渗流固结问题的数值分析及粘性土—砂井地基固结模拟
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摘要
采用竖向排水体加快软土地基固结的排水预压法,是广泛应用于港口工程、房屋建筑以及道路建设中的地基加固技术。在这些竖向排水体中,砂井最为普遍,最具有代表性。砂井如何影响粘性土地基在预压排水过程中的变化,属于典型的渗流固结问题,对地基加固工程的设计和分析具有重要意义。粘性土-砂井地基的研究,涉及到渗流固结问题的一些基本课题,如粘性土渗流固结模型的建立和分析求解。数值法是解决这些课题的重要手段,采用数值模拟的办法分析粘性土-砂井地基固结行为,也越来越受到国内外学术界和工程界的重视。最近十几年,我国基础工程建设,特别是高速公路建设突飞猛进,粘性土-砂井地基的堆载预压、真空预压以及联合预压技术随之得到深入的实践和发展。然而,由于分析方法研究与推广的相对滞后,工程人员仍然没有摆脱主要依靠经验和传统解析法对砂井地基预压工程进行设计和分析的局面,遇到不少难以解决的问题,发展科学实用的工程计算方法,特别是数值模拟技术,就成为一种迫切的需求。然而,粘性土-砂井地基固结的模拟,存在诸多的困难,包括如何确定粘性土应力应变行为、渗透性的复杂特征、如何概化砂井地基模型、如何刻画砂井分布“细而密”的空间特征、如何处理预压条件等等。
在上述背景下,本文的研究目的就是对渗流固结基本课题进行深入、系统的数值分析,提炼出最新的认识,帮助解决粘性土-砂井地基固结的基础问题,同时,对现有的粘性土-砂井地基固结模型和数值计算方法进行整理和改进,使之更合理、更具有实用价值。
本文提出了单井固结新的理论模型,即一维双重介质模型,并得到在完整砂井条件下的解析解,该解析解为完整砂井等应变固结的普遍解。一维双重介质模型克服了现有单井固结理论将砂井周围土体简单划分为涂抹区和非涂抹区的缺点,适用于砂井周围土体渗透系数任意分布的情况,减少了单井固结分析的不确定性参数。更重要的是,该模型最大限度的降低了砂井地基模型的空间维数,却不失合理性。
利用一维双重介质模型,本文首次实现了砂井地基固结的一维有限差分模拟(自编模拟程序为SDDM),得到非完整砂井地基单井固结的固结度曲线。通过分析对比,证明现有的非完整砂井单井固结的各种近似计算公式都存在系统的误差。其中,国内方法和谢康和改进法计算结果偏大,Hart法计算结果在固结前期比较准确,在固结后期显著偏小,谢康和改进法的误差随着砂井贯入地基深度的减小而增大。
以一维双重介质模型为基础,本文对砂井“等效”处理为砂沟的数值模拟方法做出了改进。改进方法严格以渗流固结偏微分方程的等价性为等效原则,不人为加大竖向排水体的间距,适用于非完整砂井条件,只需要转换含砂井区土体水平渗透系数和砂沟的渗透系数两个参数。这些特点使改进方法比现有“等效”转换方法更严密,更能够反映实际砂井
Vertical drains is available to expedite the consolidation of cohesive soil and extensivelyapplied over the world in construction engineering of ports, buildings and roadways to reinforcesoft foundations. Sand Drain, is the most representative one in the group of vertical drains. It issuggested to call the foundation with cohesive soil layers improved by sand drains CohesiveSoil-Sand Drained Foundation (CSDF).
It is important to research the behavior of sand drains in preloading processing for design andanalysis of ground improving programs. As a typical problem of consolidation process, theinvestigation of seepage and transformation of CSDF is involved with a set of basic questions toconsolidation, such as the models and solutions of consolidation for cohesive soil. Numericalmethod is one of the significant techniques for these problems and has been applied to modelingthe CSDF. In resent years, the reinforcement technique of CSDF, preloading under embankment,vacuum or coupled vacuum-embankment, has been deeply developed in China for constructionof highways. However, the engineers still do their work mainly with experience and classicalanalytical methods since the absence of new efficacious solutions and, the programs are alwaysled to some difficult situations. They help the conditions could be changed.
So, it's valuable to develop scientific technique for simulation of CSDF and helpful to dealwith complex conditions for engineers.
Here the purposes are presented for the paper: to analysis the normal consolidation behavior ofcohesive soil with numerical methods and abstract new points of view;to develop the modelsand numerical solutions of CSDF. Research results have been established in the dissertation forthese purposes.
Uniform model in nonlinear is suggested to be combined with nonlinear constitutive modeland nonlinear seepage model. The basic nonlinear model of consolidation is shown in thediagram as following:
In which H-V is the elasto-plastic element, H|N is the visco-elastic element and V|H is thevisco-plastic element for the soil framework, and W is the pore water with Darcy flow ornon-Darcy flow. A common formula is proposed to describe the volume strain speed in onedimensional condition:bPbucccdtautaPttvpvve??ε v =??εz=1 ??+2??+1+2+1εv+2εv+1ε+In which P is the loading on soil column and u is the excess pore pressure, a1、a2、b1、b2、c1、c2、c3、d are parameters involved to the state of effective stress and strain.It is also suggested to write the differential equations of Biot consolidation in matrix form asfollowing:[ K ]δ ? [C]u=F? [C ]T δ& +[G]u=nβu&+QDefinitions are:[K]=[B]T[D][B] —— stiffness matrix[G]=[C]T[H][C]/γw —— seepage matrix[C] —— coupled matrixF —— forces for equilibriumQ —— flux for equilibriumAnd δ is the displacement of soil framework, β is the coefficient for compressibility of water.The advantage of this kind of expression is that it make the analytical equations are very closeto the FEM equations of Biot consolidation.A new theoretical model of consolidation for single sand drain is developed and called as oneDimensional Dual-medium Model (DDM). Rigorous solution is given for fully penetrating sanddrain with the new model as a normal solution for the case of equal vertical strain. DDM hassome advantages related to the theory developed before: the area of soil around sand drain is notsimply marked as smear zone and non-smear zone and the model is available for any distributionof soil permeability;the number of uncertain parameters is decreased from two to only one. Themore important characteristic of DDM, is reducing mostly of the modeling dimensions forCSDF.The consolidation of CSDF is simulated firstly with one dimensional finite differencemodeling based on the DDM and curves of consolidated degree U for non-fully penetrating sanddrain is obtained. According to the comparing of results, systemic error is found in approximatedformulations proposed before for non-fully penetrating sand drain. The U calculated by China'sand Xie's(1987) methods are too great and the Hart's method is suitable at the beginning ofconsolidation and much too small at last. The errors of Xie's(1987) method increases withdecreases of length of sand drains.The commonly used technique for numerical modeling of sand drains, replacing sand drainsby equivalent sand grooves or named as sand walls, is modified based on DDM. Equivalence of
differential equations is strictly satisfied in the modified method and the distance betweenvertical drains is not magnified yet done in the others. Also the modified method is available fornon-fully penetrating sand drains.Difference format of non-linear consolidation equations is developed in the case of onedimension and microcomputer code NLCL is established for modeling process.This program is applied to analysis non-linear effects in K0 consolidation, including Davis'smodel, non-Darcy model and rheological model.For Davis's model, the non-linear dimensionless consolidation equation is simulated:vcTUZUUUZk??? [ ? ??]=1?with conditions:①Tv=0, U=1;②Z=0, U=1+P/σ′z0 = n +1;③Z=1,U=n+1In which ck is the attenuation index of conductivity with increase of effective stress and n=P/σ′z0is the degree of loading related to initial effective stress。The equation had been investigated byDavis(1965) in analytical solution while ck=1 but modeling results are obtained with any cases ofck and n in the paper.The effects of two possible types of non-Darcy flow are discussed according to modelingresults. An important new conclusion for the existence of non-Darcy flow in soil columnexperiments is presented from reviewing of the analysis given by previous researcher. Twomistakes are found in previous analysis of soil column experiments: (1) transformation of soilframework was neglected during unsteady flow of pore water;(2) the effects of observationinstruments to the results of experiment were neglected. The mistakes are associated with aviewpoint that the phenomenon in soil column experiments shows non-Darcy flow in clay.However, it's proved in this paper, with analytical solution and water pressure-tube modifiednumerical modeling, that linear theory of consolidation with Darcy flow has enough satisfiabilityto the soil column experiment presented by Feng(1995) and by contraries non-Darcy flow'ssolutions are not agree to the data observed.Finite element method (FEM) for Biot consolidation and finite difference method (FDM) forTerzaghi consolidation of soft foundations with sand grooves are studied in the paper. Modelingcode CFEM2 and CFDM2 are developed to accomplish the simulation work.It's found from repeatedly modeling test that, to ensure the convergence of iterative solutionprocedure for coupled numerical equations of consolidation, time steps should be restricted as:max2()[()]Cv?t >sω ?LWhere ?t is the length of time step, ?L is the scale of elements in network and Cv is thecoefficient of consolidation. The function s(ω) gives a value greater than zero and increasingwith increase of the acceleration factor ω.The applicability of FEM and FDM is discussed for soft foundations. Compared to the results
of FEM, excess pore pressure is greater and settlement is smaller which given by FDM. Thedifferences of settlement on the surface arises from drains is too great with FDM. The viewpointis proposed that FEM is more reasonable than FDM since it deal with the transformation andexcess pore pressure at the same time considering coupled consolidation but settlement valuesare indirectly obtained from excess pore pressure with FDM. However, FDM is simple forapplication and available to the cases of precision demand less than 10%.Some common characteristics have been shown in modeling of soft foundations with sandgrooves such as the stiffness of shallow pre-consolidated soils do influence to the settlement offoundation remarkably.The PDSS ( Plane Deformation & Spatial Seepage) modeling method which proposed by Xiein 1987 is suggested to be the most valuable procedure for simulation of CSDF.The paper has developed Xie's PDSS model at some aspects as following:1.Distribution of sand drains is not directly treated while the grid patterns are triangular inXie's model. To solve the problem, three symmetric sections are selected to excavate themodeling space:and the space is divided into spatial elements with 6 nodes or 8 nodes.2. A sand drain is located on a series of element in column with the shape of square in Xie'smodel. It is pointed out that the procedure is not correct for radial flow around sand drains andwould bring errors of excess pore pressure at the degree of 7.3% or more. In the paper, atechnique developed in hydrogeology for well flow in numerical analysis of groundwater isintroduced to the modeling of sand drains. The modified numerical flow model in threedimensions is successful to consider the smear action and well resistance effect.3. The place of nodes in the FEM network are fixed in Xie's model. It is not proper for thestrain of soil is always larger than 10% and great settlement would reshape the foundation. So,the location of grid nodes are moveable in modified model.4. Also the effects of large and uneven settlement to actual load distribution on the surface offoundations are neglected in Xie's model and considered in the paper.xySection 1Section 2Section 3 Drains
5. The model is improved to be able to accomplish inverse programs. The values of parametersinput to the model are auto-optimal adjusted by compare the modeling results with observeddata.A C++ code program FDSPACE is designed according to above modifies for FEM modelingof CSDF.The program has been successfully used to a case study of preloading project for softfoundation beneath highway embankment in Guangdong province, China. The foundation wasdrained by sand drains 21.5 m long and preloaded by combined vacuum and embankment. Largesettlement more than 3.0m had been observed in about 2.5 years.Liner elastic model, Duncan-Chang's model and Cambridge Clay model and models forsecondary compression are introduced to clay layers in the ground. The optimal values of 54parameters are obtained from inverse procedure. It is indicated that Cambridge Clay model is themost satisfactory one among the models. Liner elastic model is not able to simulate the plasticpatterns of cohesive soils.It has been predicted that the post-construction subsidence of the foundation would be over to0.5m in 10 years.The development of seepage fields and stress fields in the foundation are investigatedaccording to the modeling results and flux of pore water in the sand drains are analyzed.The actions of some factors to consolidation of CSDF under vacuum and embankmentcombined preloading, such as length and interval of sand drains, depth and thickness of sandylayers, are studied in the paper. It's indicated that the sand drains have functions at large depthmore than 20m in the ground. Compared to preloading by embankment only, preloading bycombined vacuum and embankment is beneficial to shorten construction period of embankmentfilling and decrease the post-construction subsidence without failing on steady of the foundation.The research results presented above have advantages for studies of numerical methods in soilmechanics and for practices of soft ground improvement techniques.
在上述背景下,本文的研究目的就是对渗流固结基本课题进行深入、系统的数值分析,提炼出最新的认识,帮助解决粘性土-砂井地基固结的基础问题,同时,对现有的粘性土-砂井地基固结模型和数值计算方法进行整理和改进,使之更合理、更具有实用价值。
本文提出了单井固结新的理论模型,即一维双重介质模型,并得到在完整砂井条件下的解析解,该解析解为完整砂井等应变固结的普遍解。一维双重介质模型克服了现有单井固结理论将砂井周围土体简单划分为涂抹区和非涂抹区的缺点,适用于砂井周围土体渗透系数任意分布的情况,减少了单井固结分析的不确定性参数。更重要的是,该模型最大限度的降低了砂井地基模型的空间维数,却不失合理性。
利用一维双重介质模型,本文首次实现了砂井地基固结的一维有限差分模拟(自编模拟程序为SDDM),得到非完整砂井地基单井固结的固结度曲线。通过分析对比,证明现有的非完整砂井单井固结的各种近似计算公式都存在系统的误差。其中,国内方法和谢康和改进法计算结果偏大,Hart法计算结果在固结前期比较准确,在固结后期显著偏小,谢康和改进法的误差随着砂井贯入地基深度的减小而增大。
以一维双重介质模型为基础,本文对砂井“等效”处理为砂沟的数值模拟方法做出了改进。改进方法严格以渗流固结偏微分方程的等价性为等效原则,不人为加大竖向排水体的间距,适用于非完整砂井条件,只需要转换含砂井区土体水平渗透系数和砂沟的渗透系数两个参数。这些特点使改进方法比现有“等效”转换方法更严密,更能够反映实际砂井
Vertical drains is available to expedite the consolidation of cohesive soil and extensivelyapplied over the world in construction engineering of ports, buildings and roadways to reinforcesoft foundations. Sand Drain, is the most representative one in the group of vertical drains. It issuggested to call the foundation with cohesive soil layers improved by sand drains CohesiveSoil-Sand Drained Foundation (CSDF).
It is important to research the behavior of sand drains in preloading processing for design andanalysis of ground improving programs. As a typical problem of consolidation process, theinvestigation of seepage and transformation of CSDF is involved with a set of basic questions toconsolidation, such as the models and solutions of consolidation for cohesive soil. Numericalmethod is one of the significant techniques for these problems and has been applied to modelingthe CSDF. In resent years, the reinforcement technique of CSDF, preloading under embankment,vacuum or coupled vacuum-embankment, has been deeply developed in China for constructionof highways. However, the engineers still do their work mainly with experience and classicalanalytical methods since the absence of new efficacious solutions and, the programs are alwaysled to some difficult situations. They help the conditions could be changed.
So, it's valuable to develop scientific technique for simulation of CSDF and helpful to dealwith complex conditions for engineers.
Here the purposes are presented for the paper: to analysis the normal consolidation behavior ofcohesive soil with numerical methods and abstract new points of view;to develop the modelsand numerical solutions of CSDF. Research results have been established in the dissertation forthese purposes.
Uniform model in nonlinear is suggested to be combined with nonlinear constitutive modeland nonlinear seepage model. The basic nonlinear model of consolidation is shown in thediagram as following:
In which H-V is the elasto-plastic element, H|N is the visco-elastic element and V|H is thevisco-plastic element for the soil framework, and W is the pore water with Darcy flow ornon-Darcy flow. A common formula is proposed to describe the volume strain speed in onedimensional condition:bPbucccdtautaPttvpvve??ε v =??εz=1 ??+2??+1+2+1εv+2εv+1ε+In which P is the loading on soil column and u is the excess pore pressure, a1、a2、b1、b2、c1、c2、c3、d are parameters involved to the state of effective stress and strain.It is also suggested to write the differential equations of Biot consolidation in matrix form asfollowing:[ K ]δ ? [C]u=F? [C ]T δ& +[G]u=nβu&+QDefinitions are:[K]=[B]T[D][B] —— stiffness matrix[G]=[C]T[H][C]/γw —— seepage matrix[C] —— coupled matrixF —— forces for equilibriumQ —— flux for equilibriumAnd δ is the displacement of soil framework, β is the coefficient for compressibility of water.The advantage of this kind of expression is that it make the analytical equations are very closeto the FEM equations of Biot consolidation.A new theoretical model of consolidation for single sand drain is developed and called as oneDimensional Dual-medium Model (DDM). Rigorous solution is given for fully penetrating sanddrain with the new model as a normal solution for the case of equal vertical strain. DDM hassome advantages related to the theory developed before: the area of soil around sand drain is notsimply marked as smear zone and non-smear zone and the model is available for any distributionof soil permeability;the number of uncertain parameters is decreased from two to only one. Themore important characteristic of DDM, is reducing mostly of the modeling dimensions forCSDF.The consolidation of CSDF is simulated firstly with one dimensional finite differencemodeling based on the DDM and curves of consolidated degree U for non-fully penetrating sanddrain is obtained. According to the comparing of results, systemic error is found in approximatedformulations proposed before for non-fully penetrating sand drain. The U calculated by China'sand Xie's(1987) methods are too great and the Hart's method is suitable at the beginning ofconsolidation and much too small at last. The errors of Xie's(1987) method increases withdecreases of length of sand drains.The commonly used technique for numerical modeling of sand drains, replacing sand drainsby equivalent sand grooves or named as sand walls, is modified based on DDM. Equivalence of
differential equations is strictly satisfied in the modified method and the distance betweenvertical drains is not magnified yet done in the others. Also the modified method is available fornon-fully penetrating sand drains.Difference format of non-linear consolidation equations is developed in the case of onedimension and microcomputer code NLCL is established for modeling process.This program is applied to analysis non-linear effects in K0 consolidation, including Davis'smodel, non-Darcy model and rheological model.For Davis's model, the non-linear dimensionless consolidation equation is simulated:vcTUZUUUZk??? [ ? ??]=1?with conditions:①Tv=0, U=1;②Z=0, U=1+P/σ′z0 = n +1;③Z=1,U=n+1In which ck is the attenuation index of conductivity with increase of effective stress and n=P/σ′z0is the degree of loading related to initial effective stress。The equation had been investigated byDavis(1965) in analytical solution while ck=1 but modeling results are obtained with any cases ofck and n in the paper.The effects of two possible types of non-Darcy flow are discussed according to modelingresults. An important new conclusion for the existence of non-Darcy flow in soil columnexperiments is presented from reviewing of the analysis given by previous researcher. Twomistakes are found in previous analysis of soil column experiments: (1) transformation of soilframework was neglected during unsteady flow of pore water;(2) the effects of observationinstruments to the results of experiment were neglected. The mistakes are associated with aviewpoint that the phenomenon in soil column experiments shows non-Darcy flow in clay.However, it's proved in this paper, with analytical solution and water pressure-tube modifiednumerical modeling, that linear theory of consolidation with Darcy flow has enough satisfiabilityto the soil column experiment presented by Feng(1995) and by contraries non-Darcy flow'ssolutions are not agree to the data observed.Finite element method (FEM) for Biot consolidation and finite difference method (FDM) forTerzaghi consolidation of soft foundations with sand grooves are studied in the paper. Modelingcode CFEM2 and CFDM2 are developed to accomplish the simulation work.It's found from repeatedly modeling test that, to ensure the convergence of iterative solutionprocedure for coupled numerical equations of consolidation, time steps should be restricted as:max2()[()]Cv?t >sω ?LWhere ?t is the length of time step, ?L is the scale of elements in network and Cv is thecoefficient of consolidation. The function s(ω) gives a value greater than zero and increasingwith increase of the acceleration factor ω.The applicability of FEM and FDM is discussed for soft foundations. Compared to the results
of FEM, excess pore pressure is greater and settlement is smaller which given by FDM. Thedifferences of settlement on the surface arises from drains is too great with FDM. The viewpointis proposed that FEM is more reasonable than FDM since it deal with the transformation andexcess pore pressure at the same time considering coupled consolidation but settlement valuesare indirectly obtained from excess pore pressure with FDM. However, FDM is simple forapplication and available to the cases of precision demand less than 10%.Some common characteristics have been shown in modeling of soft foundations with sandgrooves such as the stiffness of shallow pre-consolidated soils do influence to the settlement offoundation remarkably.The PDSS ( Plane Deformation & Spatial Seepage) modeling method which proposed by Xiein 1987 is suggested to be the most valuable procedure for simulation of CSDF.The paper has developed Xie's PDSS model at some aspects as following:1.Distribution of sand drains is not directly treated while the grid patterns are triangular inXie's model. To solve the problem, three symmetric sections are selected to excavate themodeling space:and the space is divided into spatial elements with 6 nodes or 8 nodes.2. A sand drain is located on a series of element in column with the shape of square in Xie'smodel. It is pointed out that the procedure is not correct for radial flow around sand drains andwould bring errors of excess pore pressure at the degree of 7.3% or more. In the paper, atechnique developed in hydrogeology for well flow in numerical analysis of groundwater isintroduced to the modeling of sand drains. The modified numerical flow model in threedimensions is successful to consider the smear action and well resistance effect.3. The place of nodes in the FEM network are fixed in Xie's model. It is not proper for thestrain of soil is always larger than 10% and great settlement would reshape the foundation. So,the location of grid nodes are moveable in modified model.4. Also the effects of large and uneven settlement to actual load distribution on the surface offoundations are neglected in Xie's model and considered in the paper.xySection 1Section 2Section 3 Drains
5. The model is improved to be able to accomplish inverse programs. The values of parametersinput to the model are auto-optimal adjusted by compare the modeling results with observeddata.A C++ code program FDSPACE is designed according to above modifies for FEM modelingof CSDF.The program has been successfully used to a case study of preloading project for softfoundation beneath highway embankment in Guangdong province, China. The foundation wasdrained by sand drains 21.5 m long and preloaded by combined vacuum and embankment. Largesettlement more than 3.0m had been observed in about 2.5 years.Liner elastic model, Duncan-Chang's model and Cambridge Clay model and models forsecondary compression are introduced to clay layers in the ground. The optimal values of 54parameters are obtained from inverse procedure. It is indicated that Cambridge Clay model is themost satisfactory one among the models. Liner elastic model is not able to simulate the plasticpatterns of cohesive soils.It has been predicted that the post-construction subsidence of the foundation would be over to0.5m in 10 years.The development of seepage fields and stress fields in the foundation are investigatedaccording to the modeling results and flux of pore water in the sand drains are analyzed.The actions of some factors to consolidation of CSDF under vacuum and embankmentcombined preloading, such as length and interval of sand drains, depth and thickness of sandylayers, are studied in the paper. It's indicated that the sand drains have functions at large depthmore than 20m in the ground. Compared to preloading by embankment only, preloading bycombined vacuum and embankment is beneficial to shorten construction period of embankmentfilling and decrease the post-construction subsidence without failing on steady of the foundation.The research results presented above have advantages for studies of numerical methods in soilmechanics and for practices of soft ground improvement techniques.
引文
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