Elastic waves in non-Newtonian (Maxwell) fluid-saturated porous media

Abstract

This paper studies the elastic waves in non-Newtonian (Maxwell) fluid-saturated porous media with the nonzero boundary slip velocity for pore size distribution. The coefficient bF _m (ω) that measures the deviation from Poiseuille flow friction in such media is presented. Based on this coefficient, we investigate the properties of elastic waves by calculating their phase velocities and attenuation coefficients as functions of frequency and the behaviour of the dynamic permeability. The study shows that the pore size distribution removes oscillations in all physical quantities in the non-Newtonian regime. Consideration of the nonzero boundary slip effect in non-Newtonian (Maxwell) fluid-saturated porous media results in (a) an overall increase of the dynamic permeability, (b) an increase of phase velocities of fast Biot waves and shear waves except in the low frequency domain and an overall increase of phase velocity of slow Biot waves and (c) an overall increase of the attenuation of three Biot waves in the intermediate frequency domain except in the deeply non-Newtonian regime. The study also shows that the attenuation coefficient of slow Biot waves is small in the deeply non-Newtonian regime at higher frequency, which encourages us to detect slow Biot waves in oil-saturated porous rock.     

基于Biot-喷射流统一模型Maxwell流体饱和孔隙介质中的弹性波

推广Biot-Tsiklauri声学模型的同时借鉴Dvorkin和Nur的工作,建立了具有任意孔径分布并顾及喷射流动机制的非牛顿流体饱和孔隙介质声学模型,研究了非牛顿流体(Maxwell流体)饱和孔隙介质中的弹性波的衰减和频散特性.着重讨论充孔隙Maxwell流体的非牛顿流效应对弹性波的频散和衰减的影响.研究表明,饱和流体的非牛顿流效应和喷射流动机制均是引起弹性波波频散和衰减的重要因素.依据非牛顿流体(Maxwell流体)饱和各向同性孔隙介质的Biot-喷射流声学模型,喷射流动只影响纵波的频散和衰减,而饱和流体的非牛顿流效应不仅影响纵波,而且还影响横波的频散和衰减.     

多孔介质BISQ模型中的慢纵波

着重研究了多孔介质BISQ模型中慢纵波的基本特性.给出了BISQ模型下慢纵波速度 和衰减的低频近似公式.与Biot理论对比，BISQ模型中慢纵波的衰减随频率降低急剧增大， 且随喷射流长度的减小而增加；相速度随喷射流长度的减小而增加，其低频极限值不是零； 孔隙流体位移与固相骨架位移之比的幅值随喷射流长度的增加而减小，其相位特点与Biot模 型预测的不同；在流体与孔隙介质的边界上可产生更大的渗流.为对比，同时也给出快纵波 的行为.依据BISQ模型可推断：非黏滞流体饱和孔隙介质中不存在喷射流机理；BISQ模型中 关键词： 多孔介质 喷射流 慢纵波 动力协调     

Borehole guided waves in a non-Newtonian (Maxwell) fluid-saturated porous medium

<正>The property of acoustic guided waves generated in a fluid-filled borehole surrounded by a non-Newtonian (Maxwell) fluid-saturated porous formation with a permeable wall is investigated.The influence of non-Newtonian effects on acoustic guided waves such as Stoneley waves,pseudo-Rayleigh waves,flexural waves,and screw waves propagations in a fluid-filled borehole is demonstrated based on the generalized Biot-Tsiklauri model by calculating their velocity dispersion and attenuation coefficients.The corresponding acoustic waveforms illustrate their properties in time domain.The results are also compared with those based on generalized Biot's theory.The results show that the influence of non-Newtonian effect on acoustic guided wave,especially on the attenuation coefficient of guided wave propagation in borehole is noticeable.     

Rayleigh wave propagation along the boundary of a non-Newtonian fluid-saturated porous medium

The Frenkel-Biot theory is used to study the reflection of elastic waves from the boundary of a non-Newtonian (Maxwell) fluid-saturated porous medium. The velocity and attenuation of a Rayleigh surface wave propagating along the boundary of the medium are determined. Two models of a fluid-saturated porous medium are used for calculation: with pore channels of a fixed diameter and with a lognormal distribution of pore channels in size. The results of calculations show that, when the fluid in the porous medium is characterized by a small Deborah number (i.e., exhibits non-Newtonian properties), the velocity of Rayleigh waves exhibits a considerable frequency dispersion. The results also suggest that, in principle, it is possible to estimate the Deborah number from the measured frequency dispersion of the Rayleigh wave velocity.     

Acoustic diagnosis for porous medium with circular cylindrical pores

Acoustic transmission coefficient and phase velocity of a Lucite slab with circular cylindrical pores with a nonrigid pore frame were experimentally and theoretically investigated. For theoretical investigation a new phenomenological model, the modified Biot-Attenborough (MBA) model, was proposed. The MBA model takes into account both the first kind and the second kind of waves introduced by Biot. It also separately considers viscous and thermal effects with three new phenomenological parameters: boundary, phase velocity, and impedance parameters. The theoretical estimation with three phenomenological parameters shows reasonably good agreement with the experimental data. The physical characteristics of porous medium such as porosity and pore size can be inversely analyzed in terms of the acoustic data such as the transmission coefficient and phase velocity as the functions of porosity and frequency. This makes acoustic diagnosis possible for noninvasively investigating physical characteristics of porous media such as bones and ocean sediments.     

Fast compressional wave attenuation and dispersion due to conversion scattering into slow shear waves in randomly heterogeneous porous media

Within the viscosity-extended Biot framework of wave propagation in porous media, the existence of a slow shear wave mode with non-vanishing velocity is predicted. It is a highly diffusive shear mode wherein the two constituent phases essentially undergo out-of-phase shear motions (slow shear wave). In order to elucidate the interaction of this wave mode with propagating wave fields in an inhomogeneous medium the process of conversion scattering from fast compressional waves into slow shear waves is analyzed using the method of statistical smoothing in randomly heterogeneous poroelastic media. The result is a complex wave number of a coherent plane compressional wave propagating in a dynamic-equivalent homogeneous medium. Analysis of the results shows that the conversion scattering process draws energy from the propagating wave and therefore leads to attenuation and phase velocity dispersion. Attenuation and dispersion characteristics are typical for a relaxation process, in this case shear stress relaxation. The mechanism of conversion scattering into the slow shear wave is associated with the development of viscous boundary layers in the transition from the viscosity-dominated to inertial regime in a macroscopically homogeneous poroelastic solid.     

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density–elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcy’s law for fluid flow and micro–poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.     

Cancellous bone analysis with modified least squares Prony's method and chirp filter: phantom experiments and simulation

The presence of two longitudinal waves in porous media is predicted by Biot's theory and has been confirmed experimentally in cancellous bone. When cancellous bone samples are interrogated in through-transmission, these two waves can overlap in time. Previously, the Modified Least-Squares Prony's (MLSP) method was validated for estimation of amplitudes, attenuation coefficients, and phase velocities of fast and slow waves, but tended to overestimate phase velocities by up to about 5%. In the present paper, a pre-processing chirp filter to mitigate the phase velocity bias is derived. The MLSP/chirp filter (MLSPCF) method was tested for decomposition of a 500 kHz-center-frequency signal containing two overlapping components: one passing through a low-density-polyethylene plate (fast wave) and another passing through a cancellous-bone-mimicking phantom material (slow wave). The chirp filter reduced phase velocity bias from 100 m/s (5.1%) to 69 m/s (3.5%) (fast wave) and from 29 m/s (1.9%) to 10 m/s (0.7%) (slow wave). Similar improvements were found for 1) measurements in polycarbonate (fast wave) and a cancellous-bone-mimicking phantom (slow wave), and 2) a simulation based on parameters mimicking bovine cancellous bone. The MLSPCF method did not offer consistent improvement in estimates of attenuation coefficient or amplitude.     

Analysis of multiscale scattering and poroelastic attenuation in a real sedimentary rock sequence

Compressional waves in heterogeneous permeable media experience attenuation from both scattering and induced pore scale flow of the viscous saturating fluid. For a real, finely sampled sedimentary sequence consisting of 255 layers and covering 30 meters of depth, elastic and poroelastic computer models are applied to investigate the relative importance of scattering and fluid-flow attenuation. The computer models incorporate the known porosity, permeability, and elastic properties of the sand/shale sequence in a binary medium, plane layered structure. The modeled elastic scattering attenuation is well described by stochastic medium theory if two-length scale statistics are applied to reflect the relative thickness of the shale layers when compared to the sand layers. Under the poroelastic Biot/squirt flow model, fluid-flow attenuation from the moderate permeability (10(-14) m2) sands may be separated in the frequency domain from the attenuation due to the low permeability (5 x 10(-17) m2) shale layers. Based on these models, the overall attenuation is well approximated by the sum of the scattering attenuation from stochastic medium theory and the volume weighted average of the attenuations of the sequence member rocks. These results suggest that a high permeability network of sediments or fractures in a lower permeability host rock may have a distinct separable attenuation signature, even if the overall volume of high permeability material is low. Depending on the viscosity of the saturating fluid, the magnitude of the flow-based attenuation can dominate or be dominated by the scattering attenuation at typical sonic logging frequencies (approximately 10 kHz).     

Estimation of critical and viscous frequencies for Biot theory in cancellous bone

The use of Biot theory for modelling ultrasonic wave propagation in porous media involves the definition of a "critical frequency" above which both fast and slow compressional waves will, in principle, propagate. Critical frequencies have been evaluated for healthy and osteoporotic cancellous bone filled with water or marrow, using data from the literature. The range of pore sizes in bone gives rise to a critical frequency band rather than a single critical frequency, the mean of which is lower for osteoporotic bone than normal bone. However, the critical frequency is a theoretical concept and previous researchers considered a more realistic "viscous frequency" above which both fast and slow waves may be experimentally observed. Viscous frequencies in bone are found to be several orders of magnitude greater than calculated critical frequencies. Whereas two waves may well be observed at all ultrasonic frequencies for water-filled cancellous bone at 20 degrees C, it is probable megahertz frequencies would be needed for observation of two waves in vivo.     

Effect of interfacial slip on the kinematic and dynamic parameters of elastic waves in a fluid-saturated porous medium

The method proposed by Bedford, Costley, and Stern (in 1984) is used to derive the expressions for the drag and virtual mass coefficients involved in the equations of the acoustics of fluid-saturated porous media taking into account the interfacial slip. Special consideration is given to the case of gas-filled pores, which allows one to obtain the expression for the isothermal slip factor in an explicit form by solving the Boltzmann kinetic equation. It is shown that, for longitudinal waves of the first kind and transverse waves, the effect of the interfacial slip on their velocities is small. The presence of the interfacial slip leads to an increase in the attenuation coefficients of these waves, but the corresponding calculated values prove to be much smaller than the measured ones. For the longitudinal waves of the second kind, the effect of the interfacial slip on their kinematic and dynamic parameters is considerable and can be estimated experimentally.     

Bifurcation of the Biot slow wave in a porous medium

The propagation of the Biot slow wave in a fluid-saturated porous medium at low frequencies is investigated by asymptotic methods. It is proven that the Biot wave has a bifurcation behavior depending on its wave number. The bifurcation occurs in a neighborhood of the critical value k(cr), which depends on the permeability of a medium and the viscosity of a fluid. The P2 wave is fully attenuated if its wave number is smaller than k(cr) and it becomes propagatory with wave numbers bigger than k(cr). Asymptotic formulas for the phase velocity and attenuation of the Biot wave are derived.     

利用井中偶极弯曲波反演渗透率的影响因素及灵敏度分析

本文以各向同性饱和流体孔隙介质充液井孔中声场传播理论为基础,系统考察了影响偶极弯曲波特征的主要因素,分析了快慢地层条件下弯曲波衰减和速度对不同参数的敏感度,对比结果表明,弯曲波的速度基本不受地层渗透率的影响,其衰减对渗透率的敏感性要远高于速度。弯曲波中心频率会随着波的衰减而发生偏移,说明利用弯曲波衰减和频移特征联合反演地层渗透率是可行的。通过对比快慢地层下的弯曲波衰减和速度灵敏度曲线,发现在慢速地层中反演渗透率的结果要好于快速地层。同时,本文也指出在低孔低渗储层中,还可以考虑利用弯曲波衰减识别气层。     

含少量气泡流体饱和孔隙介质中的弹性波

考虑孔隙流体中含有少量气泡,且气泡在声波作用下线性振动,研究声波在这种孔隙介质中的传播特性.本文先由流体质量守恒方程和孔隙度微分与流体压力微分的关系推导出了含有气泡形式的渗流连续性方程;在处理渗流连续性方程中的气体体积分数时间导数时,应用Commander气泡线性振动理论导出气体体积分数时间导数与流体压强时间导数的关系,进而得到了修正的Biot形式的渗流连续性方程;最后结合Biot动力学方程求得了含气泡形式的位移场方程,便可得到两类纵波及一类横波的声学特性.通过对快、慢纵波的频散、衰减及两类波引起的流体位移与固体位移关系的考察,发现少量气泡的存在对快纵波和慢纵波的传播特性影响较大.     

Effect of porous surface layer on wave propagation in elastic cylinder immersed in fluid

To study the damage to an elastic cylinder immersed in fluid, a model of an elastic cylinder wrapped with a porous medium immersed in fluid is designed. This structure can both identify the properties of guided waves in a more practical model and address the relationship between the cylinder damage degree and the surface and surrounding medium. The principal motivation is to perform a detailed quantitative analysis of the longitudinal mode and flexural mode in an elastic cylinder wrapped with a porous medium immersed in fluid. The frequency equations for the propagation of waves are derived each for a pervious surface and an impervious surface by employing Biot theory. The influences of the various parameters of the porous medium wrapping layer on the phase velocity and attenuation are discussed. The results show that the influences of porosity on the dispersion curves of guided waves are much more significant than those of thickness, whereas the phase velocity is independent of the static permeability. There is an apparent "mode switching" between the two low-order modes. The characteristics of attenuation are in good agreement with the results from the dispersion curves. This work can support future studies for optimizing the theory on detecting the damage to cylinder or pipeline.     

Features of acoustic waves in media with large porosity values in the framework of the Biot theory

The Biot theory is used to study the properties of acoustic waves in fluid-saturated porous media with large porosity values. Materials are considered for which a large elasticity modulus of the matrix (skeleton) is comparable in size to the large fluid pore saturation modulus.     

Wave-induced fluid flow in random porous media: attenuation and dispersion of elastic waves

A detailed analysis of the relationship between elastic waves in inhomogeneous, porous media and the effect of wave-induced fluid flow is presented. Based on the results of the poroelastic first-order statistical smoothing approximation applied to Biot's equations of poroelasticity, a model for elastic wave attenuation and dispersion due to wave-induced fluid flow in 3-D randomly inhomogeneous poroelastic media is developed. Attenuation and dispersion depend on linear combinations of the spatial correlations of the fluctuating poroelastic parameters. The observed frequency dependence is typical for a relaxation phenomenon. Further, the analytic properties of attenuation and dispersion are analyzed. It is shown that the low-frequency asymptote of the attenuation coefficient of a plane compressional wave is proportional to the square of frequency. At high frequencies the attenuation coefficient becomes proportional to the square root of frequency. A comparison with the 1-D theory shows that attenuation is of the same order but slightly larger in 3-D random media. Several modeling choices of the approach including the effect of cross correlations between fluid and solid phase properties are demonstrated. The potential application of the results to real porous materials is discussed.     

流体/准饱和多孔介质中伪Scholte波的传播特性

研究流体/多孔介质界面Scholte波的传播特性对于水下勘探、地震工程等领域具有重要意义.本文基于Biot理论和等效流体模型,采用势函数方法,推导了描述有限厚度流体/准饱和多孔半空间远场界面波的特征方程和位移、孔压计算公式.在此基础上,分别以砂岩和松散沉积土为例,研究了流体/硬多孔介质和流体/软多孔介质两种情况下,可压缩流体层厚度和多孔介质饱和度对伪Scholte波传播特性的影响.结果表明：多孔介质软硬程度显著影响界面波的种类、相速度、位移和水压力分布;有限厚度流体/饱和多孔半空间界面处伪Scholte波相速度与界面波波长和流体厚度的比值有关;孔隙水中溶解的少量气体对剪切波的相速度的影响不大,对压缩波相速度、伪Scholte波相速度和孔隙水压力分布影响显著.     

孔隙介质中快纵波的衰减特性和动力协调现象

依据Biot理论并采用前人的骨架模量孔隙度关系,计算了弹性波在孔隙介质中的衰减曲线,发现快纵波存在零衰减点.为解释这种现象,分析了孔隙流体相对于骨架运动的特点.结果表明,存在快纵波动力协调介质,快纵波在这种介质中传播时,流体和骨架之间无宏观相对位移,因此无论频率多高,流体粘滞系数多大,快纵波都不衰减.一个引申的结论是,在动力协调的渗透性孔隙介质中,即使存在双电层,快纵波也不产生流动电势.