Consolidation of sensitive clay with vertical drain

The coefficient of consolidation is one of the most important parameters that control the rate of consolidation. Conventional consolidation theories assume that the coefficient of consolidation is constant during the whole consolidation process. In the case of sensitive clay, the coefficient of consolidation is strongly dependent on the level of effective stress of clay. With the dissipation of pore water pressure and the increase of effective stress, the soil structure of the upper subsoil is gradually destroyed downwards and its coefficient of consolidation becomes smaller. At the same time, the coefficient of permeability of the vertical drains drops down because of the kinking or bending effect. The destructured upper subsoil and the kinking of the vertical drain hinder the dissipation of the pore pressure in the lower subsoil. This paper presents a model to describe the above important phenomena during the consolidation of sensitive clay with vertical drain. The solution for proposed model can be obtained by extending the closed‐form solution of the consolidation of double‐layered ground with vertical drain by the interactive method introducing the concept of the moving boundary and the reduction of discharge capacity of vertical drain. The numerical results obtained from the finite element method package PLAXIS (Ver. 7.2) are adopted to compare those obtained from the present algorithm. Moreover, the rationality of the moving boundary is explained by the distributions of the excess pore water pressure in natural soil zone along the radial direction. Wenzhou airport project is taken as a case study in this paper. The results for the sensitive soil with decaying sand drain agree very well with the in situ measured data. The calculated results can properly explain two general phenomena observed in the consolidation of soft sensitive soil ground with vertical drains: one is that the time to achieve the same consolidation degree is much longer under heavy loading than that under light loading; the other is that the consolidation speed is much slower in the lower subsoil than in the upper subsoil. Finally, it is indicated that the vertical drains can decrease the hindrance effect of the destructured subsoil. Copyright © 2007 John Wiley & Sons, Ltd.     

A new analytical model for consolidation with multiple vertical drains

Various analytical theories of consolidation for soils with vertical drains have been proposed in the past. Most conventional theories are based on a cylindrical unit cell that contains only a single vertical drain. This paper described a new analytical model where a vertical drain located at the centre (the ‘inner vertical drain’) and is surrounded by two or three vertical drains (the ‘outer vertical drains’), the number of which depends on whether the configuration is triangular or rectangular. Both types of drains are combined into a cylindrical unit cell, and the water is assumed to flow both inwards to the inner vertical drain and outwards to the outer vertical drains distributed around the circumference. The outer radial boundary of the unit cell is regarded as a permeable boundary, with a drainage capacity of two or three separate vertical drains for triangular and rectangular configurations, respectively. The smear effects and the drainage resistances for both the inner and outer vertical drains are considered in the analysis as well. In this way, the equations governing the consolidation process with multiple vertical drains are derived, and the corresponding analytical solutions are obtained for instantaneously loading, ramp loading and multi‐stage of instantaneously loading and multi‐stage of ramp loading. The present solutions are finally compared with several conventional solutions for a single vertical drain in the literature. The results show that the present model predicts the same average degree of consolidation as conventional models do, which verifies the correctness of this new model. Finally, the settlement predicted by the present solution is compared with the measured settlement from a field test at the Port of Brisbane, Australia, which shows a good agreement between them. Copyright © 2016 John Wiley & Sons, Ltd.     

Consolidation of double‐layered ground with vertical drains

Vertical drains are usually installed in subsoil consisting of several layers. Due to the complex nature of the problem, over the past decades, the consolidation properties of multi‐layered ground with vertical drains have been analysed mainly by numerical methods. An analytical solution for consolidation of double‐layered ground with vertical drains under quasi‐equal strain condition is presented in this paper. The main steps for the computation procedure are listed. The convergence of the series solution is discussed. The comparisons between the results obtained by the present analytical method and the existing numerical solutions are described by figures. The orthogonal relation for the system of double‐layered ground with vertical drains is proven. Finally, some consolidation properties of double‐layered ground with vertical drains are analysed. Copyright © 2001 John Wiley & Sons, Ltd.     

Three-dimensional finite element analysis of spatially variable PVD improved ground

A stochastic approach that investigates the effects of soil spatial variability on stabilisation of soft clay via prefabricated vertical drains (PVDs) is presented and discussed. The approach integrates the local average subdivision of random field theory with the Monte Carlo finite element (FE) technique. A special feature of the current study is the investigation of impact of spatial variability of soil permeability and volume compressibility in the smear zone as compared to that of the undisturbed zone, in conjunction with uncoupled three-dimensional FE analysis. A sensitivity analysis is also performed to identify the random variable that has the major contribution to the uncertainty of the degree of consolidation achieved via PVDs. The results of this study indicate that the spatial variability of soil properties has a significant impact on soil consolidation by PVDs; however, the spatial variability of soil properties in the smear zone has a dominating impact on soil consolidation by PVDs over that of the undisturbed zone. It is also found that soil volume compressibility has insignificant contribution to the degree of consolidation estimated by uncoupled stochastic analysis.     

成层竖向排水井地基固结分析

实际工程中竖井地基具有成层性,有时竖井也并未完全打穿软土层。在竖井打设区满足竖井等应变固结理论、下卧层满足一维太沙基固结理论假设的基础上,将现有竖井地基固结理论推广到成层未打穿竖井地基情况。利用边界条件和竖直向连续条件,确定该系统的正交关系,并给出了其固结解答,该解具有广泛的适用性。通过对竖井打设区和下卧层层数的变化,即可获得现有关于简单未打穿竖井地基的固结解答。将该解答编制成应用程序,对一算例进行了分析。结果表明,平均固结度按孔压定义和按变形定义是不相同的,硬表层的存在会加快其下土层的固结。     

任意附加应力下竖井径竖向耦合固结等应变解

竖向排水体包括砂井和塑料排水板已广泛应用于加速软黏土地基的固结。不同形式的轴对称条件下含竖向排水体均质各向同性地基的固结解析解答也相继被提出,而各种解答均假定地基中附加应力在竖向排水体的深度范围内是均匀分布的。显然,当竖向排水体深度与地面堆载区宽度的比值相对较高,或竖向排水体的影响区域靠近地面堆载区的边缘时,这个假定并不适用。为此,在现有径竖向耦合固结等应变解的基础上,提出任意分布的地基附加应力条件下的解答,并分析附加应力分布形式和线性加载速率对地基固结的影响。结果表明,均匀分布的附加应力条件下的解答低估了地基的固结速率,而瞬时加载条件下的解答则高估了地基的固结速率。     

Reliability-based semi-analytical solution for ground improvement by PVDs incorporating inherent (spatial) variability of soil

The design of soil consolidation via prefabricated vertical drains (PVDs) has been traditionally carried out deterministically and thus can be misleading due to the ignorance of the uncertainty associated with the inherent (spatial) variation of soil properties. To treat such uncertainty in the design process of soil consolidation by PVDs, stochastic approaches that combine the finite element method with the Monte Carlo technique (FEMC) have been usually used. However, such approaches are complex, computationally intensive and time consuming. In this paper, a simpler reliability-based semi-analytical (RBSA) method is proposed as an alternative tool to the complex FEMC approach for soil consolidation by PVDs, considering soil spatial variability. The RBSA method is found to give similar results to those obtained from the FEMC approach and can thus be used with confidence in practice.     

Analytical models for consolidation of combined composite ground improved by impervious columns and vertical drains

In recent years, a new technique of ground improvement, which involves the combined use of impervious column and vertical drains, has been proposed and utilized in many field projects to accelerate consolidation and increase bearing capacity of soft soil ground. To cover the possible distribution patterns of impervious columns and vertical drains, 2 analytical models, including Model A with outward flow and Model B with inward flow within the soils, are proposed to predict the consolidation of combined composite ground by considering the following factors: (1) disturbance effects of both impervious columns and vertical drains, (2) the well resistance of vertical drains, and (3) time‐variant loadings. The average degrees of consolidation predicted by the proposed analytical models are compared with several existing solutions and then against the measured data in the literature. The consolidation behavior of a combined composite ground is investigated by the proposed analytical solutions. The results show that the combined use of impervious columns and vertical drains can remarkably accelerate the consolidation rate of soft soils compared with the single use of either of them. The average degrees of consolidation predicted by both analytical models agree well with the measured data. Compared with Model B, Model A usually predicts a faster consolidation rate because of a shorter drainage path. Many factors can influence consolidation behavior of combined composite ground, such as loading scheme, distribution patterns and the disturbance effects of impervious columns and vertical drains, and compression modulus ratio of impervious column to soil.     

砂墙地基二维固结自由应变解

将含竖向排水体地基的三维固结变形问题等效为平面应变问题进行数值分析时,砂墙地基二维固结解析解答是合理确定其等效固结计算参数的重要依据。为辨析现有砂墙地基等应变固结近似解答的适用性,针对微单元土体严格的二维固结微分方程,考虑对地基固结有重要影响的井阻作用,以及涵盖完全透水和不完全透水的更一般边界面排水条件,推求得到了其自由应变解答。并与现有解答进行对比分析,同时,分析了泊松比效应以及水平和竖向排水对地基固结的影响。结果表明,现有砂墙地基的等应变固结解答虽然近似,但已有足够精确;砂墙地基以水平向固结为主,竖向固结几乎可以忽略不计;地基固结速率随着泊松比的增大而增大,在将竖向排水体等效为砂墙时,应考虑其作用影响。     

复杂荷载作用下考虑下卧层三维渗流的未打穿竖井地基固结分析

针对任意复杂变荷载作用下未打穿的竖井地基,通过在竖井底面以下土层中设置虚拟竖井,使其能够合理考虑下卧层土体三维渗流问题,运用Laplace变换,求得频域内竖井地基的固结解。通过Laplace逆变换,得到了任意荷载作用下竖井地基的平均固结度、孔压消散曲线、沉降曲线。结合具体算例,对影响竖井地基固结的主要影响因素进行了详细分析,并将现有未打穿竖井地基平均固结度近似计算方法的精度和适用的范围进行了对比,得到了一些有益于工程实践的结论。     

Consolidation behavior of soft deposits considering the variation of prefabricated vertical drain discharge capacity

This paper is primarily focused on why and how to consider the varied discharge capacity during simulations of consolidation via prefabricated vertical drains (PVDs) for soft soil deposits. First, the existing studies regarding discharge capacity are summarized and discussed. These studies conclude that the discharge capacity of PVDs at sites vary with the confining pressure and consolidation time. Next, a series of analytical solutions that consider the variation of discharge capacity with ground depth, consolidation time or both simultaneously are presented and compared. Applications of these solutions and of the newly introduced parameters are described. Then, a well-documented case history on ground treatment with PVDs is analyzed, in which the parameters related to the time-dependent discharge capacity were obtained from laboratory tests. A comparison between a classical solution and the newly presented method indicates that consideration of the varied discharge capacity in the consolidation theory can better predict the consolidation process of PVD-improved ground.     

考虑井阻时空变化的竖井地基非线性固结解析解

软土固结过程中展现出明显的非线性压缩和渗透特性,同时竖井的淤堵效应常导致井阻在固结过程中随深度和时间不断演化,但目前能考虑井阻随时空演化的竖井地基非线性固结解析解还很鲜见。通过引入孔隙比与有效应力及孔隙比与渗透系数间的半对数模型描述了土体的非线性固结特性,建立了能同时考虑井阻随时空变化和涂抹影响的竖井地基非线性固结模型,并采用分离变量法获得了固结模型的解析解。将特定参数下固结解的计算结果与实测数据、已有的竖井地基固结解答进行了对比分析以验证其可靠性。最后,对竖井地基的非线性固结性状开展了大量计算分析。结果表明：竖井渗透系数随深度线性衰减越明显则地基固结速率越慢;外荷载一定时,随着软土压缩指数c_c与渗透率指数c_k之比的增大,竖井地基固结速度减慢;在c_c /c_k值不变的情况下,外荷载增加,地基固结速率加快。在涂抹区的3种径向渗透系数变化模式中,抛物线变化模式下的地基固结速度最快,线性变化模式下的地基固结速度次之,恒定模式下的地基固结速度最慢,且这种性状并不因为考虑井阻变化或土体非线性固结特性而发生改变。     

Finite element consolidation analysis of soils with vertical drain

This paper presents a finite element procedure for the analysis of consolidation of layered soils with vertical drain using general one‐dimensional (1‐D) constitutive models. In formulating the finite element procedure, a Newton–Cotes‐type integration formula is used to avoid the unsymmetry of the stiffness matrix for a Newton (Modified Newton) iteration scheme. The proposed procedure is then applied for the consolidation analysis of a number of typical problems using both linear and non‐linear soil models. Results from this simplified method are compared with those from a fully coupled consolidation analysis using a well‐known finite element package. The average degree of consolidation, excess porewater pressure and average vertical effective stress are almost the same as those from the fully coupled analysis for both the linear and non‐linear cases studied. The differences in vertical effective stresses are tolerable except for the values near the vertical drain boundaries. The consolidation behaviour of soils below a certain depth of the bottom of vertical drain is actually one‐dimensional for the partially penetrating case. Therefore, there are not much differences in whether one uses a one‐dimensional model or a three‐dimensional model in this region. The average degree of consolidation has good normalized feature with respect to the ratio of well radius to external drainage boundary for the cases of fully penetrating vertical drain using a normalized time even in the non‐linear case. Numerical results clearly demonstrate that the proposed simplified finite element procedure is efficient for the consolidation analysis of soils with vertical drain and it has better numerical stability characteristics. This simplified method can easily account for layered systems, time‐dependent loading, well‐resistance, smear effects and inelastic stress–strain behaviour. This method is also very suitable for the design of vertical drain, since it greatly reduces the unknown variables in the calculation and the 1‐D soil model parameters can be more easily determined. Copyright © 2000 John Wiley & Sons, Ltd.     

Vertical drain consolidation analysis in one,two and three dimensions

A new method, the η or ‘eta’ method, for modeling consolidation by vertical and horizontal drains is presented. The approach is applicable in one, two and three dimensional as well as axisymmetric cases. Material and geometry properties are familiar from unit cell vertical drain analysis and are consistent across dimensions. An uncoupled finite element method (FEM) program is used to test the efficacy of the new approach. Because drains are not explicitly modeled in the finite element mesh, mesh complexity and computational time are greatly reduced. Unlike existing plane strain matching methods there is no special transformation of permeability or drain properties. The analyses conducted indicate that the η method provides an efficient and consistent means of modeling drains in any dimension.     

成层软黏土地基的固结沉降计算分析

针对实际工程中软黏土地基的成层特性及竖井有时并未打穿软土层情况,采用太沙基一维固结理论和巴伦固结理论的基本假定,利用固结度联系孔压与变形,提出一种简化成层软黏土地基的固结沉降计算方法,对竖向排水地基的固结度及沉降等进行预测。对计算的几个关键问题进行探讨：如固结系数的反演选择、竖井未打穿土层时,竖井底部成层地基固结度计算、压缩土层厚度选取等问题。在上述理论的基础上,用FORTRAN语言编写排水竖井固结程序CONSOL,在工程实践中得到应用,取得良好的效果。     

固结排水中排水带通水量的影响及选择

井阻作用影响地基的固结速率,排水带的通水量与井阻作用有直接关系,过小通水量的排水带在地基中产生较大的井阻,通水量较大的排水带又使生产成本提高。笔者分析了通水量对地基固结的影响,并提出根据被加固地基的性状来选择通水量的方法。建议用公式 来确定所使用排水带的容许通水量下限值。     

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.     

基于非等温管道流竖井地基热排水固结模拟

针对新技术热排水固结法,采用非等温管道流模拟竖井中U型导热管的传热过程,考虑温度对竖井扰动区和未扰动区渗透性的影响,在COMSOL Multiphysics有限元软件中进行二次开发,建立了竖井地基热排水固结法的有限元模型。以热排水固结软基处理原型试验为例,重点分析了模型耦合、部分耦合和不耦合情况下软土地基的固结度。结果表明,相对于传统排水固结法的不耦合模型,部分耦合模型下因温度产生的孔压增量延缓了地基固结的发展,固结速率有所减慢;耦合模型下温度虽也产生一定的孔压增量,但温度有效地改善了竖井涂抹区土的渗透特性,地基的固结速率加快,固结周期缩短,与试验结果一致。     

A new simplified Hypothesis B method for calculating the consolidation settlement of ground improved by vertical drains

Vertical drains are widely used in soft ground improvements to accelerate the consolidation process. This paper develops a new simplified Hypothesis B method for calculating the consolidation settlement of a soil layer improved by vertical drains under the instant and ramp loadings. As a comparison, the traditional Hypothesis A method is also used to calculate the settlement. Then, a fully coupled finite element consolidation analysis is utilized to examine and verify this simplified method and Hypothesis A method. For the instant loading, Carrillo‐Barron method and Zhu‐Yin method are used to obtain the average degree of consolidation for vertical drain system. Typical parameters, such as over‐consolidation ratio (OCR), smear zone, and space ratio of vertical drains, are considered. It is found that the calculation results from the new simplified method in this study agree well with finite element simulations, and relative errors are in the range of 0.1% to 12.3%. Comparatively, there are obvious differences between the calculated results from Hypothesis A method and finite element results. Carrillo‐Olson method and Zhu‐Yin method are utilized to obtain the average degree of consolidation for the vertical drain system to consider the ramp loading. Equivalent time is determined from half of the construction period to calculate the creep compression under the ramp loading. The accuracy of this simplified Hypothesis B method using both Carrillo‐Olson method and Zhu‐Yin method is acceptable with the relative errors less than 9.4%.     

考虑幂函数渗透压缩变化的双层砂井地基固结性状

采用Barron轴对称固结及大变形固结问题的某些简化与假定,推导建立了砂井地基大变形固结控制方程,利用建立的双层砂井地基大变形固结方程及编制的计算程序,通过引入软土渗透系数、有效应力与孔隙比之间的幂函数关系k =ced与e=a( )b,对瞬时加载下双层砂井地基固结性状进行算例计算。结果表明：（1）双层软土幂函数渗透关系及压缩关系中诸参数对双层砂井地基固结性状有重要影响：随着两层软土幂函数渗透关系中参数c1、c2的增加（渗透性增加）、或幂函数压缩关系中参数a1、a2的增加,各土层水平径向与竖向孔隙比减小更快,沉降发展速率与超静孔压消散速率也相应增加,且沉降发展速率快于孔压消散速率。（2）两层土在分界面处的孔隙比及平均超静孔压均出现明显的突变,将沿深度分布曲线分成形状不同的两段,表现出不同的固结性状。