Biot's Dynamic Theory Research Articles

Staggered finite volume algorithms for dynamic analysis of saturated soil with low permeability and incompressible pore fluid by using OpenFOAM

Biot dynamic theory forms the basis of dynamic analysis for soil-pore fluid coupling, whose formulations can be simplified by u-p approximation for low frequency phenomena without loss of accuracy. In the past decades, fully-coupled approach and segregated method were two common strategies for Finite Volume (FV) solutions to Biot dynamic theory. However, both methods encounter problems that limit their applicability when soil permeability and pore fluid compressibility reduce to a relatively low level. Under this circumstance, fully-coupled approach requires mixed formulation to satisfy Babuška-Brezzi condition, which complicates numerical implementation. On the other hand, segregated method suffers from divergence and instability, and shows poor performance in computation efficiency. In this article, two staggered FV algorithms incorporating operator splitting technique are devoted to overcoming the aforementioned shortcomings, and they are implemented on the open-source, broadly used and versatile platform for FV calculation, OpenFOAM. Through comparing the computed results of the two algorithms with solutions in the literature, both of the proposed algorithms are shown to reasonably reflect the dynamic responses of porous media. By conducting stability analyses based on the numerical cases, the two algorithms are shown to enlarge the stability region and enable larger time steps for computation.

Dynamic behaviour of layered transversely isotropic poroelastic media subjected to rectangular harmonic loads

AbstractThis paper aims to investigate the dynamic behaviour of layered transversely isotropic (TI) poroelastic media subjected to rectangular harmonic loads. First, governing equations of a medium layer derived from Biot's dynamic theory of porous media and constitutive equations of TI materials are converted into the ordinary differential domain by means of integral transform techniques. Next, by introducing boundary and load conditions, the extended precise integration method (PIM) is applied to obtain the solution for a multi‐layered system. Then, after an inverse integral treatment, the numerical results in the physical domain are verified with published work, and some examples are designed to unravel the effect of transverse isotropy, load frequencies and stratification on the dynamic behaviour of media.

Sound insulation properties in low-density, mechanically strong and ductile nanoporous polyurea aerogels

Aerogels are quasi-stable, nanoporous, low-density, three-dimensional assemblies of nanoparticles. In this paper, an extremely high sound transmission loss for a family of ductile polyurea aerogels (e.g., over 30dB within 1 to 4kHz at bulk density 0.25g/cm3 and 5mm thickness) is reported. The fundamental mechanisms behind the aerogel acoustic attenuations are investigated. Sharing striking similarities with acoustic metamaterials, initially, aerogels are studied via a one-dimensional multi degree-of-freedom mass-spring system. Different effects such as spring constant disparity are investigated in regards to the structural vibration wave transmission loss. Results are given for different configurations consistent with the aerogel nano/microstructures. A significant wave attenuation is observed by considering a random spring distribution. In the next step towards modeling such a complex hierarchical and random structural material, the continuum Biot's dynamic theory of poroelasticity is implemented to analyze the experimental sound transmission loss results. In this framework, a two-dimensional plane strain analysis is considered for the interaction of a steady state time-harmonic plane acoustic wave with an infinite aerogel layer immersed in and saturated with air. The effects of bulk density and thickness on the aerogel sound transmission loss are elucidated. By comparing the theoretical results with the experimental observations, this study develops a qualitative/quantitative basis for the dynamics of the aerogel nanoparticle network as well as the air flow and solid vibroacoustic interactions. This basis provides a better understanding on the overall acoustic properties of the aerogels that might also be helpful in the design of the future hierarchical materials.

Seismoelectric couplings in a poroelastic material containing two immiscible fluid phases

A new approach of seismoelectric imaging has been recently proposed to detect saturation fronts in which seismic waves are focused in the subsurface to scan its heterogeneous nature and determine saturation fronts. Such type of imaging requires however a complete modelling of the seismoelectric properties of porous media saturated by two immiscible fluid phases, one being usually electrically insulating (for instance water and oil). We combine an extension of Biot dynamic theory, valid for porous media containing two immiscible Newtonian fluids, with an extension of the electrokinetic theory based on the notion of effective volumetric charge densities dragged by the flow of each fluid phase. These effective charge densities can be related directly to the permeability and saturation of each fluid phase. The coupled partial differential equations are solved with the finite element method. We also derive analytically the transfer function connecting the macroscopic electrical field to the acceleration of the fast P wave (coseismic electrical field) and we study the influence of the water content on this coupling. We observe that the amplitude of the co-seismic electrical disturbance is very sensitive to the water content with an increase in amplitude with water saturation. We also investigate the seismoelectric conversions (interface effect) occurring at the water table. We show that the conversion response at the water table can be identifiable only when the saturation contrasts between the vadose and saturated zones are sharp enough. A relatively dry vadose zone represents the best condition to identify the water table through seismoelectric measurements. Indeed, in this case, the coseismic electrical disturbances are vanishingly small compared to the seismoelectric interface response.

Dynamic response to fluid extraction from a poroelastic half‐space

SUMMARYIn this study, the dynamic response of a poroelastic half‐space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re‐formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full‐space fundamental singular solution for a point sink is also derived using potential methods. The mirror‐image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half‐space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

AbstractThe vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 John Wiley & Sons, Ltd.

Dynamic Stress Concentrations in Lined Twin Tunnels within Fluid-Saturated Soil

An exact analysis for three-dimensional dynamic interaction of monochromatic seismic plane waves with two lined circular parallel tunnels within a boundless fluid-saturated porous elastic medium is presented. The novel features of Biot dynamic theory of poroelasticity along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to obtain a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which two identical tunnels, lined with concrete and embedded within water-saturated soils of distinct frame properties (i.e., soft or stiff soils), are insonified by plane fast compressional or shear waves at end-on incidence. The basic dynamic field quantities such as the hoop and axial stress amplitudes are evaluated and discussed for representative values of the parameters characterizing the system. The effects of formation material type, angl...

Scattering of plane wave by circular-arc alluvial valley in a poroelastic half-space

Based on the complex variable function method, a new approach for solving the scattering of plane P wave by circular-arc alluvial valley in poroelastic half-space is developed in the paper. In this analysis, the poroelastic half-space and the circular-arc valley are modeled as poroelastic medium based on Biot's dynamic theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Here, the large circle assumption is applied to simulate the boundary conditions at the half-space boundary. The stresses and pore pressures are obtained by using the boundary conditions and continuous conditions of the poroelastic half-space and the circular-arc alluvial valley. Numerical results show that the dynamic stresses concentration and pore pressures concentration are mainly relative to the wave shape of incidence, angle of incidence, dimensionless frequency of incident wave, stiffness and pore ratio of the poroelastic half-space and valley.

Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation

An exact analysis for two-dimensional dynamic interaction of monochromatic progressive plane compressional and shear seismic waves with a permeable circular tunnel lining of circumferentially varying wall thickness buried in a boundless porous elastic fluid-saturated formation is presented. The novel features of Biot dynamic theory of poroelasticity in conjunction with the translational addition theorems for cylindrical wave functions, along with the appropriate wave field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. The analytical results are illustrated with numerical examples in which an air-filled and water-saturated permeable tunnel lining of variable wall thickness, embedded within water-saturated surrounding formations of distinct frame properties (soft, stiff, and stiff viscoelastic soil), is insonified by fast compressional or shear waves at selected angles of incidence. The effects of liner eccentricity, interface permeability, formation material type, incident wave frequency, and angle of incidence on the hoop stress amplitude are evaluated and discussed for representative values of the parameters characterizing the system. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.

Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation

Acoustic radiation and the dynamic field induced by a cylindrical source of infinite extent, undergoing angularly periodic and axially-dependent harmonic surface vibrations, while eccentrically suspended in a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation, are analyzed in an exact manner. This configuration, which is a realistic idealization of an acoustic logging tool suspended in a fluid-filled borehole within a permeable surrounding formation, is of practical importance with a multitude of possible applications in seismo-acoustics. The formulation utilizes the novel features of Biot dynamic theory of poroelasticity along with the translational addition theorem for cylindrical wave functions to obtain a closed-form series solution. The basic dynamic field quantities such as the resistive and the reactive components of the modal acoustic radiation impedance load on the source in addition to the radial and transverse stresses induced in the surrounding formation by an eccentric pulsating/oscillating cylinder in a water-filled borehole within a water-saturated Ridgefield sandstone medium are evaluated and discussed. Special attention is paid to the effects of source eccentricity, excitation frequency, and mode of surface oscillations on the modal impedance values and the dynamic stresses. Limiting cases are considered and good agreements with available solutions are obtained.

Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites

Interaction of time harmonic fast longitudinal and shear incident plane waves with an elliptical fiber embedded in a porous elastic matrix is studied. The novel features of Biot dynamic theory of poroelasticity along with the classical method of eigen-function expansion and the pertinent boundary conditions are employed to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the non-orthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A MATHEMATICA code is developed for computing the Mathieu functions in terms of complex Fourier coefficients which are themselves calculated by numerically solving appropriate sets of eigen-systems. The analytical results are illustrated with numerical examples in which an elastic fiber of elliptic cross section is insonified by a plane fast compressional or shear wave at normal incidence. The effects of fiber cross sectional ellipticity, angle of incidence (fiber two-dimensional orientation), and incident wave polarization (P, SV, SH) on dynamic stress concentrations are studied in a relatively wide frequency range. Limiting cases are considered and fair agreements with well-known solutions are established.

Seismic responses of a hemispherical alluvial valley to SV waves: a three-dimensional analytical approximation

An analytical solution to the three-dimensional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expansion technique. The hemispherical alluvial valley with saturated soil deposits is simulated with Biot's dynamic theory for saturated porous media. The following conclusions based on numerical results can be drawn: (1) there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model; (2) the normalized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles, the dimensionless frequency of the incident SV waves and the porosity of sediments; (3) with the increase of the incident angle, the displacement distributions become more complicated; and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases.

Diffraction of sound by a poroelastic cylindrical absorber near an impedance plane

Acoustic scattering from an infinitely long fluid-saturated porous elastic circular cylinder located near a planar boundary with locally varying surface impedance is studied. The formulation utilizes the novel features of Biot dynamic theory of poroelasticity, the appropriate wave field expansions, the classical method of images and the translational addition theorem for cylindrical wave functions along with a simple local surface reaction model involving a complex amplitude wave reflection coefficient applied to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with a numerical example in which a cylindrical plastic foam absorber is located near a layer of foam material set on an impervious rigid wall. The numerical results reveal the important effects of incident wave frequency, angle of incidence, interface local surface reaction, cylinder poroelasticity and position on the acoustic field quantities. The proposed model can lead to a better understanding of acoustic scattering from two-dimensional near-interface porous absorbers or targets which are commonly encountered problems in outdoor acoustics, noise control engineering, and ocean engineering. The presented solution could eventually be used to validate those found by numerical approximation techniques.

Harmonic wave diffraction by two circular cavities in a poroelastic formation

This study investigates the dynamic interaction of time harmonic plane waves with a pair of parallel circular cylindrical cavities of infinite length buried in a boundless porous elastic fluid-saturated medium. The novel features of Biot dynamic theory of poroelasticity along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which two empty cavities are insonified by a fast compressional or a shear wave at end-on incidence. The basic dynamic field quantities such as the hoop stress amplitude and the radial displacement of the elastic frame are evaluated and discussed for representative values of the parameters characterizing the system. The effects of the proximity of the two cavities, the incident wave frequency and type are examined. Particular attention has been focused on multiple scattering interactions in addition to the slow wave coupling effects which is known to be the primary distinction of the scattering phenomenon in poroelasticity from the classical elastic case. Limiting case involving two empty cylindrical cavities in an elastic solid is considered and excellent agreement with a well-known solution is established.

An analytical solution for three-dimensional diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits

An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot's dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot's dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.

Acoustic performance of a multi-layer close-fitting hemispherical enclosure

The recent literature on prediction methods for determination and assessment of acoustic performance of noise control enclosures is briefly reviewed. The novel features of Biot dynamic theory of poroelasticity along with Havriliak-Negami model for viscoelastic material behaviour are subsequently employed to develop an exact acousto-elastic formulation for prediction of soundproof characteristics of an airtight close-fitting multi-layered hemispherical enclosure containing porous and/or viscoelastic damping materials. The analytical model considers the enclosure and the encapsulated sound source as a system in which the full interaction between the structural vibration and the internal cavity pressure field are rigorously taken into account. The classical method of separation of variables, the appropriate field expansions, the pertinent boundary conditions, and the translational addition theorems for spherical wave functions are used to find a closed-form solution in form of infinite series. The effects of enclosure structure, source size, excitation frequency, source eccentricity and surface velocity distribution on acoustic performance of the enclosure are examined. Numerical results reveal the superior performance of an ideal double-walled enclosure in comparison with a foam-cored or a polymer-cored enclosure. Furthermore, as the source size (eccentricity) decreases (increases), acoustic performance of the enclosure deteriorates. In particular, the smallest source eccentricity effects are generally noted for a polymer-core enclosure panel. Moreover, it is found that the use of foam lining in enclosure panel design can effectively compensate for deterioration of enclosure acoustic performance due to source eccentricity. The limiting case is considered and fair agreement with a well-known solution is shown.

Scattering of plane P waves by circular-arc alluvial valleys with saturated soil deposits

This paper presents an analytical solution for two-dimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow saturated soil deposits. The solution is based on Biot's dynamic theory for saturated porous media, and derived by employing Fourier–Bessel series expansion technique. In this analysis, soil deposits in the circular-arc valley are modeled as saturated porous media based on Biot's dynamic theory, and the circular-arc valley is assumed to be imbedded in an infinite half-space, filled with elastic single-phase media. Numerical results from this solution show that the amplitudes of displacement at the surface of an alluvial valley are mainly relative to the angle of incidence, the dimensionless frequency of incident P wave, the degree of saturation and porosity of soil deposits, and the stiffness and Poisson's ratio of the solid skeleton of the soil deposits. Furthermore, the proposed solution is compared with the previous solution, in which the soil deposit was modeled as an elastic single-phase solid.

An Analytic Solution for the Three-Dimensional Diffraction of Plane SV-Waves by Hemispherical Canyon in a Fluid-Saturated Porous Medium Half Space

On the basis of the Biot dynamic theory, an analytic solution of three-dimensional scattering of incident plane SV waves by hemispherical canyons in a fluid saturated porous media half space is derived. The Fourier-Bessel series expansion technique is used to get the analytic solution for the first time. The results of surface displacement amplitudes are presented, the effects of frequency and angle of incident waves on the surface displacement amplitudes are discussed. These results are compared with those from the same situation but with an elastic one-phase medium. We conclude that (1) there is a significant difference between the previous single-phase model and the present two-phase model; (2) the normalized displacements on the free surface of alluvial valley are mainly dependent on the incident wave angles, the dimensionless frequency of the incident SV-waves; (3) with the increases of the incident angle, frequency, the displacements become more complicated; and the amplification of the displacement on the free surface of alluvial valley is more prominent than that modeled by single-phase.

Scattering of plane SV waves by cylindrical canyons in saturated porous medium

On the basis of Biot dynamic theory, an analytic solution of two-dimensional scattering and diffraction of plane SV waves by circular cylindrical canyons in a half space of saturated porous media is presented in this paper for the first time. The solution is obtained by employing the Fourier–Bessel series expansion technique. Parametric studies had been carried out, which includes: the angle of incidence, the frequency of the incident SV wave, the porosity of saturated porous medium and the stiffness and Poisson's ratio of the solid-skeleton. All the outcomes are useful for the seismic analysis of the surface topography conditions.

An Analytical Solution for the Diffraction of Plane P-Waves by Three-Dimensional Hemispherical Canyon in a Fluid-Saturated Porous Media Half Space

On the basis of the Biot dynamic theory, an analytic solution of three-dimensional scattering and diffraction by hemispherical canyon “ in a fluid saturated porous media half space under plane P-waves incidence is presented. The Fourier-Bessel series expansion technique is used to get the analytic solution for the first time. The results of surface displacement amplitudes are presented, and the effects of frequency and angle of incident waves on the surface displacement amplitudes are discussed. These results are compared with those from the same situation but with an elastic one-phase media and two-dimensional circular cylindrical canyons in two-phase media, respectively.