ( j ) ( j 1) ( j ) R T ( j) T ( j ) ( j ) R ( j ) ( j 1) ( j ) ( r ) - E ( r 12 ( j ) ( j 1) ( j ) ( r ) - E ( r 22 ( j) c c ) ) 1 ( j 1) , - E 12 - E 22 ( j 1) ( j 1) ( j ) ( j ) ( j ) ( r ) E ( r ) 11 ( j ) ( j ) ( j ) ( r ) E ( r ) 21 (A-5.2) . (A-5.3) . (A-5.4) A-6 DERIVATION OF THE GENERALIZED R/T MATRICES
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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
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3.9 CONCLUSIONS A semi-analytical a
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Seismic wave attenuation includes i
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Under this model, the high frequenc
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the amplitude. In this study, I con
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and the variance to be 2 S 0 (
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o dl 18 ( f S f R ) / B 2 - 92 -
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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
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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
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
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: where, ( j ) k 2 ( j) e 32 ( j )
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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