Large-strain numerical solution for coupled self-weight consolidation and contaminant transport considering nonlinear compressibility and permeability

Based on the one-dimensional (1D) consolidation equation and advection-dispersion transport equation, this paper presents a large-strain numerical solution for coupled self-weight consolidation and contaminant transport in saturated deforming porous media considering nonlinear compressibility and permeability relationships. The finite difference method is used to solve the governing equations for consolidation and transport. The proposed numerical solution for consolidation accounts for vertical strain, soil self-weight, and nonlinearly changing compressibility and hydraulic conductivity during consolidation. The solution for solute transport accounts for advection, diffusion, mechanical dispersion, linear and nonlinear equilibrium sorption, and porosity-dependent effective diffusion coefficient. The proposed numerical solution is verified against a self-weight consolidation field tank test, an analytical solution in the literature, and the CST1 numerical model. Using the verified solution, a series of parametric study is conducted to investigate the effect of several important parameters on the contaminant transport process for confined disposal of dredged contaminated sediments. The results indicate that the consolidation process and contaminant transport process induced by soil self-weight- can be very different from those induced by the more traditional external surcharge loading. Treating the self-weight loading as traditional external surcharge loading can underestimate the rate of contaminant outflow, especially in the early times. The compressibility and permeability relationships of sediment and the type of loading (i.e., self-weight loading versus external surcharge loading) can all significantly affect the contaminant transport process for confined disposal of dredged contaminated sediment.     

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media. The flow equation is solved by the mixed finite element method, and the transport equation is approximated by a discontinuous Galerkin method. The scheme is continuous in time and a priori hp error estimates is presented.     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation

The microbial degradation of organic contaminants in the subsurface holds significant potential as a mechanism for in-situ remediation strategies. The mathematical models that describe contaminant transport with biodegradation involve a set of advective–diffusive–reactive transport equations. These equations are coupled through the nonlinear reaction terms, which may involve reactions with all of the species and are themselves coupled to growth equations for the subsurface bacterial populations. In this article, we develop Eulerian–Lagrangian localized adjoint methods (ELLAM) to solve these transport equations. ELLAM are formulated to systematically adapt to the changing features of governing partial differential equations. The relative importance of retardation, advection, diffusion, and reaction is directly incorporated into the numerical method by judicious choice of the test functions that appear in the weak form of the governing equation. Different ELLAM schemes for linear variable–coefficient advective–diffusive–reactive transport equations are developed based on different operator splittings. Specific linearization techniques are discussed and are combined with the ELLAM schemes to solve the nonlinear, multispecies transport equations. © 1995 John Wiley & Sons, Inc.     

Large scale nonlinear deterministic and stochastic optimization: Formulations involving simulation of subsurface contamination

Once subsurface water supplies become contaminated, designing cost-effective and reliable remediation schemes becomes a difficult task. The combination of finite element simulation of groundwater contaminant transport with nonlinear optimization is one approach to determine the best well selection and optimal fluid withdrawal and injection rates to contain and remove the contaminated water. Both deterministic and stochastic programming problems have been formulated and solved. These tend to be large scale problems, owing to the simulation component which serves as a portion of the constraint set. The overall problem of combined groundwater process simulation and nonlinear optimization is discussed along with example problems. Because the contaminant transport simulation models give highly uncertain results, quantifying their uncertainty and incorporating reliability into the remediation design results in a class of large stochastic nonlinear problems. The reliability problem is beginning to be addressed, and some strategies and formulations involving chance constraints and Monte Carlo methods are presented.     

Advection of methane in the hydrate zone: model,analysis and examples

A two‐phase two‐component model is formulated for the advective–diffusive transport of methane in liquid phase through sediment with the accompanying formation and dissolution of methane hydrate. This free‐boundary problem has a unique generalized solution in L¹; the proof combines analysis of the stationary semilinear elliptic Dirichlet problem with the nonlinear semigroup theory in Banach space for an m‐accretive multi‐valued operator. Additional estimates of maximum principle type are obtained, and these permit appropriate maximal extensions of the phase‐change relations. An example with pure advection indicates the limitations of these estimates and of the model developed here. We also consider and analyze the coupled pressure equation that determines the advective flux in the transport model. Copyright © 2015 John Wiley & Sons, Ltd.     

Mathematical Analysis and Numerical Simulation of a Nonlinear Thermoelastic System

Abstract

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Contaminant Transport with Adsorption in Dual-Well Flow

Numerical approximation schemes are discussed for the solution of contaminant transport with adsorption in dual-well flow. The method is based on time stepping and operator splitting for the transport with adsorption and diffusion. The nonlinear transport is solved by Godunov's method. The nonlinear diffusion is solved by a finite volume method and by Newton's type of linearization. The efficiency of the method is discussed.     

On the numerical analysis of finite element and dirichlet-to-neumann methods for nonlinear exterior transmission problems

We provide the numerical analysis of the combination of finite elements and Dirichlet-to-Neumann mappings (based on boundary integral operators) for a class of nonlinear exterior transmission problems whose weak formulations reduce to Lipschitz-continuous and strongly monotone operator equations. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, coupled with the Laplace equation in the corresponding unbounded exterior part. A discrete Galerkin scheme is presented by using linear finite elements on a triangulation of the domain, and then applying numerical quadrature and analytical formulae to evaluate all the linear, bilinear and semilinear forms involved. We prove the unique solvability of the discrete equations, and show the strong convergence of the approximate solutions. Furthermore, assuming additional regularity on the solution of the continuous operator equation, the asymptotic rate of convergence O(h) is also derived. Finally, numerical experiments are presented, which confirm the convergence results.     

Numerical Simulation of Two-Phase Flow through Heterogeneous Porous Media

A mixed finite element method is combined to finite volume schemes on structured and unstructured grids for the approximation of the solution of incompressible flow in heterogeneous porous media. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a nonlinear degenerate diffusion–convection equation arising in modeling of flow and transport in porous media.     

Iterative solution of the drift-diffusion equations

The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Influence of the saturation–suction relationship in the formulation of non-saturated soil consolidation models

The main scope of this paper is to present a fully coupled numerical model for isothermal soil consolidation analysis based on a combination of different stress states. Being originally a non-symmetric problem, it may be straightforward reduced to a symmetric one, and general guidelines for the conditions in which this reduction may be carried out, are addressed. Non-linear saturation–suction and permeability-suction functions were incorporated into a Galerkin approach of the non-saturated soil consolidation problem, which was solved using the finite element method.In order to validate the model, various examples, for which previous solutions are known, were solved. The use of either a strongly non-linear and non-symmetric formulation or a simple symmetric formulation with accurate prediction in deformation and pore-pressures is extremely dependent on the soil characteristic curves and their derivatives and this aspect is taken into account in the present mathematical approach. The emergent coupling effects may be easily uncoupled in the computer model by merely recasting some coefficients of the discrete equation system.     

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.     

Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.     

Hybrid Finite Element Method for Two-phase Miscible Displacement in Porous Media

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Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

Efficient Numerical Methods for an Anisotropic,Nonisothermal, Two-Phase Transport Model of Proton Exchange Membrane Fuel Cell

We carry out model and numerical studies for a three-dimensional, anisotropic, nonisothermal, two-phase steady state transport model of proton exchange membrane fuel cell (PEMFC) in this paper. Besides fully addressing the conservation equations of mass, momentum, species, charge and energy equations arising in the PEMFC, we present some efficient numerical methods for this model to achieve a fast and convergent nonlinear iteration, comparing to the oscillatory and nonconvergent iteration conducted by commercial flow solvers or in-house codes with standard finite element/volume method. In a framework of a combined finite element-upwind finite volume method, Kirchhoff transformation plays an important role in dealing with the discontinuous and degenerate water diffusivity in its transport equation. Preconditioned GMRES solver together with Newton’s linearization scheme make the entire numerical simulation more efficient. Three-dimensional numerical simulations demonstrate that the convergent physical solutions can be attained within 30 steps. Numerical convergence tests are also performed to verify the efficiency and accuracy of the presented numerical algorithms and techniques.     

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.