Rock Mechanics and Rock Engineering

, Volume 49, Issue 12, pp 4681–4698 | Cite as

Estimation Criteria for Rock Brittleness Based on Energy Analysis During the Rupturing Process

Original Paper

Abstract

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.

Keywords

Brittleness 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)

a

The weight coefficient of each mineral (Eq. 2)

i

The types of brittle minerals (Eq. 2)

j

The types of all minerals (Eq. 2)

M

The mineral content (volume fraction) (Eq. 2)

σc, σt

Uniaxial compressive strength and tensile strength of rock material (Eq. 3)

σci

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)

εTOT

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)

dWf

The fracture energy (Eq. 8)

dWue

The unloading elastic energy (Eq. 8)

dWd

The dissipation energy of pre-peak stage (Eq. 8)

dWx

The extra energy required (type I behavior) or the excess energy released(type II behavior) (Eq. 9)

dWe

The total elastic energy accumulated in the rock specimen when reaching the peak strength (Eq. 10)

dWt

The energy which unconsumed or converts into other forms (Eq. 10)

σP

The peak strength of the rock specimen under compression (Eq. 10)

σR

The residual strength; σi represents the function of pre-peak curve (Eq. 10)

E

Elastic modulus of stress–strain curve (Eq. 11)

M

Post-peak modulus of stress–strain curve (Eq. 11)

σi

The function of pre-peak curve (Eq. 13)

εp

The strain corresponding to the peak strength (Eq. 13)

1 Introduction

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.

$$b_{1} = \frac{{V_{\text{Quartz}} }}{{V_{\text{Quartz}} + V_{\text{Carbonate}} + V_{\text{Clay}} }}$$
(1)
$$b_{2} = \frac{{\sum\nolimits_{i = 1}^{m} {a_{i} M_{i} } }}{{\sum\nolimits_{j = 1}^{m} {a_{j} M_{j} } }}$$
(2)
where VQuartz, VCarbonate and VClay are the contents of quartz, carbonate and clay, respectively; a is the weight coefficient of each mineral; i denotes the type of brittle mineral; j denotes all minerals; and M is the mineral content (volume fraction).

2.2 Evaluation Indices Based on Mechanical Parameters

The brittleness of a rock is closely related to its mechanical properties. Hucka and Das (1974) studied the relationship between rock brittleness and uniaxial compressive and tensile strength based on experimental data. They proposed a ratio (b3) to characterize rock brittleness and believed that with a higher ratio between compressive strength and tensile strength, internal microcracks were more likely to occur and brittle fractures within the rock were more likely to be produced. Altindag and Guney (2010) also studied the relationship between rock brittleness and rock strength and established functions b4 and b5 to describe rock brittleness. However, Mikaeil et al. (2011) fitted and analyzed experimental data obtained from rock mechanics tests and found that the relationship between the brittleness indices did not appear to be a linear one based on strength parameters and rock brittleness.
$$b_{3} = \frac{{\sigma_{\text{c}} }}{{\sigma_{\text{t}} }}\quad b_{4} = \frac{{\sigma_{\text{c}} \sigma_{\text{t}} }}{2}\quad b_{5} = \sqrt {\frac{{\sigma_{\text{c}} \sigma_{\text{t}} }}{2}}{.}$$
(3)
From theoretical analysis and some tentative calculations, Wang et al. (2014) obtained the intrinsic relationship between the initial stress of brittle rock and the brittleness indices based on the ratio between compressive strength and tensile strength of rock, so they simply used the initial value of stress in the stress–strain curve to represent brittleness without taking the corresponding strain of the initial stress into consideration (b6b8). This method has yielded contradictory results with respect to those calculated from the perspective of energy; in addition, the results are not monotonic and continuous.
$$b_{6} = \frac{{8\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}\quad b_{7} = \frac{{\sigma_{\text{c}} \sigma_{\text{ci}} }}{16}\quad b_{8} = \sqrt {\frac{{\sigma_{\text{c}} \sigma_{\text{ci}} }}{16}}{.}$$
(4)
Rickman et al. (2008) proposed an index (b9) to determine the rock brittleness based on the elastic modulus and Poisson’s ratio, which had shown a high level of field applicability. However, statistical analysis of field data obtained from the hydraulic fracturing development of Barnet Shale is needed to calculate this index, and the index does not consider the fracture mechanism of the rock. Many laboratory experiments have indicated that b9 has limitations, especially that the weights of the elastic modulus and Poisson’s ratio in the index are yet to be verified.
$$b_{9} = \frac{1}{2}\left( {\frac{{E - E_{\hbox{min} } }}{{E_{\hbox{max} } - E_{\hbox{min} } }} + \frac{{v - v_{\hbox{max} } }}{{E_{\hbox{min} } - E_{\hbox{max} } }}} \right){.}$$
(5)

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

The stress–strain curves obtained from rock compression tests are a direct reflection of the mechanical behavior of rock, which can reveal the internal mechanism of rock rupture. Therefore, the stress–strain curves serve as a qualitative and quantitative evaluation approach for evaluation of rock brittleness. The stress–strain curves obtained during the whole test from loading to specimen failure can quantitatively describe both the brittle characteristics of different rocks under the same stress state and the brittleness of the same rock under different stress states. Several different brittleness indices have been established based on the complete stress–strain curves of the rock material. Hucka and Das (1974) believed that the ratio of elastic strain to total strain b10 could be used to characterize shale brittleness. With a decrease in brittleness, an obvious yield platform would appear in the curve before the peak stress.
$$b_{10} = \frac{{\varepsilon_{\text{el}} }}{{\varepsilon_{\text{tot}} }}{.}$$
(6)
Bishop (1967) suggested that the characteristics of complete stress–strain curves could reflect the brittleness of rock. They pointed out that brittleness is closely related to the difference between the peak strength and residual strength and thus proposed brittleness index b11 (Eq. 7). Subsequently, Hajiabdolmajid and Kaiser (2003) suggested using the peak strain and residual strain as the brittleness parameters and established the corresponding brittleness index b12.
$$b_{11} = \frac{{\sigma_{\text{p}} - \sigma_{\text{r}} }}{{\sigma_{\text{p}} }}\quad b_{12} = \frac{{\varepsilon_{\text{p}} - \varepsilon_{\text{r}} }}{{\varepsilon_{\text{p}} }}{.}$$
(7)

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

The entire process of rock failure, from loading to failure, is always accompanied by energy conversion and energy exchange with the outside. The failure of rock material is an outcome of the dissipation and release of energy. Studies of rock deformation and failure can more accurately reveal the nature of the rock mass failure when viewed from the perspective of energy. Zuo et al. (2014); Wang (2014) analyzed the deformation and failure process of brittle rock based on energy transformation. They found that the rock had stored a considerable amount of energy and thus achieved a very unstable state when the peak strength was reached; there was a fall in strength from the peak to the residual strength; after the residual strength was reached, the rock reached a steady state of energy transformation. The characteristics of energy transformation in rock failure can indicate the corresponding rupture behavior of the rock specimen. Energy transformation during rock failure falls into three stages: energy accumulation, energy dissipation and energy conversion or release (Fig. 1). There are two types of stress–strain curves that correspond to two failure types for rock materials: type I and type II. The energy change differs greatly for these two types of failure, especially at the energy conversion or release stage (Fig. 2).
Fig. 1

Energy at each stage of the stress–strain curve

Fig. 2

Schematic diagrams of energy conversion during rock failure: (a) energy conversion during type I failure process; (b) energy conversion during type II 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.

Shear failure of the rock material does not develop instantaneously, but extends through the specimen with time. The shear resistance and ensuing displacement on the future shear plane appear to differ greatly from place to place. The failure process is characterized by three different zones (Tarasov and Potvinb 2013): Zone A, the fracture head, where the crack initially appears and is ready to expand; Zone B, the core frictional zone, which is situated after the fracture head and in which full friction takes place and Zone C, the intact zone, where the rock material is not yet broken. The shear resistance in this zone is provided by cohesion of rock material. When the fracture propagates ahead, the zone ahead will be substituted by the adjacent zone behind, during which process the cohesive strength will be replaced by frictional resistance. During the fracturing process of Zone A, the fracture mechanism plays a crucial role in the features of transformation from cohesive to frictional strength. The crack propagation under triaxial compression falls into four stages, as illustrated in Fig. 3a. In stage 1, the fracture head first appears, and the shear resistance of the rock specimen is provided only by cohesion from the intact zone. In stage 2, as the crack continues to propagate, the core frictional zone appears. In stage 3, the core frictional zone gradually expands and the intact zone gradually decreases. The force provided to resist the failure of the rock specimen shifts from cohesive to the frictional strength from the fracture plane, which explains the reduction in the bearing capacity of the specimen from a microscopic view. In stage 4, the shear resistance is entirely provided by the frictional strength, from which a complete shear rupture along the plane is consequently formed.
Fig. 3

Schematic diagram of the mechanism of development of shear fractures: (a) four stages of shear rupture development in a specimen at triaxial compression; (b) fracture head (fracture initiation zone) of type I and type II failure behavior; (c) the variation of shear resistance of type I and type II failure behavior; (d) Stress-strain curve of type I and type II failure behavior

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

A reasonable rock brittleness index should both possess sufficient physical meaning and be able to describe the whole range from plasticity to brittleness. The brittleness will affect the continuous fracturing process of the rock material. In defining and measuring brittleness, the mechanical properties of either the pre-peak or the post-peak stage should not be considered separately, but instead be regarded as a whole failure process. The brittleness index should represent both the ability of the rock material to resist non-elastic deformation before failure and the change in its bearing capacity after brittle failure; it should describe both the weakening of the elasto-plasticity of the material and the strengthening of the elasto-brittleness. From the discussion in Sect. 3.1, the smaller the dissipation energy during the pre-peak stage and the greater the ability to self-sustain the fracture during the post-peak stage, the greater the brittleness will be. The occurrence of self-sustaining fracturing is determined by the difference between dWe and dWf. Therefore, based on the discussion above, we propose one possible definition of absolute brittleness: the ability of rock material to completely self-sustain fracture propagation. As illustrated in Fig. 4, the stress–strain curve of absolute brittle rock should have the following characteristics: (1) before reaching the peak stress, there is almost no yield platform in the stress–strain curve; (2) the post-peak modulus (M) equals the elastic modulus (E) and (3) the elastic energy accumulated in the rock specimen is little transformed into fracture energy of the post-peak stage and thus the specimen is characterized by completely self-sustaining fracturing. From this, we can obtain the corresponding definition of absolute ductility and its characteristics: (1) the dissipation energy during the pre-peak stage and the fracture energy during the post-peak stage become infinite; (2) the yield platform and the post-peak curve join together to become one straight horizontal line and (3) there is no failure point in the curve, i.e., with increasing loading the material remains in a state of deformation. The variation law of stress–strain curve over the whole scale (Fig. 4) can be used to describe the variation of rock brittleness. The next objective is to find parameters that can quantitatively describe the change.
Fig. 4

Change in the degree of brittleness represented by B1 and B2 based on the energy changes of stress–strain curves

Tarasov (2010) found that brittle rocks are able to sustain fracture propagation. He proposed two brittleness indices that did not take into account the influence of dissipation energy during the pre-peak stage. Herein, we establish two new brittleness indices, B1 and B2, based on the variation of energy in the stress–strain curves combined with the whole scale (Fig. 4) of the transformation from absolute ductility to absolute brittleness described above. The new brittleness indices are defined as follows:
$$B_{1} = \frac{{{\text{d}}W_{\text{f}} + {\text{d}}W_{\text{d}} }}{{{\text{d}}W_{\text{ue}} + {\text{d}}W_{\text{d}} }}$$
(8)
$$B_{2} = \frac{{{\text{d}}W_{\text{x}} }}{{{\text{d}}W_{\text{ue}} + {\text{d}}W_{\text{d}} }}$$
(9)
where
$${\text{d}}W_{\text{ue}} = {\text{d}}W_{\text{e}} - {\text{d}}W_{\text{t}} = \frac{{\sigma_{\text{P}}^{2} - \sigma_{\text{R}}^{2} }}{2E}$$
(10)
$${\text{d}}W_{\text{f}} = {\text{d}}W_{\text{ue}} - {\text{d}}W_{\text{x}} = \frac{{\left( {\sigma_{\text{P}}^{2} - \sigma_{\text{R}}^{2} } \right)\left( {M - E} \right)}}{2EM}$$
(11)
$${\text{d}}W_{\text{x}} = \frac{{\sigma_{\text{P}}^{2} - \sigma_{\text{R}}^{2} }}{2M}$$
(12)
$${\text{d}}W_{\text{d}} = \int_{o}^{{\varepsilon_{\text{p}} }} {\sigma_{i} } {\text{d}}\varepsilon_{i} - \frac{{\sigma_{\text{P}}^{2} }}{2E}$$
(13)
in which dWf is the fracture energy; dWe is the total elastic energy accumulated in the rock specimen when it reaches peak strength; dWue is the unloading elastic energy; dWt represents the energy that is unconsumed or has been converted into other forms; dWx represents the extra energy required (the defined by yellow dotted lines for type I behavior) or the excess energy released (the yellow area for type II behavior); dWd is the dissipation energy of the pre-peak stage; σp is the peak strength of the rock specimen under compression; σr is the residual strength; σi represents the function of the pre-peak curve; εp is the strain corresponding to the peak strength; E is the elastic modulus (unloading elastic modulus); and M is the post-peak modulus.

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

To discuss further the correctness of the brittleness indices proposed in this paper and their application to different rocks, a series of triaxial compression tests are carried out on red sandstone, black shale and granite. The test equipment is a rigid servo-controlled triaxial compression test machine with a stiffness of 30 MN/mm, a loading capacity of 2500 kN and an upper limit of confining pressure of 200 MPa (Fig. 5). It has been proposed that axial strain control is impossible for type II behavior because the stress–strain curve does not display a monotonic increase in strain; therefore, the axial–circumferential strain control suggested by International Society of Rock Mechanics (ISRM) is adopted in the test. At the beginning of the test, the loading mode of axial strain control is adopted. When the specimen displays obvious circumferential deformation, the load mode is changed to circumferential strain control. This measurement allows the strain to increase monotonically even if the specimen does not deform axially. The suggested circumferential strain rate is 10−4 strain/s. To compare the brittleness characteristics of different rock materials, the rock specimens should be of consistent size and shape. Previous studies suggested that the diameter of cylindrical specimens in triaxial compression tests should be 20–60 mm and the aspect ratios should be 2–3. Such variation in size is quite small to cause an appreciable size effect. In this study, the specimen size is Φ25 mm × 60 mm (Fig. 6). The strain gauge should be installed at both ends of the cylindrical specimen as far as possible.
Fig. 5

Test equipment and gages

Fig. 6

Rock specimens used in the tests. Left to right red sandstone, black shale and granite

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.

As illustrated in Fig. 7, the mineral compositions of the red sandstone, granite and black shale are analyzed by X-ray diffraction. In the red sandstone: the contents of quartz and feldspar are 31.3 and 8.1 %, respectively, and the total content of clay minerals, such as montmorillonite, chlorite, illite and kaolinite, is 30.9 %. Montmorillonite and illite make up 10.3 and 11.4 % of the rock, respectively. The granite specimens are mainly composed of quartz (27.7 %) and feldspar (55.6 %); the amounts of biotite and hornblende are 10.7 and 4.87 %, respectively, and there are almost no clay minerals in the granite. The black shale consists dominantly of quartz and clay minerals with small amounts of potassium feldspar, plagioclase, calcite, dolomite and pyrite. The amounts of quartz, feldspar and clay minerals are 54.9, 8.64 and 28.38 %, respectively; the proportion of carbonate is small, approximately 6.9 %. Granite has the highest levels of quartz and feldspar and almost no clay minerals, whereas the red sandstone has the lowest content of brittle minerals and the highest clay mineral content. Because of the strong brittleness and high strength of quartz and feldspar and the strong plasticity and low strength of clay minerals, we can conclude that the granite should have the highest strength and strongest brittleness, and the strength and brittleness of red sandstone should be the lowest.
Fig. 7

Mineral composition of the red sandstone, granite and black shale

Prior to the triaxial compression test, the other basic mechanical parameters of tested rock materials are measured. For the red sandstone, the density is 2.42 g/cm3, the average tensile strength is 13.98 MPa, the average cohesion is 8.34 MPa and the average internal friction angle is 39.6°. The granite has a density of 2.69 g/cm3, an average tensile strength of 15.05 MPa, an average cohesion of 18.96 MPa and an average internal friction angle of 57.2°. The black shale has a density of 2.67 g/cm3, but because of the presence of bedding planes, the tensile strength, cohesion and internal friction angle have remarkable discreteness. When shale fails along the bedding planes, its average tensile strength, cohesion and internal friction angle are 4.03 MPa, 5.14 MPa and 39.5°, respectively. In contrast, when the rock fails perpendicular to the bedding planes, the average tensile strength, cohesion and internal friction angle are 7.03 MPa, 15.42 MPa and 54.8°. Table 1 displays the data from triaxial compression tests of red sandstone, granite and black shale.
Table 1

Data of the triaxial compression test on red sandstone, granite and black shale

Rock type

Compressing pressure (MPa)

Peak strength (MPa)

Residual strength (MPa)

Average elastic modulus (GPa)

Average Poisson’s ratio

Elastic strain

Peak strain

Residual strain

Red sandstone

0

95.5

48.2

13.24

0.18

0.00602

0.00751

0.0101

 

30

122.2

53.1

16.36

0.24

0.00741

0.0101

0.0152

 

60

169.4

76.7

20.88

0.28

0.00762

0.0125

0.0197

 

90

252.8

132.3

20.13

0.26

0.0133

0.0234

0.0384

 

120

309.7

227.3

23.58

0.36

0.0117

0.0245

0.0404

Granite

0

137.3

63.4

50.44

0.22

0.00351

0.00491

0.00603

 

30

225.1

64.7

53.92

0.20

0.00449

0.00601

0.00674

 

60

342.2

75.2

65.19

0.24

0.00602

0.0066

0.00644

 

90

437.4

78.6

65.23

0.25

0.00831

0.00853

0.00762

 

120

468.3

98.4

60.89

0.24

0.00778

0.00889

0.01092

Black shale

0

78.1

24.4

18.35

0.25

0.00242

0.0035

0.00401

 

30

129.3

41.5

22.34

0.21

0.0035

0.00415

0.0036

 

60

162.7

51.1

23.17

0.22

0.00402

0.00475

0.00333

 

90

197.2

74.9

24.25

0.28

0.00611

0.00635

0.00382

 

120

237.6

110.2

26.14

0.33

0.00741

0.0077

0.00451

5.3 Effect of Confining Pressure on Rock Brittleness

From analysis of various brittleness evaluation standards, Holt et al. (2011) believed that the stress condition of the reservoir must be considered in brittleness evaluation of rocks based on experimental data from triaxial compression tests. They concluded that brittleness decreased with increasing confining pressure. Moreover, research by Paterson and Wong (2005) and Zuo et al. (2014) showed that under confining pressure, the decrease of brittleness is characterized by two features: (1) the plasticity at pre-peak stage increases and (2) the stress after the peak decreases at a slower rate. However, some recent studies have found that, for different rocks, the pattern in which brittleness decreases with the increase of confining pressure only applies in particular ranges of confining pressure, whereas for other values of confining pressure, the brittleness of the material may increase with rising confining pressure (Tarasov 2010, 2011; Tarasov and Randolph 2011). Similar patterns were observed in the tests in this paper. As illustrated in Figs. 8, 9, and 10, an increase in the confining pressure can lead to uncommon behavior of rock material within different ranges of confining pressure. Specifically, the rock behavior can switch from type I to type II and then to type I again (Fig. 9), indicating that the rock brittleness undergoes the following changes: as the confining pressure begins to increase, the brittleness of rock material begins to rise; when the confining pressure increases to a certain value, the rock brittleness reaches its maximum value, but subsequently the rock brittleness decreases with the continued increase of confining pressure.
Fig. 8

Stress–strain curves of red sandstone under different confining pressures

Fig. 9

Stress–strain curves of black shale under different confining pressures

Fig. 10

Stress–strain curves of granite under different confining pressures

From Fig. 8, all the stress–strain curves of red sandstone under different confining pressures display type I rock behavior; the yield platform during the pre-peak stage increases and the post-peak modulus (M) decreases with increasing confining pressure. Li et al. (2012) proved that the brittle failure of rock material is caused by the non-uniform stress distribution within the material; local fractures are formed randomly everywhere within the material, which consequently leads to the spatial distribution of fractures. They also found that the more completely the rock specimen became broken and the larger the number of fracture planes, the greater the brittleness of the rock material. In terms of the number of cracks, splitting failure and double-shear failure can produce more cracks than single-shear failure and demonstrate a higher level of brittleness. The failure patterns of red sandstone are plotted in Fig. 12. Under low confining pressure, split-shear composite failure of the specimens occurs, indicating that the red sandstone has lower brittleness. With increases in the confining pressure, the failure patterns of the specimen are mainly shear fractures. Both the stress–strain curves and failure patterns of red sandstone indicate that the brittleness of the red sandstone is weak and that the degree of brittleness decreases with rising confining pressure. The brittleness of red sandstone is calculated using the indices B1 and B2 established in this paper (Fig. 11). The results of theoretical calculation of the brittleness decrease of red sandstone are consistent with the experimental results.
Fig. 11

Brittleness of different rocks under different values of the confining pressure indices B1 and B2

Fig. 12

Fracture patterns of red sandstone under different confining pressures

The stress–strain curves of black shale show different trends (Fig. 9). The curve exhibits weak brittleness in the absence of confining pressure. With increasing confining pressure, the post-peak modulus (M) gradually increases and the curve transforms to type II behavior. However, when the confining pressure reaches 120 MPa, the curve switches back to type I behavior. This result means that the brittleness of shale rises under low confining pressure but falls after the maximum brittleness is reached. For granite (Fig. 10), the specimens show weak brittleness when the confining pressure is less than 30 MPa and the brittleness increases sharply with increasing confining pressure. The trend of increasing brittleness is also apparent in the failure patterns of granite specimens (Fig. 13): under low confining pressure, splitting-shear composite failure appears, becoming the dominant failure pattern with rising confining pressure. In conclusion, for different rocks, with rising confining pressure, the transformation from brittleness to ductility is not monotonic but instead an initial increase and later decrease. Moreover, this change is more likely to occur at higher confining pressure for rocks with stronger brittleness. Using the black shale (Fig. 9) as an example, its stress–strain curves show the entire pattern of brittleness increasing and then decreasing, whereas the red sandstone and granite only show part of the pattern, either the decreasing or the increasing stage of brittleness, respectively. This pattern is consistent with the brittleness range calculated by B1 and B2 (Fig. 11). Based on this pattern, we predict the complete brittleness curves of red sandstone and granite, which are plotted as blue dotted lines in Fig. 11. Zhang and Gao (2012, 2015a, b) studied energy variations in the failure process of coal, sandstone and granite based on mechanical tests. The experimental results indicated that the stress–strain curves of granite displayed type II behavior, but coal with strong plasticity exhibited only type I behavior. This finding is consistent with the experimental results obtained in this paper. Therefore, the brittleness evaluation indices established herein are reliable for describing the brittleness variation of rocks under different confining pressures.
Fig. 13

Fracture patterns of granite under different confining pressures

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.

Figures 14 and 15 show the morphology of the fracture surfaces and debris of the three kinds of rock material. As illustrated in Fig. 14, the debris from red sandstone specimens is granular and of small size, which indicates that most blocks between the fracture surfaces collapse and are crushed during the loading process of red sandstone. More energy is needed to maintain fracture propagation in this situation, which is typical of type I rock behavior. In contrast, after fracturing of shale and granite, the debris appears to be sheet- or strip-shaped, which accounts for the fan structure that appears in the fracture zones of the specimens (Fig. 3). Figure 14 also indicates that fractures in black shale and granite are able to self-sustain propagation and that these rocks have stronger brittleness. Figure 15 shows SEM images (400×) of the fracture surfaces of the three rock types under a confining pressure of 60 MPa. The fracture surface of the red sandstone specimen is granular, whereas the black shale and granite display a lamellar structure with a small number of attached grains. In addition, the blocks from the fracture process of the sandstone specimen (Fig. 14a) appear collapsed, in good agreement with the macroscopic observations. The fractures of black shale (Fig. 14b) and granite (Fig. 14c) appear fan-shaped, as discussed above, and these rocks exhibit type II rock behavior.
Fig. 14

Debris from different types of rock. a Red sandstone. b Black shale. c Granite

Fig. 15

SEM images of the fracture surfaces of different types of rock. a Red sandstone. b Black shale. c Granite

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.

Figure 16 illustrates the fracture patterns of black shale specimens with different loading directions. Shear fracture occurs dominantly in the shale with expansion of the bedding plane when α is relatively large. When α = 45°, shear failure occurs on the specimen along the bedding plane; when α = 75° or 90°, longitudinal multi-splitting failure occurs in most specimens, indicating strong brittleness. The smaller the value of α, the greater the brittleness of the black shale. Thus, analysis of shale brittleness using only site seismic data or mineral composition is inaccurate. Figure 17 shows the stress–strain curves for different values of α. When α is less than or equal to 45°, the rock behavior is type II, indicating that smaller α means stronger brittleness of the specimen. The brittleness variation demonstrated by the stress–strain curves is consistent with that in the fracture patterns of the shale specimens.
Fig. 16

Fracture patterns of black shale specimens with different loading directions

Fig. 17

Stress–strain curves of shale specimens with different loading directions

The brittleness of shale in different directions can be calculated using B1 and B2, as illustrated in Fig. 18. With an increase in the inclination angle of the bedding plane, the brittleness of shale specimens alters from weak to medium; brittleness reaches its highest level when α approaches 90°. This pattern can be explained by fractures within the shale propagating along the bedding planes more easily when α is small. In addition, because of the lamellar structure of shale grains and the development of bedding planes, a fan structure is likely to be formed, which leads to a smaller fracture energy during the fracture process. The fracture energy has its minimum value when α = 90°; the specimen at this time possesses a stronger ability to self-sustain fracture propagation, and thus a stronger brittleness.
Fig. 18

Brittleness of black shale in different directions based on indices B1 and B2

6 Comparison of Indices B1 and B2 with Other Brittleness Indices

The brittleness evaluation criteria established in this paper depend on the energy balance of the pre-peak and post-peak phases of the rock failure process, so they can describe the two types of rock behavior (type I and type II) in the form of a continuous, monotonic and unambiguous scale of brittleness. The criteria are based on sound physical principles reflecting the degree of post-peak instability. In this section we analyze whether other existing criteria can provide unambiguous characterization of rock brittleness representing post-peak instability.
  1. 1.
    First, the brittleness of three rock materials under no confining pressure are calculated using indices b3b5. 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 b3b5 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 b3b5, these indices are not unambiguous for characterizing the brittleness of rock.
    Table 2

    The brittleness of red sandstone, granite and black shale calculated by b3b5

    Type

    b3

    Brittleness grade

    b4

    b5

    Brittleness grade

    Red sandstone

    6.83

    Weak brittle

    667.55

    25.83

    Super high brittle

    Granite

    9.12

    Weak brittle

    2066.37

    45.46

    Super high brittle

    Black shale

    11.11

    Moderate brittle

    549.04

    23.43

    High brittle

     
  2. 2.
    Indices b6b8 are modified by the crack initiation stress σci with respect to b3b5, 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 b7b8 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 b7b8 indicate that the red sandstone is a rock material with super-strong brittleness under high confining pressure, contrary to the facts. As for b3b5, indices b6b8 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 b6b8

    Type

    Compressing pressure (MPa)

    Peak strength (MPa)

    σci

    σc/σci

    b6

    Brittleness grade

    b7

    b8

    Brittleness grade

    Red sandstone

    0

    95.5

    36.29

    0.38

    21.1

    Brittle

    216.6

    14.7

    Moderate brittle

     

    30

    122.2

    51.32

    0.42

    19.1

    Brittle

    391.9

    19.8

    Brittle

     

    60

    169.4

    76.23

    0.45

    17.8

    Brittle

    807.1

    28.4

    Super high brittle

     

    90

    252.8

    136.51

    0.54

    14.8

    Moderate brittle

    2156.9

    46.4

    Super high brittle

     

    120

    309.7

    151.75

    0.49

    16.3

    Brittle

    2937.4

    54.2

    Super high brittle

    Granite

    0

    137.3

    60.41

    0.44

    18.2

    Brittle

    518.4

    22.8

    High brittle

     

    30

    225.1

    85.53

    0.38

    21.1

    Brittle

    1203.4

    34.7

    Super high brittle

     

    60

    342.2

    157.41

    0.46

    17.4

    Brittle

    3366.6

    58.1

    Super high brittle

     

    90

    437.4

    223.07

    0.51

    15.7

    Brittle

    6098.3

    78.1

    Super high brittle

     

    120

    468.3

    192.00

    0.41

    19.5

    Brittle

    5619.7

    74.9

    Super high brittle

    Black shale

    0

    78.1

    26.55

    0.34

    23.5

    Brittle

    129.6

    11.3

    Moderate brittle

     

    30

    129.3

    47.84

    0.37

    21.6

    Brittle

    386.6

    19.6

    Brittle

     

    60

    162.7

    68.33

    0.42

    19.1

    Brittle

    694.9

    26.3

    Super high brittle

     

    90

    197.2

    100.57

    0.51

    15.7

    Brittle

    1239.6

    35.2

    Super high brittle

     

    120

    237.6

    106.92

    0.45

    17.8

    Brittle

    1587.8

    39.8

    Super high brittle

     
  3. 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.
    Fig. 19

    Brittleness of red sandstone, black shale and granite calculated using b10 and b11

    Fig. 20

    Brittleness of red sandstone, black shale and granite calculated using b12

     

7 Discussion

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.

8 Conclusions

  1. a.

    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.

     
  2. b.

    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.

     
  3. c.

    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.

     
  4. d.

    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.

     

Notes

Acknowledgments

The research was supported by Natural Science for Youth Foundation of China (No. 51504068).

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Department of Petroleum EngineeringNortheast Petroleum UniversityDaqingChina

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