Investigation of mechanical properties of bedded shale by nanoindentation tests: A case study on Lower Silurian Longmaxi Formation of Youyang area in southeast Chongqing, China

Fig. 1. Nano-indentation load-displacement curve diagram (revised according to reference [12]).

The hardness can be calculated by Eq. (1)^[15]:

(1)$H=\frac{{{p}_{\max }}}{{{A}_{\text{c}}}}$

The projected area A_c is the function of the contact depth h_c:

(2)${{A}_{\text{c}}}=24.5h_{\text{c}}^{\text{2}}$

There’s a relationship between h_c and h_max:

(3)${{h}_{\max }}-{{h}_{\text{c}}}=\varepsilon \frac{{{p}_{\max }}}{S}$

(4)${{h}_{\text{c}}}=\frac{2({{\nu }_{\text{E}}}-1)}{2{{\nu }_{\text{E}}}-1}{{h}_{\max }}$

Thus, the reduced modulus E_r can be calculated by Eq. (5) with contact area and depth given:

(5)${{E}_{\text{r}}}=\frac{\sqrt{\text{ }\!\!\pi\!\!\text{ }}S}{2\beta \sqrt{{{A}_{\text{c}}}}}$

The reduced modulus reflects the composite deformation of indenter and rock, the rock Young’s modulus E can be expressed as Eq. (6):

(6)$\frac{1}{{{E}_{\text{r}}}}=\frac{1-{{\nu }^{2}}}{E}+\frac{1-\nu _{\text{i}}^{\text{2}}}{{{E}_{\text{i}}}}$

Rock fracture toughness represents the potential intrinsic ability of rock to withstand a given stress field at the tip of a crack and to resist against progressive crack extension^[16]. The energy analysis and indentation shape methods can be used to calculate rock fracture toughness by nano-indentation experiment data. In this study, the former was used. Yang et al.^[17] proposed that the fracturing energy can be calculated by the relations between pure plasticity U_pp and irreversible energy U_ir and expressed as Eq. (6):

(7)${{U}_{\text{c}}}={{U}_{\text{ir}}}-{{U}_{\text{pp}}}={{U}_{\text{t}}}-{{U}_{\text{e}}}-{{U}_{\text{pp}}}$

The relation between pure plasticity U_pp and total energy U_t can be written as Eq. (7):

(8)$\frac{{{U}_{\text{pp}}}}{{{U}_{\text{t}}}}=1-\left[ \frac{1-3{{\left( \frac{{{h}_{\text{f}}}}{{{h}_{\max }}} \right)}^{2}}+2{{\left( \frac{{{h}_{\text{f}}}}{{{h}_{\max }}} \right)}^{3}}}{1-{{\left( \frac{{{h}_{\text{f}}}}{{{h}_{\max }}} \right)}^{2}}} \right]$

The total energy U_t can be computed by:

(9)${{U}_{\text{t}}}=\int_{0}^{{{h}_{\text{max}}}}{{{p}_{\text{c}}}\text{d}{{h}_{\text{c}}}}=\frac{{{p}_{\text{max}}}{{h}_{\text{max}}}}{3}$

To calculate the fracture toughness, the critical energy release rate G_c needs to be worked out first by Eq. (10) below:

(10)${{G}_{\text{c}}}=\frac{\partial {{U}_{\text{c}}}}{\partial {{A}_{\text{c}}}}=\frac{{{U}_{\text{c}}}}{{{A}_{\text{max}}}}$

The maximum contact area A_m can be calculated by:

(11)${{A}_{\max }}=24.5h_{\max }^{2}$

The fracture toughness can be computed by the Eq. (12):

(12)${{K}_{\text{c}}}=\sqrt{{{G}_{\text{c}}}{{E}_{\text{r}}}}$

1.2. Samples preparation

Four shale samples were drilled from full size cores at depth of 2 010 m of Silurian Longmaxi shale at Youyang area, Southwest of Chongqing, China. The cylinder samples parallel and vertical to the bedding plane were cut (25 mm in dimeter and 55 mm-60 mm long). Then, they were cut into standard cylinder samples (25 mm in dimeter and 50 mm long) for conventional uni-axial compressive test labelled as VA, VB, HA and HB, and samples for nanoindentation (25 mm in dimeter and 10 mm long) labelled as V1, V2, H1 and H2.

As local pores and bond interfaces between different minerals can potentially affect the accuracy of nanoindentation experiments, the surface of samples were carefully polished by SiC sand paper of different grit sizes from 600 to 1 200 first, and then by argon ion machine. Moreover, the digital Zeta-20 machine was used to scan the roughness of surface randomly to assure the roughness met the requirement for nanoindentation test. Then the samples were ultrasonic cleaned and treated with acetone to remove dust^[18,19]. The RoqSCAN equipment was used for mineral composition analysis, and Scanning Electron Microscopy (SEM), Back scattered Electron Imaging (BSE) and Energy Dispersive Spectroscopy (EDS) can be obtained simultaneously to map the mineral distribution of each indents. Additionally, SEM was secondly performed on the rock surface after nanoindentation to get the indentation fracture morphology.

The nanoindentation experiments were performed on a Nanoindenter (Agilent Technologies G200) with load resolution of 50 nN, standard peak load of 500 mN, displacement resolution of less than 0.01 nm in the z direction and 1 μm in the x and y direction. Moreover, the least indentation depth for each nanoindentation experiment was greater than 500 μm, with a travel range of 100 mm×100 mm. The tip radius of curvature for Berkovich diamond was less than 20 nm.

1.3. Experimental process

The experiment was divided into three steps.

(1) The indenter approached the sample surface at the loading rate of 30 nm/s, sudden increase of load shown by the testing system meant the indenter contacted the sample surface already, then the load and indentation depth were recorded automatically. The maximum load of each indentation was set at 500 mN. The force was held constant at the maximum indentation for 15 seconds.

(2) Prior to the nanoindentation test, the tip area function was calibrated by several indentations on a standard fused quartz sample. The nanoindentation tests were performed on a 4 by 8 grid at 15 μm spacing between indentations (The total individual indentations were 128 for all four samples). By this test method, a group of hardness, Young’s modulus and fracture toughness were obtained for each indentation to calculate the average mechanical parameters of each sample. EDS was used to analyze the sample surface and position special minerals, then the mechanical properties of special minerals were calculated.

(3) The compression tests on the samples were performed on a triaxial testing apparatus that could independently apply confining stress and axial stress. The macroscale results of uniaxial compression tests were compared with that obtained by nanoindentation.

2. Results

2.1. Mineral composition and microstructure of shale

BSE scanning results of shale samples from Longmaxi Formation show that the samples have good cementation of particles, few large pores and only a small number of small pores (Fig. 2a). The quality fraction and area fraction of mineral composition and mineral analysis statistics were obtained by EDS mapping data (Table 1). The results show that the rock is mainly composed of quartz minerals (quartz, feldspar, potassium feldspar), carbonate minerals (calcite, dolomite), clay minerals (illite, kaolinite, chlorite), but also contains a small amount of saponite, dolomite, pyrite and amorphous substances difficult to distinguish.

Fig. 2.

Fig. 2. Analysis and test results of vertical bedding components of shale mineral.

Table 1 Analysis results of shale composition.

Parameter		Mass fraction/%	Area fraction/%
Quartz class	Quartz	27.38	26.92
	Albite	21.52	23.94
	Potash feldspar	1.91	2.20
Carbonate	Calcite	4.47	5.12
Carbonate	Bitter spar	1.64	2.12
Clay class	Illite	24.77	20.15
	Kaolinite	2.13	1.22
	Chlorite	2.10	1.77
Else		9.88	12.86

SEM results of gird matrix indentations and single nano indentation (Fig. 3a, 3b) show that there are some intrinsic pores and micro cracks in the samples. Thus, in the nanoindentation process, the increase of load with the rise of displacement would generate fractures. The propagation of fractures would be affected by the micro-defects of the sample, and would likely to go along the edges of existent pores or microcracks. Due to the heterogeneity of the sample, if the indentation location is just on hard materials like carbonate, and below it is soft material like clay, the so-called “base effect” (clay gathering) could occur, which can affect the nanoindentation results.

Fig. 3.

Fig. 3. Grid matrix indentations SEM image (a) and single indentation SEM image (b) after nano indentation.

2.2. Load-displacement curve

The Load-displacement curve is base to analyze deformation behavior and mechanical parameters of nanoindentation. The load-displacement curves obtained are not smooth in certain stages, and there are some “pop in” behaviors at the loading stage (Fig. 4a). The main reason for “pop in” is that when the load approaches to the yield strength of rock, fractures occur. As the fracture tip begins to expand inward rock, and the process continues until load stops. In addition, the heterogeneity of the shale also could cause “pop in” behavior, especially indentation fractures contact with internal defects such as micro-cracks, pores, or interface between hard material and soft materials such as kerogen.

Fig. 4.

Fig. 4. Load - displacement curves with “pop in” (a) and normal (b) nano indentation.

There are three stages in the loading process during nanoindentation experiment: elastic deformation, elastoplastic deformation and plastic deformation. The differences in rock mechanical properties result in different durations of the stages. Fig. 4a shows that the sample enters the elastoplastic phase from elastic phase at the displacement of 1 000 nm, while Fig. 4b shows that the sample enters elastoplastic phase from elastic phase after the displacement of 2 000 nm. During the loading process, the curvature of the load-displacement curve will increase rapidly in the elastic and elastoplastic stages, which reflects the rapid elastic deformation when nano-indenter contacts shale surface. When the load reaches the yield strength of the rock, elastoplasticity occurs. When the indentation load exceeds the strength of the rock, permanent deformation would cause nanoindentation fractures.

2.3. Test results of mechanical parameters on nano scale

According to the load-displacement curve, the mechanical parameters such as hardness and elastic modulus of rock can be calculated by using equations (1) and (6) respectively, and the fracture toughness of rock can be calculated by equation (12). Due to the large amount of data in nanoindentation experiment, the averaging method or the deconvolution method is usually used to process the mechanical parameters. The deconvolution method needs to distinguish the rock components clearly and may have multiple solutions. Therefore, the averaging method was adopted for data analysis in this work. Statistics on 128 groups of mechanical parameters show (Table 2) the samples with parallel bedding planes have an average elastic modulus of 45.21 GPa (standard deviation of 5.59 GPa), average hardness of 2.43 GPa (standard deviation of 0.56 GPa), and average fracture toughness of 2.68 MPa•m^1/2 (standard deviation of 0.33 MPa•m^1/2). The samples with vertical bedding planes have an average elastic modulus of 43.60 GPa (standard deviation of 4.03 GPa), average hardness of 2.85 GPa (standard deviation of 0.78 GPa), and average fracture toughness of 2.37 MPa•m^1/2 (the standard deviation of 0.66 MPa•m^1/2). The samples with parallel bedding planes have slightly higher elastic modulus and fracture toughness than those with vertical bedding planes. At nano scale, as the indentation points are on different minerals, the mechanical parameters tested would fluctuate somewhat. Therefore, in order to obtain representative and reliable mechanical parameters, more indentation points need to be tested to mitigate the uncertainty caused by rock heterogeneity.

Table 2 Mechanical parameters by average method from nanoindentation test.

Coring angle	Elastic modulus/GPa				Hardness/GPa				Fracture toughness/(MPa·m^1/2)
Coring angle	Minimum	Maximum	Average	Standard deviation	Minimum	Maximum	Average	Standard deviation	Minimum	Maximum	Average	Standard deviation
Parallel bedding	35.61	59.47	45.21	5.59	1.58	3.89	2.43	0.56	2.02	3.41	2.68	0.33
Vertical bedding	37.41	53.80	43.60	4.03	1.79	4.58	2.85	0.78	0	2.94	2.37	0.66

With the help of EDS mineral calibration results, the mechanical parameters of specific minerals can be obtained after the nanoindentation test, which provides parameters for the upscaling model. The mechanical test results of specific minerals obtained in this study are in good agreement with the experimental results of other researchers^[9,10] (Table 3). Due to the low content of minerals such as chlorite and pyrite in the samples, no relevant indentation data was recorded. In addition to nanoindentation on specified minerals, nanoindentation was also conducted on the interface between quartz and clay minerals. The results show that dolomite and quartz have the highest elastic modulus of 115 GPa and 99 GPa, respectively; the elastic modulus of clay is the lowest, that of kaolinite and illite are 55 GPa and 62 GPa, respectively; and that of the mixed mineral of quartz/clay matrix is about 59 GPa.

Table 3 Mechanical parameters of specific minerals from nanoindentation.

Mineral classification	Mineral	Elasticity modulus/GPa
Mineral classification	Mineral	Mean value of this paper	Data in literature [9-10]
Clay class	Kaolinite	55	56
Clay class	Illite	62	66
Carbonate	Calcite	79	85
Carbonate	Dolomite	115	125
Quartz class	Quartz	99	100
Quartz class	Albite	72	68
Else	Pyrite	/	250-310
Quartz/clay matrix interface		59	/

3. Discussion

3.1. Correlation between different mechanical parameters at nano scale

From linear regression, there is a positive correlation between Young’s modulus and hardness of shale samples with parallel and vertical beddings. The correlation coefficients are 0.57 and 0.58 for rock samples with parallel and vertical beddings, respectively (Fig. 5). There is also a positive correlation between Young’s modulus and fracture toughness of shale samples with parallel and vertical beddings. The correlation coefficients are 0.54 and 0.51 for them respectively (Fig. 6). However, there is poor linear relationship between hardness and fracture toughness with correlation coefficients of 0.31 and 0.36 (Fig. 7). That’s because the calculation of hardness depends on the contact projection area and load, but the fracture toughness calculation uses the sum of the shaded areas as input. The differences in the inputs for computation result in discrepancy. Therefore, to reduce the discrepancy, the contact depth, contact stiffness and residual depth must be calculated more accurately besides increasing indentation points tested.

Fig. 5.

Fig. 5. The relationship between elastic modulus and hardness of samples with parallel bedding (a) and vertical bedding (b).

Fig. 6.

Fig. 6. The relationship between elastic modulus and fracture toughness of samples with parallel bedding (a) and vertical bedding (b).

Fig. 7.

Fig. 7. The relationship between hardness and fracture toughness of samples with parallel bedding (a) and vertical bedding (b).

3.2. Distribution of different mechanical parameters at nano scale

Nanoindentation results show that there are some fluctuation in Young’s modulus, hardness and fracture toughness of the same sample. Take Young’s modulus of rock samples with parallel beddings for instance, the maximum value is 59.47 GPa, while the minimum value is 35.61 GPa. The maximum value is 1.67 times of the minimum value. The reason is the heterogeneity of the samples. Therefore, the test results of nanoindentation need to be analyzed statistically to find their uncertainty. Previous studies on rock mechanical parameters distribution show Weibull model can effectively characterize heterogeneity and mechanical response features of rock under external force. More importantly, the parameters in this model can be easily measured^[20,21]. The Weibull model can be expressed as:

(13)${{P}_{\text{w}}}=1-\exp \left[ -{{\left( \frac{X-{{X}_{0}}}{{{X}_{i}}} \right)}^{m}} \right]$

where P_w is the probability that the mechanical parameter value is less than X, X_iis the characteristic mechanical parameter. X₀ is generally set at 0, representing the minimum variable value. The reliability parameter m named Weibull modulus, reflects the scatter of mechanical parameters. The higher the m, the lower the discreteness of the parameter will be.

The statistical results obtained from nanoindentation experiments were arranged in numerical order to get the probability of each experiment result no more than X (Table 4). The m value was worked out by least square method (LSM) (Figs. 8-10). The m of three mechanical parameters are all small, indicating the test results of nanoindentation are high in discreteness, among them, the m of hardness is the smallest, indicating hardness has the highest discreteness. The root causes of the data discreteness are: Firstly, the different size, shape and arrangement of mineral particles on the surface and in the interior of rocks, and different positions of indentation lead to differences in experimental results. Secondly, when the fracture from indentation propagate, mineral particles with large size or strong strength could change the local stress field distribution, leading to deviation and blockage of indentation fracture, and thus jump in experimental data. Thirdly, the calculation of hardness only considers the indentation shaded area, but the arrangement of hard and soft substances in rock interior can possibly create “base” effect, affecting the accuracy of calculated projection shaded area, and consequently the precision of hardness calculated. Moreover, the m value of samples with vertical beddings are all lower than that of samples with parallel beddings, which indicates that the samples with vertical beddings have higher heterogeneity than those with parallel beddings.

Table 4 Statistics on m values of nano indentation mechanical parameters.

Coring angle	m value of fracture toughness	m of the elastic modulus	m of hardness
Parallel bedding	3.73	2.99	1.18
Vertical bedding	2.12	2.27	1.01

Fig. 8.

Fig. 8. Hardness Weibull curve of samples with parallel bedding (a) and vertical bedding (b).

Fig. 9.

Fig. 9. Weibull curve of elastic modulus of shale samples with parallel bedding (a) and vertical bedding (b).

Fig. 10.

Fig. 10. Weibull fracture toughness curve of shale samples with parallel bedding (a) and vertical bedding (b).

3.3. Upscaling model of mechanical parameters from nano scale to micro scale

During the nanoindentation test, it’s easy to indent on different materials without the aid of EDS mapping, thus resulting in discrete test results. To get more accurate bulk mechanical parameters of rock, it’s necessary to increase the number of indents in the same rock surface. By comparing the nanoindentation results of different minerals, the mechanical parameters of different minerals in the same sample differ widely. But the whole rock can be regarded as a composite media composed of three-phases, the high strength minerals (dolomite, et al.), medium strength minerals (quartz, albite, calcite, et al.) and low strength minerals (clay, organic matter, etc.). Moreover, the three types of minerals can be regarded as isotropic at the nano scale. By joining the volume fraction of different phases and the composition mechanical model, the mechanical properties of bulk rock by upscaling from nano- scale to centimeter scale can be obtained. It should be noted that there are possible nano and micro pores in clay or kerogen, thus the calculation of mechanical properties of bulk rock should consider natural fracture, nano and micro pores.

In this study, the Mori-Tanaka approach was used to upscale mechanical parameters from nano scale to micro scale. The shear and bulk modulus of rock can be calculated as^[22,23]:

(14)${{K}_{\text{M}}}=\frac{\sum{{{f}_{r}}\frac{{{k}_{r}}}{3{{k}_{r}}+4{{\mu }_{\text{low}}}}}}{\sum{\frac{{{f}_{r}}}{3{{k}_{r}}+4{{\mu }_{\text{low}}}}}}$ (r=0, 1, 2)

(15)${{G}_{\text{M}}}=\frac{\sum{\frac{{{f}_{r}}{{\mu }_{r}}}{{{\mu }_{\text{low}}}(9{{k}_{\text{low}}}+8{{\mu }_{\text{low}}})+6{{\mu }_{r}}({{k}_{\text{low}}}+2{{\mu }_{\text{low}}})}}}{\sum{\frac{{{f}_{r}}}{{{\mu }_{\text{low}}}(9{{k}_{\text{low}}}+8{{\mu }_{\text{low}}})+6{{\mu }_{r}}({{k}_{\text{low}}}+2{{\mu }_{\text{low}}})}}}$

(16)${{k}_{r}}=\frac{{{E}_{r}}}{3\left( 1-2{{\nu }_{r}} \right)}$

(17)${{\mu }_{r}}=\frac{{{E}_{r}}}{2\left( 1+2{{\nu }_{r}} \right)}$

where r=0 represents carbonate minerals such as dolomite et al., r=1 denotes quartz minerals such as quartz and albite et al., r=2 represents clay minerals.

The minerals with the highest strength and medium strength can be obtained directly by combining nanoindentation data and EDS mapping. However, the low strength minerals should consider porosity of rock (The porosity of rock assumed as 3.5% in this study), since nano and micro pores would affect the results of nanoindentation test. The shear and bulk modulus of low strength mineral can be computed by:

(18)${{k}_{\text{low}}}=\frac{4\left( 1-\phi \right){{k}_{\text{s}}}{{\mu }_{\text{s}}}}{4{{\mu }_{\text{s}}}+3\phi {{k}_{\text{s}}}}$

(19)${{\mu }_{\text{low}}}=\frac{\left( 1-\phi \right){{\mu }_{\text{s}}}}{1+6\phi \frac{{{k}_{\text{s}}}+2{{\mu }_{\text{s}}}}{9{{k}_{\text{s}}}+8{{\mu }_{\text{s}}}}}$

The Young’s modulus of bulk rock can be obtained by three-phase minerals and relating volume fractions:

(20)${{E}_{\text{HOM}}}=\frac{9{{K}_{\text{M}}}{{G}_{\text{M}}}}{3{{K}_{\text{M}}}+{{G}_{\text{M}}}}$

Fig. 11.

Fig. 11. The petrophysical model (a) converted to equivalent mechanical parameter model (b).

To further demonstrate the accuracy of the upscaling model, uniaxial compression tests were performed on cylinder shale samples to get stress vs. strain curve and calculate the Young’s modulus (Fig. 12).

Fig. 12.

Fig. 12. Shale uniaxial compression stress-strain curve.

Table 5 shows mechanical parameters obtained from three different methods. We can see that the calculated values from the upscale model have a good agreement with those obtained by uniaxial compression tests, but the calculated results by the averaging approach of nanoindentation data are higher in values. The mechanical properties are different at different scale and this is because test samples at larger size have more natural micro pores and fractures, thus the increase of load can induce fracture extension from natural damage area and grow as interconnected fractures. Although the upscale model provides a good tool to characterize bulk mechanical properties and potentially reduce the number of nanoindentation, the combination of the nanoindentation and mineral mapping technique at the rock surface is necessary to get reliable mechanical properties of each phase.

Table 5 Upscaling elastic modulus results.

Coring angle	Young’s Modulus by compression test/GPa	Young’s Modulus by upscaling model/GPa	Young’s Modulus by averaging approach/GPa
Parallel beddings	41.12	45.21	48.52
Vertical beddings	39.15	43.60	45.25

4. Conclusions

A total of 128 nanoindentation experiments were carried out on Longmaxi shale samples with parallel and vertical beddings. The hardness, Young’s modulus and fracture toughness of the shale samples were obtained by load-displacement curve. Additionally, mechanical parameters of samples with parallel bedding are slightly higher than those of samples with vertical bedding. There is a good linear correlation between Young’s modulus and hardness, and between Young’s modulus and fracture toughness.

The results of mechanical parameters at nano scale show certain discreteness and conform to Weibull distribution. The hardness data has the highest discreteness. The discreteness of mechanical parameters not only is related to the heterogeneity of shale, but also is affected by occasional and regional fracture propagation resistance due to non-uniform mineral distribution. Besides, taking shaded area below displacement vs. load curve only as parameter in calculation is another reason of the discreteness.

The combination of BSE and EDS mapping was used to analyze the mechanical properties of certain minerals in nanoindentation measurements. A micro-mechanical model consisting of three-phase media (high-strength minerals, medium-strength minerals and low-strength minerals) was proposed. The Mori-Tanaka model was used to upscale data from nano scale to micro scale, and the results were compared with results from averaging method and uniaxial compression test, which shows the calculated values from the upscale model are in good agreement with the results from uniaxial compression test, but lower than those from nano-scale calculation. This is because the samples have natural micro pores and fractures etc, leading to differences in their mechanical properties.

Acknowledgements

Special thanks to Professor YANG Jijin and Professor DU Zhongming from Institute of Geology and Geophysics at Chinese Academy of Sciences for their help with the experiments.

Nomenclature

A_c—the area of contact projection under the corresponding load, m²;

A_max—maximum contact projection area under corresponding load, m²;

E—Young’s modulus of the sample, Pa;

E_HOM—equivalent elastic modulus of the rock at centimeter scale, Pa;

E_i—elastic modulus of Berkovich indenter, 1 114 GPa;

E_r—elastic modulus of different minerals, Pa;

E_r—reduced Young’s modulus, Pa;

f_r—volume fraction of different minerals, %;

G_c—critical energy release rate, N/m;

G_M—bulk modulus, Pa;

h_c—contact depth, m;

h_f——residual depth, m;

h_L—initial displacement of the holding point, m;

h_max—maximum displacement, m;

H—hardness, Pa;

K_low—equivalent volume modulus for low strength minerals, Pa;

K_c—fracture toughness, Pa·m^1/2;

K_M—shear modulus, Pa;

k_s—the proportion of clay minerals, %;

k_r—volume modulus of different mineral particles, Pa;

m—Weibull modulus, dimensionless;

p_c—load, N;

p_max—maximum load, N;

P_w—probability of lower than the random variable;

S—contact stiffness, N/m;

U_c—fracture energy, N·m;

U_e—elastic energy, N·m;

U_ir—irreversible energy, N·m;

U_pp—energy lost during pure plastic stage, N·m;

U_s—nanoindentation absolute energy, N·m;

U_t—total fracture energy, N·m;

X—random variable;

X₀—the minimum parameter value of the random variable;

X_i—characteristic parameters;

β—a constant related to the geometry of indenter;

ϕ—shale porosity, %;

μ_low—equivalent shear modulus for low-strength minerals, Pa;

μ_s—shear modulus of clay minerals, Pa;

μ_r—shear modulus of different minerals, Pa;

ν—Poisson’s ratio of sample, dimensionless;

ν_E—elastic energy ratio, dimensionless;

ν_i—Poisson’s ratio of Berkovich indenter, 0.77;

ν_r—Poisson’s ratio of different minerals, dimensionless;

ε—constant, 0.75.

The authors have declared that no competing interests exist.

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In view of the difficulty to get mechanical characteristics of shale reservoirs, a quantitative evaluation method based on micro-indentation test technology was proposed to research the meso-mechanical properties of shale. Through micro-indentation test of shale outcrop samples from the Cambrian Longmaxi Formation in the Changning area, Sichuan Basin, the meso-mechanical properties of shale were analyzed, and the relationship between the macro-scale and meso-scale shale mechanical properties was evaluated. The analysis results of micro-indentation tests show that the mesoscopic elastic modulus and indentation hardness are heterogeneous in distribution. The comparison of macro and meso experiments shows that the statistical mean value of meso-elastic modulus is approximate to the value of the macro-elastic modulus. The relationship between composition and mechanical properties of shale was obtained based on the packing density model theory. The result shows that mesoscopic elastic modulus and indentation hardness increase nonlinearly with the increase of the packing density. Based on the hardness-packing density model, the reverse analysis of cohesion and friction angles shows that the mesoscopic value is slightly smaller than the macroscopic value. The micro-indentation test technology can evaluate shale meso-mechanical properties and predict the macro-mechanical properties effectively.

[7]

DONG G

, CHEN

A comparative experiment investigate of strength parameters for Longmaxi shale at the macro- and mesoscales

International Journal of Hydrogen Energy, 2017,42(31):20082-20091.

DOI:10.1016/j.ijhydene.2017.05.240 URL [Cited within: 1]

The strength parameters of rock is important to the storage and development of hydrogen gas and natural gas. The rock strength parameters is the basis of the establishment of storage space and wellbore, especially for the shale formation. However, the strength parameters and indentation mechanism of shale are unclear. It is necessary to research the strength parameters and dynamic indentation of shale from the multiscale perspective. A comparative experimental study of shale indentation at the macro- and mesoscales was conducted. The experimental investigate determined the four stages of shale indentation at the mesoscale. With increasing loading rate at the same load value, indentation depth decreases, and the critical fracture load of shale increases. The result shows that the indentation load can improve the rate of penetration, but the shale strength increases slightly at the same time, which suppresses the energy release of the shale fragmentation. The probability of indentation fragmentation occurs 2 3 times at the mesoscales is greater than 4 times the value at the macroscale. The macro averages of shale hardness and elastic modulus are close to the mesoscale values. The differences are 7% and 5.5%, respectively. The study results can be used in the storage and development of hydrogen gas and the optimization design of drilling tools.

[8]

BOBKO

, ULM F

The nano-mechanical morphology of shale

Mechanics of Materials, 2008,40(4):318-337.

DOI:10.1016/j.mechmat.2007.09.006 URL [Cited within: 1]

Shale, the sealing formations in most hydrocarbon reservoirs, is made of highly compacted clay particles of sub-micrometer size, nanometric porosity and different mineralogy. In this paper, we propose and validate a technique to identify the nano-mechanical morphology of such a nanocomposite material. In particular, by means of a massive nanoindentation campaign at two different scales on a large range of shale materials, we show that the highly compacted plate- or sheet-like clay particles have a distinct nano-mechanical morphology with no privileged orientation of the particle-to-particle contact surface, as evidenced by a mechanical percolation threshold of 0 0.5. Furthermore, the nanoindentation results provide strong evidence that the nano-mechanical elementary building block of shales is transversely isotropic in stiffness, and isotropic and frictionless in strength. These observations lead to a sphere-like mechanical morphology for visibly plate- or sheet-like clay particles. The contact forces between the sphere-like particles activate the intrinsicly anisotropic elastic properties within the clay particles and the cohesive bonds between the clay particles. The mechanical stiffness and strength properties of porous clay scale with the clay packing density toward a unique set of shale-invariant material properties. The determination of mechanical microstructure and invariant material properties are of great importance for the development of predictive microporomechanical models of the stiffness and strength properties of shale. The approach presented here also applies to other chemically and mechanically complex materials exhibiting nanogranular behavior.

[9]

SHUKLA, P, KUMAR

, CURTIS

, et al. Nanoindentation studies on shales. San Francisco: 47th US Rock Mechanics/Geomechanics Symposium, 2013.

[Cited within: 2]

[10]

KUMAR

, CURTIS M

, GUPTA

, et al.

Estimation of elastic properties of organic matter in Woodford shale through nanoindentation measurements

SPE 162778-MS, 2012.

DOI:10.2118/162778-MS URL [Cited within: 2]

Shales are one of the most heterogeneous and complex natural materials found. Recent spike in the activities in shale gas and oil plays has been possible through horizontal drilling and hydraulic fracturing, which requires better understanding ofmechanical properties. Complexities associated with elastic properties of shale are amplified with presence of wide range of organic fraction present in them. There is a need to understand the mechanical properties of organics and their associatedimpact on bulk mechanical properties.

[11]

LIU

, OSTADHASSAN

Microstructural and geomechanical analysis of Bakken shale at nanoscale

Journal of Petroleum Science and Engineering. 2017,153:133-144.

DOI:10.1016/j.petrol.2017.03.039 URL [Cited within: 1]

With the development in production from shale oil and shale gas in North America during the last decade, more studies are being conducted in order to improve our knowledge of the shale characteristics. In this paper, samples from Upper and Middle Bakken Formation, which is an oil-bearing shale formation, were collected and analyzed. Permeability, porosity and saturation of the samples were studied in the lab. 2D XRD and EDX were used to study the mineral compositions, and FESEM was used to characterize the pore structure at micro and nanoscale. Implementing the image analysis method, the pore structure and pore size distributions (PSD) of the samples at nanoscale were quantified. In addition, nanoindentation method, which is a novel technique to investigate the geomechanical behavior of rocks, was applied to quantify the mechanical properties of the shale samples including Young's modulus, hardness, and fracture toughness at nanoscale.

[12]

LI W

, SAKHAEE-POUR

Macroscale Young’s Moduli of shale based on nanoindentations

SPWLA V57N6A3, 2016.

[13]

BLOCK M

, BONIJOLY

, HOFMANN

, et al. Shale geomechanics: A nano-indentation application. Amsterdam RAI, The Netherlands: 76th EAGE Conference & Exhibition, 2014.

[14]

LIU

, OSTADHASSAN

, BUBACH

Applications of nano-indentation methods to estimate nanoscale mechanical properties of shale reservoir rocks

Journal of Natural Gas Science and Engineering, 2016,35:1310-1319.

DOI:10.1016/j.jngse.2016.09.068 URL [Cited within: 1]

61Nano-indentation method was used to analyze the mechanical properties of shale from Bakken Formation.61Abnormal phenomena from the nano-indentation curves were reported and their causes were explained.61Elastic modulus and Hardness values of samples at nanoscale were calculated and their relationships with mineral compositions and microstructures were investigated.61Energy analysis method was used to quantify the nanoscale rock fracture toughness.

[15]

OLIVER W

, PHARR G

An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments

Journal of Materials Research, 1992,7(6):1564-1583.

DOI:10.1557/JMR.1992.1564 URL [Cited within: 1]

[16]

ZHANG

Guangqing

, CHEN

Mian

, JIN

Yan

, et al.

Measurement and interpretation of shale toughness under confining pressures

Journal of Engineering Geology, 2004,12(4):431-435.

URL [Cited within: 1]

Based on the fracture mechanics theory under confining pressure conditions, the authors test rock fracture toughness using the thick-wall cylinder specimens with double-wing fractures and different clay contents under various confining pressures. The authors then use the finite element model to interpret the toughness results. The regression analysis based on a number of experiments indicates that the relationships of fracture toughness with confining pressure and shale contents are linear with small errors. The experimental procedures and interpreting method presented in this paper assure a better accuracy for determination of fracture toughness for hydraulic fracturing than that obtained from experiments without confining pressures, and are thus favorable for improvement of hydraulic fracturing performance.

[17]

YANG T

, LI Z

, CHENG C

Scaling relationships for indentation measurements

Philosophical Magazine A, 2002,82(10):1821-1829.

DOI:10.1080/01418610208235693 URL [Cited within: 1]

Using dimensional analysis and finite-element calculations, several relationships that relate features of indentation loading and unloading curves to the hardness, the elastic modulus and the work of indentation are extended to conical indentation in elastic-plastic solids with various angles. These relationships provide new insights into indentation measurements. They may also be useful to the interpretation of results obtained from instrumented indentation experiments.

[18]

WEN

, GONG J

, PENG Z

, et al.

Analysis of continuous stiffness data measured during nanoindentation of titanium films on glass substrate

Materials Chemistry and Physics, 2011,125(3):500-504.

DOI:10.1016/j.matchemphys.2010.10.026 URL [Cited within: 1]

An empirical function was proposed to describe the continuous stiffness curves, i.e., versus h (where Er is the reduced modulus, H is the composite hardness and h is the indenter penetration depth), measured with nanoindentation tests for titanium films on glass substrate. By analyzing the variations of the parameters included in this empirical equation with film thickness, the physical meanings of this empirical equation were discussed. It was shown that the mechanical properties of the substrate and the film may be extracted from such analyses.

[19]

DELAFARGUE

. Material invariant properties of shales: Nanoindentation and microporoelastic analysis. Cambridge: Massachusetts Institute of Technology, 2004.

[20]

LUO

Rong

, ZENG

Yawu

, CAO

Yuan

, et al.

Research on influence of inhomogeneity degree on mechanical parameters of inhomogeneous rock

Rock and Soil Mechanics, 2012,33(12):3788-3794.