An efficient iterative method for solving the governing equation of water infiltration problems in unsaturated soils

One of the significant problems in geo-environmental and geotechnical engineering is the unsaturated flow of soil in unsaturated soils. The model of this phenomenon in porous media is governed by the Richards equation. In this paper a new, efficient, iterative method is used to handle the Richards equation. This new technique is obtained from the variational iteration method by a simple reconstruction that is the Laplace iteration method (LIM). In order to evaluate the efficiency and accuracy of the solutions obtained by the proposed method, two representative examples were investigated. The obtained results show that the Laplace iteration method is a very effective method, simplifies the difficulty of classical techniques and is quite accurate for systems of partial differential equations.     

Solution of the unsaturated soil moisture equation using repeated transforms

An alternative method of solution for the linearized ‘theta‐based’ form of the Richards equation of unsaturated flow is developed in two spatial dimensions. The Laplace and Fourier transformations are employed to reduce the Richards equation to an ordinary differential equation in terms of a transformed moisture content and the transform variables, s and ξ. Separate analytic solutions to the transformed equation are developed for initial states which are either in equilibrium or dis‐equilibrium. The solutions are assembled into a finite layer formulation satisfying continuity of soil suction, thereby facilitating the analysis of horizontally stratified soil profiles. Solution techniques are outlined for various boundary conditions including prescribed constant moisture content, prescribed constant flux and flux as a function of moisture change. Example solutions are compared with linearized finite element solutions. The agreement is found to be good. An adaptation of the method for treating the quasilinearized Richards equation with variable diffusivity is also described. Comparisons of quasilinear solutions with some earlier semi‐analytical, finite element and finite difference results are also favourable. Copyright © 2001 John Wiley & Sons, Ltd.     

非饱和土壤Richards方程入渗求解探讨

根据理查兹(Richards)方程求解非饱和土壤垂直一维入渗率历时规律,因缺乏土壤水分特性函数K(θ)、D(θ)的一般有效公式,而采用与实际相差甚远的定值或线性函数假设,使成果隐含过多经验成分或需要反复迭代计算.笔者在文献[1]中提出的非饱和土壤水分特性函数的一般有效基础公式,用分离变量法直接求解Richards方程,用土壤水动力学方法分别导出了含有渗前土湿因子的新入渗公式.它摆脱了同类入渗公式对水分函数过于简单的概化假设和经验成分.为放弃迄今水文学产流计算中惯用的下渗容量曲线"截首留尾"的经验做法,以直接使用含有渗前土湿因子的下渗公式提供了理论依据,从而将有利于产流计算方法的改进和计算精度的提高.     

Explicit approximate analytical solutions of seepage‐deformation in unsaturated soils

An approximate analytical solution is presented for the coupled seepage and deformation problem of unsaturated soils. Because of the matric suction dependence of both saturation and permeability coefficient, the coupled governing equations are strongly nonlinear. To obtain an analytical solution, these coupled governing equations are linearized and analytically solved for a specified saturation using the eigenfunction method. Then, the obtained analytical solutions are extended to the entire saturation range. Comparison between the current solution and the previous theoretical solution indicates that the proposed solution yields excellent results. Due to its analytical nature, the proposed procedure can be effectively used to obtain the solution of the coupled seepage and deformation of unsaturated soils.     

非饱和土入渗的数值模拟

入渗引起土中孔隙水压力的变化,是影响非饱和土性状的最重要的原因之一。因此,正确地分析和模拟非饱和土入渗过程具有重要意义。通过对一维和二维降雨入渗问题的研究,展示了湿润锋面、地下水位面及孔隙水压力分布随时间的变化,使得对入渗问题的认识更加直观和清楚。     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.     

Theoretical study and simulation of moisture transport in unsaturated porous media by periodic homogenization

In this work, the macroscopic Richards equation for moisture transport is established in unsaturated porous media using periodic homogenization. By performing dimensional analysis on microscopic equations of moisture transfer, dimensional numbers characterizing moisture transport appear. The application of the asymptotic homogenization leads to the classical Richards equation, which is justified rigorously this way. Moreover, we obtain an accurate definition of the homogenized diffusion tensor of moisture involving the geometric properties of the microstructure and known transport properties of the material. A different behavior for the transport of water vapor between hygroscopic and super‐hygroscopic region is revealed. Finally, a simple 2D example where an analytical solution exists is addressed. Copyright © 2014 John Wiley & Sons, Ltd.     

非饱和土层一维固结特性分析

在Fredlund非饱和土的一维固结理论的基础上进行假设,由得出的液相及气相的控制方程、Darcy定律及Fick定律,采用Laplace 变换、逆变换等数学方法得到了大面积均布瞬时加载下表面为透水透气面、底面为不透水和不透气面的非饱和土层一维固结时间域内的超孔隙水压力、超孔隙气压力及土层沉降的解析解;应用典型算例,分析了不同气、水渗透系数比情况下土体超孔隙水压力、超孔隙气压力消散及土层沉降随时间的变化规律以及不同时间超孔隙水压力、超孔隙气压力消散随深度的变化规律。将得出的结果退化成相应的饱和土的解与太沙基饱和土固结理论结果比较,验证了其正确性。     

非饱和带有限分析数值模拟的误差分析

由于描述包气带水分运移的Richard方程具有高度非线性,常用数值方法求解。有限分析法可以较好地保持原有问题的物理特性,近年来被广泛应用于环境及农业工程领域。基于不同的原理,目前有两种有限分析法模拟非饱和带水分运移问题,即基于局部线性化有限分析法和基于Kirchhoff变换有限分析法。为探讨两者在求解非饱和带水分运移问题时数值表现的差异性,采用已有解析解对两种方法进行初步验证;使用两个数值实验比较两种方法的计算精度;应用实测数据验证Kirchhoff变换有限分析法。相比于局部线性化有限分析法,Kirchhoff变换有限分析法能够更好地控制误差及获得更高精度的数值解。该研究对于完善非饱和带水分运移有限分析数值模拟的理论与方法有重要意义。     

Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils

An analytical solution to 1D coupled water infiltration and deformation is derived using a Fourier integral transform. Exponential functional forms are used to represent the hydraulic conductivity–pore‐water pressure relationship and the soil‐water characteristic curve. Fredlund's incremental‐linear constitutive model for unsaturated soils is adopted. The analytical solution considers arbitrary initial pore‐water pressure distributions and flux and pressure boundary conditions. The corresponding analytical solutions to coupled steady‐state problems are also obtained. The analytical solutions demonstrate that the coupling of seepage and deformation plays an important role in water infiltration in unsaturated soils. In the early stages of infiltration, the difference between uncoupled and coupled conditions becomes marked over time, and in late stages, the difference caused by the coupling effects diminishes toward the steady state. The difference between the uncoupled and coupled conditions increases with decreasing desaturation coefficient (α). Pore‐water pressure or deformation changes caused by the coupling effects are mainly controlled by the degree of soil volume change due to a change in soil suction (H). The smaller the absolute value of H, the greater the effect of coupling on the infiltration and deformation. The ratio of rainfall intensity to saturated permeability (q/k_s) also has a strong influence on the coupled seepage and deformation. Copyright © 2008 John Wiley & Sons, Ltd.     

非饱和土中水平入渗方程显式求解

在求解非饱和态土中水分入渗问题时,水力函数是体积含水率或者吸力的函数,致使其控制方程呈现出强非线性的特征,进而使得其求解变得十分困难。基于水分在土体介质中流动耗时取极值路径的选择这一假定,引入时间泛函,基于变分法原理将水平入渗问题转化为泛函极值问题。通过求解Euler–Lagrange方程,结合边界条件,得到非线性瞬态水平入渗问题的显式解析解。结合Brooks-Corey型水力函数,显式地求解出该类型非饱和态土的体积含水率发展分布规律。通过计算4种不同类型土体的水平入渗规律,将求解结果与已有结果以及数值结果进行对比,验证了该方法的有效性。结果表明:体积含水率分布与位置距离和湿润峰距离比值呈幂函数关系,指数取决于土-水特征曲线的形状参数;初始条件与边界条件会对体积含水率分布造成不同程度的影响。     

The role of pore water in the mechanical behavior of unsaturated soils

Deformation and failure of soils are governed by the stresses acting on the soil skeleton. The isotropic stress acting on the soil skeleton can be divided into two components. One is the stress component which is transmitted through the soil skeleton. This skeleton stress is influenced by the pore water (bulk water) in the soil. The other is the internal stress component which does not contribute to equilibrium with a given external force. The internal stress is induced by the capillary tension of meniscus water clinging to the contact point of soil particles and acts so as to connect the soil particles tightly. Therefore, in modeling the stress and strain relations for unsaturated soils, it is of much importance to quantitatively evaluate how the pore water exists in the soil. This paper discusses the role of pore water on the mechanical behaviour of the soil. In particular, the significance of the water retention curve is emphasized from a mechanical viewpoint. Essential features required in modeling of the constitutive relations for unsaturated soils are discussed and presented.     

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.     

Isogeometric analysis for unsaturated flow problems

Unsaturated flow problems in porous media often described by Richards’ equation are of great importance in many engineering applications. In this contribution, we propose a new numerical flow approach based on isogeometric analysis (IGA) for modeling the unsaturated flow problems. The non-uniform rational B-spline (NURBS) basis is utilized for spatial discretization whereas the stable implicit backward Euler method for time discretization. The nonlinear Richards’ equation is iteratively solved with the aid of the Newton–Raphson scheme. Owing to some desirable features of an efficient numerical flow approach, major advantages of the present formulation involve: (a) numerical oscillation at the wetting front can be avoided or facilitated, simply by using either an h-refinement or a lumped mass matrix technique; (b) higher-order exactness can be obtained due to the nature of the IGA features; (c) the approach is straightforward to implement and it does not need any transformation, e.g., Kirchhoff transformation or filter algorithm; and (d) in contrast to the Picard iteration scheme, which forms linear convergences, the proposed approach can however yield quadratic convergences by using the Newton–Raphson method for solving resultant nonlinear equations. Numerical model validation is analyzed by solving a three-dimensional unsaturated flow problem in soil, and its derived results are verified against analytical solutions. Numerical applications are then studied by considering three extensive examples with simple and complex configurations to further show the accuracy and applicability of the present IGA.     

Analytical solution for one‐dimensional consolidation of unsaturated soils using eigenfunction expansion method

This paper introduces an exact analytical solution for governing flow equations for one‐dimensional consolidation in unsaturated soil stratum using the techniques of eigenfunction expansion and Laplace transformation. The homogeneous boundary conditions adopted in this study are as follows: (i) a one‐way drainage system of homogenous soils, in which the top surface is considered as permeable to air and water, whereas the base is an impervious bedrock; and (ii) a two‐way drainage system where both soil ends allow free dissipation of pore‐air and pore‐water pressures. In addition, the analytical development adopts initial conditions capturing both uniform and linear distributions of the initial excess pore pressures within the soil stratum. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained based on the proposed boundary conditions. Besides, the Laplace transform method is adopted to solve the first‐order differential equations. Once equations with transformed domain are all obtained, the final solutions, which are proposed to be functions of time and depth, can be achieved by taking an inverse Laplace transform. To verify the proposed solution, two worked examples are provided to present the consolidation characteristics of unsaturated soils based on the proposed method. The validation of the recent results against other existing analytical solutions is graphically demonstrated. Copyright © 2013 John Wiley & Sons, Ltd.     

Coupled solution of heat and moisture flow in unsaturated clay barriers in a repository geometry

An analytical solution in the Laplace transform domain is obtained for the transient heat and moisture transport in an unsaturated clay buffer with a geometry simulating repository conditions. A numerical inversion scheme based on Crump's method is used to obtain the time‐domain solution. The coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low‐permeability buffers. The practical background is based on the case of an engineering bentonite barrier placed in a drift excavated in rock in the context of underground disposal of high‐level radioactive waste. Parametric study has been performed to assess the effects of dimensionless geometry and material parameters on flow field. Despite the simplified assumptions required in order to obtain analytical expressions, the results incorporate the main mechanisms involved in the coupled thermo‐hydraulic (T–H) problem, and they may be eventually used for validation purposes. Copyright © 2006 John Wiley & Sons, Ltd.     

Three‐dimensional transient thermo‐hydro‐mechanical fundamental solutions of unsaturated soils

The governing differential equations of unsaturated soils considering the thermo‐poro‐mechanical behaviour consist of equilibrium, moisture air and heat transfer equations. In this paper at first, following some necessary simplifications, the thermal three‐dimensional fundamental solution for an unsaturated deformable porous medium with linear elastic behaviour in Laplace transform domain is presented. Subsequently, the closed‐form time domain fundamental solutions are derived by analytical inversion of the Laplace transform domain solutions. Then a set of numerical results are presented, which demonstrate the accuracies and some salient features of the derived analytical transient fundamental solutions. Finally, the closed‐form time domain fundamental solution will be verified mathematically by comparison with the previously introduced corresponding fundamental solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Based on the consolidation theory raised by Fredlund, the solutions for the equal-strain consolidation of unsaturated foundation with the prefabricated vertical drain considering smear effect and drain resistance are analytically formulated in this paper. Firstly, governing equations for excess pore pressures (i.e., excess pore-air and pore-water pressures) under the equal-strain hypothesis are derived with the introduction of radial boundary conditions. Afterwards, the obtained coupled equations are solved by applying general integration, decoupling process, and Fourier sine series expansion. The smear coefficients and factors of drain resistance corresponding to air and water phases are both captured explicitly in the final solutions. Furthermore, the degenerated solutions are employed to verify the reliability of the current solutions. Finally, a parametric study is conducted to study the consolidation characteristics of the proposed foundation model against modeling sizes (S and N), smear coefficients (α_a and α_w), and drain resistance factors (G_a and G_w).     

均质非饱和土一维稳态流水头垂直分布的解析解

渗流场水头分布计算是进行渗流量和渗流水力坡降计算的基础,准确、有效地求取渗流场水头分布是渗流计算的关键环节。对均质非饱和土体一维稳态流的流动方程进行分析,考虑到渗透系数是与基质吸力相关的函数,通过数学变换,给出了稳定渗流场的解析通式,并基于渗透性函数中的Gardner模型,给出了非饱和土一维稳态流水头垂直分布的解析解。该解析通式表明,均质非饱和土一维稳态流水头垂直分布主要受地表水头、深度和流动率3个因素控制。分别计算了一维稳态蒸发条件下粉土和黏土两种典型土类水头沿垂直方向的分布。计算结果表明：稳态蒸发条件下粉土层和黏土层内的水头分布表现出相似的变化规律,即自地表至地下水位处随着土层深度的增加,水头分布呈现出加速递减的趋势;在相同的蒸发条件下,对于相同深度处的黏土和粉土而言,黏土层内水头更高些;对同一种土类而言,在较大的蒸发状态下同一深度处土层内水头更高。反之,则较低。     

水蒸气增湿非饱和黄土热湿迁移规律研究

温度梯度、含水率梯度和压力梯度对非饱和土体中的水热迁移有重要影响。用非饱和黄土填筑室内模型,通入0.10MPa高温水蒸气,研究蒸汽压梯度、温度梯度和含水率梯度耦合作用下的水热迁移规律。结果表明:在水蒸气迁移范围之内,土体的升温速率、增湿速率和增湿程度较大,以蒸汽传热和压力梯度驱动的水蒸气增湿为主。而在水蒸气未到位置,以温度梯度引起的热传导和含水率梯度与温度梯度耦合驱动的水热迁移为主;水蒸气迁移时,受土颗粒阻碍和蒸汽压消散的影响,土体升温速率、增湿速率和温度传导速率均随径向距离的增大而减小,且增湿速率小于升温速率;水蒸气增湿土体的最大含水率接近最优含水率,增湿效果较好,可有效提高一定范围内土体的压实性能;基于模型试验边界条件,确定含湿毛细多孔介质中二维热湿迁移方程的一组代数显式特解,并以20 cm测点为例,将温度和含水率实测值和计算值进行对比分析。研究结果可为非饱和黄土水-汽-热耦合传输规律和水蒸气增湿新技术研究提供理论支持。