My PhD Thesis, PDF 3MB - Stanford University

More documents

Recommendations

Info

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

Chapter 2 Elastic Waves in Radially Multi-layered Media In this chapter, a new method based on generalized reflection and transmission coefficients is proposed to calculate synthetic seismograms in radially multi-layered media without vertical variation (media with vertical variation are considered in Chapter 3). This method can be used to efficiently simulate full waveform acoustic logging, crosswell seismic profiling, and single borehole profiling in situations where we need to consider borehole effects. The formulation is tested by comparing the numerical results with available previous work, showing excellent agreement. Because of the use of the normalized Hankel functions and the normalization factors, this new algorithm for computing seismograms is numerically stable even for the high frequency problem. Thus, we can simulate some models that can not be solved by other methods. An example of this is given in Section 2.5.1. In Chapter 5, I will apply this method for acoustic full waveform logging to investigate the effects of complex invaded zones on the geometrical spreading and attenuation estimation for P waves. 2.1 INTRODUCTION Sonic logging in cased boreholes is useful for evaluating the quality of the cement job. Although acoustic logging in open boreholes is common, effort is also being made - 25 -
to measure formation properties from acoustic logs run in cased boreholes. A cased borehole and near borehole alterations, e.g., mudcake and invaded zones, can be modeled as a radially layered medium. A fluid-filed open borehole is also a radially layered medium. Thus, study of elastic waves in a radially layered medium is useful for understanding and interpreting full wave form sonic logs. Tubman et al. (1984) used the Thomson-Haskell method to study this problem. In this study we propose an alternative approach, i.e., the generalized reflection and transmission (R/T) coefficients method. The generalized R/T coefficients method is widely used in modeling elastic waves in vertically layered media because of its computational stability and efficiency over the Thomson-Haskell method, especially for high frequency problems (see, e.g., Luco & Apsel, 1983; Kennett, 1983; Chen, 1993). Yao & Zheng (1985) derived a set of formulas for computing synthetic seismograms in a radially layered borehole environment using the generalized R/T coefficients method, but they did not conduct numerical tests to check their formulation. A set of alternative formulations are derived in this chapter, which are more stable and efficient than the previous methods for numerical computation. In this chapter, I introduce the concepts of modified and generalized reflection and transmission matrices for radially layered media and derive a recursive scheme to calculate them; then, I determine wave fields using the R/T matrices. To check the formulation, I compare the calculated results with that of Tubman et al. (1984) for a four-layer cased borehole model and those of Baker (1984) for borehole models with invaded zones. As special applications, I give two numerical examples in single borehole profiling. - 26 -
Page 1 and 2: SEISMOGRAM SYNTHESIS IN COMPLEX BOR
Page 3 and 4: 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: simultaneously determine both phase
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 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
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 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
show all

My PhD Thesis, PDF 3MB - Stanford University

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?