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.

Multi-Attribute Seismic/Rock Physics Approach to Characterizing Fractured Reservoirs

“…Both approaches allow for different angular distributions of fractures. has shown that to first order the slip-interface model of Schoenberg and Muir (1989) is mathematically equivalent to that of . In other words, for any distribution of Schoenberg-Muir fractures, we can find some distribution of Hudson penny shaped cracks that gives the same anisotropic effective elastic medium, and vice versa.…”

“…The second modeling approach, popularized by Schoenberg and Muir (1989), represents fractures as planes of discontinuous slip or very thin planar zones that are more compliant than the unfractured rock. The planes extend infinitely far.…”

Quasi‐static analysis of elastic behavior for some systems having higher fracture densities

“…Achieving this goal will require somewhat delicate analysis, and so we are careful not to attempt to push it beyond the realm where it is obviously still valid. To implement the quasi-static averaging method, we make use of the early work of Backus [3], as well as more recent work of Schoenberg and Muir [4]. (Also see Milton [22].)…”

“…One goal of the present work is to develop such analytical methods. To accomplish these goals, generalizations of the Backus [3] and Schoenberg and Muir [4] approaches are used to analyze layered systems, whose layers are intrinsically anisotropic due to any physical and/or mechanical mechanism. The pertinent mechanism considered here is due entirely to locally aligned fractures.…”

Theoretical Methods for Wave Propagation across Jointed Rock Masses

“…The negative aspects of this approach (White 1983;Schoenberg and Muir 1989;Pyrak-Nolte et al 1990b;Zhao J et al 2006;Zhao XB et al 2006a, b) are due to the simplification of the discontinuous rock mass to an equivalent medium. This approach cannot work well in representing media where the fractures are relatively large and sparsely spaced (with spacing of the order of, or larger than, a seismic wavelength) (PyrakNolte et al 1990b).…”