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

Appendix A A-1 S WAVE POTENTIAL IN RADIALLY SYMMETRIC MEDIA The displacement vector u can be represented by the P wave potential and the S wave potential : u e [ r r ] e [ r r z z r - 151 - ] e [ z z 1 ( r ) ] , (A-1.1) r r where ( r , z ) and (r , z ) are independent of . Since there is no SH wave, the displacement in e direction is zero. It can be seen from (A-1.1) that displacements in e r and e z directions are independent of r and z in this radially symmetric case. Therefore, we may choose r = z = 0. letting , we can write the S wave potential as e ( r , z ) . (A-1.2)
A-2 ELASTODYNAMIC EQUATION IN TERMS OF POTENTIALS The elastodynamic equation for isotropic media is ??u (x , t ) f ( 2 ) u ( x , t ) u ( x , t ) , (2.1 again) A point explosion source ( t ) is located at r=0 and z=0, which satisfies ( r ) ( z ) f [ ( t ) r ] , (A-2.1) where relationship ( x ) ( y ) (z ) (r ) ( z ) / r (Bracewell, 1986, Page 85) is used. The displacement u can be represented in terms of potentials, i.e., u (r , z, t ) , (2.2 again) where and are P wave and S wave potentials, respectively. Substituting equations (2.2) into the elastodynamic equation (2.1), we obtain Here, vector identities: ( 2 ) 2 2 ?? ( r ) ( z ) , r (A-2.2a) ?? 0 . (A-2.2b) 2 ( ) ( ) , ( ) 0 , - 152 -
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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
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
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: Parker, K., Lerner, R. & Waag, R.,
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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