Abstract

Different boundary conditions applied to finite-size, heterogeneous rocks result in different average or apparent elastic stiffnesses. In statics, the apparent stiffness tensors that correspond to the homogeneous stress and homogeneous strain boundary conditions provide the lower and upper bounds for the true or effective stiffness tensor. It appears that similar bounds cannot be established for experiments utilizing wave propagation. We present static and dynamic computations of the apparent properties of stacks of horizontal isotropic layers. The effective stiffness tensors of such media are known to be vertically transversely isotropic (VTI) and are given by Backus theory for statics in the limit of zero thicknesses of the constituent layers. We verify the theoretical static bounds for these effective stiffnesses with the finite-element method. The existing stiffness bounds, however, do not lead to bounds for the effective Thomsen anisotropic coefficients. In particular, we show that any static measurement on rocks composed of a finite number of isotropic layers underestimates Thomsen coefficients ϵ and γ and is likely to overestimate δ. The apparent stiffnesses of a stack of layers derived from wave-propagation experiments can deviate from Backus theory and depend on details of the wave signature used to estimate them. Our numerical tests indicate that the apparent stiffnesses do not converge to their effective, Backus-predicted values when the ratio of layer thicknesses to the minimum wavelength goes to zero; however, the differences are usually insignificant for practice.

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