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.     

基于最小作用原理的Richards方程变分解

经典的Richards入渗控制方程属于偏微分方程,具有强烈的非线性,难以求得解析解。以入渗时间为最小作用量,基于Richards方程建立关于入渗路径的时间泛函,将考虑重力项的非饱和土垂直入渗问题转化为泛函极值问题,并构造等价的Euler-Lagrange方程进行求解。计算结果表明,扩散系数D（?）与概化湿润锋距离具有函数关系,当扩散系数D（?）形式已知时,可求得最优路径下湿润锋处含水率、较远处湿润锋最小含水率、土壤含水率最大熵分布3个问题,并基于最优路径检验了本研究条件下,Boltzmann变换和线性变换求解Richards方程的精度。求解过程未引进新变量化简Richards方程,不改变原方程结构,因此其解具有普遍性,可作为非饱和土力学计算的一个补充。     

Chloride transfer in cement‐based materials. Part 1. Theoretical basis and modelling

In this first part of the work, we develop macroscopic models for migration and diffusion–migration of ionic species in saturated porous media, based on periodic homogenization. The prior application is chloride transport in cementitious materials. The dimensional analysis of Nernst–Planck equation lets appear to dimensionless numbers characterizing the ionic transfer in the porous medium. Using experimental data obtained from electrodiffusion tests on cement‐based materials (Part II), these dimensionless numbers are linked to the perturbation parameter ?. For a strong imposed electrical field, the asymptotic expansion of Nernst–Planck equation leads to a macroscopic model where the migration is predominant. For a weak imposed electrical field or in natural diffusion, we obtain a macroscopic model coupling diffusion and migration at the same order. In both models, the expression of the homogenized diffusion tensor is identical and only involves the geometric properties of the material microstructure. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of actual evapotranspiration by numerical modeling of water flow in the unsaturated zone: a case study in Buenos Aires, Argentina

A method is presented to estimate actual evapotranspiration (ET_A) from potential evapotranspiration (ET_P) by numerical modeling of water flow in the unsaturated zone. Water flow is described by the Richards equation with a sink term representing the root water uptake. Evaporation is included in the model as a Neumann boundary condition at the soil surface. The Richards equation is solved in a one-dimensional domain using a mixed finite element method. The values of ET_A are obtained by applying a water stress factor to ET_P to account for soil moisture changes during the simulation period. The proposed numerical model is used to estimate ET_A in an experimental plot located in a flatland area in Buenos Aires (Argentina). Numerical results show that the proposed model is a useful tool for evaluating evapotranspiration under different scenarios.     

蒸发条件下土壤水分响应模拟

基于有限元(FEM)和改进的积分型Richards方程解法(IRE方法)对蒸发条件下5种土体土壤水分响应进行了研究.数值实验结果表明:在土壤表面潜在蒸发量0.50 cm/d的情况下,5种土体土壤含水率变化曲线均呈现单拐点两阶段的特点,拐点出现在地表下20cm左右,拐点上部区域曲线曲率大于下部区域,两阶段的划分以15d左右为界,前阶段比后阶段的土壤水分变化快;蒸发模拟结果很好的证明了蒸发三阶段理论.总蒸发量和下边界排水量与土壤结构密切相关,而总水量变化量和变化率与土壤质地有关.IRE方法与FEM模拟结果基本一致,解法相对简单,模拟结果可靠性高.     

Holocene sediment properties of the East Greenland and Iceland continental shelves bordering Denmark Strait (64–68°N), North Atlantic

The oceanographic Polar Front separates the East Greenland and Iceland margins. Surface water temperatures across Denmark Strait vary by 8–12 °C and represent one of the steepest oceanographic gradients on earth. The East Greenland margin is a polar environment, with extensive sea‐ice cover and calving glacier margins; in contrast, the Iceland shelf is much more temperate, and freshwater run‐off is a key component in land–ocean sediment transfers. Average sediment properties from these two contrasting climate and oceanographic continental shelf environments are compared in the spatial domain at 13 sites; the data represent the last 10 000 radiocarbon years of `normal' marine sedimentation for the two regions. The two regions have similar average rates of sediment accumulation (around 43·5 cm kyr<sup>?1</sup>), so that this key variable is factored out in explaining any differences in sediment properties. Dry sediment density, moisture content, hygroscopic moisture, total organic carbon and carbonate contents, mass magnetic susceptibility and the percentages of sand and silt are compared focusing on: (1) median values for sediment properties; and (2) downcore variability, measured by the coefficient of variation (CV). There are significant differences in all but one (hygroscopic moisture) of the sediment properties between Iceland and East Greenland; in four cases, the sense of the differences was not as predicted. In terms of downcore variation (CV), no difference was found between the two regions, nor between the 13 sites, whereas there are some significant differences between the variables. Carbonate and mass magnetic susceptibility have the largest spreads, and moisture content and dry sediment density are the least variable. Protocols are developed to identify the `type core' in a regional series of sites. The results indicate a need to develop a regional perspective on sediment properties, both as inputs to models of sedimentary processes in different polar/arctic environments, and as an indication of which sediment properties might be best suited for palaeoenvironmental downcore time series.     

Macroscopic diffusivity in concrete determined by computational homogenization

Effective moisture and chloride ion diffusivity coefficients for concrete are determined by computational homogenization, where concrete is modeled on the mesoscale as a heterogenous three‐phase composite material. By imposing moisture and chloride ion gradients on a representative volume element, effective macroscale properties are obtained through finite element analysis. A parametric study of the effects of the ballast content was carried out. The numerical results correspond well with an estimate of the Hashin–Shtrikman type available in the literature. The computational homogenization strategy proposed here also includes the interfacial transition zone, and its influence on the effective diffusivity coefficients is assessed. Copyright © 2012 John Wiley & Sons, Ltd.     

土壤湿度观测、模拟和估算研究

总结土壤湿度的观测手段和土壤湿度数据集建立的现状,详细阐述与土壤湿度模拟有关的方程离散化求解、物理和生化过程、陆面过程模式比较和陆面模式参数优化等方面的研究进展;综述估算土壤湿度廓线的数据同化方法,仔细比较集合卡曼滤波(EnKF)和四维变分(4-D Var)2类目前流行的同化算法,并对估算土壤湿度廓线的研究工作进行全面评估;最后,对土壤湿度观测、模拟和同化中需继续努力的方向进行了思索和展望。     

非饱和土壤水分函数解析与Richards方程入渗新解

从土壤微观统计学角度出发，用各向均一无结构土壤“切片”孔径级配函数构建的统计毛管束模型和土壤水分分布依赖的力学原理，论证了理查兹方程中导水率、基质势、扩散率等水分函数和入渗率随充水度，土壤基质而变的机理、规律以及两者之间的内在联系，给出了相应的计算公式和可操作的参数确定方法。把迄今水分函数研究的纯经验相关现状提高到理论分析水平，使建立理论公式成为现实。     

A study on iterative methods for solving Richards’ equation

This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards’ equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards’ equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L-scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L-scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L-scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

Effect of soil disturbance on recharging fluxes: case study on the Snake River Plain,Idaho National Laboratory,USA

Soil structural disturbance influences the downward flow of water that percolates deep enough to become aquifer recharge. Data from identical experiments in an undisturbed silt-loam soil and in an adjacent simulated waste trench composed of the same soil material, but disturbed, included (1) laboratory- and field-measured unsaturated hydraulic properties and (2) field-measured transient water content profiles through 24 h of ponded infiltration and 75 d of redistribution. In undisturbed soil, wetting fronts were highly diffuse above 2 m depth, and did not go much deeper than 2 m. Darcian analysis suggests an average recharge rate less than 2 mm/year. In disturbed soil, wetting fronts were sharp and initial infiltration slower; water moved slowly below 2 m without obvious impediment. Richards’ equation simulations with realistic conditions predicted sharp wetting fronts, as observed for disturbed soil. Such simulations were adequate for undisturbed soil only if started from a post-initial moisture distribution that included about 3 h of infiltration. These late-started simulations remained good, however, through the 76 d of data. Overall results suggest the net effect of soil disturbance, although it reduces preferential flow, may be to increase recharge by disrupting layer contrasts.     

非饱和土路基毛细作用的数值与解析方法研究

通过建立非饱和土毛细作用的孔隙分布的分形模型,推导获得了非饱和土毛细水的最大上升高度,同时基于Richards水分运动微分方程,引入边界及初始条件,基于Laplace变换,得到了毛细作用下非饱和土路基湿度变化的解析解;同时引入算例,将所提出的解析方法计算结果与未简化参数的数值计算结果进行了对比分析;最后考虑不同因素的影响进行了非饱和土路基毛细作用下的湿度变化分析。分析结果表明：解析求解获得的路基湿度变化趋势和未进行参数简化的数值法求解结果基本一致,证明解析解法是合理可信的;路基填筑的初始含水率越大,填土的初始吸力越小,毛细水上升的高度及湿度变化量也相应越小;透水性能较好的路基填土毛细水上升速度较快,但上升高度较小,毛细水可以在较短时间内上升到最大高度;路基的填土类型不同,路基在毛细作用下的湿度变化状态也不同,需要针对不同的填土路基进行相应的防排水措施。     

Green-Ampt模型渗透系数取值方法研究

格林-安姆普特（Green-Ampt）模型原理简单、使用方便,在浅层滑坡的降雨入渗分析中有很大的应用潜力。渗透系数是Green-Ampt模型中的一个主要参数,该渗透系数并不一定等于土体的饱和渗透系数。通过与理查茲（Richards）方程求解进行比较,研究了Green-Ampt模型中渗透系数的取值方法。研究发现,为获得与Richards方程相同的入渗量计算结果,需对饱和渗透系数进行修正,该修正系数的大小与入渗深度和土体类型有关。对于文中研究的土体,当Green-Ampt模型中渗透系数取为0.7倍饱和渗透系数时,由Green-Ampt模型计算的孔隙水压力分布与Richards方程计算结果最为接近,建议Green-Ampt模型中渗透系数修正系数取0.7。     

P-S波反射系数的一种小角度近似

从Aki和Richards在弱反差条件下给出的P-S波反射系数近似公式出发,提出了一种在小角度入射范围内的P-S波反射系数近似公式.为了得到该近似公式,对原公式中的角度项进行了二项式展开,直接利用入射角代替入射角的正弦.不同模型的计算表明,该近似在小角度入射范围内相当于Aki和Richards的原近似公式,在大角度时有一定改善.新的近似公式形式简单、涉及的弹性参数较少,容易应用到反演当中.另外,利用一阶小角度近似结果可以对横波阻抗进行反演.     

Steady infiltration from buried point source into heterogeneous cross‐anisotropic unsaturated soil

The paper presents the analytical solution for the steady‐state infiltration from a buried point source into two types of heterogeneous cross‐anisotropic unsaturated half‐spaces. In the first case, the heterogeneity of the soil is modelled by an exponential relationship between the hydraulic conductivity and the soil depth. In the second case, the heterogeneous soil is represented by a multilayered half‐space where each layer is homogeneous. The hydraulic conductivity varies exponentially with moisture potential and this leads to the linearization of the Richards equation governing unsaturated flow. The analytical solution is obtained by using the Hankel integral transform. For the multilayered case, the combination of a special forward and backward transfer matrix techniques makes the numerical evaluation of the solution very accurate and efficient. The correctness of both formulations is validated by comparison with alternative solutions for two different cases. The results from typical cases are presented to illustrate the influence on the flow field of the cross‐anisotropic hydraulic conductivity, the soil heterogeneity and the depth of the source. Copyright © 2004 John Wiley & Sons, Ltd.     

A wind tunnel study of aeolian sand transport on a wetted sand surface using sands from tropical humid coastal southern China

This article reported a wind tunnel test of sediment transport related to surface moisture content and wind velocity using sands from tropical humid coastal area. A 1 mm-thick portion of surface sand was scraped using a self-made sediment sampler, and the gravimetric moisture content was determined. Sand transport was measured via a standard vertical sand trap with a 60 cm height. The result shows that the sand transport profile above the wet surface can be expressed with an exponential equation. In general, the influence of moisture content on sand transport profile mainly focuses on the bottom of the blowing sand cloud. Meanwhile, with moisture content increased, total sand transport dropped, and a relatively larger proportion is transported at greater heights. The vertical movement of particles on higher moisture surface (0.587% < M < 1.448%) is more sensitive to moisture content variation as compared to those on low wet surface (M < 0.587%), total sand transport rate tends to be rather low (0.99 g cm⁻¹ s⁻¹) when M > 1.448%. The total sand transport rate varying with moisture content is divided into three regions of differing gradient at the moisture contents of 0.587 and 1.448%. The gradient of the curve reflected the different influences of the various water forms in surface sediments. The higher moisture surface (M > 1.448%) merely functions as a transport plain for the saltation material. Surface moisture content was the dominant control factor for saltation activity between the moisture contents of 0.587 and 1.448%, wind velocity could resume control saltation after the surface dried to the extent (M < 0.587%).     

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.     

One‐dimensional contaminant transport through a deforming porous medium: theory and a solution for a quasi‐steady‐state problem

Contaminant migration through soil is usually modelled mathematically using the dispersion–advection equation. This type of model finds application when planning the remediation of contaminated land, predicting the movement of polluted groundwater and designing engineered landfills. Usually the analysis assumes that the porous media through which the contaminant migrates is stationary. However, the construction of landfills on clay soils means that the soil beneath the landfill will undergo time‐dependent deformation as the soil consolidates. To date, there are no published data on the effect a deforming porous media may have on contaminant transport beneath a landfill; indeed, there appears to be no theory of contaminant migration through a deforming soil. In this paper, a one‐dimensional theory of contaminant migration through a saturated deforming porous media is developed based on a small and large strain analysis of a consolidating soil and conservation of contaminant mass. By selection of suitable parameters, the new transport equation reduces to the familiar one‐dimensional dispersion–advection equation for a saturated soil with linear, reversible, equilibrium controlled sorption of the contaminant onto the soil skeleton. Analytic solutions to a quasi‐steady‐state contaminant transport problem for a deforming media are presented, and a preliminary assessment made of the potential importance of soil deformation on the results of a contaminant migration analysis. Copyright © 2000 John Wiley & Sons, Ltd.