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In the first part of “An Experimental Investigation of Shale Mechanical Properties..." was developed:

3 Elastic Moduli of Shale—Theory

4 Elastic Moduli—Results and Discussion

The CIU test was performed under cyclic loading–reloading, followed by a 4-MPa cyclic amplitude on a sample drilled perpendicular to the bedding (θ=0°). The idea was to see more of the elastic response during such a small cycle due to non-linearity (Fig. 15c). The test sample for the cyclic test was taken from the same core block used in the previous monotonic samples tested. The cyclic test included 3–4 unloading–reloading cycles. The cycle for each step was 5 MPa during the triaxial phase, with a cycling amplitude of 4 MPa. The PP during consolidation was 10 MPa, and the confining stress was 25 MPa. In the triaxial phase, the strain rate was set to 2x10-7 s-1. The test required 4 days to reach yielding.

The postmortem analysis of the sample showed a localization of the deformation in a shear band inclined at an angle of θ=45° to the horizontal bedding plane (Fig. 15a). Under the cyclic triaxial test, the estimated axial stress and the PP at failure was approximately 49 and 16.3 MPa, respectively, which appeared to be fairly consistent with the corresponding established monotonic triaxial test, where the measured values were 46.5 and 17.2 MPa, respectively. Due to cyclic loading, the PP increased 10 % higher than in the monotonic tested samples (Fig. 15g). The slope of the unloading–reloading cycles at different stress levels showed stiffer material when compared with only loading once (Fig. 15e, g). In this particular case, the shale stiffness under cyclic triaxial test conditions was approximately 50 % higher than in the monotonic triaxial test. This increase may be due to an irreversible change in the microstructure of the rock. An irreversible strain (plastic strain) was observed. However, the degradation effects are negligible here because we performed only one cycle and were still well below the peak strength. We cycled at approximately 50 % of the peak stress.

In the cyclic triaxial test condition, an exception was observed in the data for calculating the Poisson’s ratio. The two Poisson’s ratios were similar under the monotonic triaxial testing, whereas they were completely different in cyclic triaxial test conditions (Fig. 15f). For this particular case, ν1 and ν2 were determined to be 0.6 and 0.46, respectively, for the cyclic test. The possible reasons for such dissimilarity may be the shale heterogeneity or the material deformation state under the cyclic stress state (see Fig. 15c, d). For the static test, the Poisson’s ratios were 0.52 and 0.53, respectively. It was also noticed that the calculated PP under the cyclic triaxial test was lower than for the monotonic triaxial test (Fig. 15g). This study confirmed that the PP plays an adverse role in the determination of the lower stiffness in monotonic tests compared with the cyclic tests. The conclusion is that the PP development is a critical parameter under the CIU tests that can specifically control material stiffness in clay-dominant samples.

There are many factors, i.e., induced cracks and their orientation, partial saturation, material heterogeneity and anisotropy, plasticity, magnitudes of the loading–reloading cycles, strain rate, etc., that could all influence the geomechanical elastic properties of shale. A detailed discussion of the elastic response due to cyclic loading has been made by Fjær et al. (2008, p. 267), (2011), Holt et al. (2011), and Niandou et al. (1997). From their analysis, the non-elastic behavior during loading is largely dependent on the stress history of the rock.

Fig. 15 Comparative studies to evaluate the mechanical properties of shale under cyclic and monotonic test conditions. Postfailure samples a cyclic test and b monotonic test, c stiffness curves, d, e E calculation,

f variations of Poisson’s ratio, and g PP development and stiffness. All samples were drilled perpendicular to bedding (θ=0°).

A comparative study of the material deformations under shearing on different bedding samples is presented in Fig. 16. This analysis is interpreted as the peak deviation with respect to axial or radial deformation. Basically, such an analysis helps provide a clear picture of weak bedding in terms of the stiffness. It is observed that the variation of the peak stress varies significantly with the bedding plane orientation. Under the CIU test matrix, the peak deviation is largest for samples drilled parallel to the bedding. The same analysis indicated larger radial deformations but relatively lower peak deviations for samples at 0° (18 %) and 45 (8 %) bedding and, thus, showed lower stiffness. In the case of the CID tests, the maximum axial deformation (35 %) was observed for samples drilled perpendicular to the bedding, whereas it was 20 % for samples drilled parallel to the bedding. Lower stiffness was measured for samples tested under the CID conditions com- pared with samples tested under the CIU conditions.

Fig. 16 Stress–strain relationship under different bedding planes

For an ideal undrained test condition, the effective stress path should be vertical because there is no change in the mean effective stress (zero change in the total volume). The total stress path will then be inclined, and the horizontal change in the mean total stress will be equal to the PP change. However, for the drained case, the PP was constant and the total stress path inclined 3–1 in the p0–q diagram (Fig. 17b). This resulted in an inclination of 3–1 for the effective stress path. The shear strength for samples at θ=90° was greater than that for samples at θ=0°. For the CIU tests, the total stress path was different from the effective stress path due to the building of the PP. Because the PP was constant for the CID tests, both the total and effective stress paths were identical.

Due to the inherent anisotropic nature of clay platelets (both texturally and mechanically), one would intuitively expect shale to be anisotropic. Figure 17a shows the stress paths for two samples at the same effective confining pressure. One of the samples was drilled with the sample axis along the bedding and the other was drilled with the sample axis normal to the bedding. The sample that was drilled parallel to the bedding was both stiffer and stronger than that drilled normal to the bedding (Fig. 17a). With increasing deviatoric stress, the mean effective stress decreased more for the sample at θ=0° due to material contraction. At this position, the pore volumes decreased and led to an increase in the PP with increasing shearing. However, when shearing passed a certain level, i.e., at 15 MPa for this particular case, the material then started to dilate or create cracks or microfractures. These attributes accelerated and caused increasing pore volume and decreasing PP. Samples at 0° bedding had both contraction and dilation tendencies, whereas for the 90° bedding, onlyparallel to the bedding (θ=0°). Therefore, wells drilled at high deviation, i.e., closer to horizontal, and also at larger depths will, thus, be more susceptible to potential stability problems due to the anisotropy.

There are several factors that may affect the degree of anisotropy. One would expect that the larger the clay content, the larger the degree of anisotropy, but this also depends on the mineral type. Both porosity and depth are important. Anisotropy effects were not, however, the main subject of this study. The need for a complete description of the anisotropy of the mechanical properties of shale will depend on the application. Generally, the acoustic properties of shale are significantly influenced by the heterogeneous nature of shale. Many authors have already implicitly analyzed this issue (Horsrud et al. 1994, 1998; Holt et al. 1996; Lockner and Stanchits 2002; Fjær et al. 2008; Sarout et al. 2007).

Fig. 17 Stress path and material anisotropy effects under a CIU response and b CID response.

The non-elastic behavior of tested shale through the unloading–reloading cycle is presented in Sect. 5. The stress path behavior is observed to be different between the cyclic and static tests, and the elastic moduli vary significantly. However, the non-elastic behavior of the tested shale under the undrained situation is governed by the PP development. The PP response during the testing of shales may indicate whether the sample is fully saturated. To quantify the PP response (DPf), the Skempton parameters A and B can be defined, as suggested by Skempton (1954):

For triaxial loading conditions, the mean effective stress can be expressed as

with the change in the mean effective stress during triaxial loading yielding Δθ = ΔP

For the poroelastic case where kf<<ks, AB can be expressed by elastic constants (Fjær et al. 2008), as suggested by The analytical expression (Eq. 3) can be used to explain the nonelastic response of the tested shale because this equation can now be related to the stress paths of the p0 –q plots (Figs. 17a, 18).

where φ is the fractional porosity, α is the Biot coefficient (=1-kfr/ks), kfr is the bulk modulus of the framework, and kf is the fluid modulus.

The analytical expression (Eq. 3) can be used to explain the non-elastic response of the tested shale because this equation can now be related to the stress paths of the p’ – q plots (Figs. 17a, 18). With φkfr<< αkf, AB = 1/3 from Eq. 2 and ΔP’=0; thus, the curve in the undrained triaxial test is vertical. This result is commonly referred to as the ‘weak frame limit’, which is the common assumption for any soil. According to Skempton (1954), for a soil in the elastic case, it can be shown that A = 1/3 and B = 1. In this study, no vertical stress path was found, the weak frame assumption did not hold, and AB ≠ 1/3.

If AB<1/3, from Eq. 2, ΔP’=(-m) ΔPf, where m is a multiplier factor and is less than 1. As a result, the stress path curves will tilt to the right, as shown in Fig. 18a (only loading parallel to bedding). In most cases, the stress path tilts slightly (i.e., at bedding angles of 45° and 60°) or more inclined (i.e., 0°) to the left (see Fig. 18). This behavior can be justified if ΔP’< 0.

Examining Eq. 3, the only solution to this equation is if kf>ks, which is not realistic. This type of behavior, thus, indicates that the rock is no longer elastic. This is, for instance, the case for a normally consolidated material such as shale. Several authors also presented the PP response in sandstone-based rock strength in different ways (Skempton 1954; Horsrud et al. 1994, 1998; Lockner and Stanchits2002; Fjær et al. 2008). However, in the end, the same conclusion was noted.

The Mohr–Coulomb failure criterion satisfies linear elastic condition. It is simpler than other models but, at the same time, the most widely used criterion in the oil industry. The material failure parameter, i.e., cohesion and uniaxial compressive strength (UCS), can be interpreted from the triaxial tests with existing mathematical expressions. For example, the τ–p’ space interprets cohesion directly, while the β=π/4+φ/2 space provides the UCS. Sometimes, the q–p' space is used, but it provides neither the cohesion nor the UCS directly. To use the q–p' space to evaluate the material failure parameters for the Mohr–Coulomb model, the parameter relation is rearranged:

Here, q is the deviatoric stress, S0 is the inherent shear strength or cohesion, ϕ is the internal friction angle, and σ is the mean effective stress. The parameter UCS (C0) and the failure angle (β) are related through

These equations determine the Mohr–Coulomb model input parameters (ϕ, β, C0). The peak stress values in Fig. 18 falls essentially on a straight line. In soil mechanical terms, this is a projection of the Hvorslev surface onto the p’–q plane (for a specific given volume). Auniform clay sample obeying the critical state theory would follow the Hvorslev surface up to the critical state line. Overconsolidated and cemented rocks will eventually behave in a non-uniform manner. This was clearly the case for the samples shown in Fig. 18. When approaching the peak stress value, localization took place, and shear bands developed, which eventually formed a macroscopic shear plane through the sample. The behavior after the peak was, thus, in this case, more dependent on the characteristics typical of a rock (e.g., cementation) rather than a soil. In addition, the dotted circles on the stress paths indicate rapid PP development, which are due to strain rate effects or to existing cracking. The straight lines in Fig. 18 could be translated into a Mohr–Coulomb failure criterion, providing an extrapolated UCS, failure angle, and friction angle. A set of Mohr–Coulomb failure model data is shown in Fig. 18. The friction angle appears to be high for the CIU tests for the relatively soft shale. Similar analysis was performed for different bedding plane orientations and at different confining pressures. The observed results are presented in Fig. 18. Depending on the bedding plane, Fig. 18 implies that the UCS varied between 9.3 and 10.5 MPa, and the failure angle varied between 23.4° and 27.9°. In the case of North Sea shale, studied by Horsrud et al. (1994), the UCS varied between 6 and 77.5 MPa, and the failure angle (β) ranged between 48° and 60°. The Young’s modulus (E) correlated with a UCS of 6.55E (R2 = 0.99). Aadnøy et al. (2009) reported that E for green river shale in Canada varied between 60 and 160 GPa.

Fig. 18 CIU test of Pierre-1 shale, showing the Mohr–Coulomb failure lines from laboratory tests and estimating Mohr–Coulomb failure parameters; a loading parallel to bedding (θ=90°), b loading perpendicular to bedding (θ=0°), c loading 45 to the normal to bedding (θ=45°), and d loading 60° to the normal to bedding (θ=60°). The anomalies marked by the dotted circles were caused by the test performances.

It is difficult to explain the exact reasons behind the high frictional angles obtained in the CIU tests. In the CIU tests, we worked with Pierre-1 shale, which has high heterogeneity. We believe that we obtained a higher frictional angle due to the frictional behavior. However, partial saturation could also lead to a high frictional angle (Sønstebø and Horsrud 1996; Schmitt et al. 1994). We cannot guarantee that the test samples for this outcrop achieved 100 % saturation. We carefully attempted to obtain good saturation with brine as a contacting pore fluid, as reported in Sect.

2.1. The sample should also improve its saturation during the mechanical loading. Some of the apparent plastic behavior may be related to the low permeability and PP development during loading. The Mohr–Coulomb failure parameters and the necesary correlations developed through this study are presented in Fig. 19. Both the CIU and CID test results are presented. The 90° bedding samples seem to have the strongest correlation, whereas the 45° bedding samples have the weakest correlation.

Fig. 19 Mohr–Coulomb failure lines connected for different bedding planes. The different colors are indications of the respective failure trend lines of the different bedding plane samples.

The rock strength behavior of the shale tested under drained and undrained conditions is considered to be a challenging and costly task. The most obvious application of this study is to supply data sets for numerical borehole stability modeling in shale. The mechanical properties of shale are demanding parameters not only for drilling engineering purposes but also in the geomechanical field. For example, the Poisson’s ratio is often used within geophysics and sanding prediction. Experimental results and theoretical considerations have shown that the Poisson’s ratio is not a single-valued, well-defined parameter for a given rock. The same observations were observed for measurements of the Young’s modulus. Young’s moduli for drained/undrained conditions varied largely with the stress level and with the amplitude and duration of the applied stress changes. Because the Poisson’s ratio represents the relationship between the P and S waves in the identification of lithology from the seismic data and the AVO analysis, finding a suitable value of this parameter is vital, particularly at interbedded formations.

In recent years, the shale-oil and shale-gas potential represents new unconventional assets where shale-related information could be valuable for other researchers working in this field. The same aspect is true for the underground storage of CO2. Our analysis will be useful for such issues. As a result, rock failure analysis was included in this work. The contribution relating to caprock failure data and shale characterization could be used for updating existing borehole stability analysis models. Obviously, improved shale property information under drained and undrained test conditions would improve model efficiency and accuracy. Very few studies have considered such extensive test matrices in shale. Moreover, this study provides some correlations between the material friction angle and the UCS. Because the UCS can be calculated from the sonic logs, such correlation can be implemented more readily. This correlation may be useful in the field of petroleum engineering with more testing under field conditions.

A dedicated testing program for Pierre-1 shale has provided a valuable database for shale properties, especially for weaker shale, which may cause borehole-stability problems. The correlations developed in this study can be used as an engineering tool to provide more reliable and more continuous estimates of the mechanical properties of shale, keeping in mind that the validity of the correlations should be verified when used in other geological and geophysical areas. Other sources of uncertainty also exist (e.g., core damage effects, temperature effects, etc.) and should be a focus in further work.

This study supplied data sets to implement an anisotropic material model that was used to simulate shale rocks. An isotropic model cannot simulate the real material behavior and, thus, should not be considered adequate. The anisotropic behavior is primarily due to the bedding planes that are formed, and because a borehole can be made at any angle to those planes, one should be able to simulate the real anisotropic behavior at any angle to the bedding plane. Thus, care should be taken when simulating such rock masses. The elastic moduli of such masses should never be considered as constant in all directions; instead, as the tests showed, we must expect them to vary considerably. This study also showed the dependency of the material on the confining stress. Obviously, the correct confining stress must be derived and used to simulate the real behavior as closely as possible.

Transverse isotropic shale has generally been studied experimentally using triaxial tests subjected to globally axisymmetric loading states, although the true stress states will not be axisymmetric due to the bedding plane inclination, inhomogeneities in the specimens, and end-effects. This study showed that the planes of weakness in bedded rocks could lead to severe borehole collapse problems.

The key findings of this study were elaborately presented in every section. However, the most crucial observations are listed here:

- Cyclic triaxial tests provided approximately 6 % higher rock strength, 50 % higher stiffness, and 10 % higher PP development than the only monotonic triaxial loading test. PP controls the rock stiffness under the CIU tests.
- Poisson’s ratio effects in the samples drilled parallel to bedding are more vital than at any other sample orientation. A large variation in Poisson’s ratios was found for drained and undrained samples drilled at the same bedding angle in transverse isotropic material. For the undrained fluid flow condition, the Poisson’s ratios were observed in the tested shale rocks to be larger than
- 0.5 and varied between 0.3 and 0.75. However, for the drained case, the maximum limit of Poisson’s ratios was 0.2. Therefore, in the case of shale, merely assuming a constant Poisson’s ratio is a risky approach.
- The elastic moduli are non-linear functions of the confining pressure but are also dependent on the effective stress. Ep (loading parallel to bedding) was higher than Et (loading perpendicular to bedding). The estimated values of Et and Ep were 0.65 and 1.55 GPa, respectively. The estimated mean relative difference of the E for these two samples (in absolute value) was approximately 58 %. A significant difference between these two moduli indicates a strong heterogeneous nature.
- The elastic moduli for drained and undrained test conditions varied largely. The drained Young’s modulus was approximately 48 % of the undrained value. The drained Poisson’s ratios were, on average, 40 % or lower than the undrained value. These mechanical properties were significantly impacted by the bedding plane orientation and the confinement pressure.
- The bedding plane, anisotropy, and material heterogeneity are three prime factors to achieve accurate estimation of the material failure and development of the PP.
- The dilation behavior is significant under the drained test conditions. At low confinement pressure, the failure is brittle, with a sudden loss of strength and a transition from compressive to dilatants volumetric strain during failure and postfailure.
- The time dependency of parameters is significant due to the low permeability of shale.

The authors thank the Department of Petroleum Engineering and Applied Geophysics, Norwegian University of Science and Technology, for their support and providing the permission to write this paper. We would like to express our appreciation to Ole Kristian Søreide and Per Horsrud of STATOIL, and Jørn Stenebra˚ten, Erling Fjær, and Olav-Magner Nes of SINTEF Petroleum Research for their contribution to the discussions of critical issues in this work. We especially thank Konstantinos Kalomoiris. His help greatly improved our work. A thorough peer review and constructive comments by the reviewers are also appreciated and acknowledged. It further helped us to improve the quality of the paper. We thank STATOIL for providing funds for the experimental investigation of borehole stability. In addition, we appreciate and acknowledge the extensive laboratory work that was performed and partially funded by SINTEF Petroleum Research.

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