1 Introduction

Rocks in nature generally contain initial micro-defects such as cracks and pores. The failure of brittle rocks under compression loading is progressive, including the crack initiation, growth, and coalescence (Brace et al. 1966; Chen et al. 2021; Zhang et al. 2020a, 2019, 2020b). The study of progressive failure is very important for the stability analysis of residual coal pillar, goaf, gas storage and borehole wall (Ma et al. 2020, 2023). Rock type, mineral particle size, crack and pore distribution influence rock failure (Byun et al. 2015; Patel and Martin 2020; Yin et al. 2019; Zhang et al. 2011; Zhao et al. 2020). Nicksiar and Martin (2014) found that the content of hard minerals and the size of mineral particles affect the crack initiation stress (CI), through statistics of the uniaxial compressions of different rock types including igneous, sedimentary, and metamorphic. Still, rock types and mineral composition do not affect the ratio of CI to Peak (CI/Peak). In addition, experiments show that the ratio of CD to Peak (CD/Peak), as an intrinsic property of the low-porosity rock, can be used to describe the failure process in rock engineering (Xue et al. 2013). These conclusions are all from the experiments under uniaxial compression or low confining pressure. There are few reports on CI/Peak and CD/Peak for high confining pressure. Theoretical analysis (Liu et al. 2017), numerical simulation (Nicksiar and Martin 2014) and laboratory experiments (Hoek and Martin 2014) were carried out to study the factors influencing the crack initiation (CI) and damage stresses (CD). But these results are mainly about the qualitative analysis of rock types, mineral particle size and shape. There are a few experimental studies about influence of initial cracks on the failure process. Patel and Martin (2020) investigated the role of initial crack volume on the failure of granite using bonded particle model. They found that cracks among the mineral grains and associated void space played a major role in CI for uniaxial compression. Yin et al. (2019) studied the relationship between initial cracks and failure process of schist quantitatively for uniaxial compression using the scanning electron microscope. Our experiments report the impact of initial cracks on failure process under high confining pressure based on acoustic waves.

The wave velocities are related to the microstructural characteristics and stress state of rocks (Fortin et al. 2007; Shapiro 2003; Wang 2001). Specifically, in the failure process under compression, the growth of cracks and the collapse of pore structures are the primary mechanisms for reducing the elastic wave velocity for crystalline and porous rocks (Scott Jr et al. 1993). The effective medium theory is useful for linking elastic modulus to composition matrix, fluid, cracks, and pores of rocks. Microstructural can be back analyzed based on the relationship between elastic wave velocity and elastic modulus. Le Ravalec and Guéguen (1996) calculated the equivalent elastic properties of an isotropic material using an extended differential self-consistent model which considers the effects of circular pores and cracks. The effective elastic properties of materials containing cracks of various shapes can be calculated based on the noninteraction approximation model, which assumes crack and pore distributions are random (Sayers and Kachanov 1995; Walsh 1965). In addition, Sayers and Kachanov (1995) proved that the noninteraction approximation model is the most appropriate for both isotropic and aligned cracks and remains accurate for high crack density when assuming locations of crack centers are homogenous.

This paper reports experiments of 121 sedimentary rocks from deep reservoirs. We carry out elastic wave velocity (\({V}_{p}, {V}_{s}\)) tests before triaxial compression with confining pressure of 12–85 MPa. The density of initial cracks is calculated using elastic wave velocities. The effect of confining pressure and initial crack density on CI and CD in failure process is investigated with these experiments, providing a reference for analyzing the surrounding rock stability of underground space, deep drilling, and others.

2 Samples isotropy from wave velocities

The initial crack density is of key importance in rock failure but is difficult to measure. Thus, this has given rise to characterize crack density with wave velocities under the isotropic assumption. We will discuss the sample isotropy in this section as a prerequisite for back-analyzing crack density in Sect. 5.1.

Experimental samples are from the Junggar Basin (southern and northwest margins) in Xinjiang. These samples are from sandstone, mudstone, and conglomerate reservoirs in the oil field. Cored samples (25 × 50mm) are dried in a vacuum oven at 60 °C for 48 h. There are 121 samples cored along the wellbore (V) and the wellbore radius (H) in Fig. 1a.

Fig. 1
figure 1

a Experimental samples and coring orientations, b the characteristic stresses in stress–strain curves, c the CD corresponds to the turning point of the volumetric strain, and the CI is determined using the LSR method, d the CC is determined with the ASR method

Full size image

P- and S-wave velocities (\({V}_{p}, {V}_{s}\)) are measured at room temperature without confining pressure. Anisotropic parameters of P- and S-waves are calculated using elastic wave velocities, as Eq. (1) (Thomsen 1986),

$$\left\{\begin{array}{c}{\gamma }_{s}=\frac{\left|{V}_{sH}^{2}-{V}_{sV}^{2}\right|}{2{V}_{sV}^{2}}\\ {\gamma }_{p}=\frac{\left|{V}_{pH}^{2}-{V}_{pV}^{2}\right|}{2{V}_{pV}^{2}}\end{array}\right.$$
(1)

where \({\gamma }_{s}\) and \({\gamma }_{p}\) are anisotropic parameters of S-and P-wave velocities. \({V}_{sH}\) and \({V}_{pH}\) are S-and P-wave velocities of samples cored along the wellbore radius. \({V}_{sV}\) and \({V}_{pV}\) are S-and P-wave velocities of samples cored along the wellbore.

We use the mean values of S- and P-wave velocities of samples cored along the wellbore (\({{V}_{sV}}^{m}, {{V}_{pV}}^{m}\)) and wellbore radius (\({{V}_{sH}}^{m}, {{V}_{pH}}^{m}\)) to calculate the anisotropy of samples (Table 1). Anisotropic parameters shows that the mudstone b is more anisotropic than other samples in Table 1. The P-wave anisotropy parameters are 0.033–0.053 and 0.132–0.047 for sandstone, 0.088–0.089 and 0.052–0.068 for conglomerate, and 0.042 and 0.029 for mudstone a. These anisotropic parameters are obviously lower than 0.16–0.66 and 0.17–0.72 of anisotropic sandstone and mudstone in Sun (2019). We can conclude that samples show less anisotropy and can be assumed as isotropic except mudstone b. Meanwhile, Table 2 shows petrophysical properties of samples.

Table 1 Mean velocities of P- and S-wave and sample anisotropy
Full size table
Table 2 Petrophysical properties of the different sandstone, mudstone and conglomerate
Full size table

3 The progressive failure for different rock types

The progressive failure, characterized by CI/Peak and CD/Peak, is inherent property of rock resulting from statistics of uniaxial experiments (Zafar et al. 2022; Shirole et al. 2020), which is unverified for rocks under high confining pressure. In this section, a big data from experiments of different rock types is used to fit CI/Peak and CD/Peak for high confining pressure, investigating difference in progressive failure between uniaxial and triaxial tests.

3.1 The determination of CI and CD

In this paper, we use the MTS816 to carry out triaxial compression experiments with confining pressures which are 12–85 MPa. The loading rate is 0.04% axial strain, and stress–strain curves are obtained. Martin (1993) divided the progressive failure process of brittle rock into five stages, (1) initial crack closure, (2) linear elastic deformation, the start point of this stage is defined as the crack closure stress (CC), (3) crack initiation and stable growth begin at the crack initiation stress (CI), (4) crack unstable growth and coalescence occurs between the crack damage stress (CD) and the peak stress (Peak), (5) failure and post-peak behavior. The characteristic stresses are captured in stress–strain curves as shown in Fig. 1b. The CD corresponds to the turning point of the volumetric strain, the CI is determined using the LSR method (Nicksiar and Martin 2013a; Wen et al. 2018) (Fig. 1c), and the CC is determined with the ASR (Rong et al. 2018; Zhang and Tang 2020) method (Fig. 1d).

3.2 The CI/Peak and CD/Peak ratios

Under triaxial compression conditions, the CI/Peak and CD/Peak for 121 sedimentary rock samples are drawn in Fig. 2. The CI and CD have a linear relationship with the Peak according to the linear least squares fitting. The results illustrate that CI/Peak under confining pressures of 12–24 MPa, 40–52 MPa and 66–85 MPa are 0.579, 0.637, and 0.623 (As shown in Fig. 2a). Figure 2b shows that CD/Peak are 0.913, 0.929 and 0.938 for 12–24 MPa, 40–52 MPa and 66–85 MPa confining pressures from our experiments. The CI/Peak and CD/Peak obviously change for different confining pressures, and we will discuss this phenomenon in detail in Sect. 4.

Fig. 2
figure 2

a Relationship between the initiation stress and the peak stress of sedimentary rocks under different confining pressures, b Relationship between the damage stress and the peak stress of sedimentary rocks under different confining pressures (the solid line is the regression line; the dotted line is the 95% confidence interval; black marker is sandstone; red marker is mudstone; blue marker is conglomerate)

Full size image

Results in publications (Nicksiar and Martin 2013a; Xue et al. 2013) show that the CI/Peak and CD/Peak are independent of rock types under uniaxial and triaxial compression for low confining pressure. To further ascertain whether different rock types have similar CI/Peak and CD/Peak for high confining pressure, the fitting results of sandstone, mudstone, and conglomerate are shown in Fig. 2. The R2 of the regression lines of all data sets are above 0.97, indicating that the data of sandstone, mudstone and conglomerate are all in good agreement with fitted lines. Therefore, the CI/Peak and CD/Peak of sedimentary rock are also independent of rock types for high confining pressure.

Table 2 shows that average grain size of mudstone is the smallest, followed by sandstone and conglomerate. In Fig. 2, data points with different grain sizes are distributed in a relatively narrow range around the fitting lines. This proves that grain size has a negligible effect on the CI/Peak and CD/Peak for triaxial compression, which agrees with result from Nicksiar and Martin (2013a) and Xue et al. (2013). They found that grain size is not a significant factor in controlling the CI/Peak and CD/Peak based on uniaxial compression tests. Therefore, the CI/Peak and CD/Peak can be used to evaluate the progressive failure of rock neglecting the rock types and grain sizes.

Grain size has less effect on the CI/Peak and CD/Peak, but will influence the absolute values of CI, CD and Peak. Figure 3 presents higher CI, CD and Peak of conglomerate than mudstone and sandstone as increase of grain size. The increasement of CI, CD are nearly equal to that of Peak varying from mudstone, sandstone and conglomerate. CI/Peak and CD/Peak are almost unchanged on account of the same increasement of CI, CD and Peak.

Fig. 3
figure 3

The crack initiation stress, the crack damage stress and the peak stress of sedimentary rocks with different grain sizes (The grain size is smallest for mudstone, it is medium for sandstone and largest for conglomerate)

Full size image

4 Effect of confining pressure on CI/Peak and CD/Peak

The effect of confining pressure on progressive failure from previous publications (Nicksiar and Martin 2013b; Xue et al. 2013; Li et al. 2020; Huang et al. 2021; Wen et al. 2018) are inconsistent resulting from absence of experiments of high confining pressure. We investigate the evolution of CI/Peak and CD/Peak for different confining pressures from 0 to 85 MPa in this part.

Wen et al. (2018) concluded that CI/Peak linearly increases with confining pressure. However, Nicksiar and Martin (2013b) believed that the confining pressure has no significant inhibition on the initiation of new cracks and has little influence on CI/Peak. As mentioned above, the confining pressure in this paper ranges from 12 to 85 MPa while it is mainly 0 – 20 MPa used by Nicksiar and Martin (2013b). Triaxial experiments from our work further support that high confining pressure strongly affects CI/Peak. Figure 4a shows that CI/Peak increase from 0.42–0.57 to 0.63–0.67 with confining pressures from 0–24 to 40–85 MPa for sedimentary rocks.

Fig. 4
figure 4

a Effects of confining pressure on CI/Peak and CD/Peak for sedimentary rocks including sandstone, mudstone and conglomerate. b Effects of confining pressure on CI/Peak and CD/Peak for sandstone (The data in the red circle is lower than the mean value at 68–85 MPa; The data in the blue circle is higher than the mean value at 40–52 MPa). c Effects of confining pressure on CI, CD and Peak for sandstone

Full size image

The moderate increasement of confining pressures significantly enhanced the CD/Peak for sedimentary rocks. CD/Peak is 0.91 for 12–24 MPa in Fig. 4a, which is obviously higher than 0.73–0.78 for 0 MPa obtained by Xue et al. (2013). Yang (2016) concluded similar results that the ratio of CD/Peak increases from 0.71 to 0.95 with 0 MPa to 35 MPa. For high confining pressure, there is slightly increase from 0.91 to 0.93 when confining pressure increase from 12–24 MPa to 68–85 MPa.

More specifically, the CD/Peak and CI/Peak of sandstone are illustrated in Fig. 4b. Figure 4b presents that CD/Peak and CI/Peak do not monotonically increase with confining pressure, which decrease by 10.4% and 4.2% when confining pressure increases from 40–52 MPa to 68–85 MPa. For 68–85 MPa, most data points are near the lower bound of the confidence interval, while most data points are near the upper bound for 40–52 MPa. These results indicate that the downward trend is more significant than average for most samples. This downward trend is significant for sandstone, but for mudstone and conglomerate, more experiments need to be conducted in the future.

Figure 4c shows that CI, CD and Peak increase differently with the increase of confining pressure for sandstone. CI and CD increases by 59% and 39% respectively, both above the 34% of Peak when confining pressure increases from 12–24 MPa to 40–52 MPa. The high increasement of CI and CD results in growth of CI/Peak and CD/Peak. The CI/Peak and CD/Peak shows downward trend for 68–85 MPa, due to the lower rise of CI and CD than Peak. Figure 4c presents that CI is almost unchanged (0.5%) but Peak increases by 11%. The results suggest that there is a critical point for confining pressure to inhibit microcrack cracking, and 40–52 MPa could be the threshold value of sandstone.

5 Relationship between initial crack density and stress ratios

The crack distribution controls progressive failure process, which is sensitive to confining pressure. However, the evaluation of crack density has remained challenge. In this part, acoustic waves are adopted to back-analyze initial crack density of sandstone based on the equivalent medium model, and thus the effect of initial crack density on CI/Peak and CD/Peak is discussed.

5.1 Initial crack densities of the samples

The elastic properties of isotropic rocks depend partly on pore (spherical) and crack (coin-shaped) distribution. The initial cracks distributed in rock account for about 1% of porosity, but they were the dominant factor affecting elastic properties (Nicolas et al. 2017b). In addition, the extension and sliding of cracks between or within mineral grains were the primary microscopic mechanism of a progressive failure process (Patel and Martin 2020). Byun et al. (2015) proved that the coin-shaped crack assumption was suitable for dry rocks. The elastic wave velocities are measured in the dry condition in this paper, so that initial crack properties are back-analyzed for coin-shaped cracks in this paper.

Equations (2)–(5) present the relationship between elastic modulus and the crack density \(\omega\). (Mallet et al. 2013) based on isotropic assumption of samples.

$$\left\{\begin{array}{c}\frac{{E}_{0}}{{E}^{*}}=1+\left(1+\frac{3}{5}\left(1-\frac{{v}_{0}}{2}\right)\left(\frac{\delta }{1+\delta }-1\right)\right)h\omega \\ \frac{{G}_{0}}{{G}^{*}}=1+\left(1+\frac{2}{5}\left(1-\frac{{v}_{0}}{2}\right)\left(\frac{\delta }{1+\delta }-1\right)\right)\frac{h\omega }{1+{v}_{0}}\end{array}\right.$$
(2)

where \({E}_{0}\), \({G}_{0}\) are young`s modulus and shear modulus of rock matrix, \({E}^{*}\), \({G}^{*}\) the effective young`s modulus and the effective shear modulus, \({v}_{0}\) the effective Poisson’s ratio, \(h\) the geometric parameters of a coin-shaped crack in Eq. (3), \(\delta\) a dimensionless parameter characterizing the stress state inside rock in Eq. (4), \(\omega\) the scalar crack density of the rock in Eq. (5).

$$h=\frac{16(1-{v}_{0}^{2})}{9(1-{v}_{0}/2)}$$
(3)
$$\delta =(1-{v}_{0}/2)\frac{{E}_{0}\xi }{{K}_{f}}h$$
(4)

In Eq. (4), \({K}_{f}\) is the bulk modulus of the fluid inside the rock, \({K}_{f}\) is 1 for dry case in this paper, \(\xi =c/a\) the average width to length ratio of the cracks, and \(c\) and \(a\) the half-width and half-length of the cracks. We use Budiansky and O’Connell’s (1976) (Nicolas et al. 2017a) definition of crack density \(\omega\) in Eq. (5).

$$\omega =\frac{1}{V}{\sum }^{N}{a}_{i}^{3}$$
(5)

where \(V\) is the volume of REV element, \(N\) the number of cracks in REV, \({a}_{i}\) the half-length of the \(i\) th crack.

The effective elastic modulus also can be calculated using elastic wave velocities as Eqs. (6) and (7).

$${G}^{*}={{V}_{s}}^{2}\rho$$
(6)
$${K}^{*}={{V}_{p}}^{2}\rho -\frac{4}{3}{G}^{*}$$
(7)

\({E}^{*}=\frac{9{K}^{*}{G}^{*}}{3{K}^{*}+{G}^{*}}\), where \({E}^{*}\) is the effective young`s modulus, \({K}^{*}\) the effective bulk modulus, \({G}^{*}\) the effective shear modulus, \(\rho\) the density of rock.

Substituting Eqs. (6) and (7) into Eq. (2), the relationship between the initial crack properties and the elastic wave velocities can be established. The parameters of initial cracks are back-analyzed using the P-wave velocities obtained in Sect.  2. Mallet et al. (2013) and Ghabezloo (2015) used wave velocities to back-analyze the crack density of glass, and the number of back-analyzed cracks was consistent with the measurements. It is worth noting that the crack density (\(\omega\)) in the above model is dimensionless in a statistical sense.

The young`s modulus of matrix \({E}_{0}\) is about 75 GPa (Fortin et al. 2007; Qin et al. 2022), the effective poison ratio \({v}_{0}\) is about 0.2 for sandstone (Mavko et al. 2020; Paterson and Wong 2005). In this paper, initial crack density \(\omega\) of sandstone (Appendix A) is back-analyzed using the least-squares method.

5.2 Effects of initial crack density on stress ratios

In Fig. 5, CI/Peak and CD/Peak of sandstone show a decreasing trend with the increase of the initial crack density for 12–24 MPa and 68–85 MPa, and they decrease more significantly for 12–24 MPa. However, it appears that CI/Peak and CD/Peak are relatively insensitive to the initial crack density for 40–52 MPa. These results confirm the effect of initial crack density on stress ratios is affected by confining pressure.

Fig. 5
figure 5

Effects of initial crack density on CI/Peak (a) and CD/Peak (b) for sandstone

Full size image

The effect of the initial crack density is weakened when the confining pressure increases from 12–24 to 40–52 MPa. The effect is enhanced when confining pressure increases from 40–52 to 68–85 MPa, see Fig. 5. For 12–24 to 40 – 52 MPa, the confining pressure significantly increase the CI/Peak and CD/Peak so that eliminate weakened effect from initial crack density. The initial crack density becomes dominate factor and reduces the CI/Peak and CD/Peak when the influence of confining pressure is weakened above critical point. Figure 4c shows that critical confining pressure is 40–52 MPa for sandstone.

Patel and Martin (2020) used PFC simulation to study the effect of initial cracks on rock mechanical properties of granite. A microfracture was created in the model by reducing the tensile strength of all elements in a flat-jointed contact to zero. They found that the model with higher initial crack density can accommodate more lateral extension among the grains and delay the crack initiation. Hence CI/Peak shows an increasing trend (from 0.29 to 0.41) with increasing initial crack density without confining pressure. However, our triaxial experiments statistically suggest a reversed trend that CI/Peak decreases with initial crack density for 12–24 MPa. The confining pressure inhibits lateral extension and accelerate crack initiation, making the significantly difference in influence of initial crack density between uniaxial and triaxial compression tests. We can infer that confining pressure is the main factor rather than the initial crack density for low confining pressure from the huge difference, although it is between granite simulations and sandstone experiments. It is an interesting topic proving this inference by experiments for future study.

6 The dependence of CD stress on crack properties

The initiation and propagation of stress-induced cracks can be analyzed using linear elastic fracture mechanics, involving micromechanical parameters such as the porosity, friction coefficient of initial crack, crack density, crack dimension and fracture toughness (Ashby and Sammis 1990; Baud et al. 2014; Ding et al. 2017; Li et al. 2018; Nicolas et al. 2017a; Wong et al. 1996; Zhang and Ma. 2021; Zhang and Ma. 2021; Zhang et al. 2024). The sliding of initial cracks among grains is the primary mechanism of crack initiation and extension under high confining pressure in experiments (Liu et al. 2017; Huang et al. 2021). And the crack density and friction-slip properties of cracks have a marked impact on the CI and CD. We will use the wing crack model (Ashby and Sammis 1990), which can describe the properties and interaction of cracks, to study the dependence of CD on initial cracks.

Firstly, we adopt the three-dimensional damage mechanic model (Ashby and Sammis 1990) to describe the cracking process with the axial load \({\sigma }_{1}\) for different confining pressure \({\sigma }_{3}\) as Eq. (8).

$${S}_{1}=-\left({C}_{2}\sqrt{{\left(A+\beta /\alpha \right)}^{3}}-{S}_{3}\left({C}_{1}B+{C}_{3}{A}^{2}\right)\right)/B$$
(8)

where \({S}_{1}\) and \({S}_{3}\) are defined as \({S}_{1}={\sigma }_{1}\sqrt{\pi a}/{K}_{{\text{IC}}}\) and \({S}_{3}={\sigma }_{3}\sqrt{\pi a}/{K}_{{\text{IC}}}\).

In Eq. (8), the constants are defined as \(A=\sqrt[3]{D/{D}_{0}}-1\), \(B=1+2{{A}^{2}D}_{0}^{2/3}/\left(1-{D}^{2/3}\right)\), \({C}_{1}={A}_{3}/{A}_{1}\), \({C}_{2}={\pi }^{2}{\alpha }^{3/2}/{A}_{1}\), \({C}_{3}=2{\alpha }^{2}{\pi }^{2}/{A}_{1}\).


where \({A}_{1}=\pi \sqrt{\beta /3}\left({\left(1+{\mu }^{2}\right)}^{1/2}-\mu \right)\), \({A}_{3}={A}_{1}\left({\left(1+{\mu }^{2}\right)}^{1/2}+\mu \right)\left({\left(1+{\mu }^{2}\right)}^{1/2}-\mu \right)\).

Specific parameters of sandstone are determined from publication of Ashby and Sammis (Ashby and Sammis 1990), the dimensionless coefficients \(\alpha\) and \(\beta\) are 0.7 and 0.45, the half-length of the initial crack \(a\) is 0. 25 mm based on grain size in Table 2, the friction coefficient of the initial crack \(\mu\) is 0.6. The fracture toughness \({K}_{{\text{IC}}}\) of sandstone samples are obtained with semi-disk test, which are \(0.76{\text{MPa}}\sqrt{{\text{m}}}\).

The initial damage factor \({D}_{0}\) and the damage factor \(D\) during loading are defined as,

$${D}_{0}=\frac{4}{3}\pi {\left(\alpha a\right)}^{3}\omega$$
(9)
$$D=\frac{4}{3}\pi {\left(l+\alpha a\right)}^{3}\omega$$
(10)

The \(\omega\) is defined in Eq. (5), then the initial damage factor \({D}_{0}\) can be obtained.

Furthermore, the critical wing crack length for the damage stress threshold is Eq. (11) (Li et al. 2018).

$${l}_{{\text{lim}}}=\alpha a\left({{D}_{0}}^{-1/3}{\alpha }^{-1}-1\right)$$
(11)

Substituting Eq. (11) into Eq. (8), we can obtain the relationship between crack damage stress (CD is the value of axial stress \({\upsigma }_{1}\) corresponding to the strat of crack unstable growth) and crack density (\(\omega\)). The crack damage stress of sandstone calculated with the above model are shown in Fig. 6 for different confining pressure (20, 45 and 80 MPa). We find that model results agree well with experiments when initial crack densities are less than 1. Meanwhile, CD decreases rapidly with the initial crack density.

Fig. 6
figure 6

Effects of initial crack properties on crack damage stress for sandstone

Full size image

However, model results for high initial crack density (\(\omega >1\)) are significantly lower than the experimental data. As shown in Fig. 6, model results with an increasing friction coefficient of initial crack (from 0.6 to 0.75) are consistent with experiments for samples with high initial crack density. This fact shows that damage stress increases with friction coefficient of cracks when crack density is high. Samples treated above 1000 °C show the same trend. They have higher crack density and concurrently increased friction coefficient of cracks, due to mineral melting and pores filling examined by XRD and SEM (Yi et al. 2021). Zhang et al. (2020c) found that the friction coefficient is the dominant factor affecting the failure process of water-saturated sandstone after introducing SC−CO2. We can conclude that the crack damage stress is also sensitive to friction coefficient when the crack density is high.

7 Conclusions

In this paper, the effect of initial crack density and high confining pressure on progressive failure are investigated based on triaxial compressions of 121 sedimentary rocks. Several conclusions can be drawn as follows:

The confining pressure significantly affects CI/Peak which increases from 0.42–0.57 to 0.63–0.67 with confining pressures from 0–20 to 12–85 MPa. CD/Peak increases from 0.73–0.78 to 0.91–0.95 with confining pressures from 0 to 85 MPa, concluding from experiments of sedimentary rocks.

Specifically, results from sandstone presents that CD/Peak and CI/Peak do not monotonically increase with confining pressure, which decrease by 10.4% and 4.2% when confining pressure increases from 40–52 to 68–85 MPa. There is a critical point for confining pressure to inhibit microcrack cracking, and 40–52 MPa could be the threshold value of sandstone.

The effect of initial crack is sensitive to confining pressure. That is weakened when the confining pressure increases from 12–24 to 40–52 MPa but enhanced when the confining pressure increases from 40–52 to 68–85 MPa. The initial crack density becomes dominate factor and reduces the CI/Peak and CD/Peak when the influence of confining pressure is weakened above critical point.

Results from damage mechanic model agree well with experiments when initial crack density is low, and CD decreases rapidly with the initial crack density. CD is also sensitive to friction coefficient of crack when crack density is high. The enhancement of friction coefficient to CD is obvious.