Stress-dependent anisotropy in transversely isotropic rocks: Comparison between theory and laboratory experiment on shale

Abstract

Understanding the effect of stress and pore pressure on seismic velocities is important for overpressure prediction and for 4D reflection seismic interpretation. A porosity-deformation approach (originally called the piezosensitivity theory) and its anisotropic extension describe elastic moduli of rocks as nonlinear functions of the effective stress. This theory assumes a presence of stiff and compliant parts of the pore space. The stress-dependent geometry of the compliant pore space predominantly controls stress-induced changes in elastic moduli. We show how to apply this theory to a shale that is transversely isotropic (TI) under unloaded conditions. The porosity-deformation approach shows that components of the compliance tensor depend on exponential functions of the principal components of the effective stress tensor. In the case of a hydrostatic loading of a TI rock, only the diagonal elements of this tensor, expressed in contracted notation, are significantly stress dependent. Two equal shear components of the compliance will depend on a combination of two stress exponentials. Exponents of the stress exponentials are controlled by components of the stress-sensitivity tensor. This tensor is an important physical characteristic directly related to the elastic nonlinearity of the porous rock. We simplify the porosity-deformation theory for TI rocks and provide corresponding explicit equations. We apply this theory to ultrasonic measurements on saturated shale samples from the North Sea. We show that the theory explains the compliance tensor, anellipticity, and three anisotropic parameters under a broad range of loads.

Parameterization of elastic stress sensitivity in shales

“…Departure from unity is observed even in the group with the values from 0.7 to 1.1. This suggests that the ratio B ¼ 1, which corresponds to the model of Ciz and Shapiro (2009), may not be adequate for shales.…”

“…Their model is based on the bimodal distribution of pore compliances and superposition of deformation fields caused by closure/shape change of these two groups of soft and stiff pores under applied stress. Ciz and Shapiro (2009) use this approach to build a model of experimental shale measurements from the North Sea previously reported by Hornby (1998) and show a good model fit to the measurements. However in the isotropic limit, the model of Ciz and Shapiro (2009) has been shown to be equivalent to the isotropic version of the model of Sayers (1999) with exponential dependency of excess compliances on pressure and with an additional assumption that normal and tangential compliances of each grain contact are equal (Gurevich et al, 2009).…”

“…Ciz and Shapiro (2009) use this approach to build a model of experimental shale measurements from the North Sea previously reported by Hornby (1998) and show a good model fit to the measurements. However in the isotropic limit, the model of Ciz and Shapiro (2009) has been shown to be equivalent to the isotropic version of the model of Sayers (1999) with exponential dependency of excess compliances on pressure and with an additional assumption that normal and tangential compliances of each grain contact are equal (Gurevich et al, 2009). This assumption may not be physically adequate, especially for water-saturated rocks at ultrasonic frequencies, because water in intergranular pores and microfractures should strongly reduce normal, but not tangential, compliance.…”

Overburden geomechanical effects on 4D seismic response – influences of stiffness contrast, anisotropy, and geometry

“…Resulting effective stress changes will alter the seismic properties of the rocks involved (Hornby, 1998;Sayers, 2002;Prioul et al, 2004;Ciz & Shapiro, 2009). This makes time lapse seismic an important tool to monitor subsurface alterations (e.g., Landrø 2001) and thereby improving recovery or storage capacity by optimal placement of new wells.…”

Estimation of stress-dependent anisotropy from P-wave measurements on a spherical sample

“…This data set was used by Prioul et al (2004) to propose a nonlinear rock physics model to predict stress-dependent elasticity parameters for transverse isotropic shale under isotropic and anisotropic stress fields. This data set was used by Ciz and Shapiro (2009) to address the stress-induced anisotropy for transversely isotropic elasticity using an anisotropic extension of the porosity deformation formalism (Shapiro, 2003). Wang (2002aWang ( , 2002b proposed a single core plug measurement of P-wave at normal, parallel, and 45 to the bedding plane and also two shear waves along the bedding plane to measure the elasticity parameters for a transversely isotropic rock sample.…”