American Mineralogist, Volume 96, pages 125–137, 2011
First-principles calculation of the elastic moduli of sheet silicates and their application to shale anisotropy B. Militzer,1,2,* H.-R. Wenk,1 S. Stackhouse,1 and L. Stixrude3 1
Department of Earth and Planetary Science, University of California, Berkeley, California 94720, U.S.A. 2 Department of Astronomy, University of California, Berkeley, California 94720, U.S.A. 3 Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K.
Abstract The full elastic tensors of the sheet silicates muscovite, illite-smectite, kaolinite, dickite, and nacrite have been derived with first-principles calculations based on density functional theory. For muscovite, there is excellent agreement between calculated properties and experimental results. The influence of cation disorder was investigated and found to be minimal. On the other hand, stacking disorder is found to be of some relevance for kaolin minerals. The corresponding single-crystal seismic wave velocities were also derived for each phase. These revealed that kaolin minerals exhibit a distinct type of seismic anisotropy, which we relate to hydrogen bonding. The elastic properties of a shale aggregate was predicted by averaging the calculated properties of the contributing mineral phases over their orientation distributions. Calculated elastic properties display higher stiffness and lower p-wave anisotropy. The difference is likely due to the presence of oriented flattened pores in natural samples that are not taken into account in the averaging. Keywords: Elasticity, clay, ab initio calculations, sheet silicates, seismic anisotropy
Introduction Sheet silicates are among the minerals with highest elastic anisotropy. Aggregates containing oriented sheet silicates, such as schists and shales, also display very high anisotropies and this has important implications for interpreting seismic travel times through such rocks. Shales are the most abundant rocks in sedimentary basins and relevant for petroleum deposits (e.g., Johansen et al. 2004) as well as for carbon sequestration (Chadwick et al. 2004). Their physical properties are of great importance in exploration geophysics (e.g., Mavko et al. 1998; Wang et al. 2001). Elastic properties of shales have been measured with ultrasound techniques (e.g., Hornby 1998; Hornby et al. 1994; Johnston and Christensen 1995), because of their importance for seismic prospecting of hydrocarbon deposits (e.g., Banik 1984). In metals and igneous and metamorphic rocks, anisotropic aggregate properties can be satisfactorily predicted by averaging the single-crystal elastic properties over the orientation distribution (Bunge 1985; Barruol and Kern 1996; Mainprice and Humbert 1994). In the case of shale, anisotropy can be calculated with this method, but absolute values of predicted elastic stiffness coefficients are over a factor of two higher than the measured results (Valcke et al. 2006; Voltolini et al. 2009; Wenk et al. 2008). Currently only empirical models are available to describe the elasticity of shales that do not rely on measured microstructural properties (e.g., Bayuk et al. 2007; Ougier-Simoni et al. 2008; Ponte Castaneda and Willis 1995; Sayers 1994). There are various reasons why the simple averaging schemes are limited in * E-mail:
[email protected] 0003-004X/11/0001–125$05.00/DOI: 10.2138/am.2011.3558
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the case of shales. One reason is that porosity and grain contacts are inadequately accounted for. Another reason is the uncertainty in the single-crystal elastic properties of sheet silicates (Chen and Evans 2006). Despite their importance, the single-crystal elastic moduli of clay minerals have not been measured experimentally, because of technical difficulties associated with the small grain size. There is also considerable uncertainty in atomic force microscopy (Prasad et al. 2002). For illite, elastic constants of the analog mineral, muscovite, have traditionally been used (Vaughan and Guggenheim 1986; Zhang et al. 2009) but considerable uncertainty arises from using such analogies. For kaolinite, values from first-principles calculations exist for the ideal structure (Sato et al. 2005; Mercier and Le Page 2008). However, the structure of kaolin minerals in natural samples is expected to differ considerably from the ideal kaolinite. Moreover, other equally important sheet silicates have not yet been examined by experiment or theory, including illite-smectite. In view of this, in the present work, we have calculated the single-crystal elastic properties of illite-smectite and the kaolin minerals kaolinite, dickite, and nacrite, which have structures more similar to those found in nature. In addition, to establish and test our method, we have computed the elastic properties of muscovite, for which analogous experimental data exists. The computed elastic properties are then used to predict the elastic properties of the classic Kimmeridge Clay shale (age 151–156 Ma) from Hornby (1998) for which experimental elastic properties are available, as well as orientation distributions of component phases (Wenk et al. 2010).
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Computational methods We used the Vienna ab initio simulation package (VASP) (Kresse and Hafner 1993; Kresse and Furthmüller 1996) to perform density functional calculations using either the local density approximation (LDA) or the Perdew-Burke-Ernzerhof (Perdew et al. 1996) generalized gradient approximation (GGA) in combination with the projector augmented-wave method (Blöchl 1994). The wave functions of the valence electrons were expanded in a plane-wave basis with an energy cut-off of 586 eV or higher and the Brillouin zone was sampled using 4 × 2 × 2 and 6 × 4 × 4 k-points grids (Monkhorst and Pack 1976) depending on the size of the unit cell. All structures were relaxed until all forces were
