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Modeling elastic wave velocities in porous media : frequency-dependent effects of heterogeneity at the pore- and patch-scale Taylor, Richard

Abstract

The prediction of relationships between elastic wave velocities in a porous medium and the properties of the fluid and solid constituents therein is a longstanding problem in geophysical exploration. Previous authors have shown that such relationships depend on both the wave frequency and degree of heterogeneity present. This frequency dependence arises via the state of relaxation of the pore fluids. At sufficiently low frequencies, deformation-induced flow leads to equilibration of the fluid pressure, and the pore fluids are said to be in a "relaxed" state. At sufficiently high frequencies, there is insufficient time for equilibration to occur, and the fluids are said to be "unrelaxed". Current models of elastic wave velocities in porous media are, for the most part, confined to either the relaxed limit (e.g., the poroelastic Biot-Gassmann theory) or the unrelaxed limit (e.g., inclusion-based effective medium theory). In this thesis we incorporate an explicit description of the relaxation mechanism into inclusion-based effective medium theory, so as to extend the theory toward the relaxed limit. Analysis of the mechanisms of relaxation leads to a description of the effective elastic behavior of the porous medium in terms of effective complex elastic moduli of the medium's constituents. Previous authors have identified two distinct scales of fluid distribution heterogeneity: the pore scale and the patch scale. Accordingly, we treat these scales separately, describing relaxation in terms of Poiseuille flow at the pore scale, and Darcy's law at the patch scale. The results of our analyses are effective medium theories that provide a consistent approach to the prediction of elastic wave velocities, as well as attenuation due to the relaxation mechanism, over a broad range of frequencies and length scales of heterogeneity. In particular, our model is applicable to the regime where the pore fluids are in a state of relaxation intermediate between the completely relaxed and completely unrelaxed end members.

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