My PhD Thesis, PDF 3MB - Stanford University

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Chapter 3 Elastic Waves in Complex Radially Symmetric Media The generalized reflection and transmission coefficients method with normal mode expansion is developed to calculate the elastic wave field for acoustic logging and crosswell seismic simulation in complex radially symmetric media. This method simulates models which consist of arbitrary cylindrical layers with each cylindrical layer having arbitrary layers in the vertical direction. In practical applications, boreholes that have casing with perforations, source arrays, and boreholes embedded in layered media can be treated as models of this type. Faults can be approximated as symmetric structures. The simulation based on this method gives direct waves, reflections, transmissions, tube waves, tube wave conversions generated at casing perforations or horizontal interfaces, and the radiation pattern of a source in a complicated borehole. This method is semi-analytical. Compared with purely numerical approaches, such as finite difference and finite element methods, this semi-analytical method is faster and avoids the grid dispersion and inaccurate handling of fluid-solid interfaces. Moreover, this method can calculate the wave field at large distances in the formation. The scale difference between the small borehole diameter and a large formation extent, which causes numerical difficulties in purely numerical methods, is not a problem in this approach. Examples are presented to verify the formulation and show the applicability of this method. In Chapter 4, this simulation method is applied to test a new attenuation - 55 -
measurement technique: seismic attenuation tomography using the frequency shift method. 3.1 INTRODUCTION For most borehole simulations (Chen et al., 1996; Tubman et al., 1984; Baker, 1984; Chapter 2 of this thesis), a borehole is modeled as cylindrical layers. In an actual borehole, however, the medium within a cylindrical layer usually is not homogeneous. Its elastic parameters often change with depth. Pai et al. (1993) simulated the electromagnetic induction logging for models of this kind. In acoustic logging problems we have to solve the elastodynamic equation considering both P and S waves. Therefore, the solutions for the elastic problem are more complex. For the elastic wave simulation, Bouchon (1993) and Dong et al. (1995) used a boundary element method for open and cased boreholes embedded in layered media, respectively. In this chapter, the generalized reflection and transmission coefficient method is further developed to calculate more realistic models, such as multi-layered boreholes (casing and invaded zones) embedded in multi-layered formation, symmetric faults in the formation, and a perforation or joint in a casing. Though this semi-analytical approach can only handle radially symmetric models, it is more efficient (less memory and computing time) than purely numerical approaches. A general configuration of the problem considered in this chapter is shown in Figure 3.1, where the formation consists of several vertically heterogeneous cylinders, the source is located at the center of the borehole, and receivers are located either inside or outside the borehole. Although this model looks complicated, it still exhibits axial symmetry. Taking advantage of this symmetry we can derive a set of formulas to - 56 -
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SEISMOGRAM SYNTHESIS IN COMPLEX BOR
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I certify that I have read this dis
Page 5 and 6: developed in this thesis do not hav
Page 7 and 8: Finally, I would like to thank my w
Page 9 and 10: 2.4.2 Cased borehole 2.4.3 Borehole
Page 11 and 12: 6.4.3 Attenuation estimation from t
Page 13 and 14: 3.4b Seismograms calculated using t
Page 15 and 16: List of Tables 2.1 Model parameters
Page 17 and 18: Seismogram synthesis can help us un
Page 19 and 20: 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: Two special examples for single bor
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 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
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profile curves within the tomograms
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Figure 4.10 The attenuation and vel
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Chapter 5 Acoustic Attenuation Logg
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decreases with the increasing offse
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Assume that the intrinsic attenuati
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2 i 0 ( f f ) i 2 R ( f ) df i
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Figure 5.3b Micro-seismograms calcu
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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 -
Page 163 and 164:
( 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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