An analytical model for stress-induced anisotropy of a cracked solid

Abstract

One of the main causes of azimuthal anisotropy in sedimentary rocks is anisotropy of tectonic stresses in the earth's crust. In this paper we analytically derive the pattern of seismic anisotropy caused by application of a small anisotropic stress. We first consider an isotropic 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). 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 small compressive stress (isotropic or anisotropic), the density of cracks along a particular plane is reduced in proportion to the normal stress traction acting on that plane.In particular, if the stress is a uniaxial compression along the x axis, then the density of cracks normal to x axis will reduce most, while the density of cracks parallel to x axis will not reduce at all. This effect is modelled using Sayers-Kachanov (1995) non-interactive approximation. The results of this derivation show that such anisotropic crack closure yields elliptical anisotropy, regardless of the value of the compliance ratio B. It also predicts the ratio of anisotropy parameters ε / γ as function of the compliance ratio B and Poisson's ratio of the unstressed rock. These results are useful for differentiating stress-induced anisotropy from fracture-induced anisotropy. Conversely, if the cause of anisotropy is known, then the anisotropy pattern allows one to estimate P-wave anisotropy from S-wave anisotropy.