Estimation Criteria for Rock Brittleness Based on Energy Analysis During the Rupturing Process
Brittleness is one of the most important mechanical properties of rock: it plays a significant role in evaluating the risk of rock bursts and in analysis of borehole-wall stability during shale gas development. Brittleness is also a critical parameter in the design of hydraulic fracturing. However, there is still no widely accepted definition of the concept of brittleness in rock mechanics. Although many criteria have been proposed to characterize rock brittleness, their applicability and reliability have yet to be verified. In this paper, the brittleness of rock under compression is defined as the ability of a rock to accumulate elastic energy during the pre-peak stage and to self-sustain fracture propagation in the post-peak stage. This ability is related to three types of energy: fracture energy, post-peak released energy and pre-peak dissipation energy. New brittleness evaluation indices B1 and B2 are proposed based on the stress–strain curve from the viewpoint of energy. The new indices can describe the entire transition of rock from absolute plasticity to absolute brittleness. In addition, the brittle characteristics reflected by other brittleness indices can be described, and the calculation results of B1 and B2 are continuous and monotonic. Triaxial compression tests on different types of rock were carried out under different confining pressures. Based on B1 and B2, the brittleness of different rocks shows different trends with rising confining pressure. The brittleness of red sandstone decreases with increasing confining pressure, whereas for black shale it initially increases and then decreases in a certain range of confining pressure. Granite displays a constant increasing trend. The brittleness anisotropy of black shale is discussed. The smaller the angle between the loading direction and the bedding plane, the greater the brittleness. The calculation B1 and B2 requires experimental data, and the values of these two indices represent only relative brittleness under certain conditions. In field operations, both the relative brittleness and the brittleness obtained from seismic data or mineral composition should be considered to gain a more comprehensive knowledge of the brittleness of rock material.
KeywordsBrittleness Type II rock behavior Fracture mechanism Energy change Anisotropy of brittleness
List of Symbols
- VQuartz, VCarbonate, VClay
The content of quartz, carbonate and clay content, respectively (Eq. 1)
The weight coefficient of each mineral (Eq. 2)
The types of brittle minerals (Eq. 2)
The types of all minerals (Eq. 2)
The mineral content (volume fraction) (Eq. 2)
- σc, σt
Uniaxial compressive strength and tensile strength of rock material (Eq. 3)
Initiation stress of rock material (Eq. 4)
- E, v
The elastic modulus and Poisson’s ratio based on the seismic data (Eq.5)
- Emax, Emin
The maximum and minimum values of the Elastic modulus (Eq. 5)
- vmax, vmin
The maximum and minimum values of the Poisson’s ratio (Eq. 5)
- εel, εpl
The elastic strain and plastic strain at the pre-peak stage of stress–strain curves (Eq. 6)
Total strain of pre-peak stage (Eq. 6)
- σp, σr
The peak strength and the residual strength of the whole stress–strain curve (Eq. 7)
- εp, εr
The peak strain and the residual strain of the whole stress–strain curve (Eq. 7)
The fracture energy (Eq. 8)
The unloading elastic energy (Eq. 8)
The dissipation energy of pre-peak stage (Eq. 8)
The extra energy required (type I behavior) or the excess energy released(type II behavior) (Eq. 9)
The total elastic energy accumulated in the rock specimen when reaching the peak strength (Eq. 10)
The energy which unconsumed or converts into other forms (Eq. 10)
The peak strength of the rock specimen under compression (Eq. 10)
The residual strength; σi represents the function of pre-peak curve (Eq. 10)
Elastic modulus of stress–strain curve (Eq. 11)
Post-peak modulus of stress–strain curve (Eq. 11)
The function of pre-peak curve (Eq. 13)
The strain corresponding to the peak strength (Eq. 13)
In the development of shale gas, the brittleness of the rock plays a significant role in the stability of the borehole wall and is also key in selection of high-quality shale reservoirs and the design of the hydraulic fracturing scale. Rock burst phenomena are also closely related to rock brittleness, because it is a crucial parameter to judge whether a rock burst occurs and the likelihood of its occurrence. Therefore, the brittleness of rock material in these applications is an indispensable factor that must be considered in deep rock engineering and the development of unconventional resources.
At present, there is still no widely accepted definition of brittleness in related fields. Morley (1944) and Hetenyi (1966) defined brittleness as the loss of plasticity of materials. Ramsay (1967) argued that when the cohesion of rock was destroyed, the material exhibited brittle failure characteristics. Obert and Duvall (1967) believed that brittleness was a feature describing the failure behavior of rock materials when the yield strength of rock is reached or exceeded. George (1995) defined brittleness as the ability of rock to deform continuously without producing permanent deformation when the rock material is subject to sufficient stress to produce micro-cracks. Goktan and Yilmaz (2005) defined rock brittleness as a rupture tendency without noticeable deformation under low stress. Li et al. (2012) believed that brittleness was a comprehensive property of rock materials: the ability to generate local damage and to develop spatial fractures under an internal non-uniform stress distribution caused by the inherent heterogeneity of the rock. Some evaluation indices to characterize rock brittleness have been proposed. Honda and Sanada (1956) put forward an index in terms of the difference of hardness and firmness to characterize the material brittleness. Jarvie et al. (2007) and Rickman et al. (2008) established a set of indices to represent the brittleness of rock based on the percentage of the contents of brittle minerals. Bishop (1967) thought that the brittleness of rock materials should be obtained directly from mechanical tests. Altindag and Guney (2010) discussed the relationship between rock brittleness and strength: they managed to characterize the brittleness of rock using the tensile strength and compressive strength. Based on the stress–strain curves obtained from rock compression tests, Hajiabdolmajid and Kaiser (2003) suggested defining brittleness in terms of the peak strain and residual strain. Tarasov and Potvinb (2013) obtained the complete stress–strain curve of rock by triaxial compression tests and established a corresponding brittleness index based on the energy balance of the post-peak stage of the curve. They believed that the rock brittleness obtained from compression tests is the ability of rock to maintain macroscopic damage in the post-peak stage. Most of these indices were proposed for specific issues applicable to different subjects, but their calculated results are not continuous and monotonic. From close analysis of previous research, a scientific and applicable rock brittleness index should possess the following features: an adequate physical basis; the capability of describing the entire range of rock behavior from absolute plasticity to absolute brittleness; and the ability to measure rock brittleness monotonically and continuously.
A reasonable index should fully consider the rupture process as a whole. In this paper, we propose two brittleness indices B1 and B2 based on the analysis of the energy transformation of pre-peak and the post-peak stages of the stress–strain curve. These indices are able to describe both the energy transformation of brittle rupture and the entire scope of rock behavior from absolute plasticity to absolute brittleness monotonically and continuously. The changing pattern of rock brittleness is analyzed based on triaxial compression tests on red sandstone, black shale and granite under different confining pressures. We further discuss the effects of the anisotropic characteristics of black shale on brittleness. The results indicate that these two indices work better than previously defined indices to describe the brittleness of different rock materials, especially for anisotropic rock materials.
2 Evaluation Indices of Rock Brittleness
2.1 Evaluation Indices Based on Mineral Composition
The mineral composition can significantly influence the mechanical properties of rock materials, so there should be a direct correlation between mineral composition and brittleness. Jarvie et al. (2007) believed that the content of quartz in rock material could affect rock brittleness, and so defined b1 as the content of quartz in rock to calculate rock brittleness. Rickman et al. (2008) analyzed the mineral composition of the Barnett Shale using X-ray diffraction and laser induced breakdown spectral (LIBS). The results proved that rock brittleness was positively proportional to the content of quartz and inversely proportional to the content of clay. Brittleness also changed within a moderate range with an increase in the carbonate content. Buller et al. (2010) suggested using the ratio of brittle minerals to the total amount of minerals to describe rock brittleness. However, evaluation indices based on the brittle mineral composition do not consider rock diagenesis, which has a great influence on brittleness. The brittleness of rock materials with similar mineral compositions that have experienced different diagenetic processes may differ substantially. Moreover, there is still no universal standard for the weight of each brittle mineral. Despite their simplicity and convenience, brittleness indices based on mineral composition lack a physical basis and may well yield contradictory results.
2.2 Evaluation Indices Based on Mechanical Parameters
Parameters such as the elastic modulus, Poisson’s ratio and mineral compositions can be conveniently obtained from field data, and field evaluation of shale brittleness has been conducted by combining these two indices (Perez and Marfurt 2013; Sun et al. 2013; Yang et al. 2013; Liu et al. 2014). However, Jin et al. (2014) believed that brittleness indices based on a combination of elastic parameters and mineral composition could not serve as a design standard for hydraulic fracturing. Huang et al. (2015) analyzed the brittleness sensitivity of anisotropic elastic parameters and established a new brittleness index based on the elastic parameters; however, this index disregards the rock fracture mechanism and its physical basis.
2.3 Evaluation Indices Based on Rock Stress–Strain Curves
Li et al. (2012) redefined the brittleness of shale based on fracture characteristics and the mechanism of brittle fracture. They found that: (1) brittleness was an integrated property of rock material that was influenced by the material heterogeneity and the external conditions of the test; (2) the characteristics of stress–strain curves during the pre-peak and post-peak stages were key to characterizing shale brittleness and (3) the ability to resist inelastic deformation before failure and the rate of loss of bearing capacity after failure were the main features of the mechanical behavior of shale brittleness. Zhou et al. (2014a, b) studied the brittleness of rock based on triaxial compression tests on different rocks. The test results showed that the yield platform appeared in the stress–strain curves of these rock specimens with lower brittleness before the peak strength, and the brittle behavior of a rock specimen was related to the characteristic of the post-peak stage of the stress–strain curve. Therefore, the characteristics of both the pre-peak and post-peak stage of the stress–strain curves should be combined to gain a more comprehensive understanding of rock brittleness.
3 Energy During the Rock Failure Process
3.1 Energy Change Before Peak Strength
During increasing loading, the deformation of rock initially rises and later transforms from elastic to plastic deformation. Subsequently, internal damage within the rock begins to appear, after which the damage zone expands and the number of new micro-cracks gradually increases. This process dissipates a portion of the energy. Before the peak strength, most of the absorbed energy is stored in the form of strain energy in the specimen. During the beginning stage of loading, most of the strain energy absorbed by the rock specimen is transformed into internal elastic strain energy and a small amount is dissipated. After the compaction phase, the dissipation energy increases little. When the peak strength is about to be reached, the dissipation energy begins to rise. In Fig. 2, point A is the peak stress and dWe (red) represents the elastic energy accumulated in the rock material before the peak stress, which is the energy source for and physical basis of rock fracture and failure. dWd (green) represents the dissipation energy of the pre-peak stage. Many previous studies have shown that the more gentle the slope of the yield platform is, the larger the green area, the greater the dissipation energy before failure and the stronger the plasticity of the rock. Zhou et al. (2014a, b) carried out a series of triaxial compression tests on granite under different confining pressures and discussed the brittle fracture mechanism of granite. The results indicated that the extent of plastic deformation was an important factor in whether brittle failure occurred. Wang et al. (2014) studied the energy change during the brittle fracture process. They found that the smaller the proportion of the plastic deformation is, the smaller the dissipation energy before the peak and more easily brittle fracture could occur. All these studies have proved that rock brittleness is closely related to the energy during the pre-peak stage of the stress–strain curve, which, in terms of energy, explains the physical meaning of the brittleness index b10.
3.2 Energy During the Post-peak Stage
Under compression, the stress–strain curves of the post-peak stage may show two different behaviors: type I and type II behavior. He et al. (1990) proved the existence of type II behavior of rock using a spring model. They thought that failure localization accounted for the type II behavior of rock. Pan et al. (1999) pointed out that the lower the value of the softening modulus, the more brittle the rock and the more likely type II behavior of rock is to occur. As illustrated in Fig. 2, the post-peak stage of the stress–strain curve is described by the post-peak modulus M (M = dσ/dε). The blue and gray areas represent, respectively, the rock fracture energy (dWf) and the unconsumed portion of energy (dWt) after failure of the rock specimen. For type I rock behavior (Fig. 2a), the post-peak modulus is negative, which means that the elastic energy accumulated in the rock material is not sufficient to maintain the entire fracture process (the red area is smaller than the blue area). Loading must continue to generate additional energy for failure of the rock to occur (the area defined by the yellow dotted line in the diagram). For type II rock behavior (Fig. 2b), the post-peak modulus is positive. The elastic energy accumulated in the rock specimen is sufficient to maintain the whole failure of the rock material (the red area is larger than the blue area), and rock failure is accompanied by some energy release (yellow area). In this case, the rock displays self-sustaining fracturing. Zhang et al. (2010) proved that rock materials with higher brittleness would enter the non-quasi-static and self-sustained deformation state during the post-peak stage, based on triaxial compression tests of granite. Tarasov and Potvinb (2013) also demonstrated that the ability of rocks to sustain macroscopic fracturing during the post-peak stage was a salient brittle feature of rock materials. The mechanisms of both types of behavior of rock materials are analyzed in detail below.
When the shear rupture in the fracture head appears, an array of short tensile cracks along the future fracture plane will be formed in front of the crack tips (Reches and Lockner 1994; Reches 1999). As illustrated in Fig. 3b, this forms the general structure of the shear fracture represented by an echelon of blocks separated by tensile cracks, which is generally called the “domino” structure or Ortlepp shears (King and Sammis 1992; Ortlepp 1997; Aswegen 2008). The displacement of the fracture plane will lead to rotation of these domino blocks along the fault. Figure 3b displays the distribution of shear resistance along the fracture head. For rocks with type I behavior (Fig. 3b, left), some blocks are formed in front of the fracture head, providing a significant amount of resistance of the rock specimen to shear rupture. However, as fracture propagation continues, the blocks collapse with rotation and break into smaller pieces, leading to a gradual shift of shear resistance from cohesion to friction. The collapse and crushing of blocks within the fracture head will absorb large amounts of energy because the development of shear fractures requires displacement of the rock specimen along the fracture plane. However, rocks with type II (Fig. 3b, right) behavior have different characteristics: the rotating blocks can withstand rotation without being crushed. The consecutive formation and rotation of the blocks in type II forms a fan-shaped structure in the fracture head. In the left-hand part of the fan-shaped structure encircled by green lines, the rotation of the blocks under the effect of normal stress will provide an active force, which is advantageous for maintaining the crack propagation process. Figure 3c shows the variation of shear resistance at the fracture head for both types of rock behavior. The blue part of the graph represents the active forces (negative resistance) generated by the fan-shaped area, which is favorable for extension and propagation of the fracture head. The blocks in the core frictional zone have completed their rotation, and the friction they are now providing becomes the normal residual one. The fan-shaped structure under active force can move spontaneously as a wave with very small shear resistance: this is known as the self-balancing mechanism. The resistance in the structure to fracture propagation depends on the tensile strength of the material, because the tensile strength is closely related to the consecutive formation of blocks in front of the fracture head. This structure possesses a very energy-efficient shear fracture mechanism because it generates a very small amount of shear resistance during the whole fracture process.
The type I stress–strain curve in Fig. 3d is the typical classic mode of rock shear behavior, which is unlike type II as illustrated in the right curve in Fig. 3d. The bearing capacity of the specimen during stage 3 can be lower than that during stage 4 because of the fact that the aggregate shear resistance of both sides of the fan-shaped structure may be close to zero, which means that fracture propagation fracture will not require a relatively large sum of external load and thus decreases the bearing capacity of the specimen. Therefore, the specimen will be more likely to experience a sudden decrease in axial stress. The greater the number of dominos involved, the greater the sudden decrease in axial stress will be, so the longer the length of zone A, the smaller the shear resistance at stage 3. This mechanism can be adopted to explain the extremely brittle behavior of rock material.
The fracture mechanism for different rock materials differs greatly in terms of energy change during the fracture process and the brittleness behavior. Figure 2 illustrates the dynamic process during post-peak stage of energy transformation from elastic energy accumulated in the rock material into fracture energy; the figure also displays the energy balance of the whole fracturing process. Point A represents the peak stress; the specimen fractures completely at point B. The blue area in the post-peak stage represents the fracture energy, the gray area represents the remaining energy and the yellow region denotes energy that has been converted into other forms. When the rock material goes beyond the peak stress, the elastic energy accumulated in the rock specimen (dWe) turns into fracture energy (dWf). For rock with type I behavior, dWe is not sufficient to maintain the whole fracture process of rock material, so loading must be continued to sustain fracture propagation, as indicated by the area with dashed yellow outline. However, the self-equilibrating mechanism endows type II rock behavior with extremely small fracture energy (blue area in Fig. 2). The elastic energy accumulated in the rock specimen is adequate to maintain the entire fracture process, indicating that rocks with type II behavior are close to absolute brittleness and that self-sustaining fracturing is a characteristic of brittle fracturing of rock material. This mechanism can better explain the rock burst phenomenon of brittle granite and the instant formation of fractures during hydraulic fracturing of shale. Additionally, the mineral composition and structure of the rock are key factors in the mechanism of different types of rock fracture.
4 Rock Brittleness Indices Based on Energy
On the basis of brittleness indices B1 and B2, quantitative descriptions of absolute ductility, absolute brittleness and the scale of brittleness between them can be obtained, which are the absolute ductility, ductility, transitional section, weak brittleness, medium brittleness, strong brittleness and absolute brittleness, respectively. In the curve of absolute ductility, the pre-peak dissipation energy and post-peak fracture energy tend toward infinity (leftmost illustration in Fig. 4). In macroscopic view, this feature represents continuous, permanent elastic deformation of the specimen under external loading. The rightmost illustration in Fig. 4 shows the curve of absolute brittleness. There is no dissipation energy and fracture energy in the whole curve, which indicates that the specimen experiences complete elastic deformation during the pre-peak stage and the fracturing of the specimen is completely self-sustaining during the post-peak stage. For rock material with a brittleness level between ductility and medium brittleness (dWx = 0), dWx denotes the additional energy required to sustain fracturing of the specimen, whereas for rock material with brittleness that is stronger than medium brittleness, dWx denotes the extra energy released after the peak. We also find that the greater the proportion of the energy dWx, the closer the rock material is to absolute brittleness. Moreover, the transformation from ductility to brittleness is accompanied by a change in the rock behavior from type I (blue rectangle) to type II (pink rectangle). The ranges of B1 and B2 are (0, +∞) and (−∞, 1), respectively. The transition is continuous and monotonic, meaning that the brittleness indices proposed in this paper can appropriately describe the whole process and the degree of embrittlement of rock materials. In addition, the brittleness indices proposed herein also possess some characteristics present in other brittleness indices, such as the variation of yield platform (b10) and the amount of decrease and rate of post-peak stress (b11 and b12).
5 Experimental Analysis of Brittleness Indices B1 and B2
5.1 Experimental Procedure
5.2 Petrographic Characteristics and Basic Physico-mechanical Properties of the Tested Rock Materials
The red sandstone specimens were obtained from a Cretaceous red sandstone sequence in the Sichuan Basin, China, and are muddy fine sandstone. The porosity of the red sandstone specimens is 8.45–14.21 %, average 11.53 %. The granite specimens are gray porphyritic coarse-grained granite from the Yanbian area of Jilin Province, China. The particle size is 0.2–7 mm, and the rock is massive in structure. The porosity of the granite specimens is 0.56 %–0.84 %, average 0.72 %. The black shale specimens are from the Longmaxi Formation of Sichuan Province, China. The Longmaxi Formation is a lower Paleozoic marine shale reservoir with large thickness. The lithofacies of the black shale is silica shale, and the rock contains obvious bedding planes. The black shale has an average porosity of 5.17 % and possesses both low porosity and low permeability. The black shale has a high organic carbon content (total organic carbon 1.3–7.4 %, average 4.52 %), and a vitrinite reflectance (R0) of 2.0–4.77 %. The black shale is highly mature to over-mature.
Data of the triaxial compression test on red sandstone, granite and black shale
Compressing pressure (MPa)
Peak strength (MPa)
Residual strength (MPa)
Average elastic modulus (GPa)
Average Poisson’s ratio
5.3 Effect of Confining Pressure on Rock Brittleness
5.4 Analysis of the Morphology of Rock Fracture Surfaces
Energy transformation accompanies the entire process of rock deformation and failure. When rock material is subjected to an external force, mechanical energy will be constantly conveyed into the rock. Part of the mechanical energy is converted into elastic energy accumulated in the rock; the rest is dissipated as damage and plastic energy. With increasing loading, microcracks begin to appear inside the rock material. After reaching the failure strength, the internal elastic energy will be converted into the kinetic energy of rock fragmentation or other forms of energy. There are three major factors affecting whether and to what extent brittle failure occurs. The brittleness and strength of minerals play a key role in determining the specific surface energy and shear fracture energy required by microcracks to initiate and extend inside the grain particles in the form of tensile rupture or shear rupture. In contrast, the structure of the grain particles will determine the failure patterns of the microcracks, which may be transgranular fracture, intergranular fracture or fracture along the joints of microcracks. To analyze further the specific relations between surface morphology and brittleness, we perform both microscopic observations and a macroscopic analysis.
5.5 Anisotropy of Shale Brittleness
In previous studies, it was thought that brittleness was an inherent, invariant property of rock materials based on seismic data or mineral composition. However, for anisotropic natural shale, the characteristics of elastic deformation and brittle fracturing are different in different directions; therefore, the brittleness of shale in different directions should also be anisotropic. Luan et al. (2014) thought that the elastic parameters of anisotropic rocks are also anisotropic, and discussed the anisotropy of shale brittleness based on the brittleness index b10. When the loading direction is parallel to the bedding plane, the brittleness of the specimen is markedly higher than the brittleness under vertical conditions. This result was further verified by Holt et al. (2015) using triaxial compression tests, which demonstrated that the increase in rock brittleness was accompanied by the occurrence of multiple longitudinal splitting fractures, and when the bedding plane and the fracture surface share the same direction, the brittleness of the anisotropic rock material was highest. In this study, we carry out a series of triaxial compression tests on black shale under a confining pressure of 60 MPa in five directions: α = 0°, 15°, 45°, 75° and 90°. The angle α is the included angle between the bedding plane and the horizontal. We discuss the anisotropy of black shale brittleness in detail and verify that the indices established herein can be used to demonstrate the brittleness anisotropy.
6 Comparison of Indices B1 and B2 with Other Brittleness Indices
- 1.First, the brittleness of three rock materials under no confining pressure are calculated using indices b3–b5. The brittleness of granite, as calculated by b3, is weaker than that of black shale, indicating that the granite is a weak-brittleness rock material (Table 2). This result and the high brittleness characteristics of granite observed in the test are contradictory. For the brittleness results from b4 and b5, the red sandstone is more brittle than shale and is super-strong brittle rock, in contradiction to the experimental results. Because of the lack of data on the tensile strength of rock under different confining pressures, the sensitivity of b3–b5 to confining pressure cannot be obtained. However, a large number of previous studies have demonstrated that, compared with tensile strength, the compressive strength of rock material is more sensitive to confining pressure, i.e., when the confining pressure increases, the compressive strength increases more. In this case, the brittleness gradually increases. But we found in the experiment that the brittleness of red sandstone decreases with increasing confining pressure, whereas the brittleness of black shale first increases and then decreases with increasing confining pressure. This analysis shows that, because of the inadequate sufficient physical basis of b3–b5, these indices are not unambiguous for characterizing the brittleness of rock.Table 2
The brittleness of red sandstone, granite and black shale calculated by b3–b5
Super high brittle
Super high brittle
- 2.Indices b6–b8 are modified by the crack initiation stress σci with respect to b3–b5, and so can be used to calculate rock brittleness under different confining pressures. The results are provided in Table 3. With increasing confining pressure, the patterns of variation of the three kinds of rock material for b6 are not clear, and the brittleness levels of the three kinds of rocks under different confining pressure are mostly brittle. This is caused by the large influence of error of σci on the calculation results, and it is difficult to obtain a relatively accurate crack initiation stress from the experimental data. The b7–b8 values of the three rock types increase with rising confining pressure, indicating that the rock brittleness increases with confining pressure, which is not in conformity with the experimental results. The calculated results of b7–b8 indicate that the red sandstone is a rock material with super-strong brittleness under high confining pressure, contrary to the facts. As for b3–b5, indices b6–b8 also lack a sufficient physical basis, leading to their inability to characterize the brittleness of different rock materials unambiguously.Table 3
The brittleness of red sandstone, granite and black shale calculated by b6–b8
Compressing pressure (MPa)
Peak strength (MPa)
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
Super high brittle
- 3.We also calculated the brittleness of tested rock materials using indices b10 and b11, as illustrated in Fig. 19. When the confining pressure is 30 MPa, the results of b10 for red sandstone and black shale are almost identical, indicating that their brittleness, as defined by the strain characteristics in the pre-peak stage, is the same; however, the values of their post-peak brittleness index b11 are significantly different. Similarly, when the confining pressure is 60 MPa, the granite and the red sandstone have the same post-peak brittleness index b11, but their pre-peak brittleness index b10 is markedly different. These two cases indicate that two curves with the same stress or strain characteristics during the pre-peak stage may have different brittleness characteristics in the post-peak stage, so self-contradictory rock brittleness results may be obtained. Figure 20 displays the calculated results of b12: the variation in the brittleness of the three rock types with confining pressure is similar to the experimental results and the calculation results for B1 and B2. This is because the difference between the peak strain and the residual strain can reflect the magnitude of rupture energy of the post-peak stage. However, for confining pressures of 90 and 120 MPa, the b12 values of red sandstone are 0.641 and 0.648, respectively. This result indicates that the brittleness of red sandstone exhibits only a very small decrease with increasing pressure, but the test results show that when the confining pressure increases from 90 to 120 MPa, the brittleness experiences a relatively large reduction. This is because b12 can only characterize the brittleness of the post-peak stage, and ignores the pre-peak brittleness change. Brittle failure of rock is the result of the combined action of energy dissipation before the peak and energy release after the peak; therefore, the rock brittleness cannot be unambiguously characterized using only pre-peak stage or post-peak stage parameters.
Unlike some mechanical parameters that represent only a single aspect of rock behavior, such as the elastic modulus and Poisson’s ratio, brittleness is an integrated description of rock mechanical behavior. There are currently two ways to obtain rock brittleness: laboratory tests and well-logging data. The brittleness values obtained through tests are relative ones and measured under specific loading conditions and are related to the experimental conditions and properties of the rock material such as size, heterogeneity and anisotropy. In contrast, the brittleness values from logging data are obtained by just a single-step calculation that treats brittleness as a constant and invariant property regardless of the in situ conditions or mechanical properties of the rock material. For example, the brittleness calculated using the previously defined indices b1 and b2 neglected the effects of bedding planes and the anisotropy of rock materials. Diao (2013) performed a comparative analysis of two evaluation methods that consider rock mechanics and mineral composition; by combining these two methods with their experimental studies, they believed that the evaluation of rock brittleness was greatly limited when calculated using a single method under specific conditions. Therefore, they combined the two methods mentioned above to determine the brittleness of rock materials. Liu and Sun (2015) obtained similar results. In this study, the indices based on energy analysis of the stress–strain curves can properly rate the sensitivity of brittleness to confining pressure and the anisotropy of rock material. In addition, the physical meaning of the rupture mechanism of rock materials can be better revealed from the viewpoint of energy. Therefore, we base our study on different mechanical properties of rock and the mechanism of brittle failure. A more effective and objective evaluation of rock brittleness will be obtained if well-logging data and the relative brittleness results obtained from laboratory experiments can be fully analyzed and synthesized into the results in this paper.
The energy transformation during the rupturing process of rock materials and the principles and applicability of some extant standards for estimating rock brittleness are analyzed. A more reasonable evaluation index of rock brittleness should include: (1) the ability of the material to resist inelastic deformation before failure; (2) the extent and rate to which the bearing capacity decreases after brittle failure; (3) the weakening of elasto-plasticity and strengthening of elasto-brittleness of the rock material; (4) the whole process of brittleness variation from plasticity to brittleness; in addition, the evaluation results should be continuous and monotonic.
The brittleness indices established in this paper based on energy analysis of the stress–strain curves of rock rupture are able to describe: (1) the brittle characteristics reflected by other brittleness indices and (2) the entire embrittlement process of rock materials continuously and monotonically. Type II rock behavior, which represents a typical feature of strong brittleness, is characterized by low fracture energy and a strong ability to self-sustain fracture propagation.
Alterations in the brittleness patterns of red sandstone, black shale and granite under different confining pressures differ under triaxial compression tests: (1) the brittleness of red sandstone is weak and the degree of brittleness decreases with increasing confining pressure; (2) the brittleness of shale rises under low confining pressure and decreases after the maximum brittleness has been reached and (3) the brittleness of granite is weak under low confining pressure and increases sharply with rising confining pressure.
The triaxial compression test results of black shale under different inclination angles of the bedding plane indicate that the brittleness of anisotropic black shale significantly differs in different directions. With increases in the inclination angle of the bedding plane, the dominant failure pattern of the shale alters from shear failure to longitudinal multi-splitting failure. The brittleness reaches its highest level when the inclination angle of the bedding plane approaches 90°. In conclusion, the anisotropy of rock material brittleness should be properly considered when evaluating brittleness.
The research was supported by Natural Science for Youth Foundation of China (No. 51504068).
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