Dynamic response of saturated layered half-space with different hydraulic interface conditions

Summary An analytic study of the response of saturated layered half-space under surface point loading is presented, with particular attention paid to the effect of hydraulic interface condition. A simplified formulation is adopted to describe the motion of the saturated porous medium, and a simple and direct approach is presented which does not involve the conventional Helmholtz decomposition in poroelasticity. Integral solutions for surface displacements are derived for different hydraulic interface conditions, and verified by reducing to well-known results of the Lamb's problem for a conventional one-phase medium. Numerical results for surface displacements in the near field are given, and the influence of the interface hydraulic condition on the displacements is shown. The present formulation possesses potential applications in practical engineering in solving various related source problems associated with uniform or layered saturated soil deposits. Received 30 September 1997; accepted for publication 14 July 1998     

Simulation of Coupled Viscous and Porous Flow Problems

Theoretical and numerical formulations are presented for the conjugate problem involving incompressible flow and flow in a saturated porous medium. The major focus of the work is the development of a generally applicable finite element method for the simulation of both fixed interface and evolving porous interface problems. The available alternatives for coupling Darcy and non-Darcy models to the Navier-Stokes equations have been studied and evaluated in a mixed finite element framework. Questions regarding convergence of the finite element method for porous flow models have been addressed. Numerical experiments on simple flow geometries have revealed the shortcomings of both the Darcy and Brinkman models. Application of the more realistic models to practical, multidimensional, flow studies has also been demonstrated.     

一维流体饱和粘弹性多孔介质层的动力响应

本文研究了不可压流体饱和粘弹性多孔介质层的一维动力响应问题。基于粘弹性理论和多孔介质理论,在流相和固相微观不可压、固相骨架服从粘弹性积分型本构关系和小变形的假定下,建立了不可压流体饱和粘弹性多孔介质层一维动力响应的数学模型,利用Laplace变换,求得了原初边值问题在变换空间中的解析解,并利用Laplace逆变换的Crump数值反演方法,得到原动力响应问题的数值解。数值研究了饱和标准线性粘弹性多孔介质层的动力响应,分析了固相位移、渗流速度、孔隙压力及固相有效应力等的响应特征。结果表明,与不可压流体饱和弹性多孔介质相同,不可压流体饱和粘弹性多孔介质中亦只存在一个纵波,并且固相骨架的粘性对动力行为有显著的影响。     

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.     

Lamb's integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil,drainage is common.In this paper,the saturated porous soil with a two-phase saturated medium is simulated,and Lamb's integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot's equation and Betti's theorem(the reciprocal theorem).According to the basic solution to Biot's equation,Green's function Gij and three terms of Green's function G4i,Gi4,and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented.The displacement field with drainage,the magnitude of drainage,and the pore pressure of the center explosion source are obtained in computation.The results of the classical Sharpe's solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared.Good agreement is observed.     

Natural convection flow with solidification between two vertical plates filled with a porous medium

Quasi-steady solidification between two vertical flat plates filled with a saturated porous medium has been investigated. The medium is homogeneous and isotropic. The convection flow of liquid takes place in the porous medium in the variable space between the two walls. One of the vertical walls is set to a temperature lower than the solidification temperature of the medium and therefore a frozen crust is formed on this wall. The second wall has a high temperature then the fusion temperature of the medium. The problem has been simplified by assuming laminar flow and the Brinkman and the Oberbeck–Bousinesq’s approximations. The results are presented in terms of the velocity for different properties of the porous medium. Various velocities are displayed in dependence of the Rayleigh and Darcy numbers. The study indicates that asymmetric boundary conditions have an important effect on the temperature and flow field. In addition, the growth of the thickness of the frozen layer with time has been derived from a simple analytical solution of the interface energy equation.     

Features of deformation of poroelastic media with low structural strength

Many natural and technological processes are associated with deformation and fracture of saturated or being saturated poroelastic media. Among such processes one can mention fluid soaking through a dam, fluid inflow to the cracks of hydraulic fracture, polishing using porous materials and special fluids, flow in catalytic pellets. All these processes are accompanied by deformation and fracture of a matrix with fluid flow. The effects at the interface porous body–fluid are essential for the processes.The specific features of deformation of poroelastic media with low structural strength are considered in this paper. The compressibility of the matrix skeleton is larger as compared to the compressibility of the saturating fluid in such media.It is shown that the oozing of the fluid at the surface of the poroelastic medium occurs in the consolidated flow regime under the action of `fluid piston' like loads if the structural strength of the medium is low. This result is obtained for both plane (deformation of a layer or halfinfinite medium) and centrally symmetric (deformation of a sphere) problems.     

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

Stability Analysis of Bioconvection of Gyrotactic Motile Microorganisms in a Fluid Saturated Porous Medium

Despite a large number of publications on bioconvection in suspensions of motile microorganisms, bioconvection in a fluid saturated porous medium is a relatively new area of research. This paper is motivated by experimental research by Kessler (1986) who established that a porous medium prevents the development of convection instability in algal suspensions. This suggests that there may exist a critical value of the permeability of a porous medium. If the permeability is smaller than critical, the system is stable and bioconvection does not develop. If the permeability is larger than critical, bioconvection may develop. This paper presents a model of bioconvection of gyrotactic motile microorganisms in a fluid saturated porous medium. The focus of this research is the determination of the critical value of permeability of a porous medium by a linear stability analysis. A simple but elegant analytical solution for the critical Darcy number is obtained.     

Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.     

Enhancement of shock waves in a porous medium saturated with a liquid containing soluble-gas bubbles

Evolution of a moderate-intensity shock wave and its enhancement after reflection from a rigid surface embedded in a porous medium are studied experimentally. The medium is saturated with a liquid that has bubbles of a soluble gas. A physical mechanism of shock wave enhancement in a saturated porous medium is proposed. Experimental data on the amplitude and velocity of reflected waves are compared with results of theoretical modeling. The process of gas bubble dissolution behind a shock wave is studied.     

On the Validity of Darcy's Law for Stable High-Concentration Displacements in Granular Porous Media

Recently, it has been suggested that Darcy's Law might not be applicable for modelling miscible, density-dependent flow in porous media. To investigate this, three sets of careful laboratory column experiments were performed on coarse and medium sands, consisting of upward displacement of water by sodium chloride solutions with concentrations ranging from 5 to 200g/l. Data on salt concentrations and water pressures were collected in horizontal transects along the flow direction. Salt concentration data were also collected in the influent and exit lines. The experimental data were analysed using a simplified approach based on Darcy's Law alone, applied with the assumption of a sharp interface. Darcy's Law was used to estimate porous medium permeability by fitting predictions to experimental data. Consistent estimates of permeability were obtained for each set of experiments. The results indicate that Darcy's Law adequately describes high concentration displacements through saturated coarse- and medium-grained porous media.     

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

Wave propagation in a porous elastic medium saturated by two immiscible fluids is investigated. It is shown that there exist three dilatational waves and one transverse wave propagating with different velocities. It is found that the velocities of all the three longitudinal waves are influenced by the capillary pressure, while the velocity of transverse wave does not at all. The problem of reflection and refraction phenomena due to longitudinal and transverse wave incident obliquely at a plane interface between uniform elastic solid half-space and porous elastic half-space saturated by two immiscible fluids has been analyzed. The amplitude ratios of various reflected and refracted waves are found to be continuous functions of the angle of incidence. Expression of energy ratios of various reflected and refracted waves are derived in closed form. The amplitude ratios and energy ratios have been computed numerically for a particular model and the results obtained are depicted graphically. It is verified that during transmission there is no dissipation of energy at the interface. Some particular cases have also been reduced from the present formulation.     

海洋地震工程流固耦合问题统一计算框架——————不规则界面情形

海底地震动场及海洋声场的模拟中,需要考虑复杂海床介质及海底地形的影响,涉及到海水、饱和海床、弹性基岩之间的相互耦合.传统的方法分别采用声波方程描述理想流体、Biot方程描述饱和海床、弹性波方程描述基岩,分别进行空间离散和界面耦合, 十分不便.本文基于理想流体、固体分别为饱和多孔介质的特殊情形(孔隙率分别为1和0),由饱和多孔介质的Biot方程可退化得到理想流体的声波方程和固体的弹性波方程.然后, 以饱和多孔介质方程为基础, 经集中质量有限元离散,严格考虑不同孔隙率的饱和多孔介质在不规则界面的耦合条件,通过求解法向和切向界面力的途径,建立了不同孔隙率的饱和多孔介质耦合情形的求解方法,将流体、固体、饱和多孔介质间的耦合问题纳入到统一计算框架,并编制了相应的三维并行分析程序.考虑海水--弹性基岩、海水--饱和海床--弹性基岩体系中凹陷地形情形,采用本文提出的统一计算框架, 结合透射边界条件,分析了P波入射时的动力反应, 并通过结果是否满足界面条件,验证了该统一计算框架的有效性以及并行计算的可行性.      

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.     

固体、流体多相孔隙介质中的波动理论及其数值模拟的进展

简要回顾了两相或三相多孔介质中波动理论和数值模拟的研究历史和现状，着重介绍了用有限元方法对饱水土层中波动进行数值模拟的研究进展，并讨论了有待进一步研究的问题．     

Boundary conditions for porous solids saturated with viscous fluid

Boundary conditions are derived to represent the continuity requirements at the boundaries of a porous solid saturated with viscous fluid. They are derived from the physically grounded principles with a mathematical check on the conservation of energy. The poroelastic solid is a dissipative one for the presence of viscosity in the interstitial fluid. The dissipative stresses due to the viscosity of pore-fluid are well represented in the boundary conditions. The unequal particle motions of two constituents of porous aggre~ gate at a boundary between two solids are explained in terms of the drainage of pore-fluid leading to imperfect bonding. A mathematical model is derived for the partial connec- tion of surface pores at the porous-porous interface. At this interface, the loose-contact slipping and partial pore opening/connection may dissipate a part of strain energy. A numerical example shows that, at the interface between water and oil-saturated sandstone, the modified boundary conditions do affect the energies of the waves refracting into the isotropic porous medium.     

Numerical modeling of formation of a gas hydrate in a finite-length porous bed purged by a gas

The process of formation of a gas hydrate in a finite-length porous medium partially saturated with water, which is purged by a cold gas, is studied. The influence of the initial parameters of the porous medium and purging conditions on the evolution of hydrate saturation and temperature is examined.     

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.