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

To determine the sensitivity of propagation path to the choice of r, we performed a series of stability tests using the same model band that produced Figures 4.16 and 17. We added a 2-cm element to one tip at a variable kink angle of α, and plotted the relationship between α and the angle at which the next increment of propagation was predicted to occur (β) for r = 1, 5 and 10 mm, and the bracketing remote differential stress magnitudes (σ d = σ11 r -σ22 r ) of zero and 0.5 (Figure 4.18). The response in every case is to correct the deviation such that β is negative for positive α along a nearly linear trend. When the remote stress state is isotropic, |β| ≤ |α| and symmetry with σ11 r is nearly restored with the next increment of propagation. When σ d is the maximum of 0.5, |β| ≥ |α| and the next increment of propagation over-corrects for the original deviation. The effect of varying r at the two stress states is also opposite: for σ d = 0, |β| varies directly with r; for σ d = 0.5, |β| varies inversely with r. Discarding r = 1 mm as unrealistically small for the reasons stated above, we observe that β never varies by more than 4° for a given α between r = 5 mm and r = 10 mm. Due to this limited effect, and to preserve the plausibility of the linear elastic continuum assumption, we chose r = 5 mm for all simulations. Finally, as the model uses linear segments to approximate CB paths that generally are smoothly curving, we examined the stability of the propagation direction as a function of the dislocation element length used. Naturally, the shorter the elements, the better the model fit to the ideal (i.e. smooth) path. As a practical matter of computational efficiency, however, longer elements are desirable. We performed a second suite of kink-angle stability tests for tip-element lengths of 2, 6 and 10 cm at r = 5 mm (Figure 4.19). Again, in every case β acts to oppose α along a nearly linear trend, with the response being strongest for σ d = 0.5. The sense of the effect at the two stress states is also opposite, with |β| varying directly with element length for σ d = 0.5, and indirectly for σ d = 0. The spread in the directional variability is about three times greater at σ d = 0.5 and is equal to about 85% of α. These results are entirely consistent with those reported in previous studies of propagation path stability for segmented opening-mode cracks using the kinked-tip approach (Broberg, 1987; Cotterell and Rice, 1980; Melin, 1983, 1987; Thomas and Pollard, 1993), which indicate that the misorientation of successive 110
Correction angle (β) in degrees -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 r=1mm, Sx=.5Sy r=5mm, Sx=.5Sy r=10mm, Sx=.5Sy r=1mm, Sx=Sy r=5mm, Sx=Sy r=10mm, Sx=Sy D n 0 0 5 10 15 20 25 30 35 40 45 Kink angle (α) in degrees Figure 4.18. Relationship between the angle of incremental deviation from symmetric propagation (kink angle α) to the subsequent angle of incremental path correction (β) as a function of both r and the remote state of principal stress (Sy = σ1, Sx = σ2). Note that the sign of β is negative for positive α in all cases. Correction angle (β) in degrees -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 2cm, Sx=.5Sy 6cm, Sx=.5Sy 10cm, Sx=.5Sy 2cm, Sx=Sy 6cm, Sx=Sy 10cm, Sx=Sy 0 0 5 10 15 20 25 30 35 40 45 Kink angle (α) in degrees Figure 4.19. Relationship between the angle of incremental deviation from symmetric propagation (kink angle α) to the subsequent angle of incremental path correction (β) as a function of both element length and the remote state of principal stress (Sy = σ1, Sx = σ2). Note that the sign of β is negative for positive α in all cases, and that geometric formulation of problem is as shown in Figure 4.18. 111 D n σ 1 α Ds β + − D n σ 2
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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
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 120 and 121: plastic compaction, as suggested by
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) (
Page 170 and 171: 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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