Using Nanoindentation to Characterize the Mechanical and Creep Properties of Shale: Load and Loading Strain Rate Effects.

The mechanical and creep properties of shale strongly influence artificial hydraulic fracturing, wellbore stability, and the evaluation of reservoir performance in shale gas exploration. This study characterized these mechanical and creep properties at the microscale through nanoindentation tests and evaluated their dependence on the indentation test parameters, specifically, the indentation load and the loading strain rate. The mechanical parameters (the Young's modulus and hardness) of shale were strongly influenced by the magnitude of an indentation load (2-400 mN). Both parameters decreased sharply as the load increased from 2 to 200 mN; they then remained relatively stable at loads of 200-400 mN, suggesting that large indentation loads (200-400 mN) can be used to detect the mechanical responses of bulk shale. In contrast, both parameters increased slightly as the loading strain rate increased from 0.005 to 0.1 s-1. The indentation creep (C IT), related to creep behavior, and the creep strain rate sensitivity (m), related to the creep mechanism of shale, both increased with increasing the indentation load, whereas they decreased with increasing the loading strain rate. This demonstrates that increasing the load or decreasing the loading strain rate can increase creep deformation in shale during nanoindentation creep testing. The values of m varied from 0.040 to 0.124 under different loading conditions, suggesting that dislocation power-law creep may be the main mechanism controlling creep in shale. This study standardizes the testing parameters for the characterization of the mechanical properties of shale by nanoindentation testing and also advances our understanding of the deformation mechanisms of shale at the microscale.

Introduction

Shale gas is an important unconventional energy resource in China. It is produced and stored in organic-rich shales that have extremely low permeability and low porosity.[1] Therefore, artificial fracturing (mainly employing hydraulic power) must be applied to extract the gas from the tight shale rock.[2] Successful fracturing depends heavily on the reliable determination of the mechanical properties of a shale reservoir.[3−5] A further important consideration is the creep behavior of shale, which can weaken the scale of artificially induced fractures after a period of time.[6] Therefore, the quantitative characterization of the mechanical properties and creep behavior of shale should greatly aid the planning and operation of hydraulic fracturing.

Conventional uniaxial/triaxial compression tests can reveal the macroscopic mechanical properties and creep behavior of shale.[7−9] However, these tests are generally time-consuming and require high-quality samples comprising a large volume free from fracture, which is generally difficult to obtain in field sampling.[10−12] Therefore, nanoindentation has been introduced as an alternative technique to characterize the mechanical properties of shale.[13−17] This technique uses only small sample volumes (cuttings are acceptable) and can quickly generate large amounts of mechanical data. It therefore greatly aids the analysis of the micromechanical properties of shale; examples include the calculation of the mechanical parameters of individual phases in shales, such as minerals[16−19] and organic matter,[14,15,20−23] the relationships between mineralogy and bulk mechanics of shale,[14,15,24] the elastic anisotropy of shale,[17,25,26] and the methods for the prediction of the macromechanical properties of shale.[23,26,27] Besides, nanoindentation can also be used to study shale creep.[28] Wick reported that nanoindentation gave results that were comparable with those of conventional uniaxial compression testing for creep behavior; thus, they believed that nanoindentation measurements were reliable in determining the creep behavior of shales.[29] Recent research has focused on the effects of shale composition[30−33] and temperature[34] on its nanoindentation creep behavior, creep models for shale,[35] and the relationship between the creep parameters obtained from nanoindentation and triaxial creep tests.[23,31,36] Nanoindentation testing has proved useful for both the characterization of mechanical properties and evaluating creep behavior.

Shale is highly heterogeneous, being composed of various hard minerals (e.g., quartz, feldspar, pyrite, calcite, and dolomite), soft clays, and organic matter, and containing many pores and other defects.[19,31,37,38] Its mechanical responses at different indentation positions will therefore vary. As such, a single indentation cannot provide sufficient information about the whole (composite) mechanical properties of shale. To solve this problem, previous researchers have taken the average value of a large number of indentation data, which are usually obtained by grid indentation to homogenize the mechanical parameters of the shale.[12,14−16,30,39] The principle used in testing is that when the maximum indentation depth (hmax) produced by an indentation load is much larger than the particle size of the individual phases, the average mechanical properties of the composite material can be obtained.[40] However, there is no standardized method for the nanoindentation testing of materials with complex microstructures such as shale, especially regarding the characteristic indentation load beyond which clay-rich shale responds homogeneously.

Additionally, nanoindentation is essentially depth-sensitive: the volume range acted on by the indenter at different depths into the sample varies, potentially producing different mechanical responses and different mechanical data. Previous materials science studies have demonstrated the universality of the indentation size effect (ISE), which is manifested as the measured elastic modulus and hardness increasing with decreasing load (or indentation depth).[41,42] The measured creep behavior of metals also shows significant ISE,[43−50] as the creep strain rate sensitivity (m) and hardness can become smaller as the indentation displacement deepens,[43−48,50] whereas some metallic films have exhibited the opposite behavior.[49] Furthermore, the mechanical and creep properties of polymer materials were reported to depend strongly on the indentation loading rate, with a higher loading strain rate generally leading to higher values of hardness and lower creep strain rate sensitivity.[48] However, some exceptions show opposite trends for hardness[51] and creep strain rate sensitivity[52] when indenting metallic materials. As for geological samples, Shi et al. discussed the impact of the loading rate on the mechanical properties of shale,[12] but its effect on creep behavior was not involved. In addition, a negative dependence on the indentation size and the loading rate was found for the mechanical properties of solid bitumen.[12] Although much work has been done in metal materials science, the effects of the load and the loading strain rate on the mechanical properties and creep behavior of shale are still not fully understood.

In general, the main purpose of this study was to use nanoindentation to explore the effects of the indentation load and the loading strain rate on the mechanical properties and creep behavior of shale at the microscale. We examine the creep strain rate sensitivity through the characterization of the creep behavior of shale under different indentation loads and loading strain rates and discuss a potential deformation mechanism. These results could aid understanding of the influence of loading conditions on the mechanical properties and creep behavior of shale, which is an important area of research in both field and laboratory settings.

Sample and Experimental Methods

Sampling and Sample Preparation

The shale sample used for nanoindentation testing was collected from the Lower Silurian Longmaxi Formation, Sichuan Basin, South China (Figure ). X-ray diffractometry analysis showed that it was composed of quartz (34.4%), feldspar (13.6%), carbonates (4.5%), pyrite (1.5%), and clay minerals (45.9%).

Figure 1

Figure 1. (a) The bulk Longmaxi shale core and (b) the polished shale sample. (Taken by the author).

Nanoindentation testing requires the sample surface to be quite flat[53] and therefore carefully polished. The sample was first cut to ∼10 mm length ×5 mm width ×5 mm height, and then cast into a resin. The exposed surface of the shale was perpendicular to the bedding plane. Various silicon carbide abrasive papers (mesh numbers from 50 to 2000 grit) were used sequentially to polish the sample. Furthermore, surface polishing was performed using a series of alumina grit pads (9, 3, and 1 μm). Finally, an IM4000 argon ion mill operated at 4.0 kV and a 5° angle, which was produced by Hitachi High-Tech Global Corporation, was used for polishing for 2 h to achieve a completely flat surface. Atomic force microscopy (AFM) was used to determine the mean roughness (root mean square roughness, Rq) from two different areas (30 × 30 μm2) of the shale surface, yielding a value of 67.9 ± 10.9 nm. Figure shows a tomographic AFM image of the sample (Dimension 3100, Bruker Nano/Veeco, CA).

Figure 2

Three-dimensional topographic image of shale rock from AFM (area 30 × 30 μm2, Rq = 60.1 nm).

Mechanical Testing

Introduction to Nanoindentation

This study used a commercial nanoindentation apparatus (NHT[3] nanoindenter, Anton Paar Company) equipped with a diamond Berkovich indenter. The stiffness threshold was 500 N/m. The indenter applied loads of 0.1–500 mN as it was pressed into the sample surface.[53] Before the nanoindentation test, a standard fused silicon specimen was employed to calibrate the tip shape. The stiffness threshold was 500 N/m and its spring compliance was 0.571 mm/N. The resolutions of the load and displacement settings were 20 nN and 0.01 nm, respectively.

Figure a outlines a schematic diagram of the nanoindentation. Complete indentation included applying a loading force to the sample surface and holding it for some time prior to being unloaded. Figure b shows a typical P–h curve, which is divided into three stages (i.e., loading, holding, and unloading).

Figure 3

Schematic of a (a) nanoindenter,[30] and (b) schematic of a typical nanoindentation curve, P: the applied load, Pmax: the peak load, S: the contact stiffness, h: the indentation displacement, hmax: the maximum indentation displacement, We: elastic work, and Wp: plastic work.

By using a continuous mechanical model, the hardness (H) and reduced Young’s modulus (Er) of the material can be obtained from the P–h curvewhere S is obtained from the initial slope of the unloading curve, which represents the stiffness of the material, β is a constant that depends on the geometry of the indenter (for a common Berkovich indenter, β = 1.034), and Ac is the projected contact area, which is calculated from the depth of contact (Hc).[53,54]

Furthermore, based on the reduced Young’s modulus and Poisson’s ratio, the Young’s modulus of the material can be defined aswhere Ei and νi are the Young’s modulus and Poisson’s ratio of the diamond indenter, respectively. For the diamond indenter, Ei = 1140 GPa and νi = 0.07. E and ν are the Young’s modulus and Poisson’s ratio of the samples prepared for testing, respectively. For the shale sample, the Poisson’s ratio is assumed to be 0.30.[26] Shale rock with a Poisson’s ratio between 0.05 and 0.30 would have an 8% uncertainty in the Young’s modulus measured using nanoindentation.[14]

Procedure for Testing Mechanical Properties

Constant load tests assessed the influence of the indentation load on mechanical properties, and tests at the constant strain rate explored the effect of the loading strain rate. Specifically, constant load tests used loads of 2, 5, 10, 20, 50, 100, 200, 300, and 400 mN, all at a loading strain rate of 0.05 s–1. The tests at the constant strain rate applied strain rates of 0.005, 0.010, 0.030, 0.070, and 0.100 s–1 during the loading stage with a maximum indentation load of 300 mN. The holding time for studying the shale’s mechanical properties was 2 s. For all the indentation tests, the indenter approached and was retracted from the sample surface at a rate of 2000 nm/min within a distance of 2000 nm of the sample surface.

As mentioned in the introduction, this work characterized the mechanical properties of shale by averaging a large amount of nanoindentation data. Obtaining the data relied on grid indentation. Figure a shows the schematic diagram of grid nanoindentation, and Figure b–d shows typical microscope images of a matrix of 100 indents (at a load of 300 mN) on the shale sample.

Figure 4

(a) Schematic of the nanoindentation grid. (b–d) Typical optical microscopy images of the shale sample with a matrix of 100 indentations made at a load of 300 mN; the photography used an eyepiece with 40× magnification and objective lenses with (b) 5×, (c) 50×, and (d) 20× magnification.

There are some specific requirements for grid nanoindentation, including the selection of the representative elementary area (REA) of the shale matrix[27] and the minimum distance between neighboring indentations.[55] The REA refers to the smallest area that can be analyzed to represent the bulk shale in all respects. The minimum distance between neighboring indentations should be at least 10 times the indentation depth in order to avoid mechanical interference between adjacent indentations.[55] The minimum REA in shale is 250 μm × 250 μm,[27] which provides a basis for selecting the minimum indentation number. Therefore, grid indentation should apply at least 200 individual indentations at loads of 2, 5, and 10 mN, and at least 100 individual indentations for loads of 20, 50, 100, 200, 300, and 400 mN to cover the REA of the sample surface in each case. The distances between neighboring indentations were 20 μm for loads of 2–10 mN, 50 μm for loads of 20–300 mN, and 60 μm for a load of 400 mN.

Creep Testing

Creep Test Procedure

Shale creep can be investigated by measuring the displacement–time curve during the holding stage, in which the applied load remains constant and the indentation displacement increases. There are three stages in a macroscale creep test: transient (or primary), steady state (or secondary), and tertiary (or accelerating).[56] Most nanoindentation cases show only the first two stages because the tertiary stage means that creep deformation is accelerating to failure, which is difficult to observe.[28,35]

Similar to the mechanical characterization, indentations at the constant load and the constant strain rate were used to investigate the effects of the load and the loading strain rate on the creep behavior of the shale. Dynamic mechanical analysis model testing was applied during the loading stage. This involved small monochromatic oscillations modulating the indenter load. The main difference between the investigation of the mechanical properties and creep behavior was the duration of the holding stage. Specifically, the creep tests used a holding time of 1200 s. The selected dwell time was much greater than most of those previously reported, which helped to identify any possible change in the creep mechanism during testing.[57]

Calculation of Creep Strain Rate Sensitivity (m)

Creep strain rate sensitivity (m) is a power-law exponent reflecting the flow performance of a material and can indicate the deformation mechanism.[11,48,50,58] The smaller the m value, the more resistance to creep deformation. It is determined from the steady-state stage of the creep test. The following power-law relationship is usually used to describe steady-state creep behavior[59]where and σ are the steady-state creep strain rate and the applied stress, respectively; A is a constant; R is the molar gas constant; Q is the activation energy for a thermally activated process; T is the temperature; and n is the power-law exponent (creep stress exponent). All the experiments were conducted at room temperature, which could be considered relatively constant, allowing eq to be described aswhere C1 is a temperature-dependent material constant. For the nanoindentation creep test via a Berkovich indenter, and σ can be calculated using their equations as follows[57]where h and t are the instantaneous indentation depth and the indentation time, respectively, and P is the instantaneous applied indentation load.

Equation can be rewritten as

Therefore, the value of m can be calculated by taking the logarithm on both sides of eq . Its value is the reciprocal of n:[48,60]

Results and Discussion

Mechanical Properties

Mechanical Properties at Various Loads

Figure a shows typical load–displacement curves resulting from a series of tests with increasing loads (2, 5, 10, 20, 50, 100, 200, 300, and 400 mN). Indentation testing comprises loading, holding, and unloading stages. The loads of 200–400 mN show similar curves for the different stages, suggesting consistent mechanical responses at these loads. Figure b shows the relationship between the indentation load and the resulting displacements. Specifically, the average displacement increased from 199.44 ± 70.85 nm at a load of 2 mN to 4507.66 ± 388.73 nm at a load of 400 mN (Table ). An increase in the indentation load results in an increase in the indentation depth.

Figure 5

Typical curves for (a) load versus displacement and (b) displacement versus load. Plots of (c)Young’s modulus and (d) hardness at indentation loads of 2–400 mN at a loading strain rate of 0.05 s–1.

Table 1

Statistical Results for Nanoindentation Mechanical Properties Measured by Different Test Methodsa

		hmax (nm)		hardness (GPa)		Young’s modulus (GPa)
Pmax (mN)	loading strain rate (s–1)	ave.	std.	ave.	std.	ave.	std.
2	0.05	199.44	70.85	4.37	5.13	55.82	23.90
5	0.05	362.84	122.95	3.69	4.66	49.92	22.15
10	0.05	526.72	151.76	3.00	3.10	47.91	14.90
20	0.05	797.57	175.26	2.25	2.01	44.30	9.99
50	0.05	1338.85	240.43	1.86	1.82	40.91	8.22
100	0.05	2032.31	275.40	1.33	0.94	37.22	5.74
200	0.05	3095.08	347.72	1.08	0.34	33.15	4.48
300	0.05	3850.71	390.53	1.04	0.28	32.10	3.94
400	0.05	4507.66	388.73	0.98	0.20	31.39	3.54
300	0.005	3945.21	365.62	0.96	0.23	30.91	3.56
300	0.01	3952.53	484.52	0.96	0.26	32.36	3.28
300	0.03	3998.32	421.01	0.97	0.39	32.29	2.96
300	0.07	3649.10	263.51	1.13	0.21	35.03	3.61
300	0.1	3702.93	334.24	1.08	0.25	33.96	4.06

Pmax: maximum load and hmax: maximum indentation depth.

Figure c,d shows the load profiles of the mean Young’s modulus and hardness, respectively. The curves show two sections: a sharp decrease at loads of 2–200 mN and a relatively flat section at loads of 200–400 mN. Specifically, the mean Young’s modulus varied from 55.82 GPa at a load of 2 mN to 33.15 GPa at a load of 200 mN, whereas hardness ranged from 4.37 to 1.08 GPa at these loads. At loads of 200–400 mN, the mean Young’s modulus varied slightly between 33.15 and 31.39 GPa, whereas hardness ranged from 1.08 to 0.98 GPa (Table ). This indicates that with increasing load, the area and volume acted on by the indentation increased continuously, so the indentation response gradually approached that of the whole shale. This result can also be derived from the decreasing standard deviations of mechanical data with the increasing indentation load. Therefore, loads of 200–400 mN can be used to characterize the mechanical properties of the bulk shale. Alternatively, the large standard deviations at relatively low loads (2–10 mN) indicate that indentations penetrated only a single mineral phase (quartz or clay minerals). Consequently, averaging data from nanoindentations at low loads cannot appropriately represent the whole shale sample.

Accuracy of Estimates of Mechanical Properties

Indentation testing in regions with predominantly hard or soft minerals would give different mechanical properties. To better understand the mechanical properties of shale at the microscale, we chose for detailed analysis two representative points (labeled a and b) from the grid indentations. Figure shows hard minerals can be easily observed, while soft minerals, mainly clay, are continuously distributed in the shale, acting as a framework in which hard mineral grains are embedded.[19] The optical microscopy image of point a indicates that the residual impression was in a region of mainly soft minerals, while point b appeared to be mainly in hard minerals (Figure ).

Figure 6

Optical microscopy images of two representative indentation points: a in mainly soft minerals and b in mainly hard minerals. Indentation was at a strain rate of 0.05 s–1 with a maximum load of 400 mN.

Figure shows that the diameters of the two representative indentations were >20 μm, which exceeds the size of most single mineral phases. The size of the residual impression for the region of soft minerals was larger than that for the region of hard minerals. Point a had lower hardness (0.87 GPa) and Young’s modulus (30.49 GPa) than point b (1.48 and 34.67 GPa, respectively). The mean hardness and Young’s modulus of the grid nanoindentation were 1.04 ± 0.28 and 32.11 ± 3.94 GPa, respectively. The mechanical parameters of the two regions are comparable and very close to the mean values. Therefore, it is acceptable to take the average value of a large number of indentation data to derive mechanical parameters representative of the whole shale.

A “pop-in” phenomenon was observed during the loading stage of nanoindentation (Figure ), characterized by a sudden increase in displacement. This has been attributed to crack propagation or microcracks and pores at the contacts among minerals in the shale.[61]

Figure 7

Typical nanoindentation curves for various loads during nanoindentation testing of the Longmaxi Shale sample. (a) Normal P–h curve and P–h curves with the “pop-in” phenomenon during the loading stage for loads of (b) 200, (c) 300, and (d) 400 mN.

To investigate the effects of the “pop-in” phenomenon on the results of nanoindentation, the following power-law functions were used to fit the loading and unloading curves[61]where P, h, and hf are the applied load, displacement, and residual displacement, respectively. K, n, α, and m are constants derived from fitting the loading and unloading curves.

Table lists curve fitting parameters for the loading and unloading stages of typical nanoindentation curves. The results indicate that the power-law functions fit both curves very well, consistent with previous findings.[61] This implies that the theory of nanoindentation can be used to calculate the shale’s mechanical parameters.

Table 2

Curve Fitting Analysis for the Loading and Unloading Stages

		loading stage			unloading stage
	Pmax (mN)	K	n	R2	α	m	R2
a	200	0.000056	1.862	0.9997	0.0026	1.7129	0.9901
b	200	0.0011	1.4675	0.9973	0.0035	1.6824	0.9892
c	300	0.00016	1.743	0.9985	0.0261	1.4044	0.9805
d	400	0.0078	1.2792	0.9986	0.00039	1.9206	0.9885

For indentation loads of 200–400 mN, 100 indentations were used to give statistically accurate average values for the mechanical parameters. Figure a,c shows the relationship between the mean Young’s modulus and the number of indentations at loads of 300 and 400 mN. It suggests that the mean Young’s modulus and standard deviation changed slightly when there were more than 30 indentations. For an allowed error of 3% in the mean Young’s modulus, a matrix of 36 indentations (6 × 6 grid) is adequate to predict the mechanical properties of the bulk shale sample (Figure b,d). Previous studies have used less than 36 indentations to characterize the mechanical properties of bulk shale,[12,30,35,39] whereas few studies have provided direct evidence supporting the choice of the number of nanoindentations. Overall, the abovementioned findings indicate that at least 36 indentations at loads of 200–400 mN can be used to obtain the mechanical properties of shale.

Figure 8

Statistical analysis of the number of indentations: the mean Young’s modulus and standard deviation with respect to the number of indentations at loads of (a) 300 and (c) 400 mN, and the error (|μ–μ100|/μ100) of the mean Young’s modulus with respect to the number of indentations at loads of (b) 300 and (d) 400 mN. Note: μ represents the mean Young’s modulus for n indentations, and μ100 represents the mean Young’s modulus for 100 indentations.

Mechanical Properties at Various Loading Strain Rates

Given the abovementioned findings, a matrix of 36 indentations was used to obtain average mechanical data for shale at each indentation loading strain rate with a constant load of 300 mN. Figure a plots the corresponding typical load–displacement curves, which are almost uniform in shape. Figure b shows the variation in indentation displacement for different loading strain rates. Increasing the loading strain rate generally decreases the average displacement. Specifically, it decreased from 3945.21 ± 365.62 nm at a loading strain rate of 0.005 s–1 to 3702.93 ± 334.24 nm at a loading strain rate of 0.1 s–1. This phenomenon has also been found in a previous study.[12]

Figure 9

Typical curves for (a) load versus displacement and (b) displacement versus loading strain rate. Plots of (c) Young’s modulus and (d) hardness at indentation loading strain rates of 0.005–0.1 s–1 and a load of 300 mN.

Figure c,d shows a generally positive relationship between the mean Young’s modulus and hardness with respect to loading strain rates (0.005–0.1 s–1). Specifically, the mean Young’s modulus varied slightly between 30.91 ± 3.56 and 35.03 ± 3.61 GPa, and the mean hardness ranged from 0.96 ± 0.23 to 1.13 ± 0.25 GPa (Table ). As the loading strain rate increased, it might have produced a higher flow stress and a larger induced compressive stress, which increased the deformation volume of the material and expanded the plastic zone size.[12] This accounts for the high loading strain rate causing a reduction in surface deformation displacement (Figure b). Consequently, pressure hardening improved the contact hardness and the compressive yield stress with an increasing loading strain rate.[12]

Notably, as the loading strain rate increased, the mean Young’s modulus and hardness increased only by 9.8% and 12.5%, respectively. Compared with the variations seen under different loads, altering the loading strain rate did not substantially change the Young’s modulus or hardness, suggesting that they were more sensitive to the indentation load than to the loading strain rate under the considered methodology and experimental conditions.

Creep Properties

Based on the results in Section , the influence of the indentation load on shale creep was studied using indentation loads of 200, 300, and 400 mN at a constant loading strain rate of 0.05 s–1, and by applying loading strain rates of 0.005, 0.010, 0.030, 0.070, and 0.100 s–1 at a constant load of 300 mN. This could eliminate the effect of the load on the bulk shale. The typical load–displacement curves and creep displacement–time curves for the different indentation loads and different loading strain rates are shown in Figures a,b and 11a,b, respectively. The corresponding measured parameters of the ratio of elastic work to total work (We/Wt) and the indentation creep rate (CIT) are displayed in Figures c,d and 11c,d. The We/Wt ratio represents the elastic part of indentation work; a higher value indicates a greater contribution of elastic deformation. CIT was calculated by dividing the maximum displacement during the loading stage by the creep displacement during the holding stage. A higher CIT value indicates greater creep deformation. Statistical results for these parameters are provided in Table .

Figure 10

Typical curves for (a) load versus displacement and (b) creep displacement versus time. Plots of (c) We/Wt and (d) CIT at indentation loads of 200, 300, and 400 mN at a loading strain rate of 0.05 s–1.

Figure 11

Typical curves for (a) load versus displacement and (b) creep displacement versus time. Plots of (c) We/Wt and (d) CIT at loading strain rates of 0.005, 0.01, 0.03, 0.05, 0.07, and 0.1 s–1 at a load of 300 mN.

Table 3

Statistical Results for Nanoindentation Creep Tests by Different Methodologiesa

		hmax (nm)		CIT (%)		We (pJ)		Wt (pJ)		Wp (pJ)		We/Wt (%)		creep strain rate sensitivity (m)
Pmax (mN)	loading strain rate (s–1)	ave.	std.	ave.	std.	ave.	std.	ave.	std.	ave.	std.	ave.	std.	ave.	std.
200	0.05	3070	239	13.06	2.72	56846	7541	265663	32277	208817	32908	21.7	3.7	0.047	0.025
300	0.05	4344	240	14.17	3.28	113178	11356	572566	64135	459388	58208	19.9	1.8	0.052	0.037
400	0.05	4726	357	17.62	2.43	157460	14813	850639	77591	693179	72922	18.6	2.0	0.066	0.036
300	0.005	4595	329	17.17	3.77	118686	12888	683686	69586	565000	65218	17.4	1.9	0.124	0.030
300	0.01	4527	424	17.86	4.78	118422	15122	640071	80872	521649	71521	18.6	1.8	0.070	0.034
300	0.03	4414	408	15.14	5.67	117789	15483	579884	71959	462095	69494	20.5	2.9	0.067	0.041
300	0.07	4567	424	12.57	1.94	105585	7847	566893	64458	461308	59696	18.8	1.6	0.040	0.020
300	0.1	4563	332	12.61	2.56	113571	10018	579341	59630	465770	56956	19.8	2.2	0.041	0.022

Pmax: the maximum load; hmax: the maximum indentation depth; CIT: the ratio of creep displacement during the holding stage to the maximum displacement during the loading stage; We: elastic work; Wp: plastic work; and Wt: total work.

Variation in Creep Behavior with Indentation Load

Figure a shows the load–displacement curves at each stage for various loads. Figure b shows creep displacement with respect to creep time (for 1200 s), showing two main stages: an initial transient stage (the first ∼600 s) and a subsequent steady-state stage (the following 600 s). During the transient period, creep displacement increased with gradually decreasing acceleration, and the variation in the creep rate (the slope of the curve) is large.[30] During the steady-state stage, the creep displacement increased linearly with the creep time, and the slope of the curve for a large indentation load appears relatively steep, while the creep rate remains nearly constant.[30] The We/Wt ratio decreased with increasing indentation load (Figure c), dropping from 21.7% at 200 mN to 18.6% at 400 mN. The CIT was positively correlated with the indentation load (Figure d), increasing from 13.06% at 200 mN to 17.62% at 400 mN, suggesting that the creep of shale increases with an increasing load.

Creep Behaviors with Respect to Loading Strain Rate

Figure a,b shows typical load–displacement curves and creep time–displacement curves for different loading strain rates (0.005, 0.010, 0.030, 0.070, and 0.100 s–1) and a total creep time of 1200 s. A lower loading strain rate led to greater creep displacement for a given holding time. According to the variation in creep displacement in Figure b, the creep–time curves also show a two-stage upward trend: an initial rapid rise within the first ∼600 s, followed by a slow linear increase. This variation conforms to established creep stages identified in previous studies.[30]

The We/Wt ratio increased as the strain rate increased up to 0.05 s–1 (Figure c), after which it slightly decreased. Overall, the generally positive trend for the We/Wt ratio shows that the shale may have undergone more elastic deformation with an increasing loading strain rate. The CIT value was negatively related to the loading strain rate, decreasing from 17.17% at a loading strain rate of 0.005 s–1 to 12.61% at a rate of 0.1 s–1, suggesting that the creep capability of the shale weakened with an increasing loading strain rate.

It is noted that the accurate measurement of mechanical parameters only needs several seconds of holding time.[62] The mean mechanical parameters (hardness and Young’s modulus) for a holding time of 1200 s were less than those for a holding time of 2 s. For example, the mean hardness varied from 1.04 ± 0.28 GPa to 0.77 ± 0.10 GPa, and the mean Young’s modulus decreased from 32.10 ± 3.94 GPa to 25.92 ± 2.37 GPa with an increasing holding time. That is, both parameters decreased as the creep time increased. Creep was possibly responsible for the decrease (consistent with previous findings[35,63]) owing to an increase in the creep time producing greater displacement, which would result in lower measured mechanical parameters.

Creep Mechanism of Shale

Calculation of the Creep Strain Rate Sensitivity (m) of Shale

The empirical formula was used here to fit the creep displacement–time curves[50,64,65]where h is the creep displacement and t is the creep time. Parameters a, b, c, d, and e are constants obtained by fitting nanoindentation creep data sets. Representative plots of experimental creep displacement versus time and the corresponding curves fitted using this equation (loading strain rate = 0.01 s–1, Pmax = 300 mN) are shown in Figure a. The values of and σ, calculated by eqs and 5, are plotted with respect to creep time in Figure b,c. The former decreased significantly from 6.31 × 10–4 to 5.71 × 10–5 s–1 during the first ∼600 s, and then remained relatively invariable at around the 10–5 order of magnitude during the later steady-state stage during the last 600 s (Figure b). In comparison, σ gradually decreased during the first ∼600 s before declining linearly. Therefore, the creep strain rate sensitivity (m) was taken from the steady-state stage. A logarithmic plot of the creep strain rate () and stress (σ) based on eq (Figure d) gave the value of m. During the steady-state stage of creep, the logarithmic value of and σ decreased correspondingly from the upper right corner to the lower left with a good linear relationship (Figure d). The red triangle in Figure d represents the fitting region and the data used to calculate m as 0.070.

Figure 12

(a) Displacement–time, (b) creep strain rate–time, (c) stress–time, and (d) logarithmic stress versus logarithmic creep strain rate curves during the creep stage at a loading strain rate of 0.01 s–1 and a load of 300 mN. The holding time was 1200 s during the creep stage.

For further nanoscale investigation of the creep behavior of bulk shale, we chose two representative indentations (labeled points 1 and 2) from the grid for detailed analysis. Table lists information about both indentations. Point 1 was indented into mainly soft minerals, as demonstrated by the low hardness and Young’s modulus, and high hmax and creep displacement. In contrast, the optical microscopy image indicates that point 2 was in mainly hard minerals; the hardness and Young’s modulus were high, and hmax and creep displacement were low (Figure ; Table ).

Table 4

Summary of the Indentation Settings and Mechanical and Creep Properties for Two Points

points	Pmax (mN)	loading strain rate (s–1)	hmax (nm)	hardness (GPa)	Young’s modulus (GPa)	creep displacement (nm)	m
point 1	300	0.05	4619.7	0.66	25.77	461.1	0.066
point 2	300	0.05	3129.5	1.71	31.89	417.4	0.046

Figure 13

Optical microscopy images of points (a) 1 and (b) 2. Curves of (c) load versus displacement and (d) creep displacement versus creep time during the creep stage for both points at the strain rate of 0.05 s–1 and a maximum load of 300 mN.

The m value of point 1 was a little larger than that of point 2 (Table ), which suggests that the region rich in soft minerals was more likely to creep. However, the small difference in m values indicates that taking an average value from many points can give a representative characterization of the shale’s creep behavior.

Effects of Indentation Load and Loading Rate on Creep Strain Rate Sensitivity

Figure shows the creep behavior of shale to be positively correlated with the indentation load but negatively correlated with the loading strain rate. The calculated m value was not fixed, but changed with the load and the loading strain rate. With the increasing indentation load, it increased slightly from 0.047 at a load of 200 mN to 0.066 at 400 mN (Table ; Figure a). A generally slight positive dependence of m on the indentation load is commonly observed for metals and bitumen, which can be interpreted using a “shear transformation zone” (STZ) or “free volume” model.[44,66−68] In the STZ model, the atomic-scale STZs consist of a large free volume site with adjacent atoms, and they are considered to be the basic shear units. They collectively deform under applied shear stress, resulting in shear bands, which may further induce the local softening of the material. It is generally considered that with an increasing indentation load, the STZ or “free volume” will continue to increase, the power to resist creep deformation will reduce, and the creep strain rate sensitivity will increase.[44,66]

Figure 14

Creep strain rate sensitivity, m, (a) at different indentation loads with a loading strain rate of 0.05 s–1 and (b) with respect to the loading strain rate at a load of 300 mN. Mean values with error bars (one standard deviation) are plotted.

In contrast to the slight increase of m with the increasing load, its value fell from 0.124 to 0.040 with the increasing loading strain rate (Table ). The generally small variation in m suggests only a weak negative dependence on the loading strain rate (Figure b), which is consistent with the variation of CIT (Figure d). This suggests that a lower loading strain rate leads to proportionally stronger creep behavior of the shale. Similar results have also been found for metals at room temperature.[48,69] A possible explanation of this result is that a higher loading strain rate can lead to higher hardness (as observed in Figure ) and hence smaller m.[48] Another possible reason is that a higher loading strain rate can cause more heterogeneous grain structures and higher local stress during the loading period, which can effectively inhibit the absorption and emission of dislocations at grain boundaries during the subsequent creep period. This finally leads to a lower m at the steady-state stage of the creep.[48]

Creep Mechanism Predicted from Creep Strain Rate Sensitivity

Possible creep mechanisms during the nanoindentation creep experiments have been established: dislocation power-law creep, diffusional creep, and grain boundary sliding.[48,64,70,71] The m values of 0.040–0.124 observed here for Longmaxi shale are relatively small, and are therefore not attributed to grain-boundary sliding. Given the high creep strain rate and deep holding depth, grain boundary diffusion and tip–sample atomic diffusion can also be eliminated.[64] Previous studies have indicated that when m = 1, the creep is dominated by the diffusion of atoms via lattice diffusion or Coble creep,[48] when m = 0.5, the creep is mainly controlled by grain boundary sliding,[70] and when m < 0.33, dislocation power-law creep is the dominant creep mechanism.[64,71] Therefore, the small values of m seen here suggest dislocation power-law creep as the dominant creep mechanism for shale. As such, due to the high load (200–400 mN) and shear stress, the sample surface subjected to the indenter underwent strong shear flow, and gliding dislocations met obstacles such as precipitates and lattice atoms, and attempted to cut through or bypass them.[72] This dislocation movement may have dominated the deformation behavior of the shale. Ma et al. found that the creep stress exponent (n = 1/m) for quartz, feldspar, and mica in granite was above 3, and suggested that dislocation creep was the dominant deformation mechanism for these minerals in granite at low temperatures and high strain rates.[73] The m value observed here is very close to those previously reported for Eagle Ford, Wolfcamp, and Woodford marine shales,[30] suggesting a consistent creep mechanism for different shales.

Furthermore, the comparison of the m value for the shale with that of other rocks or their components could provide useful results. Table summarizes m values for different rocks and rock components as reported in the literature.

Table 5

Comparison of m Values From Previous Works

reference	rock or its components	n	m	test technique
Brantut et al. 2012	granite, basalt, and sandstone	5.5∼34	0.03–0.18	Triaxial creep tests
Rybacki et al. 2017	Posidonia shale	∼4.8	∼0.21
Naumann et al. 2007	Opalinus Clay	∼5	∼0.2
Gupta et al. 2018	Woodford, Wolfcamp, and Eagle Ford shale	5.6–6.9	0.14–0.17	nanoindentation creep tests
Ma et al., 2021	quartz in granite	10.5–13.5	0.07–0.10
Ma et al., 2021	feldspar and mica in granite	>3	<0.33
Liu et al., 2019	solid bitumen	81–1250	0.0008–0.0123
this work	Longmaxi shale	10–30	0.040–0.124

The results were obtained primarily via macroscopic creep testing (i.e., triaxial compression testing)[74−76] and nanoindentation creep tests.[11,30,73] None of the studies provided a value of m above 0.33. The lowest was 0.0008–0.0123 for solid bitumen, and the highest was 0.21 for Posidonia shale; therefore, dislocation creep may be the dominant mechanism in all these cases. The m values of shales from nanoindentation creep tests are very close to those of Posidonia shale from triaxial creep testing, which suggests that nanoindentation testing is useful for predicting the macroscopic creep behavior of shale. Further research should compare the creep behavior during nanoindentation and traditional triaxial testing.

Shale is a complex multiphase Earth material, and explaining its physical deformation mechanism remains challenging. Traditional macroscopic tests have suggested that a range of mechanisms may drive creep in shale, including subcritical crack growth, pressure solution, frictional slip, internal grain deformation, grain compaction and rearrangement, grain sliding, and the collapse of clay floccules.[31] The present nanoindentation test results indicate that dislocation creep may be the dominant deformation mechanism for shale. However, dislocations here may have been produced by the compaction and rearrangement of grains and their motion through frictional sliding. As dislocations are too small to be directly observable by optical microscopy, future observation by high-resolution transmission electron microscopy should provide direct evidence at the nanoscale, as it can examine the variation of fine textures in shale before and after indentation, including the boundaries of dislocations. Overall, the present findings provide quantitative guidance for revealing the micromechanism of shale creep behavior and may provide a baseline to investigate the creep behavior of other rocks at elevated temperatures.

Limitations and Suggestions for Further Work

Although the effects of the indentation load and the loading strain rate on m have been described above, the large standard deviation of its values, especially for different indentation loads, does not indicate that the conclusions are highly reliable. The variations in m observed in the present study may have been due to the following reasons:

The holding time of the indentation experiment was a major factor owing to the limited thermal stability of the nanoindenter: the relatively short duration of most indentation tests (minutes rather than days or months) typically used to assess tensile creep might have influenced the results.[57]

Primary creep might have significantly affected the observed creep behavior.[31]

Using a heterogeneous and anisotropic sample (i.e., shale) during the test might have induced complex creep behavior. The creep curve with the “pop-in” phenomenon and/or data with a relatively large thermal drift must be discarded.[33,57]

Shale is a type of fine-grained sedimentary rock composed of a variety of minerals (clay, quartz, feldspar, pyrite, and carbonate) and organic matter. Each component has its own creep behavior. Of these components, organic matter and clay minerals are highly flexible and thus susceptible to creep deformation. A nanoscale investigation of the creep behavior of shale can help elucidate its creep deformation at a larger scale. Therefore, our objective for forthcoming work is to determine the dominant mechanism controlling the creep deformation of shale at a fine scale.

Furthermore, the influences of fluid, pressure, and temperature were not considered here. Therefore, the determined mechanical properties and creep behavior of shale do not fully represent those in the subsurface. It is reported that the friction coefficient between clay minerals and organic matter would increase with increasing temperature, thus altering the creep behavior of shales.[34] In addition, during the hydraulic fracturing procedure of a shale gas formation, the interaction between fracturing fluids and shale rock may affect its creep behavior. Considering the millimeter size of shale and the relatively short test duration, our creep test was carried out without involving the effects of fluids. Details of in situ shale reservoir conditions (i.e., temperature and fluids) should be considered in a future study.

Finally, although previous research has shown that the elastic modulus determined by indentation testing is similar to that obtained from traditional triaxial compression testing, nanoindentation technology is quite different from macroscale compression testing in terms of boundary conditions and loading geometry.[31] Moreover, for heterogeneous shale, the Young’s modulus from the upscaling model by mineral fraction is close to that obtained by uniaxial compressive testing,[26,39] whereas there have been inconsistencies among values from averaged nanoindentation data, the upscaling model by deconvolution analysis and macroscale methods.[14,27,39] In addition, creep behavior measured by microindentation can predict that measured by the macroscopic uniaxial compressive testing of similar composite materials such as concrete.[77] However, creep behaviors from triaxial testing and micromechanic-based homogenization schemes from nanoindentation experiments show some discrepancies, which may be attributed to the differences in creep strain values and spatial scales for the heterogeneous shale between these two methods.[31,36] At present, it remains unclear whether nanoindentation results are comparable with those from traditional triaxial testing. Therefore, the relationship between the results obtained from these two methods should be explored.

Conclusions

This study evaluated the mechanical parameters (the Young’s modulus and hardness) and creep behavior of Longmaxi shale via nanoindentation testing and investigated their sensitivities to the indentation load and the loading strain rate.

Nanoindentation tests using indentation loads of 2–400 mN obtained information on the mechanical properties of bulk shale. The mechanical properties significantly decreased as load increased from 2 to 200 mN, but remained relatively stable at 200–400 mN, suggesting that the applied methodology and experimental conditions can influence the measured mechanical properties of bulk shale.

The mechanical properties were more sensitive to the indentation load than to the loading strain rate. Increasing the indentation load (from 2 to 400 mN) initially decreased the Young’s modulus and hardness. In contrast, both parameters increased slightly as the loading strain rate increased from 0.005 to 0.100 s–1.

Both the indentation creep (CIT) and the creep strain rate sensitivity (m) were positively dependent on the indentation load, whereas they were negatively dependent on the loading strain rate.

Values of m varied from 0.040 to 0.124, suggesting that dislocation power-law creep is the main mechanism controlling deformation of the shale.

In summary, optimizing the indentation testing method for shale will provide fundamental experimental data for better understanding and predicting shale mechanical deformation and fracturing performance. However, because of the heterogeneity and anisotropy of shale, the mechanism controlling its creep deformation remains unclear. Further experimental and theoretical research in the field is still required to understand the various factors influencing the creep mechanism, including the mineral compositions of the shale and in situ reservoir conditions (i.e., temperature and fluid).