Numerical Laplace-Fourier transform inversion technique for layered-soil consolidation problems: I. Fundamental solutions and validation

A numericl method for solving consolidation problems of layered soils is developed. Starting from the governing differential equations for the coupled poro-elastic medium, the governing partial differential equations are reduced to ordinary differential equations by means of the appropriate displacement functions and Laplace-Fourier transformation. Once the fundamental solution in the transformed domain has been found, the solution in the physical domain is obtained by numerically inverting the transformations. A series of soil consolidation problems have been solved and validated against existing solutions in order to compare the feasibility and the accuracy of the present technique.     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.     

多层横观各向同性地基轴对称固结的传递矩阵解

从横观各向同性地基轴对称Biot固结的基本方程出发,通过关于t的Laplace变换和关于r的Hankel变换,得到关于z的一阶常微分方程组。然后,对变换域内的基本未知量进行线性化处理,建立了变换域内的基本状态变量在z = 0处和任意深度处z的显式关系。利用传递矩阵法,结合层间连续性条件和边界条件,得到了多层横观各向同性地基的Biot固结轴对称问题的解答。该解答能避免随着层数增加而需要求解大型方程组的困难,明显地提高了计算效率。     

Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Based on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one‐dimensional consolidation of single‐layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions psi₁ and ψ₂, where ψ₁ and ψ₂, respectively, satisfy two second‐order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Consolidation by vertical drains under time‐dependent loading

A solution for the consolidation by vertical drains under time‐dependent loading is presented in this paper. Considering the well resistance and the smear action, the simultaneous basic partial differential equations of the consolidation by vertical drains are obtained for the arbitrary loading method. However, the impulse function method cannot be directly applied to obtain the solution. The partial differential equations and the solution conditions that satisfy the impulse function method are obtained after some mathematical processing. The solution for the consolidation by vertical drains under time‐dependent loading is obtained by virtue of the impulse function method and the solution under instantaneous loading. The solutions under single ramp loading and multi‐ramp loading are obtained and the feasibility of Carrillo's method under time‐dependent loading is discussed. Further, the characteristics of the consolidation by vertical drains under instantaneous loading and time‐dependent loading are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

竖向简谐荷载下二维层状饱和地基的解析层元解

基于Biot固结理论,运用解析层元方法求解竖向简谐荷载作用下二维层状饱和地基的动力响应问题。从直角坐标平面应变问题控制方程出发,通过Fourier-Laplace变换将偏微分方程组转化为常微分方程组,求解得到单层饱和地基的解析层元。结合层间连续条件和边界条件,组装得到多层饱和地基的总刚度矩阵方程,进而求得变换域内的解。借助Fourier-Laplace逆变换的数值积分方法,获得平面应变动力问题在物理域内的解,编制了相应的计算程序,其计算结果与已有文献结果吻合较好。通过算例分析了荷载圆频率、荷载作用深度及地基成层性对地基竖向位移的影响。计算结果表明：随荷载圆频率的增大,地基竖向位移先增加后减小;地基竖向位移在荷载作用点处呈现波峰,且受表层土性的影响较大。     

The response of an elastic half‐space under a momentary shear line impulse

The response of an ideal elastic half‐space to a line‐concentrated impulsive vector shear force applied momentarily is obtained by an analytical–numerical computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. The shear force is concentrated along an infinite line, drawn on the surface of the half‐space, while being normal to that line as well as to the axis of symmetry of the half‐space. An exact loading model is introduced and built into the computational method for this shear force. With this model, a compatibility exists among the prescribed applied force, the geometric decay of the shear stress component at the precursor shear wave, and the boundary conditions of the half‐space; in this sense, the source configuration is exact. For the transient boundary‐value problem described above, a wave characteristics formulation is presented, where its differential equations are extended to allow for strong discontinuities which occur in the material motion of the half‐space. A numerical integration of these extended differential equations is then carried out in a three‐dimensional spatiotemporal wavegrid formed by the Cartesian bicharacteristic curves of the wave characteristics formulation. This work is devoted to the construction of the computational method and to the concepts involved therein, whereas the interpretation of the resultant transient deformation of the half‐space is presented in a subsequent paper. Copyright © 2003 John Wiley & Sons, Ltd.     

Analytical layer‐element method for non‐axisymmetric consolidation of multilayered soils

A numerically efficient and stable method is developed to analyze Biot's consolidation of multilayered soils subjected to non‐axisymmetric loading in arbitrary depth. By the application of a Laplace–Hankel transform and a Fourier expansion, the governing equations are solved analytically. Then, the analytical layer‐element (i.e. a symmetric stiffness matrix) describing the relationship between generalized displacements and stresses of a layer is exactly derived in the transformed domain. Considering the continuity conditions between adjacent layers, the global stiffness matrix of multilayered soils is obtained by assembling the inter‐related layer‐elements. Once the solution in the Laplace–Hankel transformed domain that satisfies the boundary conditions has been obtained, the actual solution can be derived by the inversion of the Laplace–Hankel transform. Finally, numerical examples are presented to verify the theory and to study the influence of the layered soil properties and time history on the consolidation behavior. Copyright © 2011 John Wiley & Sons, Ltd.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis for temperature and pressure fields in process of hydrate dissociation by depressurization

An improved axisymmetric mathematic modeling is proposed for the process of hydrate dissociation by depressurization around vertical well. To reckon in the effect of latent heat of gas hydrate at the decomposition front, the energy balance equation is employed. The semi‐analytic solutions for temperature and pressure fields are obtained by using Boltzmann‐transformation. The location of decomposition front is determined by solving initial value problem for system of ordinary differential equations. The distributions of pressure and temperature along horizontal radiate in the reservoir are calculated. The numeric results indicate that the moving speed of decomposition front is sensitively dependent on the well pressure and the sediment permeability. Copyright © 2010 John Wiley & Sons, Ltd.     

Semi-analytical solutions of two-dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

The study presents semi-analytical solutions of two-dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two-dimensional plane strain consolidation equations in the form of two-order partial differential equations with three variables are firstly converted to two-order partial differential equations with two variables, which are similar to those of one-dimensional consolidation problem. The four-order ordinary differential equations about excess pore-air and excess pore-water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two-dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air-water and lateral-vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.     

基于同伦分析法的非饱和土层固结问题的级数解

由于非饱和土的渗透系数是基质吸力的函数,使得控制方程带有强非线性的特征,进而使得控制方程的解析求解变得十分困难。同伦分析法对级数基函数和辅助线性算子的选择具有更大的自由性、灵活性,且收敛性的控制和调节更加容易实现,求解强非线性微分方程时在选择线性算子以及辅助参数上具有明显的优势。因此,针对非饱和土固结方程的非线性特征,对于处于地表浅层的非饱和土层,假设孔隙气压力为大气压力,在Richard经验公式与非饱和土一维固结理论的基础上,推导了非饱和一维固结无量纲控制方程;应用同伦分析法,通过选取适当的初始猜测解与辅助参数,将该非线性方程转换为线性的微分方程组并求解得到固结问题的级数解。此外,以压实高岭土为研究对象,在收集相关试验参数基础之上,将由同伦分析法求得的固结问题的近似解析解与有限差分法数值结果相对比,分析结果验证了解析解的正确性。     

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.     

非均匀地基与均匀地基一维固结解答之间的转换关系

基于一维固结理论,研究了土层物理力学性质沿深度非均匀连续变化的固结问题。首先,利用分离变量法,获得了渗透系数和压缩系数随深度按指数函数变化的非均匀地基一维固结方程的解析表达式; 然后与Terzaghi固结理论的经典解答进行了比较,获得了两种解答之间的相似转换关系。这样,非均匀地基固结问题就可以用相同荷载以及边界条件下的均匀地基固结的经典解线性表示。因此,非均匀地基固结问题的求解转化为对相似转换系数的计算。该系数集中反映土层非均匀性对地基固结的影响,从而为解决非均匀地基一维固结问题提供了便捷途径。     

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.     

互补算法在一维非线性固结求解中的应用

以一维非线性固结为例,介绍了互补算法在固结问题求解中的应用。将地基土体的压缩特性曲线进行分段线性拟合并引入控制变量,得到土体孔隙比与有效应力间的统一关系式,并获得压缩曲线中的互补关系;详细推导了以孔隙比与有效应力为变量的一维非线性固结微分方程,通过对其进行差分求解,并结合压缩曲线中的互补条件,使得不同时刻的非线性固结问题转化为标准的线性互补问题。这样,非线性固结问题可通过在线性固结差分法中嵌入互补模型算法进行求解。该法的合理性通过与普通迭代算法进行对比获得证明。在此基础上,对一维固结问题线性与非线性解答的差异及影响非线性计算的因素进行了分析。计算中发现,互补模型算法在计算效率方面优于普通迭代法;线性计算程序中嵌入互补模型算法计算非线性固结在程序上容易实现;互补模型算法中通过对控制变量的求解,可判断地基土体所处的压缩状况     

Modelling of steady‐state fluid flow in 3D fractured isotropic porous media: application to effective permeability calculation

In this paper, 3D steady‐state fluid flow in a porous medium with a large number of intersecting fractures is derived numerically by using collocation method. Fluid flow in the matrix and fractures is described by Darcy's law and Poiseuille's law, respectively. The recent theoretical development presented a general potential solution to model the steady‐state flow in fractured porous media under a far‐field condition. This solution is a hypersingular integral equation with pressure field in the fracture surfaces as the main unknown. The numerical procedure can resolve the problem for any form of fractures and also takes into account the interactions and the intersection between fractures. Once the pressure field and then the flux field in fractures have been determined, the pressure field in the porous matrix is computed completely. The basic problem of a single fracture is investigated, and a semi‐analytical solution is presented. Using the solution obtained for a single fracture, Mori‐Tanaka and self‐consistent schemes are employed for upscaling the effective permeability of 3D fractured porous media. Copyright © 2012 John Wiley & Sons, Ltd.     

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

Based on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite layer analysis of consolidation. I

A method for the analysis of the consolidation of a horizontally layered soil under plane conditions is developed. The method depends upon the transformation of the governing equations by a Fourier trasform. This transformation has the effect of reducing the partial differential equations of consolidation to ordinary differential equations. The ordinary differential equations are then solved using a finite layer or finite difference approach. Once the solution in the transformed plane has been found, the actual solution is synthesized by Fourier inversion. The method leads to a considerable reduction in the amount of core storage necessary for solution and enables the solution of quite significant problems to be obtained on a mini-computer.