Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary

The semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation for unsaturated soils with a semi-permeable drainage boundary are pre- seated. Two variables are introduced to transform the two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations （PDFs）, which are easily solved by the Laplace transform method. Then, the pore-water pressure, pore-air pressure, and soil settlement are obtained in the Laplace domain. The Crump method is adopted to perform the inverse Laplace transform in order to obtain the semi-analytical solutions in the time domain. It is shown that the proposed solutions are more applicable to various types of boundary conditions and agree well with the existing solutions from the literature. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with single, double, mixed, and semi-permeable drainage boundaries. The changes in the pore-air and pore-water pres- sures and the soil settlement with the time factor at different values of the semi-permeable drainage boundary parameters are illustrated. In addition, parametric studies are con- ducted on the pore-air and pore-water pressures at different ratios （the air permeability coefficient to the water permeability coefficient） and depths.

One-dimensional nonlinear consolidation analysis of soil with continuous drainage boundary

Following the assumptions proposed by MESRI and ROKHSAR,the one-dimensional nonlinear consolidation problem of soil under constant loading is studied by introducing continuous drainage boundary.The numerical solution is derived by using finite difference method and its correctness is assessed by comparing with existing analytical and numerical solutions.Based on the present solution,the effects of interface parameters,stress ratios(i.e.,final effective stress over initial effective stress,N_(σ))and the ratio c_(c)/c_(k)of compression index to permeability index on the consolidation behavior of soil are studied in detail.The results show that,the characteristics of one-dimensional nonlinear consolidation of soil are not only related to c_(c)/c_(k)and N_(σ),but also related to boundary conditions.In the engineering practice,the soil drainage rate of consolidation process can be designed by adjusting the values of interface parameters.

Modelling of Quasi 3-D Groundwater Flow and Studying of Equivalent Drainage Boundary in Beihai Peninsula,Guangxi

This paper describes a quasi 3-D finite element model of the groundwater flow in two -aquifer system which is constructed from a sequence of aquifer flow equations coupled by leakage terms representing flow through the aquitard . It is applied to evaluate the maximum rate of groundwater resources exploited from the coastal aquifer without seawater intrusion . The main task in this model is to determine the drainage boundary of the aquifer extending under the sea . The information of the boundary can be obtained from the fluctuations of the groundwater level caused by sea-tide fluctuations . A new idea, Equivalent Drainage Boundary (EDB), is proposed and the corresponding methods , determining the EDB, are developed with tidal fluctuations data observed in boreholes . The quasi 3-D model and the methods determining EDB have been applied to the aquifer system of Beihai peninsula , Guangxi Autonomous Region of China for calculating the available groundwater resources .

A general solution for vertical-drain consolidation with impeded drainage boundaries

An analytical solution is derived from the generalized governing equations of equal-strain consolidation with vertical drains under multi-ramp surcharge preloading. The hydraulic boundary conditions at both top and bottom of the consolidating soil are modelled as impeded drainage. The impeded drainage is described by using the third type boundary condition with a characteristic factor of drainage efficiency. Fully drained and undrained boundary conditions can also be modelled by applying an infinite and a zero characteristic factor, respectively. Simultaneous radial and vertical flow conditions are considered, together with the effects of drain resistance and smear. An increase in total stress due to multi-ramp loading is reasonably modelled as a function of both time and depth. A solution to calculate excess pore-water pressure at any arbitrary point in soil is derived, and the overall average degree of consolidation is obtained. It shows that the proposed solution can be used to analyze not only vertical-drain consolidation but also one-dimensional consolidation under either one-way or two-way vertical drainage conditions. The characteristic factors of drainage efficiency of top and bottom boundaries have a potentially important influence on consolidation. The boundary may be considered fully drained when the characteristic factor is greater than 100 and fully undrained when the characteristic factor is less than 0.1. The stress distribution along depth induced by the surcharge loading has a limited effect on the overall average degree of consolidation.

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Analysis of one-dimensional consolidation of fractional viscoelastic saturated soils under continuous drainage boundary conditions

This paper presents the one-dimensional(1D)viscoelastic consolidation system of saturated clayey soil under continuous drainage boundaries.The fractional-derivative Merchant(FDM)model has been introduced into the consolidation system to simulate the viscoelasticity.Swartzendruber’s flow law is also introduced to describe the non-Darcian flow characteristics simultaneously.The generalized numerical solution of the 1D consolidation under continuous boundaries is given by the finite difference scheme.Furthermore,to illustrate the effectiveness of the numerical method,two simplified cases are compared against the current analytical and numerical results.Finally,the effects of boundary parameters and model parameters on the viscoelastic consolidation were illustrated and discussed.The results indicated that the boundary parameters have a significant influence on consolidation.The larger the values of boundary parameters,the faster the whole dissipation of the excess pore-water pressure and soils’settlement rate.Fractional-order and viscosity parameter have little effect on consolidation,which are primarily significant in the middle and late consolidation phases.With the increase of the modulus ratio,the whole consolidation process becomes faster.Moreover,considering Swartzendruber’s flow delays the consolidation rate of the soil layer.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED SOILS WITH IMPEDED BOUNDARIES UNDER TIME-DEPENDENT LOADINGS

On the basis of Terzaghi's one-dimensional consolidation theory, the variation of effective stress ratio in layered saturated soils with impeded boundaries under time-dependent loading was studied. By the method of Laplace transform, the solution was presented. Influences of different kinds of cyclic loadings and impeded boundaries conditions were discussed. Through numerical inversion of Laplace transform, useful illustrations were given considering several common time-dependent loadings. Pervious or impervious boundary condition is just the special case of the problem considered here. Compared with average index method,the results from the method illustrated are more accurate.

连续排水边界下考虑指数渗流的软土一维非线性固结分析

为研究考虑边界排水的时效特性、变荷载、非达西渗流以及土体非线性特性的饱和软土固结问题,通过引入连续排水边界,建立考虑指数渗流和土体非线性的固结方程。利用有限差分法得到变荷载及瞬时荷载下基于连续排水边界考虑指数渗流的软土一维非线性固结解答,通过渗流形式退化和边界退化的方法,将所得解答与达西渗流下的解答及传统排水边界下的解答对比验证了该文差分法解答的可靠性。基于所得解答,详细分析了界面参数、指数渗流及土体非线性等因素对固结特性的影响。结果表明:界面参数取值越大则边界排水能力越大,边界排水性能较差时提高界面参数取值能有效提高边界排水能力;边界排水性能主要影响土体固结前期,而指数渗流主要影响固结后期;实际工程中如果关注固结的整个过程,则需要考虑边界排水性能的影响,但如果只关注固结后期的沉降值,则可忽略边界排水性能的影响。

PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS

We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.

考虑变荷载和变边界的软土大应变非线性固结分析

为了准确预测实际土层的固结速率,在Gibson大应变固结理论基础之上,考虑土体的非线性压缩渗透特性和土层表面随时间改变的变排水边界条件,建立考虑变荷载和变边界的软土大应变非线性固结模型。在此基础上,获得单级加载和多级加载条件下大应变非线性固结模型的显式解析解,利用获得的解析解开展计算以分析界面参数、荷载取值对土层固结性状的影响。结果表明:外荷载对土体内部超静孔压消散的影响表现为随着荷载的增大,超静孔压消散得越快,但其不会影响土体最终完成固结的时间;参数I_(c)(α-2)取值对土体固结性状的影响表现为固结速率随着参数I_(c)(α-2)取值的变化而变化。值得注意的是,描述土体非线性特性的参数取值对固结性状的影响在变边界下显现的固结性状会出现延迟现象,不同参数下的变边界更能反映土体边界的真实透水性情况,在实际工程中可以根据超静孔压实际的消散速率反演界面参数,以便能准确预估土层的固结进程。

Modflow中关键参数取值对矿井涌水量预测的影响研究

针对鄂尔多斯地区矿井开采过程中的涌水问题,以核桃峪井田为研究对象,对区域水文地质条件进行梳理,确定合理的概念模型和边界范围,建立了Visual Modflow数值模型,应用模型的排水沟边界模块,预测响应工作面采掘进度的矿井涌水量变化。并对排水沟水力传导系数C与含水层渗透系数K两个影响因子进行了不确定性分析,确定C的取值对工作面涌水量预测的影响更显著。根据临近的雅店井田两个工作面实际涌水量对C值进行反演,确定研究区域C值范围并代入核桃峪井田工作面预测中。结果表明:工作面涌水量预测值会随着C值取值增大而增大,当C取值约6.5×10^(-5)时,雅店井田工作面涌水量接近实际涌水量。基于该值计算出核桃峪井田各工作面矿井涌水量情况,待5个工作面回采结束后,矿井涌水量总量达到2697.2 m^(3)/h。

变透水边界与起始坡降下软土路基固结模型

研究目的:路基沉降问题是引起铁路、公路等基础设施发生工程事故的主要原因。为提高软土地区路基土沉降预测水平,本文考虑土体的非达西渗流效应、土体边界透水能力的变化以及变形非线性特征,建立相应的分段线性化固结模型。其中,起始坡降渗透模型用来反映土体的非达西渗流过程,连续排水边界用来描述土体边界透水性能的变化,e-lg(av0)关系用来表示土体的非线性变形特征。然后,应用一些算例分析探讨边界透水性能变化、土体非线性变形以及非达西效应下土体的固结行为,以期为软土路基工程工后沉降预测提供理论支撑。研究结论:(1)土体的非达西效应越强(即起始坡降值越大),边界透水性能越差(即连续排水界面参数越小),相同时刻下的固结滞后现象越明显;(2)起始坡降值能同时影响固结过程及最终结果;(3)连续排水边界和起始坡降下初始压缩系数对沉降过程的影响比较显著;(4)本文固结模型可模拟深厚软土路基的沉降过程,相关结论可应用于深厚软土的堆载排水加固过程。

广东省深圳河综合治理思考

深圳河作为深圳和香港地区的界河,在国家大力推进生态文明建设的背景下,为深化深港协同实现高质量发展提供水务保障。介绍了深圳河的概况,总结了当前存在的关键问题,提出了当前深圳河需从流域统筹、防洪排涝等方面着手,建立深港协同、河湾联动、洪涝共治的治理策略及实施方案,以期为界河的治理提供参考。

Analytical solutions for consolidation of stone column composite foundations considering time-dependent boundary and loading

In response to the existing consolidation theory for stone column composite foundations which cannot consider the time-dependent loading and the well resistance effect of stone columns under time-dependent boundaries,a consolidation model that can reflect these characteristics is developed in this study,and the corresponding analytical solutions are obtained under permeable top surface with permeable bottom surface(PTPB)and permeable top surface with impermeable bottom surface(PTIB),respectively.In addition,the reliability of the proposed solutions is verified by comparing them with existing analytical solutions.Extensive calculations are then performed by the proposed solutions to analyze the consolidation behaviors of stone column composite foundations under time-dependent boundaries,the results show that the interface parameters have a large effect on the distribution of excess pore water pressure(EPWP)along the depth;for projects with longer construction time,the permeability of the top and bottom surfaces of the composite foundation has a smaller effect on the average consolidation rate.Finally,the proposed solution is applied to the settlement calculation in an actual engineering project,and the theoretical results show a general agreement with the measured data by considering the influence of the interface parameters.

Transient Pressure Behavior of a Well Located between a Constant Pressure Boundary and a Sealing Fault

Pressure transient analysis has been used to evaluate performance of a well located between one sealing fault and one constant pressure boundary. Type curves were generated by determining 1) dimensionless pressure and 2) rate of change of dimensionless pressure drop with respect to dimensionless time. When the well is located closer to the no flow boundary, both sets of type curves have three distinct slopes. These slopes characterize: 1) flow in an infinite reservoir, 2) presence of the no flow, and 3) the constant-pressure boundaries. When the well is closer to the constant pressure boundary, the type curves show two distinct slopes. These correspond to: 1) flow in an infinite reservoir, and 2) the presence of a constant pressure boundary. The type curves can be used to match actual pressure drawdown data and determine the drainage area and relative well location with respect to physical boundaries.

连续排水边界下成层陆域吹填地基一维固结解析解

新近吹填地基的固结效应十分显著,地基土内超静孔隙水压力的消散将引起地表的整体沉降,对吹填地基上基础工程的稳定性产生不利影响。基于连续排水边界,综合考虑吹填土的自重和下伏原状海床的成层特性,建立了吹填土自重驱动下的吹填地基-原状海床一维固结方程,并基于本征函数法推导了孔隙水压力响应、固结度等的解析解。通过边界条件和模型的退化,验证了解答的正确性。利用该解答讨论了吹填土自重、成层地基模量、时间因数、地表附加荷载等时空条件对吹填地基固结沉降的影响。研究结果表明:(1)吹填土自重是吹填地基固结的驱动因素,在实际工程中不可忽略;(2)土体渗透系数对超静孔隙水压力的影响较为复杂,某一层土的渗透性改变后,对其上覆和下伏土层的超静孔隙水压力将产生相反的影响;(3)土层的体积压缩系数对超静孔隙水压力的影响十分显著,且深部土体的体积压缩系数影响更大;(4)地表堆载等引起的附加荷载会减缓超静孔隙水压力的消散。

细管径虹吸流速计算参数分析与修正计算方法

高扬程边坡虹吸排水会释放大量气泡,为防止虹吸过程产生空气积累破坏虹吸排水的长效性,需要采用d≤5 mm管径形成段塞流。由于管径小且有气液二相流,已有虹吸流速计算可能出现较大误差,影响虹吸排水系统设计的合理性。针对细管径虹吸流速计算误差大的问题,通过分析虹吸流动过程特征来确定流速计算参数,将巴尔公式引入沿程水头损失计算中,并开展足尺物理模型试验。确定适用于虹吸流速迭代运算的沿程水头损失系数计算公式,提出利用断面平均水头损失反映边界层作用影响的圆管层流流速计算方法,在巴尔公式中引入折减系数降低临界雷诺数区的流速计算误差。物理模型试验的测试结果表明,提出的修正计算方法提高了虹吸流速的计算精度,可为工程应用提供参考。

考虑边界排水时间效应的软土一维非线性固结近似解答

为了考虑实际工程中,土层边界的排水性状是一个随时间连续变化的过程,基于经典的非线性压缩和非线性渗透关系,研究考虑排水边界时间效应时固结系数随时间变化的土体一维非线性固结问题。通过分离变量法和Laplace变换技术,获得该固结问题的近似解答;再将该文解退化并与既有差分解答进行对比,从而验证近似解答正确性。基于所得解,探讨了压缩指数与渗透指数比值C_(c)/C_(k)、最终有效应力与初始有效应力比值N_(q)及界面参数α、β对土体固结特性的影响。结果表明:在连续排水条件下,土体内的不排水对称面的位置与界面参数有关,并且随时间会发生变化;当C_(c)/C_(k)>1时,在固结前期,土体的沉降速率随着N_(q)值的增大而增大,而在固结后期则相反。排水边界的界面参数取值对反映边界排水速率影响很大,因此工程中计算地基固结速率时有必要确定界面参数取值。

变荷载下考虑起始水力坡降与连续排水边界的固结解

采用连续排水边界的必要性及软黏土中水的渗流存在着起始水力坡降(i0)的现象已逐渐被认识,但变荷载下同时考虑连续排水边界条件和起始水力坡降的一维固结解析解还鲜见报道。针对外荷载随时间逐渐增加的实际情况,建立变荷载下同时考虑连续排水边界和起始水力坡降的固结模型。采用傅里叶变换及拉普拉斯变换获得模型的近似解析解,利用该解答分析动边界移动规律、超静孔隙水压力随时间的消散规律及平均固结度的增长规律。结果表明:在加载速率不变的情况下,起始水力坡降下排水面透水情况对固结性状的影响与其在达西定律下相同,透水情况越好,孔压消散速率越快;透水情况越差,超静孔隙水压力消散速度越慢。与完全透水边界条件下相比,连续排水条件下起始水力坡降i0对固结性状影响无明显改变,i0越大,移动边界到达土层底部的时间越长,固结完成时土中残留的超静孔压越大,按孔压定义的平均固结度越小;i0值越小,移动边界到达土层底部的时间越短,固结完成时土中残留的超静孔压越小,按孔压定义的平均固结度越大。在起始水力坡降和边界排水条件不变的情况下,随着加载时间的增加,超静孔隙水压力峰值越小,超静孔隙水压力达到峰值的时间越长,但加载时间对超静孔压残留值及按孔压定义的最终平均固结度无影响。

连续排水边界下成层地基一维非线性固结解析解

为了研究边界排水时间效应对固结过程的影响,引入连续排水边界研究成层地基一维非线性固结问题。通过引入孔隙比与有效应力之间的非线性关系,建立了瞬时荷载作用下成层地基固结问题的控制方程,进而获得按沉降定义和按超静孔隙水压力定义的平均固结度以及成层地基沉降的解析解。通过与有限差分解析解对比以及现有解析解对比验证了所得解析解的正确性。基于所得解,通过详细的参数分析研究了成层地基一维非线性固结特性。结果表明,排水界面参数对成层地基的固结性状有很大影响,固结速率随界面参数的增大而增大。在连续排水边界条件下,按沉降定义的平均固结度随有效应力比的增大而增大,而按超静孔隙水压力定义的平均固结度随有效应力比的增大而减小。