Modelling elastic anisotropy of dry rocks as a function of applied stress
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We propose an analytical model for seismic anisotropy caused by the application of an anisotropic stress to an isotropic dry rock. We first consider an isotropic, linearly elastic medium (porous or non-porous) permeated by a distribution of discontinuities with random (isotropic) orientation (such as randomly oriented compliant grain contacts or cracks). The geometry of individual discontinuities is not specified. Instead, their behaviour is defined by a ratio B of the normal to tangential excess compliances. When this isotropic rock is subjected to a compressive stress (isotropic or anisotropic), the specific surface area of cracks aligned parallel to a particular plane is reduced in proportion to the normal stress traction acting on that plane. This effect is modelled using the Sayers-Kachanov non-interactive approximation, which expresses the effect of cracks on the elastic compliance tensor as an integral over crack orientations. This integral is evaluated using the Taylor expansion of the stress dependency of the specific surface area of the cracks. This allows the analytical solution previously derived for small anisotropic stresses to be extended to large stresses. Comparison of the model predictions with the results of laboratory measurements shows a reasonable agreement for moderate magnitudes of uniaxial stress (up to 50 MPa).While the model contains five independent parameters, the variation of the anisotropy pattern (which can be expressed by the ratios of Thomsen’s anisotropy parameters ε, δ and γ) with normalized stress is controlled by only two parameters: Poisson’s ratio ν of the unstressed rock and the compliance ratio B. The model predicts that the ε/γ ratio depends on both ν and B but varies only mildly with stress, while the ε/δ ratio varies between 0.8–1.1 in a wide range of values of ν and B. The latter observation implies that the anisotropy remains close to elliptical even for larger stresses (within the assumptions of the model). The proposed model of stress-induced anisotropy may be useful for differentiating stress-induced anisotropy from that caused by aligned fractures. Conversely, if the cause of seismic anisotropy is known, then the anisotropy pattern allows one to estimate P-wave anisotropy from S-wave anisotropy.
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