Decomposition of a compliance tensor for fractures and transversely isotropic medium

Abstract

ABSTRACTThe purpose of this paper is to develop an analytical method to decompose an observed anisotropic compliance tensor into two transversely isotropic (TI) tensors that are associated with layers and fractures. Specifically, the fracture parameters and the TI background medium parameters are obtained from a given monoclinic compliance tensor. Here the set of parallel fractures and TI medium can be arbitrarily oriented in ; they are not constrained to vertical and horizontal directions respectively. First the summation of the two TI tensors, which represent fractures and layerings, is obtained in order to have the form of the resultant monoclinic medium. The orientation of each TI medium is represented by one Euler angle once the mirror plane normal of the monoclinic tensor is determined. This is because the mirror plane normal of the monoclinic medium is perpendicular to the rotation axes of the two TI tensors. Thus a layered medium with one set of parallel fractures is represented by nine parameters; two for rotationally symmetric fractures, five for a generic background TI medium and two Euler angles for the orientations of the structures. The decomposition problem, which is to find these nine parameters from a given monoclinic compliance tensor with thirteen parameters, is solved in the paper. Finally, the decomposition method is extended to media with three structures, namely two sets of fractures and layerings whose rotation axes lie in the same plane.