2.1 Materials and Methods
In this section we perform uniaxial compression of a shale and sandstone sample. The shale sample is an Opalinus clay originating from the Mont-Terri Underground Research Laboratory operated by NAGRA (Switzerland). Details on the mineralogy and key petrophysical characteristics of this shale can be found in Sarout et al. (
2014). It had a diameter of 38 mm, height of 85.37 mm and a dry mass of 242.61 g. The sandstone sample originates from the Jurassic eolian Nugget sandstone formation originating from a quarry in Utah. Its depositional environment confers to the Nugget reservoir intrinsic anisotropy supported by alternating sand granulometry along the bedding planes. It had a diameter of 38 mm, height of 84.76 mm and a dry mass of 226.77 g. Both samples were dried at
\(105^{\circ }\)C for 48 h prior to testing and then weighed. The ultrasonic wave velocities were measured during the loading to quantify their change with the stress-induced damage.
The specially-designed ultrasonic transducer locations are shown in Fig.
2. The numbered circles show the location of P-wave transducers and the numbered rectangles show the location and orientation of the S-wave transducers. Table
1 summarizes the ultrasonic velocities measured where we are using the notation:
\(V_{i/j}\) to denote the speed of an ultrasonic wave propagating in the
\(x_i\)-direction with particle displacements in the
\(x_j\)-direction (longitudinal if
\(i=j\) and shear if
\(i\ne j\)).
\(V_{ij/kl}\) are quasi-longitudinal or quasi-transverse waves propagating in the direction (
\(e_i + e_j\)) with particle displacements in a plane parallel to the plane formed by
\(x_k\) and
\(x_l\).
When measuring ultrasonic wave speeds we took multiple measurements along different paths for the same expected wave speed according to the material’s symmetry axes under damage: the samples initial axis of isotropy (
\(x_1\)) and the loading axis (
\(x_3\)). For example, in Table
1, when we measured the P-wave speed in the
\(x_1\)-direction, 4 transducers are used (transducers 1,3,5 and 7 shown in Fig.
2), and hence we obtained 4 different measurements of the P-wave in the
\(x_1\)-direction. This enabled us to calculate the standard error from these different measurements of the same expected wave speed. It is expected that there would be some natural variability in rock samples and using multiple transducers gives more confidence in using only one sample for shale and sandstone in this study. For each of the samples, and at each stress condition, we recorded a large number of ultrasonic wave velocities along multiple ray paths directions/positions within the sample, hence the large amount of data, and the confidence in them for the interpretations provided. The initial CT scans in Figs.
3 and
4 also show the degree of homogeneity of the materials. Ultrasonic investigations of materials undergoing uniaxial loading (Castellano et al.
2017; Marguéres and Meraghni
2013) often rely on single or a limited number of samples due to the complexity of the experimental set-up and the post-processing of the data. Mondal et al. (
2020b) showed using numerical simulations of damage that the overall peak stress for a model of sandstone undergoing uniaxial loading did not exhibit much variation with different realizations of rock heterogeneity. We expect that during the initial stages of damage before the final failure that there would be less variability in the materials response, and that the rock heterogeneity may have a bigger impact on the final stages of macrocracking.
We apply uniaxial compression at a constant axial strain rate of
\(10^{-5} \text{ s}^{-1}\) to the samples where the applied stress in the
\(x_3\)-direction is coaxial with the bedding plane of the sample. We consider two cases: assume that the sample is initially transverse isotropic with an axis of isotropy in the
\(x_1\)-direction in Appendix
B, or assume that the sample is initially orthotropic with the material axes aligned with the axes shown in Fig.
2 (as well as aligned with the applied uniaxial compressive stress). This means that in both cases the material axes of the damage tensor and damaged stiffness tensor are the same as the original undamaged shale’s material axes, and coaxial with the applied principal stresses. In the case of an initially transverse isotropic shale with an axis of isotropy in the
\(x_1\)-direction we assume that the loading in the
\(x_3\)-direction could break the symmetry in the
\(x_2\)–
\(x_3\)-plane. This is assumed because Curie’s principle states that the physical effects are at least as symmetric as the causes and we expect uniaxial compression in the
\(x_3\)-direction to either cause the sample to retain its transverse isotropy or become orthotropic (Rasolofosaon
1998).
Measurements of the ultrasonic velocities beyond the elastic regime of deformation can reveal the progression of damage and if a change in the anisotropy has occurred. Once the ultrasonic velocities are measured the coefficients of the damaged elastic stiffness tensor can be calculated from the well-known Christoffel’s equations for elastic wave propagation (Van Buskirk et al.
1981):
$$\begin{aligned} {\tilde{E}}_{11}= & {} \rho V_{1/1}^2, \nonumber \\ {\tilde{E}}_{22}= & {} \rho V_{2/2}^2 ,\nonumber \\ {\tilde{E}}_{33}= & {} \rho V_{3/3}^2 ,\nonumber \\ {\tilde{E}}_{44}= & {} \rho V_{2/3}^2 = \rho V_{3/2}^2, \nonumber \\ {\tilde{E}}_{55}= & {} \rho V_{1/3}^2 = \rho V_{3/1}^2, \nonumber \\ {\tilde{E}}_{66}= & {} \rho V_{1/2}^2 = \rho V_{2/1}^2, \end{aligned}$$
(1)
where
\(\rho \) is the density of the shale or sandstone. The above equations hold when the ultrasonic velocities are recorded along the sample’s material symmetry axes. However the off-axis ultrasonic measurements of the quasi P-wave velocities gives us the observed group velocities (
\(V^{ij/ij}_\phi \)) measured at a ray path angle,
\(\phi \), of
\(45^{\circ }\) between the material axes
i and
j. Since determination of
\({\tilde{E}}_{12} ,{\tilde{E}}_{13},\) and
\({\tilde{E}}_{23}\) requires knowledge of the phase velocity,
\(V^{ij/ij}_\theta \), it is necessary to solve for the eigenvalues of the Christoffel equations (Tsvankin
2001). Eigensolutions to the Christoffel equation for orthotropic materials are given by setting the following determinants to zero:
$$\begin{aligned} \left| \begin{array}{cc} {\tilde{E}}_{11} \sin ^2\theta + {\tilde{E}}_{55} \cos ^2\theta - \rho (V^{13/13}_\theta )^2 &{} ({\tilde{E}}_{13} + {\tilde{E}}_{55})\sin \theta \cos \theta \\ ({\tilde{E}}_{13} + {\tilde{E}}_{55})\sin \theta \cos \theta &{} {\tilde{E}}_{55} \sin ^2\theta + {\tilde{E}}_{33} \cos ^2\theta - \rho (V^{13/13}_\theta )^2 \end{array} \right|= & {} 0, \nonumber \\ \left| \begin{array}{cc} {\tilde{E}}_{22} \sin ^2\theta + {\tilde{E}}_{44} \cos ^2\theta - \rho (V^{23/23}_\theta )^2 &{} ({\tilde{E}}_{23} + {\tilde{E}}_{44})\sin \theta \cos \theta \\ ({\tilde{E}}_{23} + {\tilde{E}}_{44})\sin \theta \cos \theta &{} {\tilde{E}}_{44} \sin ^2\theta + {\tilde{E}}_{33} \cos ^2\theta - \rho (V^{23/23}_\theta )^2 \end{array} \right|= & {} 0, \nonumber \\ \left| \begin{array}{cc} {\tilde{E}}_{11} \sin ^2\theta + {\tilde{E}}_{66} \cos ^2\theta - \rho (V^{12/12}_\theta )^2 &{} ({\tilde{E}}_{12} + {\tilde{E}}_{66})\sin \theta \cos \theta \\ ({\tilde{E}}_{12} + {\tilde{E}}_{66})\sin \theta \cos \theta &{} {\tilde{E}}_{66} \sin ^2\theta + {\tilde{E}}_{22} \cos ^2\theta - \rho (V^{12/12}_\theta )^2 \end{array} \right|= & {} 0. \end{aligned}$$
(2)
The group velocities (
\(V_\phi ^{ij/ij}\)) are related to the phase velocities (
\(V_\theta ^{ij/ij}\)) by (Berryman
1979; Thomsen
1986; Tsvankin
2001; Dewhurst and Siggins
2006):
$$\begin{aligned} V_\phi ^2= & {} V_\theta ^2 + \left( \frac{\partial V_\theta }{\partial \theta } \right) ^2 \nonumber \\ \tan \phi= & {} \frac{V_\theta \sin \theta + \frac{\partial V_\theta }{\partial \theta }\cos \theta }{V_\theta \sin \theta - \frac{\partial V_\theta }{\partial \theta }\cos \theta } . \end{aligned}$$
(3)
where the observed group angle is
\(\phi =45^{\circ }\) for our ultrasonic measurements and we are measuring the group velocities in the
\(x_1\)–
\(x_2\),
\(x_1\)–
\(x_3\) and
\(x_2\)–
\(x_3\) planes, and the sample’s initial transverse isotropic axis of symmetry is
\(x_1\) and the uniaxial loading direction is
\(x_3\).
Equations (
2) and (
3) give us three nonlinear equations to solve for
\(V_\theta ^{ij/ij}, {\tilde{E}}_{ij}\) and
\(\theta \) for each
\(ij = 12,13\) or 23. These variables
\(V_\theta ^{ij/ij}, {\tilde{E}}_{ij}\) and
\(\theta \) were solved using the nonlinear solver,
vpasolve, in matlab for each measured off-axis group velocity recorded. Once the damaged stiffness tensor,
\({\tilde{E}}\), has been calculated using Eqs. (
1)–(
3) we can calculate the associated damage tensor and elastic moduli. Details of the derivation for the damaged tensor are given in Appendix
A.
2.2 CT Scans of Shale and Sandstone Samples Before and After Loading
Figures
3 and
4 show the X-Ray Computed Tomography images of the two samples obtained before and after loading along the
\(x_3\) axis (shown as
Z-axis here). The slices are reported along the main planes investigated with our velocity arrays - i.e.
X–
Y (
\(x_1\)–
\(x_2\)),
X–
Z (
\(x_1\)–
\(x_3\)) and
Y–
Z (
\(x_2\)–
\(x_3\)) planes. Please note that the axis of isotropy for both samples (
\(x_1\)) is defined as
X and is in blue for the shale sample Fig.
3, and is in red for the sandstone sample in Fig.
4.
The CT image for the shale sample in Fig.
3 shows that the predominant mode of fracturing is extension fractures between the bedding planes or possibly some sliding of the bedding layers, and that in the microcracking stage it is likely that the shale sample retained its transverse isotropy. Calculation of the Young’s and shear moduli in the following Sect.
3.1 in Fig.
6 using ultrasonic measurements confirm this hypothesis as well. However there is more ambiguity in interpreting the change in the Poisson’s ratios in Fig.
7 which may be because the initial magnitudes of the Poisson’s ratio for the shale sample more closely resemble an orthotropic material symmetry initially in Fig.
5b. However the ultrasonic measurements for the shear and Young’s moduli more closely resemble the assumed transverse isotropic material symmetry in Fig.
5a with axis of isotropy in the
\(x_1\) direction.
The CT image for the sandstone sample in Fig.
4 shows some initial extension or shear fracturing between the bedding planes before the final failure of shear fracturing at an angle to the loading as observed commonly in isotropic samples. This mixed mode of fracturing might be more likely for the sandstone because overall it is less anisotropic than the shale sample.