Tube wave signatures in cylindrically layered poroelastic media computed with spectral method

Abstract

SUMMARY This paper describes a new algorithm based on the spectral method for the computation of Stoneley wave dispersion and attenuation propagating in cylindrical structures composed of fluid, elastic and poroelastic layers. The spectral method is a numerical method which requires discretizationofthestructurealongtheradialaxisusingChebyshevpoints.Toapproximatethe differential operators of the underlying differential equations, we use spectral differentiation matrices. After discretizing equations of motion along the radial direction, we can solve the problem as a generalized algebraic eigenvalue problem. For a given frequency, calculated eigenvaluescorrespondtothewavenumbersofdifferentmodes.Theadvantageofthisapproach is that it can very efficiently analyse structures with complicated radial layering composed of different fluid, solid and poroelastic layers. This work summarizes the fundamental equations, followed by an outline of how they are implemented in the numerical spectral schema. The interface boundary conditions are then explained for fluid/porous, elastic/porous and porous interfaces. Finally, we discuss three examples from borehole acoustics. The first model is a fluid-filledboreholesurroundedbyaporoelasticformation.Thesecondconsidersanadditional elastic layer sandwiched between the borehole and the formation, and finally a model with radially increasing permeability is considered.