Visco-Elastic Waves in Rock-Like Composites

Abstract

The subject of this paper is the treatment of rocks - and, especially, fluid-saturated and partially saturated reservoir rocks, as composite visco-elastic media. By this we mean to study and partially answer the question of how the effective material (frequency-dependent and complex-valued stiffness/density) parameters can be estimated from a knowledge of the constituents of the rocks, their volume fractions, the statistical distribution of sizes, shapes, orientations and positions of the individual particles (minerals of quartz, clay, etc.) and cavities (pores, cracks, etc.); in addition to parameters related to the fluid and its ability to flow, at the scale of the microstructure as well as that of the wavelength (assumed to be long compared to the scale-size of the microstructure). Our approach is to develop and combine a theory of stochastic waves with established results for the micromechanics of defects in solids, as well as state-of-the-art models of wave-induced fluid flow. Specifically, we first derive an exact formal expression for the effective material parameters in terms of a dynamic T-matrix for the material, which satisfies a single integral equation of the Lippmann-Schwinger type (known from quantum scattering theory), but formulated in an abstract vector space, associated with the combination of the strain and velocity fields into a more general state vector Ψ. Inclusions-based models are developed on the basis of standard many-body techniques, known from the static T-matrix approach as well as nuclear collision theory. The t-matrix of a low-aspect-ratio spheroidal crack is expressed in terms of the familiar displacement discontinuity parameters of Hudson, via the so-called K-tensor, which is of interest in itself, for example, when connecting cracks to pores (in the presence of multiple solid constituents) on the basis of an expression for the t-matrix of a communicating cavity. The present theory can in principle be used beyond the Rayleigh limit, but explicit estimates of the effective material parameters have so far been derived only under the assumption that (scattering attenuation can be ignored) the wavelength is large compared to the scale-size of a representative volume element. Starting with the dynamic equations of motion, we show that the behaviour of the mean wave in the Rayleigh limit is indeed determined by the effective stiffness tensor associated with a static theory of composites, in conjunction with the spatially averaged density for the heterogeneous material as a whole. Thus, we have provided justification to the procedure we used in a series of related papers, where we started out with the static equilibrium condition and employed the elastic/visco-elastic correspondence principle. Numerical examples (dealing with the effects of randomly oriented cracks on the isotropic velocity and attenuation spectra of a dual porosity model of clay-sand mixtures, and the effects of spatial distribution on the anisotropic attenuation spectra of fully aligned cracks that are partially saturated with two different fluids) will be provided in order to complement those in our earlier papers.