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

plastic compaction, as suggested by the elliptical band profiles, then the assumption of at least quasi-elastic behavior inside the bands on the time scale of propagation might be justified (Sternlof et al., 2005). Despite these uncertainties, the distribution of Dn measured for the 25-m-long CB profile featured by Sternlof et al. (2005) can be used to estimate an approximate effective elastic stiffness (Eb) for the anticrack elements (Figure 4.16). For Eb = 0 (i.e. a perfect, traction-free anticrack), the model calculates a distribution of Dn two orders of magnitude too high (~10 cm versus ~1 mm in the middle and ~1 cm versus ~.1 mm at the tips). The addition of even a small amount of element stiffness, however, drastically reduces Dn, and we found that setting Eb = 15 (3% of Es) produced a slight over estimate. Of course, the measured distribution of Dn could be matched exactly by independently varying Eb for every element, but the benefits of such specificity do not justify the effort, particularly as the assumption of elastic behavior is dubious at best. For our immediate purposes, it is enough that the near-tip magnitudes of Dn calculated by the model generally coincide with those measured for natural bands, since these displacement discontinuities, in concert with any Ds accommodated, dictate the stress perturbations produced outside the band. The next aspect of essential model calibration involves choosing the characteristic distance r at which σθθ max will be calculated. As demonstrated by Sternlof et al. (2005), because the model represents CBs as trains of constant displacement dislocations, stress magnitudes calculated in the very near-tip field (r < 0.1 mm) vary as 1/r (Figure 4.17). For r ranging from about 0.5 mm to 5 mm, the calculated stresses vary along a more a crack-like 1/√r trend. Beyond 5mm, the BEM model-generated stresses basically coincide with those calculated analytically using the Eshelby inclusion method (Eshelby, 1959; Mura, 1987), which we consider the most accurate representation of the ideal, isolated CB based on the field data, and which predicts a normalized σθθ max of about 8.75 as r → 0. The stress singularity generated by the model at r = 0 has no significance in physical reality, both because the nature of deformation at the tip is actually plastic, and because the assumption of homogeneous, isotropic elasticity cannot be presumed to hold as r drops below the grain size of the sandstone. In fact, with an average grain size of ~0.2 mm, any r less than about 5 mm becomes suspect (Amadei and Stephansson, 1997). 108
Distribution of Dn in millimeters 2.5 2 1.5 1 0.5 Eb=10 Eb=12.5 Eb=15 Eb=17.5 Eb=20 Field data 0 −15 −10 −5 0 5 10 15 Distance in meters from center of band profile Figure 4.16. Distributions of closing-mode displacement discontinuity (Dn) predicted by the BEM model as a function of varying internal element stiffness (Eb) for a 25-m-long band trace. Best-fit ellipse to Dn data as calculated from field measurements shown by thick black curve. MPa 10 4 10 3 10 2 10 1 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Distance from tip, r (m) Figure 4.17. Log-log plot of the distribution of σ11 with distance from the model compaction band tip: BEM anticrack solution (solid line), Eshelby inclusion solution (dashed line), crack-like contours of 1/sqrt(r) (dotted lines), dislocation-like contours of 1/r (dash-dot lines). At the distance r = 1 cm from the tip, the two solutions effectively coincide. As r decreases, the Eshelby solution approaches the 1/sqrt(r) distribution, although it remains finite. The BEM anticrack solution, which is composed of a step-wise distribution of constant closing-mode displacement discontinuity elements, goes to infinity as 1/r (from Sternlof et al., 2005). 109
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STRUCTURAL GEOLOGY, PROPAGATION MEC
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Abstract Low-porosity, low-permeabi
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delivered with fortitude, humor, su
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Chapter 3—Energy-release model of
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List of Illustrations Figure A. Cov
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Figure A. Cover photo that accompan
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likely present in subsurface sandst
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hooking-tip interactions—using th
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and sandstone, my co-authors—Moha
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the hard data from which accurate p
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Although the Aztec sandstone experi
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Waterpocket Fault 36 o 26‘ N 0 1
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Cenozoic Mesozoic Paleozoic Quatern
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3.3. Deformation The Aztec also has
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There also are relatively high-angl
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(e) (f) (c) 500µm compaction band
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dihedral angle of 80° or more, and
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5. Compaction band orientations Ori
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n = 20 M n = 20 P n = 22 B n = 20 R
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were encountered, giving the dihedr
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Eichhubl et al., 2004) did not lend
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P (a) (c) S P S Figure 1.10. Stereo
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possible, and use these to better c
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1.5(ρgz) (b) WEST Willow Tank Uppe
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9. Acknowledgements My sincere than
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The term compaction band (CB) was c
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Pollard, 2002). The particular util
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Hue-based image analysis using MATL
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a direct genetic relationship (Hill
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Compaction band fin Depositional be
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suggests—that to first approximat
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Porosity Porosity 0.3 0.25 0.2 0.15
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pore-clogging clay—due presumably
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500 µm Figure 2.9. Electron backsc
Page 70 and 71: (a) (b) σ 3 σ 1 x 3 x 1 x 1 (c) u
Page 72 and 73: 6. Elastic properties Despite a lon
Page 74 and 75: Comparison of the two approaches es
Page 76 and 77: term in (6a) and (6c) begins to dom
Page 78 and 79: MPa MPa 80 70 60 50 40 30 20 10 0 0
Page 80 and 81: 2002) and use a BEM approach (Crouc
Page 82 and 83: MPa 10 4 10 3 10 2 10 1 10 −5 10
Page 84 and 85: concentration of quartz plasticity
Page 86 and 87: conducted on well-cemented sandston
Page 88 and 89: ~ 62 m compaction band trend 500µm
Page 90 and 91: Sternlof et al. (2005) have suggest
Page 92 and 93: 2005). It consists of a long (infin
Page 94 and 95: ⎡ ∆ ⎤ ⎧ p 1 ∆ ⎛ M ⎞
Page 96 and 97: which point the inelastic strain ha
Page 98 and 99: they can exert significant effects
Page 100 and 101: interact is inversely proportional
Page 102 and 103: (a) (b) (c) (d) Figure 4.3. Typical
Page 104 and 105: Viewed individually, CB traces tend
Page 106 and 107: (a) (b) (c) (d) (e) (f) Figure 4.7.
Page 108 and 109: 1973; Mardon, 1988; Peck et al., 19
Page 110 and 111: undamaged host rock. That CBs canno
Page 112 and 113: 0.5 0.4 0.3 0.2 0.1 0.5 0.4 0.3 0.2
Page 114 and 115: Normalized stress magnitude 3 2.5 2
Page 116 and 117: helps to explain why the oblique ap
Page 118 and 119: and ts = G·ds + H·dn (2) where tn
Page 122 and 123: To determine the sensitivity of pro
Page 124 and 125: segments will self-correct to provi
Page 126 and 127: 6.3. Approaching tip interactions A
Page 128 and 129: 0.2 0.15 0.1 0.05 0 −0.05 −0.1
Page 130 and 131: 0.2 0.15 0.1 0.05 0 −0.05 −0.1
Page 132 and 133: the hooking patterns commonly obser
Page 134 and 135: (a) (b) (c) σ1 compaction band max
Page 136 and 137: σ 3 σ 2 σ 1 Figure 4.24. Schemat
Page 138 and 139: NV UT CA AZ Park Road Map Detail Pa
Page 140 and 141: compaction band Figure 5.2. Typical
Page 142 and 143: 3. Computational method The methodo
Page 144 and 145: compaction band A 5 mm A‘ compact
Page 146 and 147: 4. Application to the Aztec sandsto
Page 148 and 149: Permeability (mD) 10 4 10 3 10 2 10
Page 150 and 151: B‘ A‘ A c f m c m c f c f c f c
Page 152 and 153: equivalent of the Aztec sandstone,
Page 154 and 155: effective permeability represents a
Page 156 and 157: NV UT CA AZ Park Road Map Detail Pa
Page 158 and 159: Commonly from ~1 mm to ~1.5 cm in t
Page 160 and 161: Although systematic arrays of DBs p
Page 162 and 163: to 2 m, with both sets in a cross-h
Page 164 and 165: y identical blocks all subject to t
Page 166 and 167: contiguous areas. This type of char
Page 168 and 169: (0,b) n 3 (0,0) f 2 n 2 n 1 (a,0) (
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6.1. Parallel Effective permeabilit
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By contrast, band-parallel effectiv
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Relative Effective Permeability 1.0
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(a) (b) (c) 0 meters 3 = 10-2 kb /k
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Though grossly systematic, anastomo
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168
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100 meters 5 meters NV UT CA AZ Par
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compaction band trend 500µm Figure
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Cover Outcrop Band N 100 meters 20
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(CBs) and the matrix rock in all si
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The finite volume discretization fo
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isotropic) can lead to numerical di
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5. Simulations Three sets of 2-D si
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Normalized ∆P ∆P ratio (n/p) 4.
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P P P P I I P P Case 1 P P Case 2 I
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Case 1 Case 1 Case 1 (a) (b) (c) In
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5.2.2. Discussion The elliptical pa
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5.3.1. Results With the regional gr
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(a) (b) Leak 200 meters Regional gr
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anticipated sustainable pumping rat
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Finally, there is the issue of fore
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200
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Bakke, S., and Øren, P. E., 1997,
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Carpenter, D. G., and Carpenter, J.
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Eichhubl, P., Taylor, W. L., Pollar
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Karimi-Fard, M., Durlofsky, L. J.,
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-, 1987, Fracture from a straight c
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Shipton, Z. K., and Cowie, P. A., 2
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Wong, T. F., David, C., and Zhu, W.
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