Explicit infiltration equations and the Lambert W-function

The Green and Ampt infiltration formula, as well as the Talsma and Parlange formula, are two-parameter equations that are both expressible in terms of Lambert W-functions. These representations are used to derive explicit, simple and accurate approximations for each case. The two infiltration formulas are limiting cases that can be deduced from an existing three-parameter infiltration equation, the third parameter allowing for interpolation between the limiting cases. Besides the limiting cases, there is another case for which the three-parameter infiltration equation yields an exact solution. The three-parameter equation can be solved by fixed-point iteration, a scheme which can be exploited to obtain a sequence of increasingly complex explicit infiltration equations. For routine use, a simple, explicit approximation to the three-parameter infiltration equation is derived. This approximation eliminates the need to iterate for most practical circumstances.     

On disturbances in a viscoelastic rod (of variable cross-section) of Reiss type placed in a magnetic field

Summary The equations of mechanical motion and electromagnetic equations of Maxwell have been used to investigate the disturbances in a viscoelastic rod of Reiss type placed in a magnetic field.     

粘弹性介质中地震体波Q值的测定方法

本文以常用的几种粘弹性体模拟地球介质的粘弹性,建立了该种介质中的波动方程组;在运用合理的近似处理方法基础上,得到了基本解;提出了校正其它衰减因素的功率谱切比雪夫拟合方法,由此给出了粘性Q值的概念和算法,最后讨论了该方法的可行性。     

Application of a Numerical Inverse Laplace Integration Method to Surface Loading on a Viscoelastic Compressible Earth Model

Normal mode approaches for calculating viscoelastic responses of self-gravitating and compressible spherical earth models have an intrinsic problem of determining the roots of the secular equation and the associated residues in the Laplace domain. To bypass this problem, a method based on numerical inverse Laplace integration was developed by Tanaka et al. (2006, 2007) for computations of viscoelastic deformation caused by an internal dislocation. The advantage of this approach is that the root-finding problem is avoided without imposing additional constraints on the governing equations and earth models. In this study, we apply the same algorithm to computations of viscoelastic responses to a surface load and show that the results obtained by this approach agree well with those obtained by a time-domain approach that does not need determinations of the normal modes in the Laplace domain. Using the elastic earth model PREM and a convex viscosity profile, we calculate viscoelastic load Love numbers (h, l, k) for compressible and incompressible models. Comparisons between the results show that effects due to compressibility are consistent with results obtained by previous studies and that the rate differences between the two models total 10–40%. This will serve as an independent method to confirm results obtained by time-domain approaches and will usefully increase the reliability when modeling postglacial rebound.     

基于非均匀各向同性介质的黏弹性波正演数值模拟

本文基于广义线性黏弹性体,通过引入黏弹性拉梅常量,提出拉梅差异矩阵,给出了非均匀各向同性介质的黏弹性波简化方程.拉梅差异矩阵具有和物性矩阵相似的形式,与弹性、黏弹性物性矩阵之间具有特定的数量关系,体现了黏弹性与弹性物性参数之间的差异程度,可以大大简化方程.并利用该简化方程对SEG/EAGE二维盐丘模型进行了正演数值模拟,将模拟得到的单炮记录和声波、Carcione黏弹性波方程模拟得到的单炮记录进行了对比分析.结果表明该简化方程模拟得到的单炮记录不仅具有较好的效果,而且比Carcione黏弹性波方程计算效率高,与理论分析基本相符.     

On the propagation of disturbances in a composite viscoelastic layer in a magnetic field

Summary The propagation of disturbances in a composite viscoelastic layer when subjected to an impulsive force in the presence of a primary magnetic field, has been investigated in this note. The mechanical displacement has been derived from the electro-magnetic equations of Maxwell, the equation of motion and the stress-strain relations of the materials considered and has been computed for small values of time.     

Stability condition of finite difference solution for viscoelastic wave equations

The stability problem is a very important aspect in seismic wave numerical modeling. Based on the theory of seismic waves and constitutive equations of viscoelastic models, the stability problems of finite difference scheme for KelvinVoigt and Maxwell models with rectangular grids are analyzed. Expressions of stability conditions with arbitrary spatial accuracies for two viscoelastic models are derived. With approximation of quality factor Q≥?5, simplified expressions are developed and some numerical models are given to verify the validity of the corresponding theoretical results. Then this paper summarizes the influences of seismic wave velocity, frequency, size of grid and difference coefficients, as well as quality factor on stability condition. Finally the prerequisite conditions of the simplified stability equations are given with error analysis.     

Dynamic response of single-span beam bridges to a series of moving loads

The paper makes a contribution to the problem of a stream of loads crossing a single-span beam bridge. There are considered the basic load models, in the form of a stream of fixed amplitude forces, unsprung masses and viscoelastic oscillators. The matrix equations of motion of the system are formulated and discussed. The problem of dynamic stability and steady-state response of a bridge carrying a periodic stream of inertial loads is formulated and solved. The paper also includes a vibration study of a beam bridge subjected to a uniform stream of moving loads, of a limited or unlimited number of load cycles.     

适用于更广渗透率范围含泥质砂岩的广义粘弹性BISQ模型

To simultaneously take into account the Biot-flow mechanism, the squirt-flow mechanism, and the frame-viscoelasticity mechanism, a generalized viscoelastic BISQ （Biot/squirt） model is developed for wave propagation in clay-bearing sandstones based on Dvorkin＇s elastic BISQ model. The present model is extended to a wide range of permeabilities （k 〉 0.05 mD） by introducing a dimensionless correction factor for viscoelastic parameters, defined as a function of the permeability and the clay content. We describe the frame＇s stress-strain relationship of the clay-bearing sandstones by the differential constitutive equations of generalized viscoelasticity and then derive the viscoelastic-wave dynamic equations. With the assumption of a plane-wave solution, we finally yield the phase velocities and the attenuation coefficients by solving the dynamic wave equations in the frequency and wave number domain. The comparison of numerical results and experimental data shows that the generalized viscoelastic BISQ model is applicable for modeling the wave propagation in most of the sandstones mainly bearing kaolinite clay.     

Stress relaxation and attenuation of fluid－saturated sandstone at low frequency

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A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.     

Seismic analysis of structures with a fractional derivative model of viscoelastic dampers

Viscoelastic dampers are now among some of the preferred energy dissipation devices used for passive seismic response control. To evaluate the performance of structures installed with viscoelastic dampers, different analytical models have been used to characterize their dynamic force deformation characteristics. The fractional derivative models have received favorable attention as they can capture the frequency dependence of the material stiffness and damping properties observed in the tests very well. However, accurate analytical procedures are needed to calculate the response of structures with such damper models. This paper presents a modal analysis approach, similar to that used for the analysis of linear systems, for solving the equations of motion with fractional derivative terms for arbitrary forcing functions such as those caused by earthquake induced ground motions. The uncoupled modal equations still have fractional derivatives, but can be solved by numerical or analytical procedures. Both numerical and analytical procedures are formulated. These procedures are then used to calculate the dynamic response of a multi-degree of freedom shear beam structure excited by ground motions. Numerical results demonstrating the response reducing effect of viscoelastic dampers are also presented.     

The effect of a vertical temperature gradient in the lithosphere on seismic and tectonic waves

A linear analysis of the stability of the lithosphere considered as a viscoelastic layer with an equilibrium vertical gradient of temperature is carried out. The problem is solved with a complete system of linearized equations of a continuous medium represented in the dimensionless form and containing a set of dimensionless parameters that determine thermomechanical properties of the lithosphere. As a result of the stability analysis, decrements are found that give the time dependence of perturbations and correspond to high-frequency seismic waves and low-frequency tectonic waves. The frequency and velocity of seismic waves are determined by the elasticity and inertial properties of the lithosphere, and their attenuation, by viscous properties of the lithosphere. The temperature gradient existing in the lithosphere influences seismic waves very weakly. On the contrary, the pattern of tectonic waves is controlled by the temperature gradient and viscous properties, while the effect of elastic and inertial properties on these waves is negligibly small. The stability of a viscoelastic lithosphere is examined using such rheological models as the Maxwell, standard linear, and Andrade media (the frequency of tectonic waves is zero in the Maxwell medium).     

Response of piles and pile groups to travelling SH-waves

The response of single piles and pile groups under vertically and obliquely incident seismic waves is obtained using the hybrid boundary element (BEM) formulation. The piles are represented by compressible beam-column elements and the soil as a hysteretic viscoelastic half-space. A recently developed Green function corresponding to the dynamic Mindlin problem is implemented in the numerical formulation. Exact analytical solutions for the differential equations for the piles under distributed harmonic excitations are used. Treating the half-space as a three-dimensional elastic continuum, the interaction problem is formulated by satisfying equilibrium and displacement compatibility along the pile-soil interface. Solutions adopted for the seismic waves are obtained by direct integration of the differential equations in terms of amplitudes. Salient features of the seismic response are identified in several non-dimensional plots. Results of the analyses compare favourably with the limited data available in the literature.     

Seismic wave modeling in viscoelastic VTI media using spectral element method

Spectral element method (SEM) for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries. It is an advanced choice for wave simulations. Due to anelasticity of earth media, SEM for elastic media is no longer appropriate. On fundamental of the second-order elastic SEM, this work takes the viscoelastic wave equations and the vertical transversely isotropic (VTI) media into consideration, and establishes the second-order SEM for wave modeling in viscoelastic VTI media. The second-order perfectly matched layer for viscoelastic VTI media is also introduced. The problem of handling the overlapped absorbed corners is solved. A comparison with the analytical solution in a two-dimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling. Furtherly, numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media. This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.     

Quasi-static deformation of a viscoelastic half-space by a displacement dislocation

The correspondence principle is used to get the Laplace transformed solution of the problem of the quasi-static deformation of a viscoelastic half-space by a shear displacement dislocation from the corresponding static elastic results. The transformed solution is inverted for the Voigt and the Maxwell viscoelastic models. It is shown that, for a vertical dip-slip fault, the surface displacements for the viscoelastic case are identical with the elastic displacements. In the case of a vertical strike-slip fault, detailed numerical results are obtained for both a point source and a finite rectangular source. It is found that the results for the viscoelastic models differ significantly from the corresponding elastic results.     

黏弹TTI介质中旋转交错网格高阶有限差分数值模拟

以Carcione黏弹各向异性理论为基础,给出了适用于黏弹性具有任意倾斜对称轴横向各向同性介质(黏弹TTI介质)的二维三分量一阶速度-应力方程,采用旋转交错网格任意偶数阶精度有限差分格式求解该方程,并推导出了二维黏弹TTI介质完全匹配层(PML)吸收边界条件公式和相应的旋转交错网格任意偶数阶精度有限差分格式,实现了该类介质的地震波场数值模拟.数值模拟结果表明:该方法模拟精度高,边界吸收效果好,可以得到高精度的波场快照和合成记录;并且波场快照和合成记录能较好地反映地下介质的各向异性特征和黏弹性特征.     

线性连续变化非弹性介质中地震波射线轨迹与走时研究

地球内部非弹性介质广泛存在,垂向线性连续变化非弹性介质模型具有重要理论与实际意义,本文利用复速度概念推导得到了复速度随深度线性连续变化的非弹性介质中地震波射线轨迹与等时线方程.理论结果表明:在线性连续变化非弹性介质中,射线轨迹呈圆形,等时面的形状为椭圆.最后对射线轨迹与等时线方程的适用性条件进行了讨论.     

基于Kelvin-Voigt黏弹性骨架的含非饱和流体孔隙介质BISQ模型(英文)

本文综合考虑了在波传播过程中孔隙介质的三种重要力学机制——"Biot流动机制一squirt流动机制-固体骨架黏弹性机制",借鉴等效介质思想,将含水饱和度引入波动力学控制方程,并考虑了不同波频率下孔隙流体分布模式对其等效体积模量的影响,给出了能处理含粘滞性非饱和流体孔隙介质中波传播问题的黏弹性Biot/squirt(BISQ)模型。推导了时间-空间域的波动力学方程组,由一组平面谐波解假设,给出频率-波数域黏弹性BISQ模型的相速度和衰减系数表达式。基于数值算例分析了含水饱和度、渗透率与频率对纵波速度和衰减的影响,并结合致密砂岩和碳酸盐岩的实测数据,对非饱和情况下的储层纵波速度进行了外推,碳酸盐岩储层中纵波速度对含气饱和度的敏感性明显低于砂岩储层。     

Acoustic log simulation in a viscoelastic formation: Cole–Cole model

It is known that sedimentary rocks demonstrate velocity dispersion in the acoustic log frequency range. In this paper we have calculated the waveforms of sonic log for a borehole located in a viscoelastic medium. The acoustic field in the borehole has been obtained for acoustic multipole sources. To describe the viscoelastic properties of a rock we used the Cole–Cole model. This model describes the dispersion of acoustic wave velocities and quality factors in a wide frequency range. To solve the acoustic log direct problem we have applied the double integral Fourier transform (RAI method). The results obtained have shown the feasibility of S-wave velocity dispersion estimation from acoustic dipole waveform processing.