Introduction

Brittleness of rock is closely related not only to the stability of rock mass engineering, but also to the reservoir stimulation in oil and gas engineering and geothermal development. Many rock mechanical issues are closely related to rock brittleness, such as rock mass drillability (Protodyakonov 1963; Aubertin 2010; Yarali and Kahraman 2011), rock cutting sawability (Gunaydin et al. 2004; Altindag and Guney 2010), brittle rupture (Hajiabdolmajida et al. 2003; Cai 2010), rockburst prediction (Feng and Xie 2000), stability of underground structure (Cai et al. 2004), and oil and gas development (Jin et al. 2014; Wang et al. 2014; Zhang et al. 2017). A brittle rock presents some special properties, such as (1) low elongation when acted on loading, (2) fracture failure occurs at small strain, (3) ratio of compressive strength to tensile strength is high, (4) higher resilience, (5) higher percentage of brittle minerals like quartz and minimal proportion of clay minerals, (6) greater internal friction angles, (7) lower Poisson’s ratio and higher elastic modulus values, (8) exhibiting intensive failure process, fracture fully developed when fracturing occurs and rocks reach failure suddenly in an intensive way.

So far, various definitions to describe rock brittleness have been proposed by many scholars and many different influential factors have been taken into consideration, including rock mechanical parameter, mineral composition, rock hardness. Each definition has its applicability and reliability for the specific required purpose. In shale gas development field, reasonable definition of brittleness is important to shale gas recovery. Hydraulic fracturing technique has been viewed as a widely used way to gain enormous amount of gas resource (Jin et al. 2014; Zhang et al. 2017). During hydraulic fracturing treatment, hydraulic fractures interact with preexisting natural fractures and complex fracture network forms. The low pore and low permeability characteristics of shale reservoirs can be improved when formation of complex fracturing network occurs. However, many factors affect the formation of an effective fracture network, among which the formation brittleness plays a predominant role. Currently, many scholars have used brittleness to evaluate the fracability of reservoirs. Generally speaking, it has been considered that good fracturing formations should have higher brittleness (Wang and Gale 2009; Wang et al. 2016; Li et al. 2012; Chong et al. 2010; Zhang et al. 2017). However, this point is not always true, because a phenomenon of fracture barrier may occur for higher brittleness formations (Bruner and Smosna 2011; Jin et al. 2014); this is adverse to form complex fracturing network during fracturing treatment. The current brittleness definition approaches cannot reflect the formation fracability; therefore, we should seek new brittleness definition method or take into account other parameters to evaluate the fracability of formations.

In this work, plenty of UCS tests have been done for various kinds of rock to seek a new brittleness definition, which is named as crack initiation stress level (σ ci/σ c). Based on the raw experimental data, because of the close relationship between the rock brittleness and its mechanical properties, we try to explore the relations between brittleness index defined using uniaxial compressive strength, σ c, and tensile strength, σ t. Our research is mainly focused on the relationship between B19, B20, B21, B22 (brittleness index based on strength ratio), and σ ci/σ c. To the author’s knowledge, so far no experimental results have been published that describe the link between brittleness index and σ ci/σ c. The aim of this study is to seek a new brittleness definition to well reflect the fracture density after UCS test. To begin with, the methods to define rock brittleness index have been summarized from the published literature. Then, the link between B19, B20, B21, B22, and σ ci/σ c was investigated using theoretical and experimental investigation. Finally, the reliability of σ ci/σ c to reflect fracture pattern has been deeply discussed. From our results, only by uniaxial compression test can the brittleness index be established easily and quickly, and also it can be used as a preliminary index in reservoir brittleness evaluation.

Methods

Quantification of brittleness

Brittleness is one of the comprehensive properties of rock material. It is influenced by many internal and external factors. The internal factors refer to the heterogeneity of rock, including mineral particles, texture, and structure; the external factors are the non-uniform stress loading on rock under certain stress conditions, which leads to local damage and then forms multidimensional failure surfaces. After literature review, methods for determining the brittleness index are based on six approaches. These methods include (1) uniaxial compressive strength and Brazilian tensile strength, (2) stress–strain curve methods, (3) hardness test, or impact test, (4) porosity and grain size, (5) mineral composition, and (6) geophysical method. The detailed methods of brittleness definitions are summarized in Table 1. The proportion of these methods to determine brittleness is about 35.32, 42.98, 6.9, 1.4, 11.1, and 2.3%, respectively. It can be concluded that the mechanical properties hold the biggest percentage which is closely related to brittleness.

Table 1 Summary of current brittleness index definitions and testing methods
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Method for determining crack initiation stress

Many methods have been proposed to obtain the crack initiation stress, σ ci, for brittle rock, and these methods mainly include (1) volumetric strain or crack volumetric strain methods, (2) lateral strain methods, and (3) acoustic emission method. These methods were summarized in detail by Nicksiar and Martin (2012), and a new method called the LSR (lateral strain response method) was also proposed through lots of laboratory test. In this work, the calculation of σ ci used the crack volumetric strain method. As for brittle rock samples, in both uniaxial and triaxial experiments, brittle rocks experienced four typical stages: (1) crack closure stage, in the axial strain curve, it presents as the starting point of linearity; (2) crack initiation and stable crack propagation, an inflection point in the lateral strain occurs during sample deformation; (3) start of unstable crack propagation, this corresponds to the maximum volumetric strain in the volumetric strain curve; and (4) peak stage, it occurs after failure of sample and after the peak stress in the axial strain. Many researchers have proved that when a stress value exceeds the crack initiation during stress–strain responses, crack initiates and propagates (Martin and Chandler 1994; Bieniawski 1967; Brace et al. 1966; Diederichs et al. 2004). Figure 1 plots the complete stress strain curve and the crack volumetric stress strain curves for brittle rock. The crack initiation stress, σ ci, can be determined using the curves.

Fig. 1
figure 1

Crack evolution process in the stress–strain diagram (modified after Martin and Chandler 1994). Note that the axial and lateral strains are measured using experiment, and the crack volumetric strain and volumetric strain curve are calculated

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Research idea

From Table 1, most of the brittleness definitions are based on rock mechanical parameters. Among many of those parameters, the two basic rock mechanics properties of σ c and σ t can be easily obtained by simple laboratory testing; they are the two important and fundamental parameters in rock mechanics. Using those two basic rock mechanical parameters, the following four definitions are based on this understanding (as shown in Table 1):

$$B19 = \sigma_{\text{c}} /\sigma_{\text{t}} ;$$
(1)
$$B20 = \left( {\sigma_{\text{c}} - \sigma_{\text{t}} } \right)/ \, \left( {\sigma_{\text{c}} + \sigma_{\text{t}} } \right);$$
(2)
$$B21 = \sigma_{\text{c}} \cdot\sigma_{\text{t}} /2;$$
(3)
$$B22 = \left( {\sigma_{\text{c}} \cdot\sigma_{\text{t}} /2} \right)^{1/2} .$$
(4)

Crack initiation stress is also viewed as an important rock mechanical parameter; it is associated with the type of rock, mineral content, and the defects that preexist in the rock, such as micro-cracks, grain boundaries, and lattice defects. The first crack initiation criterion for brittle material was proposed by Griffith. Crack starts to initiate when KI = K IC, where K IC is named as the fracture toughness of the rock and its value can be calculated from the rock’s tensile strength in the form of \(\sigma_{\text{t}} = 6.88K_{{{\rm I}C}}\) (Sih 1974). Zhang et al. (2016) have proved that σ c = (55–82) K IC and σ c/σ t = 8–12. Cai et al. (2004) studied the relationship between crack initiation stress and strength ratio, when the half crack length equals 0.5, 1, and 1.5 mm, respectively. In the aspect of mechanical property, σ ci/σ c can be functioned as the fracture toughness, tensile strength, and compressive strength. From the analysis above, σ ci/σ c is also close related to the tensile strength and compressive strength. It can be assumed that σ ci/σ c may be expressed as a function of fracture toughness:

$$\left. {\begin{array}{*{20}l} {\sigma_{\text{ci}} = f\left( {\sigma_{\text{c}} ,\sigma_{\text{t}} } \right)} \hfill \\ {\sigma_{\text{c}} = f\left( {K_{\text{IC}} } \right)} \hfill \\ {\sigma_{\text{t}} = f\left( {K_{\text{IC}} } \right)} \hfill \\ \end{array} } \right\} \Rightarrow \sigma_{\text{ci}} /\sigma_{\text{c}} = f\left( {K_{\text{IC}} } \right).$$
(5)

As stated above, fracture toughness and σ c and σ t are closely related. Combined with σ ci/σ c and these strength parameters, we assume that there may be a certain relationship between the brittleness index and σ ci/σ c, and the relation can be expressed as:

$$\left. {\begin{array}{*{20}l} {B = f\left( {\sigma_{\text{c}} ,\sigma_{\text{t}} } \right)} \hfill \\ {K_{\text{IC}} = f\left( {\sigma_{\text{c}} ,\sigma_{\text{t}} } \right)} \hfill \\ \end{array} } \right\} \Rightarrow B = f\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right).$$
(6)

In uniaxial compression tests, σ ci and σ c are not equal. We can use the ratio of these two indices to reflect the difference under a compression test. For brittle rock, generally, σ c is not used alone to reflect the brittleness of rock, because a rock which is much more brittle may have low σ c, vice versa. To reasonably reflect the brittleness, in Table 1, we summarize the brittleness definitions using σ c and σ t. In addition, to deeply study the issue of crack initiation in brittle rock, Cai (2010) has done plenty of theoretical and experimental analyses and found that there exists a certain relation between tensile strength and crack initiation stress. Therefore, in this work, the mechanical parameters of σ c and σ ci are used to redefine the brittleness, as below:

$$B23 = \sigma_{\text{ci}} /\sigma_{\text{c}} .$$
(7)

B23 reflects the difference in heterogeneity and texture. For rock with fine structured particle, this value is generally larger than rock with medium and coarse particles (Cai 2010). In the next section, a series of experimental works have been done for different rocks to investigate the relationship between brittleness B23 and B19, B20, and B21.

Testing and test results

Tested materials

The rock samples used in this work include diorite, granite, marble, sandstone, and shale. The gray diorite was obtained from the Hot ha tal hydropower station in Xinjiang, China. It is a kind of medium- to coarse-grained rock, with an average particle size between 2 and 4 mm. The granite is obtained from 500 to 600 m depth in Alxa area of China, it is a kind of extreme heterogeneous and coarse-grained rock, and the color of the rock is pink, with its density of 2.67 g/cm3. Plenty of brittle minerals exist in the granite sample, such as k-feldspar with grain size of 2.0–20 mm is 30%, plagioclase with grain size of 2.0–6.0 mm is about 35%, and quartz with grain size of 2.0–3.0 mm is about 35%. The white marble is obtained from a stone pit, with a density of 2.51 g/cm3. The sandstone was obtained from a tight gas sandstone reservoir; it is a kind of fine-grained red brown stone, obtained from Tarim Basin, China; and the density of the sample is about 2.58 g/cm3. The shale was from Longmaxi Formation in the Lower Silurian located in Shizhu County, Chongqing, China. The shale cores, obtained from a depth about 20 m underground with alternating thin and thick layers, represent a carbonate-rich, mature gas shale. The average density of the shale samples is 2.62 g/cm3. For shale samples, cylindrical samples were in situ drilled out of the bulk material, under dry conditions with bedding plane inclinations of 0°, 15°, 30°, 45°, 60°, 75°, and 90°, which are defined as the intersection angle between the loading direction and the bedding face orientation. For the six kinds of rock samples, they were prepared according to ISRM standards with a length-to-diameter ratio of approximately 2 (e.g., uniaxial compressive test is 100 mm × 50 mm and tensile strength test is 50 mm × 25 mm) (ISRM 1981). The samples were ground to a planeness of 0.01 mm and the cylinder perpendicularity within 0.05 radians. The tolerance of planeness and cylinder perpendicularity of the tested samples met the standards of the ISRM-suggested method. Partial rocks used in the experiment are shown in Fig. 2.

Fig. 2
figure 2

Partial tested rock samples in this work. a Granite. b Marble. c Sandstone. d Shale, β = 0°. e Shale, β = 90°. f Shale orientation

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Experimental setup

The UCS tests were performed at a servo-controlled deformation apparatus (GCTS RTR-2000). The maximum applied axial force and frame stiffness are 4600 kN and 10.05 MN/mm, respectively. The apparatus can provide a maximum confining pressure of 210 MPa. The dynamic frequency of the machine is 0–10 Hz. It can test the rock sample under the maximum temperature of 200°. A LVDT gauge was installed onto the sample to measure the axial strain and lateral strain. The axial loading was carried out under strain control with loading rate of 0.06 mm/min (strain rate of 1.0e−5 s−1) for UCS test. During the uniaxial compressive test, two strain gauges measuring the axial and lateral deformation were installed on rock samples, and the axial stress, axial strain, lateral strain, and circumferential strain can be recorded simultaneously at the same sampling frequency during the whole deformation process. The tensile strength of rock is determined by Brazilian tests, and the loading rate is 0.1 mm/min. The crack initiation stress is calculated using the method proposed by Martin and Chandler (1994): crack volumetric strain approach. Because the defined brittleness index in this work is strongly related to the rock mechanical parameters, therefore, among many of the definitions in Table 1, brittleness indexes B19, B20, B21, and B22 are used to calculate the brittleness. Based on the experimental results, we further discuss the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c).

Calculation of σ ci for typical samples

From the crack volumetric strain curves, the crack initiation stress is obtained for different rock samples. We plot the complete stress strain curve and crack volumetric strain curve for diorite, granite, marble, sandstone, and shale, as shown in Fig. 3. Due to the difference in mineral composition and microscopic structural fabric, the heterogeneity characteristics for these samples are different; therefore, the crack initiation stress level is different. For these six samples, the value of crack initiation stress level is 0.55, 0.15, 0.48, 0.31, 0.60, and 0.64, respectively (Fig. 4).

Fig. 3
figure 3

GCTS RTR-2000 rock mechanic system used in the test

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Fig. 4
figure 4figure 4

Stress strain curves and crack initiation stress curves for typical samples. a Diorite sample #13; b granite sample #1; c marble sample #1; d sandstone sample #1; e shale sample with 0° orientation; f shale sample with 90° orientation

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Testing results and correlation analysis

Case 1: Medium- to coarse-grained diorite

According to the ISRM-suggested methods (ISRM 1981), the tested rock belongs to strong (80 MPa < σ c < 100 MPa) and very strong (σ c > 100 MPa) rocks. The average compressive and tensile strength were 103 and 7.66 MPa, respectively. According to the analysis in “Methods” section, σ ci is calculated by the crack volumetric strain curve. Uniaxial compression strength, Brazilian tensile strength values, and the crack initiation stress were viewed as first-hand data, and the brittleness index values of B19, B20, B21, and B22 were obtained using Eqs. (1)–(4), respectively (Table 1). Regression analysis was used to explore their relationship, and the equation with good correlations and the best-fit line was determined for each regression. Figure 5 shows the relationship between B19, B20, B21, B22, and B23 (σ ci/σ c). It is seen that there is no evident relation between the brittleness B21, B22, and B23 (i.e., there is a lower correlation coefficient of 0.627 and 0.621 that is found between them). However, there are significant relations between the brittleness values B19, B20, and σ ci/σ c. The correlation coefficients are 0.937 and 0.913, respectively. The relation follows a power function and linear function, respectively. As shown in Fig. 5, the brittleness decreased with increasing σ ci/σ c. The regression equations are as follows:

Fig. 5
figure 5

Plots of the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c) for coarse-grained diorite

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$$B19 = 7.8685\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right)^{ - 0.8333} ,\quad R^{2} = 0.937;$$
(8)
$$B20 = 0.96011 - 0.1838\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right),\quad R^{2} = 0.913.$$
(9)

Case 2: Coarse-grained granite

The rock mechanical parameters were obtained as stated above; brittleness index of B19, B20, B21, and B22 is also calculated using the method mentioned above. UCS for the samples ranges from 99 to 160 MPa, tensile strength is between 3 and 6 MPa, and crack initiation stress level ranges from 0.15 to 0.31. We used regression analysis to study their relationship, and the equation of the correlation coefficients and the best-fit line and were determined for each regression. Figure 6 shows the relationship between B19, B20, B21, B22, and B23 (σ ci/σ c). From the results, a strong correlation exists between B19, B20, and B23 (σ ci/σ c) and the correlation coefficients are 0.977 and 0.985, respectively. However, no evident relation exists between the brittleness B21, B22, and σ ci/σ c. The regression equations are:

Fig. 6
figure 6

Plots of the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c) for granite

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$$B19 = 10.761\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right)^{ - 0.624} ,\quad R^{2} = 0.977;$$
(10)
$$B20 = - \,0.991 - 0.194\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right),\quad R^{2} = 0.985.$$
(11)

Marble

Figure 7 shows the test results for the marble samples. UCS value ranges between 200 and 230 MPa. For tensile strength, the value ranges between 10 and 16 MPa. The crack initiation stress was calculated from the crack volumetric strain, and crack initiation stress level ranges from 0.48 to 0.55, with an average value of 0.51. From the regression results for B20, B21, B22, and B23 (σ ci/σ c), an obvious correlation exists between B19, B20, and B23 (σ ci/σ c) and the correlation coefficients are 0.891 and 0.903, respectively. However, no evident relation between the brittleness B21, B22, and σ ci/σ c is found. The regression equations are as follows:

$$B19 = 2.625\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right)^{ - 2.987} ,\quad R^{2} = 0.891;$$
(12)
$$B20 = 1.187 - 0.585\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right),\quad R^{2} = 0.903.$$
(13)
Fig. 7
figure 7

Plots of the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c) for marble

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Case 4: Tight sandstone

Using the same method mentioned above, the basic rock mechanical parameters and the associated brittleness index are obtained. UCS of sandstone is between 90 and 131 MPa, tensile strength obtained by Brazilian test ranges between 3 and 6 MPa, and the crack initiation stress is between 0.3 and 0.5. Using regression analysis to obtain the relationship between B19, B20, B21, B22, and B23 (σ ci/σ c). Figure 8 plots the relationship between B20, B21, B22, and B23 (σ ci/σ c). From the results, there exists a strong correlation between B19, B20, and B23 (σ ci/σ c) and the correlation coefficients are 0.823 and 0.834, respectively. However, no evident relations are found between the brittleness B21, B22, and σ ci/σ c. The regression equations are as follows:

Fig. 8
figure 8

Plots of the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c) for tight sandstone

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$$B19 = 8.347\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right)^{ - 2.987} ,\quad R^{2} = 0.823;$$
(14)
$$B20 = 0.999 - 0.223\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right),\quad R^{2} = 0.845.$$
(15)

Case 5: Shale

The UCS of shale first decreases and then increases with the increase in orientation; the changes in UCS against orientation show a “U” shape. The UCS value ranges from 84 to 192 MPa, it is the maximum for a sample with 0° orientation, and it is the minimum for a sample with 30° inclination. For tensile strength, the value ranges between 4  and 10 MPa, it also shows an obvious anisotropy. The crack initiation stress was calculated from the crack volumetric strain, and typical curves for sample with 0° and 90° orientation are shown in Fig. 4. The value of B23 for shale samples with different orientation is different, and it shows strong anisotropic characteristics. For shale sample with 30° orientation, B23 is the maximum with its value of 0.62; however, for shale sample with 90° orientation, B23 is the minimum with its value of 0.56. Regression analysis is used to explore the relationship between them, and the equations which have best correlation coefficients were determined for each regression. Figure 9 shows the relationship between B19, B20, B21, B22, and B23 (σ ci/σ c). It can be seen that there is no evident relation between the brittleness B19, B20, and σ ci/σ c. However, obvious relationships exist between the brittleness indexes B21, B22, and σ ci/σ c. The correlation coefficients are 0.711 and 0.711, respectively. The regression equations are as follows:

Fig. 9
figure 9

Plots of the relationships between B19, B20, B21, B22, and B23 (σ ci/σ c) for the anisotropic shale

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$$B21 = 7.300\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right)^{ - 1.863} ,\quad R^{2} = 0.711;$$
(16)
$$B22 = 0.999 - 0.223\left( {\sigma_{\text{ci}} /\sigma_{\text{c}} } \right),\quad R^{2} = 0.711.$$
(17)

Discussion

From the experimental results, B23 as a new brittleness index to study the rock brittleness, the correlation between brittleness indexes (i.e., B19, B20, B21, B22, and B23) defined based on σ c, σ ci, and σ t is investigated. For the UCS test, those parameters are basic and important parameters to reflect the strength and deformation characteristics of brittle rock. Because of the difference in rock mineral composition and microscopic fabric characteristics, crack initiation stress level (B23 = σ ci/σ c) presents obvious discrepancy. Experimental results show that the brittle index defined in this paper can well reflect the fabric inhomogeneity and structural heterogeneity. For the tested diorite, granite, sandstone, and marble, a correlation exists between B19, B20, and B23; however, for the anisotropic shale, an obvious correlation is found between B21, B22, and B23. These results indicate that index of B23 depends on the structural characteristics of rock. The tested shale samples are typical anisotropic material, and pronounced bedding plane develops during the diagenetic process. Due to the presence of foliation, bedding planes, cleavage, schistosity, and macro- or micro-natural fractures, the gas/oil shale is characterized by anisotropy. These structural characteristics result in the obvious anisotropy and heterogeneity characteristics, the existence of bedding face alters not only its structural characteristics, but also its mechanical properties (e.g., modulus, fracture toughness, failure modes, stress–strain behaviors, and brittleness). Compared to other relative homogenous rocks in this study, correlation among the brittleness indexes is different for shale samples (Fig. 10).

Fig. 10
figure 10

Failure morphology of typical rock samples for granite, marble, and sandstone. a Granite; B23 = 0.242. b Marble; B23 = 0.387. c Sandstone; B23 = 0.511

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The absence of universally accepted definition and measurement of brittleness has led to various methods or models for its quantification, as already summarized in Table 1. Generally, it has been long accepted that formations with high brittleness are easy to be fractured, but this assumption is not always true. Taking dolomitic limestone for example, fracture barrier occurs in this high brittleness rock, because fracture gradient in dolomitic limestone formation is higher than in shale formation, and the fracture is different to cross it (Jin et al. 2014). Therefore, it may not be reasonable for formations to be easily fractured with high brittleness and brittleness index based on rock geomechanics should reflect the fracture pattern and crack density; in other works, we should establish relationship between the brittleness index and fracturing effectiveness. Here, the proposed brittle index of B23, objectively, can better reflect the fracturing characteristics and the associated fracture pattern after experiment. For granite, marble, and sandstone, the average B23 value is 0.242, 0.387, and 0.512, respectively. Figure 11 shows the failure morphology of these three kinds of rock. It can be seen that the fracture scale is the largest for rock with a lower B23 value.

Fig. 11
figure 11

Crack morphology visualized by CT imaging. Image 2D reconstructions of shale samples with different inclination, CT scans at different positions, at 35, 50, and 65 mm, from the top of the samples

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Fig. 12
figure 12

Plot of fracture density against bedding plane orientation

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From the fracture pattern observation on sample surface alone one may not verify the reliability that B23 can reflect the fracturing effectiveness or the fracability of rock formations. We further study the fracture morphology in shale sample using high-energy X-ray CT. Figure 11 plots typical CT images, and the cracks are identified for shale with orientations of 0°, 15°, 30°, 45°, 60°, 75°, and 90°, respectively. From the observation of the CT images, the fracture scale is the maximum for sample with 90° orientation; however, it is the minimum for sample with 30° orientation. We plot the relationship between fracture density (which is defined as the ratio of total crack area to the sample section) and orientation, as shown in Fig. 11. Quantitative analysis also indicates that fracture density is larger for shale samples with 90° orientation. The relationship between the fracture density and the B23 is shown in Fig. 13, it shows that a good correlation exists between them, and the sample with larger stimulated fracture area corresponds to a lower σ ci/σ c value (Fig. 12).

Fig. 13
figure 13

Plot of fracture density against crack initiation stress level (B23)

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Conclusions

Brittleness is known as one of the most important properties of rock material. Brittleness of rock is very closely related to the stability of the rock mass engineering and reservoir engineering. Therefore, it is significant and important to conduct some related research on rock brittleness. In this work, we have conducted plenty of UCS tests for diorite, granite, marble, sandstone, and shale and a new brittleness index defined as σci/σc is used to evaluate the brittleness and fracture pattern. The link between brittleness and σ ci/σ c was explored from the theoretical and experimental analysis. The relations between the brittleness and σ ci/σ c were statistically investigated using the raw data obtained from experimental data of the five examples. Some main conclusions are drawn as follows:

  1. 1.

    Brittleness index of B23 reflects the rock anisotropy and non-homogeneity, and it is strongly related to the rock mesoscopic fabric. A strong relationship between the brittleness indexes B19, B20, B21, B22, and B23 (σ ci/σ c) was found. Due to the differences in rock fabric, the regression results are various for different rocks. It can be concluded that index of σ ci/σ c can be viewed as a promising index for the evaluation of rock brittleness.

  2. 2.

    For the studied rocks, a negative correlation exists between B19, B20, B21, B22, and B23. The results indicate that crack initiation stress level decreases with the increase in brittleness and micro-fracture would occur earlier during deformation. For sample with low σ ci/σ c value, larger fracture network density may be stimulated.

  3. 3.

    Fracture morphology description and CT images analysis indicate that rock brittleness has certain internal relationship with crack initiation stress level. For shale samples studied in this work, a good correlation has been found between the fracture density and σ ci/σ c, which is in good agreement with the CT scanning results.