Modelling of groundwater flow in heterogeneous porous media by finite element method

The HySuf‐FEM code (Hydrodynamic of Subsurface Flow by Finite Element Method) is proposed in this article in order to estimate the spatial variability of the transmissivity values of the Berrechid aquifer (Morocco). The calibration of the model is based on the hydraulic head, hydraulic conductivity and recharge. Three numerical tests are used to validate the model and verify its convergence. The first test case consists in using the steady analytical solution of the Poisson equation. In the second, the model has been compared with the hydrogeological system which is characterized by an unconfined monolayer (isotropic layer) and computed by using PMWIN‐MODFLOW software. The third test case is based on the comparison between the results of HySuf‐FEM and the multiple cell balance method in the aquifer system with natural boundaries case. Good agreement between the Hydrodynamic of Subsurface Flow, the numerical tests and the spatial distribution of the thickening of the hydrogeological system is deduced from the analysis and the interpretations of hydrogeological wells. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of multiscale finite element method in the uncertainty qualification of large-scale groundwater flow

In this article, we discuss the application of multiscale finite element method (MsFEM) to groundwater flow in heterogeneous porous media. We investigate the ability of MsFEM in qualifying the flow uncertainty. Monte Carlo simulation is employed to implement the stochastic analysis, and MsFEM is used to avoid a full resolution to the spatial variable conductivity field. Large-scale flow with high variability is investigated by inspecting the single realization as well as the probability distribution functions of head and velocity. The numerical results show that the performance of MsFEM depends on the ratio between the correlation length and the coarse element size. An accurate prediction to the velocity requires a much lower ratio than the head. The MsFEM has different convergence rates for the head and the velocity, while the convergence rates do not deteriorate as the variance grows.     

Galerkin finite element method and the groundwater flow equation: 1. Convergence of the method

This paper is concerned with the convergence of the Galerkin finite element method applied to a groundwater flow problem containing a borehole, with special reference to quadrature effects and the accuracy of the solution. It is shown that there exists an optimal quadrature rule for every choice of piecewise polynomial basis functions. Another interesting result proved here is that, in a direct application of the method the accuracy is very nearly independent of the degree of the polynomial basis functions, but strongly dependent on the distance of the borehole from the boundary if this is small.     

结构随机响应的一种改进的概率谱叠加方法

目前有一些关于多自由度结构体系的地震响应中响应峰值按降序(即由大到小的顺序)排列时的峰值统计性质的研究。本文对这些研究中的概率方法提出一些改进。由于这些改进，概率谱叠加方法在实际应用中就变得更准确和实用。本文以五层的结构模型为例，计算其在埃尔森特罗(1940)地震激励下的响应，并与时程分析结果进行比较。结果表明，改进的概率谱叠加方法能够谁确的计算较高几阶重要响应峰值的幅值。     

An adaptive local–global multiscale finite volume element method for two-phase flow simulations

Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine scale permeability variations through the calculation of specialized coarse scale basis functions. Most of the multiscale techniques presented to date employ localization approximations in the calculation of these basis functions. For some highly correlated (e.g., channelized) formations, however, global effects are important and these may need to be incorporated into the multiscale basis functions. This can be accomplished using global fine scale simulations, but this may be computationally expensive. In this paper an adaptive local–global technique, originally developed within the context of upscaling, is applied for the computation of multiscale basis functions. The procedure enables the efficient incorporation of approximate global information, determined via coarse scale simulations, into the multiscale basis functions. The resulting procedure is formulated as a finite volume element method and is applied for a number of single- and two-phase flow simulations of channelized two-dimensional systems. Both conforming and nonconforming procedures are considered. The level of accuracy of the resulting method is shown to be consistently higher than that of the standard finite volume element multiscale technique based on localized basis functions determined using linear pressure boundary conditions.     

Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow

Richards’ equation (RE) is commonly used to model flow in variably saturated porous media. However, its solution continues to be difficult for many conditions of practical interest. Among the various time discretizations applied to RE, the method of lines (MOL) has been used successfully to introduce robust, accurate, and efficient temporal approximations. At the same time, a mixed-hybrid finite element method combined with an adaptive, higher order time discretization has shown benefits over traditional, lower order temporal approximations for modeling single-phase groundwater flow in heterogeneous porous media. Here, we extend earlier work for single-phase flow and consider two mixed finite element methods that have been used previously to solve RE using lower order time discretizations with either fixed time steps or empirically based adaption. We formulate the two spatial discretizations within a MOL context for the pressure head form of RE as well as a fully mass-conservative version. We conduct several numerical experiments for both spatial discretizations with each formulation, and we compare the higher order, adaptive time discretization to a first-order approximation with formal error control and adaptive time step selection. Based on the numerical results, we evaluate the performance of the methods for robustness and efficiency.     

Variability of seismic response of soils using stochastic finite element method

The evaluation of seismic response of soil sites constitutes an important problem with respect to groundmotion amplification and soil instability because of liquefaction. The base motion generated during earthquake is a random process. In addition, the soil sites are usually homogenous with randomly varying characteristics. The uncertainties associated with the input motion and site characteristics may lead to a wide range of variability of the site response. In this paper, a Monte-Carlo based stochastic finite element method is used to study the variability of seismic response.     

An improved optimal elemental method for updating finite element models

The optimal matrix method and optimal elemental method used to update finite element models may not provide accurate results. This situation occurs when the test modal model is incomplete, as is often the case in practice. An improved optimal elemental method is presented that defines a new objective function, and as a byproduct, circumvents the need for mass normalized modal shapes, which are also not readily available in practice. To solve the group of nonlinear equations created by the improved optimal method, the Lagrange multiplier method and Matlab function fmincon are employed. To deal with actual complex structures, the float-encoding genetic algorithm (FGA) is introduced to enhance the capability of the improved method. Two examples, a 7degree of freedom (DOF) mass-spring system and a 53-DOF planar frame, respectively, are updated using the improved method.The example results demonstrate the advantages of the improved method over existing optimal methods, and show that the genetic algorithm is an effective way to update the models used for actual complex structures.     

基于MVFOSM有限元可靠度方法的结构整体概率抗震能力分析

结构整体概率抗震能力分析既属于结构体系抗力统计分析研究内容,也属于地震易损性分析研究范畴,多采用数值模拟方法.作为一种不确定性传递的近似解析分析工具,平均值一次二阶矩方法(MVFOSM)广泛地应用于结构构件抗力的统计分析,但是很难应用于结构整体抗力的统计分析,主要困难在于结构反应是基本随机变量的隐式函数,梯度信息很难得到.将结构体系抗力的统计分析和地震易损性分析结合起来,通过基于MVFOSM的有限元可靠度方法,以新一代的地震工程模拟仿真软件OpenSees为计算平台,以最大层间位移角作为结构整体抗震能力参数,对钢筋混凝土框架结构的整体概率抗震能力进行分析,并用Monte Carlo模拟法结果进行验证,从而建立了钢筋混凝土框架结构的整体概率抗震能力模型.算例分析表明,MVFOSM有限元可靠度方法的精度和效率都很高,只需要进行一次有限元分析即可较为准确地获得结构整体概率抗震能力的前二阶矩信息.     

Mass conservation in finite element groundwater models

Numerical mass balance relations are derived for common formulations of the hydraulic and species transport equations, by summing the Galerkin equations. Precise mass balance is demonstrated, provided the Galerkin equation is retained at all boundaries. The effects of quadrature, variable coefficients, transients and irregular geometry are addressed, and numerical experiments verify the algebra.     

An efficient,high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y=ln_Ks$Y=\ln K_s$. The hydraulic head h   and average pore-velocity v$\mathbf{v}$ are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v$\mathbf{v}$ computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.     

Temperature-driven fluid flow in porous media using a mixed finite element method and a finite volume method

We present a numerical scheme for the computation of conservative fluid velocity, pressure and temperature fields in a porous medium. For the velocity and pressure we use the primal–dual mixed finite element method of Trujillo and Thomas while for the temperature we use a cell-centered finite volume method. The motivation for this choice of discretization is to compute accurate conservative quantities. Since the variant of the mixed finite element method we use is not commonly used, the numerical schemes are presented in detail. We sketch the computational details and present numerical experiments that justify the accuracy predicted by the theory.     

Treatment of time derivative and calculation of flow when solving groundwater flow problems by Galerkin finite element methods

Two techniques connected with the use of the finite element Galerkin method for solving the linear parabolic differential equation describing unsteady groundwater flow in an anisotropic non-homogeneous aquifer are introduced. The first is a mode superposition technique for dealing with the time derivative which involves computing the smallest eigenvalues and associated eigenvectors of the matrices arising from the Galerkin method. It is shown how such a technique allows us to interpret the response of the groundwater level to input in terms of parallel linear reservoirs. It is further argued that if properly implemented, the technique will have computational advantages over standard finite difference methods, e.g. in the case when the input function is constant over relatively large time subintervals. The second is a technique based on so-called generalized flow formulae for calculating flow values across external or internal boundaries, posterior to obtaining the groundwater level values. The implementation of the technique in the case of linear triangular elements on an irregular grid is discussed. It is finally argued from simplified cases that, apart from guaranteeing a match with prescribed input, the technique may often be expected to give more accurate flow values than those obtained directly from the groundwater gradients.     

An efficient approach for successively perturbed groundwater models

Global optimization methods such as simulated annealing, genetic algorithms and tabu search are being increasingly used to solve groundwater remediation design and parameter identification problems. While these methods enjoy some unique advantages over traditional gradient based methods, they typically require thousands to tens of thousands of forward simulation runs before reaching optimal or near-optimal solutions. Thus, one severe limitation associated with these global optimization methods is very long computation time. To mitigate this limitation, this paper presents a new approach for obtaining, repeatedly and efficiently, the solutions of a linear forward simulation model subject to successive perturbations. The proposed approach takes advantage of the fact that successive forward simulation runs, as required by a global optimization procedure, usually involve only slight changes in the coefficient matrices of the resultant linear equations. As a result, the new solution to a system of linear equations perturbed by the changes in aquifer properties and/or sinks/sources can be obtained as the sum of a non-perturbed base solution and the solution to the perturbed portion of the linear equations. The computational efficiency of the proposed approach arises from the fact that the perturbed solution can be derived directly without solving the linear equations again. A two-dimensional test problem with 20 by 30 nodes demonstrates that the proposed approach is much more efficient than repeatedly running the simulation model, by more than 15 times after a fixed number of model evaluations. The ratio of speedup increases with the number of model evaluations and also the size of simulation model. The main limitation of the proposed approach is the large amount of computer memory required to store the inverse matrix. Effective ways for limiting the storage requirement are briefly discussed.     

An Hermitian finite element solution of the two-dimensional saturated-unsaturated flow equation

This paper describes a Galerkin-type finite element solution of the two-dimensional saturated-unsaturated flow equation. The numerical solution uses an incomplete (reduced) set of Hermitian cubic basis functions and is formulated in terms of normal and tangential coordinates. The formulation leads to continuous pressure gradients across interelement boundaries for a number of well-defined element configurations, such as for rectangular and circular elements. Other elements generally lead to discontinuous gradients; however, the gradients remain uniquely defined at the nodes. The method avoids calculation of second-order derivatives, yet retains many of the advantages associated with Hermitian elements. A nine-point Lobatto-type integration scheme is used to evaluate all local element integrals. This alternative scheme produces about the same accuracy as the usual 9- or 16-point Gaussian quadrature schemes, but is computationally more efficient.     

An elasto-visco-plastic model using the finite element method for crustal and lithospheric deformation

A novel numerical model based on solid deformation is presented in this paper. This thermo-mechanical model can simulate the tectonic evolution of crust and (lithospheric and asthenospheric) mantle under different conditions. Our implementation uses the finite element method (FEM) in order to solve the equations. As a Lagrangian approach is employed, remeshing techniques are implemented to avoid distortion problems when a certain deformation threshold is reached. The translation of the state between the old and new mesh is achieved by means of the information stored on Lagrangian particles, which minimizes the diffusion. The model is able to represent elastic, viscous and plastic behaviour inside the studied domain. Three types of creep mechanism (diffusion, dislocation and Peierls) are included. Two different quadrilateral isoparametric elements were implemented and can be employed to perform the calculations. The first one is an element with 4 nodes, selective reduced integration and a stabilization operator to diminish hourglass modes, which reduces the computational time needed. The second one has 8 nodes located in standard positions, uses full integration scheme and has no hourglass modes as it satisfies the Inf-Sup condition. Several test cases with known solutions were run to validate the different aspects of the implementation.     

Backward probabilistic model of groundwater contamination in non-uniform and transient flow

Backward location and travel time probabilities, which provide information about the former location of contamination in an aquifer, can be used to identify unknown contamination sources. Backward location probability describes the possible upgradient positions of contamination at a known time in the past, and backward travel time probability describes the time required for contamination to travel from a known upgradient location to an observation point. These probabilities are related to adjoint states of resident concentration, and their governing equation is the adjoint of a forward contaminant transport model. Using adjoint theory to obtain the appropriate governing equation, we extend the backward probability model for conservative solutes to more general non-uniform and transient flow fields. In particular, we address three important extensions, spatially-varying porosity, transient flow and temporally-varying porosity, and internal distributed sources and sinks of solute and water. For the first time we learn that forward and backward location and travel time probabilities are not necessarily equivalent to adjoint states, but are related to them. The extensions are illustrated using a vertically-integrated groundwater model, creating transient flow by a step change in pumping and using areal recharge as an internal distributed source. Both the movement and spread of probabilities are affected. With internal sources of water, there are two interpretations of backward probability, depending on whether or not the source of water is also a source of solute. The results demonstrate how the backward probability model can be applied to other, perhaps more important, non-uniform and transient flow conditions, with time- and space-varying water storage, such as time-varying pumping or unsaturated (or saturated–unsaturated) flow and transport with spatially- and temporally-varying moisture content.     

Finite element analysis of three-dimensional groundwater flow problems

A numerical method is presented for analysing either steady state or transient three-dimensional groundwater flow problems. The governing equation is formulated in terms of the finite element process using the Galerkin approach, and cubic isoparametric elements are used to simulate the flow domain as these permit accurate modelling of curved boundaries. Particular attention is paid to the time dependent movement of the phreatic surface where an iterative technique based on the replacement of the original transient problem by a discrete number of steady state problems is used to effect a solution. Furthermore, in tracing the movement of the surface use is made of the element formulation theory in order to compute the normal to the boundary.The validity of the technique is first established by analysing a radially symmetrical problem for which an alternative analytical solution is available. Finally, a general three-dimensional flow system is studied for which there is no known analytical solution. It is shown that relatively few elements are required to yield practical solutions.