4
DRILLING AND BORING OF ROCK

Introduction

In the smart drilling approach, whose development is recommended in this report, the process of local rock fracture, comminution, or other novel forms of rock removal is the first, and perhaps most crucial, step in a long succession of processes of reaching a target stratum or resource reservoir. As described in Chapter 5, mechanical fracturing of rock is still the most effective means of advancing the drill head. Thus, in the proposed ideal smart drilling approach, the drill must sense the type of rock or stratum ahead of the drill bit; recognize its resistance; and automatically adjust the drilling process in terms of rate, contact pressure, and so forth. If necessary, a smart drill will divert around a particularly difficult heterogeneity or seek alternative directions to avoid premature termination of the drilling operation.

Although identifying and evaluating rock ahead of the drill bit will require sophisticated advances in sensing and guidance, the actual fracturing, or comminution, of the rock requires a thorough understanding of the mechanical properties of rock and its response under the drill bits or cutters. Currently, there is insufficient understanding of both the interaction of cutting tools with the rock and the possible variations of such interactions based on rock type.

This chapter presents a summary of the current understanding of the structure and mechanical properties of rock, with particular emphasis on its fracture in compression, both under quasi-homogeneous stress fields and under conditions resembling the interaction of rock with the drill bit. The chapter is partly abstracted from a 1983 report by the National Materials Advisory Board (NMAB, 1983), but new developments have been added.



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4 DRILLING AND BORING OF ROCK Introduction In the smart drilling approach, whose development is recommended in this report, the process of local rock fracture, comminution, or other novel forms of rock removal is the first, and perhaps most crucial, step in a long succession of processes of reaching a target stratum or resource reservoir. As described in Chapter 5, mechanical fracturing of rock is still the most effective means of advancing the drill head. Thus, in the proposed ideal smart drilling approach, the drill must sense the type of rock or stratum ahead of the drill bit; recognize its resistance; and automatically adjust the drilling process in terms of rate, contact pressure, and so forth. If necessary, a smart drill will divert around a particularly difficult heterogeneity or seek alternative directions to avoid premature termination of the drilling operation. Although identifying and evaluating rock ahead of the drill bit will require sophisticated advances in sensing and guidance, the actual fracturing, or comminution, of the rock requires a thorough understanding of the mechanical properties of rock and its response under the drill bits or cutters. Currently, there is insufficient understanding of both the interaction of cutting tools with the rock and the possible variations of such interactions based on rock type. This chapter presents a summary of the current understanding of the structure and mechanical properties of rock, with particular emphasis on its fracture in compression, both under quasi-homogeneous stress fields and under conditions resembling the interaction of rock with the drill bit. The chapter is partly abstracted from a 1983 report by the National Materials Advisory Board (NMAB, 1983), but new developments have been added.

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Types of Rock The term ''rock" includes a great variety of material types with distinctive characteristics. Rock is one of the materials for which fracture behavior under compressive stresses has been studied most thoroughly. For example, granitic rocks can behave in a brittle manner up to a confining pressure of 1 gigapascal (GPa), whereas carbonate rocks become plastic at moderate pressures of about 100 megapascals (MPa). Extensive crystal plasticity is observed in rock salt at moderate stress at room temperature, whereas most quartz-bearing rocks do not show significant dislocation activities up to about 400°C. Rocks tend to be permeated with pores and microcavities, which were either formed during the inception of the rock or produced by its subsequent stress history. The porosity and microcavity morphology of rocks are as important as the mineralogic composition itself. Collectively, the microcavities cause nonlinear behaviors in many mechanical properties. These manifest themselves in the stress or pressure dependence of strain, velocity of sound, stress wave attenuation, and fracture behavior. Microcavities also introduce a scale effect into the prediction of mechanical behavior, and heterogeneities in the form of distribution of microcavities are the principal source of scatter in test results. Thus, this relatively readily probed characteristic of rocks may become an important indicator of mechanical properties to be sensed by the smart drill. Rock has a finite hydraulic conductivity, which implies that a portion of the void space forms an interconnected network. Petrological and geophysical evidence indicates that rocks are saturated with water to a depth of tens of kilometers. Pore fluids play a significant role in engineering applications for energy resource recovery or dam construction. The effect of pore fluids on fracture behavior can be either mechanical through pore pressure diffusion or chemical through stress corrosion (NMAB, 1983). The effect of pore fluid will be an important measurable indicator of the mechanical properties of rock relevant to drilling.

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Theoretical Models for the Behavior of Rocks in Compression Like all solids, rocks can undergo true intrinsic inelastic behavior by dislocation motion, diffusional flow, or analogous processes occurring in glassy media. In most crustal rocks of interest, however, such intrinsic inelastic behavior is exhibited only at elevated temperatures or pressures that are not encountered during mechanical drilling. In most drilling applications, rocks act as purely elastic solids, but the heterogeneities discussed above can affect their elastic behavior. For equiaxed heterogeneities, rock behavior can be readily accounted for by a variety of bounding approaches, or more elegantly by ellipsoidal inclusion models (Chow, 1978) or self-consistent models (Budiansky, 1965). Corresponding approaches that account for the effects of microcracks are also well developed (Salganik, 1973; Budiansky and O'Connell, 1976). Overviews of such self-consistent models have been given by a number of authors (Cleary and others, 1980; Haskin, 1988; Kachanov, 1992). Thus, considerable theoretical and mechanistic methodologies exist that are capable of relating elastic properties to heterogeneities in the rock that govern its fracture behavior. The apparent inelastic behavior of rocks known as clastic flow results from brittle fracture processes due to the formation and stable growth of brittle microcracks. This behavior has been dealt with in two ways. The first, a purely phenomenological approach, is used widely by civil engineers for characterizing the related clastic flow of concrete. Specific developments of this behavior by so-called deformation theories (Kupfer and Gerstle, 1973; Bazant and Tsubaki, 1980) or by incremental flow theories (Bazant and Kim, 1979; Nemat-Nasser and Shokovoh, 1980) have been discussed in the above-mentioned NMAB (1983) report. As discussed below, these formalisms are quite useful for predicting the development of shear faulting zones in rock, which is a central mechanism of rock fracture in drilling. They also find ready application in the understanding of chip formation under the drill bit. Of fundamental interest to drilling is the behavior of fractured rocks in compression. Griffith (1924), who pioneered the understanding of brittle fracture in tension, also was the first to elucidate brittle fracture in compression. He noted that in a solid containing microcracks of many orientations under triaxial compression, local tensile fracture is initiated when shear stresses produced by unequal compression displace surfaces of

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FIGURE 4.1 Extrinsic fracture in compression initiated from a large crack. (a) Stresses are concentrated at the ends of a shearing crack. (b) Microcracks extend from the shear crack parallel to the principal compression direction (NMAB, 1983). preexisting microcracks. These displacements produce tensile stresses near the tips of these microcracks, which result in their growth across the local maximum tensile stress when this stress reaches the cohesive strength (Figure 4.1a). Griffith used this model of crack initiation to show that the uniaxial compressive strength of brittle solids must be many times (about eightfold) higher than their tensile strength, and that this compressive strength could be increased monotonically by the application of a confining pressure. It later became clear that the Griffith model needed modification to account for the frictional resistance on the touching faces of the microcracks; such a modified model was provided by McClintock and Walsh (1962). Figure 4.2 shows the predictions of the models of Griffith (1924) and McClintock and Walsh (1962) compared to experimental

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measurements (Ohnaka, 1973; NMAB, 1983). Clearly, the McClintock and Walsh model tends to track the experimental data better than the original Griffith model. FIGURE 4.2 Relation between compressive strength and confining pressure for a series of rocks tested at room temperature (after Ohnaka, 1973).

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It has been recognized by many investigators (e.g., Brace and Bombolakis, 1963; Erdogan and Sih, 1963; McClintock, 1965; Babel and Sines, 1968; Nemat-Nasser and Horii, 1982; Ashby and Hallam, 1986) that when the local recracking condition is satisfied, wing cracks develop from the extremities of the initial microcracks and extend stably in the direction of principal compression as shown in Figure 4.1b. Since the continued extension of the wing cracks necessitates continued relative translation of the faces of the initial microcracks, and since the total amount of such displacement is limited by their original lengths, overall compressive failure cannot result from the limited growth of wing cracks in "parts" (individual rock pieces or fragments) with dimensions much larger than those of the initial microcracks (NMAB, 1983). Thus, it is remarkable that the agreement between experiments and the simple recracking models is as good as that presented in Figure 4.2. It is well known that brittle fracture in compression in massive parts occurs through the development of shear faults. This occurs when many developing wing cracks, aligned in a plane, begin to interact strongly en echelon and form a nucleus of a more macroscopic shear fault (Figures 4.3 a-b), which then spreads longitudinally to result in overall fracture. Although the conditions for development of the nucleus of the shear fault have not been well studied, the overall asymptotic conditions for their localization have been developed as a bifurcation model by Rudnicki and Rice (1975). They found that the critical strain at which the shear fault can develop freely depends strongly on the incremental moduli of the material containing the accumulating microcracks. The best agreement between theory and experiment, at least qualitatively, is obtained with a deformation theory approach. However, the actual conditions for fault development depend sensitively on the presence of initial imperfections, as is the case for all bifurcation phenomena (NMAB, 1983). The actual mechanism of brittle fracture in compression is a simple criterion of an equality between a crack driving force KI and a material fracture toughness KIc at the tips of the microcracks, interacting en echelon. However, the development of a shear fault that produces eventual failure or chip formation in drilling obeys a phenomenological pressure-dependent deviatoric stress criterion (NMAB, 1983). Such mechanistic understanding of the development of shear faults should be most useful in the effective control of the smart drilling process.

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FIGURE 4.3 Development of intrinsic fracture behavior by initiation and propagation of a shear fault in compression. (a) Beginning of en echelon alignment of adjacent microcracks, producing a zone of increased shear compliance. (b) Idealized spread of the high-compliance zone across the part, resulting in a shear fault (NMAB, 1983). Experimental Verification of Models Two Views of Rock Behavior Two approaches have been taken in the study of the mechanical behavior of rocks that together have contributed significantly to current understanding. The first is a global "applied mechanics" approach commonly used for engineering design and for the analysis of geologic faulting. The fracture process is taken as a discrete event without significant prior deformation and without warning. The only physical quantity of interest is the peak stress, which is of interest as an upper bound on solutions of the relevant boundary value problems. The fundamentals of this approach have been covered by many investigations including Jaeger and Cook (1979), Goodman (1980), Hoek and Brown

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(1980), and Germanovich and Cherepanov (1987). A previous report by the U.S. National Committee for Rock Mechanics of the National Research Council (NRC, 1981) also touches on this subject. The second is a "mechanistic" approach that supplements standard deformation tests with nondestructive evaluation and microscopy, aiming at a fundamental understanding of the microscopic mechanism. The evolution of microstructure is treated as a continuous process culminating in the coalescence of microfissures to form a throughgoing fault. This approach was adopted by Paterson (1978) in his extensive review of rock fracture. It is this latter approach that has led to more definitive mechanistic understanding of the clastic behavior of rocks and their eventual fracture, which is discussed in a following section. Laboratory Experiments on Clastic Flow and Fracture of Rocks As noted previously, the true plastic response of rocks by dislocation motion occurs at such high temperature and pressure as to be of no relevance to the problem of mechanical rock drilling. Many constituent minerals in rocks can, of course, undergo twinning that, for all practical purposes, is not a thermally assisted rate process but usually requires very high stresses. Moreover, twinning is a very inhomogeneous form of deformation, and in the absence of local plasticity, it can at best influence only the fracture behavior of rocks. Thus, experimental studies carried out in the laboratory have been devoted largely to understanding the complex processes of microcracking under compression that result in clastic flow, which is a forerunner of the eventual shear faulting process discussed previously. The clastic flow response of some common rocks such as Indiana limestone (Myer and others, 1992), Berea sandstone (Myer and others, 1992; Wong and others, 1992), and Carrara marble (Fredrich and others, 1989) has been investigated in the laboratory in a series of elegant experiments. These experiments have identified the key microstructural damage processes, which include fracturing of weak interface boundaries and grain boundaries; Hertzian diametral fracturing of grains; relative sliding across separated interfaces or boundaries, which act as inclined planes that "jack open" tensile wing cracks running parallel to the principal compression direction; and finally, en echelon action of interacting wing cracks that results in zones of shear faulting. In porous rock, pores may be

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crushed closed, which results in substantial permanent compression strains that locally stiffen and strengthen the rock. In laboratory simulations of borehole breakout experiments on externally pressurized thick-wall cylinders of both Indiana limestone and Berea sandstone (Ewy and Cook, 1990a,b), the localization of en echelon damage has been followed in detail, leading to local breakout (see also Dyskin and Germanovich, 1993; Dyskin and others, 1993). These breakout experiments have demonstrated the anisotropic nature of such rocks and have also demonstrated a substantial size effect on rock strength. In particular, rocks in such breakout experiments exhibit a two- to threefold increase in local tangential splitting strength over laboratory-size cylindrical rock samples undergoing fracture by vertical spalling. This size effect, which clearly can be important in the actual fracture response of rock under drill bits, is not well understood and needs further study. Compressive strength measurements of about 140 MPa, which have been obtained from the laboratory borehole breakout simulations, are quite high. Even when divided by 2.5, for example, to account for the size effect, these effective strengths are between 5 and 6 times higher than the often-quoted estimates of upper crustal effective strengths of about 10 MPa, which are commonly assumed to be proper for rock strength governed by frictional resistance (Zoback and others, 1993). Nevertheless, the strength levels measured in the laboratory may be more appropriate for understanding the local fracture strength of rock in the small volumes subjected to contact pressure by a drill bit. Clearly, these factors must be well understood to at least develop proper strength scaling relations that will be relevant in any definitive model of drilling. Effect of Confining Pressure and Temperature on Strength The brittle strength of rocks shows a strong pressure dependence because of its fractured state. It is not uncommon to achieve a tenfold increase in strength by a moderate increment in mean confining stress, as shown in Figure 4.2. Depending on the signs of the principal stresses, a sample can fail in tension or in shear. However, only shear fracture that occurs when the principal stresses are all compressive (Figure 4.4) is considered here. The fracture angle is defined as the angle between the shear fault surface and the maximum principal compressive stress 1.

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FIGURE 4.4 Relationship between shear and normal stresses and principal stresses at failure ranging from tensile, to uniaxial compression, to compression under a confining pressure (Hoek and Brown, 1980). Most of the empirical fracture criteria discussed earlier are formulated from conventional triaxial test data with the implicit assumption of the independence of the fracture phenomenon on the intermediate principal stress 2. However, Mogi (1972) concluded from his "true" triaxial tests on cube-shaped samples that this is only an approximation. He found that when the minimum compressive stress3 is kept constant, an increase in 2 results in an increase of 1 at failure (Figure 4.5a) and, furthermore, that the fracture angle decreases with increasing 2 when 3 is fixed (Figure 4.5b). The fracture surface, however, always contains the 2 direction. Ductility also is found to decrease with an increase in 2. Similar conclusions were drawn by Handin and others (1960, 1963) from torsion tests on hollow cylindrical specimens. The effect of temperature increases is to stabilize postfailure behavior (Wong, 1982). In general, in comparison with pressure, temperature has a relatively small effect on the brittle fracture of dry rocks.

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FIGURE 4.5 (a) Dependence of compressive strength 1 on transverse compressive stress 2 for different levels of the third principal compressive stress 3 from tests in triaxial compression. (b) Dependence of the fracture angle, between the shear fault plane and the direction of principal compression, on the principal stress difference from tests in triaxial compression (Mogi, 1972). Effect of Pore Fluid Experiments on sedimentary rocks by Handin and others (1963) and on crystalline rocks by Brace and Martin (1968) show that if the sample is "drained," Terzaghi's principle of effective stress should apply to fracture (Jaeger and Cook 1979). If pore fluid diffusion is relatively slow (so that pore pressure is no longer uniform), it is necessary to take into account microcavity deformation and permeability. Theories for such so-called poroelastic behavior have been developed primarily in petroleum engineering (Biot, 1941) and have been reviewed by Rice and Cleary (1976; see

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analyzed the wedge penetration process as a quarter-space loading problem, where the free surface and the vertical splitting delineate the quarter-space boundaries as shown in Figure 4.7. The normal and frictional loading of the interface between the rock and the wedge is then considered in detail, the stress distribution in the rock is calculated, and the Coulomb-Mohr criterion is applied to determine the point at which fracturing will begin. The growth direction of the chip was assumed to follow the steepest slope of the Coulomb-Mohr function. This is clearly somewhat arbitrary; the generalized slip line field approach discussed above probably should have been used instead. Other Applications of Rock Fracture Other applications of rock fracturing are of industrial interest. Hydraulic fracturing is of great interest in the improvement of oil recovery from oil-bearing rock beds. Here the problem is one of tensile fracturing of rock under a large wellbore pressure and the opposing confining pressures of the surrounding rock, made complicated by the flow processes of pore fluids. Problems of mine shaft failures and lateral displacement of large earthworks such as dams and embankments are direct applications of compressive fracturing of rock and soil flow, both obeying the general framework of shear faulting theories. The framework for the solution to such problems exists, and solutions will be urgently required in the context of the development of the smart drilling process. Priorities for R&D The rate-controlling process in drilling is, and will likely be for the foreseeable future, the rate of rock removal by the drill bit. Increasing the efficiency of the rock removal rate requires a fundamental understanding of how solid rock is fragmented by drill bits and the numerous factors that control this very inhomogeneous and local process. There are several areas in which additional research would be most beneficial in improving rock removal, particularly in the context of the smart drilling system elucidated in Chapter 2. These are broadly the following:

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In situ characterization of rock: The chemical constitution (including mineralogy and petrology), physical characteristics, microstructure, and nature and distribution of preexisting flaws (e.g., faults, joints, and bedding planes) of rock have been studied in great detail for many decades. There have been equally detailed laboratory studies of the mechanical properties of rock over a wide range of temperature and pressure to identify mechanisms for drilling and excavation. In any new thrust to capitalize on the potential of the smart drilling approach, it is essential to tailor the response of the drill to the local rock environment (e.g., temperature, pressure, fracture density, and resistance). This could be achieved if the relevant mechanical properties of the rock could be assessed in situ by making meaningful measurements of its chemical, mineralogical, elastic, and acoustic properties, and if these measurements could be used to make adjustments to the drill bit in order to optimize the rate of rock removal. Fracture processes: For drilling applications, the relevant rock behavior is purely elastic, with brittle fracture occurring under the drill bit. However, conditions under the drill bit differ significantly from the quasi-homogeneous conditions in the Earth's crust or in typical laboratory experiments. The stress field under the drill bit is highly inhomogeneous over volume elements much smaller than those encountered in the usual laboratory experiments, where rock fracture is subject to the strength-size effects discussed previously. Presently, there is inadequate information on the nature of this inhomogeneous fracture process, both qualitatively and quantitatively. The fracture process should be studied in greater detail to identify the specific factors affecting both the local driving forces of fracture and the response of the rock to these forces. These processes should be modeled to obtain useful scaling laws for drilling practice. Matching the drilling process to the physical environment: The requirements for effective drilling vary for different environments, for example, near-surface, large-scale excavation; porous and fluid-bearing rock; dense rock in the upper crust; and fractured, hot rock in geothermal reservoirs. The physical conditions and mechanical properties of rocks are well known in these environments, but there is inadequate information on how to optimize the drilling process to best suit these different requirements. 

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Physicochemical effects on rock fracture and tool wear: A potentially important avenue in the improvement of drilling practice involves the physicochemical aspects of the mechanical response of rock in surface-active fluid environments, to take advantage of the so-called Rehbinder effects. Every brittle fracture process is ultimately based on the severance of chemical bonds. There are many examples of dramatic reduction of the local work of brittle separation in active liquid environments. Although the mechanistic details of these effects are not fully understood, they are often well characterized, and some have been tried with some success in drilling. Because of their attractive potential, these effects should be given special consideration. Unconventional methods for rock removal: Rock is a brittle solid with a characteristic set of strength-impairing imperfections such as pores, weak boundaries, and microcracks. The tensile strength of unconfined rock is generally low, approaching at best 50 to 100 MPa. When confined, or under pressure, as is the case at great depths in the Earth's crust, these imperfections are rendered ineffective by being pressed shut. Under these conditions, the compressive strength of rock is high and increases with increasing pressure. Thus, drilling rock at great depth by subjecting it locally to greater pressure is probably an inefficient means of removing material. Efforts should be made to identify unconventional forms of material removal that capitalize on the low tensile strength of rocks.   A large number of unconventional methods of rock removal have been studied. When compared with mechanical drilling, these techniques have so far been found to be quite ineffective, as discussed in more detail in the next chapter. Nevertheless, the existing information on them should be reevaluated to reach firm quantitative conclusions concerning their potential, particularly for use in conjunction with mechanical drilling.  References Altiero, N. J., and Sikarskie, D. L., 1974, Fracture initiation and propagation in elastic-brittle materials subjected to compressive stress fields—An experimental study: Mechanical Research Communication, v. 1 (4), p. 225-231.

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Anand, L., 1983, Plane deformations of ideal granular materials: Journal of the Mechanics and Physics of Solids, v. 31 (2), p. 105-122. Ashby, M. F., and Hallam, S. D., 1986, The failure of brittle solids containing small cracks under compressive stress states: Acta Metallurgica, v. 34 (3), p. 497-510. Atkinson, B. K., 1982, Subcritical crack propagation in rocks: theory, experimental results and applications: Journal of Structural Geology, v. 4 (1), p. 41-56. Atkinson, C., and Craster, R. V., 1991, Plane strain fracture in poroelastic media: Proceedings of the Royal Society, v. A434, p. 605-663. Babel, H. W., and Sines, G., 1968, A biaxial fracture criterion for porous brittle materials: ASME, Journal of Basic Engineering, ser. D, v. 90 (2), p. 285-291. Bailey, C. D., Hamilton, J. M., Jr., and Pless, W. M., 1979, Acoustic emission of impact-damaged graphite-epoxy composites: Materials Evaluation, v. 37 (6), p. 43-48. Bazant, Z. P., and Kim, S. S., 1979, Plastic fracturing theory for concrete: Journal of Engineering Mechanics, v. 105 (3), p. 407-428. Bazant, Z. P., and Tsubaki, T., 1980, Total strain theory and pathdependence of concrete: Journal of Engineering Mechanics, v. 106 (6), p. 1151-1173. Benjumea, R., and Sikarskie, D. L., 1969, A note on the penetration of a rigid wedge into a nonisotropic brittle material: International Journal of Rock Mechanics and Mineral Science, v. 6 (4), p. 343-352. Biot, M. A., 1941, General theory of three-dimensional consolidation: Journal of Applied Physics, v. 12 (2), p. 155-164. Brace, W. F., and Bombolakis, E. G., 1963, A note on brittle crack growth in compression: Journal of Geophysical Research, v. 68 (12), p. 3709-3713. Brace, W. F., and Martin, R. J., III, 1968, A test of the law of effective stress for crystalline rocks of low porosity: International Journal of Rock Mechanics and Mineral Science, v. 5 (5), p. 415-426. Budiansky, B., 1965, On the elastic moduli of some heterogeneous materials: Journal Mechanics and Physics of Solids, v. 13 (4), p. 223-227. Budiansky, B., and O'Connell, R. J., 1976, Elastic moduli of dry and saturated cracked solids: International Journal of Solids Structures, v. 12 (1), p. 81-97.

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Byerlee, J. D., and Lockner, D., 1977, Acoustic emission during fluid injection into rock: Proceedings of the First Conference on Acoustic Emission/Microseismic Activity in Geological Structures and Materials, Handy, H. R., Jr., and Leighton, F. W., eds. Clausthal, FRG, Trans. Tech., p. 87-98. Cheatham, J. B., and Sikarskie, D. L., 1973, Penetration problems in rock mechanics: Annual ASME Applied Mechanics Division of Rock Mechanics Winter Symposium, Sikarskie, D. L., ed., New York, American Society of Mechanical Engineers, v. 3, p. 41-71. Chow, T. S., 1978, Effect of particle shape at finite concentration on the elastic moduli of filled polymers: Journal of Polymer Science (Polymer Physics), v. 16 (6), p. 959-970. Cleary, M. P., Chen, I.-W., and Lee, S.-M., 1980, Self consistent techniques for heterogeneous media : Journal of Engineering Mechanics, v. 106 (5), p. 861-887. Cooper, G. A., and Bernie, J., 1978, Comments on influence of chemomechanically active fluids on diamond wear driving hard rock drilling: Journal of Materials Science, v. 13 (12), p. 2716-2723. Cooper, G. A., 1979, Some observations on environmental effects when diamond drilling: inThe Science of Ceramic Machining and Surface Finishing II, Hockey, B. J., and Rice, W. R., eds., NBS Special Publication 562, Washington, D.C., U.S. Government Printing Office, p. 115-138. Detournay, E., and Cheng, A. H.-D., 1991, Plane strain analysis of a stationary hydraulic fracture in a poroelastic medium: International Journal of Solids and Structures, v. 27 (13), p. 1645-1662. Drescher, A., Kwaszczynska, K., and Mroz, Z., 1967, Statics and kinematics of granular medium in case of wedge indentation: Archiwum Mechaniki Stosowanej, v. 19, p. 99. Drucker, D. C., and Prager, W., 1952, Soil mechanics and plastic analysis or limit design: Quarterly of Applied Mathematics, v. 10, p. 157-166. Dyskin, A. V., and Germanovich, L. N., 1993, Model of rock burst caused by cracks growing near free surface: Rockbursts and Seismicity in Mines '93, Young, P., ed., Rotterdam, Balkema, p. 169-175. Dyskin, A. V., Germanovich, L. N., and Ustinov, K. B., 1993, A 2-D model of skin rock burst and its application to rock burst monitoring: in Geotechnical Instrumentation in Open Pit and Underground Mining, Szwedzicki, T., ed., Rottterdam, Balkema, p. 133-141.

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Erdogan, F., and Sih, G. S., 1963, On the crack extension in plates under plan loading and transverse shear: ASME Journal of Basic Engineering, ser. D, v. 85 (4), p. 519-527. Evans, I., and Murrell, S. A. F., 1958, Mechanical Properties of Nonmetallic Brittle Materials: New York, Interscience, 432 pp. Ewy, R. T., and Cook, N. G. W., 1990a, Deformation and fracture around cylindrical openings in rock—I. Observations and analysis of deformations: International Journal of Rock Mechanics, Mineral Science, and Geomechanical Abstracts, v. 27 (5), p. 387-407. Ewy, R. T., and Cook, N. G. W., 1990b, Deformation and fracture around cylindrical openings in rock—II. Initiation, growth and interaction of fractures: International Journal of Rock Mechanics, Mineral Science, and Geomechanical Abstracts, v. 27 (5), p. 409-427. Fredrich, J. T., Evans, B., and Wong, T.-F., 1989, Micromechanics of the brittle to plastic transition in Carrara marble: Journal of Geophysical Research, v. 94 (B4), p. 4129-4145. Germanovich, L. N., and Cherepanov, G. P., 1987, Fracture criteria for material with defects: Applied Mathematics and Mechanics, v. 51, p. 256-264. Goodman, R. E., 1980, Introduction to Rock Mechanics: New York, J. Wiley and Sons, 475 pp. Griffith, A. A., 1920, The phenomena of rupture and flow in solids: Philosophical Transactions of the Royal Society, v. A221, p. 163-198. Griffith, A. A., 1924, Theory of rupture: Proceedings of the 1st International Conference on Applied Mechanics, Delft, Holland, p. 55-63. Handin, J., Higgs, D. V., and O'Brien, J. K., 1960, Torsion of Yule marble under confining pressure: inRock Deformation, Griggs, D. T., and Handin, J., eds., GSA Memoir No. 79, Boulder, Colo., Geological Society of America, p. 245-274. Handin, J., Hager, V., Friedman, M., and Feather, J. N., 1963, Experimental deformation of sedimentary rocks under confining pressure: pore pressure tests: Bulletin of the American Association of Petroleum Geologists, v. 47 (5), p. 717-751. Hartman, H. L., 1959, Basic studies of petroleum drilling: Transactions AIME, v. 24 (1), p. 68.

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