A calculus for finely layered anisotropic media

Abstract

Matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic, anisotropic, homogeneous medium which is equivalent, in the long‐wavelength limit, to a heterogeneous distribution of fine layers, each layer itself an elastic anisotropic medium. The properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group. A reverse mapping returns the material properties of the constituent. Adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers. Addition of an inverse element—that is subtraction—provides the means to remove a set of layers from an anisotropic medium; then, if the remaining layer is a stable anisotropic medium, a valid decomposition of the original medium into anisotropic constituents is obtained. Within the group structure, eight subgroups corresponding to eight types of elastic symmetry systems may be identified, immediately yielding the symmetry of the equivalent medium, given the symmetry of its constituents. Sets of parallel fractures or aligned microcracks are also represented as group elements, allowing fractures and anisotropic rocks to be manipulated in a consistent and uniform manner. These group elements depend on at most six fracture parameters and are independent of the properties of the material in which the fractures are embedded. Multiple sets of fractures are easily taken into account by rotating (back in model space) a rock to a coordinate system appropriate to each fracture system, and then adding the appropriate fracture system group element.