Acoustic wave propagation in saturated porous media: reformulation of the Biot/Squirt flow theory

A model of wave propagation in fluid-saturated porous media is developed where the principal fluid/solid interaction mode affecting the propagation of the acoustic wave results from the conjunction of the Biot and the Squirt flow mechanism. The difference between the original Biot/Squirt (BISQ) flow theory and the new theory, which we call the reformulated BISQ, is that the average fluid pressure term appearing in the dynamic equation for a two component solid/fluid continuum is independent of squirt flow length. P-velocity and attenuation relate to measurable rock physical parameters: the Biot's poroelastic constants, porosity, permeability, pore fluid compressibility and viscosity. Modelling shows that velocity and attenuation dispersion obtained using the reformulated BISQ theory are of the same order of magnitude as those obtained using the original BISQ theory. Investigation on permeability effect on velocity and attenuation dispersion indicate that the transition zone in velocity and attenuation peak, occurring both at the relaxation frequency, shifts toward high frequency when permeability decreases. This behaviour agrees with Biot's theory prediction.     

Reflection and transmission ofSH waves in an initially stressed medium consisting of a sandy layer lying over a fluid-saturated porous solid

Biot's theory is employed to study the reflection and transmission ofSH waves in a sandy layer lying over a fluid-saturated porous solid half-space. The entire medium is considered under constant initial stress. Effects of sandiness, initial stress, anelasticity and viscosity of the interstitial fluid on the partitioning of energy are studied. In the presence of initial stress the incident wave starts attenuating when incider beyond a certain angle (depending upon the amount of initial stress), even if the medium is perfectly clastic. Anelasticity of the solid layer results in the dissipation of energy during transmission. The direction of attenuation vector of incident wave affects the dissipation energy to a large extent. Effect on partitioning of energy reverse at incidence after the critical angle. A complete account of energy returmed back to the underlying half-space and that which is dissipated in the overlying layer has been discussed analytically as well as numerically.     

Scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space, modeled using the indirect boundary element method

The scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space is modeled using the indirect boundary element method in the frequency domain. The free-field responses are calculated to determine the displacements and stresses at the surface of the canyon, and fictitious distributed loads are then applied at the surface of the canyon in the free field to calculate the Green's functions for displacements and stresses. The amplitudes of the fictitious distributed loads are determined from the boundary conditions, and the displacements arising from the waves in the free field and from the fictitious distributed loads are summed to obtain the solution. The effects of fluid saturation, boundary conditions, porosity, and soil layers on the surface displacement amplitudes and phase shifts are discussed, and some useful conclusions are obtained. It is shown that the surface displacement amplitudes due to saturation and boundary conditions, different porosities, or the presence of a soil layer can be very dissimilar, and large phase shifts can be observed. The resulting wavelengths for an undrained saturated poroelastic medium are slightly longer than those for a drained saturated poroelastic medium; and are longer for a drained saturated poroelastic medium than those for a dry poroelastic medium. As porosity increases, the wavelengths become longer; and a layered half-space produces longer wavelengths than a homogeneous half-space.     

Rayleigh modes in anisotropic,heterogeneous poroelastic layers

The generalized Rayleigh type surface waves are studied in a multilayered medium consisting of anisotropic poroelastic solid layered stack beneath a fluid layer and overlying a heterogeneous elastic solid half-space. The heterogeneity, considered, is of vertical type. The interface between solid layer and half-space is treated as an imperfect interface and suitable boundary conditions are applied thereat. The technique of transfer matrix is used to obtain the dispersion equation in compact and convenient form. Numerical results are obtained for particular models. The effects of anisotropy and heterogeneity on the surface waves speed are discussed.     

Stresses and excess pore pressure induced in saturated poroelastic halfspace by moving line load

This paper examines stresses and excess pore fluid pressure that are induced in a saturated poroelastic soil of halfspace extent by a concentrated line load. The line load is moving at a constant velocity along the surface of the poroelastic halfspace. The governing equations for the proposed analysis are based on the Biot's theory of dynamics in saturated poroelastic soils. The governing partial differential equations are solved using Fourier transforms. The solutions for the stresses and excess pore pressure are expressed in the forms of inverse Fourier transforms. The numerical results are obtained by performing the numerical inversion of the transform integrals. A parametric study is presented to illustrate the influences of the velocity of moving load and the poroelastic material parameters on the stresses and excess pore pressure. At a high velocity, the maximum values of the stresses in a poroelastic halfspace are smaller than those in an elastic solid, whilst at a low velocity the stresses in a poroelastic halfspace are larger than those in an elastic halfspace. The potential of diffusivity has an important influence on the stresses and excess pore pressure.     

Dynamic response of a multi-layered poroelastic medium

An exact stiffness matrix method is presented to evaluate the dynamic response of a multi-layered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. The system under consideration consists of N layers of different properties and thickness overlying a homogeneous half-plane or a rigid base. Fourier integral transform is used with respect to the x-co-ordinate and the formulation is presented in the frequency domain. Fourier transforms of average displacements of the solid matrix and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness (impedance) matrices describing the relationship between generalized displacement and force vectors of a layer of finite thickness and a half-plane are derived explicitly in the Fourier-frequency space by using rigorous analytical solutions for Biot's elastodynamic theory for porous media. The global stiffness matrix and the force vector of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The numerical solution of the global equation system for discrete values of Fourier transform parameter together with the application of numerical quadrature to evaluate inverse Fourier transform integrals yield the solutions for poroelastic fields. Numerical results for displacements and stresses of a few layered systems and vertical impedance of a rigid strip bonded to layered poroelastic media are presented. The advantages of the present method when compared to existing approximate stiffness methods and other methods based on the determination of layer arbitrary coefficients are discussed.     

Analysis of reservoir heterogeneity-induced amplification effect on time-lapse seismic responses of fluid substitution: A physical modelling study

Seismic wave propagation through a fluid-saturated poroelastic layer might be strongly affected by media heterogeneities. Via incorporating controlled laboratory simulation experiments, we extend previous studies of time-lapse seismic effects to evaluate the wave scattering influence of the heterogeneous nature of porous permeable media and the associated amplification effects on 4D seismic response characteristics of reservoir fluid substitution. A physical model consisted of stratified thin layers of shale and porous sandstone reservoir with rock heterogeneities was built based on the geological data of a real hydrocarbon-saturated reservoir in Northeast China. Multi-surveys data of good quality were acquired by filling poroelastic reservoir layers with gas, water and oil in sequence. Experimental observations show that reservoir heterogeneity effect causes significantly magnified abnormal responses to the fluid-saturated media. Specifically, reflection signatures of the gas-filled reservoir are dramatically deviated from those of the liquid fluid-filled reservoir, compared with ones of the homogeneous media. By removing the influences unrelated to reservoir property alterations, 4D seismic estimates of travel-time and frequency-dependent characteristic are reasonably consistent with fluid variations. Nevertheless, strong 4D amplitude difference anomalies might not correspond to the regions where fluid variations occur. We also find that 4D seismic difference attributes are evident between oil- and water-filled models, whereas significant between oil- and gas-filled models. Meanwhile, rock physics modelling results reveal the predicted 4D seismic differences are obviously smaller than those calculated from seismic observations. The results in this paper, therefore, implicate that the effect of a reservoir's heterogeneous nature might be beneficial for hydrocarbons detection as well as monitoring small variations in pore fluids.     

Time harmonic point load and dynamic contact problem of contacting fluid and poroelastic half-spaces

The dynamic response of contacting fluid and fluid-saturated poroelastic half- spaces to a time-harmonic vertical point force or a point pore pressure is investigated. The solutions are formulated using the boundary conditions at the fluid-porous medium interface. The point load solutions are then used to solve the dynamic problem of the vertical vibration of a rigid disc (both permeable and impermeable discs are included) on the surface of the poroelastic half-space. The contact problems are solved by integrating the point force and point pore pressure solutions over the contact area with unknown discontinuous force and pore pressure distributions, which are determined from the boundary conditions. The solutions are expressed in terms of dual integral equations, which are converted to Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic compliance coefficient for the cases with or without fluid overlying the poroelastic half-space are presented to show the effects of the fluid. The influence of the permeability condition of the disc on the compliance of the poroelastic half-space is investigated. The displacement, vertical stress, pore pressure in the poroelastic half-space and water pressure in the fluid half-space are also examined for different poroelastic materials and frequencies of excitation. The present results are helpful in the study of the dynamic response of foundations on the seabed under seawater.     

横观各向同性饱和土的基本方程组

两相饱和多孔介质的基本方程组和计算参数的选取在一定程度上是混乱的.本文利用连续介质力学理论,结合空间平均化方法,根据应力-应变关系、运动学关系、连续性方程及广义Darcy定律,建立了横观各向同性液体饱和多孔介质的基本方程组;通过将该基本方程组与Biot理论的基本方程组进行比较,确定了Biot方程组中的弹性常数Bi与弹性常数cij的关系,并得到了确定Biot参数mi和ri的计算公式——Biot参数与渗透率、孔隙率及黏性系数的关系;最后对几种特殊情形进行了讨论,给出了简化方程组.     

声电效应测井的有限差分模拟

本文研究声电效应测井波场的有限差分模拟算法.忽略井外地层中诱导电磁场对孔隙弹性波的影响,将求解动电耦合波方程组的问题解耦,先计算孔隙弹性波,再计算其诱导电磁场.基于轴对称柱坐标系下的速度-应力交错网格,采用时域有限差分计算井孔流体声波和井外地层孔隙弹性波.将电磁场近似看作似稳场,基于轴对称柱坐标系下的5点式有限差分网格,求解不同时刻的电位Poisson方程,计算诱导电场.结果表明:本文算法可准确模拟频率6.0 kHz的声电效应测井全波;在声波测井频率范围内,电导率、动电耦合系数和动态渗透率的低频近似对伴随电磁场的计算影响不大;地层水平界面导致伴随反射斯通利波的电场和显著的界面电磁波,后者对于探测地层界面具有潜在的应用价值.     

Propagation of seismic waves in patchy-saturated porous media: double-porosity representation

Propagation of harmonic plane waves is studied in a patchy-saturated porous medium. Patchy distribution of the two immiscible fluids is considered in a porous frame with uniform skeletal properties. A composition of two types of patches, connected through continuous paths, constitutes a double-porosity medium. Different compressibilities of pore-fluids in two porous phases facilitate the wave-induced fluid-flow in this composite material. Constitutive relations are considered with frequency-dependent complex elastic coefficients, which define the dissipative behaviour of porous aggregate due to the flow of viscous fluid in connected patches. Relevant equations of motion are solved to explain the propagation of three compressional waves and one shear wave in patchy-saturated porous solids. A numerical example is solved to illustrate dispersion in phase velocity and quality factor of attenuated waves in patchy-saturated porous materials. Role of fluid–solid inertial coupling in Darcy's law is emphasized to keep a check on the dispersion of wave velocities in the porous composite. Effects of patchy saturation on phase velocities and quality factors of attenuation are analysed using the double-porosity formulation as well as the reduced single-porosity equivalents.     

平面P波在饱和半空间中洞室周围的散射(I):解析解

利用波函数展开法给出了入射平面P波在饱和半空间中圆形洞室周围散射问题的一个解析解。半空间假定为无粘性流体饱和介质,满足Biot理论。采用一种基于实验数据的孔隙率和模量之间的线性关系来确定Biot模型中的介质参数。解答考虑了透水边界和非透水边界两种情况。对边界条件进行了数值检验,结果表明,随着级数截断项数的增大,边界残量衰减很快。解答为进一步研究入射波频率和角度、边界渗透条件、孔隙率、泊松比等参数对散射的影响奠定了基础。     

Seismic wave attenuation in fluid-saturated porous media

Contrary to the traditional view, seismic attenuation in Biot's theory of fluid-saturated porous media is due to viscous damping of local (not global) pore-fluid motion. Since substantial inhomogeneities in fluid permeability of porous geological materials are to be expected, the regions of highest local permeability contribute most to the wave energy dissipation while those of lowest permeability dominate the fluid flow rate if they are uniformly distributed. This dichotomy can explain some of the observed discrepancies between computed and measured attenuation of compressional and shear waves in porous earth. One unfortunate consequence of this result is the fact that measured seismic wave attenuation in fluid-filled geological materials cannot be used directly as a diagnostic of the global fluid-flow permeability.     

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part II: Finite element formulation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation towards infinity. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. In Part I (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 411-21), a local absorbing boundary condition for wave propagation in saturated poroelastic media has been developed. In the present paper, this boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid. Spurious reflections for oblique incident waves on the absorbing boundary contribute to the solution errors. Therefore, a spectral element method, based on classical analytical solution techniques, is used to assess the accuracy of the finite element formulation.     

On results of the surface wave analyses in poroelastic media by means of the Simple Mixture Model and the Biot model

This note is devoted to an overview of the results on the number, velocity and attenuation of surface waves in two-component saturated poroelastic media. The author's numerical results which have been obtained using the Simple Mixture Model are compared to some results of other authors who carried out either an analysis by means of the classical Biot model for poroelastic media or experiments. It is shown that the Simple Mixture Model is a simplification of the Biot model. Due to its simpler form the Simple Mixture Model is more suitable for a complex analysis which depends on several parameters. Thus, the Simple Mixture Model is the only model for which, simultaneously, either the dependence on the frequency, ω, and the dependence on the bulk permeability, π (boundary porous medium/vacuum) or the dependence on the frequency, ω, and the dependence on the surface permeability, α (boundary porous medium/fluid) has been considered.     

Effects of rainfall on soil–structure system frequency: Examples based on poroelasticity and a comparison with full-scale measurements

In this paper, a simple two-dimensional soil–structure interaction model, based on Biot's theory of wave propagation in fluid saturated porous media, is used to explain the observed increase of the apparent frequencies of Millikan library in Pasadena, California, during heavy rainfall and recovery within days after the rain. These variations have been measured for small amplitude response (to microtremors and wind excitation), for which Biot's linear theory is valid. The postulated hypothesis is that the observed increases in frequency are due to the water saturation of the soil. The theoretical model used to explore this hypothesis consists of a shear wall supported by a circular foundation embedded in a poroelastic half-space. This rigid foundation model may be appropriate only for the NS response of Millikan library. This paper presents results for the foundation stiffness, and for the system response for model parameters similar to those for Millikan library (located on alluvium with shear wave velocity of about 300 m/s). The foundation impedance matrix, foundation input motion and system response are compared for dry and fully saturated half-space, with permeable and impermeable foundation. The results show that for embedded foundations, the effects of saturation on the horizontal foundation stiffness are as significant as for the vertical stiffness, contrary to what has been known for surface foundations investigated by other authors. Further, the results suggest a 1–2% increase in system frequency of the first two modes of vibration, depending on the drainage condition along the foundation–soil interface. Such increases agree qualitatively with the observations.     

Plain strain soil–structure interaction model for a building supported by a circular foundation embedded in a poroelastic half-space

A simple theoretical model for soil–structure interaction in water saturated poroelastic soils is presented, developed to explore if the apparent building–foundation–soil system frequency changes due to water saturation. The model consists of a shear wall supported by a rigid circular foundation embedded in a homogenous, isotropic poroelastic half-space, fully saturated by a compressible and inviscid fluid, and excited by in-plane wave motion. The motion in the soil is governed by Biot's theory of wave propagation in fluid saturated porous media. Helmholtz decomposition and wave function expansion of the two P-wave and the S-wave potentials is used to represent the motion in the soil. The boundary conditions along the contact surface between the soil and the foundation are perfect bond (i.e. welded contact) for the skeleton, and either drained or undrained hydraulic condition for the fluid (i.e. pervious or impervious foundation). For the purpose of this exploratory analysis, the zero stress condition at the free surface is relaxed in the derivation of the foundation stiffness matrix, which enables a closed form solution. The implications of this assumption are discussed, based on published comparisons for the elastic case. Also, a closed form representation is derived for the foundation driving forces for incident plane (fast) P-wave or SV wave. Numerical results and comparison with the full-scale measurements are presented in the companion paper, published in this issue.     

A few remarks on Biot's model and linear acoustics of poroelastic saturated materials

This review article consists of two parts. The first part concerns the admissibility of four contributions characteristic for Biot's model of poroelastic materials: coupling of partial stresses, presence of relative accelerations in equations of motion, dependence of permeability on frequency and changes of porosity. The second part is devoted to the demonstration of those contributions in the analysis of acoustic waves. Propagation of fronts, monochromatic waves and surface waves are considered.     

双相介质分界面上弹性波的反射与透射

本文基于Biot理论,推导出Zoeppritz形式的双相介质分界面上弹性波的反射与透射公式,对单相Zoeppritz公式与双相反射系数公式进行了比较,对双相介质含油、水、气不同流体时的反射规律以及孔隙度、渗透率、饱和度等储层参数对纵波反射的影响进行了研究.数值模拟分析表明,双相与单相反射公式的主要差异在于双相介质反射公式中考虑了液相、固液耦合相弹性模量的影响;油砂、水砂、气砂岩的慢P波反射差异明显;快P波反射对孔隙度的变化敏感,饱和度次之,对渗透率和频率的变化不敏感.     

基于Biot-Squirt方程的波场模拟

Biot流动和喷射流动是含流体多孔隙介质中流体流动的两种重要力学机制,对地震波和声波的传播均产生重要影响. Dvorkin和Nur提出了同时包含Biot流动和喷射流动力学机制的统一的BISQ(Biot-Squirt)模型,基于这一模型,尽管有关弹性波在多孔隙介质中的衰减和频散问题已被广泛研究,然而,基于BISQ波传播方程的波场数值模拟至今仍未见报道. 本文从同时包含两种力学机制的孔隙弹性波方程出发,利用FCT有限差分法对含流体孔隙各向同性介质中的地震波和声波进行了数值模拟,并与基于Biot流动的Biot理论之模拟结果进行比较. 数值模拟结果表明:同时包含Biot流动和喷射流动影响的地震波和声波速度比仅包含Biot流动作用的地震波和声波速度慢,慢P波的衰减比根据Biot理论模拟的慢P波衰减更强.