Effect of bedding dip angle and dynamic load on spatial variation of microscopic failure stress of shale

Abstract

Physico-mechanical properties of shale are important parameters in evaluating the stability of potential wellbore and the design of hydraulic fracturing, which are primarily affected by their non-uniform spatial distribution of the microscopic physical–mechanical properties at particle scale. A series of constant strain rate experiments and stress-cycling experiments on shale specimens with different bedding dip angles were conducted to have a comprehensive understanding of the effect of the non-uniform distribution of microscopic failure stress on macroscopic physico-mechanical properties. According to the experimental results and Weibull distribution, we find that bedding dip angle and the dynamic load applying type affect the spatial distributions of microscopic failure stress. The values of crack damage stress (σ_cd), σ_cd/σ_ucs (peak stress), ε_cd (strain at crack damage stress), Poissons' ratio (ν), elastic strain energy (U_e) and dissipated energy (U_irr) of the specimens with more uniform distribution of microscopic failure stress are overall higher, while ε_ucs (peak strain)/ε_cd and elastic modulus (E) are lower. The dynamic load enables the spatial distributions of microscopic failure stress trend to be more homogeneous prior to the final failure with the increment of σ_cd/σ_ucs, ν, U_e and U_irr and the decrement of E.

Introduction

Shale is one of the various rock types encountered when exploring and drilling for petroleum and shale gas as well as during civil engineering project construction process^1–6. Its macroscopic physico-mechanical properties are important parameters in evaluating the stability of potential wellbore and the design of hydraulic fracturing, etc. The effects of microstructure on macroscopic physico-mechanical properties of shale have been one of the major research topics in recent years ^5,7–15. These studies indicate that the heterogeneity of the microstructure is an intrinsic composite characteristic of the shale, which is made up of various mineralogical components, particle-sizes, porosities, cracks, etc. The composite heterogeneous microstructure induces non-uniform distribution of microscopic physico-mechanical properties at particle scale, significantly affecting rock progressive failure process and macroscopic physico-mechanical properties^16–26. Therefore, it is important to investigate the relationship between the microscopic and macroscopic physico-mechanical properties of shales. However, due to the characteristics of the random distribution, the microscopic physico-mechanical properties are difficult to be quantified by analytical methods or measured by experiments as other rock and rock-like materials²⁷.

Weibull distribution has been used widely to model the statistical distribution of the microscopic physical–mechanical properties of rock or rock-like materials induced by the multiscale material heterogeneities^28–32. The microscopic physical–mechanical properties, such as element failure stress (hereinafter referred to as failure stress), σ_f, are assumed to be randomly distributed throughout the specimen elements following a Weibull probability distribution function:

$$f\left({\sigma }_f\right)=\frac{m}{{\sigma }_0}{\left(\frac{{\sigma }_f}{{\sigma }_0}\right)}^{m-1}{e}^{-{\left(\frac{{\sigma }_f}{{\sigma }_0}\right)}^m}$$ 1

where σ₀ is the mean failure strength of the elements of the specimen, m is the homogeneity index of the material. For given σ₀, an infinitely high m value corresponds to a uniform distribution of failure strength, whereas a broad distribution of failure strength is associated with a relatively low m value^33,34. A case of Weibull strength distribution for different homogeneity index m is shown in Fig. 1. It indicates that the heterogeneity of failure stress can be characterized by the Weibull distribution with a reasonable homogeneity index m.

Figure 1.

Weibull probability distribution of failure strength for different homogeneity index m.

In previous numerical modelling studies, Weibull distribution with various m values has been used as a promising statistic model to characterize the spatial variation of the microscopic physico-mechanical properties of rock and rock-like materials^35–39. One of the challenges of using Weibull distribution is to obtain the m value practically. In these studies, the m values were normally obtained through the Linear Least Squares (LLS) technique, which involves the ranking of σ_f of the elements of the modelled specimen in ascending order⁴⁰. In order to assess the spatial variation of the microscopic physical–mechanical properties of rock specimen in laboratory scale, Pan et al.⁴¹ proposed an experimental method for obtaining the m value via macroscopic physico-mechanical parameters.

What’s more, dynamic cyclic loading is an important loading type for shales near the wellbore surface and hydraulic fractured surface which are under conditions of uniaxial pressure or low confining pressure⁴². The objective of this paper is to investigate the effect of the non-uniform distribution of microscopic failure stress on macroscopic physico-mechanical properties of shales subjected to uniaxial dynamic pressures based on Weibull distribution.

Methodology

Obtaining the value of m practically is a critical issue to characterize the heterogeneous distribution of σ_f of shale specimens using Weibull distribution. Pan et al.⁴¹ constructed a quantitative relationship between m value and strain ratio (ε_ucs/ε_cd) of specimen subjected to uniaxial compression loading. As shown in Fig. 2, peak strain ε_ucs is the strain at peak stress σ_ucs (point D), crack damage strain ε_cd is the strain at crack damage stress σ_cd (point C). The value of m can be determined after the ε_ucs/ε_cd has been estimated through experimental results. It is well known that ε_ucs can be directly determined from the stress–strain curve while it is difficult for ε_cd. Therefore, volumetric strain reversal point, ε_vr, (point C₁ in volumetric strain-axial strain curve as shown in Fig. 2) is used to estimate the σ_cd. The quantitative relationship between m and ε_ucs/ε_cd is presented as follows.

Figure 2.

Stress–strain diagram showing the steps of crack development and relationship with volumetric strain-axial strain curve (modified after Martin and Chandler⁴³).

Relationship between the homogeneity index m and strain ratio ε_ucs/ε_cd

For a cylindrical core specimen of rock and rock-like material in laboratory scale, the σ_f of each element was suggested to obey a Weibull distribution to represent the inherent heterogeneity of failure strength distribution. When the specimen was subjected to uniaxial compression load, a quantitative relationship between m (m ≥ 1) and ε_ucs/ε_cd was constructed based on two-dimensional renormalization group theory by Pan et al.⁴¹ as follows.

$$\frac{{\epsilon }_{\mathrm{ucs}}}{{\epsilon }_{\mathrm{cd}}}={\left[-,\frac{1}{m\mathit{ln}\left(1,-,A_{\ast },+,B_{\ast }^{\frac{1}{C_{\ast }^m-D_{\ast }}}\right)}\right]}^{\frac{1}{m}}$$ 2

where A_* = 1.004079, B_* = 0.768013, C_* = 1.603054, and D_* = 1.131608.

However, the relationship was not established as 0.5 ≤ m < 1. Based on Eq. 8 presented in Pan et al.⁴¹, A_*, B_*, C_* and D_* for 0.5 ≤ m < 1, is extracted by the trust region regression method⁴⁴, which are 1.036838, 0.219326, 3.210927 and 0, respectively.

The analysis based on the results indicates that ε_ucs/ε_cd gradually decreases as m increases from 0.5 to 30. Thus, the m value can be obtained once the ε_ucs/ε_cd is determined.

Determination of the crack damage strain ε_cd

Volumetric strain (ε_v) is a pervasive volumetric property of the rock and rock-like material⁴⁵. For a cylindrical specimen subjected to axial loading and under small strain, the volumetric strain is given by Martin and Chandler⁴³:

$${\epsilon }_{\mathrm{v}}={\epsilon }_{\mathrm{axial}}+2{\epsilon }_{\mathrm{lateral}}$$ 3

An example of relationship between axial, lateral strain versus axial stress curve and volumetric strain versus axial strain curve in uniaxial compression is given in Fig. 2. The axial strain level where the volumetric strain reversal occurs at point C₁ (volumetric strain reversal point ε_vr) marks the beginning of Region IV and represents the onset of unstable crack growth at point C^43,46. The volumetric strain reversal occurs at point C₁ can be directly obtained in the curve. Therefore, the ε_cd can be estimated using the same strain level of ε_vr.

Experiments

In this study, constant strain rate experiments and stress-cycling experiments were both performed in a RTR-2000 triple-axis dynamic testing system. It was manufactured by GCTS Company, USA, for physico-mechanical properties testing of rocks with a maximum axial load of 2000 KN and a maximum confining pressure of 140 MPa. Constant strain rate experiments were carried out to estimate the σ_ci (crack initiation stress as shown in Fig. 2 at point B), σ_cd and σ_ucs of shale specimens with the same bedding dip angle which would be tested under dynamic loading afterwards. Stress-cycling experiments were conducted to investigate the effect of dynamic loading on heterogeneous distribution of σ_f and physico-mechanical properties.

Specimen preparation

Shale specimens with three types of bedding dip angles were cored through a shale block with different drilling angles. The block collected from Lower Silurian Longmaxi Formation of Sichuan basin in Chongqing, China, was composed of quartz, feldspar, pyrite, clay, calcite, mica, and a connected low porosity and low permeability. As shown in Fig. 3, three types of bedding dip angles (θ = 0°, 60° and 90°) made with specimen end surfaces were conducted. The specimens of each type of bedding dip angle were prepared in quintuplicates, which had been standardized by International Society for Rock Mechanics standards⁴⁷. Each specimen had a 50 mm in diameter and a length to diameter ratio of 2, and the end surfaces were flat with an accuracy of 0.02 mm. The mean acoustical properties for three types of shale specimens with different bedding dip angles were presented in Table 1. It indicates that the difference of micro-structures among the specimens were dominated by the bedding dip angles.

Figure 3.

Plots of shale specimens cored from a shale block with different bedding dip angles (θ): (a) 0°, (b) 60° and (c) 90°.

Table 1.

Mean acoustical properties for shale specimens with three types of bedding dip angles.

Bedding dip angle θ (°)	0	60	90
P-wave velocity (m/s)	4091	4362	4563

Testing procedure

The quintuplicate specimens for each type of bedding dip angle were separated into five groups named A, B, C, D and E, respectively. Specimens of group A, B and C were applied at a constant strain rate of 6.0 × 10^–4/s. The σ_ucs was directly obtained from the stress–strain curve, the σ_cd was estimated using the method presented in Sect. "Determination of the crack damage strain ε_cd" corresponding to ε_vr, and the σ_ci can be identified from the point where the horizontal section of calculated crack volumetric strain-axial curve ends (point B₂ in Fig. 2)⁴³. Specimens of group D and E were applied with two types of dynamic cyclic load, respectively, which have different stress value of the first-time unloading as shown in Fig. 4. For specimens of group D, the first-time unloading was carried out until the axial stress (σ_i) exceeded the σ_ci but lower than the σ_cd as shown in Fig. 4a. The determination procedure of σ_i of the specimen of group D with the 0° bedding dip angle was presented as an example. Firstly, the mean values of σ_ci, σ_cd and σ_ucs of specimens of groups A, B and C (θ = 0°) were 25.93, 59.93 and 98.31 MPa (see Table 2), respectively. Therefore, 35.00 MPa (larger than σ_ci = 25.93 MPa and smaller than σ_cd = 59.93 MPa) with σ_i /σ_ucs = 0.36 (larger than σ_ci /σ_ucs = 0.26 and smaller than σ_cd /σ_ucs = 0.61) was determined as the σ_i of the first-time unloading for specimen of group D (θ = 0°). However, the σ_ucs (128.16 MPa) of specimen of group D (θ = 0°) was much higher than the mean value of that of specimens of groups A, B and C (θ = 0°) as affected by the dynamic cyclic loads. As a result, the values of estimated σ_i (35.00 MPa) and σ_i/σ_ucs (0.27) were much lower than the σ_ci (48.70 MPa) and σ_ci/σ_ucs (0.38) of specimen of group D (θ = 0°). Secondly, a new test with a new specimen of group D₂ (θ = 0°) was thus carried out with higher σ_i (75.00 MPa). Following the same procedure, the reasonable values of σ_i of specimens of group D for θ = 60° and θ = 90° were determined. For specimens of group E, the axial stress σ_i of the first-time unloading was between the σ_cd and σ_ucs as shown in Fig. 4b. The values of σ_i were estimated by the same procedure as those used in specimens of group D. Subsequently, peak stress (σ_p) of each cycle of the two types of dynamic cyclic loads both had a 2 MPa increment until final failure occurred.

Figure 4.

Plots of two types of dynamic cyclic loads used in experiments. Loading–unloading procedures start after an initial stress σ_i has been applied to the value (a) larger than σ_ci and smaller than σ_cd, and (b) larger than σ_cd and smaller than σ_ucs. The subsequent peak stress (σ_p) per cycle had a 2 MPa increment until final failure occurred.

Table 2.

Experimental results of shale specimens of groups A, B, C that were obtained from the constant strain rate experiments and corresponding results of shale specimens of groups D and E.

Bedding dip angle	Specimen No	σucs (MPa)	σci (MPa)	σcd (MPa)	σci/σucs	σcd/σucs	σi (MPa)	σi/σucs
θ = 0°	Group A	104.65	26.92	62.37	0.26	0.60	–	–
	Group B	92.11	20.43	47.74	0.22	0.52	–	–
	Group C	98.16	30.43	69.69	0.31	0.71	–	–
	Group D Group D2	128.16 159.37	48.70 50.37	89.71 96.92	0.38 0.32	0.70 0.61	35.00 75.00	0.27 0.47
	Group E	96.97	34.94	80.77	0.36	0.83	81.00	0.84
θ = 60°	Group A	57.98	18.07	34.98	0.31	0.60	–	–
	Group B	53.00	15.70	28.04	0.30	0.53	–	–
	Group C	57.92	16.82	44.45	0.29	0.77	–	–
	Group D	49.51	17.75	44.52	0.36	0.90	32.00	0.65
	Group E	71.98	21.77	46.29	0.30	0.64	60.00	0.83
θ = 90°	Group A	81.79	24.76	63.29	0.30	0.77	–	–
	Group B	80.82	24.36	46.58	0.30	0.58	–	–
	Group C	72.81	6.89	39.71	0.09	0.55	–	–
	Group D	133.96	39.19	83.91	0.29	0.63	40.00	0.30
	Group E	87.71	34.64	43.07	0.39	0.49	70.00	0.80

Results

Mechanical properties of shale specimens of groups A, B and C

Table 2 first presents the experimental results of the shale specimens of groups A, B and C with different bedding dip angles (θ = 0°, θ = 60° and θ = 90°) obtained from the constant strain rate experiments, i.e., σ_ucs, σ_ci, σ_cd, σ_ci/σ_ucs, σ_cd/σ_ucs, which were used to estimate the σ_i of specimens of group D and E. Figure 5 shows the mean values of σ_ucs, σ_ci, σ_cd, σ_ci/σ_ucs and σ_cd/σ_ucs of shale specimens of groups A, B and C for each dip angle. Figure 5a indicates that the mean values of σ_ci, σ_cd, σ_ucs of shale specimens for θ = 90° were larger than that for θ = 60° but smaller than the θ = 0°, i.e., the mean values of σ_ucs for θ = 90°, θ = 60° and θ = 0° were 78.47, 56.30, 98.31 MPa, respectively. Figure 5b indicates that the mean values of σ_cd/σ_ucs did not increase significantly as bedding dip angle increased which were 061, 0.63 and 0.63, while the mean values of σ_ci/σ_ucs increased slightly as bedding dip angles increased from 0° (0.26) to 60° (0.30) and decreased as bedding dip angles increased to 90° (0.23).

Figure 5.

Plots of mean values of (a) σ_ucs, σ_ci, σ_cd and (b) σ_ci/σ_ucs, σ_cd/σ_ucs of shale specimens of groups A, B and C for each bedding dip angle.

Variation of physico-mechanical properties of shale specimens of groups D and E

The typical stress–strain curves of specimens of θ = 90° in group D and θ = 0° in group E were shown in Fig. 6. Except the estimated σ_i and σ_i/σ_ucs, σ_ucs, σ_ci, σ_cd, σ_ci/σ_ucs and σ_cd/σ_ucs were also presented in Table 2. σ_ci of group D was determined from the first cycle loading and σ_cd from the first cycle loading that the σ_p exceeded the crack damage stress using the method presented in Sect. "Determination of the crack damage strain ε_cd", while the σ_ci and σ_cd of group E was determined from the first cycle loading as the initial loading was larger than the crack damage stress. σ_ucs, σ_ci, σ_cd of specimens of groups D and E with θ = 0° were both larger than that of θ = 90°, and θ = 60° were the smallest, i.e., σ_ucs of groups D for θ = 0°, θ = 90° and θ = 60° were 159.37, 133.96 and 49.51 MPa, respectively, which were in descending order. For specimens of dip angles of θ = 0° and θ = 90°, σ_ucs, σ_ci, σ_cd of of group D were larger than that of group E, while for specimens of dip angles of θ = 60°, σ_ucs, σ_ci, σ_cd of of group D were smaller than that of group E. For example, σ_ucs of groups D and E of θ = 0°were 159.37and 96.97 MPa, for θ = 90° were 133.96 and 87.71 MPa, while for θ = 60° were 49.51 and 71.98 MPa.

Figure 6.

The stress–strain curves of specimens of (a) θ = 90° in group D and (b) θ = 0° in group E.

Figure 7 presented the experimental results of each cycle loading of shale specimens of groups D and E with different bedding dip angles, including σ_cd, σ_cd/σ_ucs, ε_cd, ε_ucs/ε_cd, elastic modulus (E), Poissons’ ratio (ν). Figure 7a and c indicate that the σ_cd and ε_cd of all specimens of group D with various dip angles increased as σ_p/σ_ucs increased before final failure as well as descending order phenomenon was observed at final failure. What’s more, for the same loading type, the values of σ_cd and ε_cd of specimens of group D with θ = 0° were the largest while the θ = 60° were the smallest. For specimens of group E, the values of σ_cd and ε_cd of θ = 0° were larger than those of θ = 60° and θ = 90°. Figure 7b and d indicate that the σ_cd/σ_ucs and ε_ucs/ε_cd of all specimens of group D with various dip angles were almost the same, which were in ascending and descending orders as σ_p/σ_ucs increased, respectively. For specimens of group E, Fig. 7b shows that the values of σ_cd/σ_ucs of various bedding dip angles decreased significantly at final failure especially for θ = 90°, and the values of θ = 0° were largest as well as θ = 90° were smallest. For the ε_ucs/ε_cd of specimens of group E, values of θ = 90° decreased at initial loading cycles and increased significantly at final failure, which were the largest and that of θ = 0° were the smallest.

Figure 7.

Evolution of (a) σ_cd, (b) σ_cd/σ_ucs, (c) ε_cd, (d) ε_ucs/ε_cd, (e) E and (f) ν for different shale specimens as σ_p/σ_ucs in ascending order.

The E and ν of each cycle loading were obtained here by Eqs. (4) and (5), respectively.

$$E=\frac{{\sigma }_{\mathrm{p}}}{{\epsilon }_{\mathrm{axial}}-{\epsilon }_{\mathrm{axial}}^{\mathrm{irr}}}$$ 4

$$\nu =-\frac{{\epsilon }_{\mathrm{lateral}}-{\epsilon }_{\mathrm{lateral}}^{\mathrm{irr}}}{{\mathrm{\epsilon }}_{\mathrm{axial}}-{\epsilon }_{\mathrm{axial}}^{\mathrm{irr}}}$$ 5

where, ${\epsilon }_{\mathrm{axial}}^{\mathrm{irr}}$ and ε_axial was the irrversible and total axial strain at the peak stress σ_p in each loading cycle respectively; ${\mathrm{\epsilon }}_{\mathit{lateral}}^{\mathit{irr}}$ and ε_lateral was the irrversible and total lateral strain at the peak stress σ_p respectively. These definiation correlated with the elatic rebound during unloading stage were more appropriate to the measurements of the elastic parameters for cyclic loading tests⁴⁸. The evolutions of E and ν of each cycle loading for specimens in group D and E as a function of the normalized stress level (σ_p/σ_ucs) in each loading cycle were presented in Fig. 7e and f. Figure 7e indicates that the E for all specimens decreased as dynamic load increased until final failure. For the same loading type, the values of E of specimens with θ = 0° were the smallest while those with θ = 90° were the largest. Figure 7f shows that the ν were in ascending order as the dynamic load increased. For the loading type of group D, the values of ν of specimens with θ = 90° were larger than those of θ = 60° and smaller than those of θ = 0°. For the loading type of group E, the values of ν of specimens among θ = 0°, θ = 60° and θ = 90° were almost the same except the values of ν of specimens with θ = 90° increased significantly at last two loading cycles.

During cycle loading process, if it assumed that there was no heat exchange, the compression energy was transferred to elastic strain energy (U_e) and dissipated energy (U_irr). U_e related to the unloading elastic modulus E, was defined as the work done by stress and strain, equalling to the area between the unloading curve and ε-axis as shown in Fig. 8. It was recoverable after unloading. U_irr was defined as the irreversible work for a loading and unloading cycle which was primarily induced by micro-crack initiation, propagation and coalescence. Figure 8 shows that it equals to the area of the shape confined by the loading, unloading curves and ε-axis in each loading cycle. Figure 9 indicates that the changes of U_e and U_irr of each cycle for all specimens in group D and E was a function of the normalized stress (σ_p/σ_ucs) in each loading cycle. U_e significantly increased as σ_p/σ_ucs increased, while U_irr had a descending order at the first loading cycle as well as increased in following cycles. The mean U_e was larger than that of U_irr.

Figure 8.

Relationship between elastic strain energy (U_e) and dissipated energy (U_irr) of rock mass unit.

Figure 9.

Evolution of U_e and U_irr of shale specimens of two groups with different dip angles as σ_p/σ_ucs in ascending order: (a) θ = 0° of group D; (b) θ = 0° of group E; (c) θ = 60° of group D; (d) θ = 60° of group E; (e) θ = 90° of group D; (f) θ = 90° of group E.

Variation of distribution of microscopic failure stress of shale specimens of groups D and E

Figure 10 presents the evolutions of m values (determined by Eq. 2) of each loading cycle of shale specimens of groups D and E with different bedding dip angles. The m values of shale specimens of groups D (θ = 0°, θ = 60° and θ = 90°) and E (θ = 0° and θ = 60°) totally had an ascending order as σ_p/σ_ucs increased as well as had a light descending order at the last one or two cycle loading before final failure. It indicates that there was a more homogeneous distribution of σ_f as σ_p/σ_ucs increased. For the shale specimen of group E (θ = 90°), the m values also increased at the initial stage, but it decreased significantly in the following cycle loading.

Figure 10.

Evolutions of m values of shale specimens of groups D and E with different bedding dip angles as σ_p/σ_ucs in ascending order: (a) θ = 0°, (b) θ = 60°, and (c) θ = 90°.

Final fracture morphology of specimens of groups A, B, C, D and E

Figure 11 shows the typical final fracture morphology of specimens of groups A, B, C, D and E with different bedding dip angles (θ = 0°, θ = 60° and θ = 90°) and various loading types, which indicates that the bedding plane was the main effect affecting the final fracture morphology. For the specimens of groups A, B, C, D and E with θ = 0°, the fracture primarily occurred along the bedding plane in horizontal direction associated with vertical direction fracture surfaces. For the specimens of groups A, B, C, D and E with θ = 60° and 90°, the dip angles of the dominated fracture surface were about 60° and 90°, which were the same as that of the bedding plane.

Figure 11.

Final fracture morphology of shale specimens with different bedding dip angles during constant strain rate experiments and dynamic cyclic loading tests at different stress levels.

Discussions

Effect of non-uniform distribution of microscopic failure stress on macroscopic physico-mechanical properties

Based on the experiment results, the effect of m value on physico-mechanical properties of the shale specimens subjected to dynamic load in laboratory scale were discussed.

Figures 12 and 13 shows the relationship between m values and σ_cd/σ_ucs, E, ν, U_e and U_irr of specimens of group D and E with various dip angles. The comparison results indicate that general speaking, σ_cd/σ_ucs, ν, U_e and U_irr increased as m values increased or there was a positive correlation between them. Conversely, E decreased as m values increased. The possible reason might be the fact that the cycles of pressure loading and unloading induce most of the low microscopic failure stresses fail first, resulting more uniform distribution of microscopic failure stresses (larger m values). In other words, the more times the cycles of pressure loading and unloading that were carried out, the more homogeneous the distribution of microscopic failure stresses. In can be observed from Fig. 7a that the uniform distribution of microscopic failure stresses delayed the threshold of crack damage stress (larger σ_cd) during the progressive failure process with larger σ_cd/σ_ucs and ε_cd (Fig. 7c). Previous study indicated more homogeneous distribution of microscopic failure stresses induced larger deformation of shale⁴⁹. E (defined as the ratio of the axial stress change to axial strain produced by the stress change) thus decreased as m values increased. On the other hand, the homogeneous distribution of microscopic failure stresses might not only induce the large axial strain but also significantly enable great deform in the direction of the radius, resulting in reduction of E. The ascending order of U_e is because that the larger the σ_p/σ_ucs, the greater the m values and axial strains, the areas between the unloading curve and ε-axis are thus larger. The phenomenon that U_irr increased as m value increased is probably due to the increment of the applied pressure stress, greater number of micro-cracks in non-uniform distribution area of microscopic failure stresses are formed, and these cracks caused more irreversible work was done. More discussion details of effect of m values on macroscopic physico-mechanical properties of specimens with different bedding dip angles can be found in following parts.

Figure 12.

The relationship between m value and σ_cd/σ_ucs, E, ν, U_e and U_irr of shale specimens of group D with various dip angles: (a) θ = 0°; (b) θ = 60°; (c) θ = 90°.

Figure 13.

The relationship between m value and σ_cd/σ_ucs, E, ν, U_e and U_irr of shale specimens of group E with various dip angles: (a) θ = 0°; (b) θ = 60°; (c) θ = 90°.

For specimens of group D, when θ = 0° (see Fig. 12a), σ_cd/σ_ucs, ν, U_e and U_irr (except 13, 24 and 30 loading cycles) were all in ascending order as m values increased, i.e., σ_cd/σ_ucs were 0.61, 0.62, 0.63, 0.63, 0.65, 0.67, 0.68, 0.69, 0.71, 0.72, 0.73, 0.75, 0.75, 0.77, 0.77, 0.78, 0.79, 0.81, 0.81, 0.82 from cycles 12 to 31, the corresponding m values were 0.99, 0.99, 1.22, 1.34, 1.65, 2.04, 2.36, 2.62, 3.07, 3.65, 4.29, 5.27, 5.64, 6.69, 7.01, 8.13, 8.99, 9.92, 10.11, 10.33, which both were in ascending order. Following the same comparison procedure, Fig. 12b indicates that the relationship between m values and σ_cd/σ_ucs, E, ν and U_e of specimens of group D with θ = 60° were the same as that of θ = 0°, the larger the m value, the larger the σ_cd/σ_ucs, ν and U_e, the smaller the E. However, U_irr decreased as m values increased. What’s more, the relationship between m values and σ_cd/σ_ucs, E (except 47 loading cycle), ν (except 47 loading cycle), U_e (except 47 loading cycles) and U_irr (except 47 loading cycles) of specimens of group D with θ = 90° were also the same as that of θ = 0°.

For specimens of group E, when θ = 0° (see Fig. 13a), the larger the m value, the larger the σ_cd/σ_ucs, ν and U_e, the smaller the E (except 1 loading cycle). For example, σ_cd/σ_ucs were 0.83, 0.85, 0.86, 0.87, 0.91, 0.90, 0.91, 0.91, 0.86 from cycles 1 to 9, the corresponding m values were 5.28, 7.13, 9.02, 10.31, 12.45, 13.02, 14.06, 15.17, 13.31. Figure 13a shows that U_irr decreased as m values increased for the first 4 loading cycles while it increased as m values increased in the following loading cycles. Following the same comparison procedure, Fig. 13b indicates that the relationship between m values and σ_cd/σ_ucs (except 5 loading cycle), E, ν, U_e and U_irr of specimens of group E with θ = 60° were the same as that of θ = 0°. Figure 13c shows that for the first 6 loading cycles as m values in ascending order, ν (except 1 loading cycle), U_e and U_irr (except 1 loading cycle) were also in ascending order while E was in descending order as well as σ_cd/σ_ucs was almost the same during these loading cycles except 1 cycle. For the cycles from 6 to 10, σ_cd/σ_ucs and E decreased as m values decreased. Conversely, ν, U_e and U_irr were in ascending order during the same loading cycles as m values decreased.

Effect of bedding dip angle on m values and macroscopic physico-mechanical properties

It can be observed from Fig. 10 that when the σ_p/σ_ucs are the same (0.9–1.0), the m values of specimens of group D with θ = 0° and θ = 90°are almost the same which are greater than those of θ = 60°, indicating that distributions of microscopic failure stresses of specimens of group D with θ = 0° and θ = 90°are more homogeneous. While for the specimens of group E, the m values of θ = 0° are the largest, those of θ = 90° are the smallest, and those of θ = 60° are between them.

Some corresponding phenomenon that physico-mechanical properties affected by bedding dip angle are also can be observed from Figs. 7 and 9, i.e., σ_cd, σ_cd/σ_ucs, ε_cd, ν, U_e and U_irr are overall proportional to m values, the larger the m values, the greater these parameter properties, while for the ε_ucs/ε_cd, E, overall negative correlation between these parameter properties and m values can be observed. For example, for the m values and σ_cd of specimens of group D with θ = 0°and θ = 90°are both larger than those of θ = 60° (see Fig. 7a), which have positive relationship. E is taken as an example of the negative relationship, the m values of specimens of group E with θ = 60° are smaller than those of θ = 0°, but are greater than those of θ = 90°. What’s more, it is interested to note that some abnormal points can be observed in some case, this condition will be verified in further studies with more specimens.

The above results were reached based on a limited number of specimens, and they are applicable only to shale specimens at the laboratory scale to have a better understanding the dynamic evolution of the micro-structure homogeneity as well as their relationship to the physico-mechanical properties.

Conclusions

Associated with a proposed equation between homogeneity index, m, and strain ratio, ε_ucs/ε_cd, Weibull distribution, a widely used statistical model characterizing the distribution of the microscopic physical–mechanical properties of rock or rock-like materials, was used to characterize the spatial variation of distribution of microscopic failure stress of shale specimens in laboratory scale subjected to dynamic load. Based on the experimental results of a series of cyclic loading tests on shale specimens with different bedding dip angles, the effect of the non-uniform distribution of microscopic failure stress on macroscopic physico-mechanical properties were analyzed. Some conclusions were apparent, and they are presented below:

1. Bedding dip angle affects the spatial distributions of microscopic failure stress of shale specimens subjected to dynamic load. The m values of specimens of group D (first-time unloading was carried out until the applied axial stress exceeded the σ_ci but lower than the σ_cd) with θ = 0° and θ = 90°are almost the same which are greater than those of θ = 60°. While for the specimens of group E (first-time unloading was carried out when the applied axial stress higher than the σ_cd), the m values of θ = 0° are the largest, those of θ = 90° are the smallest, and those of θ = 60° are between them.

2. The values of σ_cd, σ_cd/σ_ucs, ε_cd, ν, U_e and U_irr of the specimens with higher m values are overall higher while ε_ucs/ε_cd and E are lower.

3. The m values have an ascending order during the dynamic load applying process with the increment of σ_p/σ_ucs, indicating that the dynamic load enables the spatial distributions of microscopic failure stress are more homogeneous prior to the final failure.

4. General speaking, σ_cd/σ_ucs, ν, U_e and U_irr increased as m values increased of each specimen. Conversely, E decreased as m values increased.

5. The final fracture morphology of specimens of both groups D and E (θ = 0°, θ = 60° and θ = 90°) were primarily controlled by the bedding surface.

Author contributions

M.W.: Conceptualization, Investigation, Methodology, Writing-Original Draft. X.S.: Investigation, Validation, Reviewing. X.P.: Methodology, Writing-Reviewing and Editing.

Data availability

All data generated during this study are included in this published article [and its supplementary information files].

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-023-30519-w.