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1.
Efficient, robust simulation of groundwater flow in the unsaturated zone remains computationally expensive, especially for problems characterized by sharp fronts in both space and time. Standard approaches that employ uniform spatial and temporal discretizations for the numerical solution of these problems lead to inefficient and expensive simulations. In this work, we solve Richards’ equation using adaptive methods in both space and time. Spatial adaption is based upon a coarse grid solve and a gradient error indicator using a fixed-order approximation. Temporal adaption is accomplished using variable order, variable step size approximations based upon the backward difference formulas up to fifth order. Since the advantages of similar adaptive methods in time are now established, we evaluate our method by comparison with a uniform spatial discretization that is adaptive in time for four different one-dimensional test problems. The numerical results demonstrate that the proposed method provides a robust and efficient alternative to standard approaches for simulating variably saturated flow in one spatial dimension. 相似文献
2.
Modelling density driven flow problems requires an excessive computational time and/or heavy equipments due to the non-linear coupling between flow and transport equations. In this work, we develop a robust numerical model with efficient advanced approximations for both spatial and temporal discretizations in order to reduce the excessive computational requirement while maintaining accuracy. 相似文献
3.
Accurate numerical modeling of biogeochemical ocean dynamics is essential for numerous applications, including coastal ecosystem
science, environmental management and energy, and climate dynamics. Evaluating computational requirements for such often highly
nonlinear and multiscale dynamics is critical. To do so, we complete comprehensive numerical analyses, comparing low- to high-order
discretization schemes, both in time and space, employing standard and hybrid discontinuous Galerkin finite element methods,
on both straight and new curved elements. Our analyses and syntheses focus on nutrient–phytoplankton–zooplankton dynamics
under advection and diffusion within an ocean strait or sill, in an idealized 2D geometry. For the dynamics, we investigate
three biological regimes, one with single stable points at all depths and two with stable limit cycles. We also examine interactions
that are dominated by the biology, by the advection, or that are balanced. For these regimes and interactions, we study the
sensitivity to multiple numerical parameters including quadrature-free and quadrature-based discretizations of the source
terms, order of the spatial discretizations of advection and diffusion operators, order of the temporal discretization in
explicit schemes, and resolution of the spatial mesh, with and without curved elements. A first finding is that both quadrature-based
and quadrature-free discretizations give accurate results in well-resolved regions, but the quadrature-based scheme has smaller
errors in under-resolved regions. We show that low-order temporal discretizations allow rapidly growing numerical errors in
biological fields. We find that if a spatial discretization (mesh resolution and polynomial degree) does not resolve the solution,
oscillations due to discontinuities in tracer fields can be locally significant for both low- and high-order discretizations.
When the solution is sufficiently resolved, higher-order schemes on coarser grids perform better (higher accuracy, less dissipative)
for the same cost than lower-order scheme on finer grids. This result applies to both passive and reactive tracers and is
confirmed by quantitative analyses of truncation errors and smoothness of solution fields. To reduce oscillations in un-resolved
regions, we develop a numerical filter that is active only when and where the solution is not smooth locally. Finally, we
consider idealized simulations of biological patchiness. Results reveal that higher-order numerical schemes can maintain patches
for long-term integrations while lower-order schemes are much too dissipative and cannot, even at very high resolutions. Implications
for the use of simulations to better understand biological blooms, patchiness, and other nonlinear reactive dynamics in coastal
regions with complex bathymetric features are considerable. 相似文献
4.
将弹性波方程变换至Hamilton体系,构造适用于弹性波模拟的高效显式二阶辛Runge-Kutta-Nystrm(RKN)格式,运用根数理论得到此格式的阶条件方程组.通过给定系数的限定条件,得到方程的对称解.为了使时间离散误差达到极小,提出数值频率与真实频率比较,通过Taylor展开,得到关于辛系数的限定方程,求解方程组得到最小频散辛RKN格式.对比分析时间演进方程的稳定性,得到使库朗数达到极大值的限定方程,求解方程组得到最稳定辛RKN格式.发现此两种格式为同一格式.新得到的辛RKN格式不依赖于空间离散方法,为了对比的需要,选取有限差分法进行空间离散.在频散、稳定性分析中,与常见辛格式对比,从理论上分析了本文提出的格式在数值频散压制、稳定性提升等方面的优势,数值实验进一步证实了理论分析的正确性. 相似文献
5.
《Advances in water resources》1999,23(1):83-95
We present a numerical model for two-phase porous media flow, where the phases are separated by a sharp interface. The model is based on a unified pressure equation, and an advection equation for tracking a pseudo-concentration function. The zero-level set of this function defines the interface between the fluids. The finite element method is used for spatial discretization, with local grid refinements in the vicinity of the interface. Examples on applications involving moving interface and steady-state seepage problems are investigated. 相似文献
6.
In general, the accuracy of numerical simulations is determined by spatial and temporal discretization levels. In fractured porous media, the time step size is a key factor in controlling the solution accuracy for a given spatial discretization. If the time step size is restricted by the relatively rapid responses in the fracture domain to maintain an acceptable level of accuracy in the entire simulation domain, the matrix tends to be temporally over-discretized. Implicit sub-time stepping applies smaller sub-time steps only to the sub-domain where the accuracy requirements are less tolerant and is most suitable for problems where the response is high in only a small portion of the domain, such as within and near the fractures in fractured porous media. It is demonstrated with illustrative examples that implicit sub-time stepping can significantly improve the simulation efficiency with minimal loss in accuracy when simulating flow and transport in fractured porous media. The methodology is successfully applied to density-dependent flow and transport simulations in a Canadian Shield environment, where the flow and transport is dominated by discrete, highly conductive fracture zones. 相似文献
7.
F.A. Radu N. Suciu J. HoffmannA. Vogel O. Kolditz C.-H. ParkS. Attinger 《Advances in water resources》2011,34(1):47-61
This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater. 相似文献
8.
《Advances in water resources》1999,22(7):681-696
The prediction of contaminant transport in porous media requires the computation of the flow velocity. This work presents a methodology for high-accuracy computation of flow in a heterogeneous isotropic formation, employing a dual-flow formulation and adaptive gridding. The dual equations, describing the hydraulic head and the streamfunction, are numerically solved through finite element approximations. The application of classic finite-element methods requires a rather large number of nodes to represent suitably the flow in high-contrast formations. We present a mesh-adaptive approach that enhances the accuracy of the numerical flow solution for a given computational effort. We rely on an a posteriori error estimator to identify areas where refinements of the finite element mesh are needed or unrefinements are acceptable. We also demonstrate through numerical experiments that the developed methodology efficiently enhances accuracy through successive mesh adaptation. 相似文献
9.
Grid convergence in space and time of variable-density flow in fractured-porous rock is systematically assessed. Convergence of the flow simulation is attained using both uniform and adaptive time-stepping. This contrasts to variable-density flow in unfractured porous media where grid convergence variable-density flow problems is almost never achieved. At high discretization levels, the number of fingers in fractured-porous rock is no longer influenced by spatial-temporal grid discretization, which is not the case in unfractured porous media. However, similar to unfractured porous media, the number of fingers in fractured-porous media varies at low discretization levels. Simulated convective pattern and penetration depth of the dense plume in fractured rock depend more on spatial discretization than on temporal discretization. The appropriate spatial-temporal grid is then used to examine some aspects of mixed convection in fractured-porous rock, characterized by the mixed convection number M. The critical mixed convection number Mc = 46 represents the transition between forced and free convection in fractured porous media, which is much higher than Mc = 1 in unfractured porous media. Thus, for mixed convective flow problems, the value of Mc is not a sufficient indicator to predict the convective mode (free convection-forced convection), and the presence of vertical fractures must be included in the prediction of convective flow modes. 相似文献
10.
11.
In this work we study mixed finite element approximations of Richards’ equation for simulating variably saturated subsurface flow and simultaneous reactive solute transport. Whereas higher order schemes have proved their ability to approximate reliably reactive solute transport (cf., e.g. [Bause M, Knabner P. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput Visual Sci 7;2004:61–78]), the Raviart–Thomas mixed finite element method (RT0) with a first order accurate flux approximation is popular for computing the underlying water flow field (cf. [Bause M, Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27;2004:565–581, Farthing MW, Kees CE, Miller CT. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv Water Resour 26;2003:373–394, Starke G. Least-squares mixed finite element solution of variably saturated subsurface flow problems. SIAM J Sci Comput 21;2000:1869–1885, Younes A, Mosé R, Ackerer P, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J Comp Phys 149;1999:148–167, Woodward CS, Dawson CN. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37;2000:701–724]). This combination might be non-optimal. Higher order techniques could increase the accuracy of the flow field calculation and thereby improve the prediction of the solute transport. Here, we analyse the application of the Brezzi-Douglas-Marini element (BDM1) with a second order accurate flux approximation to elliptic, parabolic and degenerate problems whose solutions lack the regularity that is assumed in optimal order error analyses. For the flow field calculation a superiority of the BDM1 approach to the RT0 one is observed, which however is less significant for the accompanying solute transport. 相似文献
12.
近年来,随着地震波数值模拟对计算精度和效率的要求越来越高,间断有限元方法开始受到越来越多的关注.本文中,针对具有吸收边界条件的二维地震声波波动方程,作者提出了一种基于局部间断有限元方法的数值模拟算法.该算法在空间上使用局部间断有限元方法进行离散,在时间上采用了显式蛙跳格式.在这种时空离散的组合方式下,每个时间步上,此算法在空间剖分的每个单元上的求解计算是相互独立的,因而具有极高的并行性.通过数值算例,我们将该算法与连续有限元方法进行了比较.结果表明,本算法不仅具有对起伏构造的良好适应性,而且在计算效率和计算精度等方面,都具有优越性. 相似文献
13.
The multiscale finite element method is developed for solving the coupling problems of consolidation of heterogeneous saturated porous media under external loading conditions. Two sets of multiscale base functions are constructed, respectively, for the pressure field of fluid flow and the displacement field of solid skeleton. The coupling problems are then solved with a multiscale numerical procedure in space and time domain. The heterogeneities induced by permeabilities and mechanical parameters of the saturated porous media are both taken into account. Numerical experiments are carried out for different cases in comparison with the standard finite element method. The numerical results show that the coupling multiscale finite element method can be successfully used for solving the complicated coupling problems. It reduces greatly the computing effort in both memory and time for transient problems. 相似文献
14.
《Advances in water resources》2002,25(8-12):1175-1213
Multi-component flow in porous media involves localized phenomena that could be due to several features, such as concentration fronts, wells or geometry of the media. Our approach to treating the localized phenomena is to use high-resolution discretization methods in combination with adaptive mesh refinement (AMR). The purpose of AMR is to concentrate the computational work near the regions of interest in the flow. When properly designed, AMR can significantly reduce the computational effort required to obtain a desired level of accuracy in the simulation. Necessarily, AMR requires appropriate techniques for communication between length scales in a hierarchy. The selection of appropriate scaling rules as well as computationally efficient data structures is essential to the success of the overall method. However, the emphasis here is on the development of efficient techniques for solving linear systems that arise in the numerical discretization of an elliptic equation for the incompressible pressure field. In this paper, the combined AMR technique has been applied to a two-component single-phase model for miscible flooding. Numerical results are discussed in one-dimensional and two-dimensional. 相似文献
15.
《Advances in water resources》1998,21(5):351-362
Flow and displacement of non-Newtonian fluids in porous media occurs in many subsurface systems, related to underground natural resource recovery and storage projects, as well as environmental remediation schemes. A thorough understanding of non-Newtonian fluid flow through porous media is of fundamental importance in these engineering applications. Considerable progress has been made in our understanding of single-phase porous flow behavior of non-Newtonian fluids through many quantitative and experimental studies over the past few decades. However, very little research can be found in the literature regarding multi-phase non-Newtonian fluid flow or numerical modeling approaches for such analyses.For non-Newtonian fluid flow through porous media, the governing equations become nonlinear, even under single-phase flow conditions, because effective viscosity for the non-Newtonian fluid is a highly nonlinear function of the shear rate, or the pore velocity. The solution for such problems can in general only be obtained by numerical methods.We have developed a three-dimensional, fully implicit, integral finite difference simulator for single- and multi-phase flow of non-Newtonian fluids in porous/fractured media. The methodology, architecture and numerical scheme of the model are based on a general multi-phase, multi-component fluid and heat flow simulator — TOUGH2. Several rheological models for power-law and Bingham non-Newtonian fluids have been incorporated into the model. In addition, the model predictions on single- and multi-phase flow of the power-law and Bingham fluids have been verified against the analytical solutions available for these problems, and in all the cases the numerical simulations are in good agreement with the analytical solutions. In this presentation, we will discuss the numerical scheme used in the treatment of non-Newtonian properties, and several benchmark problems for model verification.In an effort to demonstrate the three-dimensional modeling capability of the model, a three-dimensional, two-phase flow example is also presented to examine the model results using laboratory and simulation results existing for the three-dimensional problem with Newtonian fluid flow. 相似文献
16.
《Advances in water resources》2001,24(8):843-862
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. 相似文献
17.
《Advances in water resources》1998,22(1):17-31
The Stokes problem describes flow of an incompressible constant-viscosity fluid when the Reynolds number is small so that inertial and transient-time effects are negligible. The numerical solution of the Stokes problem requires special care, since classical finite element discretization schemes, such as piecewise linear interpolation for both the velocity and the pressure, fail to perform. Even when an appropriate scheme is adopted, the grid must be selected so that the error is as small as possible. Much of the challenge in solving Stokes problems is how to account for complex geometry and to capture important features such as flow separation. This paper applies adaptive mesh techniques, using a posteriori error estimates, in the finite element solution of the Stokes equations that model flow at pore scales. Different selected numerical test cases associated with various porous geometrics are presented and discussed to demonstrate the accuracy and efficiency of our methodology. 相似文献
18.
Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock–Euler method and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock–Euler time integrator has advantages over standard time-discretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Léja points techniques make these computations efficient.The Rosenbrock-type methods use the appropriate rational functions of the Jacobian of the ODEs resulting from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples. 相似文献
19.
In this paper, a computational model for the simulation of coupled hydromechanical and electrokinetic flow in fractured porous media is introduced. Particular emphasis is placed on modeling CO2 flow in a deformed, fractured geological formation and the associated electrokinetic flow. The governing field equations are derived based on the averaging theory and the double porosity model. They are solved numerically with a mixed discretization scheme, formulated on the basis of the standard Galerkin finite element method, the extended finite element method, the level-set method and the Petrov–Galerkin method. The standard Galerkin method is utilized to discretize the equilibrium and the diffusive dominant field equations, and the extended finite element method, together with the level-set method and the Petrov–Galerkin method, are utilized to discretize the advective dominant field equations. The level-set method is employed to trace the CO2 plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. The proposed mixed discretization scheme leads to a convergent system, giving a stable and effectively mesh-independent model. The accuracy and computational efficiency of the proposed model is evaluated by verification and numerical examples. Effects of the fracture spacing on the CO2 flow and the streaming potential are discussed. 相似文献
20.
Liangsheng Shi Jinzhong Yang Lingzao Zeng 《Stochastic Environmental Research and Risk Assessment (SERRA)》2012,26(3):393-404
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. 相似文献