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

term in (6a) and (6c) begins to dominate the tip stress state, with σ11 t increasing linearly with ε11 p at a rate 1/ν ∞ times greater than σ33 t . Figure 2.12 illustrates this relationship using the putative remote stress state and elastic parameters defined above. Normal stress components at the tip are plotted as functions of ε11 p from 0 to 0.1 for two cases: Gb = 2.182Gs and νb = 0.5νs (solid lines), and Gb = 0.857Gs and νb = 2νs (dotted lines). Thus Eb = 2Es in the first case and 0.5Es in the second, while in both cases, Kb/Ks = 1.5. The two sets of lines lie virtually on top of each other over the entire range of ε11 p , indicating not only that the actual plastic strain of 0.1 dominates the state of stress at the tip, but that the elastic properties inside the model band have a negligible effect on the state of stress even as ε11 p approaches zero. The results of this analysis, however, pertain only to a point infinitesimally close to the tip of an infinitesimally thin ellipsoidal inclusion. The aspect ratio of the idealized ellipsoidal CB is small (10 -4 ) but not zero, and the approximation of the granular Aztec sandstone as a homogeneous continuum makes looking very close to the tip problematic. We therefore turn to a complete analytical solution of the Eshelby problem to match the exact physical parameters of the field-based conceptual model and extend the analysis away from the band-sandstone interface. 7.2. Eshelby inclusion model For this analysis, we use an exact, closed-form solution to the general three- dimensional Eshelby problem based on the equivalent inclusion method. The solution is coded in MATLAB® and was made available to us by Pradeep Sharma at the University of Houston. Within certain limitations, the code accepts any combination of ellipsoidal inclusion dimensions, internal and external elastic properties and remote loading to calculate the state of stress for any point outside the inclusion. The embedded layer stress results in Figure 2.12 can be reproduced to within a fraction of 1% using the Eshelby code, by inputting an axisymmetric inclusion with a 10 -7 aspect ratio and calculating the state of stress just outside (within machine precision) of the tip. To represent the field-based conceptual model, dimensions of 25 m by 25.1 m by 9.38 mm are used, along with a fixed value of ε11 p = 0.1. The orientation of the model band with respect to the remote stress field, the range of elastic parameters inside and outside 64
MPa 2000 1800 1600 1400 1200 1000 800 600 400 σ 11 200 σ22 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 p Uniaxial Plastic Strain ( ε11) Figure 2.12. Normal stress components just outside the tip of the model compaction band as linear functions of the uniaxial plastic strain from the embedded layer solution. Two scenarios are plotted: one for a band with Young’s modulus twice that of the surrounding material (solid lines) and one with Young’s modulus half that of the surroundings (dotted lines). In both cases, the band/surroundings ratio of bulk moduli is 1.5 and the lines plot virtually on top of each other. The state of stress at the tip is dominated by the plastic strain and insensitive to differences in elastic properties. 65 σ 33
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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
Page 26 and 27: Cenozoic Mesozoic Paleozoic Quatern
Page 28 and 29: 3.3. Deformation The Aztec also has
Page 30 and 31: There also are relatively high-angl
Page 32 and 33: (e) (f) (c) 500µm compaction band
Page 34 and 35: dihedral angle of 80° or more, and
Page 36 and 37: 5. Compaction band orientations Ori
Page 38 and 39: n = 20 M n = 20 P n = 22 B n = 20 R
Page 40 and 41: were encountered, giving the dihedr
Page 42 and 43: Eichhubl et al., 2004) did not lend
Page 44 and 45: P (a) (c) S P S Figure 1.10. Stereo
Page 46 and 47: possible, and use these to better c
Page 48 and 49: 1.5(ρgz) (b) WEST Willow Tank Uppe
Page 50 and 51: 9. Acknowledgements My sincere than
Page 52 and 53: The term compaction band (CB) was c
Page 54 and 55: Pollard, 2002). The particular util
Page 56 and 57: Hue-based image analysis using MATL
Page 58 and 59: a direct genetic relationship (Hill
Page 60 and 61: Compaction band fin Depositional be
Page 62 and 63: suggests—that to first approximat
Page 64 and 65: 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 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 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
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6.3. Approaching tip interactions A
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0.2 0.15 0.1 0.05 0 −0.05 −0.1
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0.2 0.15 0.1 0.05 0 −0.05 −0.1
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the hooking patterns commonly obser
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(a) (b) (c) σ1 compaction band max
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σ 3 σ 2 σ 1 Figure 4.24. Schemat
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NV UT CA AZ Park Road Map Detail Pa
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compaction band Figure 5.2. Typical
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3. Computational method The methodo
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compaction band A 5 mm A‘ compact
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4. Application to the Aztec sandsto
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Permeability (mD) 10 4 10 3 10 2 10
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B‘ A‘ A c f m c m c f c f c f c
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equivalent of the Aztec sandstone,
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effective permeability represents a
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NV UT CA AZ Park Road Map Detail Pa
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Commonly from ~1 mm to ~1.5 cm in t
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Although systematic arrays of DBs p
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to 2 m, with both sets in a cross-h
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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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