两种不同势函数下半空间饱和多孔介质中Rayleigh波求解比较

分别对"考虑两种压缩波和幅值比例系数"和"考虑一种压缩波(P1或P2波)但不考虑幅值比例系数"两种不同势函数下的半空间饱和多孔介质中Rayleigh波求解进行详细推导,理论分析表明"考虑两种压缩波和幅值比例系数"下Rayleigh波求解推导更为严密,与饱和多孔介质中存在两种压缩波的事实相一致。在研究半空间饱和多孔介质中Rayleigh波时应采用"考虑两种压缩波和幅值比例系数"的势函数。     

含孔隙介质的分层半空间表面瑞利波的衰减特性

以分层半空间内部含有一层孔隙介质为物理模型进行数值计算,研究半空间表面瑞利波的传播和衰减特性.为更加接近实际,结合瑞利波的激发特性,确定了瑞利波的主衰减曲线,并主要以此进行规律分析.针对速度递增和含低速层这两种典型的地质模型,讨论了瑞利波的传播衰减在不同地质模型下的特性,并分析了各自的规律.结果表明,在这两种模型下瑞利波的主衰减曲线都受孔隙介质所处空间位置影响产生比较明显的变化,但衰减系数极大值对应的波长与模型的表层厚度存在较明显的线性对应关系,利用这一关系,可以在实际勘探中快速得到表层介质厚度.另外,通过对比分析还可以看到,瑞利波主衰减曲线随孔隙介质的孔隙度和渗透率的变化都强于主频散曲线的变化,表明衰减曲线对孔隙度和渗透率的变化更加敏感,理论上更加适合进行介质参数反演工作.综合对比结果,我们认为瑞利波主衰减曲线中包含了更丰富的介质参数信息,如果能够有效利用,将可以提高瑞利波勘探的准确性和应用范围.     

平面S波在饱和多孔热弹性介质边界的反射问题研究

针对饱和多孔介质中热弹性波的传播特性问题,基于多孔介质理论和广义的热弹性模型,研究平面S波在饱和多孔热弹性介质边界上的反射问题.以考虑流-固耦合的饱和多孔介质波动方程和热-弹耦合的广义热弹性基本方程出发,建立饱和多孔介质的热-流-固耦合弹性波动模型.通过引入势函数并考虑自由透水和绝热的边界条件,经过理论推导最终给出在饱...     

A viscous boundary for transient analyses of saturated porous media

An absorbing boundary for saturated porous media is developed that can be used for transient analyses in the time domain. The elastic constitutive equations for the saturated porous media follow Bowen's formulation. The method consists of applying viscous tractions along the artificial boundary. The absorbing boundary behaviour is assumed linear and isotropic. Hadamard's conditions provide the speeds of the dilatational and shear waves that propagate in saturated porous media. Since these expressions are frequency independent, the intensities of the viscous tractions are evaluated in the time domain, and the two dilatational waves are accounted for. The viscous tractions are defined from the drained characteristics, assuming an infinite permeability, at variance with the traditional ‘undrained’ method based on undrained characteristics and a null permeability. Solid media and materials with low permeability are also retrieved as subcases. The results show that, at no additional cost, this ‘drained’ method is more accurate for all permeabilities than the ‘undrained’ method, which disregards the existence of the second dilatational wave. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

The influence of gas saturation and pore pressure on the characteristics of the Frenkel-Biot <Emphasis Type="Italic">P</Emphasis> waves in partially saturated porous media

For the case of a partially saturated porous medium, analysis of the conditions is carried out under which the properties of the Frenkel-Biot P waves are similar. The condition of dynamic compatibility (with fulfillment of which a wave of the first kind is propagated without attenuation) is generalized to the case of partially saturated porous media. It is found that the wave connected with the matrix deformation possesses a high attenuation coefficient in a porous medium saturated with a weakly-compressible liquid, but it is a weakly decaying wave in a gas-saturated porous medium. Asymptotic formulas for phase wave velocities are obtained within a low-frequency and high-frequency limit for the general case of a partially saturated porous medium. It is shown that in the domain of low gas saturation, the attenuation coefficient of a wave of the first kind (i.e., a wave connected with the compressibility of phases) depends on the state of the gas in porous space. The following three cases are considered: (1) the microbubbles occluded in the saturating liquid; (2) the microbubbles adsorbed on the walls of pores; and (3) the macrobubbles that completely occupy one or several pores. This characteristic can be used as the diagnostic parameter.     

Numerical analysis of bacterial transport in saturated porous media

A mathematical model to describe bacterial transport in saturated porous media is presented. Reversible/irreversible attachment and growth/decay terms were incorporated into the transport model. Additionally, the changes of porosity and permeability due to bacterial deposition and/or growth were accounted for in the model. The predictive model was used to fit the column experimental data from the literature, and the fitting result showed a good match with the data. Based on the parameter values determined from the literature experimental data, numerical experiments were performed to examine bacterial sorption and/or growth during bacterial transport through saturated porous media. In addition, sensitivity analysis was performed to investigate the impact of key model parameters for bacterial transport on the permeability and porosity of porous media. The model results show that the permeability and porosity of porous media could be altered due to bacterial deposition and growth on the solid matrix. However, variation of permeability due to bacterial growth was trivial compared with natural permeability variation. Copyright © 2005 John Wiley & Sons, Ltd.     

含黏滞流体各向异性孔隙介质中弹性波的频散和衰减

基于Biot理论,考虑液相的黏弹性变形和固液相接触面上的相对扭转,提出了含黏滞流体VTI孔隙介质模型.从理论上推导出,在该模型中除存在快P波、慢P波、SV波、SH波以外,还将存在两种新横波－慢SV波和慢SH波.数值模拟分析了6种弹性波的相速度、衰减、液固相振幅比随孔隙度、频率的变化规律以及快P波、快SV波的衰减随流体性质、渗透率、入射角的变化规律.结果表明慢SV波和慢SH波主要在液相中传播,高频高孔隙度时,速度较高;大角度入射时,快P波衰减表现出明显的各向异性,而快SV波的衰减则基本不变;储层纵向和横向渗透率存在差异时,快SV波衰减大的方向渗透率高.     

Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium

An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [Lo WC, Sposito G, Majer E. Wave propagation through elastic porous media containing two immiscible fluids. Water Resour Res 2005;41:W02025]. The dispersion equation obtained is complex-valued due to viscous dissipation resulting from the relative motion of the solid to the pore fluids. As an excitation frequency is stipulated, the dispersion equation that is a cubic polynomial is numerically solved to determine the phase speed and attenuation coefficient of Rayleigh waves in Columbia fine sandy loam permeated by an air–water mixture. Our numerical results show that, corresponding to three dilatational waves, there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10 Hz–10 kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest.     

Thin layer element method for dynamic soil-structure interaction analysis of axi-symmetric structure in submerged soil

The three-dimensional thin layer element method is formulated for the dynamic response analysis of an axi-symmetric structure in submerged soil. Biot's wave equation for fluid-filled porous medium is used in the formulation. The three-dimensional thin layer element method computes the wave numbers and their associated mode shapes, for both Rayleigh waves and Love waves in submerged soil, which define the characteristics of the waves. The submerged condition affects the characteristics of the Rayleigh waves in soil. As a result, it alters substantially the soil-structure interaction stresses if the permeability of the soil is relatively large and, to less extent, the response of the structure. The thin layer element method is far more efficient than the finite element method for analyzing the fluid-filled porous medium, yet capable of taking into account a multi-layered inhomogeneous soil.     

完全匹配层吸收边界在孔隙介质弹性波模拟中的应用

模拟弹性波在孔隙介质中传播,需要稳定有效的吸收边界来消除或尽可能的减小由人工边界引起的虚假反射. 本文在前人工作基础上,首次建立了弹性孔隙介质情况下完全匹配层吸收边界的高阶速度-应力交错网格有限差分算法,并详细讨论了完全匹配层的构建及其有限差分算法实现. 首先,本文通过均匀孔隙模型的数值解与解析解的对比,验证所提出的数值方法的正确性;然后,本文考察了完全匹配层对不同入射角度入射波和自由表面上的瑞利波的吸收性能,将完全匹配层与廖氏和阻尼吸收边界进行了对比,研究了这三种吸收边界在不同吸收厚度情况下对弹性波吸收能力. 数值结果表明,在孔隙介质中,完全匹配层作为吸收边界能十分有效地吸收衰减外行波,无论对体波还是面波,是一种高效边界吸收算法.     

Effective parameters for flow in saturated heterogeneous porous media

Effective parameters for flow in saturated porous media are obtained via Taylor-Aris-Brenner moment analysis considering both periodic as well as stationary porous medium properties. It is assumed that a slug is instantaneously introduced into an unbounded, anisotropic porous medium having a compressible matrix, and that the correlation length of the local hydraulic conductivity and specific storage fluctuations is smaller than the correlation length of hydraulic head fluctuations (gradually varying flow). It is shown that the effective specific storage is equal to its volume average. The effective hydraulic conductivity is derived by a small-perturbation analysis and it is shown to consist of its volume average and of a second term which accounts for the ‘small’ local conductivity fluctuations.     

A viscous-spring transmitting boundary for cylindrical wave propagation in saturated poroelastic media

Based on the u–U formulation of Biot equation and the assumption of zero permeability coefficient, a viscous-spring transmitting boundary which is frequency independent is derived to simulate the cylindrical elastic wave propagation in unbounded saturated porous media. By this viscous-spring boundary the effective stress and pore fluid pressure on the truncated boundary of the numerical model are replaced by a set of spring, dashpot and mass elements, and its simplified form is also given. A u–U formulation FEA program is compiled and the proposed transmitting boundaries are incorporated therein. Numerical examples show that the proposed viscous-spring boundary and its simplified form can provide accurate results for cylindrical elastic wave propagation problems with low or intermediate values of permeability or frequency content. For general two dimensional wave propagation problems, spuriously reflected waves can be greatly suppressed and acceptable accuracy can still be achieved by placing the simplified boundary at relatively large distance from the wave source.     

Pseudo Rayleigh wave in a partially saturated non-dissipative porous solid

Propagation of surface waves is studied at the pervious boundary of a porous solid saturated with a mixture of two immiscible fluids. An approach, based on continuum mixture theory, is used to derive a secular equation for the propagation of harmonic waves at the stress-free plane surface of this non-dissipative medium. Numerical analysis shows that this secular equation may not represent the propagation of true surface wave in the porous aggregate. Then, this equation is solved numerically for the propagation of pseudo Rayleigh wave or the leaky surface waves. To ensure the existence of pseudo Rayleigh wave, capillary effect between two (wetting and non-wetting) pore-fluids is related to the partial saturation. Effects of porosity and partial saturation coupled with capillary effect are observed on the phase velocity of pseudo Rayleigh waves in sandstone saturated with water-CO₂ mixture.     

黏弹性与弹性介质中Rayleigh面波特性对比研究

Rayleigh面波的频散特性可以用来研究地表浅层结构. 本文使用时域有限差分法来模拟复杂黏弹性介质中的Rayleigh面波,研究了Q值对面波频散特性的影响.文中采用旋转交错网格有限差分,以非分裂卷积形式的完全匹配层为吸收边界,推出了求解二阶位移-应力各向同性黏弹性波动方程的数值方法.为了检验数值解的精度,首先将简单模型的正演结果与解析解对比,验证了方法的正确性;然后模拟了横向缓变层状介质和含有洞穴的介质中的面波,对弹性和黏弹性介质中的面波的频散特性进行对比分析.模拟结果表明浅层Q值对面波的频散特性有显著的影响;强吸收情况下,高阶面波的能量相对低阶面波能量显著增强.     

Testing density-dependent groundwater models: two-dimensional steady state unstable convection in infinite,finite and inclined porous layers

This study proposes the use of several problems of unstable steady state convection with variable fluid density in a porous layer of infinite horizontal extent as two-dimensional (2-D) test cases for density-dependent groundwater flow and solute transport simulators. Unlike existing density-dependent model benchmarks, these problems have well-defined stability criteria that are determined analytically. These analytical stability indicators can be compared with numerical model results to test the ability of a code to accurately simulate buoyancy driven flow and diffusion. The basic analytical solution is for a horizontally infinite fluid-filled porous layer in which fluid density decreases with depth. The proposed test problems include unstable convection in an infinite horizontal box, in a finite horizontal box, and in an infinite inclined box. A dimensionless Rayleigh number incorporating properties of the fluid and the porous media determines the stability of the layer in each case. Testing the ability of numerical codes to match both the critical Rayleigh number at which convection occurs and the wavelength of convection cells is an addition to the benchmark problems currently in use. The proposed test problems are modelled in 2-D using the SUTRA [SUTRA––A model for saturated–unsaturated variable-density ground-water flow with solute or energy transport. US Geological Survey Water-Resources Investigations Report, 02-4231, 2002. 250 p] density-dependent groundwater flow and solute transport code. For the case of an infinite horizontal box, SUTRA results show a distinct change from stable to unstable behaviour around the theoretical critical Rayleigh number of 4π² and the simulated wavelength of unstable convection agrees with that predicted by the analytical solution. The effects of finite layer aspect ratio and inclination on stability indicators are also tested and numerical results are in excellent agreement with theoretical stability criteria and with numerical results previously reported in traditional fluid mechanics literature.     

Controls on event runoff coefficients in the eastern Italian Alps

Analyses of event runoff coefficients provide essential insight on catchment response, particularly if a range of catchments and a range of events are compared by a single indicator. In this study we examine the effect of climate, geology, land use, flood types and initial soil moisture conditions on the distribution functions of the event runoff coefficients for a set of 14 mountainous catchments located in the eastern Italian Alps, ranging in size from 7.3 to 608.4 km². Runoff coefficients were computed from hourly precipitation, runoff data and estimates of snowmelt. A total of 535 events were analysed over the period 1989–2004. We classified each basin using a “permeability index” which was inferred from a geologic map and ranged from “low” to “high permeability”. A continuous soil moisture accounting model was applied to each catchment to classify ‘wet’ and ‘dry’ initial soil moisture conditions. The results indicate that the spatial distribution of runoff coefficients is highly correlated with mean annual precipitation, with the mean runoff coefficient increasing with mean annual precipitation. Geology, through the ‘permeability index’, is another important control on runoff coefficients for catchments with mean annual precipitation less than 1200 mm. Land use, as indexed by the SCS curve number, influences runoff coefficient distribution to a lesser degree. An analysis of the runoff coefficients by flood type indicates that runoff coefficients increase with event snowmelt. Results show that there exists an intermediate region of subsurface water storage capacity, as indexed by a flow–duration curve-based index, which maximises the impact of initial wetness conditions on the runoff coefficient. This means that the difference between runoff coefficients characterised by wet and dry initial conditions is negligible both for basins with very large storage capacity and for basins with small storage capacity. For basins with intermediate storage capacities, the impact of the initial wetness conditions may be relatively large.     

Internal waves and temperature fronts on slopes

Time series measurements from an array of temperature miniloggers in a line at constant depth along the sloping boundary of a lake are used to describe the ‘internal surf zone” where internal waves interact with the sloping boundary. More small positive temperature time derivatives are recorded than negative, but there are more large negative values than positive, giving the overall distribution of temperature time derivatives a small negative skewness. This is consistent with the internal wave dynamics; fronts form during the up-slope phase of the motion, bringing cold water up the slope, and the return flow may become unstable, leading to small advecting billows and weak warm fronts. The data are analysed to detect ‘events’, periods in which the temperature derivatives exceed a set threshold. The speed and distance travelled by ‘events’ are described. The motion along the slope may be a consequence of (a) instabilities advected by the flow (b) internal waves propagating along-slope or (c) internal waves approaching the slope from oblique directions. The propagation of several of the observed ‘events’ can only be explained by (c), evidence that the internal surf zone has some, but possibly not all, the characteristics of the conventional ‘surface wave’ surf zone, with waves steepening as they approach the slope at oblique angles.     

Study of pore pressure variation during liquefaction using two constitutive models for sand

Numerical analyses of liquefiable sand are presented in this paper. Liquefaction phenomenon is an undrained response of saturated sandy soils when they are subjected to static or dynamic loads. A fully coupled dynamic computer code is developed to predict the liquefaction potential of a saturated sandy layer. Coupled dynamic field equations of extended Biot's theory with u–P formulation are used to determine the responses of pore fluid and soil skeleton. Generalized Newmark method is employed for integration in time. The soil behavior is modelled by two constitutive models; a critical state two-surface plasticity model, and a densification model. A class ‘B’ analysis of a centrifuge experiment is performed to simulate the dynamic response of level ground sites. The results of the numerical analyses demonstrate the capability of the critical sate two-surface plasticity model in producing pore pressures that are consistent with observations of the behavior of liquefiable sand in the centrifuge test.     

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.