Semi-analytical and numerical methods for computing transient waves in 2D acoustic / poroelastic stratified media

Wave propagation in a stratified fluid / porous medium is studied here using analytical and numerical methods. The semi-analytical method is based on an exact stiffness matrix method coupled with a matrix conditioning procedure, preventing the occurrence of poorly conditioned numerical systems. Special attention is paid to calculating the Fourier integrals. The numerical method is based on a high order finite-difference time-domain scheme. Mesh refinement is applied near the interfaces to discretize the slow compressional diffusive wave predicted by Biot's theory. Lastly, an immersed interface method is used to discretize the boundary conditions. The numerical benchmarks are based on realistic soil parameters and on various degrees of hydraulic contact at the fluid / porous boundary. The time evolution of the acoustic pressure and the porous velocity is plotted in the case of one and four interfaces. The excellent level of agreement found to exist between the two approaches confirms the validity of both methods, which cross-checks them and provides useful tools for future researches.Comment: Wave Motion (2012) XX

Dynamic response of a circular tunnel with imperfect surface interaction embedded in an elastic medium

The research work proposed here is part of a global project that aims at better characterizing a specific underground environment in the LSBB (Low Noise Inter-disciplinary Underground Science and Technology), situated in Rustrel, Vaucluse, France. The experimental environment under study is characterized by a system of galleries, several ones with a concrete layer. The first step of the methodology deals with setting up a forward problem to apprehend the geometry of the LSBB. \\In this paper, the 2D transient response of imperfect bonded circular lined pipeline lying in an elastic, homogeneous and infinite medium is studied. At first, the problem is solved in the frequency domain by using the wave function expansion method and imperfect interaction surface between elastic medium and tunnel is modeled as a linear spring. Wave propagation fields in tunnel and soil are expressed in terms of infinite series and stresses and displacements are given based on those series. By implying boundary conditions a linear equations system is obtained and the results of these equations lead to displacement and stress responses of the rock and tunnel.To solve the transient problem, the Laplace transform with respect to time is used which leads to system of linear equations in the Laplace domain. Durbin's numerical Laplace transform inversion method is employed to obtain dynamic responses. To exhibit a behavior of the responses, influences of the different parameters such as wall thickness of the tunnel is investigated. Hoop stresses and the displacements of the tunnel and rock are obtained due to acting load on the inner surface of the tunnel for selected parameters. In order to check the validity of the present work, we pay attention on the convergence of the results and also excellent agreement with previous result is achieved

Contribution to the modeling and the mechanical characterization of the subsoil in the LSBB environment

The present research work aims at better characterizing the specific underground environment of the LSBB (Low Noise Inter-Disciplinary Underground Science and Technology, Rustrel, France) using mechanical wave propagation information. The LSBB experimental environment is characterized by a system of cylindrical galleries, some of them presenting a concrete layer. In the global project, three steps are considered : firstly the construction of an efficient forward mechanical wave propagation model to calculate the displacement vector and stress tensor components; secondly a sensitivity analysis to extract the pertinent parameters in the configurations and models under study (viscoelastic or poroviscoelastic media with potential anisotropy); and lastly an inversion strategy to recover some of the pertinent parameters. In this proposal, the first step, under progress, is described. The work carried out is in the continuity of the work presented by Yi et al. (2016) [1] who studied the harmonic response of a cylindrical elastic tunnel, impacted by a plane compressional wave, embedded in an infinite elastic ground. The interface between the rock mass and the linen is an imperfect contact modeled with two spring parameters, Achenbach and Zhu (1989) [2]. We choose a semi-analytical approach to calculate the two-dimensional displacement and stress fields in order to get a fast tool, from the numerical point of view. The main steps of the theoretical approach are : use of the Helmholtz decomposition, solving the wave equation based on the separation method and the expansion in Bessel function series in the harmonic domain. The harmonic results are validated by comparison with Yi et al. (2016) [1] and new ones are presented. Moreover, the transient regime case obtained with the use of a Fourier transform on the time variable, is under progress

Axisymmetric wave propagation in multilayered poroelastic grounds due to a transient acoustic point source

International audienceThis paper deals with the study of axisymmetric wave propagation in various acoustic / porous stratified media coupling configurations. It presents the theoretical development of a semi-analytical method, its validation for a limit test-case half-space ground, and an extension to a realistic multilayered seabed, when spherical waves are emitted from a transient point source in water

Study of transient poroviscoelastic soil motions by semi-analytical and numerical approaches

International audienceIn this paper, the authors compare results obtained by semi-analytical and numerical approaches for the dynamic response of a poroviscoelastic soil under transient loads. The behaviour of the medium is governed by complete Biot formalism. The semi-analytical approach is based on Helmholtz decompositions and Fourier transforms, and yields exact solid and fluid displacements in the transformed domain. The numerical approach uses a C++ object oriented programming finite element–finite difference code. Both methods give concurring results. Moreover, influence of viscous coupling on the response of the ground and visualization of the compressional wave of the second kind are discusse

Transient solution for multilayered poroviscoelastic media obtained by an exact stiffness matrix formulation

International audienceThe authors propose a semi-analytical approach to studying wave propagation in multilayered poroviscoelastic grounds due to transient loads. The theoretical development is based on the exact stiffness matrix method for the Biot theory coupled with a matrix conditioning technique. It is developed in the wavenumber frequency domain after a Fourier transform on the surface space variables and the time variable. The usual methods yield a poorly conditioned numerical system. This is due in particular to the presence of mismatched exponential terms. In this article, increasing exponential terms are eliminated and only decreasing exponential terms remain. Consequently, the method can be applied to a large field of configurations without restriction concerning high frequencies, large Fourier transform parameters or large layer thicknesses. Validation and efficiency of the method are discussed. Effects of layering show that the layer impedance influence on solid and fluid displacements. Moreover, this approach can be of interest for the validation of numerical tool

Retrieval of the physical properties of an anelastic solid half space from seismic data

International audienceIn recent years, due to the rapid development of computation hard- and software, time domain full-wave inversion, which makes use of all the information in the seismograms without appealing to linearization, has become a plausible candidate for the retrieval of the physical parameters of the earth's substratum. Retrieving a large number of parameters (the usual case in a layered substratum comprising various materials, some of which are porous) at one time is a formidable task, so full-wave inversion often seeks to retrieve only a subset of these unknowns, with the remaining parameters, the priors, considered to be known and constant, or sequentially updated, during the inversion. A known prior means that its value has been obtained by other means (e.g., in situ or laboratory measurement) or simply guessed (hopefully, with a reasonable degree of confidence). The uncertainty of the value of the priors, like that of data noise, and the inadequacy of the theoretical/numerical model employed to mimick the seismic data during the inversion, is a source of retrieval error. We show, on the example of a homogeneous, isotropic, anelastic half-plane substratum configuration, characterized by five parameters: density, P and S wavespeeds and P and S quality factors, when a perfectly-adequate theoretical/numerical model is employed during the inversion and the data is free of noise, that the retrieval error can be very large for a given parameter, even when the prior uncertainty of another single parameter is very small. Furthermore, the employment of other load and response polarization data and/or multi-offset data, as well as other choices of the to-be-retrieved parameters, are shown, on specific examples, not to systematically improve(they may even reduce) the accuracy of the retrievals when the prior uncertainty is relatively-large. These findings, relative to the recovery, via an exact retrieval model processing noiseless data obtained in one of the simplest geophysical configurations, of a single parameter at a time with a single uncertain prior, raises the question of the confidence that can be placed in geophysical parameter retrievals: 1) when more than one parameters are retrieved at a time, and/or 2) when more than one prior are affected by uncertainties during a given inversion, and/or 3) when the model employed to mimick the data during the inversion is inadequate, 4) when the data is affected by noise or measurement errors, and 5) when the parameter retrieval is carried out in more realistic configurations

Propagation d'ondes dans un milieu poroélastique multicouches

Cette communication présente une approche semi-analytique permettant l'étude de la propagation d'ondes dans les sols  multicouches poroélastiques en régime transitoire. L'approche théorique est basée sur la méthode de matrice de raideur exacte adaptée à la théorie de Biot, développée dans le domaine des nombres d'ondes et couplée à une technique de conditionnement matriciel. Effectivement, les méthodes classiques peuvent fournir des systèmes matriciels mal conditionnés. La technique de conditionnement appliquée ici au cas poroélastique permet de traiter n'importe quelle configuration de problèmes, y compris ceux impliquant de hautes fréquences, de grandes valeurs de nombres d'ondes ou encore de fortes épaisseurs de couches. Différents types de milieux et d'excitations peuvent alors être étudiés. L'efficacité de la technique ainsi que l'influence de l'hétérogénéité multicouches  sont  présentées dans cette communication. Enfin, les techniques semi-analytiques développées peuvent servir de benchmark aux outils de simulation en poroélasticité

Characterization of a viscoelastic heterogeneous object with an effective model by nonlinear full waveform inversion

International audienceThe determination of equivalent viscoelastic properties of heterogeneous objects remains challenging in various scientific fields such as (geo) mechanics, geophysics or biomechanics. The present investigation addresses the issue of the identification of effective constitutive properties of a binary object by using a nonlinear and full waveform inversion scheme. The inversion process, without any regularization technique or a priori information, aims at minimizing directly the discrepancy between the full waveform responses of a bi-material viscoelastic cylindrical object and its corresponding effective homogeneous object. It involves the retrieval of five constitutive equivalent parameters. Numerical simulations are performed in a laboratory-scale two-dimensional configuration: a transient acoustic plane wave impacts the object and the diffracted fluid pressure, solid stress or velocity component fields are determined using a semi-analytical approach. Results show that the retrieval of the density and of the real parts of both the compressional and the shear wave velocities have been carried out successfully regarding the number and location of sensors, the type of sensors, the size of the searching space, the frequency range of the incident plane pressure wave, and the change in the geometric or mechanical constitution of the bi-material object. The retrieval of the imaginary parts of the wave velocities can reveal in some cases the limitations of the proposed approach

Etude numérique de la propagation des ondes mécaniques dans un milieu poreux en régime impulsionnel

The aim of this research consists in studying numerically the mechanical wave propagation in a two-phase porous continuum in the time domain. A finite element code allowing the simulation and the analysis of the mechanical behaviour is developed for this purpose.A preliminary study presents some general characters of the poroviscoelastic concept of medium, in the Biot theory as in mechanical wave propagation. An outcome, encountered in the literature, shows the necessity of complementary investigation using the whole two-phase porous continuum in the time domain.A finite element approach within the framework of the complete Biot theory is proposed. The characteristics of the numerical tool developed are specified. Particularly, the C++ objet oriented programming gives a low-sized solver organised in three interchangeable classes. Moreover, a previous semi-analytical workprovides the dispersion and attenuation relationships, as well as the exact determination of the wave celerities.Applications deal with porous soils: the time domain displacements of the solid and the fluid phases over and within the porous semi-infinite ground are obtained for two-dimensional problems.Parametric studies of the mechanical couplings are carried out. The compressional wave of the second kind is highlighted. A first approach of heterogeneous or partially saturated soils is also proposed.The study of three-dimensional cases is then considered. The important numerical size of this kind of problems requires the parallelization of the code. Tests on various supercomputers are carried out in order to measure the performance of parallel computing and lead to three-dimensional results.L'objectif de ce travail consiste à étudier numériquement la propagation des ondes mécaniques dans un milieu poreux continubiphasique en régime impulsionnel. Un code de calcul permettant la simulation et l'analyse du comportement mécanique est mis au point à cet effet. Une analyse préliminaire présente des généralités sur la notion de milieu poroviscoélastique, sur la théorie de Biot ainsi que sur la propagation des ondes mécaniques. Un bilan des résultats rencontrés dans la bibliographie montre la nécessité de mener une étude complémentaire sur le modèle complet du milieu poreux continu biphasique dans le domaine temporel.Une approche par éléments finis est proposée dans le cadre de la théorie générale de Biot. Les caractéristiques de l'outil numérique développé sont précisées. En particulier, la structure orientée objet donne un code compact et souple. Un travailsemi-analytique, préalablement effectué, s'intéresse aux phénomènes de dispersion, d'atténuation et à la détermination des vitesses de propagation des différentes ondes.Une modélisation bidimensionnelle permet d'obtenir les déplacements temporels des phases solide et fluide en surface et en profondeur d'un sol poreux semi-infini. Une étude paramétrique des couplages mécaniques est effectuée. La seconde onde de compression est mise en évidence. Une première approche de sols hétérogènes ou partiellement saturés est en outre proposée.L'étude de problèmes tridimensionnels est ensuite envisagée. La taille numérique importante de cette problématique nécessite alors la parallélisation du code de calcul. Des essais sur différents supercalculateurs sont réalisés pour mesurer la performance ducalcul parallèle et conduisent à des résultats tridimensionnels