One-dimensional unsteady solute transport along unsteady flow through inhomogeneous medium

The one-dimensional linear advection–diffusion equation is solved analytically by using the Laplace integral transform. The solute transport as well as the flow field is considered to be unsteady, both of independent patterns. The solute dispersion occurs through an inhomogeneous semi-infinite medium. Hence, velocity is considered to be an increasing function of the space variable, linearly interpolated in a finite domain in which solute dispersion behaviour is studied. Dispersion is considered to be proportional to the square of the spatial linear function. Thus, the coefficients of the advection–diffusion equation are functions of both the independent variables, but the expression for each coefficient is considered in degenerate form. These coefficients are reduced into constant coefficients with the help of a new space variable, introduced in our earlier works, and new time variables. The source of the solute is considered to be a stationary uniform point source of pulse type.     

Three-dimensional temporally dependent dispersion through porous media: analytical solution

A three-dimensional model for non-reactive solute transport in physically homogeneous subsurface porous media is presented. The model involves solution of the advection-dispersion equation, which additionally considered temporally dependent dispersion. The model also account for a uniform flow field, first-order decay which is inversely proportional to the dispersion coefficient and retardation factor. Porous media with semi-infinite domain is considered. Initially, the space domain is not solute free. Analytical solutions are obtained for uniform and varying pulse-type input source conditions. The governing solute transport equation is solved analytically by employing Laplace transformation technique (LTT). The solutions are illustrated and the behavior of solute transport may be observed for different values of retardation factor, for which simpler models that account for solute adsorption through a retardation factor may yield a misleading assessment of solute transport in ‘‘hydrologically sensitive’’ subsurface environments.     

Analytical solutions for advective–dispersive solute transport in double‐layered finite porous media

Analytical solutions for advection and dispersion of a conservative solute in a one‐dimensional double‐layered finite porous media are presented. The solutions are applicable to five scenarios that have various combinations of fixed concentration, fixed flux and zero concentration gradient conditions at the inlet and outlet boundaries that provide a wide number of options. Arbitrary initial solute concentration distributions throughout the media can be considered via explicit formulations or numerical integration. The analytical solutions presented have been verified against numerical solutions from a finite‐element‐based approach and an existing closed‐form solution for double‐layered media with an excellent correlation being found in both cases. A practical application pertaining to advective transport induced by consolidation of underlying sediment layers on contaminant movement within a capped contaminated sediment system is presented. Comparison of the calculated concentrations and fluxes with alternative approaches clearly illustrates the need to consider advection processes. Consideration of the different features of contaminant transport due to varying pore‐water velocity fields in primary consolidation and secondary consolidation stages is achieved via the use of non‐uniform initial concentration distributions within the proposed analytical solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.     

Asymptotic solutions for solute transport in dual velocity media

Dual velocity (often called dual porosity) models exist to describe a variety of solute transport processes. These exist for both chemical and geochemical systems. All current models reviewed in this paper can be represented by a generalized form. Characteristics of the solutions are obtained by investigating moments of the solution. Of particular interest is the simple asymptotic behavior. To verify the approach, an example problem is investigated where the exact analytical solution is compared to the asymptotic solution. It is shown that many dual velocity models can be well-represented by the inclusion of an increased dispersion term in a simpler single velocity model.     

Pathline tracing on fully unstructured control-volume grids

The trend toward unstructured grids in subsurface flow modeling has prompted interest in the issue of streamline or pathline tracing on unstructured grids. Streamline tracing on unstructured grids is problematic because a continuous velocity field is required for the calculation, while numerical solutions to the groundwater flow equations provide velocity in discretized form only. A method for calculating flow streamlines or pathlines from a finite-volume flow solution is presented. The method uses an unconstrained least squares method on interior cells and a constrained least squares method on boundary cells to approximate cell-centered velocities, which can then be continuously interpolated to any point in the domain of interest. Two-dimensional tests demonstrate that the method correctly reproduces uniform and corner-to-corner flow on fully unstructured grids. In three dimensions using regular hexahedral grids, the method agrees well with established semianalytical methods. Tests also demonstrate that the method produces physically realistic results on fully unstructured three-dimensional grids.     

非稳态热传导时层状路面体系的温度响应

利用解析层元法推导温度荷载作用下非稳态热传导时层状路面体系的温度响应解答。从热弹性理论平面应变问题的控制方程出发,借助于Laplace-Fourier积分变换,推导出单层介质及下卧半平面的精确刚度矩阵即解析层元,结合有限层法原理及边界条件,组装并求解总刚度矩阵,得到其在变换域内的解答,最后通过相应的积分逆变换得到物理域内的真实解。由于该法刚度矩阵元素中不含正指数项,计算时不会出现溢出或病态矩阵的现象。编译了相应的计算程序,所得结果与有限元模拟结果吻合较好。在此基础上,对有限深度和半平面两种假定条件下的解答进行对比分析,并分析层状路面体系中位移和温度随时间的变化趋势及沿深度的分布规律。分析表明：温度场具有一定的影响深度,超过此深度,有限深度与半平面理论解答基本一致;温度荷载的影响深度与其强度有关,强度越大,其影响深度越深。     

Transverse Dispersion of a Kinetically Sorbing Solute

A recursion formulation for the transverse spreading of a solute is developed, and under conditions of steady flow in a stratified aquifer, the transport of a linearly sorbing solute undergoing nonequilibrium sorption is studied. The effect of spatial variability in the velocity field and the sorption kinetics are modeled to see the combined effect of the two processes on the spreading of the solute injected at a point in the aquifer. The main result of this work is a transport model based on a discrete formulation that includes local dispersion and leads to nonasymptotic behavior in the spreading of the plume in a direction normal to the mean flow velocity.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

http://dx.doi.org/10.1016/j.gsf.2015.11.001

A new method, the characteristic finite element method(CFEM), was developed to simulate solute transport in a cross-fracture. The solution of this mathematical model for solute transport considered that the contribution of convection and dispersion terms was deduced using the single-step, trace-back method and routine finite element method(FEM). Also, experimental models were designed to verify the reliability and validity of the CFEM. Results showed that experimental data from a single fracture model agreed with numerical simulations obtained from the use of the CFEM. However, routine FEM caused numerical oscillation and dispersion during the calculation of solute concentration. Furthermore, in this cross-fracture model, CFEM simulation results predicted that the arrival time of concentration peak values decreased with increasing flux. Also, the second concentration peak value was obvious with the decrease of flux, which may have resulted from the convergence of solute concentrations from main, and branch, fractures.     

Comparison between numerical and analytical solution of solute transport models

The fate and movement of dissolved substances in soils and groundwater has generated considerable concern for the quality of the subsurface environment. Many analytical solutions for the partial differential equations that describe solute and pollutant movement exist. Numerical solutions are more general, and often more difficult to verify. In order to determine the model error, the examination of the ability of numerical methods compared to analytical methods is strongly recommended. The objective of the study is to make a comparison between numerical and analytical solution models for solute transport equation. In this study, the numerical solution calculated with the WAVE-model is compared with the analytical solution calculated with CXTFTT-model. The study scenarios considered variables such as compartment depth, applied flux at the top and soil dispersivity under steady-state conditions. The simulations depend on 27 solute infiltration scenarios. The solute concentrations were calculated with the WAVE-model and the CXTFIT-model for each scenario. The WAVE-model error was evaluated with three methods: absolute average maximum error, relative average maximum error and relative average area error. The study implied that the WAVE-model error increased with the increase of the compartment depth, decreasing soil dispersivity, and decrease in flux. The study leads to the recommendation to use compartment depth as thin as possible to minimise the WAVE-model error. Furthermore, it is more useful to use several numerical solution models, such as SWMS-2D model, to evaluate and examine the WAVE-model.     

A fast flux tube-based method for solute-transport simulation

A new method to calculate the transport of dissolved species in aquifers is presented. This approach is an extension of the stream tubes which are used for flow computation. The flux tubes defined here are conservative for solutes, but not for water mass. The flux tubes are first defined in a general domain and then calculated in a two-dimensional uniform flow field. The tubes?? computation is based on a parametric solution. The method is extended further in order to deal with heterogeneous media. A particle-tracking algorithm is used where the deviation of the flux-tube boundaries due to dispersion is accounted for. The approximate solution obtained by this approach is compared to classical numerical solutions given by a finite difference code (RT3D) and a finite element code (FEFLOW). This comparison was performed for several test cases with increasing complexity. The differences between the flux-tube approach and the other methods always remain small, even regarding mass conservation. The major advantage of the flux-tube approach is the ability to reach a solution quickly, as the method is hundreds to thousands of times faster than classical finite difference or finite element models.     

Effect of irregularity on torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space

The present paper is concerned with the propagation of torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space. We assume the quadratic inhomogeneity in rigidity and density in the lower half-space and irregularity is taken in the form of rectangle at the interface separating the layer from the lower half-space. The dispersion equation for torsional waves has been obtained in a closed form. Velocity equation is also obtained in the absence of irregularity. The study reveals that the presence of irregularity, initial stress, porosity, inhomogeneity and anisotropy factor in the dispersion equation approves the significant effect of these parameters in the propagation of torsional waves in porous medium. It has also been observed that for a uniform media, the velocity equation reduces to the classical result of Love wave.     

Analytical and numerical solution of the elastodynamic strip load problem

Analytical and numerical solutions of the elastodynamic problem of an instantaneous strip load on a half space are presented and compared. The analytical solution is obtained using the De Hoop–Cagniard method, and the numerical solution is obtained using the dynamic module of the finite element package Plaxis. The purpose of the paper is to validate the numerical solution by comparison with a completely analytical solution, and to verify that the main characteristics of the analytical solution are also obtained in the numerical solution. Particular attention is paid to the magnitude, the velocity, and the shape of the Rayleigh wave disturbances. Copyright © 2007 John Wiley & Sons, Ltd.     

Torsional wave propagation in non‐homogeneous layer between non‐homogeneous half‐spaces

The study of surface wave in a layered medium has their possible application in geophysical prospecting. In the present work, dispersion equation for torsional wave in an inhomogeneous isotropic layer between inhomogeneous isotropic half‐spaces has been derived. Two cases are discussed separately for torsional wave propagation in inhomogeneous layer between homogeneous and non‐homogeneous half‐spaces, respectively. Further, two possible modes for torsional wave propagation are obtained in case of inhomogeneous layer sandwiched between non‐homogeneous half‐spaces. Closed form solutions for displacement in the layer and half‐spaces are obtained in each case. The study reveals that the layer width, layer inhomogeneity, frequency of inhomogeneity, as well as inhomogeneity in the half‐space has significant effect on the propagation of torsional surface waves. Displacement and implicit dispersion equation for torsional wave velocities are expressed in terms of Heun functions and their derivatives. Effects of inhomogeneity on torsional wave velocity are also discussed graphically by plotting the dimensionless phase velocity against dimensionless and scaled wave number for different values of inhomogeneity parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

A new numerical method of considering local longitudinal dispersion in single fractures

The solutions of advection–dispersion equation in single fractures were carefully reviewed, and their relationships were addressed. The classic solution, which represents the resident or flux concentration within the semi‐infinite fractures under constant concentration or flux boundary conditions, respectively, describes the effluent concentration for a finite fracture. In addition, it also predicts the cumulative distribution of solute particle residence time passing through a single fracture under pulse injection condition, based on which a particle tracking approach was developed to simulate the local advection–dispersion in single fractures. We applied the proposed method to investigate the influence of local dispersion in single fractures on the macrodispersion in different fracture systems with relatively high fracture density. The results show that the effects of local dispersion on macrodispersion are dependent on the heterogeneity of fracture system, but generally the local dispersion plays limited roles on marodispersion at least in dense fracture network. This trend was in agreement with the macrodispersion in heterogeneous porous media. Copyright © 2013 John Wiley & Sons, Ltd.     

Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM

This paper presents a new numerical tool to model the two-dimensional contaminant transport through saturated porous media using a meshfree method, called radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. The advection–dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using 2D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Three numerical examples are presented and the results are compared with those obtained from the finite element method and analytical solutions. In order to test the practical applicability and performance of the RPIM, two case studies of contaminant transport through landfill liners are presented. A good agreement is obtained between the results of the RPIM and the field investigation data.     

线性分布基床系数弹性地基梁有限单元法改进

弹性地基梁法常用于研究土和结构的相互作用,对于均布荷载和边界条件简单的弹地基梁,采用理论解即可方便地进行计算。侧向荷载作用下桩体、嵌入式挡墙一般根据弹性地基梁理论进行分析,并假定基床系数随深度增加。对于基床系数呈线性分布或呈均匀分布但边界条件复杂的弹性地基梁理论求解困难,通常采用有限差分法或有限单元法近似求解。采用有限单元法计算线性分布基床系数弹性地基梁时,若单元划分数量不够,就存在计算精度不足的问题。采用加权余量法推导了更为精确的2节点5次位移函数和相应的单刚矩阵,得出了线性分布荷载作用下挠度的5次多项式近似解,从而实现只需划分很少的单元数,节点位移及单元内位移的分布即可达到较高的计算精度,极大地提高了计算效率,单元内力的分布可直接由位移函数导出,简化了后处理计算程序。