Reflection and Transmission of Elastic Waves from the Interface of a Fluid-saturated Porous Solid and a Double Porosity Solid

The reflection and transmission characteristics of an incident plane P1 wave from the interface of a fluid-saturated single porous solid and a fluid-saturated double porosity solid are investigated. The fluid-saturated porous solid is modeled with the classic Biot’s theory and the double porosity medium is described by an extended Biot’s theory. In a double-porosity model with dual-permeability there exist three compressional waves and a shear wave. The effects of the incident angle and frequency on amplitude ratios of the reflected and transmitted waves to the incident wave are discussed. Two boundary conditions are discussed in detail: (a) Open-pore boundary and (b) Sealed-pore boundary. Numerical results reveal that the characteristics of the reflection and transmission coefficients to the incident angle and the frequency are quite different for the two cases of boundary conditions. Properties of the bulk waves existing in the fluid-saturated porous solid and the double porosity medium are also studied.     

A mixture theory analysis for the surface-wave propagation in an unsaturated porous medium

The mixture theory is employed to the analysis of surface-wave propagation in a porous medium saturated by two compressible and viscous fluids (liquid and gas). A linear isothermal dynamic model is implemented which takes into account the interaction between the pore fluids and the solid phase of the porous material through viscous dissipation. In such unsaturated cases, the dispersion equations of Rayleigh and Love waves are derived respectively. Two situations for the Love waves are discussed in detail: (a) an elastic layer lying over an unsaturated porous half-space and (b) an unsaturated porous layer lying over an elastic half-space. The wave analysis indicates that, to the three compressional waves discovered in the unsaturated porous medium, there also correspond three Rayleigh wave modes (R1, R2, and R3 waves) propagating along its free surface. The numerical results demonstrate a significant dependence of wave velocities and attenuation coefficients of the Rayleigh and Love waves on the saturation degree, excitation frequency and intrinsic permeability. The cut-off frequency of the high order mode of Love waves is also found to be dependent on the saturation degree.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

Propagation and Evolution of Wave Fronts in Two-Phase Porous Media

An investigation is made into the propagation and evolution of wave fronts in a porous medium which is intended to contain two phases: the porous solid, referred to as the skeleton, and the fluid within the interconnected pores formed by the skeleton. In particular, the microscopic density of each real material is assumed to be unchangeable, while the macroscopic density of each phase may change, associated with the volume fractions. A two-phase porous medium model is concisely introduced based on the work by de Boer. Propagation conditions and amplitude evolution of the discontinuity waves are presented by use of the idea of surfaces of discontinuity, where the wave front is treated as a surface of discontinuity. It is demonstrated that the saturation condition entails certain restrictions between the amplitudes of the longitudinal waves in the solid and fluid phases. Two propagation velocities are attained upon examining the existence of the discontinuity waves. It is found that a completely coupled longitudinal wave and a pure transverse wave are realizable in the two-phase porous medium. The discontinuity strength of the pore-pressure may be determined by the amplitude of the coupled longitudinal wave. In the case of homogeneous weak discontinuities, explicit evolution equations of the amplitudes for two types of discontinuity waves are derived.     

On the existence of the low-frequency surface waves in a porous mediumSur l'existence des ondes de surface basse fréquence en milieux poreux

The existence and propagation of the surface waves at a vacuum/porous medium interface are investigated in the low frequency range. Two types of surface waves are shown to be possible: the generalized Rayleigh wave, which always exists, and the Stoneley wave, which exists for a limited range of wave numbers. Moreover, within the k-domain of existence the Stoneley wave cannot appear for certain values of elastic parameters of the solid phase. The bifurcation behavior of both the Stoneley wave and the Biot (P2) bulk wave, depending on the wave number, is revealed. The asymptotic formulas for the phase velocities of the surface waves are derived. To cite this article: I. Edelman, C. R. Mecanique 332 (2004).     

Three-dimensional green water velocity on a model structure

Flow kinematics of green water due to plunging breaking waves impinging on a simplified, 3D model structure was investigated in the laboratory. Two breaking wave conditions were tested: one with waves impinging on the vertical wall of the model at still water level, and the other with waves impinging on the horizontal deck surface. The bubble image velocimetry (BIV) technique was used to measure flow velocities. Measurements were taken on both vertical and horizontal planes. Evolution of green water flow kinematics in time and space was revealed and was found to be quite different between the two wave conditions, even though the incoming waves are essentially identical. The time history of maximum velocity is demonstrated and compared. In both cases, the maximum velocity occurs near the green water front and beneath the free surface. The maximum horizontal velocity for the deck impinging case is 1.44C with C being the wave phase speed, which is greater than 1.24C for the wall impingement case. The overall turbulence level is about 0.3 of the corresponding maximum velocity in each wave condition. The results were also compared with 2D experimental results to examine the 3D effect. It was found that the magnitude of the maximum vertical velocity during the runup process is 1.7C in the 3D model study and 2.9C in the 2D model study, whereas the maximum horizontal velocity on the deck is similar, 1.2C in both 3D and 2D model studies.     

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids.The propagation of three longitudinal waves is represented through three scalar potential functions.The lone transverse wave is presented by a vector potential function.The displacements of particles in different phases of the aggregate are defined in terms of these potential functions.It is shown that there exist three longitudinal waves and one transverse wave.The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated.For the presence of viscosity in pore-fluids,the waves refracted to the porous medium attenuate in the direction normal to the interface.The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a nonsingular system of linear algebraic equations.These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave.The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model.The conservation of the energy across the interface is verified.The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed.     

On dispersive propagation of surface waves in patchy saturated porous media

Frequency-dependent velocity and attenuation for Rayleigh-wave propagation along a vacuum/patchy saturated porous medium interface are investigated in the low frequency band (0.1–1000 Hz). Conventional patchy saturation models for compressional waves are extended to account for Rayleigh wave propagation along a free surface. The mesoscopic interaction of fluid and solid phases, as a dominant loss mechanism in patchy saturated media, significantly affects Rayleigh-wave propagation and attenuation. Researches on the dispersion characteristics at low frequencies with different gas fractions in patchy saturated media also demonstrate a strong correlation between the Rayleigh-wave mode and the fast compressional wave. Especially, the strongest attenuation with the maximum value of 1/Q$1/Q$ for Rayleigh waves are obtained in the frequency range of 1–200 Hz. Numerical results show that the significant dependence of velocity and attenuation on frequencies and gas fractions presents a distinctive dynamical response of Rayleigh waves in the time domain.     

On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

非平整、多孔介质海底上波浪传播的复合方程

为了反映近岸区域实际存在的多孔介质海底效应，并且考虑到波浪在刚性海底上传播模型的 最新研究进展，运用Green第二恒等式建立了波浪在非平整、多孔介质海底上传播的复合方 程. 假设水深和多孔介质海底层厚度均由两种分量组成：慢变分量，其水平变化的长度尺度大于 表面波的波长；快变分量，其水平变化的长度尺度与表面波的波长等阶，但其振幅小于表面 波的振幅. 另外，多孔介质层下部边界的快变分量比水深的快变分量小1个量级. 针对水体层和多孔介质层，选择Green第二恒等式方法给出了波浪传播和渗透的复合方程， 它在交接面上满足压力和垂直渗透速度的连续性条件,可充分考虑波数变化的一般连续性， 并包含了某些著名的扩展型缓坡方程.     

Rarefaction shock waves in porous media that accumulate CO2, CH4, N2

In this paper, a porous medium whose core is built from a solid substance is considered. Such a medium can accumulate liquid and gaseous substances absorbing on the surface of the solid body and diffusing within its molecular lattice. The medium forms a so-called solid solution. A change in the values of the thermodynamic parameters of this medium leads to phase transitions in these substances. Besides, some other associated phenomena may occur. There are given the results of laboratory analyses concerning the relation between the specific volumeV of the solid solution formed by the coal with CO₂, CH₄, N₂ dissolved in it and its confining pressurep. This relation indicates the possibility of the propagation of a rarefaction shock waves in such a medium.A mathematical model of the generation of such waves and its specific applications to describe the initial boundary conditions affected during the experiments presented. The results of these experiments have confirmed the adopted model, particularly those aspects that are concerned with the specific sliced structures of the coal medium with the accumulated CO₂ and N₂, through which a rarefaction shock wave has passed (Figs. 10a, 10b, 11a, and 11b). The presented model explains the phenomena of sudden massive rock-and-gas outbursts, occurring in nature, e.g. as a result of disturbing the primary equilibrium of the rocks in the Earth crust which form a solid solution, by underground mining exploitation.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

The shock-wave structure in a two-velocity mixture of compressible media with different pressures

The problem of the shock-wave structure in a mixture of two compressible media with different velocities and pressures of components is considered. The problem is reduced to solving a boundary-value problem for two ordinary differential equations that describe the velocity relaxation and pressure equalization of the components. Using methods of the qualitative theory of dynamic systems on a plane, the existence and uniqueness of four types of waves are shown: (a) fully dispersed waves; (b) frozen-dispersed waves; (c) dispersed-frozen waves; (d) frozen waves of two-front configuration. A chart of solutions of the corresponding flow types is constructed in the plane of the following parameters: the initial velocity of the mixture and the initial volume concentration of one of the components. The numerical calculations conducted illustrate the obtained analytical structures of the shock wave. It is shown that the results obtained using the suggested mathematical model are in agreement with experimental data on the dependence of the velocity of the dispersed shock wave on the equilibrium pressure behind the shock-wave front for a mixture of silica sand and water. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 10–19, March–April, 1998.     

The Peculiarities of Linear Wave Propagation in Double Porous Media

The features of propagation of longitudinal and transverse waves (LW and TW) in fractured porous medium (FPM) saturated with liquid are investigated by methods of multiphase mechanics. The mathematical model of FPM accounting for inequality of velocities and pressures of liquid in pores and fractures, liquid mass exchange and nonstationary interaction forces is developed. Processes of monochromatic wave propagation are studied. The dispersion relation is obtained and the effect of model parameters on wave propagation is analysed. It is established that one transverse and three longitudinal waves propagate in FPM saturated with liquid. The fastest LW is a deformational wave and the two others are filtrational. Filtrational waves attenuate much stronger than deformational and transverse waves. Distinction of velocities and pressures in liquid in various pore systems provides an explanation for the existence of the two filtrational waves in porous medium with two different characteristic sizes of pores.     

Difference between the longitudinal Frenkel-Biot waves in water-and gas-saturated porous media

The problem of the propagation of longitudinal Biot waves in a porous medium saturated with a weakly compressible liquid (water) or a gas is considered theoretically. The frequency dependence of the phase velocities and damping coefficients is investigated numerically. It is shown that for a certain relationship between the parameters of the porous medium and the saturating fluid there is a “critical” frequency at which the properties of longitudinal waves of both kinds are identical. An analytical expression for this “critical” frequency is obtained. It is shown that for a gas-saturated porous medium, at a certain frequency, in both longitudinal waves the relative gas-matrix motion changes type. Assuming that the saturating-gas behavior corresponds to an adiabatic equation of state, an estimate is obtained for the threshold pore pressure necessary for the restructuring of the relative motion. The wave associated with matrix deformation is shown to have a high damping coefficient in a porous medium saturated with a weakly compressible liquid (water in the case considered) but to be only weakly damped in a gas-saturated porous medium.     

Two-dimensional unstructured elastic model for acoustic pulse scattering at solid-liquid interfaces

Abstract. A numerical model to simulate elastic waves and acoustic scattering in two spatial dimensions has been developed and thoroughly tested. The model universally includes elastic solids and liquids. The equations of motion are written in terms of stresses, displacements and displacement velocities for control volumes constructed about the nodes of a triangular unstructured grid. The latter conveniently supports various geometries with complex external and internal boundaries separating sub-domains of different elastic properties. Theoretical dispersion for zero mode symmetric () and antisymmetric () waves in a plate has been reproduced numerically with high accuracy, thus verifying the method and code. Comparison of simulated acoustic pulse scattering at water-immersed steel plate with the respective experiments reveals a very good agreement in such delicate features as excitation of the surface (A) wave. The numerical results explain the peculiar location of the surface wave relative to the other ones in experimental registrations. Examples of acoustic pulse interactions with curvilinear metallic shells in water demonstrate flexibility of the method with respect to complex geometries. Potential applications as well as some directions for further improvement to the technique are briefly discussed. Received 5 September 2002 / Accepted 25 November 2002 Published online 4 February 2003 RID="*" ID="*"Permanent address: Ioffe Physical-Technical Institute, 26 Polytekhnicheskaya, 194021 St. Petersburg, Russia Correspondence to: P. Voinovich (e-mail: vpeter@scc.ioffe.ru)     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.