We develop a rock physics model based on nonlinear elasticity that describes the dependence of the effective stiffness tensor as a function of a 3D stress field in intrinsically anisotropic formations. This model predicts the seismic velocity of both P-and S-waves in any direction for an arbitrary 3D stress state. Therefore, the model overcomes the limitations of existing empirical velocity-stress models that link P-wave velocity in isotropic rocks to uniaxial or hydrostatic stress.
To validate this model, we analyze ultrasonic velocity measurements on stressed anisotropic samples of shale and sandstone. With only three nonlinear constants, we are able to predict the stress dependence of all five elastic medium parameters comprising the transversely isotropic stiffness tensor. We also show that the horizontal stress affects vertical S-wave velocity with the same order of magnitude as vertical stress does. We develop a weak-anisotropy approximation that directly links commonly measured anisotropic Thomsen parameters to the principal stresses. Each Thomsen parameter is simply a sum of corresponding background intrinsic anisotropy and stress-induced contribution. The stress-induced part is controlled by the difference between horizontal and vertical stresses and coefficients depending on nonlinear constants. Thus, isotropic rock stays isotropic under varying but hydrostatic load, whereas transversely isotropic rock retains the same values of dimensionless Thomsen parameters. Only unequal horizontal and vertical stresses alter anisotropy. Since Thomsen parameters conveniently describe seismic signatures, such as normal-moveout velocities and amplitude-variation-with-offset gradients, this approximation is suitable for designing new methods for the estimation of 3D subsurface stress from multicomponent seismic data.