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

segments will self-correct to provide a reasonable, zig-zag approximation to the ideal path, certainly for |β| less than ~10° so long as σ d ≠ 0. In all the simulations presented below, we therefore adopted fixed values of Eb = 15 and r = 5 mm, and use 10 cm propagation elements so long as β < 10°. Insofar as necessary to maintain numerical stability, and capture details of strong curvature and the close approach of adjacent bands, we reduced the added element length to 2 cm. We also verified that at no point in any simulation did Dn on any element become positive (i.e. opening-mode). Finally, given the current lack of data on a critical σθθ threshold for propagation to occur, we used a model threshold of σθθ max > σ11 r = 1. In practice, the normalized magnitude of σθθ max never dropped below about 1.25. 6.2. Symmetric propagation In order to examine the tendency for CBs to form symmetric (orthogonal) to σ11 r , we simulated the propagation of a short, incipient band (three elements, 6 cm) over a range of misalignment angles α (relative to the global coordinate system) for each of the remote stress states σ d = 0.05, 0.25 and 0.5 (Figure 4.20). Not surprisingly, the results indicate that the tendency of a misaligned incipient CB to propagate into symmetry with the remote stress field decreases as σ d → 0. For a perfectly isotropic state of remote stress, all band trends are preferred equally. For the maximum probable remote differential stress (σ d = 0.5), the band trend orthogonal to σ11 r is strongly preferred. Even at σ d = 0.25 (σ22 r = 0.75σ11 r ), the symmetric orientation is clearly preferred, particularly if one allows for the fact that 6 cm greatly exaggerates the length a misaligned incipient CB would attain without starting to curve. Another factor to consider in assessing the tendency toward symmetric propagation is how σθθ max at the tips of the incipient model CB varies with α (Figure 4.20d). Even for σ d = 0.05, a slight but appreciable increase of 5.4% in σθθ max is realized as α goes from 90° to 0°. For σ d = 0.25 and 0.5, the increase is 33.3% and 100%, respectively. These results demonstrate that, even as σ d → 0, the propagation of incipient bands oriented more nearly orthogonal to σ11 r is preferred, likely stranding those oriented at higher α. 112
Distance in meters Distance in meters 0.15 0.1 0.05 0 −0.05 −0.1 0.1 0.05 0 −0.05 −0.1 σ 1 α σ 2 (a) 1 deg. 5 deg. 15 deg 30 deg. 45 deg. 60 deg. 75 deg. 85 deg. 89 deg. −0.15 −0.15 −0.1 −0.05 0 0.05 Distance in meters 0.1 0.15 0.15 (c) 1 deg. 5 deg. 15 deg 30 deg. 45 deg. 60 deg. 75 deg. 85 deg. 89 deg. −0.15 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Distance in meters Distance in meters 0.15 0.1 0.05 0 −0.05 −0.1 (b) 1 deg. 5 deg. 15 deg 30 deg. 45 deg. 60 deg. 75 deg. 85 deg. 89 deg. −0.15 −0.15 −0.1 −0.05 0 0.05 Distance in meters 0.1 0.15 3 2.8 (d) Maximum tangential stress 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 Diff. stress=0.5 Diff. stress=0.25 Diff. stress=0.05 1 0 10 20 30 40 50 60 70 80 90 Misalignment angle (α) in degrees Figure 4.20. Propagation of an incipient anticrack band as a function of misalignment angle α from the symmetric orientation for normalized remote differential stress magnitudes of 0.5 (a), 0.25 (b) and 0.05 (c). A schematic of the model geometry is shown in the upper left of plot (a). Each propagation simulation started from an initially straight band 6 cm long and centered at the origin. Plot (d) shows the normalized value of the maximum tangential stress at the tip of the initial band as a function of α and the remote differential stress magnitude. 113
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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
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(a) (b) σ 3 σ 1 x 3 x 1 x 1 (c) u
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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
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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 120 and 121: plastic compaction, as suggested by
Page 122 and 123: To determine the sensitivity of pro
Page 126 and 127: 6.3. Approaching tip interactions A
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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) (
Page 170 and 171: 6.1. Parallel Effective permeabilit
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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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