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岩石破坏

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岩石破坏 International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 Damage in a rock core caused by induced tensile stress and its relation to differential strain curve analysis K. Sakaguchi*, W. Iino1, K. Matsuki Department of Geoscience and Technology,...
岩石破坏
International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 Damage in a rock core caused by induced tensile stress and its relation to differential strain curve analysis K. Sakaguchi*, W. Iino1, K. Matsuki Department of Geoscience and Technology, Graduate School of Engineering, Tohoku University, 01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan Accepted 8 April 2002 Abstract A finite element analysis was carried out to analyze the distribution of tensile stress within and below a long HQ core stub for 77 in situ stress conditions. The maximum tensile stress experienced by the core along the axis during boring under in situ stress was accumulated in an equal-area stereonet for a central part of the cross-section. The maximum tensile stress accumulated for a central area of less than about 60% of the total cross-sectional area was concentrated in a certain direction, which was nearly the same direction as the minimum principal stress for all of the stress conditions, except those in which the minimum principal stress (s3) was equal to the intermediate principal stress (s2). When s2 ¼ s3; the direction of the cumulative maximum tensile stress lay approximately in the plane of s2 ¼ s3; which is perpendicular to the maximum principal stress. Based on the assumption that a penny-shaped crack is produced normal to the maximum tensile stress in proportion to the magnitude of such stress, the crack density in the core was analyzed by calculating strain under hydrostatic pressure as in differential strain curve analysis (DSCA). The maximum principal crack density in the central part of the core was much greater than the intermediate and minimum principal crack densities, excluding special cases in which s2 ¼ s3: The direction of the maximum crack density was similar to that of the accumulated maximum tensile stress. Thus, the direction of the maximum crack density obtained by DSCA predicts the direction of the minimum principal stress rather than that of the maximum principal stress, if the distribution of pre-existing microcracks before stress relief is isotropic and if additional microcracks are produced only by tensile stress during boring under in situ stress. To verify this, crack parameters were measured by DSCA for two cores of quartz-diorite, which were taken by overcoring when the hemispherical-ended borehole technique was used to measure in situ stress. The directions of the maximum crack parameters measured by DSCA were nearly the same as that of the minimum principal stress for one of the cores. For the other core, for which the magnitudes of the intermediate and minimum principal stresses were close to each other and, accordingly, the direction of the minimum principal stress was uncertain, the direction of the maximum crack density estimated by damage analysis under the assumption that s2 ¼ s3 coincided with the directions of the maximum crack parameters measured by DSCA. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Crack density; Tensile stress; DSCA; Damage of rock core; Boring under stress 1. Introduction Differential strain curve analysis (DSCA) is a method for estimating in situ stress by measuring the distribu- tions of microcracks in an oriented core taken from depth [1]. Since this method is simple and can be applied to great depth, many researchers have used DSCA to estimate three-dimensional directions of principal stres- ses and the ratios of their magnitudes [2–6]. DSCA is based on the assumption that the stress relief in the core produces microcracks perpendicular to the principal stresses in proportion to the magnitude of each principal stress [1]. Carlson and Wang [7] showed that microcrack porosity in a rock core increases with the mean in situ stress, which may indirectly support the above assump- tion. Charlez et al. [8] proposed a rheological relaxation model to explain microfracturing induced by stress relief. However, this assumption has not yet been sufficiently verified. Furthermore, DSCA assumes that *Corresponding author. Fax: +81-22-217-7381. E-mail addresses: saka@rock.earth.tohoku.ac.jp (K. Sakaguchi), matsuki@rock.earth.tohoku.ac.jp (K. Matsuki). 1Present address: Fuji Photo Film Co., Ltd. 1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 2 ) 0 0 0 3 9 - 4 rocks have no effective microcracks, which is not always valid. It is well known that core discing occurs when the principal stresses perpendicular to the core axis are high [9]. This phenomenon is believed to occur due to tensile stress near the bottom of the core stub [10–12]. Li and Schmitt [13] showed that tensile stress could produce another type of crack, such as a petal fracture, under different in situ stress conditions. Therefore, if a macroscopic fracture is not created, the tensile stress induced by boring may produce microfractures in a rock core according to the in situ stress, and consequently the core may be more or less damaged. Evaluating damage in the rock core is also important for understanding how the mechanical properties of rock samples from deep underground differ from those in situ. In this study, we analyzed the history of the tensile principal stress experienced by an HQ core during boring under in situ stress by using finite element method (FEM) to calculate the tensile stress distribu- tions within and below a long HQ core under a general state of in situ stress. First, the maximum tensile stress experienced by the core along the axis was accumulated in an equal-area stereonet for a central part of the cross- section. Next, based on the assumption that the maximum tensile stress produces a penny-shaped crack in proportion to the magnitude of such stress at each position in the cross-section, the crack density in the core was evaluated. Our analysis showed that the direction of the maximum crack density is nearly the same as that of the minimum principal stress rather than that of the maximum principal stress, contrary to the assumption in DSCA. Finally, by using two rock cores taken by overcoring when in situ stress was measured by the hemispherical-ended borehole technique, we measured the crack parameters in the specimens by DSCA and compared the principal directions with those of in situ stress. For both cores, the directions of the maximum crack parameters measured by DSCA were consistent with the results obtained by the damage analysis. 2. Analysis of tensile principal stress 2.1. Finite element method A finite element code for an axisymmetric body subjected to non-axisymmetric boundary conditions [14] was used to calculate stresses in the model shown in Fig. 1. Fig. 1(b) is an enlarged diagram of part (b) in Fig. 1(a). In this stress analysis, boundary conditions were expressed with a Fourier series and the results obtained for each component were superimposed to obtain the stress distributions in an arbitrary vertical cross-section (Fig. 1). The core was an HQ (core radius/ borehole radius=0.6612) with a length of 4 times the core radius (R). The tip of the core bit was assumed to be a semi-torus. The z coordinate (core axis=Z-axis) was set to be zero at the beginning of the semi-circular cross-section of the bit (Fig. 1(b)). Since the core was sufficiently long, no appreciable stress was produced in the upper part of the core. The model was composed of 957 6-point and 8-point elements with 2688 nodes. Poisson’s ratio was assumed to be 0.25. A tensile principal stress analysis was carried out for a region within and below the core stub, shown by thick lines in Fig. 1(b), which is referred to the analyzed region below. As shown in Fig. 2, the entire horizontal cross-section of the core was divided into 168 equal-area elements for the analyzed region (z=R ¼ �4 to 4); 7 for the radial direction (from r1 to r7; see Fig. 1(b) and Fig. 2) and 24 for the circumferential direction (from y1 to y24; see Fig. 2), and the stresses at the center of gravity for each element were analyzed. Thus, the stress in each element is equivalent in integrating the effects on the horizontal cross-section of the core. The circumferential coordinate (y) was taken from the X -axis to the Y -axis, as shown in Fig. 2. The maximum principal stress lies within the ZX -plane, as will be described later. If all of the principal in situ stresses are in compres- sion, the maximum (s1) and intermediate (s2) principal stresses normalized by the mean stress (sm) lie within the triangular area shown in Fig. 3. To consider general in situ stress conditions, 11 sets of normalized principal stress magnitudes were adopted for the analysis, as shown with open circles in Fig. 3. The magnitudes of the normalized principal stresses are summarized in Table 1. The direction of s1 was limited to within the ZX -plane and the inclination angle of s1 from the core axis (f1) was varied from 01 to 901 in steps of 301 (Fig. 4). The direction of s2 was varied in the plane perpendicular to s1 by increasing the angle (g2) from the Y -axis towards the X -axis from 01 to 901 in steps of 301. As a result, the analysis was performed for 77 in situ stress conditions. 2.2. Analysis of maximum tensile stress Fig. 5 shows examples of the tensile principal stress distribution along the core axis at certain positions in the horizontal cross-section (s1=sm ¼ 1:5; s2=sm ¼ 1; s3=sm ¼ 0:5; f1 ¼ 601; g2 ¼ 301). The tensile principal stress (st1) and the z coordinate are normalized by the mean in situ stress (sm) and the core radius (R), respectively. As a horizontal cross-section approaches the bottom of the core, the tensile principal stress increases to a maximum near the bottom of the core and decreases to be negligible at z=R greater than 3. This figure also shows an example of the history of the tensile principal stress experienced by the core at an arbitrary position in the horizontal cross-section until the boring is complete, if no macroscopic fractures are created. The K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380368 same tendency in the tensile stress distribution was observed for all of the in situ stress conditions used in this study. We examined the maximum tensile stress experienced by the core along the core axis at each position in the horizontal cross-section, since the damage due to tensile stress may be determined by the maximum tensile stress (st max) and, accordingly, the Fig. 1. Mesh diagram used in FEM analysis for (a) the whole model and (b) an enlarged diagram for part (b) in (a). Fig. 2. Points at which tensile stress was analyzed in an arbitrary horizontal cross-section of a core. Fig. 3. Magnitudes of principal in situ stresses normalized by the mean stress (s1=sm; s2=sm) used in FEM analysis. K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 369 damage in the core may be governed by the distribution of st max accumulated for the cross-section of the core. To obtain the cumulative distribution of st max; an equal-area stereonet was divided into 20� 20 subdivi- sions, and the magnitude of st max was accumulated in one of these subdivisions according to the direction. The results are shown as a contour diagram. The procedure for determining a contour diagram is as follows: (1) Prepare an equal-area stereonet with 20� 20 subdivisions (Xi; Yj) (upper hemisphere)(Fig. 6(a)), and initialize the cumulative value Sij for each subdivision: Sij ¼ 0: (2) For r ¼ r1 to r2 and for y ¼ y1 to y24: (i) Find the maximum tensile stress with respect to the z coordinate. Let the direction of the maximum tensile stress be (l, m, n) and the magnitude be st max (Fig. 6(b)). (ii) Calculate the inclination angle relative to the core axis (fe) and the azimuth (ye) of the direction (l; m; n): tan fe ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ m2 p n and tan ye ¼ m l : (iii) Calculate the position (xe; ye) of the direction (l; m; n) in the equal-area stereonet: xe ¼ ffiffiffiffi 2 p sin fe 2 cos ye and ye ¼ ffiffiffi 2 p sin fe 2 sin ye: (iv) Find the subdivision (Xi; Yj) in the stereonet in which (xe; ye) falls, and add the normalized magnitude st max=sm to Sij: Sij ¼ Sij þ st max=sm: (3) Create contour lines of Sij by using software that is commercially available. Fig. 7 explains the meaning of symbols used in a contour diagram that will be shown below. The smaller circle in the diagram indicates an inclination angle of 601 Table 1 Magnitudes of principal in situ stresses normalized by the mean stress (s1=sm; s2=sm; s3=sm) used in this study No. 1 2 3 4 5 6 7 8 9 10 11 s1=sm 1 1.5 2 2.5 3 1.25 1.5 1.5 2 2 2.5 s2=sm 1 0.75 0.5 0.25 0 1.25 1.5 1 1 0.75 0.5 s3=sm 1 0.75 0.5 0.25 0 0.5 0 0.5 0 0.25 0 Fig. 5. Examples of the distribution of normalized tensile principal stress (st1=sm) along the core axis (z=R) at certain positions in the cross-section (s1=sm ¼ 1:5; s2=sm ¼ 1; s3=sm ¼ 0:5; s1 ¼ 601; s2 ¼ 301). Fig. 4. Definitions for the directions of principal in situ stresses. Fig. 6. (a) An equal-area stereonet with 20� 20 subdivisions (upper hemisphere) and (b) the magnitude (st max) and direction (l; m; n) of the maximum tensile stress determined with respect to the z coordinate for a position in the cross-section. K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380370 from the core axis. The rectangles indicate the directions of in situ principal stresses (s1; s2; s3). The value of the outermost contour is 0.1 and the numeral shown at the bottom-right of the diagram indicates the maximum value. Note that the contour lines exist in a small area. The letter (c) indicates the classification of the cumula- tive distribution, as will be described later. Fig. 8 shows examples of the contour diagram for the cumulative distribution of normalized maximum tensile stress (st max=sm), which was accumulated for the central part of the core (r ¼ r1 and r2; 28.6% of the horizontal cross- sectional area) (s1=sm ¼ 1:5; s2=sm ¼ 1; s3=sm ¼ 0:5). The cumulative distribution of st max is clearly concen- trated nearly in the same direction as the minimum principal stress (s3). This characteristic was found with all of the in situ stress conditions used in this study, except for special cases in which the direction of s3 is not unique (s2 ¼ s3). The cumulative distribution of st max can be classified into four types according to the inclination of s3 from the core axis (f3). When f3 ¼ 01; the direction of the maximum cumulative st max is equal to that of s3 (type (a)). When 01of3o601; the azimuth of the maximum cumulative st max is almost equal to that of s3; but the inclination from the core axis is smaller than that of s3 (type (b)). When 601of3o901; the azimuth of the maximum cumulative st max is deviated from that of s3 towards the Y -axis, and the inclination from the core axis is greater than that of s3 (type (c)). When f3 ¼ 601; both types (b) and (c) appear. Fig. 8. Examples of contour diagrams in an equal-area stereonet of the cumulative maximum tensile stress normalized by the mean in situ stress (st max=sm) accumulated for r ¼ r1 and r2 (28.6% of the horizontal cross-section) (upper hemisphere, s1=sm ¼ 1:5;s2=sm ¼ 1; s3=sm ¼ 0:5). Fig. 7. Meaning of symbols in a contour diagram in an equal-area stereonet of the cumulative maximum tensile stress normalized by the mean in situ stress (st max=sm). K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 371 When f3 ¼ 901; the direction of the maximum cumula- tive st max again agrees with that of s3 (type (d)). Fig. 9 shows examples of the contour diagram of the cumulative distribution of st max=sm accumulated for the central part of the core (r ¼ r1 and r2) when s2 ¼ s3 (s1=sm ¼ 2; s2=sm ¼ s3=sm ¼ 0:5). The direction of the maximum principal stress (s1) and the plane perpendi- cular to s1 are shown in these diagrams. When the inclination of s1 from the core axis (f1) is 01, the direction of the maximum cumulative st max is dispersed in the horizontal plane (XY-plane), since the tensile stress was evaluated at every 151 in the circumferential direction. When f1 ¼ 301 or 601, the direction of the maximum cumulative st max is concentrated in nearly the same direction as the Y -axis. When f1 ¼ 901; the maximum cumulative st max is concentrated in the direction of the core axis. For all cases in which s2 ¼ s3; the direction of the maximum cumulative st max lies approximately within the plane in which s3 (=s2) exists. The vector of st max evaluated for a central part of the core was summed to determine a unique direction for the cumulative distribution of st max (denoted by sa) for all stress conditions except cases in which s2 ¼ s3: The results are shown in Fig. 10 for the principal axes of in situ stress. Figs. 10(a) and (b) show the directions of sa=sm accumulated for r1 and r2 (28.6% of the horizontal cross-sectional area) and for r1 to r4 (57.1% of the horizontal cross-sectional area), respectively. The directions of sa=sm for these different areas are similar, and are close to that of the minimum principal stress (s3) regardless of the in situ stress conditions. Since the direction of the tensile stress varies greatly at the outer part of the core [12], the direction of sa=sm was more dispersed and was less meaningful as the area considered increased. The results shown in Fig. 10 are consistent with core discing because core discing occurs due to tensile stress in the direction of the core axis when s3 is nearly in the same direction as the core axis (type (a)) [12]. We carried out a similar analysis for every tensile principal stress, instead of only for the maximum tensile stress, by accumulating the tensile stress along the core axis. The results with respect to the direction are almost Fig. 9. Examples of contour diagrams in an equal-area stereonet of the cumulative maximum tensile stress normalized by the mean in situ stress (st max=sm) accumulated for r ¼ r1 and r2 (28.6% of the horizontal cross-section) (upper hemisphere, s1=sm ¼ 2;s2=sm ¼ s3=sm ¼ 0:5). Fig. 10. Stereoplotted directions of the sums of the maximum tensile stress vectors in the principal axes of in situ stress, accumulated (a) for r ¼ r1 and r2 (28.6% of the horizontal cross-section) and (b) for r ¼ r1 to r4 (57.1%). K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380372 identical to those obtained for the maximum tensile stress described above. This means that the directions of the tensile principal stresses produced in the central part (at least for 57.1% of the horizontal cross-sectional area) of the core are fairly uniform. 3. Damage analysis of the core 3.1. Evaluation of crack parameters by DSCA In DSCA, strain gauges are glued to a rock specimen, which is usually cubic, on three surfaces perpendicular to each other (see Fig. 11) and hydrostatic pressure is applied both to the rock specimen and to fused silica as a reference material. The use of fused silica is to compensate errors due to drifts and temperature effects in the measurement of strain by taking the difference between strains of the rock specimen and those of fused silica [15]. The difference between these strains is called differential strain [15]. Fig. 11 shows a schematic of normal strain of rock specimen in a certain direction versus hydrostatic pressure. Note that the strain in this figure is not a differential strain for convenience. At low pressures, open cracks are closed, to cause a large increase in strain. The initial gradient in the diagram gives an effective linear compressibility (beff ). With an increase in pressure, the gradient gradually decreases due to the closure of cracks with smaller aspect ratios. When all of the effective cracks are closed, the gradient takes a constant value,
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