污染物在非饱和带内运移的流固耦合数学模型及其渐近解

污染物在非饱和带中运移过程是多组分多相渗流问题.在考虑气相的存在对水相影响的前提下,基于流固耦合力学理论,建立了污染物在非饱和带内运移的流固耦合数学模型.对该强非线性数学模型采用摄动法及积分变换法进行拟解析求解,得出了解析表达式.对非饱和带内的孔隙压力分布、孔隙水流速以及污染物的浓度在耦合与非耦合气相条件下的分布规律进行解析计算.对该渐近解与Faust模型的计算结果进行了对比分析,结果表明:该模型解与Faust解基本吻合,且气相作用以及介质的变形对溶质的输运过程产生较大的影响,从而验证了解析表达式的正确性和实用性.这为定量化预报预测污染物在非饱和带中迁移转化和实验室确定压力-饱和度-渗透率三者之间的关系提供了可靠的理论依据.     

Saturated-unsaturated flow in a fractured unconfined aquifer

Saturated-unsaturated flow is investigated in an unconfined aquifer containing a distribution of fractures such that on some macroscopic scale there can be defined a smooth averaged hydraulic conductivity. After constructing the conditions which relate the hydraulic properties between the unsaturated and saturated zones, it is shown that the flow in the unsaturated zone is essentially vertical. Below the water table the macropores are conduits for groundwater and the flow has a large horizontal component. Some important implications on previous models is discussed.     

A two-scale reaction-diffusion system with time-dependent microstructure

We present a two-scale model for reactive transport in a unsaturated porous medium. Reaction and diffusion of a gas dissolved in the pore water is modelled at the pore-scale, while the fast transport of the gas in the air phase is described by an averaged equation. The pore geometry is allowed to vary in time to account for temporal changes of the domain occupied by the pore water. The resulting semilinear parabolic system of PDEs is coupled in a non-standard way. After a transformation to a fixed domain, the existence of a unique weak W^1,p -solution is established by semigroup methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady Groundwater Flow with Steep Gradients: An Analytic Approach to the 2D–Problem with a Vertical Pressure Boundary Condition and Steady Recharge/Drainage

Steady groundwater flow with steep gradients in a vertical plane due to superficial recharge/drainage, inner sources/sinks and a one‐sided pressure boundary condition can be described by a 2D Poisson equation with a nonlinear free surface boundary condition. By means of conformal mapping techniques Schmitz and Edenhofer [1] derived the exact explicit solution of this problem in a horizontally infinite aquifer. Their results are extended to problems with a one‐sided vertical pressure boundary condition, modelling f. ex. the boundary between an aquifer and an adjacent free water body. According to ist simple parametrization, this approach can be applied on one hand to model various real world phenomena like river–aquifer–systems. It may on the other hand serve as a tool for investigating the exactness of numerical solutions and the range of validity of simplifying assumptions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

非饱和膨胀土的结构损伤模型及其在土坡多场耦合分析中的应用

针对原状膨胀土对气候变化反映敏感的特点和其具有胀缩性、裂隙性、超固结性等力学特性,以非饱和土力学和损伤力学为基础,建立了一个非饱和膨胀土的弹塑性损伤本构模型及相应的固结模型．把原状膨胀土看成是由未损部分和损伤部分的复合体,未损部分用非饱和土的非线性本构关系描述,损伤部分用损伤演化方程和两个屈服面（即,加载屈服面和剪切屈服面）描述．其中的损伤演化方程包括加载引起的损伤和干湿循环引起的损伤两个方面,用作者研制的CT-三轴试验确定．设计了相应的有限元程序UESEPDC;对非饱和膨胀土边坡进行了三相多场耦合问题的数值分析．分析分为4个阶段,得到了边坡在每一分析阶段的应力场、位移场、孔隙水压力场、孔隙气压力场、含水量场、基质吸力场、结构损伤演化场和塑性区扩展的动态图,较好地揭示了膨胀土边坡在开挖和气候变化条件下逐渐发生失稳滑动的现象及其机理．     

定荷载和双面排水条件下非饱和土一维固结的解析解

基于多孔介质弹性理论,结合粒间吸应力表示的有效应力原理,建立了非饱和土固结的耦合偏微分控制方程.考虑一维问题,采用Laplace积分变换,得到了定荷载和双面排水条件下非饱和土固结的解析解答.通过数值算例,分析了土体饱和度对超孔隙水压力、有效应力以及土层沉降的影响规律.结果表明,土体的初始饱和度越高,则孔隙水压力消散得越快,有效应力增加越快.     

Mathematical modelling and study of the consolidation of an elastic saturated soil with an incompressible fluid by FEM

The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the stability of the method as well as the estimation of the error by using semi-discretization in time. Finally, we then solved this one by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data.     

Classical solutions of many-dimensional elliptic–parabolic free boundary problems

We consider a free boundary problem modeling fluid flow in a partially saturated porous media. In that context an unknown function represents the pressure and satisfies an elliptic equation in the saturated domain and a quasilinear parabolic equation in the unsaturated domain. The principal part of this work is the investigation of the smoothness properties of an unknown (free) boundary between domains of ellipticity and parabolicity.      

非饱和土固结的混合物理论（Ⅰ）

非饱和土是由土粒、水、气组成的三相介质，本以混合物理论为基础研究了非饱和土的固结问题。中导出了各向异性多孔介质及非饱和土 的有效应力的理论公式，把有效应力原理和Curie对称原理作为非饱和土的两个重要的本构原理，建立了非饱和 土固结的数学模型：由25个方程求解25个未知量。在增量线性化的情况下，本模型简化为5个控制方程求解5个未知量：3个固相位移、孔隙水压力和孔隙气压力。模型中包含7个材料参数，都可由试验测定，便于工程应用，Biot理论是本模型的特例。     

On the infiltration of rain water through the soil with runoff of the excess water

This paper deals with the modelling of the rain water infiltration through the soil above the aquifer in case of runoff of the excess water. The main feature of the model lies on the correct definition of the boundary condition on the ground surface. The latter allows to estimate, after saturation, the real amount of the water that penetrates the soil and the one which runs off. The quantity playing a key role is the so-called rain pressure, defined as the pressure exerted by the rain on the soil. Although its importance is basically theoretical and it can be neglected for practical purposes, it helps understanding the real evolution of the physical problem, providing a theoretical justification of the empirical procedures.     

Existence for a time-dependent rainfall infiltration model with a blowing up diffusivity

A difficulty in the modelling of water infiltration into an unsaturated soil is due to the presence of a diffusion coefficient that blows up at the moisture saturation value. This is put in evidence in some well-known hydraulic models like those of Broadbridge and White and van Genuchten. In this paper, we obtain results concerning the existence, uniqueness and regularity properties of the solution of unsaturated water flow determined by a time-dependent rainfall, with a nonlinear flux boundary condition on the outflow boundary and a singular diffusion coefficient. Some considerations related to the possibility of saturation occurrence and the extension of the results to the model describing the infiltration into an nonhomogeneous stratified soil are finally made.     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

Strong solutions to the Richards equation in the unsaturated zone

We study the unsaturated case of the Richards equation in three space dimensions with Dirichlet boundary data. We first establish an a priori L^∞-estimate. With its help, by means of a fixed point argument we prove global in time existence of a unique weak solution in Sobolev spaces. Finally, we are able to improve the regularity of this weak solution in order to gain a strong one.     

Modeling,shape analysis and computation of the equilibrium pore shape near a PEM–PEM intersection

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: (i) the existence and uniqueness of the electrical potential, (ii) the shape derivatives of state variables and (iii) the shape differentiability of the corresponding energy and the corresponding Euler–Lagrange equation. The latter leads to a modified Young–Laplace equation on the interface. This modified equation is compared with the classical Young–Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.     

Mathematical Modeling to Simulate the Movement of Contaminants in Groundwater

The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the unsaturated zone is obtained. Indeed, the solution can be used as a tool to verify the accuracy of numerical models of water flow and solute transport. The second part of this paper, deals with how the water level in a water reserve drops due to pumping water out of a well that is some distance away.     

A One‐dimensional flow problem in porous media with hydrophile grains

We study a free boundary problem describing the propagation of the wetting front following the injection of a liquid into a porous medium with hydrophile granules. The absorption process produces a non‐local interaction with the flow so that the porosity appearing in the parabolic equation for pressure is a functional of saturation and of the free boundary. Our analysis is confined to the unsaturated regime, which is the first stage of the process. An existence theorem is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Revisiting total, matric, and osmotic suction in partially saturated geomaterials

An accurate quantification of negative pore pressure (commonly referred to as ??suction??) in the pore network is necessary for modeling the mechanical response of unsaturated geomaterials. Traditional definitions and formulations of total, matric, and osmotic suction suggest incorrect pore fluid pressures under certain conditions. In this paper, the notion of suction is revisited by deriving an expression for pore fluid pressure in a simple osmotic, capillary tube using the framework of mixture theory in conjunction with the fundamental laws of thermodynamics. Based on the derived expression for the tube, expressions are derived for total, matric, and osmotic suction for partially saturated geomaterials. Particular attention is given to osmotic suction since confusion regarding its mechanisms has apparently contributed to its misapplication in geomechanics. The new expressions derived herein adequately explain behavior that is incorrectly explained by the traditional formulations and unifies two approaches to modeling osmotic suction previously considered to be in contradiction.     

On the Description of Partially Saturated Soils by the Theory of Porous Media

In this contribution, a multi‐phase soil model based on the Theory of Porous Media (TPM) is presented. The model is fully coupled in the following constitutive phases: An elasto‐plastic or elasto‐viscoplastic solid skeleton, a materially incompressible pore‐liquid (water) and a materially compressible pore‐gas (air). The interaction of the solid skeleton and the pore‐fluids is specified by a capillary pressure‐saturation relation, whereas the mobilities of the fluid phases in the pore‐space of the solid skeleton are described by the so‐called relative permeabilities. Finally, a gravity governed initial‐boundary‐value problem solved by the FE method is presented.     

The effect of seepage on the stress–strain state of rock near a borehole

The effect of the strain-strength and seepage properties of rock and the compressibility of the percolating fluid on the dimensions of the rock fracture zones, which occur in oil and gas boreholes when the bottom hole pressure is reduced, is investigated. The seepage is considered basing on the stationary formulation of the problem, which enables the general case to be investigated. It is shown that in the case of unsteady flow, the stresses on the boundary of the rock fracture zone and, as a consequence, on its dimensions, are independent of the nature of the pressure distribution in the stratum, and are determined solely by the pressure of the percolating fluid on the boundary of this zone. It is established that an increase in the compressibility of the percolating fluid leads to an increase in the dimensions of the rock fracture zone.     

饱和粘土非线性渗流规律与径向固结

以多孔介质等效渗流概念得出粘土平均孔隙微尺度数量级范围在0.01 μm至0.1μm,与测试结果一致．实验结果表明饱和粘土微尺度孔隙渗流为非线性流．理论推导表明固液界面作用与渗透率平方根或孔隙半径成反比,固液界面相互作用是导致饱和粘土非线性渗流的重要原因．提出了精确描述饱和粘土微尺度孔隙非线性渗流基本规律的数学模型,其参数量纲明确,物理意义清楚．建立了饱和粘土非线性渗流新定律,统一描述了从较低到较高水力梯度全过程渗流曲线特征,达西定律是其特例．基于新定律,建立了饱和粘土非线性渗流定流量径向固结数学模型．以粘性边界层思想与稳态依序替换法及积分方程法,导出了粘土非线性渗流平均质量守恒方程及活动边界运动方程,给出了饱和粘土非线性渗流超孔隙水压力分布公式与平均固结度计算公式,获得了粘土层压力分布规律和平均固结度随时间变化规律．结果表明:饱和粘土非线性渗流使活动边界运动速度减小．研究结果为粘土地质工程与岩土工程应用提供了新的科学依据．达西渗流径向固结计算是新的非线性渗流固结计算的特例．