Low Static Shear Modulus Along Foliation and Its Influence on the Elastic and Strength Anisotropy of Poorman Schist Rocks, Homestake Mine, South Dakota

Abstract

We investigate the influence of foliation orientation and fine-scale folding on the static and dynamic elastic properties and unconfined strength of the Poorman schist. Measurements from triaxial and uniaxial laboratory experiments reveal a significant amount of variability in the static and dynamic Young’s modulus depending on the sample orientation relative to the foliation plane. Dynamic P-wave modulus and S-wave modulus are stiffer in the direction parallel to the foliation plane as expected for transversely isotropic mediums with average Thomsen parameters values 0.133 and 0.119 for epsilon and gamma, respectively. Static Young’s modulus varies significantly between 21 and 117 GPa, and a peculiar trend is observed where some foliated sample groups show an anomalous decrease in the static Young’s modulus when the symmetry axis (x₃-axis) is oriented obliquely to the direction of loading. Utilizing stress and strain relationships for transversely isotropic medium, we derive the analytical expression for Young’s modulus as a function of the elastic moduli E₁, E₃, ν₃₁, and G₁₃ and sample orientation to fit the static Young’s modulus measurements. Regression of the equation to the Young’s modulus data reveals that the decrease in static Young’s modulus at oblique symmetry axis orientations is directly influenced by a low shear modulus, G₁₃, which we attribute to shear sliding along foliation planes during static deformation that occurs as soon as the foliation is subject to shear stress. We argue that such difference between dynamic and static anisotropy is a characteristic of near-zero porosity anisotropic rocks. The uniaxial compressive strength also shows significant variability ranging from 21.9 to 194.6 MPa across the five sample locations and is the lowest when the symmetry axis is oriented 45° or 60° from the direction of loading, also a result of shear sliding along foliation planes during static deformation.

Appendices

Appendix A: Analytical Expression for Young’s Modulus of Rotated TI Mediums

1. Uniaxial stress is applied in the x₃-direction making σ₃₃ the only nonzero stress in the initial x₁–x₃ coordinate system.

$$\sigma =\left[\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& {\sigma }_{13}\\ {\sigma }_{12}& {\sigma }_{22}& {\sigma }_{23}\\ {\sigma }_{13}& {\sigma }_{23}& {\sigma }_{33}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& {\sigma }_{33}\end{array}\right].$$

2. Rotate the stress matrix about the x₂-axis to the x₁′–x₃′ coordinate system. The rotation matrix, R, is given from the direction cosines between the initial and prime axis.

$$R=\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{cos}90& \mathrm{cos}(\theta +90)\\ \mathrm{cos}90& \mathrm{cos}0& \mathrm{cos}90\\ \mathrm{cos}(90-\theta )& \mathrm{cos}90& \mathrm{cos}\theta \end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right],$$

$${\sigma }^{{\prime}}=R\sigma R^{-1}=\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& {\sigma }_{33}\end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& \mathrm{sin}\theta \\ 0& 1& 0\\ -\mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]=\left[\begin{array}{ccc}{\mathrm{sin}}^{2}\theta & 0& -\mathrm{sin}\theta\mathrm{cos}\theta \\ 0& 0& 0\\ -\mathrm{sin}\theta\mathrm{cos}\theta & 0& {\mathrm{cos}}^{2}\theta \end{array}\right]{\sigma }_{33}.$$

3. Rewrite the rotated stress matrix in Voigt notation and multiply the compliance matrix by the rotated stress matrix to find the strain in the prime coordinate system.

$${\varepsilon }^{{'}}=S*{\sigma }^{{'}},$$

$$\left[\begin{array}{c}{\varepsilon }_{11}^{{'}}\\ {\varepsilon }_{22}^{{'}}\\ {\varepsilon }_{33}^{{'}}\\ 2{\varepsilon }_{23}^{{'}}\\ 2{\varepsilon }_{13}^{{'}}\\ {2\varepsilon }_{12}^{{'}}\end{array}\right]=\left[\begin{array}{cccccc}{s}_{11}& {s}_{12}& {s}_{13}& 0& 0& 0\\ {s}_{12}& {s}_{11}& {s}_{13}& 0& 0& 0\\ {s}_{13}& {s}_{13}& {s}_{33}& 0& 0& 0\\ 0& 0& 0& {s}_{44}& 0& 0\\ 0& 0& 0& 0& {s}_{44}& 0\\ 0& 0& 0& 0& 0& {s}_{66}\end{array}\right]*\left[\begin{array}{c}{\mathrm{sin}}^{2}\theta\\ 0\\ {\mathrm{cos}}^{2}\theta\\ 0\\ -\mathrm{sin}\theta\mathrm{cos}\theta\\ 0\end{array}\right]{\sigma }_{33},$$

$${\varepsilon }^{^{\prime}}=\left[\begin{array}{ccc}{s}_{11}{\mathrm{sin}}^{2}\theta +{s}_{13}{\mathrm{cos}}^{2}\theta & 0& -\frac{{s}_{44}\mathrm{sin}\theta \mathrm{cos}\theta }{2}\\ 0& {s}_{12}{\mathrm{sin}}^{2}\theta +{s}_{13}{\mathrm{cos}}^{2}\theta & 0\\ -\frac{{s}_{44}\mathrm{sin}\theta \mathrm{cos}\theta }{2} & 0& {s}_{13}{\mathrm{sin}}^{2}\theta +{s}_{33}{\mathrm{cos}}^{2}\theta \end{array}\right]{\sigma }_{33}.$$

4. Rotate the strain tensor back to the initial coordinate system using the inverse rotation matrix R⁻¹.

$$\varepsilon={R}^{\text{-1}}{\varepsilon}^{\prime}R,$$

$$\varepsilon=\left[\begin{array}{ccc}\mathrm{cos}\uptheta & 0& \mathrm{sin}\uptheta \\ 0& 1& 0\\ -\mathrm{sin}\uptheta & 0& \mathrm{cos}\uptheta \end{array}\right]\left[\begin{array}{ccc}{\mathrm{s}}_{11}{\mathrm{sin}}^{2}\uptheta +{\mathrm{s}}_{13}{\mathrm{cos}}^{2}\uptheta & 0& -\frac{{\mathrm{s}}_{44}\mathrm{sin}\uptheta \mathrm{cos}\uptheta }{2}\\ 0& {\mathrm{s}}_{12}{\mathrm{sin}}^{2}\uptheta +{\mathrm{s}}_{13}{\mathrm{cos}}^{2}\uptheta & 0\\ -\frac{{\mathrm{s}}_{44}\mathrm{sin}\uptheta \mathrm{cos}\uptheta }{2} & 0& {\mathrm{s}}_{13}{\mathrm{sin}}^{2}\uptheta +{\mathrm{s}}_{33}{\mathrm{cos}}^{2}\uptheta \end{array}\right]{\upsigma }_{33}\left[\begin{array}{ccc}\mathrm{cos}\uptheta & 0& -\mathrm{sin}\uptheta \\ 0& 1& 0\\ \mathrm{sin }\uptheta & 0& \mathrm{cos}\uptheta \end{array}\right].$$

The individual components of the resulting strain matrix are written in the equations below:

$${\varepsilon }_{11}=\left({s}_{11}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta +{s}_{13}\left({\mathrm{sin}}^{4}\theta +{\mathrm{cos}}^{4}\theta \right)+{s}_{33}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta -{s}_{44}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta \right) {\sigma }_{33},$$

$${\varepsilon }_{22}=\left({\mathrm{s}}_{12}{\mathrm{sin}}^{2}\theta +{s}_{13}{\mathrm{cos}}^{2}\theta \right) {\sigma }_{33},$$

$${\varepsilon }_{33}=\left({s}_{11}{\mathrm{sin}}^{4}\theta +2{s}_{13}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta +{s}_{33}{\mathrm{cos}}^{4}\theta +{s}_{44}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta \right) {\sigma }_{33},$$

$${\varepsilon }_{23}=0,$$

$${\varepsilon }_{13}=\left(-{s}_{11}{\mathrm{sin}}^{3}\theta \mathrm{cos}\theta +{s}_{13}\left({\mathrm{sin}}^{3}\theta \mathrm{cos}\theta -\mathrm{sin}\theta {\mathrm{cos}}^{3}\theta \right)+{\mathrm{s}}_{33}\mathrm{sin}\theta {\mathrm{cos}}^{3}\theta +{s}_{44} \frac{ {(\mathrm{sin}}^{3}\theta \mathrm{cos}\theta -\mathrm{sin}\theta {\mathrm{cos}}^{3}\theta )}{2} \right) {\sigma }_{33},$$

$${\varepsilon }_{12}=0.$$

5. The Young’s modulus is determined from dividing the applied stress by strain in the same direction

$$E={\sigma }_{33}/{\varepsilon }_{33}$$

and can be written in the following convenient form by substituting the compliance matrix components.

$${\varepsilon }_{33}=\left(\frac{{\mathrm{sin}}^{4}\theta }{{E}_{1}}-\frac{2{\nu }_{31}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta }{{E}_{3}}+\frac{{\mathrm{cos}}^{4}\theta }{{E}_{3}}+\frac{{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\theta }{{G}_{13}}\right) {\sigma }_{33},$$

where for a transverse isotropic material with symmetric axis in the three directions

$${s}_{11}=\frac{1}{{E}_{1}},$$

$${s}_{13}=\frac{-{\nu }_{31}}{{E}_{3}}=\frac{-{\nu }_{13}}{{E}_{1}},$$

$${s}_{33}=\frac{1}{{E}_{3}},$$

$${s}_{44}=\frac{1}{{G}_{13}}=\frac{1}{{G}_{23}}.$$

Appendix B: Sample Locations and Borehole Diagram

Figure 17 shows the borehole diagram along the west drift on the 4850-ft depth level of SURF adapted from Morris et al. (2018) with discs that indicate the intended notch locations for hydraulic stimulation in the EGS Collab project. The testbed is comprised of eight sub-horizontal boreholes oriented around the intended stimulation zone. Sample locations are marked with a square for the planar sample groups and a circle for the folded sample groups. Sample names are assigned based on the borehole and sample orientation. The first letters (P, I, OB, or PDB) indicate the borehole and the second numbers or letters (ex: 0, 45, X, Z) indicate the sample orientation. The sample depths along the length of the borehole are provided in Table

Table 7 Sample depths measured along the borehole axis taken from core logs, photographs, and markings on the host core

7.

Appendix C: Ultrasonic Velocity and Static Young’s Modulus Table

See Tables 8 and

Table 8 Sample length, diameter, density and velocity measurements under hydrostatic, triaxial, and uniaxial stress conditions

Table 9 Static Young's modulus measurements organized by stress phase and sample

9.