Contaminant transport with non‐equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

A non‐equilibrium sorption—advection—diffusion model to simulate miscible pollutant transport in saturated–unsaturated soils is presented. The governing phenomena modelled in the present simulation are: convection, molecular diffusion, mechanical dispersion, sorption, immobile water effect and degradation, including both physical and chemical non‐equilibrium processes. A finite element procedure, based on the characteristic Galerkin method with an implicit algorithm is developed to numerically solve the model equations. The implicit algorithm is formulated by means of a combination of both the precise and the traditional numerical integration procedures. The stability analysis of the algorithm shows that the unconditional stability of the present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results illustrate good performance of the present algorithm in stability and accuracy, and in simulating the effects of all the mentioned phenomena governing the contaminant transport and the concentration distribution. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.     

The use of a mixed scheme: mixed hybrid finite elements method/finite volumes (MHFE/FV), for the modeling of contaminants transport in unsaturated porous mediums

This work is intended for the development of a numerical method to simulate flows and solute transport in multiphasical porous medium taking into consideration the interaction of solid/solute. More precisely, the studied problem is modeled by a coupled system composed of an elliptical equation (for the flow) and an equation convection–diffusion–reaction (for the transfer). Numerical simulations were realistic for two-dimensional problems confirming the stability and efficiency of the combined scheme in the characterization of a pollutant transport through an unsaturated zone of an industrial site.     

Error Estimates for a Time Discretization Method for the Richards' Equation

We present a numerical analysis of a time discretization method applied to Richards' equation. Written in its saturation-based form, this nonlinear parabolic equation models water flow into unsaturated porous media. Depending on the soil parameters, the diffusion coefficient may vanish or explode, leading to degeneracy in the original parabolic equation. The numerical approach is based on an implicit Euler time discretization scheme and includes a regularization step, combined with the Kirchhoff transform. Convergence is shown by obtaining error estimates in terms of the time step and of the regularization parameter.     

Evaluation of Containment Performance from Viewpoints of Diffusion and Dispersion for Impervious Structures at Coastal Landfill Sites

The containment performance of impervious structures is considered the most important performance, which the toxic substances are enclosed in coastal landfill sites. The containment performance is evaluated generally by the hydraulic conductivity and the thickness of impervious structures to focus on an advection. However, the leakage of toxic substances is affected by a diffusion and a dispersion. The diffusion and the dispersion are considered to be easily distinguished on the impervious structure which has the low-hydraulic conductivity. Such phenomena should be considered due to evaluate the containment performance. This is because to improve the containment performance and comprehend leakage of the toxic substances. The containment performance is evaluated from the viewpoint of the diffusion and the dispersion in this research. Concretely, the influence which the toxic substances leaks from impervious structures on the diffusion and the dispersion is evaluated by the infiltration and advection–dispersion analysis on steel-made side impervious walls. In other word, The relation between the leaking amount of toxic substances and coefficient of molecular diffusion, the relation between the travel time or the flow velocity and the leaking amount and the change in the water-level difference are considered respectively. So, the leaking amount which is influenced by a diffusion and a dispersion is considered. Consideration about to secure the containment performance of impervious structure which is restrained leakage by a diffusion and a dispersion.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

The analysis of pollutant migration through soil with linear hereditary time-dependent sorption

This paper presents a non-equilibrium sorption dispersion–advection transport model for the analysis of pollutant migration through soil. The formulation involves a convolution integral of the product of the rate of change of concentration and a time-dependent sorption coefficient, suggesting an integral transformation of the governing equations. This facilitates the primary purpose of this paper, to incorporate a time-dependent solute sorption process into a computationally efficient and accurate semi-analytic Laplace transform method. An application of the non-equilibrium sorption model for backfiguring dispersion–advection equation parameters from experimental data is presented, and the implications of non-equilibrium sorption on the design of landfill liners is explored by means of an illustrative example.     

基于单颗粒受力的均匀推移质颗粒对流和扩散特征

为揭示均匀推移质颗粒对流和扩散特性的控制因素,建立了间歇郎之万方程模型。该模型可在单颗粒尺度充分考虑颗粒的受力特性,模拟颗粒随机、间歇运动过程。通过该模型引入不同的停时分布,对模拟的大量单颗粒运动过程进行统计,从而研究均匀颗粒在大的时空尺度上的对流和扩散特征。结果表明,对于均匀颗粒,受颗粒速度分布的窄尾性限制,即便单步步长是长尾分布,也不一定产生超扩散,扩散特性由停时分布的尾部特征决定,不同分布的停时可导致欠扩散、超扩散和正常扩散。进一步与已有的、复杂程度不同的随机模型进行对比,表明忽略单步时间将影响颗粒的扩散(二阶)特性,但不影响颗粒的对流(一阶)特性,类似地可以推广到更普遍规律,即所研究随机发生的统计矩阶数越高,需要的模型越复杂。     

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.     

Analysis of subdiffusion in disordered and fractured media using a Grünwald-Letnikov fractional calculus model

The increasing applications of fractional calculus in simulating the anomalous transport behavior in disordered and fractured heterogeneous porous media has grown rapidly over the past decade. In the present study, a temporal fractional flux relationship is employed as a constitutive equation to relate the volumetric flow rate to the gradient of the pore pressure. The novelty of this paper entails interpreting the time fractional derivative operator in the flux relationship by the Grünwald-Letnikov (G-L) definition as opposed to the Caputo interpretation which has been widely considered. Subsequently, a numerical scheme based on the block-centered finite-difference discretization is formulated to handle the resulting non-linear fractional diffusion model. In addition, a linear stability analysis is successfully performed to establish the stability criterion of the developed numerical scheme. An expression for the modified incremental material balance index was derived to assess the effectiveness of the numerical discretization process. Finally, numerical experiments were performed to provide qualitative insights into the nature of pressure evolution in a hydrocarbon reservoir under the influence subdiffusion. In summary, the results establish that subdiffusion regime results in the development of higher pressure drop in the reservoir. This paper will provide a strong foundation for researchers interested in investigating anomalous diffusion phenomena in porous media.     

A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations

In this paper, we study the properties of approximate solutions to a doubly nonlinear and degenerate diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW), using a numerical approach based on the Galerkin finite element method. This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases the porous medium equation and the p-Laplacian. Diverse numerical schemes have been implemented to approximately solve the DSW equation and have been successfully applied as suitable models to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out in order to study the properties of approximate solutions. In this study, we propose a numerical approach as a means to understand some properties of solutions to the DSW equation and, thus, to provide conditions for which the use of the DSW equation may be inappropriate from both the physical and the mathematical points of view, within the context of shallow water modeling. For analysis purposes, we propose a numerical method based on the Galerkin method and we obtain a priori error estimates between the approximate solutions and weak solutions to the DSW equation under physically consistent assumptions. We also present some numerical experiments that provide relevant information about the accuracy of the proposed numerical method to solve the DSW equation and the applicability of the DSW equation as a model to simulate observed quantities in an experimental setting.     

Numerical limit analysis of three-dimensional slope stability problems in catchment areas

Risk analysis of existing slopes in catchment areas requires quantification of their stability. This quantification becomes particularly difficult when dealing with larger areas under 3D conditions and including saturated and unsaturated water flow. This paper proposes the use of an effective numerical procedure to solve three-dimensional slope stability problems in large areas subjected to pore pressure effects. This numerical approach, numerical limit analysis, utilizes the finite element method and mathematical programming techniques. Mathematical programming is needed because the basic plasticity theorems for limit analysis can be cast as optimization problems. The generated optimization problem is formulated under a second-order cone programming framework, which is known to solve large-scale problems with great computational efficiency. The main objective of this work was to determine the slope safety factor and the collapse mechanism of soils governed by the Drucker–Prager yield criterion for large-scale 3D problems including pore pressure effects. This approach is applied to an experimental catchment in the Oregon Coast Range that failed after an intense rainfall. The results were compared with a previous stability analysis of the area available in the literature that used a novel 3D limit equilibrium method.     

Comparison Between Analytical and Numerical Methods in Evaluating the Pollution Transport in Porous Media

Contaminant transport modelling in environmental engineering is generally conducted to evaluate the potential impact of contaminant migration on the subsurface environment or for interpreting tracer tests or groundwater quality data. In the past few decades a number of mathematical models have been established for evaluating the migration of pollution as indicated in the literature. This paper presents a comparison between a number of analytical and numerical models in evaluating pollution transport in soils. Three analytical models and a finite element model developed in this research are used for comparing four numerical examples under different conditions. Four cases of advection dominated problem with line source boundary, advection dominated problem with semi-line source boundary, advection–dispersion–sorption problem with line source boundary and advection–dispersion–sorption problem with semi-line source are considered. Based on the results the best analytical model that has a higher accuracy is recommended for practical applications.     

Thermal reservoir modeling in petroleum geomechanics

Thermal oil recovery processes involve high pressures and temperatures, leading to large volume changes and induced stresses. These cannot be handled by traditional reservoir simulation because it does not consider coupled geomechanics effects. In this paper we present a fully coupled, thermal half‐space model using a hybrid DDFEM method. A finite element method (FEM) solution is adopted for the reservoir and the surrounding thermally affected zone, and a displacement discontinuity method is used for the surrounding elastic, non‐thermal zone. This approach analyzes stress, pressure, temperature and volume change in the reservoir; it also provides stresses and displacements around the reservoir (including transient ground surface movements) in a natural manner without introducing extra spatial discretization outside the FEM zone. To overcome spurious spatial temperature oscillations in the convection‐dominated thermal advection–diffusion problem, we place the transient problem into an advection–diffusion–reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid‐scale/gradient subgrid‐scale method. Copyright © 2008 John Wiley & Sons, Ltd.     

Hybrid discretization of multi-phase flow in porous media in the presence of viscous,gravitational, and capillary forces

Multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces is described by advection diffusion equations with nonlinear parameters of relative permeability and capillary pressures. The conventional numerical method employs a fully implicit finite volume formulation. The phase-potential-based upwind direction is commonly used in computing the transport terms between two adjacent cells. The numerical method, however, often experiences non-convergence in a nonlinear iterative solution due to the discontinuity of transmissibilities, especially in transition between co-current and counter-current flows. Recently, Lee et al. (Adv. Wat. Res. 82, 27–38, 2015) proposed a hybrid upwinding method for the two-phase transport equation that comprises viscous and gravitational fluxes. The viscous part is a co-current flow with a one-point upwinding based on the total velocity and the buoyancy part is modeled by a counter-current flow with zero total velocity. The hybrid scheme yields C¹-continuous discretization for the transport equation and improves numerical convergence in the Newton nonlinear solver. Lee and Efendiev (Adv. Wat. Res. 96, 209–224, 2016) extended the hybrid upwind method to three-phase flow in the presence of gravity. In this paper, we present the hybrid-upwind formula in a generalized form that describes two- and three-phase flows with viscous, gravity, and capillary forces. In the derivation of the hybrid scheme for capillarity, we note that there is a strong similarity in mathematical formulation between gravity and capillarity. We thus greatly utilize the previous derivation of the hybrid upwind scheme for gravitational force in deriving that for capillary force. Furthermore, we also discuss some mathematical issues related to heterogeneous capillary domains and propose a simple discretization model by adapting multi-valued capillary pressures at the end points of capillary pressure curves. We demonstrate this new model always admits a consistent solution that is within the discretization error. This new generalized hybrid scheme yields a discretization method that improves numerical stability in reservoir simulation.     

多阶多层复杂边坡稳定性的通用上限方法

极限分析上限方法在边坡稳定性评价中受到了广泛关注,但当前所取得的解析成果尚不能直接应用于解决任意多土层分布、多台阶的广义复杂层状边坡。基于组合对数螺线的旋转破坏机制,推导了具有任意坡面几何特征、任意多土层（含非水平土/岩层）边坡的外功率统一积分表达式及相应的虚功率方程,提出了多阶多层复杂边坡稳定性的通用极限分析上限方法;为克服积分式的复杂解析计算,引入了数值积分技术。在此基础上,结合最优化方法和强度折减技术,优化求解了复杂边坡的全局稳定性安全系数及相应的临界滑动面。通过多个典型算例的验证与对比分析,表明该方法具有较高的精度和广泛适用性。最后,针对典型多阶多层边坡实例,开展了上限法的深度拓展与应用研究,其结果为广义复杂层状边坡的稳定性评价提供了新思路。     

基于数值建模方法的弹塑性固结问题解耦研究

研究了弹塑性固结问题的解耦方法。首先,在数值建模方法下得出土的弹塑性本构关系,推导了两类问题下的应力-应变关系统一矩阵式,并将数值建模方法与Biot固结理论相结合,建立了基于此本构关系的固结问题控制方程的增量形式。考虑应力路径的影响,讨论了此类液-固耦合问题的解耦条件,导出了在该条件下的扩散方程和非耦合控制方程,并编制有限元程序计算了两个典型算例,通过对比分析表明,该方法简单合理,能考虑剪胀性对固结规律的影响。     

The significance of isotope specific diffusion coefficients for reaction-transport models of sulfate reduction in marine sediments

Modeling isotopic signatures in systems affected by diffusion, advection, and a reaction which modifies the isotopic abundance of a given species, is a discipline in its infancy. Traditionally, much emphasis has been placed on kinetic isotope effects during biochemical reactions, while isotope effects caused by isotope specific diffusion coefficients have been neglected. A recent study by Donahue et al. (2008) suggested that transport related isotope effects may be of similar magnitude as microbially mediated isotope effects. Although it was later shown that the assumed differences in the isotope specific diffusion coefficients were probably overstated by one or two orders of magnitude (Bourg, 2008), this study raises several important issues: (1) Is it possible to directly calculate isotopic enrichment factors from measured concentration data without modeling the respective system? (2) Do changes in porosity and advection velocity modulate the influence of isotope specific diffusion coefficients on the fractionation factor α? (3) If one has no a priori knowledge whether diffusion coefficients are isotope specific or not, what is the nature and magnitude of the error introduced by either assumption? Here we argue (A) That the direct substitution of measured data into a differential equation is problematic and cannot be used as a replacement for a reaction-transport model; (B) That the transport related fractionation scales linearly with the difference between the respective diffusion coefficients of a given isotope system, but depends in a complex non-linear way on the interplay between advection velocity, and downcore changes of temperature and porosity. Last but not least, we argue that the influence of isotope specific diffusion coefficients on microbially mediated sulfate reduction in typical marine sediments is considerably smaller than the error associated with the determination of the fractionation factor.