Relating shear sonic anisotropy directions to stress in deviated wells

Abstract

We develop a model that relates the polarization direction of a dipole sonic fast shear wave propagating along the borehole (hereafter called FSA) to the relative deviatoric stress tensor for arbitrary well orientations. We first define stress quantities, or “subsidiary principal stress,” relevant for sonic dipole shear characterized sufficiently far away into the formation to be unaffected by borehole stress concentration. Next, we show analytically for wells oriented within principal stress planes and numerically for wells outside principal stress planes that the stress-induced FSA coincides with the maximum normal stress direction orthogonal to the borehole (“maximum subsidiary principal stress”) using a nonlinear elastic model for isotropic unstressed background media and plane wave solutions. This model is independent of stress sensitivity parameters. This result is a consequence of, first, the direct relationship between the stressellipsoid factor and the shear stiffness difference ratio, [Formula: see text], and, second, the almost elliptical nature of stress-induced orthorhombic media. Using published laboratory data relating stiffnesses to applied differential stress for isotropic unstressed limestone and sandstone rocks, the model is confirmed for most well orientations except in the vicinity of the stress nodal point within the maximum and minimum principal stress plane, due to small anellipticity as stress increases. The orientations near the nodal point are also the zones of weakest stress-induced anisotropy zones and are therefore the least likely to be measured in practice as long as a quality indicator depending on both the degree of anellipticity and the strength of the shear anisotropy is below 0.5. Finally, we present a synthetic example of the application of this model to estimate the relative deviatoric stress tensor using two wells with different orientations.

Modeling elastic anisotropy resulting from the application of triaxial stress

“…We showed that the anisotropy resulting from the application of a triaxial stress on a linearly elastic isotropic medium is ellipsoidal provided stresses are small enough, which confirms results previously obtained by Rasolofosaon (1998) and Sun and Prioul (2010). This result opens up the possibility of differentiating stress-induced anisotropy from fracture-induced anisotropy by estimating the degree of anellipticity.…”

“…Note that equations 43 and 44 agree with the relationship derived by Sun and Prioul (2010) using the third-order elasticity theory in the limit of weak anisotropy:…”

“…For Figure 3c to 3f, the anisotropy parameters ratios computed using the model derived for large stresses deviate significantly from the corresponding stress ratios because a 20% error is obtained for normalized stresses as low as 1.5. Besides, on Figure 3a and 3b, the green dashed curve represents the relationship linking γ ð1Þ , γ ð2Þ and the stress ratio derived by Sun and Prioul (2010) given in equation 46. We notice that their relationship and the one we derive, equation 44, yield results which are very close, the maximum absolute error between both of these relationships being less than 0.01 for the two examples shown in this study.…”

Frequency dependence of anisotropy in fluid saturated rocks – Part I: aligned cracks case

“…). The anisotropy resulting from the application of a uniaxial stress on an initially isotropic medium was found to be elliptical (e.g., Rasolofosaon ; Sun and Prioul ), which is characterized in terms of Thomsen () anisotropy parameters by: . In the case of aligned cracks, the dry medium is elliptical if and only if the compliance ratio B equals to unity.…”

“…Several authors used the model of aligned cracks with a stress-dependent crack density to model the effect of stress on elastic properties (e.g., Brajanovski, Gurevich, and Schoenberg 2005;Thanoon et al 2015). The anisotropy resulting from the application of a uniaxial stress on an initially isotropic medium was found to be elliptical (e.g., Rasolofosaon 1998;Sun and Prioul 2010), which is characterized in terms of Thomsen (1986) anisotropy parameters by: = δ. In the case of aligned cracks, the dry medium is elliptical if and only if the compliance ratio B equals to unity.…”