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1.
Low-Frequency Asymptotic Analysis of Seismic Reflection from a Fluid-Saturated Medium 总被引:1,自引:0,他引:1
D. B. Silin V. A. Korneev G. M. Goloshubin T. W. Patzek 《Transport in Porous Media》2006,62(3):283-305
Reflection of a seismic wave from a plane interface between two elastic media does not depend on the frequency. If one of
the media is poroelastic and fluid-saturated, then the reflection becomes frequency-dependent. This paper presents a low-frequency
asymptotic formula for the reflection of seismic plane p-wave from a fluid-saturated porous medium. The obtained asymptotic
scaling of the frequency-dependent component of the reflection coefficient shows that it is asymptotically proportional to
the square root of the product of the reservoir fluid mobility and the frequency of the signal. The dependence of this scaling
on the dynamic Darcy’s law relaxation time is investigated as well. Derivation of the main equations of the theory of poroelasticity
from the dynamic filtration theory reveals that this relaxation time is proportional to Biot’s tortuosity parameter. 相似文献
2.
《Wave Motion》2020
Scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium is considered. Wave propagation in the medium is described by Biot’s equations of poroelasticity. The scattering problem is formulated in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space in the region occupied by the inclusion. An efficient numerical method is applied to solve these equations. In the method, Gaussian approximating functions are used for discretization of the problem. For regular node grids, the matrix of the discretized problem has Toeplitz’s properties, and the Fast Fourier Transform technique can be used for the calculation of matrix–vector products. The latter accelerates substantially the process of iterative solution of the discretized problem. For material parameters of typical sedimentary rocks, the system of differential equations of poroelasticity contains a differential operator with a small parameter. As the result, the wave field in the inclusion region is split up into a slowly changing part, and boundary layer functions concentrated near the inclusion interface. The method of matched asymptotic expansions is used for the numerical solution in this case. For a spherical inclusion, the results of the numerical and matched asymptotic expansion methods are compared with a semi-analytical series solution. For a non-spherical heterogeneous inclusion, an example of the numerical solution is presented. 相似文献
3.
The paper investigates time-harmonic wave propagation in continuously stratified solids and provides the results of a reflection-transmission process generated by a layer sandwiched between homogeneous half-spaces. The layer is continuously stratified and allows for jump discontinuities at a finite number of planes. The dissipative effects are accounted for through the classical Boltzmann law of viscoelasticity. By using displacement and traction as convenient vector variables, the governing equations are considered in a vector Volterra integral equation and the solution is determined by means of a matricant. Next the matricant is applied to determine the reflection and transmission coefficients of a layer, with a generic piecewise continuous profile of the material properties. The reflection-transmission process produced by an obliquely incident wave, is considered for horizontally-polarized waves. The low-frequency approximation is derived for the reflection and transmission coefficients. Next, the high-frequency approximation is investigated by a WKB-like procedure which involves a complex valued frequency-dependent shear modulus. The displacement solution is obtained for the forward- and the backward-propagating waves in the layer along with the reflection and transmission coefficients. 相似文献
4.
In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by
a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed
of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified
Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na+ and Cl− are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem.
The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through
boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity
relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component.
A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical
parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and
microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and
the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The
purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem
considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face
contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive
behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s
law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous
effects and the reflection coefficient are computed for a range particle distances and sat concentrations. 相似文献
5.
L. A. Tkacheva 《Fluid Dynamics》2001,36(5):776-789
Plane surface wave diffraction by a floating semi-infinite plate is studied. An analytic solution of the problem is constructed by the Wiener-Hopf technique. Analytic formulas for the reflection and transmission coefficients and their shortwave and longwave asymptotics are obtained. An explicit representation for the fluid velocity potential is found. The displacement, strain, and pressure distributions over the plate are investigated as functions of a dimensionless parameter, namely, the reduced rigidity of the plate, and the asymptotic distribution is studied for long and short waves. 相似文献
6.
Surface Water Waves and Tsunamis 总被引:1,自引:0,他引:1
Walter Craig 《Journal of Dynamics and Differential Equations》2006,18(3):525-549
Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18]. 相似文献
7.
V. V. Ostapenko 《Journal of Applied Mechanics and Technical Physics》1999,40(4):647-654
A mathematical model is developed for formation and propagation of discontinuous waves caused by sliding of a shoreside landslide
into water. The model is based on the equations of a two-layer “shallow liquid” with specially introduced “dry friction” in
the low layer, which allows one to describe the joint motion of the landslide and water. An explicit difference scheme approximating
these equations is constructed, and it is used to develop a numerical algorithm for simulating the motion of the free boundaries
of both the landslide and water (in particular, the propagation of a water wave along a dry channel, incidence of the wave
on the lakeside, and flow over obstacles).
Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from
Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 109–117, July–August, 1999. 相似文献
8.
A theory is developed providing a rational framework for spatial scale- dependent fluid’s flow and heat transfer, and mass
of a component migrating with it through porous media. Introducing the assumption of a non-Brownian type motion and referring
to asymptotic expansion in powers of a small defined parameter, we develop a novel approach associated with macroscopic balance
equations obtained by averaging over a Representative Elementary Volume (REV). We prove that these equations can be decomposed
into a primary part that refers to the REV length scale and a secondary part valid at a length scale smaller than that of
the corresponding REV length. Further to our previous development, we obtain two general forms of the primary and secondary
macroscopic balance equations. One is based on the assumption that the advective flux of the extensive quantity is dominant
over that of the dispersive flux, whereas the other disregards this assumption. Moreover we also introduce the primary and
secondary macroscopic forms for the fluid heat- transfer equation. Considering a Newtonian fluid, the resulting primary Navier–Stokes
equation can vary from a nonlinear wave equation to a drag-dominant equation at the fluid–solid interface (Darcy’s law). The
secondary momentum balance equation describes a wave equation governing the concurrent propagation of the intensive momentum
and the dispersive momentum flux, deviating from their corresponding average terms. The primary macroscopic fluid heat-transfer
equation accounts for advective and dispersive heat fluxes and the secondary macroscopic heat-transfer equation involves the
simultaneous advection of heat deviating from its corresponding intensive average quantity. The primary macroscopic solute
mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic component mass balance
equation is in the form of pure advection governing migration of the deviation from the average component concentration. At
this stage, we focus on establishing the viability of the developed theory. We do this by arguing that field observations
of motion at small spatial scales are coherent with the hyperbolic characteristics of the secondary balance equations. Field
observations under natural gradient flow conditions show excessive high concentration (average of 50 mg/L) of colloids under
land irrigated by sewage effluents. We argue that this displacement of condensed colloidal parcels manifests the theoretical
findings for the smaller spatial scale. Further evidence show the accumulation of particles moving behind the front of an
emitted shockwave. We consider this as an experimental proof reinforcing the argument that colloidal migration is subject
to the action of a shockwave in the fluid and pure advection transport, governed by the respective suggested hyperbolic macroscopic
balance equations of fluid momentum and component mass at the smaller spatial scale. 相似文献
9.
Holger Steeb 《Archive of Applied Mechanics (Ingenieur Archiv)》2010,80(5):489-502
Wave propagation in fluid-saturated cancellous bone is studied on the basis of two approaches: The thermodynamic-consistent
Theory of Porous Media (TPM) and Biot’s theory. Phase velocities in the low-frequency range, calculated with the Biot-Gassmann
relations, Wyllie’s equation and the TPM, are demonstrating that a simple, so-called hybrid biphasic TPM model is able to
capture the main acoustical effects in cancellous bones. Furthermore, an extension towards high-frequency wave propagation
is discussed on the basis of the constitutive relations for the momentum exchange of the fluid and the solid phases. Further
numerical results show that, in the high-frequency (ultrasound) range a viscous correction as well as an added mass effect
(tortuosity) needs to be taken into account to explain experimentally obtained results. 相似文献
10.
The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores. 相似文献
11.
The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix
is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space.
The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques
from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration.
At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate
boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties
are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach
is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation
employing finite elements or analogous numerical procedures. 相似文献
12.
Permeability of the Fluid-Filled Inclusions in Porous Media 总被引:1,自引:0,他引:1
In this article, we propose an approach to obtain the equivalent permeability of the fluid-filled inclusions embedded into
a porous host in which a fluid flow obeys Darcy’s law. The approach consists in the comparison of the solutions for one-particle
problem describing the flow inside the inclusion, firstly, by the Stokes equations and then by using Darcy’s law. The results
obtained for spheres (3D) and circles (2D) demonstrate that the inclusion equivalent permeability is a function of its radius
and, additionally, depends on the host permeability. Based on this definition of inclusion permeability and using effective
medium method, we have calculated the effective permeability of the double-porosity medium composed of the permeable matrix
(with small scale pores) and large scale secondary spherical pores. 相似文献
13.
The response of a semi-infinite compressible fluid to a step-wise change in temperature of its boundary is investigated analytically
and numerically. Numerical results of the boundary layer structure are compared with Clarke’s analytical solution for a gas
with thermal conductivity proportional to temperature. To avoid unwanted numerical dissipation in the numerical analysis,
the space-time conservation element and solution element (CESE) method has been adopted to solve the unsteady 1-D Navier-Stokes
equations. Good agreement between analytical and numerical results has been found for the development of the thermal boundary
layer on a long time scale. Weak shock waves and expansion waves induced by the thermal boundary layer due to its compressibility,
are observed in the numerical simulation. Finally, the numerical method has been applied to the reflection of a non-linear
expansion wave and to a shock wave from an isothermal wall, thereby illustrating the effect of the boundary layer on the external
flow field. 相似文献
14.
H. Blok 《Applied Scientific Research》1967,17(4-5):331-339
Summary The longitudinal propagation and reflection of a plane electromagnetic wave in a horizontally stratified magneto-ionic medium is considered. In this case Maxwell's equations reduce to two uncoupled ordinary second-order differential equations, describing the propagation of two elliptically polarized plane waves. The electron density of the medium is assumed to vary with the vertical Cartesian coordinatez according to the Epstein law. Rigorous solutions of the relevant differential equations can be obtained either in the form of hypergeometric functions or in the form of an integral representation. The reflection coefficients of both waves are then expressed in terms of gamma functions. The following quantities are considered in detail in their dependence on the parameters involved: the modulus of the reflection coefficient, the phase delay time and the group delay time. Some numerical results are given. 相似文献
15.
In the frame of wave propagation in damaged (elastic) solids, an analytical approach for normal penetration of a plane wave through a periodic array of inclined cracks is developed. The problem is reduced to an integral equation holding over the length of each crack; approximated forms (of one-mode and low-frequency types) are then given to the kernel, so as to derive explicit formulas for the reflection and transmission coefficients. Numerical resolution of the relevant equations finally provides some graphs that are compared. 相似文献
16.
Unsteady flow over an infinite permeable rotating cone in a rotating fluid in the presence of an applied magnetic field has been investigated. The unsteadiness is induced by the time-dependent angular velocity of the body, as well as that of the fluid. The partial differential equations governing the flow have been solved numerically by using an implicit finite-difference scheme in combination with the quasi-linearization technique. For large values of the magnetic parameter, analytical solutions have also been obtained for the steady-state case. It is observed that the magnetic field, surface velocity, and suction and injection strongly affect the local skin friction coefficients in the tangential and azimuthal directions. The local skin friction coefficients increase when the angular velocity of the fluid or body increases with time, but these decrease with decreasing angular velocity. The skin friction coefficients in the tangential and azimuthal directions vanish when the angular velocities of fluid and the body are equal but this does not imply separation. When the angular velocity of the fluid is greater than that of the body, the velocity profiles reach their asymptotic values at the edge of the boundary layer in an oscillatory manner, but the magnetic field or suction reduces or suppresses these oscillations. 相似文献
17.
Radiation-conduction interaction on mixed convection from a horizontal circular cylinder 总被引:1,自引:0,他引:1
A mixed convection flow of an optically dense viscous incompressible fluid along a horizontal circular cylinder has been
studied with the effect of radiation when the surface temperature is uniform. Using appropriate transformations, the boundary
layer equations governing the flow are reduced to local nonsimilarity form. Solutions of the governing equations are obtained
employing the implicit finite difference method. Effects of varying the pertinent parameters, such as, the Planck number,
R
w the surface temperature parameter, θw and the buoyancy parameter, α on the local skin-friction and local heat transfer coefficients are shown graphically as well
as in tabular form against the curvature parameter ξ, while taking Prandtl number Pr = 1.0. It is found that an increase of
R
d,θw or α leads to increases in the values of the local skin-friction and the local rate of heat transfer coefficients. At the
stagnation point asymptotic solutions for large value of α are also obtained and the effect of the other pertinent parameters
on the formation of the flow separation are studied.
Received on 28 July 1998 相似文献
18.
Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable
media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone,
capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture
profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture
profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding
permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the
analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop
a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which
accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete
fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability.
The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the
fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased
by more than 5% was observed for flows within high permeability-fractured porous media. 相似文献
19.
Ali Nemati Hayati Mohammad Mehdi Ahmadi Soheil Mohammadi 《Transport in Porous Media》2012,93(1):13-27
Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such
a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the
medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation
in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak
shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards
by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical
method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to
explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations
in the porous medium, which is of high interest in the design of granular shock isolators. 相似文献
20.
Fluid flow at the interface of a porous medium and an open channel is the governing phenomenon in a number of processes of
industrial importance. Traditionally, this has been modeled by applying the Brinkman’s modification of Darcy’s law to obtain
the velocity profile in terms of an additional parameter known as the “apparent viscosity” or the “slip coefficient”. To test
this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close
vicinity of the permeable boundary of a porous medium. The porous medium used in the experiments consisted of a network of
continuous glass strands woven together in a random fashion.
A Hele–Shaw cell was partially filled with a fibrous preform such that an open channel flow is coupled with the Darcy flow
inside the preform through the permeable interface of the preform. The open channel portion of the Hele–Shaw cell also acts
as an ideal porous medium of known in-plane permeability which is much higher than the permeability of the fibrous porous
medium. A viscous fluid is injected at a constant flow rate through the above arrangement and a saturated and steady flow
is established through the cell. Using LDA, steady state velocity profiles are accurately measured by traversing across the
cell in the direction perpendicular to the flow. A series of experiments were conducted in which fluid viscosity, flow rate,
solid volume fraction of the porous medium and depth of the Hele–Shaw cell were varied. For each and every case in which the
conditions for Hele–Shaw approximation were valid, the depth of the boundary layer zone or the screening length inside the
fibrous preform was found to be of the order of the channel depth. This is much larger as compared to the Brinkman’s prediction
of the screening length which is of the order of √K, where K is the permeability of the fibrous porous medium. Based on this finding, we modified the boundary condition in the Brinkman’s
solution and found that the velocity profile results compared well with the experimental data for the planar geometry and
the fibrous preforms for volume fractions of 7%, 14% and 21% for Hele–Shaw cell depths of 1.6 and 3.175 mm. For a cell depth
of 4.8 cm, in which the Hele–Shaw approximation was not valid, the boundary layer thickness or the screening length was found
to be less than the mold or channel depth but was still much larger than the Brinkman’s prediction.
Received: 10 May 1996 / Accepted: 26 August 1996 相似文献