Anisotropic inversion of seismic data for stressed media: Theory and a physical modeling study on Berea Sandstone

Abstract

Nonhydrostatic stress, an often‐ignored source of seismic anisotropy, is universally present in the subsurface and may be as common as intrinsic or fracture‐induced anisotropy. Nonhydrostatic stress, applied to an initially transversely isotropic solid with vertical symmetry axis (VTI), results in an effective medium having almost orthorhombic symmetry (provided that one of the principal stresses is aligned with the symmetry axis). The symmetry planes observed in this orthorhombic medium are aligned with the orientations of the principal stresses, and anisotropic parameters (ε(1,2), δ(1,2,3), and γ(1,2)) can reveal information about the stress magnitudes. Thus, time‐lapse monitoring of changes in anisotropy potentially can provide information on temporal variations in the stress field.We use nonlinear elasticity theory to relate the anisotropic parameters to the magnitudes of the principal stresses and verify these relationships in a physical modeling study. Under the assumption of weak background and stress‐induced anisotropy, each effective anisotropic parameter reduces to the sum of the corresponding Thomsen parameter for the unstressed VTI background and the corresponding parameter associated with the nonhydrostatic stress. The stress‐related anisotropic parameters depend only on the differences between the magnitudes of principal stresses; therefore, principal stresses can influence anisotropic parameters only if their magnitudes differ in the symmetry plane in which the anisotropic parameters are defined.We test these predictions on a physical modeling data set acquired on a block of Berea Sandstone exhibiting intrinsic VTI anisotropy. Uniaxial stress, applied normal to the VTI symmetry axis, i.e., horizontally, produces an effective medium that is close to orthorhombic. We use two different methods to estimate the anisotropic parameters and study their variation as a function of stress. The first method utilizes conventional measurements of transmission velocities along the principal axes of the sample. The second method uses PP and PS reflection data acquired along seven different azimuths on the surface of the block.In accordance with theoretical predictions, the anisotropic parameters in the vertical plane normal to the stress are almost insensitive to the magnitude of the stress. In contrast, anisotropic parameters in the vertical plane of the applied stress increase approximately in a linear fashion with increasing stress. Except for the parameter δ(1), comparison of the measured values of anisotropic parameters with theoretical predictions shows satisfactory agreement.Despite some documented discrepancies, we believe that nonlinear elasticity may provide a suitable framework for estimating pore pressure and 3D stresses from seismic data.