Semi-analytical response of acoustic logging measurements in frequency domain

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 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

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.V. Darrigrand, A. Rodriguez-Rozas and D. Pardo were partially funded by the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824-P, MTM2016-76329-R (AEI/FEDER, EU), MTM2016-81697-ERC and the Basque Government Consolidated Research Group Grant IT649- 13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. A. Rodriguez-Rozas and D.Pardo were also partially funded by the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323 and the Basque Government through the BERC2014-2017 program. A. Rodriguez-Rozas acknowledges support from Spanish Ministry under Grant No. FPDI- 2013-17098. I. Muga was partially funded by the FONDECYT project 1160774. The first four authors were also partially funded by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 644202. Serge Prudhomme is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada

Análise computacional de ondas dispersivas em poços preenchidos por fluídos

Orientador: Odilon Divino Damasceno Couto JuniorDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb WataghinResumo: Neste trabalho, usamos o Método dos Elementos Finitos (MEF) para estudar a propagação de modos acústicos em um poço infinito preenchido por fluido e cercado por um sólido homogêneo e isotrópico. Primeiramente, usando uma aproximação bidimensional (2D) no plano transversal do poço, um modelo de análise modal foi proposto. Discutimos a distribuição geométrica, interpretação física e obtivemos as curvas de dispersão dos principais modos propagantes: Stoneley, flexural e quadrupolar. Os resultados foram validados por comparação direta com a solução analítica e numérica disponíveis na literatura. Após isso, analisamos o problema no domínio do tempo usando uma aproximação no plano longitudinal. Utilizamos o MEF para simular a aquisição de sinais por uma ferramenta de perfilagem composta por uma fonte de monopolo com N detectores igualmente espaçados. Os resultados foram validados aplicando processos de pós-processamento de sinais, como o Slowness Time Coherence (STC) e o Phase-based Dispersion Analysis (PBDA), para obter as curvas de dispersão dos modos. Isso nos permitiu demonstrar a equivalência de nossas análises modal e temporal. Nas duas formulações, analisamos a propagação de ondas em formações rápidas e lentas, dando atenção ao caso para o limite a baixas frequências. Demonstramos que nosso modelo pode ser efetivamente útil na interpretação de dados de baixa relação sinal-ruído e também pode ser expandida para estudar sistemas mais complexosAbstract: In this work, we use the Finite Element Method (FEM) to study the propagation of acoustic modes in an infinite fluid-filled borehole surrounded by a homogeneous and isotropic solid. First, using a two-dimensional (2D) approach in the transverse plane of the borehole, we carry out a modal analysis of the problem. We discuss the geometric distribution, physical interpretation, and obtain the dispersion curves of the main propagating modes, namely, the Stoneley, flexural, and quadrupole modes. The results are validated by direct comparison with analytical and numerical solutions available in the literature. Second, we analyze the time domain problem with a two-dimensional approximation in the borehole longitudinal plane. We use the FEM interface to simulate the signal acquisition of a logging tool composed of a monopole source with N equally spaced detectors. The results are validated by applying data post-processing methods, like the Slowness Time Coherence (STC) and Phase-Based Dispersion Analysis (PBDA), to obtain the dispersion curves of the modes. This allows us to demonstrate the equivalence of our modal and temporal analysis. In both approaches, we analyze the wave propagation in fast and slow formations, paying particular attention to the low frequency limit. We showed that our simulation model can be effectively helpful in the interpretation of low signal-to-noise experimental data and also be expanded to study more complex systemsMestradoFisica AplicadaMestre em Física135915/2018-0CNP

Goal-Oriented Adaptivity using Unconventional Error Representations

In Goal-Oriented Adaptivity (GOA), the error in a Quantity of Interest (QoI) is represented using global error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient GOA. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. While the method can be applied to a variety of problems, in this Dissertation we first focus on one-dimensional (1D) problems, including Helmholtz and steady-state convection-dominated diffusion problems. Numerical results in 1D show that for the Helmholtz problem, it is advantageous to select the Laplace operator for the alternative error representation. Specifically, the upper bounds of the new error representation are sharper than the classical ones used in both energy-norm and goal-oriented adaptive methods, especially when the dispersion (pollution) error is significant. The 1D steady-state convection-dominated diffusion problem with homogeneous Dirichlet boundary conditions exhibits a boundary layer that produces a loss of numerical stability. The new error representation based on the Laplace operator delivers sharper error upper bounds. When applied to a p-GOA, the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations. We then focus on the two- and three-dimensional (2D and 3D) Helmholtz equation. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones. When using the alternative error indicators, a naive p-adaptive process converges, whereas under the same conditions, the classical method fails and requires the use of the so-called Projection Based Interpolation (PBI) operator or some other technique to regain convergence. We also provide guidelines for finding operators delivering sharp error representation upper bounds.Basque Government Consolidated Research Group Grant IT649-1

Simulation of wireline sonic logging measurements acquired with Borehole-Eccentered tools using a high-order adaptive finite-element method

The paper introduces a high-order, adaptive finite-element method for simulation of sonic measurements acquired with borehole-eccentered logging instruments. The resulting frequency-domain based algorithm combines a Fourier series expansion in one spatial dimension with a two-dimensional high-order adaptive finite-element method (FEM), and incorporates a perfectly matched layer (PML) for truncation of the computational domain. The simulation method was verified for various model problems, including a comparison to a semi-analytical solution developed specifically for this purpose. Numerical results indicate that for a wireline sonic tool operating in a fast formation, the main propagation modes are insensitive to the distance from the center of the tool to the center of the borehole (eccentricity distance). However, new flexural modes arise with an increase in eccentricity distance. In soft formations, we identify a new dipole tool mode which arises as a result of tool eccentricity. © 2011 Elsevier Inc