My PhD Thesis, PDF 3MB - Stanford University

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

distance of the two artificial boundaries; k is the increment of wavenumber; T max is the trace length of a seismogram; V m in is the lowest velocity in a given model; and I is the imaginary part of the frequency used in the discrete wavenumber method. To examine the validity of this method, let us first consider an extreme case in which all cylindrical layers are homogeneous in the vertical direction. This problem has been well studied by several authors as summarized by Paillet and Cheng (1991). The more general algorithm in this chapter should provide an identical solution to the one which is directly solved for this special case (see Chapter 2). In fact, the more general solution obtained in this chapter can be analytically reduced to the solution directly derived for this extreme case in Chapter 2. Let k constants, then from equations (3.25) and (3.26) we obtain and ( n ( j ) ) 2 ( j ) ( k ) 2 ( j ) a ( l, n ) p ln ( n ( j ) ) 2 ( j ) ( k ) 2 ( j ) a ( l , n ) s ln 2 k n 2 k n - 71 - ( j ) ( z ) k ( j ) ( j ) ( j ) and k ( z ) =k , (3.27a) . (3.27b) Substituting these special solutions into equations (3.9) and (3.10), we can obtain identical formulas with the solutions in Chapter 2. Bouchon (1993) used the boundary element method to calculate an open borehole embedded in a layered medium. For comparison, I here use the generalized R/T matrices method to solve the same problem. Figure 3.4a shows the model configuration and parameters. The low velocity layer is 1 meter thick. The source is located 2 meters above the layer interface. The source function is a Ricker wavelet (equation 2.27) with be
peak frequency 1KHz. Figures 3.4b gives the seismograms calculated by the generalized R/T matrices method presented in this chapter. Figure 3.4c is taken from Bouchon (1993) for comparison. It can be seen that main features of Figures 3.4b and 3.4c are the same. Figure 3.4b (generalized R/T matrices method) seems have less numerical noise than Figure 3.4c (boundary element method), especially for the later arrivals. As mentioned earlier, if there is discontinuity in the vertical direction, the solution given in this chapter is approximated, i.e., the P-S and S-P conversions at horizontal interfaces are not included in this case. Considering this approximation, the agreement of the comparison is fair. The low velocity layer exhibits a significant influence on the wave field. The tube wave converts to P and S waves at the boundaries. The wave field within the low velocity zone is very complicated. - 72 -
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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
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 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: Eigenvalues ( n ( j ) ) 2 and ( n
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
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
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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
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The generalized reflection and tran
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eflection and transmission (R/T) co
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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
show all

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