Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
j jÆÅÇ[È”u j jÎÅÇ[È”uwŠ š ± ± ± 188 PARALLEL CRACKS 0 0.01 0.02 0.03 0.04 depth [m] Parallel Cracks 0 0.01 0.02 0.03 0.04 width [m] Figure 3: Randomly distributed rectilinear parallel non-intersecting thin dry cracks in homogeneous 2D-media. A part of Model 11p is shown. This section considers randomly distributed rectilinear parallel non-intersecting thin dry cracks in 2D-media (see Figure 3). The paper of Kachanov (1992) discusses three different theoretical descriptions of effective velocities in such a case. Namely, they are the “theory for non-interacting cracks (NIC)”, the “modified (or differential) self-consistent theory (MSC)” and a extension of the modified self-consistent theory (EMSC) by Sayers and Kachanov (1991). In order to give an overview we present here the effective Young's modulus Á³ ^ for the three theories: Á³µÂÄÓ ³µ˜ ^—U ´³µÉwʧ“ ³¿˜Ëš ^—U ‹|¡¢œ²ž Á³¿ÌÍÉwʧ“ ³¿˜ ^—U œ€ž where is the Young's modulus of the unfractured medium and u is the crack density. The effective velocities of SV- and P- waves predicted by this three theories for parallel cracks are plotted in Figure 4 using lines. Our numerical results for parallel cracks can ³¿˜ be seen in the same Figure using dots. Details of the models No. 8p-11p used for the experiments are displayed in Table 1. We show the calculations for two types of waves. The relative decrease of the effective velocity for one given crack density is in the following order: For SV-waves we obtain the smallest decrease followed by P- waves. The main result of the investigations in this section is the fact, that the three theories for parallel cracks can only be applied successfully to a dilute crack density. For large crack densities further research is necessary.
189 1 0.9 top: bottom: SV− waves P− waves Veff / V 0.8 0.7 0.6 non− interacting cracks mod. self − consistent Sayers & K.− technique numerical results 0.05 0.1 0.15 0.2 0.25 Crack density ρ Figure 4: Normalized effective velocity for parallel cracks versus crack density. Dots: Numerical results, Lines: Different theoretical predictions. CONCLUSIONS We present a new numerical tool to calculate effective velocities in fractured media. Finite-difference modeling of the elastodynamic wave equation is very fast and accurate. In contrast to a standard staggered grid, high-contrast inclusions do not cause instability difficulties for our rotated staggered grid. Thus, our numerical modeling of elastic properties of dry rock skeletons can be considered as an efficient and well controlled computer experiment. We propose a heuristic approach called the critical crack density (CCD) formulation. This formulation introduces the critical crack density into the modified (differential) self consistent media theory. The CCD formulation predicts effective velocities for SV-, SH- and P- waves in fractured 2D-media with intersecting rectilinear thin dry cracks. The numerical results support predictions of this new formulation. Moreover, we show that different theories for effective velocities for parallel cracks can only be applied successfully to a dilute crack density. REFERENCES Andrews, D. J., and Ben-Zion, Y., 1997, Wrinkle-like slip pulse on a fault between different materials: Journal of Geophysical Research, 102, No. B1, 553–571. Dahm, T., and Becker, T., 1998, On the elastic and viscous properties of media containing strongly interacting in-plane cracks: Pure Appl. Geophys., 151, 1–16.
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189<br />
1<br />
0.9<br />
top:<br />
bottom:<br />
SV− waves<br />
P− waves<br />
Veff / V<br />
0.8<br />
0.7<br />
0.6<br />
non− interacting cracks<br />
mod. self − consistent<br />
Sayers & K.− technique<br />
numerical results<br />
0.05 0.1 0.15 0.2 0.25<br />
Crack density ρ<br />
Figure 4: Normalized effective velocity for parallel cracks versus crack density. Dots:<br />
Numerical results, Lines: Different theoretical predictions.<br />
CONCLUSIONS<br />
We present a new numerical tool to calculate effective velocities in fractured media.<br />
Finite-difference modeling of the elastodynamic wave equation is very fast and accurate.<br />
In contrast to a standard staggered grid, high-contrast inclusions do not cause<br />
instability difficulties for our rotated staggered grid. Thus, our numerical modeling<br />
of elastic properties of dry rock skeletons can be considered as an efficient and well<br />
controlled computer experiment.<br />
We propose a heuristic approach called the critical crack density (CCD) formulation.<br />
This formulation introduces the critical crack density into the modified (differential)<br />
self consistent media theory. The CCD formulation predicts effective velocities for<br />
SV-, SH- and P- waves in fractured 2D-media with intersecting rectilinear thin dry<br />
cracks. The numerical results support predictions of this new formulation.<br />
Moreover, we show that different theories for effective velocities for parallel cracks<br />
can only be applied successfully to a dilute crack density.<br />
REFERENCES<br />
Andrews, D. J., and Ben-Zion, Y., 1997, Wrinkle-like slip pulse on a fault between<br />
different materials: Journal of Geophysical Research, 102, No. B1, 553–571.<br />
Dahm, T., and Becker, T., 1998, On the elastic and viscous properties of media containing<br />
strongly interacting in-plane cracks: Pure Appl. Geophys., 151, 1–16.
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