Multiple-Porosity Contaminant Transport by Finite-Element Method

An exponential finite-element model for multiple-porosity contaminant transport in soils is proposed. The model combines three compartments for dissolved contaminants: a primary compartment of diffusion–advection transport with nonequilibrium sorption, a secondary compartment with diffusion in rectangular or spherical soil blocks, and a tertiary compartment for immobile solutions within the primary compartment. Hence the finite-element model can be used to solve four types of mass-transfer problems which include: (1) intact soils, (2) intact soils with multiple sources of nonequilibrium partitioning, (3) soils with a network of regularly spaced fissures, and (4) structured soils. Hitherto, mobile/immobile compartments, fissured soils, and nonequilibrium sorption have been treated separately or in pairs. A Laplace transform is applied to the governing equations to remove the time derivative. A Galerkin residual statement is written and a finite-element method is developed. Both polynomial and exponential finite elements are implemented. The solution is inverted to the time domain numerically. The method is validated by comparison to analytical and boundary element predictions. Exponential elements perform particularly well, speeding up convergence significantly. The scope of the method is illustrated by analyzing contamination from a set of four waste repositories buried in fissured clay.     

Inclusion of Some Aspects of Chemical Behavior of Unsaturated Soil in Thermo/Hydro/Chemical/Mechanical Models. I: Model Development

This paper presents an investigation of the inclusion of some aspects of chemical behavior within a model of coupled thermo/hydro/chemical/mechanical behavior of unsaturated soils. In particular, multicomponent reactive chemical transport behavior is addressed. The chemical transport model is based on the advection/dispersion/reaction equation, while geochemical reactions are considered via coupling with an established geochemical speciation model. A numerical solution of the governing differential equations is achieved by the use of the Galerkin-weighted residual method for spatial discretization and an implicit backward Eulerian finite-difference method for temporal discretization. The solution of the geochemical reactions is achieved externally to the main solution procedure. Coupling between the chemical transport and geochemical models is achieved via the implementation of both sequential iterative and sequential noniterative techniques. Three application problems are then presented to demonstrate the capability of the coupled model.     

Contaminant Transport in Fissured Soils by Three-Dimensional Boundary Elements

A boundary element method for the solution of contaminant transport equations in three-dimensional (3D) fissured soils with non-equilibrium exchange is proposed. Fast transport is assumed to take place through a network of fissures which can be one-dimensional, two-dimensional or 3D space. Soil-matrix blocks act as storage sites for the contaminant, combining with sorption in the fissures, to increase retardation. The 3D boundary element method is formulated in the Laplace domain, where equations are linearized with respect to time, and the need for time stepping is hence removed. Solution is inverted back into time domain using a numerical technique. The numerical solution is validated and the scope of the method is illustrated by applying it to the study of cylindrical, rectangular, and spherical waste repositories in a fissured clay. The effects on contamination patterns of various fissure configurations as well as the shapes, dimensions, and number of repositories are assessed.     

Mathematical Formulation and Validation of a Mixed Finite Element–Finite Difference Model for Simulating Phreatic Surfaces

The phreatic surface in an unconfined aquifer exists as a movable interface between the saturated and unsaturated zones. The movement of the phreatic surface depends on recharge, hydraulic conductivity, porosity, and horizontal and vertical flows. The location of the phreatic surface helps define the variably saturated flow domain in the subsurface. The variably saturated flow process in the subsurface is described by a parabolic partial differential equation. In this equation, the hydraulic conductivity and soil moisture capacity are used as the subsurface characteristics. The location of the phreatic surface is governed by a first-order partial differential equation. The governing parabolic partial differential equation is solved using a variational finite element formulation. The first order phreatic surface equation is then solved by loosely coupling with the governing parabolic partial differential equation describing the variably saturated flow. In the present study, a two-dimensional space is used to investigate the movement of the phreatic surface in a variably saturated unconfined flow domain. Based on the time-varying solutions of hydraulic heads, the location of the phreatic surface is simulated in a finite two-dimensional space.     

Finite Analytic Model for Flow and Transport in Unsaturated Zone

The finite analytic method is employed to solve the vertical, two-dimensional subsurface flow and transport equations in an unsaturated zone. The finite analytic method treats the nonlinear coefficient terms of the governing equations as constants in the element so that linearized partial differential equations can be obtained and solved in each element. The accuracy and limitations of the numerical method are systematically explored. The flow and transport simulations are examined using a one-dimensional laboratory infiltration test and an analytical solution of a two-dimensional subsurface transport problem, respectively. In the advection-dominant, vertical, one-dimensional infiltration problem, nine spatial weighting schemes are proposed to evaluate the averaged unsaturated hydraulic conductivity in a discretized element. Among them, the geometric mean weighting scheme provides the most accurate results as compared with the infiltration data. In verification of the two-dimensional solute transport problem, the nine-node elements are placed in the interior domain, and different layers of five-node elements are placed at the boundaries to investigate if the numerical experiment setup was proper and the algorithm was accurate. The developed numerical model is then applied to an irregular-domain landfill leaching problem to reveal the features of subsurface transport in unsaturated zone. Numerical aspects to be further explored are suggested.     

Finite Difference Method for Computation of 1D Pollutant Migration through Saturated Homogeneous Soil Media

The physical processes such as advection, dispersion, and diffusion and interaction between the solution and the soil solids such as sorption, biodegradation, and retention processes have been considered in the governing equation used in the present study. Finite difference method has been adopted herein to solve the one-dimensional contaminant transport model to predict the pollutant migration through soil in waste landfill. In the finite difference technique, the velocity field is first determined within a hydrologic system, and these velocities are then used to calculate the rate of contaminant migration by solving the governing equation. A total of seven contaminants have been chosen for analysis to represent a wide variety of wastes both organic and inorganic. A computer software CONTAMINATE has been developed for solution of the contaminant transport model. Results of this study have been compared with existing analytical solution for validation of the proposed solution technique. Design charts for liners have also been developed to facilitate the designers. The liner thickness has been optimized by considering the effect of velocity of advection, dispersion coefficient, and geochemical reactions for all the contaminants of this study.     

Finite-Element Approach to the Erosion and Transport of Fine Particles in Granular Soils

The increase of the water table level presently occurring in the city of Milan led to some leakage of groundwater in underground facilities and subway tunnels. These structures, in fact, were completed about three decades ago when the water table was deeply located. To locally lower the ground water, and to eliminate or limit its flow towards the submerged openings, a series of pumping wells was placed in their vicinity. This provision, however, could lead to possible erosion of the fine fraction of the granular soil and to consequent settlements of nearby buildings. To investigate this phenomenon a finite-element approach has been developed for the analysis of the erosion and transport of the fine particles of granular soils subjected to a seepage flow. First, the continuity equation governing the problem and its finite-element formulation are discussed. Then, on the basis of the results of erosion tests presented in the literature, a law is derived that accounts for the nonlinear relationship between the total amount of eroded material, for time tending to infinity, and the velocity of seepage. Finally, this law is applied to the solution of one- and two-dimensional test examples, and some conclusions are drawn on the limits of the developed numerical model and on its possible improvement.     

Plane Strain and Axially Symmetric Consolidation in Unsaturated Soils

A method is presented for the analysis of coupled consolidation in unsaturated soils due to loading under conditions of plane strain as well as axial symmetry. The method is based on the transformation of the governing differential equations by the Fourier transform, when the soil system is deformed under plane strain conditions, or Hankel transform for problems exhibiting axial symmetry. The effect of such transformations is to simplify considerably the solution from a computational point of view. In addition, using these transformations the same differential equations can be used to analyze consolidation under both the above conditions. Results are presented to point out some aspects of the consolidation in unsaturated soils generated by the application of strip as well as circular loads.     

Two-Dimensional Simulation Model for Contour Basin Layouts in Southeast Australia. I: Rectangular Basins

Contour basin irrigation layouts are used to irrigate rice and other cereal crops on heavy cracking soils in Southeast Australia. In this study, a physically based two-dimensional simulation model that incorporates all the features of contour basin irrigation systems is developed. The model’s governing equations are based on a zero-inertia approximation to the two-dimensional shallow water equations of motion. The equations of motion are transformed into a single nonlinear advection–diffusion equation in which the friction force is described by Manning’s formula. The empirical Kostiakov equation and the quasi-analytical Parlange equation are used to model the infiltration process. The governing equations are solved by using a split-operator approach. The numerical procedure described here is capable of modeling rectangular basins; a procedure for irregular shaped basins is presented in Paper II. The model was validated against field data collected on commercial lasered contour layouts.     

Nonlinear Finite-Element Analysis Software Architecture Using Object Composition

Object composition offers significant advantages over class inheritance to develop a flexible software architecture for finite-element analysis. Using this approach, separate classes encapsulate fundamental finite-element algorithms and interoperate to form and solve the governing nonlinear equations. Communication between objects in the analysis composition is established using software design patterns. Root-finding algorithms, time integration methods, constraint handlers, linear equation solvers, and degree of freedom numberers are implemented as interchangeable components using the Strategy pattern. The Bridge and Factory Method patterns allow objects of the finite-element model to vary independently from objects that implement the numerical solution procedures. The Adapter and Iterator patterns permit equations to be assembled entirely through abstract interfaces that do not expose either the storage of objects in the analysis model or the computational details of the time integration method. Sequence diagrams document the interoperability of the analysis classes for solving nonlinear finite-element equations, demonstrating that object composition with design patterns provides a general approach to developing and refactoring nonlinear finite-element software.     

Stochastic Finite-Element Approach to Quantify and Reduce Uncertainty in Pollutant Transport Modeling

One of the important issues in simulation of contaminant transport in the subsurface is how to quantify the hydraulic properties of soil that are randomly variable in space because of soil heterogeneity. Stochastic approaches have the potential to represent spatially variable parameters, making them an appropriate tool to incorporate the effects of the spatial variability of soil hydraulic properties on contaminant fate. This paper presents development and application of a numerical model for simulation of advective and diffusive-dispersive contaminant transport using a stochastic finite-element approach. Employing the stochastic finite-element method proposed in this study, the response variability is reproduced with a high accuracy. Comparison of the results of the proposed method with the results obtained using the Monte?Carlo approach yields a pronounced reduction in the computation cost while resulting in virtually the same response variability as the Monte?Carlo technique.     

Unsaturated Soil Seepage Analysis Using a Rational Transformation Method with Under-Relaxation

The finite-element method provides a convenient and effective means for solving problems of seepage in unsaturated soils. However, convergence difficulties exist in numerical simulations of unsaturated flow analyses because of the high nonlinearity of the soil hydraulic properties. This technical note presents a combination approach consisting of a rational function transformation method and a common under-relaxation technique to solve the h-based form of Richards equation. Numerical studies show that this combined method can use a larger time step and corresponding oscillation-free mesh size to produce acceptable results and also converge to a stable solution quickly in each time step.     

Modeling Transport of Gaseous Ozone in Unsaturated Soils

The use of in-situ ozone venting is an effective and economic technology to remediate soils contaminated with organic chemicals. A model was developed in this study to describe the transport of gaseous ozone in unsaturated soils. Mass balance equations for soil organic matter, phenanthrene, and ozone were incorporated into the model. The model was found to fit the experimental data obtained from one-dimensional columns, using the previously published rate expressions for the reactions of ozone with soil organic matter and phenanthrene. However, it was observed that the initial unsteady-state conditions for the gas pressure resulted in minor deviations between the simulated and experimental results. Based on the simulated results, the reactions of ozone with phenanthrene and organic matter can be modeled as either parallel or serial reactions. As the initial distribution of contaminant would be nonuniform and some contaminant would be sorbed or trapped in immobile regions, it is recommended that in situ ozonation be ceased no sooner than when the effluent ozone concentrations begins to stabilize.     

Using Finite-Element Method to Simulate Wave Transformations in Surf Zone

In this study, a finite element method proposed by Hsu et al. in 2003 is extended to develop a numerical model for the simulation of wave transformation in the surf zone. The governing equation is the elliptic mild-slope equation including the energy dissipation of wave breaking. At the open boundaries with varying depth, the reflected waves caused by shoaling are adopted to the radiation boundary conditions. The rationality of the present numerical model is examined through the cases of offshore parallel breakwater problems. The results of calculation are in good agreement with experimental results.     

Influence of Clay Lens on Migration of Light Nonaqueous Phase Liquid in Unsaturated Zone

This study concerns the control of movement of light nonaqueous phase liquid (LNAPL) in unsaturated zone in the presence of relatively low permeability clay lens. A two-dimensional, finite-difference numerical model for the simultaneous movement of LNAPL and water through the unsaturated zone of the soil has been developed. The system is a three fluid phase system (water, LNAPL, and air) but in the derivation of the model, air was treated as an immobile phase at constant atmospheric pressure. The flow equations for LNAPL and water were cast in terms of the wetting and non-wetting fluid pressure heads, respectively. The finite-difference equations were solved implicitly with explicitly scheme using Newton-Raphson iteration with Taylor series expansion to treat nonlinearity. A physical model to represent the infiltration of kerosene above the clay lens was constructed. The numerical results were compared with those observed experimentally. The results of all tests showed that the presence of clay lens controls the vertical movement of LNAPL in heterogeneous porous medium.     

Multiple-Porosity Model for the Transport of Reactive Contaminants in Fissured Soils with Nonequilibrium Partitioning

A multiple-porosity model for the transport of reactive contaminants in fissured media with multiple-source, nonequilibrium partitioning is proposed, widening the scope of existing models. The proposed model extends the bicontinuum, dual-porosity concepts by combining five contaminant compartments: (1) mobile water in the fissures; (2) immobile water in the fissures; (3) water diffusing into the soil matrix; (4) sorption in the fissures; and (5) sorption in the soil matrix. Both instantaneous and nonequilibrium sorption are represented in the fissures. Mobile/immobile compartments, fissured soils, and nonequilibrium sorption have been hitherto treated separately or in pairs. Exchange of contaminants occurs between all compartments. Equations for the model are formulated and transformed into the Laplace domain. Solutions for the one-dimensional problem of a leaking storage tank overlying a fissured soil are found. The effects of the inclusion of various contaminant compartments and exchange parameters are analyzed through numerical experiments.     

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.     

Steady-State Diffusion–Advection by Exponential Finite Elements

Conventional finite-element solutions of the diffusion–advection equation exhibit numerical oscillations around the exact solution in the presence of strong advective transport. Stabilized methods modifying the standard Galerkin statement of the equation are usually used to remove oscillations and improve the speed of convergence of the method. This paper proposes an alternative approach, based on an unmodified Galerkin statement using a special eight-noded finite element whose interpolation functions vary exponentially, rather than polynomially, yielding a better approximation of the solution of the differential equation. In one-dimensional problems with specified concentration or flux at the inlet, the method increases the element Péclet number limit from 1 to 150. In two-dimensional problems, a significant improvement in accuracy relative to conventional polynomial elements is achieved. The method is particularly suitable for the h-adaptive schemes and can be easily incorporated into existing finite-element software through a minimal modification of their element libraries.