Unconsolidated Granular Media Research Articles

Terraced slope metasurface in granular media

SUMMARY In this work, the propagation and attenuation of vertically polarized surface waves when interacting with terraced slopes is studied experimentally and numerically. To validate the devised simulation, a laboratory-scale physical model is tested in order to examine the attenuation properties of this well-known artificial landform. The experiment involves formation of a terraced slope, in a laboratory setup, via use of an unconsolidated granular medium made of silica microbeads. This granular medium exhibits a gravity-induced power-law stiffness profile, resulting in a depth-dependent velocity profile. A piezoelectric actuator was used to excite vertically polarized surface acoustic modes localized near the surface of the medium. The three components of the particle velocity field of these modes were measured by means of a 3-D laser Doppler vibrometer. In accordance with the terraced slope, a simple inclined plane was further tested to investigate and highlight the differences in terms of wave propagation along these two different ground formations. The results of this research provide significant experimental evidence that the terraced slopes form mechanisms which attenuate low-frequency surface waves, thus acting as metasurfaces. This work suggests the use of laboratory-scale physical models to investigate the wave propagation in different landforms, which extend beyond typical horizontal ground morphologies, and which could be linked to atypical wave propagation properties, possibly even influencing propagation of seismic waves.

Geometric and material attenuation of surface acoustic modes in granular media

SUMMARYGranular materials can be used in laboratory-scale physical models to simulate and study seismic wave propagation in various unconsolidated, porous heterogeneous media. This is due to the diverse available grain configurations, in terms of their shape, size and mechanical parameters, which enable the physical and geological modelling of various complex substrates. In this work, an unconsolidated granular medium, made of silica microbeads, featuring a gravity-induced power-law stiffness profile is experimentally tested in a laboratory setting. The objective is to investigate the attenuation mechanisms of vertically polarized seismic waves traveling at the surface of unconsolidated substrates that are characterized by power-law rigidity profiles. Both geometric spreading and material damping due to skeletal dissipation are considered. The understanding of these two attenuation mechanisms is crucial in seismology for properly determining the seismic site response. An electromagnetic shaker is employed to excite the granular medium between 300 and 550 Hz, generating linear modes that are localized near the surface. A densely sampled section is recorded at the surface using a laser vibrometer. The explicit solution of the geometric attenuation law of Rayleigh-like waves in layered media is employed to calculate the geometric spreading function of the vertically polarized surface modes within the granular material. In accordance with recent studies, the dynamics of these small-amplitude multimodal linear waves can be analysed by considering the granular medium as perfectly continuous and elastic. By performing a nonlinear regression analysis on particle displacements, extracted from experimental velocity data, we determine the frequency-dependent attenuation coefficients, which account for the material damping. The findings of this work show that laboratory-scale physical models can be used to study the geometric spreading of vertically polarized seismic waves induced by the soil inhomogeneity and characterize the material damping of the medium.

Unraveling the nonlinear elastic response of materials with dynamic acousto-elastic testing

Compared to standard nonlinear ultrasonic techniques based on harmonic generation or resonance tests that provide an average nonlinearity level, Dynamic Acousto-Elastic Testing (DAET) is a pump-probe approach that allows one to track the nonlinear elastic response at each phase of the dynamic cycle, and to quantify additional features such as transient softening/weakening and hysteresis. These so-called nonlinear signatures can help us unravel the origins of the nonlinear response at the micro/mesoscopic scale. In this talk, we first review some of the DAET results obtained by our group and others in a wide range of materials from cracked metals to rocks and unconsolidated granular media. Next, we present some of our most recent results where we investigate the effect of relative humidity (RH) on the nonlinear elastic response of granular media. We use DAET on samples of glass beads under dry (∼10%), ambient (∼60%) and humid (∼100%) conditions. We find that all extracted nonlinear parameters in humid samples increases by an order of magnitude compared to those in dry samples. This is consistent with the idea that water adsorption on the grains makes the contact junctions weaker. This has great relevance when trying to quantify the nonlinearity of porous materials.

Mitigation of Rayleigh-like waves in granular media via multi-layer resonant metabarriers

In this work, we experimentally and numerically investigate the propagation and attenuation of vertically polarized surface waves in an unconsolidated granular medium equipped with small-scale metabarriers of different depths, i.e., arrays composed of one, two, and three embedded layers of sub-wavelength resonators. Our findings reveal how such a multi-layer arrangement strongly affects the attenuation of the surface wave motion within and after the barrier. When the surface waves collide with the barriers, the wavefront is back-scattered and steered downward underneath the oscillators. Due to the stiffness gradient of the granular medium, part of the wavefield is then rerouted to the surface level after overcoming the resonant array. Overall, the in-depth insertion of additional layers of resonators leads to a greater and broader band wave attenuation when compared to the single layer case.

Locally Resonant Metasurfaces for Shear Waves in Granular Media

In this paper, the physics of horizontally polarized shear waves traveling across a locally resonant metasurface in an unconsolidated granular medium is experimentally and numerically explored. The metasurface is comprised of an arrangement of subwavelength horizontal mechanical resonators embedded in a granular layer made of silica microbeads. The metasurface supports a frequency-tailorable attenuation zone induced by the translational mode of the resonators. The experimental and numerical findings reveal that the metasurface not only backscatters part of the energy but also redirects the wave front underneath the resonators, leading to a considerable amplitude attenuation at the surface level, when all the resonators have similar resonant frequency. A more complex picture emerges when using resonators arranged in a so-called graded design, e.g., with a resonant frequency increasing or decreasing throughout the metasurface. Unlike the mechanism observed in a bilayered medium, shear waves localized at the surface of the granular material are not converted into bulk waves. Although a detachment from the surface occurs, the depth-dependent velocity profile of the granular medium prevents the mode conversion and the horizontally polarized shear wave front returns to the surface. The outcomes of our experimental and numerical studies allow for understanding the dynamics of wave propagation in resonant metamaterials embedded in vertically inhomogeneous soils and, therefore, may be valuable for improving the design of engineered devices for ground-vibration and seismic wave containment.

Hybridization of Guided Surface Acoustic Modes in Unconsolidated Granular Media by a Resonant Metasurface

Elastic metasurfaces can manipulate the propagation of surface acoustic waves (SAWs) for applications as wave filters, including barriers for seismic surface waves---earthquake protection, that is. Efforts so far have focused on homogenous media, in crude approximation of the real world. The authors study the dynamics of an elastic metasurface in an unconsolidated granular medium, which presents an inhomogeneous stiffness profile. The metasurface's resonance hybridizes the lowest-order SAW and down-converts all higher-order SAWs, preventing the surface-wave delocalization seen in homogeneous media. These findings could impact protective designs for inhomogeneous stratified soils.

Large scale metasurfaces for seismic waves control

Elastic metamaterials are artificial composites with subwavelength resonant particles hosted in a medium able to manipulate the propagation of elastic waves. When the resonant particles are placed at the free surface of the medium to form a resonant “metasurface,” the localization mechanism and the direction of surface waves can be fully controlled. In this talk, we discuss the use of resonant metasurfaces to control the propagation of vertically and horizontally polarized surface waves and their possible application for seismic waves mitigation. By combining analytical, numerical, and experimental studies, we describe the interaction of Rayleigh waves with a metasurface of vertical resonators and design large-scale resonant barriers to deviate damaging seismic Rayleigh waves into the medium bulk. Additionally, we investigate the effect of material stratification on the metasurface dynamics by analyzing the propagation of surface waves in unconsolidated granular media with depth-dependent stiffness profile. Finally, we describe the interaction of Love waves guided by a stratified medium with a metasurface of horizontal resonators and design large-scale resonant metalenses to redirect their propagation.

Anisotropic surface-wave characterization of granular media

In unconsolidated granular media, the state of stress has a major effect on the elastic properties and wave velocities. Effective media modeling of granular packs shows that a vertically oriented uniaxial state of stress induces vertical transverse isotropy (VTI), characterized by five independent elastic parameters. The Rayleigh wave velocity is sensitive to four of these five VTI parameters. We have developed an anisotropic Rayleigh wave inversion that solves for four of the five independent VTI elastic parameters using anisotropic Thomson-Haskell matrix equations that can be used to determine the degree of anisotropy and the state of stress in granular media. The proposed method incorporates an initial model search guided by rock-physics models and dispersion curve inversion that uses a gradient-based solver. We carried out the inversion on a synthetic example and a real data set in which we inverted for in situ elastic moduli of unconsolidated sands at very low effective stress. The synthetic data example results showed that we are able to correctly estimate the granular media’s effective elastic moduli. When the inversion was carried out with real data, the results indicate that the sand at very low effective stress is elastically anisotropic and that it is stiffer along the vertical axis than the horizontal axis. Furthermore, our results are in agreement with past studies related to granular media.

Nonlinear coda wave interferometry: Characterizing damage in complex solids

In this talk, we report results on the nonlinear interactions of ultrasonic coda waves, in reverberating or multiple scattering mesoscopic solid media. Using the method of coda wave interferometry (CWI), we analyze the effect of mixing a coda wave with an additional lower frequency pump wave. The extracted CWI parameters, known to be highly sensitive to small geometric or elastic modifications of the tested medium, are shown to be pump-amplitude dependent and to capture finely the results of the nonlinear interactions. Although nonlinear self-action effects with coda waves have been reported in unconsolidated granular media, they are difficult to implement in cracked solids or concrete. Instead, the reported nonlinear CWI class of methods (NCWI) shows robustness, a high sensitivity, and has been applied successfully to various complex media and structures. We show through several examples on « model » media (cracked glass plates) and on concrete structures, that NCWI can be useful for the nondestructive evaluation of complex solids that are strongly scattering at usual probing frequencies. Preliminary results and prospects in nonlinear elastic properties imaging and quantitative evaluation with NCWI are discussed.

Nonlinear Coda Wave Interferometry: Detecting, quantifying, and locating damage in complex solids

In this talk, we report results on the nonlinear interactions of ultrasonic coda waves with lower pump waves, in reverberating or multiple scattering mesoscopic solid media. Using the method of coda wave interferometry (CWI), we analyze the effect of mixing a coda wave with an additional lower frequency pump wave. The extracted CWI parameters, known to be highly sensitive to small geometric or elastic modifications of the tested medium, are shown to be pump-amplitude dependent and to capture finely the results of the nonlinear interactions. Although nonlinear self-action effects with coda waves have been reported in unconsolidated granular media, they are difficult to implement in cracked solids or concrete. Instead, the reported nonlinear CWI class of methods (NCWI) shows robustness, a high sensitivity, and has been applied successfully to various complex media and structures. We show through several examples on « model » media (cracked glass plates) and on concrete structures, that NCWI can be useful for the nondestructive evaluation of complex solids that are strongly scattering at usual probing frequencies. Preliminary results and prospects in nonlinear elastic properties imaging and quantitative evaluation with NCWI are discussed.

Pore Space Connectivity and the Transport Properties of Rocks

Pore connectivity is likely one of the most important factors affecting the permeability of reservoir rocks. Furthermore, connectivity effects are not restricted to materials approaching a percolation transition but can continuously and gradually occur in rocks undergoing geological processes such as mechanical and chemical diagenesis. In this study, we compiled sets of published measurements of porosity, permeability and formation factor, performed in samples of unconsolidated granular aggregates, in which connectivity does not change, and in two other materials, sintered glass beads and Fontainebleau sandstone, in which connectivity does change. We compared these data to the predictions of a Kozeny-Carman model of permeability, which does not account for variations in connectivity, and to those of Bernabe et al . (2010, 2011) model, which does [Bernabe Y., Li M., Maineult A. (2010) Permeability and pore connectivity: a new model based on network simulations, J. Geophys. Res. 115 , B10203; Bernabe Y., Zamora M., Li M., Maineult A., Tang Y.B. (2011) Pore connectivity, permeability and electrical formation factor: a new model and comparison to experimental data, J. Geophys. Res. 116 , B11204]. Both models agreed equally well with experimental data obtained in unconsolidated granular media. But, in the other materials, especially in the low porosity samples that had undergone the greatest amount of sintering or diagenesis, only Bernabe et al. model matched the experimental data satisfactorily. In comparison, predictions of the Kozeny-Carman model differed by orders of magnitude. The advantage of the Bernabe et al. model was its ability to account for a continuous, gradual reduction in pore connectivity during sintering or diagenesis. Although we can only speculate at this juncture about the mechanisms responsible for the connectivity reduction, we propose two possible mechanisms, likely to be active at different stages of sintering and diagenesis, and thus allowing the gradual evolution observed experimentally.

Experimental study of nonlinear ultrasonic behavior of soil materials during the compaction

In this paper, the nonlinear ultrasonic behavior of unconsolidated granular medium – soil during the compaction is experimentally studied. The second harmonic generation technique is adopted to investigate the change of microstructural void in materials during the compaction process of loose soils. The nonlinear parameter is measured with the change of two important environmental factors i.e. moisture content and impact energy of compaction. It is found the nonlinear parameter of soil material presents a similar variation pattern with the void ratio of soil samples, corresponding to the increased moisture content and impact energy. A same optimum moisture content is found by observing the variation of nonlinear parameter and void ratio with respect to moisture content. The results indicate that the unconsolidated soil is manipulated by a strong material nonlinearity during the compaction procedure. The developed experimental technique based on the second harmonic generation could be a fast and convenient testing method for the determination of optimum moisture content of soil materials, which is very useful for the better compaction effect of filled embankment for civil infrastructures in-situ.

Asymmetric Acoustic Propagation of Wave Packets Via the Self-Demodulation Effect.

This Letter presents the experimental characterization of nonreciprocal elastic wave transmission in a single-mode elastic waveguide. This asymmetric system is obtained by coupling a selection layer with a conversion layer: the selection component is provided by a phononic crystal, while the conversion is achieved by a nonlinear self-demodulation effect in a 3D unconsolidated granular medium. A quantitative experimental study of this acoustic rectifier indicates a high rectifying ratio, up to 10^{6}, with wide band (10kHz) and an audible effect. Moreover, this system allows for wave-packet rectification and extends the future applications of asymmetric systems.

A fractional calculus approach to the propagation of waves in an unconsolidated granular medium

Our study builds on the work of Buckingham [JASA (2000)] which employed a grain-shearing (GS) model to describe the propagation of elastic waves in saturated, unconsolidated granular materials. He ensemble averages the random stick-slip process which follows the velocity gradient set up by the wave. The stick-slip process is due to the presence of micro-asperities between the contact surfaces of the grains. This is a strain hardening process which is represented by a time-dependent coefficient in the Maxwell element, besides the coefficient also gives the order of the loss term in the wave equations. We find that the material impulse response derived from the GS model is similar to the power-law memory kernel of fractional calculus. The GS model then gives two equations; a fractional Kelvin-Voigt wave equation for the compressional wave and a fractional diffusion equation for the shear wave. These equations have already been analyzed extensively in the framework of fractional calculus. Since the Kelvin-Voigt model is used in biomechanics of living tissue, we believe the GS theory could offer insights into ultrasound and elastography as well. The overall goal is to understand the role of different material parameters which affect wave propagation.

Analysis of laboratory data on ultrasonic monitoring of permeability reduction due to biopolymer formation in unconsolidated granular media

We show how to estimate the fluid permeability changes due to accumulated biopolymer within the pore space of a granular material using laboratory measurements of overall permeability, together with various well-known quantitative measures (e.g., porosity, specific surface area, and formation factor) of the granular medium microstructure. The main focus of the paper is on mutual validation of existing theory and a synthesis of new experimental results. We find that the theory and data are in good agreement within normal experimental uncertainties. We also establish quantitative empirical relationships between seismic and/or acoustic attenuation and overall permeability for these same systems.

Physical modelling of a surface-wave survey over a laterally varying granular medium with property contrasts and velocity gradients

Laboratory experiments using laser-based ultrasonic techniques can be used to simulate seismic surveys on highly controlled small-scale physical models of the subsurface. Most of the time, such models consist in assemblies of homogeneous and consolidated materials. To enable the physical modelling of unconsolidated, heterogeneous and porous media, the use of granular materials is suggested here. We describe a simple technique to build a two-layer physical model characterized by lateral variations, strong property contrasts and velocity gradients. We use this model to address the efficiency of an innovative surface-wave processing technique developed to retrieve 2-D structures from a limited number of receivers. A step by step inversion procedure of the extracted dispersion curves yields accurate results so that the 2-D structure of the physical model is satisfactorily reconstructed. The velocity gradients within each layer are accurately retrieved as well, confirming current theoretical and experimental studies regarding guided surface acoustic modes in unconsolidated granular media.

A new method for acoustic characterization of particulate materials: preliminary results

We describe a new method of measuring acoustic properties of particulate (unconsolidated) materials and present preliminary results of its application to studying artificial granular media. The results demonstrate promising prospects for the new method. Earlier unknown features of a dynamic response of unconsolidated granular medium are revealed. Experimental observations of slow relaxation in such media are performed.

Nonlinear elastic properties of a model one-dimensional granular unconsolidated structure

The characteristic features of the propagation of finite-amplitude elastic waves in a model one-dimensional unconsolidated granular medium are investigated. The model medium is represented by a linear chain of 80 steel balls with a diameter of 6.5 mm each, this chain being preliminarily loaded with an external static force F. The elastic properties of the model are analyzed. The theoretical dependences of the coefficients of elasticity of the second, third, and fourth orders on the force F are obtained. The experimental setup is described. The results of studying the nonlinear effects, namely, the higher harmonic generation and the wave generation at combination frequencies, which accompany the acoustic wave propagation in the chain, are presented. For the chain of balls under study, a structural phase transition from the 1D structure to a 2D one is observed with an increase in the external compression force F applied to the balls. The results of the study are analyzed using the Hertz theory of contact interactions.

Rock physics modeling of static and dynamic reservoir properties—a heuristic approach for cemented sandstone reservoirs

Rock physics models for fluid and stress dependency in reservoir rocks are essential for quantification and interpretation of 4D seismic signatures during reservoir depletion and injection. However, our ability to predict the sensitivity to pressure from first principles is poor. The current state-of-the-art requires that we calibrate the pressure dependence of velocity with core measurements. A major challenge is the fact that consolidated rocks often break up during coring, and hence the stress sensitivity is likely to be overpredicted in the laboratory relative to the in-situ conditions (Furre et al., 2009). For unconsolidated sands, acquisition of core samples is not very feasible due to the friable nature of the sediments. One physical model that has been applied to predict pressure sensitivity in unconsolidated granular media is the Hertz-Mindlin contact theory. Several authors (Vernik and Hamman, 2009, among others) have suggested empirical models with fitting parameters that correlate with microcrack intensity, soft porosity, and aspect ratio of the rock, and feasibility studies can be undertaken based on assumptions about these parameters. These models may not be easy to use for poorly to moderately consolidated sandstones with contact cement, where crack parameters and aspect ratios are difficult to quantify.

Elasticity profile of an unconsolidated granular medium inferred from guided waves: Toward acoustic monitoring of analogue models

The monitoring of materials elastic parameters during geological analogue experiments would highly improve the interpretation of involved processes. It would more particularly help the implementation of joint analytical or numerical analyses. We propose laser-Doppler vibrometry as a probing tool for the systematic characterisation of analogue models. We therefore develop an experimental set-up to record small-scale seismic lines at the surface of an unconsolidated granular medium in laboratory conditions. Pressure-wave first arrival times and surface-wave dispersion are then inverted for a one-dimensional Pressure- and Shear-wave velocity structure. Inferred profiles appear to match previously thoroughly estimated properties of the probed medium, thus validating our approach.