土壤水流与溶质耦合运移问题的混合元-迎风广义差分法研究

本文研究了一维非饱和土壤水流与溶质耦合运移问题的数学模型, 建立了求其数值解的守恒混合元-迎风广义差分格式. 对非线性土壤水分入渗方程, 采用守恒混合元法进行离散模拟, 同时得到了土壤含水量和水分通量; 而对对流-扩散形式的溶质运移方程, 利用迎风的广义差分法离散求解. 且分析了解的存在唯一性, 并讨论了误差估计. 最后给出数值算例, 模拟结果表明利用本文格式来求解非饱和土壤水流与溶质耦合运移问题是可靠的, 且该格式具有稳定性和可实用性.     

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.     

Uncertainty analysis of transport of water and pesticide in an unsaturated layered soil profile using fuzzy set theory

Incomplete information is notoriously common in planning soil and groundwater remediation. For making decisions groundwater flow and transport models are commonly used. However, uncertainty in prediction arises due to imprecise information on flow and transport parameters like saturated/unsaturated hydraulic conductivity, water retention curve parameters, precipitation and evapo-transpiration rates as well as factors governing the fate of pollutant in soil like dispersion, diffusion, degradation and chemical transformation. Different methods exist for quantifying uncertainty, e.g. first and second order Taylor’s Series and Monte-Carlo method. In this paper, a methodology based on fuzzy set theory is presented to express imprecision of input data, in terms of fuzzy number, to quantify the uncertainty in prediction. The application of the fuzzy set theory is demonstrated through pesticide (endosulfan) transport in an unsaturated layered soil profile. The governing partial differential equation along with fuzzy inputs, results in a non-linear optimization problem. The solution gives complete membership functions for flow (suction head) and pesticide concentration in soil column.     

Excess pore water pressure due to ground surface erosion

The Laplace transform is applied to solve the groundwater flow equation with a boundary that is initially fixed but that starts to move at a constant rate after some fixed time. This problem arises in the study of pore water pressures due to erosional unloading where the aquifer lies underneath an unsaturated zone. We derive an analytic solution and examine the predicted pressure profiles and boundary fluxes. We calculate the negative pore water pressure in the aquifer induced by the initial erosion of the unsaturated zone and subsequent erosion of the aquifer.     

Error estimates for an Euler implicit-mixed finite element scheme for reactive transport in saturated/unsaturated soil

We present convergence results for a fully discrete scheme based on the mixed finite element (MFE) method and an one-step Euler implicit (EI) method for simulating reactive solute transport in saturated/unsaturated soil. The results considered the low regularity of the solution of the degenerate parabolic equation describing the water flow in porous media. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Gas absorption with first order chemical reaction in a laminar falling film over a reacting solid wall

In the present work, a general case of gas absorption with first order irreversible chemical reaction in a liquid film, for laminar flow over a solid wall, has been analyzed theoretically. First order chemical reaction between the diffused solute and the wall is also considered. Laplace transform followed by power series method has been applied to solve the governing equations. Thereafter, the obtained analytical solution of the developed general model has been successfully verified by an explicit numerical scheme. The general model has also been reduced to six simplified cases, tackled by previous workers and an excellent agreement in the solutions is observed. Moreover, the results are validated by the experimental data available in the literature. The obtained concentration profiles in both the phases have been used to find the absorption rates and enhancement factor.     

Thermo-elastodynamic response of a spherical cavity in saturated poroelastic medium

In this paper, an attempt has been made to investigate the thermo-hydro-elastodynamic response of a spherical cavity in isotropic saturated poroelastic medium when subjected to a time dependent thermal/mechanical source. The fully coupling thermo-hydro-elastodynamic model is presented on the basis of equations of motion, fluid flow, feat flow and constitutive equation with effective stress and temperature change. Solutions of displacement, temperature and stresses are obtained by using a semi-analytical approach in the domain of Laplace transform. Numerical results are also performed for portraying the nature of variations of the field variables, i.e. the coefficient of thermo-osmosis, the permeability. In addition, comparisons are presented with the corresponding partially thermo-hydro-elastodynamic model and thermo-elastodynamic model to ascertain the validity and the difference between these models.     

双重介质中溶质径向运移微分方程组的精确解

本文对溶质径向运移问题综合了数学模型,考虑了非均衡线性吸附作用和介质的双重性质以及溶质的衰变.在第一类边值条件下,用Laplace变换求得了严格的解析解.用FORTRAN程序在DJS-040机上对无量纲化的问题解进行了计算.求出了浓度的分布和变化,讨论了有实际意义的各种极限情况并给出了相应的解,通过数值分析,得出了几点有价值的结论.     

堆浸工艺水动力学研究

利用无量纲的形式推导出堆浸工艺中通过非饱和堆我动区和滞留区的溶质运移模型,通过最小二乘法确定模型参数,并就两种不同供液情形下的解析解进行了讨论.该模型可用来研究堆浸中溶质浓度的变化规律.     

A two-dimensional finite volume method for transient simulation of time- and scale-dependent transport in heterogeneous aquifer systems

In this paper, solute transport in heterogeneous aquifers using a modified Fokker-Plauck equation (MFPE) is investigated. This newly developed mathematical model is characterised with a time-, scale-dependent dispersivity. A two-dimensional finite volume quadrilateral mesh method (FVQMM) based on a quadrilateral background interpolation mesh is developed for analysing the model. The FVQMM transforms the coupled non-linear partial differential equations into a system of differential equations, which is solved using backward differentiation formulae of order one through five in order to advance the solution in time. Three examples are presented to demonstrate the model verification and utility. Henry's classic benchmark problem is used to show that the MFPE captures significant features of transport phenomena in heterogeneous porous media including enhanced transport of salt in the upper layer due to its parameters that represent the dependence of transport processes on scale and time. The time and scale effects are investigated. Numerical results are compared with published results on the same problems.     

Transient elastodynamic three-dimensional problems in cracked bodies

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

Generalized alternating-direction collocation methods for parabolic equations. II. Transport equations with application to seawater intrusion problems

The alternating-direction collocation (ADC) method combines the attractive computational features of a collocation spatial approximation and an alternating-direction time marching algorithm. The result is a very efficient solution procedure for parabolic partial differential equations. To date, the methodology has been formulated and demonstrated for second-order parabolic equations with insignificant first-order derivatives. However, when solving transport equations, significant first-order advection components are likely to be present. Therefore, in this paper, the ADC method is formulated and analyzed for the transport equation. The presence of first-order spatial derivatives leads to restrictions that are not present when only second-order derivatives appear in the governing equation. However, the method still appears to be applicable to a wide variety of transport systems. A formulation of the ADC algorithm for the nonlinear system of equations that describes density-dependent fluid flow and solute transport in porous media demonstrates this point. An example of seawater intrusion into coastal aquifers is solved to illustrate the applicability of the method. An alternating-direction collocation solution algorithm has been developed for the general transport equation. The procedure is analogous to that for the model parabolic equations considered by Celia and Pinder [2]. However, the presence of first-order spatial derivatives requires special attention in the ADC formulation and application. With proper implementation, the ADC procedure effectively combines the efficient equation formulation inherent in the collocation method with the efficient equation solving characteristics of alternating-direction time marching algorithms. To demonstrate the viability of the method for problems with complex velocity fields, the procedure was applied to the problem of density-dependent flow and contaminant transport in groundwaters. A standard example of seawater intrusion into coastal aquifers was solved to illustrate the applicability of the method and to demonstrate its potential use in practical problems.     

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.     

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 path probability approach to the solute transport problem

An alternative solute transport algorithm, termed the path probability approach, is presented. The method may be more effective than typical finite methods in some numerical solution problems. The method differs from predominant numerical techniques in that the conventional conservation differential is replaced by a series of equations defining solute migration in terms of a finite number of representative particle position histories and corresponding probabilities. The governing physical assumptions applied in model development are conditionally consistent with those of conventional Lagrangian and Eulerian conceptualizations, but mathematical expression and development of the problem differ from traditional approaches. The method represents dispersion orthogonal to the primary axis of advective transport explicitly, according to arbitrary user-defined probability functions; longitudinal dispersion effects result from the existence of velocity shear effects along the primary axis of movement. It is possible to incorporate Fickian dispersive processes, and thereby reproduce results obtainable with traditional stream tube models; but non-Fickian alternatives can also be explored. Simulation results of the path probability model are compared to analogous results from a representative finite difference model for a hypothetical test flow channel. The comparison demonstrated the ability of the new model to effectively generate simulation results with a computational effeciency considerably higher than conventional techniques under the conditions tested.     

Modeling non-Darcian flow and solute transport in porous media with the Caputo–Fabrizio derivative

In this study, the non-Darcian flow and solute transport in porous media are modeled with a revised Caputo derivative called the Caputo–Fabrizio fractional derivative. The fractional Swartzendruber model is proposed for the non-Darcian flow in porous media. Furthermore, the normal diffusion equation is converted into a fractional diffusion equation in order to describe the diffusive transport in porous media. The proposed Caputo–Fabrizio fractional derivative models are addressed analytically by applying the Laplace transform method. Sensitivity analyses were performed for the proposed models according to the fractional derivative order. The fractional Swartzendruber model was validated based on experimental data for water flows in soil–rock mixtures. In addition , the fractional diffusion model was illustrated by fitting experimental data obtained for fluid flows and chloride transport in porous media. Both of the proposed fractional derivative models were highly consistent with the experimental results.     

A Laplace transform technique for the analytical solution of a diffusion- convection equation over a finite domain

A Laplace transform technique has been utilized to obtain two different analytic solutions to a single diffusion-convection equation over a finite domain. One analytic solution is continuous at both ends of the domain of interest, while the other solution is discontinuous at the origin. This difference in the two solutions is explained. An application of the Laplace transform technique to a more complex system of equations, on a finite domain, is noted and an error apparent in a previous paper is corrected.     

Modeling dissolution in a filtration flow

The article considers mathematical models that describe in various approximations dissolution in a moving filtration flow. A more complete new model is developed and its results are compared with experimental data. The new model reflects the main features of mineral leaching by sulfuric-acid solutions. Self-similar solutions are obtained describing the motion of the solution leading front that displaces the natural water and the trailing front that tracks the progress of the active dissolution region. The effective dissolution zone width and the dissolution zone transport velocity are determined as functions of the filtration velocity, porosity, saturated solution concentration, and other parameters of the medium. The model is applied to construct the longitudinal concentration distribution of the leached substance for a one-dimensional constant-velocity flow. __________ Translated from Prikladnaya Matematika i Informatika, No. 21, pp. 30–47, 2005.