Semi-analytical response of acoustic logging measurements in frequency domain

Abstract

This work proposes a semi-analytical method for simulation of the acoustic response of multipole eccentered sources in a fluid-filled borehole. Assuming a geometry that is invariant with respect to the azimuthal and vertical directions, the solution in frequency domain is expressed in terms of a Fourier series and a Fourier integral. The proposed semi-analytical method builds upon the idea of separating singularities from the smooth part of the integrand when performing the inverse Fourier transform. The singular part is treated analytically using existing inversion formulae, while the regular Email addresses: ignacio.muga@ucv.cl (Ignacio Muga), dzubiaur@gmail.com (David Pardo), pawel@ices.utexas.edu (Pawe l J. Matuszyk), cverdin@uts.cc.utexas.edu (Carlos Torres-Verdín) part is treated with a FFT technique. As a result, a simple and effective method that can be used for simulating and understanding the main physical principles occurring in borehole-eccentered sonic measurements is obtained. Numerical results verify the proposed method and illustrate its advantages.

Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing

“…In Section 3, we have obtained a description of the solution of problem (10) by means of the Fourier modes (8). Now, we are interested in the application of the inverse Fourier transform (9) to recover the solution of (10) through those Fourier modes. The main drawback is that the Fourier mode û0 behaves singularly at ξ = 0, which complicates standard numerical integration.…”

“…The main drawback is that the Fourier mode û0 behaves singularly at ξ = 0, which complicates standard numerical integration. To deal with this, we apply a classical idea from previous papers (see, for similar techniques, [9,4,3]), which consists in characterizing the singular part of û0 (denoted hereafter by ûsing 0 ), and remove it from û0 in order to define a regular part ûreg 0 := û0 − ûsing 0 . This regular part admits standard numerical Fourier inversion techniques; while the singular part is treated using analytical Fourier inversion formulae.…”

“…This regular part admits standard numerical Fourier inversion techniques; while the singular part is treated using analytical Fourier inversion formulae. The global Fourier inversion is reconstructed using linearity of the integral transform in (9).…”

“…Several works can be found in the literature regarding the derivation of analytical or semianalytical solutions in cylindrical domains. For example, [9] derives semi-analytical solutions for an elasto-acoustic problem set in a borehole involving solid and fluid subdomains. In the same way, [15] derives the Green's function for the Helmholtz equation set in an infinite cylinder with Robin boundary conditions.…”

Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids

“…But they are feasible if additional assumptions are made on the resistivity distribution (see e.g. [26] or [19]). For simplicity of computations, we focus on the case where the resistivity distribution is invariant along one space direction, say y.…”