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Anisotropic creep characteristics and mechanism of shale under elevated deviatoric stress
Journal Pre-proof Anisotropic creep characteristics and mechanism of shale under elevated deviatoric stress Cunbao Li, Jun Wang, Heping Xie PII:
S0920-4105(19)31091-5
DOI:
https://doi.org/10.1016/j.petrol.2019.106670
Reference:
PETROL 106670
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 7 December 2018 Revised Date:
14 September 2019
Accepted Date: 7 November 2019
Please cite this article as: Li, C., Wang, J., Xie, H., Anisotropic creep characteristics and mechanism of shale under elevated deviatoric stress, Journal of Petroleum Science and Engineering (2019), doi: https://doi.org/10.1016/j.petrol.2019.106670. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Anisotropic Creep Characteristics and Mechanism of Shale under Elevated Deviatoric Stress Cunbao Li*
1
Jun Wang2 , Heping Xie3
Abstract: The time-dependent characteristics of anisotropic shale are important for accurately predicting a reservoir’s behavior over a long period of time. To study the transverse isotropic creep characteristics of shale, a series of shale creep tests using specimens with 4 different bedding layer orientations (0◦ , 45◦ , 75◦ and 90◦ ) under multiple levels of deviatoric stress were conducted. The experimental results indicate that shale presents creep behavior even when the deviatoric stress is relatively low. Anisotropy has significant influences on the creep deformation and steady creep rate. When the bedding plane inclined angle is 45◦ and under the same deviatoric stress, the creep deformation and steady creep rate are the highest, while the smallest creep deformation and steady creep rate are generated when the bedding plane orientation is 90◦ . Time-dependent deformation appears the tertiary creep stage when the deviatoric stress is crack damage threshold, which implies that crack damage stress can be regarded as the shale’s long term strength. The steady creep rate increases exponentially with increasing deviatoric stress as the stress is larger than the crack initiation threshold. The rationality of the empirical creep law is evaluated. It is concluded that the empirical creep model can only fit the existing creep data, but it is almost impossible to predict the creep deformation. The mechanism for generating anisotropic creep behavior of shale is analyzed in detail, and three basic creep patterns for shale are proposed based on the action mechanism of the stress tensor components. A general methodology for proposing the anisotropic creep model is finally suggested. Keywords: anisotropic creep; shale; bedding layer orientation; empirical creep law; basic creep patterns
1 Introduction The commercialized exploitation of shale gas has changed the global energy pattern to some extent. The anisotropic mechanical properties of shale, including its strength, deformation, failure pattern, brittleness, and crack propagation, have been extensively investigated [1, 2, 3, 4, 5, 6]. However, studying the time-dependent deformation of shale is also of importance for accurately evaluating the long period mechanical behaviors of reservoirs. Failing to address the anisotropic creep properties of shale may lead to non-negligible deviations in predicting the in situ stress [7], which has a significant role in designing hydraulic fracturing and optimizing drilling. Some studies [8, 9, 10] state that creep is one reason for the unavoidable significant decline in the shale gas production rate after the initial reservoir stimulation. In general, most shale gas field data indicate that the production rates generally decrease 45% – 55% after the first 5–6 months of stimulation, and the first 3-year average decrease in productivity rates ranges between 77% and 89%. Creep can make the cracks narrower and the reservoir’s permeability 1
*Corresponding author; Institute of Deep Earth Science and Green Energy, Shenzhen University; Institute of New Energy and Low-carbon Technology, Sichuan University, Sichuan University; Email: [email protected]; [email protected] 2 Institute of Deep Earth Science and Green Energy, Shenzhen University; MOE Key Laboratory of Deep Underground Science and Engineering, Sichuan University 3 Institute of Deep Earth Science and Green Energy, Shenzhen University; MOE Key Laboratory of Deep Underground Science and Engineering, Sichuan University
1
smaller. To accurately predict reservoir behavior and performance, it is essential to study the time-dependent characteristics of shale. The characteristics of creep for anisotropic rock have been studied primarily through experiments. As early as 1939, Griggs [11] utilized mechanical creep facilities to study the timedependent behavior of shale; although Tiggers [11] insisted that there were some unknown errors in the tests that made the experimental results difficult to understand, the tests demonstrated that shale possessed evident creep deformation. Nishihara [12] improved the experimental equipment designed by Tiggers and proved that increasing moisture content and stress could enhance the creep deformation. Since then, however, few papers focused on the shale creep properties [13] until the past 12 years thanks to the prosperity of shale gas exploitation. Sone [7], Sone and Zoback[14, 15], Yang and Zoback [16] used uniaxial and triaxial creep experiments of reservoir shale to analyze the influence of clay content on the shale creep behavior, and the results indicated that a higher clay content leads to larger creep deformation, and shale indentation creep tests also supported this conclusion [17]. Shale creep behaviors can also be enhanced in the presence of fluid [18, 19], under a shear loading path [20] and at elevated temperature and stress [7, 21, 22, 23]. However, the application of deviatoric stress in the published creep experimental results ignores the importance of crack initiation threshold or crack damage threshold. As the deviatoric stress is larger than the crack initiation threshold but lower than the crack damage threshold, the microcracks in the rock would be in a stable growth stage; if the deviatoric stress is larger than the crack damage threshold, the microcracks would be in an unstable growth stage [24, 25]. The understanding of shale creep behavior under crack initiation threshold or crack damage threshold would guide evaluations of the long-term stability of shale. Because of the sedimentary structural features, i.e., bedding plane, of shale, the shale creep is anisotropic. Due to the time-consuming nature of anisotropic shale creep experiments, relatively few studies have addressed this issue [15, 26, 27]. Those results implied that the creep deformation when the bedding layer orientation was horizontal was higher than that when the bedding plane orientation was vertical. However, in most of the published experimental data, only when the bedding plane orientation is 0◦ or 90◦ are studied; as bedding layer orientation that is neither parallel nor perpendicular to the loading direction, little detailed information of anisotropic creep behavior can be obtained. The main purpose of this research is to investigate the anisotropic creep behaviors of shale. A series of triaxial creep experiments on shale specimens with 4 bedding layer orientations under multiple deviatoric stress levels are performed. Then, the creep properties of shale with considering crack initiation stress and crack damage stress are presented in detail. The effects of bedding layer orientations on creep behaviors, including deformation and steady creep rate, are analyzed. The empirical equation to describe the anisotropic creep behavior of shale is illustrated in detail, and suggestions regarding the rationality of the empirical equations are presented. Finally, the mechanical mechanism of anisotropic creep is emphasized, and the methodology for theoretically modeling the creep anisotropy based on the anisotropic creep mechanism is also discussed.
2 Shale properties and anisotropic creep experiment methodology 2.1
Anisotropic shale specimen preparation and experimental apparatus 2
In this work, the shale samples were obtained from the outcrop of Lower Silurian Longmaxi Formation in Pengshui county, Chongqing, China. Fig. 1 obvious macroscopic bedding planes. The shale dry density is around 2.7 g/cm3 , and its porosity is 4.7% – 5.2%. The mineral composition and proportion of the shale samples were characterized by X-ray diffusion (XRD) analysis, which mainly observed quartz, clay minerals, albite, calcite and dolomite (Table 1). The clay mineral contained Illite, Kaolinite, Chlorite and Interstratified Illite / Smectite (Table 2). Based on a series of compression experiments with different bedding layer inclination angles [4], the tested shale is transversely isotropic and the sedimentary plane can be regarded as isotropic plane (Fig. 2). Table 1: Mineral composition of the Longmaxi shale Mineral composition
Quartz
Albite
Calcite
Orthoclase
Dolomite
Pyrite
Clay
Fraction (%)
37.0
10.9
7.7
3.2
5.1
1.9
34.2
Table 2: Clay mineral composition of the Longmaxi shale Clay Mineral Interstratified Illite Kaolinite Chlorite composition Illite /Smectite Fraction (%)
66.0
3.0
16.0
15.0
To avoid the influence of water on shale mechanical behavior, specimens with 4 different bedding plane inclined angles (β = 0◦ , 45◦ , 75◦ and 90◦ , as shown in Fig. 2 (a)) were drilled without using a water-based coolant. The cylindrical specimens were finally prepared with a diameter of 50 mm and a height of 100 mm, respectively. The end faces of the cylindrical samples were carefully processed to keep the two end faces with a roughness within ±0.05 mm. Both of the end faces were perpendicular to the specimen axis with a discrepancy of less than ±0.25◦ and parallel to each other. Creep experiments were conducted using a servo-controlled geomaterial rheological apparatus (Fig. 3 (a)) with confining pressure in the range of 0–100 MPa and maximum axial force range of 2000 kN. The axial deformation can be recorded by one pair of linear variable displacement transducers with a maximum range of 6 mm, and the radial deformation can be measured by the annular extensometer wrapped around the tested specimens with a maximum range of 3 mm (Fig. 3 (b)). The control system and measurement system of the rheological experimental apparatus are stable and have high accuracy, which is critical for long-term creep tests.
2.2
Anisotropic creep experiment methodology
To investigate the effects of anisotropy on shale creep behavior, creep tests were performed with multistep loadings by using specimens with 4 different bedding plane orientations (0◦ , 45◦ , 75◦ and 90◦ , the definition of bedding layer orientation β is defined in Fig. 2 (b)). The reason of selecting those 4 angles is that the failure pattern and mechanism of shale are similar when the bedding plane inclination angles are 15◦ , 30◦ and 45◦ , while the failure mechanisms are the 3
1s ple ditables pl. pdf Figure 1: Shale outcrop
0°
45°
75° β 90°
(a)
(b)
2s ple ditables pl. pdf Figure 2: (a) Definition of global coordinate system and specimen preparation diagram of with different bedding plane inclined angles; (b) definition of the bedding plane inclined angle.
4
same when the bedding plane inclination angles are 60◦ and 75◦ [28].A confinement of 50 MPa, which was determined by the in situ stress situation of the Longmaxi shale gas reservoir, was applied in all the experiments. The creep tests began with a relatively low stress level, which was then increased in steps. When the bedding plane orientations were 0◦ and 90◦ (specimen numbers are 0-50 and 90-50, respectively, where 0 or 90 represents the inclination angle of the bedding layer and 50 represents the confining pressure), deviatoric stress levels of 20, 40, 60 , 80, 100 and 120 MPa were used. When the bedding plane orientations were 45◦ and 75◦ (specimen numbers are 45-50 and 75-50, respectively), deviatoric stress levels of 60, 80, and 120 MPa were adopted to contrast with the experimental results of specimens 0-50 and 90-50 and to reduce the test time. To investigate the creep behavior when the deviatoric stresses are crack initiation stress and crack damage stress, which have been determined by the author’s previous research [29], these two specific stresses were applied on specimens 45-50 and 75-50. For specimen 45-50, the crack initiation stress adopted in this study was 190 MPa and the crack damage stress was 205 MPa, and for specimen 75-60, these two specific stresses were 190 MPa and 223 MPa. The reason that the crack initiation stresses for specimens 45-50 and 75-50 are the same is that the influences of bedding plane inclined angle on the crack initiation stress is little [29]. In each deviatoric stress level, the creep test time was selected as 48 hours according to the findings that 1-day creep tests was enough for understanding the long-term mechanical behaviors of shale [27]. The confining pressure remained as a constant in each deviatoric stress level and during loading deviatoric stress. The details of the specimens, confining pressure and stress level are summarized in Table 3. During the loading period, the confining pressure and deviatoric stress were increased at rates of 3 MPa/min and 2 MPa/min, respectively. The radial and axial deformations of the shale specimens were recorded every 1 second in the loading and initial stage of primary creep and after that the data was recorded every 5 seconds.
LVDT
specimen 煤 样
Annular extensometer
3s ple ditables pl. pdf Figure 3: (a) Rock rheological experimental apparatus; (b) positions of specimen and measurement equipment. 5
Table 3: Summaries of the confining pressure and deviatoric stress level in the multistep anisotropic creep tests (σci and σcd represent the crack initiation stress and crack damage stress, respectively) Specimen Bedding plane Confining Deviatoric stress (MPa) 4th 5th 6th number orientation(◦ ) pressure(MPa) 1st 2nd 3rd 0-50 0 50 20 40 60 80 100 120 45-50 45 50 60 80 120 190 (σci ) 205 (σcd ) \ 75-50 75 50 60 80 120 190 (σci ) 223 (σcd ) \ 90-50 90 50 20 40 60 80 100 120
3 Anisotropic creep experimental results of shale 3.1
General observation of creep characteristics when the deviatoric stress is smaller than the crack initiation stress
Several typical creep curves of deformed Longmaxi shale with different bedding layer orientations under multistep deviatoric stress are illustrated in Fig. 4. The axial deformation shown here includes the strain induced by increasing axial load and the strain caused by creep effect. It should be noted that the deformation curves in this paper are plotted with all the recorded data. However, the noises of the data is not as serious as that presented by Sone and Zoback [15]. As shown in the Figure 5, even creep curve is enlarged, the curve does not present many noises. The reason is that the experimental apparatus used in this study is specially designed for creep test. The device can record data more stable for the long-time creep test. The famous MTS 815 rock mechanics testing system is also used by us to do shale creep test with considering high temperature. However, the data has serious noises, as shown in Figure 6. As shown in Fig. 4, all the specimens exhibit time-dependent deformation irrespective of the magnitudes of deviatoric stress. All the time-dependent deformation curves have two stages: primary creep stage, where the strain increase per unit time experienced a rapid decrease, and secondary creep stage, where the creep rate keeps constant. No tertiary creep behavior is observed in the tests because the deviatoric stress is not higher than the crack damage stress. The duration of the primary creep stage ranges from 4 hours to 15 hours based on the analysis of the variation of creep rate. The higher the deviatoric stress is, the longer is the duration of the primary creep stage. During the creep strain measurement under any different deviatoric stress level, there is no sign of the termination of creep deformation; however, when the deviatoric stress is low, the steady creep rate is small (Fig. 7). Note that this observation is different from the results of some argillaceous rock or shale [30, 31], in which secondary creep can only be observed when the deviatoric stress exceeds crack initiation threshold or crack damage threshold. The reasons for this difference may be the accuracy of the experimental apparatus or the mineral content and distribution of the tested rock. For a rock specimen containing multilevel cracks and under constant deviatoric stress, “there are always some sites of high stress concentration leading to the initiation and propagation of micro-cracks and inducing creep deformation [9]. Fig. 7 presents the relationship between the steady creep rate and deviatoric stress. One can see with increasing deviatoric stress, the steady creep rate increases almost linearly regardless of the bedding plane inclination angle. Fig. 8 illustrates the creep deformation under the same deviatoric stress and different bedding layer orientations. This figure shows that the 6
deviatoric stress has significant influences on the magnitude of time-dependent deformation. The correlation between the creep deformation and deviatoric stress could be described by a linear function. The anisotropic features of the steady creep rate and deformation will be analyzed in the following section.
0.008
0.008
(a) 0°
0.007
0.006
40 MPa
0.005 60 MPa
0.004
80 MPa
0.003
Axial srain
Axial strain
0.006
(b) 45°
0.007
20 MPa
0.005 0.004 0.003
60 MPa
0.002
100 MPa
0.002
80 MPa
0.001
120 MPa
0.001
120 MPa
0
0 0
10
20 30 Time (hours)
40
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(c) 75°
0.008
10
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50
(d) 90°
0.006
0.007
20 30 Time (hours)
0.005
20 MPa
0.004
40 MPa
0.005
Axial strain
Axial strain
0.006 0.004 0.003 0.002
60 MPa 80 MPa 120 MPa 30 40
0.001 4E-18 -0.001 0
10
20
60 MPa
0.003
80 MPa 0.002 100 MPa 0.001
50
Time (hours)
120 MPa
0 0
10
20 30 Time (hours)
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4s ple ditables pl. pdf Figure 4: Shale creep deformation under multiple levels of deviatoric stress with different bedding layer orientations: (a) β = 0◦ ; (b) β = 45◦ ; (c) β =75◦ ; and (d) β = 90◦ .
3.2
Effects of anisotropy on shale creep behavior
To investigate the effects of bedding plane on the creep characteristics in detail, the timedependent deformations of shale samples with 4 different bedding plane orientations (β = 0◦ , 45◦ , 75◦ and 90◦ ) under the same deviatoric stress are depicted in Fig. 9. As illustrated in Fig. 9 (a), the instantaneous strain β , where the subscript β represents the bedding plane orientation, satisfies 90◦ < 75◦ < 45◦ < 0◦ . This result occurs because the apparent Young’s modulus Eβ of the transversely isotropic shale satisfies E90◦ > E75◦ > E45◦ > E0◦ . Fig. 9 also presents that under the same deviatoric stress condition, the primary creep deformations and creep rate when the bedding layer orientations are 45◦ and 75◦ are more apparent than those of the other two bedding layer orientations. This phenomenon can be explained more clearly by Fig. 7, which reveals that anisotropy has significant impacts on the steady creep rate. As shown in Fig. 7, considering that the deviatoric stress is 120 MPa, when the bedding layer inclination angle is 45 ◦ , the steady creep rate is the highest, which is 3.5 times higher than the smallest steady creep rate generated by the shale specimen with the 7
50
0.008
0.0048
0.007
45°
45°
0.0047 Axial strain
Axial strain
0.006 0.005 0.004 0.003
60 MPa 80 MPa 120 MPa
0.002 0.001
0.0046 0.0045 0.0044 80 MPa 0.0043 0.0042
0 0
10
20 30 Time (hours) (a)
40
0
50
10
20 30 Time (hours)
40 (b)
5s ple ditables pl. pdf Figure 5: Creep deformation curves obtained by TOP geomaterial rheological apparatus have little noise: (a) creep deformation curve plotted with all recorded data; (b) enlarged creep curve plotted with all recorded data
0.00355 0.00350
Axial strain
0.00345 0.00340 0.00335 0.00330 0.00325 0.00320 0.00315 0
5
10
15 20 25 Time (hours)
30
35
40
6s ple ditables pl. pdf Figure 6: Creep deformation curve of shale obtained by MTS 815 rock mechanics testing system has obvious noises
8
50
1.4E-04 0-50
1.2E-04 steady creep rate ( /day)
45-50 1.0E-04 75-50 8.0E-05
90-50
6.0E-05 4.0E-05 2.0E-05
0.0E+00 0
20
40 60 80 Deviatoric stress (MPa)
100
120
7s ple ditables pl. pdf Figure 7: Steady creep rate under multiple levels of deviatoric stress with different bedding layer orientations
6.0E-04 y = 4E-06x + 0.0001 R² = 0.9957
Creep strain
5.0E-04
y = 4E-06x + 6E-05 R² = 0.9998
4.0E-04
0-50
y = 2E-06x + 0.0001 R² = 0.9974
3.0E-04
45-50
2.0E-04
75-50 1.0E-04
y = 1E-06x + 0.0001 R² = 0.8816
90-50
0.0E+00 0
20
40
60
80
100
120
Deviatoric stress (MPa)
8s ple ditables pl. pdf Figure 8: Shale creep deformation under multiple levels of deviatoric stress and different bedding layer orientations. bedding plane orientation of 90◦ . The steady creep rate as the bedding plane orientation is 75◦ is the second highest, while the steady creep rate as the bedding plane orientation is 0◦ is the 9
4.0E-03
5.0E-03
3.5E-03
4.5E-03
3.0E-03
4.0E-03
2.5E-03
Axial strain
Axial strain
third highest in the same stress situation. The anisotropic characteristics of the primary creep and steady creep rate described above results in the creep strain of shale are completely different when the specimen’s bedding layer orientations vary, which is clearly plotted in Fig. 8. This figure presents that the creep strain of specimen 45-50 is the largest and reaches 3.5 × 10−4 when the deviatoric stress is 60 MPa, which is 1.4 times larger than that of specimen 0-50. This difference in creep strain leads to the axial strain of specimen 45-50 exceeding the axial strain of specimen 0-50, as shown in Fig. 9 (a); thus, the initial axial strain of specimen 45-50 when applying deviatoric stress (80 MPa and 120 MPa) is higher than that of specimen 0-50, which is illustrated in Fig. 9 (b) and (c).
3.5E-03
0-50 45-50 75-50 90-50
2.0E-03 1.5E-03
0-50 45-50 75-50 90-50
3.0E-03 2.5E-03
1.0E-03
2.0E-03 0
10
20 30 40 Time (hours) (a) Deviatoric stress is 60 MPa
50
0
10
20 30 40 Time (hours) (b) Deviatoric stress is 80 MPa
8.0E-03
Axial strain
7.0E-03 6.0E-03 5.0E-03 0-50 45-50 75-50 90-50
4.0E-03 3.0E-03 2.0E-03 0
10
20 30 40 50 Time (hours) (c) Deviatoric stress is 120 MPa
9s ple ditables pl. pdf Figure 9: Shale creep deformation under the same deviatoric stress and different bedding layer orientations. .
3.3
Creep behavior when the deviatoric stresses are crack initiation stress and crack damage stress
Fig. 10 shows the creep curves of deformed shale specimens when the bedding layer orientations are 45◦ and 75◦ under the effects of crack initiation stress and crack damage stress. To enhance the contrast, the typical time-dependent deformations when the deviatoric stress (60, 80 and 120 MPa) is lower than the crack initiation stress are also plotted in Fig. 10. When the deviatoric stress is crack initiation stress, the primary creep deformation is considerably larger than that 10
50
when the stress is 60 MPa, 80 MPa or 120 MPa regardless of the bedding layer inclination angle, and there is no tertiary creep stage. However, a tertiary creep stage is observed when the deviatoric stress is crack damage stress in both bedding layer orientations, which reveals that crack damage stress can be regarded as the long-term strength of shale. The normalization method is used to describe the variation trend of steady creep rate with increasing the deviatoric stress. Here the steady creep rate when the bedding plane inclination angles are 45◦ and 75◦ are normalized separately. In each bedding plane inclination angle, the highest strain rate is set as 1, then the normalized values of the other steady creep rates are calculated. Fig. 11 illustrates the normalized steady creep rates under 5 different deviatoric stresses (60 MPa, 80 MPa, 120 MPa, crack initiation stress and crack damage stress) and 2 bedding plane inclination angles (45◦ and 75◦ ). As shown, the normalized steady creep rate raises exponentially with increasing deviatoric stress. The normalized creep rates of specimens 45-50 and 75-50 when the deviatoric stresses are crack damage stress are approximately 7.7 times and 10.6 times higher, respectively, than those when the deviatoric stress is 20 MPa. The slope of the tangent line to the normalized steady creep rate - deviatoric stress curve is extraordinarily steep when the deviatoric stress arrives at the crack damage stress, which indicates the upcoming failure of the specimen.
0.02 (a) 45° 60 MPa
Axial srain
0.015
80 MPa 120 MPa
0.01
190 MPa 205 MPa
0.005
0 0
10
20 30 Time (hours)
40
50
0.02 (b) 75° Axial strain
0.015 60 MPa 80 MPa
0.01
120 MPa 190 MPa
0.005
223 MPa 0 0
10
20 30 Time (hours)
40
50
10s ple ditables pl. pdf Figure 10: Creep deformation as the deviatoric stresses are 60, 80, and 120 MPa; crack initiation stress; and crack damage stress: (a) bedding layer orientation is 45◦ ; (b) bedding layer orientation is 75◦ .
11
1.0
Normalized steady creep rate
0.8
45-50 75-50
0.6
0.4
0.2
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80 120 160 Deviatoric stress (MPa)
200
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11s ple ditables pl. pdf Figure 11: Normalized steady creep rate as the deviatoric stresses are 60, 80, and 120 MPa; crack initiation stress; and crack damage stress.
4 Discussion 4.1
Rationality of empirical laws to describe the anisotropic creep deformation of shale
Because of the simplicity of empirical laws, the empirical creep model is widely used to describe the creep deformation from primary creep [27, 23, 15]. The most popular equation adopted in the published studies is written as (t) = σBtn
(1)
where (t) is the strain induced by creep at time t; B and n are material parameters. Thus, the creep compliance function J(t) can be defined as J(t) = (t)/σ = Btn
(2)
Eq. (2) implies that the relationship between creep compliance function J(t) and time t in double logarithmic coordinates is linear. The creep compliance function of shale, which is calculated based on the experimental data and Eq. (2), is shown in Fig. 12. However, the trend observed in Fig. 12 indicates that the linear function cannot describe the variation of the creep compliance function with time in double logarithmic coordinates, particularly the instantaneous deformation immediately after applying the load. Therefore, Eqs. (1) and (2) may not be a good function to describe creep characterization. To better depict the overall process of time-dependent deformation, the following creep compliance function is suggested: J(t) = J1 + Btn
(3)
where J1 is the asymptotic instantaneous compliance. The reason for introducing the asymptotic instantaneous compliance J1 is that geomaterial typically exhibits creep, even for extremely short load durations. The introduction of J1 is important for accurately describing the creep strain, particularly the primary creep stage [32, 12
-4.28
log (J)
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log (time)
12s ple ditables pl. pdf Figure 12: Variation of creep compliance defined in Eq. 2 with time in double logarithmic coordinates (taking specimen 75-50 when the deviatoric stress is 60 MPa as an example) 33]. The value of J1 can be easily determined from the asymptote of the creep compliance function - time curve in the logarithmic space when the time tends to zero. Eq. (3) indicates that the relationship between J − J1 and time t is a linear function in the double logarithmic scale. Based on the algorithm presented above, the variation of J − J1 with timein the double logarithmic space is presented in Fig. 13. As shown, the log(J − J1 ) varies linearly with log(t) in the entire process of time-dependent deformation, which implies that Eq. (3) is better than Eq. (2) for depicting the creep deformation.
-5.2
-5.3
log (J - J1)
-5.4
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13s ple ditables pl. pdf Figure 13: Variation of creep compliance defined in Eq. 3 with time in double logarithmic coordinates (taking specimen 75-50 when the deviatoric stress is 60 MPa as an example) According to the anisotropic creep experimental data and Eq. (3), the asymptotic instantaneous compliance at multiple levels of deviatoric stress and different bedding layer orientations is shown in Fig. 14. As shown, asymptotic instantaneous compliance increases with increasing 13
deviatoric stress. Moreover, the asymptotic instantaneous compliance is enhanced by increasing the deviatoric stress and is dependent on the bedding plane orientations. As the bedding plane orientation is 90◦ , the asymptotic instantaneous compliance is the smallest compared with the other cases when the deviatoric stresses are the same. The growth rate of the asymptotic instantaneous compliance when the bedding plane orientation is 45◦ is the fastest. Material parameters B and n in Eq. (3) are obtained by fitting the time-dependent deformation curve (Fig. 4) in double logarithmic coordinates (Fig. 13) and are presented in Figs. 15 - 16. One can note that the scatter of material parameters B and n is high. It is hardly possible to determine any relations between material parameters with bedding layer orientation or deviatoric stress. Therefore, Eq. (3) may be good at fitting the existing experimental data, but it is impossible to predict the anisotropic creep behavior when the bedding layer orientations are random if there are only creep data for a few bedding plane orientations. For example, this research has performed creep tests for 4 different bedding plane inclination angles (0◦ , 45◦ , 75◦ and 90◦ ), and although Eq. (3) can fit those experimental data, it is absolutely impossible to evaluate the creep behavior when the bedding plane inclination angles are different from the 4 bedding plane orientations that have been tested. The essential reason for the lack of predictability of Eq. (3) is that empirical creep laws cannot consider the anisotropic creep mechanism. Empirical creep laws are mathematical fitting formulas with almost no physical meaning. The ideal anisotropic creep model that can fit and predict creep deformation should consider the anisotropic creep mechanism, which will be presented as follows.
6.0E-05
J1( / MPa)
5.0E-05
4.0E-05
0-50 45-50 75-50
3.0E-05
90-50
2.0E-05 0
20
40 60 80 Deviatoric stress (MPa)
100
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14s ple ditables pl. pdf Figure 14: Asymptotic instantaneous compliance under multiple levels of deviatoric stress with different bedding layer orientations
4.2
Basic creep pattern for anisotropic creep behavior
The bedding layer is the most important structural feature of shale. Because of the existence of the bedding layer, the mechanical behavior,including deformation, strength and failure pattern, of shale presents significant anisotropy. Assume that the principal stress tensor σij of shale representative volume element in the material coordinate system o − xyz (Fig. 17) is written as 14
4.0E-06
0-50 45-50 75-50
3.0E-06
B
90-50
2.0E-06
1.0E-06
0.0E+00 0
20
40 60 80 Deviatoric stress (MPa)
100
120
15s ple ditables pl. pdf Figure 15: Values of parameter B in Eq. (3) under multiple levels of deviatoric stress with different bedding layer orientations
0.40
0.35
n
0.30
0.25
0.20
0-50 45-50
0.15
75-50 90-50
0.10 0
20
40 60 80 Deviatoric stress (MPa)
100
120
16s ple ditables pl. pdf Figure 16: Values of parameter n in Eq. (3) under multiple levels of deviatoric stress with different bedding layer orientations
σxx 0 0 σij = 0 σyy 0 0 0 σzz
(4)
where i, j = x, y, z = Cartesian subscripts. Given that shale can be assumed to be a transversely isotropic material, it is generally more convenient to describe the mechanical properties of shale by mathematical equations in the local coordinate system o0 − x0 y 0 z 0 , as shown in Fig. 17. According to the transformation of coordinates, the principal stress tensor σij defined in Eq. (4) can be reformulated in the local coordinate system o0 − x0 y 0 z 0 as
15
z' o x o' x'
z y y'
α
17s ple ditables pl. pdf Figure 17: Schematic diagram of coordinate transformation of transverse isotropic shale
σxx 0 0 σyy cos2 (α) + σzz sin2 (α) σyy sin(α) cos(α) − σzz sin(α) cos(α) (5) σij0 = 0 0 σyy sin(α) cos(α) − σzz sin(α) cos(α) σyy sin2 (α) + σzz cos2 (α) where α is the rotation angle of the local coordination system o0 − x0 y 0 z 0 around the x0 axis, as presented in Fig. 17. Because of the presence of the bedding plane, the component of the stress tensor σij0 will result in different effects. For example, the stiffness in the z 0 direction is weaker than that in the x0 or y 0 direction; thus, when the magnitudes of σx0 0 x0 or σy0 0 y0 and σz0 z0 are the same, the time-dependent deformation in the z 0 direction should be larger. This is why the creep strain when the bedding layer orientation is 0◦ is higher than that when the bedding plan orientation is 90◦ (Fig. 8). Note that the bedding layer is a weak plane, and its resistance to deformation and strength is quite small. If there is a shear component σy0 0 z0 acting on the bedding plane, the creep deformation should be much larger compared with the deformation of the same size stress component acting on the shale matrix, which leads to the creep strain and creep rate are the highest when the bedding layer orientation is 45◦ , as clearly shown in Fig. 7, Fig. 8 and Fig. 9. According to the above analysis, the effects of the stress tensor components on the creep should be taken into account separately. Therefore, three basic creep patterns for the creep of a transversely isotropic material are proposed, as presented in Fig. 18. The first case is that the action direction of the stress tensor component is normal to the bedding layer; the second case is that the action direction of the stress tensor component is parallel to the bedding layer; and the third case is that the shear component of the stress tensor acts along the bedding layer and perpendicular to the bedding layer normal direction. All the other creep cases will be the combinations of these three basic creep patterns. It is suggested that an anisotropic creep theoretical model with a solid physical foundation may be better based on the three basic creep patterns. To propose such a theoretical anisotropic creep model, two steps can be adopted. The first step is to deduce the theoretical equation to calculate the time-dependent deformation under a single stress component, and this step can combine with damage mechanics to consider the coupling between creep and damage. The second step is to form the creep model under general loading conditions based on the principle of superposition. 16
σ1
σ1
σ1
σ1
σ1
σ1
σ1
σ1 (a) First case
(b) Second case
(c) Third case
18s ple ditables pl. pdf Figure 18: Three basic creep patterns for the creep of a transverse isotropic material
5 Conclusions Anisotropic creep tests on Longmaxi shale considering 4 different bedding layer orientations (0◦ , 45◦ , 75◦ , and 90◦ ) under multiple levels of deviatoric stress are conducted. The anisotropic creep characterizations and mechanism of shale are analyzed in detail. According to the current analysis and discussions, the following conclusions can be drawn: 1. Longmaxi shale presents primary creep and secondary creep even when the deviatoric stress is lower than crack initiation stress. The time-dependent deformation and steady creep rate increase linearly with increasing deviatoric stress when the stress is less than 120 MPa. 2. Anisotropy has significant effects on the creep strain and steady creep rate. The creep strain and steady creep rate reaches the maximum when the bedding layer orientation is 45◦ and under the same stress level, while the creep strain and steady creep rate generated are the minimum when the bedding plane orientation is 90◦ . 3. Crack damage stress can be regarded as the long-term strength of shale. The steady creep rate raises exponentially with increasing stress as the deviatoric stress is not lower than the crack initiation stress. 4. The empirical creep model can only fit the existing creep data, but it is almost impossible to predict the creep deformation because it has little physical foundation and does not take any creep mechanism into account. 17
5. The components of the stress tensor will lead to different time-dependent deformations when the material is anisotropic. Three basic creep patterns for shale are proposed based on the action mechanism of the stress tensor components. The general methodology for proposing an anisotropic creep model is presented. Acknowledgment: The authors wish to thank the financial support from the National Natural Science Foundation of China (No.51804203) and Sichuan University (No.20826041C4082, No.0060304153002).There is no conflict of interest between the authors.
References [1] Mokhtari, M., Bui, B.T., Tutuncu, A.N. 2014. Tensile failure of shales: impacts of layering and natural fractures. In SPE Western North American and Rocky Mountain Joint Meeting. Society of Petroleum Engineers. [2] Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D. 1997. Laboratory investigation of the mechanical behaviour of tournemire shale. International Journal of Rock Mechanics and Mining Sciences. 34(1):3–16. [3] Masri,M., Sibai,M., Shao,J.F., Mainguy,M. 2014. Experimental investigation of the effect of temperature on the mechanical behavior of Tournemire shale. International Journal of Rock Mechanics and Mining Sciences. 70,185–191. [4] Li, C.B., Bazant, Z.P., Caner, F.C., Chau, V.T. 2017. Spherocylindrical microplane constitutive model for shale and other anisotropic rocks. Journal of the Mechanics and Physics of Solids, 133:155–178. [5] He, B., Xie, L. Z., Li, F. X. 2017. Anisotropic mechanism and characteristics of deformation and failure of Longmaxi shale. Sci Sin-Phys Mech Astron, 47: 114611. [6] Heng,S., Guo,Y.T., Yang,C.H., Daemen,J.J.K., Li,Z. 2015. Experimental and theoretical study of the anisotropic properties of shale. International Journal of Rock Mechanics and Mining Sciences. 74:58–68. [7] Sone, H. 2012. Mechanical Properties of Shale Gas Reservoir Rocks, and Its Relation to the In-situ Stress Variation Observed in Shale Gas Reservoirs. PhD thesis, Stanford University. [8] Asala, H., Ahmadi, M., Taleghani, A. D. 2016. Why re-fracturing works and under what conditions. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. [9] Viet T. Chau, Cunbao Li, Saeed Rahimi-Aghdam, Bažant, Z. P. 2017. The Enigma of Large-Scale Permeability of Gas Shale: Preexisting or Frac-Induced?. Journal of Applied Mechanics-ASCE, 84(6): 061008. [10] Chhatre, S. S., Braun, E. M., Sinha, S., Determan, M. D., Passey, Q. R., Zirkle, T. E., Wood, A. C., Boros, J. A., Berry, D. W., Leonardi, S. A. 2015. Steadystate stress-dependent permeability measurements of tight oil-bearing rocks. Petrophysics. 56(02):116–124. [11] Griggs, D. 1939. Creep of rocks. The Journal of Geology, 47(3): 225-251. [12] Nishihara M. 1952. Creep of shale and sandy-shale. The Journal of the Geological Society of Japan, 58(683): 373-377. 18
[13] Horsrud, P., Holt, R. M., Sonstebo, E. F., Svano, G., Bostrom, B. 1994. Time dependent borehole stability: Laboratory studies and numerical simulation of different mechanisms in shale. Rock Mechanics in Petroleum Engineering: Society of Petroleum Engineers. [14] Sone, H., Zoback, M. D. 2013. Mechanical properties of shale-gas reservoir rocks - Part 2: Ductile creep, brittle strength, and their relation to the elastic modulus. Geophysics, 78(5): D390-D399. [15] Sone, H. and Zoback, M. D. 2014. Time-dependent deformation of shale gas reservoir rocks and its long-term effect on the in situ state of stress. International Journal of Rock Mechanics and Mining Sciences, 69:120–132. [16] Yang, Y., Zoback, M. 2016. Viscoplastic Deformation of the Bakken and Adjacent Formations and Its Relation to Hydraulic Fracture Growth. Rock Mechanics and Rock Engineering, 49(2): 689-698. [17] Kabir, P., Ulm, F.J., Akono, A.T. 2017. Rate-independent fracture toughness of gray and black kerogen-rich shales. Acta Geotechnica, 12(6): 1-21 [18] Guindon L. 2015. Viscoelastic Creep of Eagle Ford Shale: Investigating Fluid/Shale Interaction. Journal of Canadian Petroleum Technology, 54(3): 142-143 [19] Almasoodi, M. M., Abousleiman, Y. N., Hoang, S. K. 2014. Viscoelastic creep of eagle ford shale: Investigating fluid-shale interaction. In SPE/CSUR Unconventional Resources Conference–Canada. Society of Petroleum Engineers. [20] Yang, S.Q. and Cheng, L. 2011.Non-stationary and nonlinear visco-elastic shear creep model for shale. International Journal of Rock Mechanics and Mining Sciences, 48(6):1011–1020. [21] Chang, C., Zoback, M. D. 2009. Viscous creep in room-dried unconsolidated Gulf of Mexico shale (I): Experimental results. Journal of Petroleum Science and Engineering, 69(3-4), 239-246. [22] Geng Z, Bonnelye A, Chen M, Jin Y, Dick P, David C, Fang X, Schubnel A. 2017. Elastic Anisotropy Reversal During Brittle Creep in Shale. Geophysical Research Letters, 44(21): 1-9 [23] Rybacki, E., Herrmann, J., Wirth, R., Dresen, G. 2017. Creep of Posidonia Shale at Elevated Pressure and Temperature. Rock Mechanics and Rock Engineering, 50(12): 31213140 [24] Nicksiar, M., Martin, C. 2012. Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mechanics and Rock Engineering 45(4): 607-617 [25] Cai, M., Kaiser, P. K., Tasaka, Y., Maejima, T., Morioka, H., Minami, M. 2004. Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 41(5): 833-847. [26] Liu, Z., Xie, S., Shao, J.F., Conil, N. 2015. Effects of deviatoric stress and structural anisotropy on compressive creep behavior of a clayey rock. Applied Clay Science, 114:491–496. [27] Rassouli F. S., Zoback M. D. 2018. Comparison of Short-Term and Long-Term Creep Experiments in Shales and Carbonates from Unconventional Gas Reservoirs. Rock Mechanics and Rock Engineering, 51(7): 1995–2014. 19
[28] Li C. B. Theoretical research on 3D volumetric fracturing based on anisotropic microplane model. 2017. Doctoral thesis of Sichuan University. [29] Li, C.B., Xie, H.P., Xie, L.Z. 2017. Experimental and theoretical study on the shale crack initiation stress and crack damage stress. Journal of China Coal Society, 42(4):969–976. [30] Fabre, G., Pellet, F. 2006. Creep and time-dependent damage in argillaceous rocks. International Journal of Rock Mechanics and Mining Sciences 43(6): 950-960. [31] Li, Y.W., Ahmad, G. 2012. Creep behavior of Barnett, Haynesville, and Marcellus shale. 46th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association. [32] Jirásek, M., Bazant, Z.P. 2002. Inelastic analysis of structures. John Wiley and Sons. [33] Bažant ZP, Jirásek M. 2018. Creep and Hygrothermal Effects in Concrete Structures. Springer.
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Highlights： 1. Anisotropic creep behaviors of shale are presented in detail. 2. Rationality of the empirical creep law for shale is evaluated. 3. Anisotropic creep mechanism is analyzed. 4. Three basic creep patterns for shale are proposed.
S0920-4105(19)31091-5
DOI:
https://doi.org/10.1016/j.petrol.2019.106670
Reference:
PETROL 106670
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 7 December 2018 Revised Date:
14 September 2019
Accepted Date: 7 November 2019
Please cite this article as: Li, C., Wang, J., Xie, H., Anisotropic creep characteristics and mechanism of shale under elevated deviatoric stress, Journal of Petroleum Science and Engineering (2019), doi: https://doi.org/10.1016/j.petrol.2019.106670. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Anisotropic Creep Characteristics and Mechanism of Shale under Elevated Deviatoric Stress Cunbao Li*
1
Jun Wang2 , Heping Xie3
Abstract: The time-dependent characteristics of anisotropic shale are important for accurately predicting a reservoir’s behavior over a long period of time. To study the transverse isotropic creep characteristics of shale, a series of shale creep tests using specimens with 4 different bedding layer orientations (0◦ , 45◦ , 75◦ and 90◦ ) under multiple levels of deviatoric stress were conducted. The experimental results indicate that shale presents creep behavior even when the deviatoric stress is relatively low. Anisotropy has significant influences on the creep deformation and steady creep rate. When the bedding plane inclined angle is 45◦ and under the same deviatoric stress, the creep deformation and steady creep rate are the highest, while the smallest creep deformation and steady creep rate are generated when the bedding plane orientation is 90◦ . Time-dependent deformation appears the tertiary creep stage when the deviatoric stress is crack damage threshold, which implies that crack damage stress can be regarded as the shale’s long term strength. The steady creep rate increases exponentially with increasing deviatoric stress as the stress is larger than the crack initiation threshold. The rationality of the empirical creep law is evaluated. It is concluded that the empirical creep model can only fit the existing creep data, but it is almost impossible to predict the creep deformation. The mechanism for generating anisotropic creep behavior of shale is analyzed in detail, and three basic creep patterns for shale are proposed based on the action mechanism of the stress tensor components. A general methodology for proposing the anisotropic creep model is finally suggested. Keywords: anisotropic creep; shale; bedding layer orientation; empirical creep law; basic creep patterns
1 Introduction The commercialized exploitation of shale gas has changed the global energy pattern to some extent. The anisotropic mechanical properties of shale, including its strength, deformation, failure pattern, brittleness, and crack propagation, have been extensively investigated [1, 2, 3, 4, 5, 6]. However, studying the time-dependent deformation of shale is also of importance for accurately evaluating the long period mechanical behaviors of reservoirs. Failing to address the anisotropic creep properties of shale may lead to non-negligible deviations in predicting the in situ stress [7], which has a significant role in designing hydraulic fracturing and optimizing drilling. Some studies [8, 9, 10] state that creep is one reason for the unavoidable significant decline in the shale gas production rate after the initial reservoir stimulation. In general, most shale gas field data indicate that the production rates generally decrease 45% – 55% after the first 5–6 months of stimulation, and the first 3-year average decrease in productivity rates ranges between 77% and 89%. Creep can make the cracks narrower and the reservoir’s permeability 1
*Corresponding author; Institute of Deep Earth Science and Green Energy, Shenzhen University; Institute of New Energy and Low-carbon Technology, Sichuan University, Sichuan University; Email: [email protected]; [email protected] 2 Institute of Deep Earth Science and Green Energy, Shenzhen University; MOE Key Laboratory of Deep Underground Science and Engineering, Sichuan University 3 Institute of Deep Earth Science and Green Energy, Shenzhen University; MOE Key Laboratory of Deep Underground Science and Engineering, Sichuan University
1
smaller. To accurately predict reservoir behavior and performance, it is essential to study the time-dependent characteristics of shale. The characteristics of creep for anisotropic rock have been studied primarily through experiments. As early as 1939, Griggs [11] utilized mechanical creep facilities to study the timedependent behavior of shale; although Tiggers [11] insisted that there were some unknown errors in the tests that made the experimental results difficult to understand, the tests demonstrated that shale possessed evident creep deformation. Nishihara [12] improved the experimental equipment designed by Tiggers and proved that increasing moisture content and stress could enhance the creep deformation. Since then, however, few papers focused on the shale creep properties [13] until the past 12 years thanks to the prosperity of shale gas exploitation. Sone [7], Sone and Zoback[14, 15], Yang and Zoback [16] used uniaxial and triaxial creep experiments of reservoir shale to analyze the influence of clay content on the shale creep behavior, and the results indicated that a higher clay content leads to larger creep deformation, and shale indentation creep tests also supported this conclusion [17]. Shale creep behaviors can also be enhanced in the presence of fluid [18, 19], under a shear loading path [20] and at elevated temperature and stress [7, 21, 22, 23]. However, the application of deviatoric stress in the published creep experimental results ignores the importance of crack initiation threshold or crack damage threshold. As the deviatoric stress is larger than the crack initiation threshold but lower than the crack damage threshold, the microcracks in the rock would be in a stable growth stage; if the deviatoric stress is larger than the crack damage threshold, the microcracks would be in an unstable growth stage [24, 25]. The understanding of shale creep behavior under crack initiation threshold or crack damage threshold would guide evaluations of the long-term stability of shale. Because of the sedimentary structural features, i.e., bedding plane, of shale, the shale creep is anisotropic. Due to the time-consuming nature of anisotropic shale creep experiments, relatively few studies have addressed this issue [15, 26, 27]. Those results implied that the creep deformation when the bedding layer orientation was horizontal was higher than that when the bedding plane orientation was vertical. However, in most of the published experimental data, only when the bedding plane orientation is 0◦ or 90◦ are studied; as bedding layer orientation that is neither parallel nor perpendicular to the loading direction, little detailed information of anisotropic creep behavior can be obtained. The main purpose of this research is to investigate the anisotropic creep behaviors of shale. A series of triaxial creep experiments on shale specimens with 4 bedding layer orientations under multiple deviatoric stress levels are performed. Then, the creep properties of shale with considering crack initiation stress and crack damage stress are presented in detail. The effects of bedding layer orientations on creep behaviors, including deformation and steady creep rate, are analyzed. The empirical equation to describe the anisotropic creep behavior of shale is illustrated in detail, and suggestions regarding the rationality of the empirical equations are presented. Finally, the mechanical mechanism of anisotropic creep is emphasized, and the methodology for theoretically modeling the creep anisotropy based on the anisotropic creep mechanism is also discussed.
2 Shale properties and anisotropic creep experiment methodology 2.1
Anisotropic shale specimen preparation and experimental apparatus 2
In this work, the shale samples were obtained from the outcrop of Lower Silurian Longmaxi Formation in Pengshui county, Chongqing, China. Fig. 1 obvious macroscopic bedding planes. The shale dry density is around 2.7 g/cm3 , and its porosity is 4.7% – 5.2%. The mineral composition and proportion of the shale samples were characterized by X-ray diffusion (XRD) analysis, which mainly observed quartz, clay minerals, albite, calcite and dolomite (Table 1). The clay mineral contained Illite, Kaolinite, Chlorite and Interstratified Illite / Smectite (Table 2). Based on a series of compression experiments with different bedding layer inclination angles [4], the tested shale is transversely isotropic and the sedimentary plane can be regarded as isotropic plane (Fig. 2). Table 1: Mineral composition of the Longmaxi shale Mineral composition
Quartz
Albite
Calcite
Orthoclase
Dolomite
Pyrite
Clay
Fraction (%)
37.0
10.9
7.7
3.2
5.1
1.9
34.2
Table 2: Clay mineral composition of the Longmaxi shale Clay Mineral Interstratified Illite Kaolinite Chlorite composition Illite /Smectite Fraction (%)
66.0
3.0
16.0
15.0
To avoid the influence of water on shale mechanical behavior, specimens with 4 different bedding plane inclined angles (β = 0◦ , 45◦ , 75◦ and 90◦ , as shown in Fig. 2 (a)) were drilled without using a water-based coolant. The cylindrical specimens were finally prepared with a diameter of 50 mm and a height of 100 mm, respectively. The end faces of the cylindrical samples were carefully processed to keep the two end faces with a roughness within ±0.05 mm. Both of the end faces were perpendicular to the specimen axis with a discrepancy of less than ±0.25◦ and parallel to each other. Creep experiments were conducted using a servo-controlled geomaterial rheological apparatus (Fig. 3 (a)) with confining pressure in the range of 0–100 MPa and maximum axial force range of 2000 kN. The axial deformation can be recorded by one pair of linear variable displacement transducers with a maximum range of 6 mm, and the radial deformation can be measured by the annular extensometer wrapped around the tested specimens with a maximum range of 3 mm (Fig. 3 (b)). The control system and measurement system of the rheological experimental apparatus are stable and have high accuracy, which is critical for long-term creep tests.
2.2
Anisotropic creep experiment methodology
To investigate the effects of anisotropy on shale creep behavior, creep tests were performed with multistep loadings by using specimens with 4 different bedding plane orientations (0◦ , 45◦ , 75◦ and 90◦ , the definition of bedding layer orientation β is defined in Fig. 2 (b)). The reason of selecting those 4 angles is that the failure pattern and mechanism of shale are similar when the bedding plane inclination angles are 15◦ , 30◦ and 45◦ , while the failure mechanisms are the 3
1s ple ditables pl. pdf Figure 1: Shale outcrop
0°
45°
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(b)
2s ple ditables pl. pdf Figure 2: (a) Definition of global coordinate system and specimen preparation diagram of with different bedding plane inclined angles; (b) definition of the bedding plane inclined angle.
4
same when the bedding plane inclination angles are 60◦ and 75◦ [28].A confinement of 50 MPa, which was determined by the in situ stress situation of the Longmaxi shale gas reservoir, was applied in all the experiments. The creep tests began with a relatively low stress level, which was then increased in steps. When the bedding plane orientations were 0◦ and 90◦ (specimen numbers are 0-50 and 90-50, respectively, where 0 or 90 represents the inclination angle of the bedding layer and 50 represents the confining pressure), deviatoric stress levels of 20, 40, 60 , 80, 100 and 120 MPa were used. When the bedding plane orientations were 45◦ and 75◦ (specimen numbers are 45-50 and 75-50, respectively), deviatoric stress levels of 60, 80, and 120 MPa were adopted to contrast with the experimental results of specimens 0-50 and 90-50 and to reduce the test time. To investigate the creep behavior when the deviatoric stresses are crack initiation stress and crack damage stress, which have been determined by the author’s previous research [29], these two specific stresses were applied on specimens 45-50 and 75-50. For specimen 45-50, the crack initiation stress adopted in this study was 190 MPa and the crack damage stress was 205 MPa, and for specimen 75-60, these two specific stresses were 190 MPa and 223 MPa. The reason that the crack initiation stresses for specimens 45-50 and 75-50 are the same is that the influences of bedding plane inclined angle on the crack initiation stress is little [29]. In each deviatoric stress level, the creep test time was selected as 48 hours according to the findings that 1-day creep tests was enough for understanding the long-term mechanical behaviors of shale [27]. The confining pressure remained as a constant in each deviatoric stress level and during loading deviatoric stress. The details of the specimens, confining pressure and stress level are summarized in Table 3. During the loading period, the confining pressure and deviatoric stress were increased at rates of 3 MPa/min and 2 MPa/min, respectively. The radial and axial deformations of the shale specimens were recorded every 1 second in the loading and initial stage of primary creep and after that the data was recorded every 5 seconds.
LVDT
specimen 煤 样
Annular extensometer
3s ple ditables pl. pdf Figure 3: (a) Rock rheological experimental apparatus; (b) positions of specimen and measurement equipment. 5
Table 3: Summaries of the confining pressure and deviatoric stress level in the multistep anisotropic creep tests (σci and σcd represent the crack initiation stress and crack damage stress, respectively) Specimen Bedding plane Confining Deviatoric stress (MPa) 4th 5th 6th number orientation(◦ ) pressure(MPa) 1st 2nd 3rd 0-50 0 50 20 40 60 80 100 120 45-50 45 50 60 80 120 190 (σci ) 205 (σcd ) \ 75-50 75 50 60 80 120 190 (σci ) 223 (σcd ) \ 90-50 90 50 20 40 60 80 100 120
3 Anisotropic creep experimental results of shale 3.1
General observation of creep characteristics when the deviatoric stress is smaller than the crack initiation stress
Several typical creep curves of deformed Longmaxi shale with different bedding layer orientations under multistep deviatoric stress are illustrated in Fig. 4. The axial deformation shown here includes the strain induced by increasing axial load and the strain caused by creep effect. It should be noted that the deformation curves in this paper are plotted with all the recorded data. However, the noises of the data is not as serious as that presented by Sone and Zoback [15]. As shown in the Figure 5, even creep curve is enlarged, the curve does not present many noises. The reason is that the experimental apparatus used in this study is specially designed for creep test. The device can record data more stable for the long-time creep test. The famous MTS 815 rock mechanics testing system is also used by us to do shale creep test with considering high temperature. However, the data has serious noises, as shown in Figure 6. As shown in Fig. 4, all the specimens exhibit time-dependent deformation irrespective of the magnitudes of deviatoric stress. All the time-dependent deformation curves have two stages: primary creep stage, where the strain increase per unit time experienced a rapid decrease, and secondary creep stage, where the creep rate keeps constant. No tertiary creep behavior is observed in the tests because the deviatoric stress is not higher than the crack damage stress. The duration of the primary creep stage ranges from 4 hours to 15 hours based on the analysis of the variation of creep rate. The higher the deviatoric stress is, the longer is the duration of the primary creep stage. During the creep strain measurement under any different deviatoric stress level, there is no sign of the termination of creep deformation; however, when the deviatoric stress is low, the steady creep rate is small (Fig. 7). Note that this observation is different from the results of some argillaceous rock or shale [30, 31], in which secondary creep can only be observed when the deviatoric stress exceeds crack initiation threshold or crack damage threshold. The reasons for this difference may be the accuracy of the experimental apparatus or the mineral content and distribution of the tested rock. For a rock specimen containing multilevel cracks and under constant deviatoric stress, “there are always some sites of high stress concentration leading to the initiation and propagation of micro-cracks and inducing creep deformation [9]. Fig. 7 presents the relationship between the steady creep rate and deviatoric stress. One can see with increasing deviatoric stress, the steady creep rate increases almost linearly regardless of the bedding plane inclination angle. Fig. 8 illustrates the creep deformation under the same deviatoric stress and different bedding layer orientations. This figure shows that the 6
deviatoric stress has significant influences on the magnitude of time-dependent deformation. The correlation between the creep deformation and deviatoric stress could be described by a linear function. The anisotropic features of the steady creep rate and deformation will be analyzed in the following section.
0.008
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4s ple ditables pl. pdf Figure 4: Shale creep deformation under multiple levels of deviatoric stress with different bedding layer orientations: (a) β = 0◦ ; (b) β = 45◦ ; (c) β =75◦ ; and (d) β = 90◦ .
3.2
Effects of anisotropy on shale creep behavior
To investigate the effects of bedding plane on the creep characteristics in detail, the timedependent deformations of shale samples with 4 different bedding plane orientations (β = 0◦ , 45◦ , 75◦ and 90◦ ) under the same deviatoric stress are depicted in Fig. 9. As illustrated in Fig. 9 (a), the instantaneous strain β , where the subscript β represents the bedding plane orientation, satisfies 90◦ < 75◦ < 45◦ < 0◦ . This result occurs because the apparent Young’s modulus Eβ of the transversely isotropic shale satisfies E90◦ > E75◦ > E45◦ > E0◦ . Fig. 9 also presents that under the same deviatoric stress condition, the primary creep deformations and creep rate when the bedding layer orientations are 45◦ and 75◦ are more apparent than those of the other two bedding layer orientations. This phenomenon can be explained more clearly by Fig. 7, which reveals that anisotropy has significant impacts on the steady creep rate. As shown in Fig. 7, considering that the deviatoric stress is 120 MPa, when the bedding layer inclination angle is 45 ◦ , the steady creep rate is the highest, which is 3.5 times higher than the smallest steady creep rate generated by the shale specimen with the 7
50
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0.0048
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5s ple ditables pl. pdf Figure 5: Creep deformation curves obtained by TOP geomaterial rheological apparatus have little noise: (a) creep deformation curve plotted with all recorded data; (b) enlarged creep curve plotted with all recorded data
0.00355 0.00350
Axial strain
0.00345 0.00340 0.00335 0.00330 0.00325 0.00320 0.00315 0
5
10
15 20 25 Time (hours)
30
35
40
6s ple ditables pl. pdf Figure 6: Creep deformation curve of shale obtained by MTS 815 rock mechanics testing system has obvious noises
8
50
1.4E-04 0-50
1.2E-04 steady creep rate ( /day)
45-50 1.0E-04 75-50 8.0E-05
90-50
6.0E-05 4.0E-05 2.0E-05
0.0E+00 0
20
40 60 80 Deviatoric stress (MPa)
100
120
7s ple ditables pl. pdf Figure 7: Steady creep rate under multiple levels of deviatoric stress with different bedding layer orientations
6.0E-04 y = 4E-06x + 0.0001 R² = 0.9957
Creep strain
5.0E-04
y = 4E-06x + 6E-05 R² = 0.9998
4.0E-04
0-50
y = 2E-06x + 0.0001 R² = 0.9974
3.0E-04
45-50
2.0E-04
75-50 1.0E-04
y = 1E-06x + 0.0001 R² = 0.8816
90-50
0.0E+00 0
20
40
60
80
100
120
Deviatoric stress (MPa)
8s ple ditables pl. pdf Figure 8: Shale creep deformation under multiple levels of deviatoric stress and different bedding layer orientations. bedding plane orientation of 90◦ . The steady creep rate as the bedding plane orientation is 75◦ is the second highest, while the steady creep rate as the bedding plane orientation is 0◦ is the 9
4.0E-03
5.0E-03
3.5E-03
4.5E-03
3.0E-03
4.0E-03
2.5E-03
Axial strain
Axial strain
third highest in the same stress situation. The anisotropic characteristics of the primary creep and steady creep rate described above results in the creep strain of shale are completely different when the specimen’s bedding layer orientations vary, which is clearly plotted in Fig. 8. This figure presents that the creep strain of specimen 45-50 is the largest and reaches 3.5 × 10−4 when the deviatoric stress is 60 MPa, which is 1.4 times larger than that of specimen 0-50. This difference in creep strain leads to the axial strain of specimen 45-50 exceeding the axial strain of specimen 0-50, as shown in Fig. 9 (a); thus, the initial axial strain of specimen 45-50 when applying deviatoric stress (80 MPa and 120 MPa) is higher than that of specimen 0-50, which is illustrated in Fig. 9 (b) and (c).
3.5E-03
0-50 45-50 75-50 90-50
2.0E-03 1.5E-03
0-50 45-50 75-50 90-50
3.0E-03 2.5E-03
1.0E-03
2.0E-03 0
10
20 30 40 Time (hours) (a) Deviatoric stress is 60 MPa
50
0
10
20 30 40 Time (hours) (b) Deviatoric stress is 80 MPa
8.0E-03
Axial strain
7.0E-03 6.0E-03 5.0E-03 0-50 45-50 75-50 90-50
4.0E-03 3.0E-03 2.0E-03 0
10
20 30 40 50 Time (hours) (c) Deviatoric stress is 120 MPa
9s ple ditables pl. pdf Figure 9: Shale creep deformation under the same deviatoric stress and different bedding layer orientations. .
3.3
Creep behavior when the deviatoric stresses are crack initiation stress and crack damage stress
Fig. 10 shows the creep curves of deformed shale specimens when the bedding layer orientations are 45◦ and 75◦ under the effects of crack initiation stress and crack damage stress. To enhance the contrast, the typical time-dependent deformations when the deviatoric stress (60, 80 and 120 MPa) is lower than the crack initiation stress are also plotted in Fig. 10. When the deviatoric stress is crack initiation stress, the primary creep deformation is considerably larger than that 10
50
when the stress is 60 MPa, 80 MPa or 120 MPa regardless of the bedding layer inclination angle, and there is no tertiary creep stage. However, a tertiary creep stage is observed when the deviatoric stress is crack damage stress in both bedding layer orientations, which reveals that crack damage stress can be regarded as the long-term strength of shale. The normalization method is used to describe the variation trend of steady creep rate with increasing the deviatoric stress. Here the steady creep rate when the bedding plane inclination angles are 45◦ and 75◦ are normalized separately. In each bedding plane inclination angle, the highest strain rate is set as 1, then the normalized values of the other steady creep rates are calculated. Fig. 11 illustrates the normalized steady creep rates under 5 different deviatoric stresses (60 MPa, 80 MPa, 120 MPa, crack initiation stress and crack damage stress) and 2 bedding plane inclination angles (45◦ and 75◦ ). As shown, the normalized steady creep rate raises exponentially with increasing deviatoric stress. The normalized creep rates of specimens 45-50 and 75-50 when the deviatoric stresses are crack damage stress are approximately 7.7 times and 10.6 times higher, respectively, than those when the deviatoric stress is 20 MPa. The slope of the tangent line to the normalized steady creep rate - deviatoric stress curve is extraordinarily steep when the deviatoric stress arrives at the crack damage stress, which indicates the upcoming failure of the specimen.
0.02 (a) 45° 60 MPa
Axial srain
0.015
80 MPa 120 MPa
0.01
190 MPa 205 MPa
0.005
0 0
10
20 30 Time (hours)
40
50
0.02 (b) 75° Axial strain
0.015 60 MPa 80 MPa
0.01
120 MPa 190 MPa
0.005
223 MPa 0 0
10
20 30 Time (hours)
40
50
10s ple ditables pl. pdf Figure 10: Creep deformation as the deviatoric stresses are 60, 80, and 120 MPa; crack initiation stress; and crack damage stress: (a) bedding layer orientation is 45◦ ; (b) bedding layer orientation is 75◦ .
11
1.0
Normalized steady creep rate
0.8
45-50 75-50
0.6
0.4
0.2
0.0 0
40
80 120 160 Deviatoric stress (MPa)
200
240
11s ple ditables pl. pdf Figure 11: Normalized steady creep rate as the deviatoric stresses are 60, 80, and 120 MPa; crack initiation stress; and crack damage stress.
4 Discussion 4.1
Rationality of empirical laws to describe the anisotropic creep deformation of shale
Because of the simplicity of empirical laws, the empirical creep model is widely used to describe the creep deformation from primary creep [27, 23, 15]. The most popular equation adopted in the published studies is written as (t) = σBtn
(1)
where (t) is the strain induced by creep at time t; B and n are material parameters. Thus, the creep compliance function J(t) can be defined as J(t) = (t)/σ = Btn
(2)
Eq. (2) implies that the relationship between creep compliance function J(t) and time t in double logarithmic coordinates is linear. The creep compliance function of shale, which is calculated based on the experimental data and Eq. (2), is shown in Fig. 12. However, the trend observed in Fig. 12 indicates that the linear function cannot describe the variation of the creep compliance function with time in double logarithmic coordinates, particularly the instantaneous deformation immediately after applying the load. Therefore, Eqs. (1) and (2) may not be a good function to describe creep characterization. To better depict the overall process of time-dependent deformation, the following creep compliance function is suggested: J(t) = J1 + Btn
(3)
where J1 is the asymptotic instantaneous compliance. The reason for introducing the asymptotic instantaneous compliance J1 is that geomaterial typically exhibits creep, even for extremely short load durations. The introduction of J1 is important for accurately describing the creep strain, particularly the primary creep stage [32, 12
-4.28
log (J)
-4.29
-4.30
-4.31
-4.32 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
log (time)
12s ple ditables pl. pdf Figure 12: Variation of creep compliance defined in Eq. 2 with time in double logarithmic coordinates (taking specimen 75-50 when the deviatoric stress is 60 MPa as an example) 33]. The value of J1 can be easily determined from the asymptote of the creep compliance function - time curve in the logarithmic space when the time tends to zero. Eq. (3) indicates that the relationship between J − J1 and time t is a linear function in the double logarithmic scale. Based on the algorithm presented above, the variation of J − J1 with timein the double logarithmic space is presented in Fig. 13. As shown, the log(J − J1 ) varies linearly with log(t) in the entire process of time-dependent deformation, which implies that Eq. (3) is better than Eq. (2) for depicting the creep deformation.
-5.2
-5.3
log (J - J1)
-5.4
-5.5
-5.6
-5.7
-5.8
-5.9 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
log (time)
13s ple ditables pl. pdf Figure 13: Variation of creep compliance defined in Eq. 3 with time in double logarithmic coordinates (taking specimen 75-50 when the deviatoric stress is 60 MPa as an example) According to the anisotropic creep experimental data and Eq. (3), the asymptotic instantaneous compliance at multiple levels of deviatoric stress and different bedding layer orientations is shown in Fig. 14. As shown, asymptotic instantaneous compliance increases with increasing 13
deviatoric stress. Moreover, the asymptotic instantaneous compliance is enhanced by increasing the deviatoric stress and is dependent on the bedding plane orientations. As the bedding plane orientation is 90◦ , the asymptotic instantaneous compliance is the smallest compared with the other cases when the deviatoric stresses are the same. The growth rate of the asymptotic instantaneous compliance when the bedding plane orientation is 45◦ is the fastest. Material parameters B and n in Eq. (3) are obtained by fitting the time-dependent deformation curve (Fig. 4) in double logarithmic coordinates (Fig. 13) and are presented in Figs. 15 - 16. One can note that the scatter of material parameters B and n is high. It is hardly possible to determine any relations between material parameters with bedding layer orientation or deviatoric stress. Therefore, Eq. (3) may be good at fitting the existing experimental data, but it is impossible to predict the anisotropic creep behavior when the bedding layer orientations are random if there are only creep data for a few bedding plane orientations. For example, this research has performed creep tests for 4 different bedding plane inclination angles (0◦ , 45◦ , 75◦ and 90◦ ), and although Eq. (3) can fit those experimental data, it is absolutely impossible to evaluate the creep behavior when the bedding plane inclination angles are different from the 4 bedding plane orientations that have been tested. The essential reason for the lack of predictability of Eq. (3) is that empirical creep laws cannot consider the anisotropic creep mechanism. Empirical creep laws are mathematical fitting formulas with almost no physical meaning. The ideal anisotropic creep model that can fit and predict creep deformation should consider the anisotropic creep mechanism, which will be presented as follows.
6.0E-05
J1( / MPa)
5.0E-05
4.0E-05
0-50 45-50 75-50
3.0E-05
90-50
2.0E-05 0
20
40 60 80 Deviatoric stress (MPa)
100
120
14s ple ditables pl. pdf Figure 14: Asymptotic instantaneous compliance under multiple levels of deviatoric stress with different bedding layer orientations
4.2
Basic creep pattern for anisotropic creep behavior
The bedding layer is the most important structural feature of shale. Because of the existence of the bedding layer, the mechanical behavior,including deformation, strength and failure pattern, of shale presents significant anisotropy. Assume that the principal stress tensor σij of shale representative volume element in the material coordinate system o − xyz (Fig. 17) is written as 14
4.0E-06
0-50 45-50 75-50
3.0E-06
B
90-50
2.0E-06
1.0E-06
0.0E+00 0
20
40 60 80 Deviatoric stress (MPa)
100
120
15s ple ditables pl. pdf Figure 15: Values of parameter B in Eq. (3) under multiple levels of deviatoric stress with different bedding layer orientations
0.40
0.35
n
0.30
0.25
0.20
0-50 45-50
0.15
75-50 90-50
0.10 0
20
40 60 80 Deviatoric stress (MPa)
100
120
16s ple ditables pl. pdf Figure 16: Values of parameter n in Eq. (3) under multiple levels of deviatoric stress with different bedding layer orientations
σxx 0 0 σij = 0 σyy 0 0 0 σzz
(4)
where i, j = x, y, z = Cartesian subscripts. Given that shale can be assumed to be a transversely isotropic material, it is generally more convenient to describe the mechanical properties of shale by mathematical equations in the local coordinate system o0 − x0 y 0 z 0 , as shown in Fig. 17. According to the transformation of coordinates, the principal stress tensor σij defined in Eq. (4) can be reformulated in the local coordinate system o0 − x0 y 0 z 0 as
15
z' o x o' x'
z y y'
α
17s ple ditables pl. pdf Figure 17: Schematic diagram of coordinate transformation of transverse isotropic shale
σxx 0 0 σyy cos2 (α) + σzz sin2 (α) σyy sin(α) cos(α) − σzz sin(α) cos(α) (5) σij0 = 0 0 σyy sin(α) cos(α) − σzz sin(α) cos(α) σyy sin2 (α) + σzz cos2 (α) where α is the rotation angle of the local coordination system o0 − x0 y 0 z 0 around the x0 axis, as presented in Fig. 17. Because of the presence of the bedding plane, the component of the stress tensor σij0 will result in different effects. For example, the stiffness in the z 0 direction is weaker than that in the x0 or y 0 direction; thus, when the magnitudes of σx0 0 x0 or σy0 0 y0 and σz0 z0 are the same, the time-dependent deformation in the z 0 direction should be larger. This is why the creep strain when the bedding layer orientation is 0◦ is higher than that when the bedding plan orientation is 90◦ (Fig. 8). Note that the bedding layer is a weak plane, and its resistance to deformation and strength is quite small. If there is a shear component σy0 0 z0 acting on the bedding plane, the creep deformation should be much larger compared with the deformation of the same size stress component acting on the shale matrix, which leads to the creep strain and creep rate are the highest when the bedding layer orientation is 45◦ , as clearly shown in Fig. 7, Fig. 8 and Fig. 9. According to the above analysis, the effects of the stress tensor components on the creep should be taken into account separately. Therefore, three basic creep patterns for the creep of a transversely isotropic material are proposed, as presented in Fig. 18. The first case is that the action direction of the stress tensor component is normal to the bedding layer; the second case is that the action direction of the stress tensor component is parallel to the bedding layer; and the third case is that the shear component of the stress tensor acts along the bedding layer and perpendicular to the bedding layer normal direction. All the other creep cases will be the combinations of these three basic creep patterns. It is suggested that an anisotropic creep theoretical model with a solid physical foundation may be better based on the three basic creep patterns. To propose such a theoretical anisotropic creep model, two steps can be adopted. The first step is to deduce the theoretical equation to calculate the time-dependent deformation under a single stress component, and this step can combine with damage mechanics to consider the coupling between creep and damage. The second step is to form the creep model under general loading conditions based on the principle of superposition. 16
σ1
σ1
σ1
σ1
σ1
σ1
σ1
σ1 (a) First case
(b) Second case
(c) Third case
18s ple ditables pl. pdf Figure 18: Three basic creep patterns for the creep of a transverse isotropic material
5 Conclusions Anisotropic creep tests on Longmaxi shale considering 4 different bedding layer orientations (0◦ , 45◦ , 75◦ , and 90◦ ) under multiple levels of deviatoric stress are conducted. The anisotropic creep characterizations and mechanism of shale are analyzed in detail. According to the current analysis and discussions, the following conclusions can be drawn: 1. Longmaxi shale presents primary creep and secondary creep even when the deviatoric stress is lower than crack initiation stress. The time-dependent deformation and steady creep rate increase linearly with increasing deviatoric stress when the stress is less than 120 MPa. 2. Anisotropy has significant effects on the creep strain and steady creep rate. The creep strain and steady creep rate reaches the maximum when the bedding layer orientation is 45◦ and under the same stress level, while the creep strain and steady creep rate generated are the minimum when the bedding plane orientation is 90◦ . 3. Crack damage stress can be regarded as the long-term strength of shale. The steady creep rate raises exponentially with increasing stress as the deviatoric stress is not lower than the crack initiation stress. 4. The empirical creep model can only fit the existing creep data, but it is almost impossible to predict the creep deformation because it has little physical foundation and does not take any creep mechanism into account. 17
5. The components of the stress tensor will lead to different time-dependent deformations when the material is anisotropic. Three basic creep patterns for shale are proposed based on the action mechanism of the stress tensor components. The general methodology for proposing an anisotropic creep model is presented. Acknowledgment: The authors wish to thank the financial support from the National Natural Science Foundation of China (No.51804203) and Sichuan University (No.20826041C4082, No.0060304153002).There is no conflict of interest between the authors.
References [1] Mokhtari, M., Bui, B.T., Tutuncu, A.N. 2014. Tensile failure of shales: impacts of layering and natural fractures. In SPE Western North American and Rocky Mountain Joint Meeting. Society of Petroleum Engineers. [2] Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D. 1997. Laboratory investigation of the mechanical behaviour of tournemire shale. International Journal of Rock Mechanics and Mining Sciences. 34(1):3–16. [3] Masri,M., Sibai,M., Shao,J.F., Mainguy,M. 2014. Experimental investigation of the effect of temperature on the mechanical behavior of Tournemire shale. International Journal of Rock Mechanics and Mining Sciences. 70,185–191. [4] Li, C.B., Bazant, Z.P., Caner, F.C., Chau, V.T. 2017. Spherocylindrical microplane constitutive model for shale and other anisotropic rocks. Journal of the Mechanics and Physics of Solids, 133:155–178. [5] He, B., Xie, L. Z., Li, F. X. 2017. Anisotropic mechanism and characteristics of deformation and failure of Longmaxi shale. Sci Sin-Phys Mech Astron, 47: 114611. [6] Heng,S., Guo,Y.T., Yang,C.H., Daemen,J.J.K., Li,Z. 2015. Experimental and theoretical study of the anisotropic properties of shale. International Journal of Rock Mechanics and Mining Sciences. 74:58–68. [7] Sone, H. 2012. Mechanical Properties of Shale Gas Reservoir Rocks, and Its Relation to the In-situ Stress Variation Observed in Shale Gas Reservoirs. PhD thesis, Stanford University. [8] Asala, H., Ahmadi, M., Taleghani, A. D. 2016. Why re-fracturing works and under what conditions. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. [9] Viet T. Chau, Cunbao Li, Saeed Rahimi-Aghdam, Bažant, Z. P. 2017. The Enigma of Large-Scale Permeability of Gas Shale: Preexisting or Frac-Induced?. Journal of Applied Mechanics-ASCE, 84(6): 061008. [10] Chhatre, S. S., Braun, E. M., Sinha, S., Determan, M. D., Passey, Q. R., Zirkle, T. E., Wood, A. C., Boros, J. A., Berry, D. W., Leonardi, S. A. 2015. Steadystate stress-dependent permeability measurements of tight oil-bearing rocks. Petrophysics. 56(02):116–124. [11] Griggs, D. 1939. Creep of rocks. The Journal of Geology, 47(3): 225-251. [12] Nishihara M. 1952. Creep of shale and sandy-shale. The Journal of the Geological Society of Japan, 58(683): 373-377. 18
[13] Horsrud, P., Holt, R. M., Sonstebo, E. F., Svano, G., Bostrom, B. 1994. Time dependent borehole stability: Laboratory studies and numerical simulation of different mechanisms in shale. Rock Mechanics in Petroleum Engineering: Society of Petroleum Engineers. [14] Sone, H., Zoback, M. D. 2013. Mechanical properties of shale-gas reservoir rocks - Part 2: Ductile creep, brittle strength, and their relation to the elastic modulus. Geophysics, 78(5): D390-D399. [15] Sone, H. and Zoback, M. D. 2014. Time-dependent deformation of shale gas reservoir rocks and its long-term effect on the in situ state of stress. International Journal of Rock Mechanics and Mining Sciences, 69:120–132. [16] Yang, Y., Zoback, M. 2016. Viscoplastic Deformation of the Bakken and Adjacent Formations and Its Relation to Hydraulic Fracture Growth. Rock Mechanics and Rock Engineering, 49(2): 689-698. [17] Kabir, P., Ulm, F.J., Akono, A.T. 2017. Rate-independent fracture toughness of gray and black kerogen-rich shales. Acta Geotechnica, 12(6): 1-21 [18] Guindon L. 2015. Viscoelastic Creep of Eagle Ford Shale: Investigating Fluid/Shale Interaction. Journal of Canadian Petroleum Technology, 54(3): 142-143 [19] Almasoodi, M. M., Abousleiman, Y. N., Hoang, S. K. 2014. Viscoelastic creep of eagle ford shale: Investigating fluid-shale interaction. In SPE/CSUR Unconventional Resources Conference–Canada. Society of Petroleum Engineers. [20] Yang, S.Q. and Cheng, L. 2011.Non-stationary and nonlinear visco-elastic shear creep model for shale. International Journal of Rock Mechanics and Mining Sciences, 48(6):1011–1020. [21] Chang, C., Zoback, M. D. 2009. Viscous creep in room-dried unconsolidated Gulf of Mexico shale (I): Experimental results. Journal of Petroleum Science and Engineering, 69(3-4), 239-246. [22] Geng Z, Bonnelye A, Chen M, Jin Y, Dick P, David C, Fang X, Schubnel A. 2017. Elastic Anisotropy Reversal During Brittle Creep in Shale. Geophysical Research Letters, 44(21): 1-9 [23] Rybacki, E., Herrmann, J., Wirth, R., Dresen, G. 2017. Creep of Posidonia Shale at Elevated Pressure and Temperature. Rock Mechanics and Rock Engineering, 50(12): 31213140 [24] Nicksiar, M., Martin, C. 2012. Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mechanics and Rock Engineering 45(4): 607-617 [25] Cai, M., Kaiser, P. K., Tasaka, Y., Maejima, T., Morioka, H., Minami, M. 2004. Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 41(5): 833-847. [26] Liu, Z., Xie, S., Shao, J.F., Conil, N. 2015. Effects of deviatoric stress and structural anisotropy on compressive creep behavior of a clayey rock. Applied Clay Science, 114:491–496. [27] Rassouli F. S., Zoback M. D. 2018. Comparison of Short-Term and Long-Term Creep Experiments in Shales and Carbonates from Unconventional Gas Reservoirs. Rock Mechanics and Rock Engineering, 51(7): 1995–2014. 19
[28] Li C. B. Theoretical research on 3D volumetric fracturing based on anisotropic microplane model. 2017. Doctoral thesis of Sichuan University. [29] Li, C.B., Xie, H.P., Xie, L.Z. 2017. Experimental and theoretical study on the shale crack initiation stress and crack damage stress. Journal of China Coal Society, 42(4):969–976. [30] Fabre, G., Pellet, F. 2006. Creep and time-dependent damage in argillaceous rocks. International Journal of Rock Mechanics and Mining Sciences 43(6): 950-960. [31] Li, Y.W., Ahmad, G. 2012. Creep behavior of Barnett, Haynesville, and Marcellus shale. 46th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association. [32] Jirásek, M., Bazant, Z.P. 2002. Inelastic analysis of structures. John Wiley and Sons. [33] Bažant ZP, Jirásek M. 2018. Creep and Hygrothermal Effects in Concrete Structures. Springer.
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Highlights： 1. Anisotropic creep behaviors of shale are presented in detail. 2. Rationality of the empirical creep law for shale is evaluated. 3. Anisotropic creep mechanism is analyzed. 4. Three basic creep patterns for shale are proposed.