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.     

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.     

多层弹性半空间中的地震波（一）

为了了解地震震源和地球介质的性质,很有必要对地震波的辐射、传播和衰减问题作仔细的分析。作为一种近似,可以暂且忽略地球的曲率,把传播地震波的地球介质视为多层半空间。为简便起见,地震波的衰减问题另作考虑。这样,便需要研究多层、均匀、各向同性和完全弹性半空间中地震震源辐射的地震波传播问题。 用哈斯克尔（Haskell）矩阵法解多层介质中弹性波的传播问题是很方便的。如果     

Viscoelastic approximation of poroelastic media for wave scattering problems

Herein, we propose a simplified and approximate method for solving dynamic problems in poroelastic media. The method is based on the definition of equivalent-viscoelastic materials that have the same wave numbers as poroelastic media. The viscoelastic approximation was applied to study the scatter of plane compressional waves by a spherical poroelastic inhomogeneity. In this wave scattering problem, the poroelastic and equivalent-viscoelastic solutions gave almost identical results far away from the inhomogeneity. The proposed method extends our existing numerical and analytical methods for poroelastic media. It is useful to derive approximate analytical solutions which can be applied for assessing the damping characteristics of small amplitude waves in saturated soils.     

层状不均匀介质中合成地震图的反射率法

本文应用传播函数矩阵的方法把Fuchs和Müller计算合成地震图的反射率法推广到层状不均匀介质中。在高频条件下,每层的基本矩阵用WKBJ渐近解表示,同时考虑到射线具有迴折点的情形。实例表明,这一算法是可行的,与原方法相比,在计算层状不均匀介质中的合成地震图方面可节省大量的计算机时。     

利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播

烃类储集层是一种复合多相介质，在固体颗粒的空隙中含有气体或液体. 研究弹性波在该类地层中的传播规律对于油气勘探开发，特别对于全波列声波测井有重要意义. 为了提高孔隙弹性介质数值模拟的计算效率，本文采用改进显式交错网格有限差分算法取代常用的空间域四阶和时间域二阶的速度 - 应力有限差分算法，算法的空间域为八阶、时间域为二阶. 虽然计算的时间步长略小于空间域四阶的情形，但高阶有限差分算法可以选择较粗糙的网格，因此补偿了计算的低效；同时高阶交错网格有限差分算法的空间频散性比低阶算法小. 利用该算法计算了一个两层模型的波场，同时还模拟了等效弹性和孔隙弹性模型中波的传播. 结果表明慢波及其影响明显，尽管慢波衰减很快，但被某一界面反射后，转换形成的P波和S波仍以正常的方式传播，且比慢波衰减小.     

Discussion of “development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation” by Chinmoy Kolay and James M. Ricles

The time integration method proposed by Kolay and Ricles, which was claimed to be both explicit and unconditionally stable, is shown to be implicit in the sense of requiring the factorization of an effective stiffness matrix where an explicit method needs no solver. Its original derivation procedure employed discrete control theory concepts, which are in fact, equivalent to conventional recurrence relation concepts aiming to match its spectral properties with those of the three-parameter optimal/generalized-α method, thus giving rise to an implicit method within the class of linear multistep methods. It is shown that the resulting method possesses several added computational drawbacks due to its derivation procedure, such as additional effective stiffness inversions and a degraded order of accuracy in general.     

Convergence analyses of different modeling schemes for generalized Lippmann–Schwinger integral equation in piecewise heterogeneous media

For wave propagation simulation in piecewise heterogeneous media, Gaussian-elimination-based full-waveform solutions to the generalized Lippmann–Schwinger integral equation (GLSIE) are highly accurate, but involved with extremely time-consuming computations because of the very large size of the resulting boundary–volume integral equation matrix to be inverted. Several flexible approximations to the GLSIE are scaled in an iterative way to adapt numerical solutions to the smoothness of heterogeneous media in terms of incident wavelengths, with a great saving of computing time and memory. Among various typical iterative schemes to the GLSIE matrix, the generalized minimal residual method (GMRES) is an efficient approach to reduce the computational intensity to some degree. The most efficient approximation can be obtained using a Born series, as an alternative iterative solution, to both the boundary-scattering and volume-scattering waves, leading to the Born-series approximation (BSA) scheme and the improved Born-series approximation (IBSA) scheme. These iteration schemes are validated by dimensionless frequency responses to a heterogeneous semicircular alluvial valley, and then applied to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments, compared with the full-waveform numerical solution, indicate that the convergence rates of these methods decrease gradually with increasing velocity perturbations. The comparison also shows that the BSA scheme has a faster convergence than the GMRES method for velocity perturbations less than 10 percent, but converges slowly and even hardly achieves convergence for velocity perturbations greater than 15 percent. The IBSA scheme gives a superior performance over the other methods, with the least iterations to achieve the necessary convergence.     

Dynamic stiffness and kinematic response of single piles in inhomogeneous soil

The effect of soil inhomogeneity on dynamic stiffness and kinematic response of single flexural elastic piles to vertically-propagating seismic SH waves is explored. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous stratum. A layered soil with piece-wise homogeneous properties is introduced to approximate the continuous inhomogeneity in the realm of a Beam-on-Dynamic-Winkler-Foundation model. The problem is treated numerically by means of a layer transfer-matrix (Haskell–Thompson) formulation, and validated using available theoretical solutions and finite-element analyses. The role of salient model parameters such as pile-head fixity conditions, pile-to-soil stiffness ratio, surface-to-base shear wave velocity ratio and rate of inhomogeneity is elucidated. A new normalization scheme for inertial and kinematic response of such systems is presented based on an average Winkler wavenumber. With reference to long piles in moderately inhomogeneous soils, results indicate that: (a) kinematic pile response is essentially governed by a single dimensionless frequency parameter accounting for pile-to-soil stiffness ratio, pile slenderness and soil inhomogeneity and (b) definition of a characteristic pile wavelength allows an approximate estimation of pile elastodynamic response for preliminary design or analysis. Issues related to domain discretization and Winkler moduli are discussed.     

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.     

A fundamental solution of a multilayered half-space due to an impulsive ring source

The fundamental solutions of axisymmetric elastodynamic problem for the multilayered half-space due to an impulsive ring source acting within a layered elastic media are derived in time domain with the aid of Laplace–Hankel mixed transform and transfer matrix techniques. In addition, an effective numerical procedure, which utilizes the fast Hankel transform algorithm, is also proposed to calculate these solutions. Illustrative examples have been given to demonstrate that the fundamental solutions can be readily evaluated and the numerical results are of high accuracy. The present solutions can be directly applied to determine the transient wave fields caused by a seismic source and show the potential application to the elastodynamic problems solved by the boundary element method.     

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.     

两相介质近场波动模拟的一种解耦有限元方法

本文将求解近场波动问题的一种解耦技术推广到两相介质,得到了一种求解两相介质近场波动问题的直接解耦方法,包括集中质量有限元模型、时域显式积分格式和局部人工边界条件. 首先应用加权残数法,并依据波动模拟的精度要求,得到了两相介质集中质量有限元模型. 然后,结合两相介质中波动的衰减特性,实现了透射边界在两相介质近场波动中的运用. 最后,通过数值实验,并与解析解对比,验证了本文方法的有效性.     

Finite difference modeling of seismic wave propagation in monoclinic media

Reported in the present paper are the results of the study of propagation of SH waves in the plane of mirror symmetry of a monoclinic multilayered medium with displacement normal to the plane. Dispersion equation has been obtained analytically ussing Haskell’s matrix method, while the finite-difference method has been employed to model the SH-wave propagation to study its phase and group velocities. The stability analysis has been carried out to minimize the exponential growth of the error of finite difference approximation in order to make the finite difference method stable and convergent. Further, variations of phase velocity with respect to both wave number and dispersion parameter for different stability ratios in monoclinic media have been examined and shown graphically. The effect of change of stability ratio on the group velocity of the wave propagation has been also investigated. Likewise, the effects of change of dispersion parameter on phase velocity and the variation of frequency with increase of wave number have been graphically represented and discussed.     

A discontinuous Galerkin method for seismic wave propagation in coupled elastic and poroelastic media

Numerical simulation in coupled elastic and poroelastic media is important in oil and gas exploration. However, the interface between elastic and poroelastic media is a challenge to handle. In order to deal with the coupled model, the first-order velocity–stress wave equations are used to unify the elastic and poroelastic wave equations. In addition, an arbitrary high-order discontinuous Galerkin method is used to simulate the wave propagation in coupled elastic–poroelastic media, which achieves same order accuracy in time and space domain simultaneously. The interfaces between the two media are explicitly tackled by the Godunov numerical flux. The proposed forms of numerical flux can be used efficiently and conveniently to simulate the wave propagation at the interfaces of the coupled model and handle the absorbing boundary conditions properly. Numerical results on coupled elastic–poroelastic media with straight and curved interfaces are compared with those from a software that is based on finite element method and the interfaces are handled by boundary conditions, demonstrating the feasibility of the proposed scheme in dealing with coupled elastic–poroelastic media. In addition, the proposed method is used to simulate a more complex coupled model. The numerical results show that the proposed method is feasible to simulate the wave propagation in such a media and is easy to implement.     

Review of the Anisotropic Interface Ray Propagator: Symplecticity,Eigenvalues, Invariants and Applications

A review of the 6 × 6 anisotropic interface ray propagator matrix in Cartesian coordinates and within the framework of the Hamiltonian formalism shows that there is one unique propagator satisfying the symplectic property. This is essential, since the symplecticity furnishes an exact inverse, while an eigenvalue analysis indicates that the propagator may be arbitrarily ill-conditioned. As such, the symplectic interface propagator naturally connects to symplectic ray integration algorithms for smooth media, designed to maintain accuracy. Moreover, several ray invariants for smooth media remain invariant across interfaces. It is straightforward to derive expressions for the interface propagator, both explicit and implicit. Symplecticity is equivalent to the condition that the propagator preserves the eikonal constraint across the interface. The symplectic interface propagator complies with phase matching of the incident and reflected/transmitted ray field, and is therefore in accordance with the earlier derived 4 × 4 matrix in ray-centred coordinates. The symplectic property is related to the symmetry of the second derivative matrix of the reflected/transmitted traveltime field. Thanks to the analytic expression of the symplectic interface propagator, relating interface curvature directly to second derivatives of traveltimes observed at a datum level, numerous applications are available in the area of processing and inversion.     

Boundary element method applied to linear soil–structure interaction on a heterogeneous soil

An extension of the boundary element method to heterogeneous domains composed of horizontal layers is here proposed. It includes a numerical computation of the corresponding Green's functions, thanks to an inverse Hankel transform of the closed form solutions obtained in the spectral domain with suitable variables derived from displacements and stress vectors to obtain the decoupling between P–SV and SH waves. Transmission and reflection operators are introduced to avoid the problem of overflowing exponentials met with in Thomson–Haskell matrices. Applications are given in the soil–structure interaction field to compute the impedances of surface and embedded circular foundations resting on a viscoelastic halfspace.     

Rocking vibrations of a rigid embedded foundation in a poroelastic soil layer

An approximate analytical method is presented for the dynamic response of a rigid cylindrical foundation embedded in a poroelastic soil layer under the excitation of a time-harmonic rocking moment. The soil underlying the foundation base is represented by a single-layered poroelastic soil based on rigid bedrock while the soil along the side of the foundation is modeled as an independent poroelastic stratum composed of a series of infinitesimally thin layers. The accuracy of the present solution is verified by comparisons with existing solutions obtained from other researchers. Numerical results for the rocking dynamic impedance and dynamic response factor are presented to demonstrate the influence of nondimensional frequency of excitation, poroelastic soil layer thickness, depth ratio of the foundation and internal friction of the poroelastic soil.     

The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid

This paper discusses surface displacements, surface strain, rocking, and energy partitioning during reflection-of-plane waves in a fluid-saturated poroelastic half-space. The medium is modeled by Biot's theory, and is assumed to be saturated with inviscid fluid. A linear porosity-modulus relation based on experimental data on sandstones is used to determine the material parameters for Biot's model. Numerical results in terms of angle of incident waves and Poisson's ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the amount of solid frame stiffness controls the response of a fluid-saturated porous system. A poroelastic medium with essentially dry-frame stiffness behaves like an elastic medium, and the influence of pore fluid increases as dry-frame stiffness is reduced. The effects of a second P-wave become noticeable in poroelastic media with low dry-frame stiffness.     

完全匹配层在数值模拟孔隙介质声波传播中的应用

The nonsplitting perfectly matched layer （NPML） absorbing boundary condition （ABC） was first provided by Wang and Tang （2003） for the finite-difference simulation of elastic wave propagation in solids. In this paper, the method is developed to extend the NPML to simulating elastic wave propagation in poroelastic media. Biot＇s equations are discretized and approximated to a staggered-grid by applying a fourth-order accurate central difference in space and a second-order accurate central difference in time. A cylindrical twolayer seismic model and a borehole model are chosen to validate the effectiveness of the NPML. The results show that the numerical solutions agree well with the solutions of the discrete wavenumber （DW） method.