A Green's function approach to the study of effective anisotropic properties of the Barnett Shale

Abstract

ABSTRACTThe purpose of this paper is to derive the Green's function of an anisotropic elastic medium and validate it with the effective stiffness tensor of Barnett Shale. We have derived the frequency‐dependent Green's function by using the spectral theorem for matrices, thus simplifying the process of computing Green's function and obtaining analytical solutions. Evaluating the inverse of the Green–Christoffel tensor is an essential part of computing the Green's function. Based on the degeneracy of the eigenvalues of the Green–Christoffel tensor, the inverse of the Green–Christoffel tensor is expressed in the form of partial fractions. Consequently, the stiffness tensors from the measurement of core samples of Barnett Shale are used to validate the Green's function. We use the generalized singular approximation method of effective medium theory to model the effective stiffness of the core samples from microstructural properties. The generalized singular approximation method also allows us to compute the theoretical stiffness tensor of the Barnett Shale for porosity variations. The behaviour of the Green's function, which reflects the behaviour of the media, is studied in the static, low‐ and high‐frequency domains and under different physical parameters. It is observed that the variations of crack‐induced porosity produce different trends in Green's function for vertically transverse isotropic and horizontally transverse isotropic media. Thus, the variation of porosity is observed to be influential in differentiating between transverse isotropic media that have inclusions. The Green's function results presented in this paper have direct applications in the construction of synthetic seismograms for unbounded transversely isotropic media.