An anisotropic fractured poroelastic effective medium theory

Abstract

Effective medium theories are used to describe the overall properties of a material containing cracks that are on a scale length much less than a wavelength, during the propagation of seismic waves. These theories are in general derived for an elastic medium with or without some form of fluid connection between the cracks. The theories are adequate for describing the properties of a material with low matrix porosities, such as carbonates, but provide a poor approximation once the matrix porosity has increased sufficiently to play an important role in determining the material properties of the effective medium, such as in sandstones. The behaviour of wave propagation within a poroelastic medium differs significantly from that in a purely elastic medium. To this end, a poroelastic model is required. Assuming that the saturated matrix material can be described by Biot's equations of poroelasticity, the method of smoothing can be used to develop an effective medium theory to first order in crack volume density in much the same way as Hudson theory uses the method for an elastic medium; the cracks are assumed to be on a scale much greater than the microscale that describes the porous matrix and much less than the macroscale, characterized by the wavelength of propagating waves. By this approach we derive effective poroelastic constants for a matrix permeated with ellipsoidal inclusions of a differing material, where the whole system is saturated with a single fluid.