Dynamic response to fluid extraction from a poroelastic half‐space

In this study, the dynamic response of a poroelastic half‐space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re‐formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full‐space fundamental singular solution for a point sink is also derived using potential methods. The mirror‐image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half‐space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.     

Displacement ring load Green's functions for saturated porous transversely isotropic tri‐material full‐space

Based on the Biot's poroelastic theory and using scalar potential functions both the ring load and point load displacement Green's functions for a transversely isotropic saturated porous full‐space composed of an upper half‐space, a finite thickness middle layer and a lower half‐space is analytically presented for the first time. It is assumed that each region consists of a different transversely isotropic material. The equations of poroelastodymanics in terms of the solid displacements and the pore fluid pressure are uncoupled with the help of two scalar potential functions, so that the governing equations for the potential functions are either a second order wave equation or a repeated wave‐heat transfer equation of sixth order. With the aid of Fourier expansion with respect to circumferential direction and Hankel integral transforms with respect to the radial direction in cylindrical coordinate system, the response is determined in the form of line integrals in the real space, followed by theorem of inverse Hankel integral transforms. The solutions degenerate to a single phase elastic material, and the results are compared with previous studies, where an excellent agreement may be observed with the results provided in the literature. Some examples of displacement Green's functions are finally given to illustrate the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Wave scattering and diffraction of subsurface cavities in layered half-space for incident SV-P and SH waves

A novel approach is proposed to deal with the problem of wave scattering and diffraction of subsurface cavities embedded in stratified half-space. The subsurface cavity with complex surroundings is treated as a substructure. The continuity condition at the interface between the substructure and the far field of stratified half-space is maintained by applying a free-field approach. As the boundary of the free-field ground is regular, the construction of the dynamic matrices and the evaluation of the wave input on the interface become considerably easier. Based on the previous work with some improvement, a novel approach for evaluation of Green's functions in stratified half-space is presented. The wave equation is decoupled into the one for SV-P wave components and the other one for SH wave component. The precise integration technique ensures high accuracy of the solution of wave equations. The layer merging technique and the dual form equation make it possible to obtain Green's function in closed-form solution of matrix equations. Numerical examples validate accuracy and efficiency of the proposed approach.     

Plane strain deformation of a multi-layered poroelastic half-space by surface loads

The Biot linearized quasi-static theory of fluid-infiltrated porous materials is used to formulate the problem of the two-dimensional plane strain deformation of a multi-layered poroelastic half-space by surface loads. The Fourier-Laplace transforms of the stresses, displacements, pore pressure and fluid flux in each homogeneous layer of the multi-layered half-space are expressed in terms of six arbitrary constants. Generalized Thomson-Haskell matrix method is used to obtain the deformation field. Simplified explicit expressions for the elements of the 6 × 6 propagator matrix for the poroelastic medium are obtained. As an example of the possible applications of the analytical formulation developed, formal solution is given for normal strip loading, normal line loading and shear line loading.     

Generalized poroelastic wellbore problem

This paper presents a novel analytical solution to the transient, z‐dependent, and asymmetric problem of an infinite wellbore drilled into a fluid‐saturated porous medium. The formulations are based on Biot's linear theory of poroelasticity, in which the dependency of poroelastic field variables to spatial coordinates as well as time domain is considered in the most general form. This gives flexibility to the solution in cases that cannot be analyzed using the conventional plane strain or symmetric models. One such case is when calculating the stress variations around an inclined wellbore where the far‐field stresses are acting over a finite vertical section. The results of our solution to this case with a three‐dimensional state of far‐field stress are used to analyze the stability of inclined wellbores passing through abnormally stressed formations. The presented solution is capable of finding expressions for fundamental solutions with stress or flow boundary conditions at the wellbore. These solutions are here adopted to analyze the pressure disturbances generated by multiprobe formation tester, a standard wireline device that is designed for downhole fluid sampling as well as estimating the directional permeabilities of subsurface earth formations. A comparison with the conventional solution for the relevant pressure diffusion equation indicates that the poroelastic effect is relatively significant in relation to the transient response of the pore pressure. Further, it is shown that the finite dimensions of sink probe would, to a great extent, contribute to the formation's pore pressure variations at its immediate proximity. Copyright © 2013 John Wiley & Sons, Ltd.     

层状饱和土体中排桩对简谐荷载隔振效果分析

根据Biot理论,利用已有的传递、透射矩阵法得到层状饱和土体内部受竖向圆形分布荷载作用下的基本解,再由Muki虚拟桩法,建立了频域内层状土-桩的第二类Fredholm积分方程。通过离散方法求解积分方程,得到了评价隔振效果的振幅比。与已知文献结果比较,验证了方法的正确性。数值结果表明：对于同种类型的振源,采用相同的隔振系统,在上软下硬的多层土体中的隔振效果比在上硬下软的土层中要好。采用较长的桩或刚度较大的桩,或桩之间的距离加密都可得到好的隔振效果。     

Quasi-static deformation due to two-dimensional seismic sources embedded in an elastic half-space in welded contact with a poroelastic half-space

The Biot linearized theory of fluid saturated porous materials is used to study the plane strain deformation of a two-phase medium consisting of a homogeneous, isotropic, poroelastic half-space in welded contact with a homogeneous, isotropic, perfectly elastic half-space caused by a two-dimensional source in the elastic half-space. The integral expressions for the displacements and stresses in the two half-spaces in welded contact are obtained from the corresponding expressions for an unbounded elastic medium by applying suitable boundary conditions at the interface. The case of a long dip-slip fault is discussed in detail. The integrals for this source are solved analytically for two limiting cases: (i) undrained conditions in the high frequency limit, and (ii) steady state drained conditions as the frequency approaches zero. It has been verified that the solution for the drained case (ω → 0) coincides with the known elastic solution. The drained and undrained displacements and stresses are compared graphically. Diffusion of the pore pressure with time is also studied.     

半空间饱和土中圆形衬砌对弹性压缩波的散射

借助于Biot 波动理论和弹性波的传播理论,采用复变函数和多级坐标法,对半空间饱和土中圆形衬砌结构对弹性稳态压缩波的散射问题进行求解和分析。利用一个半径很大的圆弧来逼近半空间直边界,将待解问题转化为稳态弹性压缩波在一个大圆孔和一个弹性衬砌结构的散射问题。通过引入势函数,将饱和土的Biot波动方程和衬砌的弹性波动方程解耦成Helmholtz 方程,借助复变函数级数展开便可以预先写出该组Helmholtz方程的通解。然后,通过引用复变量,把饱和土和衬砌结构中的应力、位移及孔压用设定的势函数表示出来,再利用半空间饱和土和衬砌结构的连续性条件和近似直边界的圆弧边界和衬砌内边界的边界条件求解出该组势函数的特解。最后,利用势函数的特解,得到饱和土中的位移,应力和孔压及衬砌结构的位移和应力;变换不同的参数求解衬砌结构内外边界的动应力和孔压的集中系数,通过对算例结果的分析得出一系列有益的结论。     

Fundamental solutions for a fluid‐saturated,transversely isotropic,poroelastic solid

The fundamental solutions were obtained for step‐like point forces acting in three orthogonal directions and an instantaneous fluid point source in a fluid‐saturated, porous, infinite solid of transversely isotropic elasticity and permeability. After expressing the governing equations in the form of matrix in the Laplace space, we employed Kupradze's method together with the triple Fourier transforms. This method reduces the simultaneous partial differential equations with respect to three displacement components and a pore fluid pressure to a differential equation in terms of only one potential scalar function, which can be operationally solved in the transformed space. After the Laplace inversion of the potential, the residue theorem was applied to its Fourier inverse transform with respect to one of the transformation variables. The Fourier transforms with respect to two other variables were rewritten into the Hankel transforms. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional time‐harmonic fundamental solutions for a fluid‐saturated poroelastic half‐space with partially permeable free surface

By virtue of a pair of scalar potentials for the displacement of the solid skeleton and the pore fluid pressure field of a saturated poroelastic medium, an alternative solution method to the Helmholtz decomposition is developed for the wave propagation problems in the framework of Biot's theory. As an application, a comprehensive solution for three‐dimensional response of an isotropic poroelastic half‐space with a partially permeable hydraulic free surface under an arbitrarily distributed time‐harmonic internal force field and fluid sources is developed. The Green's functions for the poroelastic fields, corresponding to point, ring, and disk loads, are reduced to semi‐infinite complex‐valued integrals that can be evaluated numerically by an appropriate quadrature scheme. Analytical and numerical comparisons are made with existing elastic and poroelastic solutions to illustrate the quality and features of the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Influence functions for stress and displacement discontinuity elements in an anisotropic medium

Influence functions, that permit us to determine stresses and displacements at an arbitrary point in an infinite, homogeneous, linear elastic, anisotropic medium due to different three-dimensional (3-D) stress or displacement discontinuities distributed on infinite, flat, band-type elements, are presented. Any straight-line segment on the band, which is perpendicular to its infinite side, has the same distribution of the discontinuities. Along with the functions, their Taylor series approximations are also provided. The last can be useful to analyse stresses and displacements at points distant from the elements. The functions allow us to avoid procedures of numerical integration in the Indirect Boundary Element Method and/or the Displacement Discontinuity Method computer codes that are able to solve complete plane-strain problems with 3-D boundary conditions for an elastic, anisotropic medium. © 1997 by John Wiley & Sons, Ltd.     

Simulation of the crosshole method in isotropic and anisotropic media

The crosshole seismic method was simulated using a finite element model with a diagonal mass matrix and a direct integration of the equations of motion in the time domain. The results were compared to those of a more efficient but also more restricted formulation using discrete Green's functions. The effects of the type of excitation, the shape of the applied pulse and the position of the receiver with respect to the source on the shape of the recorded motions were investigated for isotropic and cross anisotropic soil deposits. The computed times of arrival of the different waves were compared to those predicted using curved ray path theory to assess the accuracy of this much simpler procedure as a means to interpret the experimental data and determine the soil properties. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamic torsional response of an end bearing pile in saturated poroelastic medium

A comprehensive analytical solution is developed in this paper to investigate the torsional vibration of an end bearing pile embedded in a homogeneous poroelastic medium and subjected to a time-harmonic torsional loading. The poroelastic medium is modeled using Biot’s two-phased linear theory and the pile using one-dimensional elastic theory. By using the separation of variables technique, the torsional response of the soil layer is solved first. Then based on perfect contact between the pile and soil, the dynamic response of the pile is obtained in a closed form. Numerical results for torsional impedance of the soil layer are presented first to portray the influence of wave modes, slenderness ratio, pile–soil modulus ratio and poroelastic properties. A comparison with the plane strain theory is performed. The selected numerical results are obtained to analyze the influence of the major parameters on the torsional impedance at the level of the pile head. Finally, the dynamic torsional impedance of this study is compared with that for floating pile in elastic soil.     

Wave propagation in a general anisotropic poroelastic medium: Biot’s theories and homogenisation theory

Anisotropic wave propagation is studied in a fluid-saturated porous medium, using two different approaches. One is the dynamic approach of Biot’s theories. The other approach known as homogenisation theory, is based on the averaging process to derive macroscopic equations from the microscopic equations of motion. The medium considered is a general anisotropic poroelastic (APE) solid with a viscous fluid saturating its pores of anisotropic permeability. The wave propagation phenomenon in a saturated porous medium is explained through two relations. One defines modified Christoffel equations for the propagation of plane harmonic waves in the medium. The other defines a matrix to relate the relative displacement of fluid particles to the displacement of solid particles. The modified Christoffel equations are solved further to get a quartic equation whose roots represent complex velocities of the four attenuating quasi-waves in the medium. These complex velocities define the phase velocities of propagation and quality factors for attenuation of all the quasi-waves propagating along a given phase direction in three-dimensional space. The derivations in the mathematical models from different theories are compared in order to work out the equivalence between them. The variations of phase velocities and attenuation factors with the direction of phase propagation are computed, for a realistic numerical model. Differences between the velocities and attenuations of quasi-waves from the two approaches are exhibited numerically.     

Analytical and computational procedure for solving the consolidation problem of layered soils

One‐dimensional consolidation analysis of layered soils conventionally entails solving a system of differential equations subject to the flow conditions at the bounding upper and lower surfaces, as well as the continuity conditions at the interface of every pair of contiguous layers. Formidable computational efforts are required to solve the ensuing transcendental equations expressing the matching conditions at the interfaces, using this method. In this paper, the jump discontinuities in the flow parameters upon crossing from one layer to the other have been systematically built into a single partial differential equation governing the space–time variation of the excess pore pressure in the entire composite medium, by the use of the Heaviside distribution. Despite the presence of the discontinuities in the coefficients of the differential equation, a closed‐form solution in the sense of an infinite generalized Fourier series is obtained, in addition to which is the development of a Green's function for the differential problem. The eigenfunctions of the composite medium are the coordinate functions of the series, obtained computationally through the application of the extended equations of Galerkin. The analysis has been illustrated by solving the consolidation problem of a four‐layer composite, and the results obtained agree very well with the results obtained by previous researchers. Copyright © 2013 John Wiley & Sons, Ltd.     

移动荷载作用下饱和土中单桩的动力响应

根据Biot动力理论,采用Fourier和Hankel变换方法得到了半空间饱和土受移动载荷及土体内受垂直简谐载荷作用下频域内基本解。根据虚拟桩法,得到了移动载荷作用下桩基的第2类Fredholm积分方程,并应用IFFT方法得到时间、空间域内单桩的动力响应。数值结果表明,移动荷载会引起桩身的负摩擦力;桩身最大轴力、孔压随移动荷载速度增加而增大;此外,在桩上端部会出现孔压集中现象。     

Finite element analysis for a finite conductivity fracture in an infinite poroelastic medium

In the technology of oil recovery, oil production rate can be increased by generation of a vertical conductive fracture adjacent to the well-bore. In this paper the seepage flow and isothermal deformation in both the oil formation and the fracture are studied by modelling the formation as a two-dimensional infinite poroelastic medium and the conductive fracture as a one-dimensional poroelastic material, saturated by a one-phase compressible fluid. The plane strain condition is employed. Solutions for a growing conductive fracture and a stationary conductive fracture in the infinite medium are obtained by means of the finite element method based on a variational principle for the formation which can impose the governing equations of the fracture. Infinite elements are used outside the finite element domain. Numerical results indicate that the injection rate, the applied pressure and the crack mouth opening displacement at the well-bore oscillate during the propagation of the conductive fracture. The production rate of a well with the conductive fracture is compared with that of a well without the conductive fracture. Finally, a new definition of the conductivity coefficient for the conductive fracture is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

衬砌隧道中移动轴向激励作用下两相多孔介质动力响应

为了研究衬砌隧道中地铁振动在饱和地层中的传播情况,采用移动轴向激励模拟技术,建立了隧道-衬砌-两相介质的动力分析模型;利用波函数展开法、傅里叶变换法等,推导了频域内衬砌隧道移动轴向激励作用下两相多孔介质动力响应的解析解,并给出了两相多孔介质临界速度的经验公式;通过离散快速傅里叶逆变换得到了时-空域内两相介质的动力响应。结果表明：无衬砌隧道中移动的轴向常激励作用下,两相介质临界速度只与介质的剪切模量和密度有关,且数值接近其剪切波速的1.1倍;对于衬砌隧道,介质的临界速度随着衬砌剪切模量的增大而增大,随着衬砌密度的增大而减小,衬砌对振动向介质中的传播有一定的削弱作用,衬砌剪切模量与介质剪切模量相差越大,削弱越明显;动力响应频率越接近激振频率,其幅值越大,所对应的临界速度越小。     

Interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic medium

This paper presents a rigorous analysis for the static interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic half‐space under vertical, horizontal, and torsional forces individually applied at the top of pile. The inhomogeneity is specified with the exponential variation of shear modulus along depth of the embedding medium, and the Poisson's ratio is assumed to be constant. By means of a set of Green's functions for pile and soil medium and satisfying the compatibility conditions between the 2 interacting media, the formulation is reduced to coupled Fredholm integral equations. Using the adaptive‐gradient elements, capable of capturing the singular stress transfer at both ends of the pile, a numerical procedure is developed and utilized for evaluating the relevant integral equations and studying the inhomogeneity effect on the soil‐pile interaction responses. The analysis results have been validated for different soil‐pile modulus ratios under axial load and for a Poisson's ratio of 0.3 under lateral load. The procedure does not consider the nonlinear behavior of the soil medium or plastic yielding in the pile section, and the impact of the unreliable results for the case of high Poisson's ratio is not examined.     

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

The vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 John Wiley & Sons, Ltd.