Static and dynamic behavior of isotropic or anisotropic media, including rocks, are conveniently described with a second-order stiffness matrix using the Voigt notation, which linearly relates stress changes to strains. However, experimental and field observations indicate that the dynamic stiffness of rocks is stress dependent. We investigate a model proposed by Fuck and Tsvankin employing a constitutive third-order elastic tensor to describe the non-linear strain sensitivity of the stiffness. By using laboratory measurements of strains and ultrasonic P- and S-wave velocities in multiple directions, we were able to invert for all the third-order parameters. We used the third-order elastic tensor to model changes in ultrasonic velocities and investigated the impact of different third-order tensor component optimization schemes on the accuracy of the velocity estimates. To our knowledge this is the first fit of dynamic stiffness of a shale to a third order constitutive model that is not restricted to isotropic strain sensitivity.


Elastic properties of most of the earth crust rocks are direction-dependent (e.g. Thomsen, 1986), which makes anisotropy an important factor for rock stiffness and seismic wave velocity analysis. The static and dynamic properties of an anisotropic medium are usually represented by a second-order elastic (SOE) matrix Cij, which linearly relates stress changes with corresponding strains (Fjör et al., 2008).

However, experimental and field data suggest that the dynamic stiffness of rocks is stress-dependent (e.g. Johnson and Rasolofosaon, 1996), i.e. the relationship between stress and deformation is non-linear. This behavior can be described in terms of stiffening grain contacts (Bachrach and Avseth, 2008; Mindlin, 1949; Walton, 1987) or with the use of crack-based models (Budiansky and O'Connell, 1976; Fjaer, 2006; Hudson, 1981). Alternatively, we can use higher order constitutive models (e.g. Prioul et al., 2004), which until now have not been systematically investigated for sedimentary rocks.

Our aim was to derive a third-order elastic (TOE) tensor cijk which we could use to approximate the dynamic behavior of transversely isotropic shales under different stress state development scenarios and verify it using laboratory data collected on shale samples. This approach is based on a fully physical strain-dependent third-order constitutive model for which we assume vertical transverse isotropy (VTI, i.e. the symmetry axis normal to the plane of isotropy) of stiffnesses and the applied stresses, which allowed us to limit the number of model parameters. Contrary to previous studies (Fuck et al., 2009; Johnson and Rasolofosaon, 1996; Prioul et al., 2004), we do not assume isotropic strain sensitivity of the velocities, which may seem to be an oversimplification for inherently anisotropic materials like shales. Velocity changes are measured in multiple directions for different stress changes. These data are used to determine the elastic parameters of two different nonlinear models and to evaluate their performance

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