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

helps to explain why the oblique approach of one band to the flank of another rarely leads to intersection, as the local direction of σθθ max can never be parallel to the band flank and thus cannot lead the propagating tip on a direct collision course with the adjacent band flank. Finally we note that, unlike opening-mode cracks, which generally are modeled as shear traction-free boundary surfaces pressed open by an internal fluid pressure that constitutes an independently variable driving force for propagation (e.g. Cotterell and Rice, 1980), anticrack CBs are compaction structures whose propagation can only be driven by near-tip compressive stress concentrations generated as a consequence of previous compaction, which must ultimately result from the failure of grain-scale flaws within the ambient remote stress field (Sternlof et al., 2005). This amounts to a stringent constraint on initiation and propagation that renders the dense variety of band patterns pervading the Aztec sandstone all the more remarkable. 5. Numerical model In order to examine the stress and strain fields generated by CBs and model their propagation and interaction as anticracks, we developed a MATLAB® code based on the displacement discontinuity boundary element method of Crouch and Starfield (1983). In our formulation of this 2-D, linear elastic plane strain model, which corresponds well with the character of the field data and interpretations described above, we consider an infinite whole space subjected to uniform principal stresses at infinity and represent the bands as segmented traces of linear boundary elements (Figure 4.15). Across each of these elements, two independent and uniformly distributed components of displacement discontinuity, or dislocation, can be accommodated: Dn (normal) and Ds (shear). The displacements realized on each element affect those on all others and depend on the remote loading, as well as the traction boundary conditions (normal and shear) which must be specified for every element. In order to determine the two unknown components of displacement for all N elements, while matching the specified traction boundary conditions, requires the formulation and solution of a traction boundary value problem consisting of a linear system of 2N equations that can be summarized in matrix notation as: tn = E·dn + F·ds (1) 104
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 (a) −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 4.14. Schematic illustration of how one anticrack band tip influences the propagation path of an approaching tip through the action of the near-tip shear stress field it generates. (a) A band entering from the upper right first experiences a slight positive shear stress which causes its path to diverge. When it enters the zone stronger, negative shear stress, the approaching band turns to follow a strongly convergent path. (b) A band entering from the upper left at first strongly diverges and then weakly converges. In both cases, if the band approached from the bottom side, the resulting propagation paths would mirror those shown. r σ22 x 1 x 2 Figure 4.15. Schematic illustration of the essential aspects of the displacement discontinuity boundary element method model. (b) 105 r σ11 D n D s 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4
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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
Page 66 and 67: pore-clogging clay—due presumably
Page 68 and 69: 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 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 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
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contiguous areas. This type of char
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(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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