Finite‐difference modeling of viscoelastic materials with quality factors of arbitrary magnitude

Abstract

To minimize acoustic noise, designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties. Efficient finite‐difference modeling of viscoelastic materials is a topic of current research. To model viscoelastic materials in the time domain through finite differences efficiently, one needs to replace the time convolution, which enters in the stress–strain relations, by a set of first‐order differential equations. This procedure is equivalent to computing a rational approximation of a certain form to the frequency‐dependent complex modulus of viscoelasticity. Known schemes for computing such approximations are designed to treat materials with low attenuation, such as underground formations, but fail to produce accurate or even physically meaningful results for highly attenuative materials. We propose a novel scheme that allows one to construct, for a given frequency range, a uniformly optimal rational approximation for the most widely used model of materials with constant quality (Q‐) factors of arbitrary magnitude. We present the proof of convergence and demonstrate it on numerical finite‐difference examples. These examples also demonstrate the effective transparency of a simple tool modeled as a pipe of highly viscoelastic material. For frequency‐dependent quality factors we present a modified numerical scheme to compute a nearly optimal rational approximation of the viscoelastic modulus.