Geomechanical Characterization of Marcellus Shale

Abstract

Understanding the reservoir conditions and material properties that govern the geomechanical behavior of shale formations under in situ conditions is of vital importance for many geomechanical applications. The development of new numerical codes and advanced multi-physical (thermo-hydro-chemo-mechanical) constitutive models has led to an increasing demand for fundamental material property data. Previous studies have shown that deformational rock properties are not single-value, well-defined, linear parameters. This paper reports on an experimental program that explores geomechanical properties of Marcellus Shale through a series of isotropic compression (i.e. σ ₁ = σ ₂ = σ ₃) and triaxial (i.e. σ ₁ > σ ₂ = σ ₃) experiments. Deformational and failure response of these rocks, as well as anisotropy evolution, were studied under different stress and temperature conditions using single- and multi-stage triaxial tests. Laboratory results revealed significant nonlinear and pressure-dependent mechanical response as a consequence of the rock fabric and the occurrence of microcracks in these shales. Moreover, multi-stage triaxial tests proved to be useful tools for obtaining failure envelopes using a single specimen. Furthermore, the anisotropic nature of Marcellus Shale was successfully characterized using a three-parameter coupled model.

Appendix: Vertical Transversely Isotropic Three-Parameter Coupling Model

The linear elastic VTI model for a sample with the z-axis being the axis of symmetry can be expressed in terms of five independent parameters, with the compliance matrix as follows:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\varepsilon_{xy} } \\ {\varepsilon_{yz} } \\ {\varepsilon_{xz} } \\ \end{array} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{1}{{E_{\rm h} }}} & { - \frac{{\nu_{\rm hh} }}{{E_{\rm h} }}} & { - \frac{{\nu_{\rm vh} }}{{E_{\rm v} }}} \\ { - \frac{{\nu_{\rm hh} }}{{E_{\rm h} }}} & {\frac{1}{{E_{\rm h} }}} & { - \frac{{\nu_{\rm vh} }}{{E_{\rm v} }}} \\ { - \frac{{\nu_{\rm vh} }}{{E_{\rm v} }}} & { - \frac{{\nu_{\rm vh} }}{{E_{\rm v} }}} & {\frac{1}{{E_{\rm v} }}} \\ \end{array} } & {} \\ {} & {\begin{array}{*{20}c} {\frac{1}{{G_{\rm vh} }}} & {} & {} \\ {} & {\frac{1}{{G_{\rm vh} }}} & {} \\ {} & {} & {\frac{{2\left( {1 + \upsilon_{\rm hh} } \right)}}{{E_{\rm h} }}} \\ \end{array} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\sigma_{xy} } \\ {\sigma_{yz} } \\ {\sigma_{xz} } \\ \end{array} } \\ \end{array} } \right\}$$

(4)

In the context of triaxial space, Eq. 4 is reduced to:

$$\left\{ {\begin{array}{*{20}c} {\delta \varepsilon_{\rm a} } \\ {\delta \varepsilon_{\rm r} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {1/E_{\rm v} } & { - 2\upsilon_{\rm vh} /E_{v} } \\ { - 2\upsilon_{\rm vh} /E_{v} } & { - \left( {1 - \upsilon_{\rm hh} } \right)/E_{\rm h} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta \sigma_{\rm a} } \\ {\delta \sigma_{\rm r} } \\ \end{array} } \right\}$$

(5)

Note that this compliance matrix is not symmetric since the strain increment and the stress quantities shown in Eq. 5 are not properly work conjugate. Recall that in the context of triaxial space the correctly chosen work-conjugate quantities are p–δε _v and q–δε _s for the volumetric and the distortional deformations, respectively (δW = σ _ij δε _ij  = pδε _v + qδε _s). Also, one can only determine E _v and ν _vh, but not E _h or ν _hh since they only appear in the composite stiffness “−(1 − ν _vh)/E _h”.

Equation 5 can be rewritten using the definitions of the triaxial strain increment and stress quantities as shown by Puzrin (2012):

$$\left\{ {\begin{array}{*{20}c} {\delta \varepsilon_{\text{v}} } \\ {\delta \varepsilon_{\text{s}} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {1/K} & { - 1/J} \\ { - 1/J} & {1/3G} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta p} \\ {\delta q} \\ \end{array} } \right\}$$

(6)

where, K stands for the bulk modulus during isotropic compression (δq = 0); G is the shear modulus for pure shear (δp = 0); and J is the coupling modulus. These three new parameters can be defined in terms of the original five VTI independent parameters (Puzrin 2012) as shown in Eqs. 7–9.

$$\frac{1}{K} = 2\frac{{1 - \nu_{\text{hh}} }}{{E_{\text{h}} }} + \frac{{1 - 4\nu_{\text{vh}} }}{{E_{\text{v}} }}$$

(7)

$$\frac{1}{J} = \frac{2}{3}\left( {\frac{{1 - \nu_{\text{hh}} }}{{E_{\text{h}} }} - \frac{{1 - \nu_{\text{vh}} }}{{E_{\text{v}} }}} \right)$$

(8)

$$\frac{1}{G} = \frac{2}{3}\left( {\frac{{1 - \nu_{\text{hh}} }}{{E_{\text{h}} }} + 2\frac{{1 + 2\nu_{\text{vh}} }}{{E_{\text{v}} }}} \right)$$

(9)

The compliance matrix in Eq. 6 is symmetric (material is elastic and satisfies the law of energy conservation), and the non-zero off-diagonal terms show the ability of the model to reproduce both coupling between volumetric and distortional effects, and the stress path dependency of stiffness (Puzrin 2012). However, this model is only correct in the context of the triaxial test, and only if the symmetry axis stays vertical. If one wants to model the transverse isotropy in a boundary value problem, the five independent parameters of the VTI model have to be used instead.