Annual Report 2000 - WIT

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‹œž Ÿ¢¡¤£¦¥ƒ§©¨ ‘˜›š ¨¤ª § § š ‹œ€ž¥ § ¨ ³µ˜ ª § §©¨ š ‹œ€ž ¥ §©¨ ·˜ ¨¤ª § § ¹ º ¡ » »¼N½©¾ ± ¾ 186 numerical results of the present study with the predicted effective velocities of a new critical crack density (CCD) formalism (Saenger, <strong>2000</strong>). The CCD formulation is valid for 2D (i.e. 3D transversely isotropic) fracturing configurations with intersecting cracks. We introduce an additional factor into results of a modified self-consistent theory (described e.g. in (Davis and Knopoff, 1995)) to include the physical behavior at the critical crack density (similar to the critical porosity described in (Mukerji et al., 1995)). We propose the following formulae for the effective elastic moduli: «¬¯®¦°²± ’‘”“•“•– ^—U «¬¯®¦° ± ´³µ“•“•– ^—U « ¬¯®¦° ± ´·§“•“•– ^„U with: utU j ¸ where is the shear modulus, is the Young's modulus, is the Poisson's ratio of the unfractured medium, ‘˜ is the crack density (as in (Kachanov, 1992)), ³¿˜ is the ·˜ u¡À u critical crack density (can be found in (Robinson, 1983) for our [ » models), is the rectilinear length of a crack and is the representative area. Now, we discuss the numerical results on effective wave velocities. They are depicted with dots in Figure 2. For comparison, the predictions of the CCD formulation described above are shown in the same Figure with lines. We show the normalized effec- ¸ tive velocities for three types of waves. The relative decrease of the effective velocity for one given crack density is in the following succession: For SH-waves we obtain the smallest decrease followed by SV-waves. For P-waves it is largest. For each wave type we perform numerical FD-calculations with different crack densities to obtain the effective velocity. For these measurements depicted in Figure 2 we use the models No. 1x,2x,3x.1-3x.6,4x,5x.1-5x.6,6x,7x (see Table 1). A final result is that our numerical simulations of P-, SV- and SH-wave velocities are in excellent agreement with the predictions of the CCD formulation.
187 1 0.8 top: SH− waves middle: SV − waves bottom: P− waves Veff / V 0.6 0.4 0.2 CCD− formulations numerical results 0.2 0.4 0.6 0.8 1 1.2 1.4 Crack density ρ Figure 2: Normalized effective velocity versus crack density. Dots: Numerical results of this study, Lines: Theoretical predictions of the CCD-formulations. The error bar denotes the standard deviation for different model realizations.
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Wave Inversion Technology WIT Annua
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Wave Inversion Technology WIT Wave
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Copyright c 2000 by Karlsruhe Unive
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i TABLE OF CONTENTS Reviews: WIT Re
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Wave Inversion Technology, Report N
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3 theory. It relates properties of
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Imaging 5
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8 two hypothetical eigenwave experi
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7 ¢¥£§¦©¨ ; < calculate sear
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7 7 searches ¡ HNJOL for ¢IHNJML
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14 Trace no. 1000 1500 2000 0.5 1.0
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16 CDP number 3100 3000 2900 2800 2
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18 REFERENCES Höcht, G., de Bazela
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20 INVERSION BY MEANS OF CRS ATTRIB
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22 For synthetic data, it is easy t
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— J J J J J J G J - J J G
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· · v 0 B 0 £ & Ä Ã & v
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31 Dürbaum, H., 1954, Zur Bestimmu
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Ã Ø Ì 0 0 Ì 4 0 Ã 0 G ' 4
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0 0 ˜ ” Z ˜ ” Z 4 4
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39 strategy by combining global and
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41 elled data is presented in Figur
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43 0.2 Distance [m] 1000 1500 2000
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45 In Figure 10 we have the optimiz
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47 Gelchinsky, B., 1989, Homeomorph
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§ ¢ ø ø for a paraxial ray, ref
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Œ § … Z ‹ Z § õ ø \ ø §
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54 As stated in Hubral and Krey, 19
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§ ó § 56 sequence. We made tes
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58 precision of the modeled input d
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60 Cohen, J., Hagin, F., and Bleist
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£ 65 The details of the theory inv
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67 of the figure. On both sides, a
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69 (a) (b) Depth (m) 600 800 Depth
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71 Langenberg, K., 1986, Applied in
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È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð
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g ý g [ ^ â g [ g g g g g â h [
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g § » ¹ [ƒŽ g 79 We now subst
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ê 81 ing was realized by an implem
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ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
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85 on a second-order approximation.
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88 kinematic (related to traveltime
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¨ º Ý È · § À · Á · Á ¼
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° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
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94 1 taper value ksi1 0 ksi2 Figure
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96 0 1 2 3 4 5 Depth [km] CMP [km]
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98 CONCLUSION As a generalization o
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100 weight during the stacking proc
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102 are not illuminated for every o
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104 obtained from Zoeppritz' equati
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106 PreSDM of Porous layer 0.358 re
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108 REFERENCES Gassmann, F., 1951,
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110 et al., 1992), Hanitzsch (1997)
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112 consecutive wavefronts). Howeve
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114 Numerical example We test the W
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116 Versteeg, R., and Grau, G., 199
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118 indicator. To extract elastic p
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€ b b 120 S where is the position
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É É É É 122 The À constant (da
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124 0 Distance [ m ] 1000 2000 3000
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126 Kosloff, D., Sherwood, J., Kore
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128
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130 penetration distances and a pot
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and » ˆ Ö Ö is the scalar hydra
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Å éÙó ó ».-.‹œì/-jì¨‹
Page 148 and 149: 136 TRIGGERING FRONTS IN HETEROGENE
Page 150 and 151: Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
Page 152 and 153: Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
Page 154 and 155: 142 300 250 200 a) eikonal equation
Page 156 and 157: 144 Shapiro, S. A., and Müller, T.
Page 158 and 159: » i » m ×]Ú ‘Œƒšj'Ÿlk hM
Page 160 and 161: 148 RESULTS Soultz-sous-Forêts Inv
Page 162 and 163: …„ƒ 150 for the smaller ellips
Page 164 and 165: 152 Figure 5: The cloud of events f
Page 166 and 167: …„ƒ Î Î ÎÄÈ‹•¨ Î-È
Page 168 and 169: 156 straightforward. The new approa
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Page 172 and 173: 160 1982). There is also the so-cal
Page 174 and 175: { ³ u ´ ´ ´ ´ -y ›-^œ ´ ´
Page 176 and 177: Ï u u ¬ 164 To summarize, we deri
Page 178 and 179: Ý ì á ä é å¤æDçzè ä é
Page 180 and 181: ø M ã = ã Ù ø ä Ú ã Ù ø
Page 182 and 183: ã Q c ã ä 170 a=10m; std.dev.=8%
Page 184 and 185: 172 Frankel, A., and Clayton, R. W.
Page 186 and 187: 174 It is well-known that inhomogen
Page 188 and 189: ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
Page 190 and 191: Kneib, 1995 285-6000 superposition
Page 192 and 193: 180 parameter: Hurst coefficient pa
Page 194 and 195: 182 REFERENCES Bourbié, T., Coussy
Page 196 and 197: 184 1999) in multiple fractured med
Page 200 and 201: j jÆÅÇ[È”u j jÎÅÇ[È”uw
Page 202 and 203: 190 Davis, P. M., and Knopoff, L.,
Page 204 and 205: 192 properties from the measured se
Page 206 and 207: 194 discussion of these problems ca
Page 208 and 209: 196 Altogether, close to 260 shotpo
Page 210 and 211: 198 Time (ms) 0 2 4 6 8 10 12 14 16
Page 212 and 213: 200 of groundwater. ACKNOWLEDGEMENT
Page 214 and 215: 202
Page 216 and 217: 204 the wavefront curvature matrix
Page 218 and 219: û ò é from ã äñ to ã î§ñ
Page 220 and 221: é 208 Table 1: Median relative err
Page 222 and 223: 210 Table 2: Median relative errors
Page 224 and 225: 212 CONCLUSIONS We have presented a
Page 226 and 227: 214 PUBLICATIONS Previous results c
Page 228 and 229: æVU ìÿå T ü ýWYXZX[ 9\]9^\ Þ
Page 230 and 231: 218 weight functions from traveltim
Page 232 and 233: 220 REFERENCES Bleistein, N., 1986,
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Page 240 and 241: 228 In this paper, we present a mor
Page 242 and 243: L • MM+,ON N N Ž ú R N N N ú
Page 244 and 245: 232 Figure 3: The evolution of a WF
Page 246 and 247: 234 Let us analyze next the computa
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236 CONCLUSIONS We have presented a
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238
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’ Ç r 240 traveltimes, a computa
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Ž Ž Ã Ž 242 Other two approxima
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“ Ã Æ ³ 244 0 0.5 [km] 0 0.5 1
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î r † Û(Û Ü Œ U Ú Ý Ü Û
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248 Ettrich, N., 1998, FD eikonal s
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ø Ð ¡ Ð÷ö Ð'ø ú Ð ¡ Ð'
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ø ü ø ø ý Þ do ø ý Ü
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ø ö ø ú ü ö ü ¡ ü
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‹ 256 transport equation. Neverth
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258 tivalued geometrical spreading
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260 .
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262
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… " ¯ ü ý +- 4=° ü ü +
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ö ô æ º Ò x ¹ v ý ñ
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268 NUMERICAL RESULTS We constructe
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¹ õ ' ' -š' ù ¹ ù ' ' -š
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£ -š' - 272 Ô- —- Figure 6: C
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274 REFERENCES Aldridge, D. F. (199
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276
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278 3. 4. True-amplitude imaging, m
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280 Research Group Hamburg (Gajewsk
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282 Stefan Lüth Robert Patzig Refr
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284 Landmark Graphics Corp. 7409 S.
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286 TotalFinaElf Exploration UK plc
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288 as research assistant at Geoeco
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290 Ingo Koglin is a diploma studen
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292 Matthias Riede received his M.S
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294 Svetlana Soukina received her d
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296 Yonghai Zhang received the Mast
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Annual Report 2000 - WIT

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