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1.
Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation, and many of these models require a relatively fine-grid discretization to accurately represent the selected process in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed. One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000. The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method can be successfully implemented to model surface water-ground water interactions. 相似文献
2.
本文以基于改进BISQ模型的二维双相各向同性介质一阶速度-应力方程为基础,推导出了曲线坐标系下对应的方程,然后采用低频散、低耗散的同位网格MacCormack有限差分法来离散方程,并采用紧致的单边MacCormack差分格式结合牵引力镜像法来施加自由地表边界条件,实现了地震波场数值模拟.曲线网格有限差分法采用贴体网格来描述自由表面,地表的网格线紧贴地形,避免了台阶近似造成的数值散射.数值模拟结果表明,在双相介质起伏自由地表和分界面处,各类波型复杂的反射透射规律可以清晰展现,曲线网格有限差分法可以精确地解决地震波在含起伏地表的双相各向同性介质中的传播问题. 相似文献
3.
《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
4.
In this paper, we deduced the corresponding first-order velocity–stress equation for curvilinear coordinates from the first-order velocity–stress equation based on the modified Biot/squirt model for a two-dimensional two-phase medium. The equations are then numerically solved by an optimized high-order non-staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free-surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite-difference scheme plus a traction-image method. In the undulating free surface and the undulating subsurface interface of two-phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear-grid finite-difference method, which uses a body-conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder-shaped grid to fit the surfaces when undulating topography is present in a two-phase isotropic medium. 相似文献
5.
6.
The nonhorizontal‐model‐layer (NHML) grid system is more accurate than the horizontal‐model‐layer grid system to describe groundwater flow in an unconfined sloping aquifer on the basis of MODFLOW‐2000. However, the finite‐difference scheme of NHML was based on the Dupuit‐Forchheimer assumption that the streamlines were horizontal, which was acceptable for slope less than 0.10. In this study, we presented a new finite‐difference scheme of NHML based on the Boussinesq assumption and developed a new package SLOPE which was incorporated into MODFLOW‐2000 to become the MODFLOW‐SP model. The accuracy of MODFLOW‐SP was tested against solution of Mac Cormack (1969). The differences between the solutions of MODFLOW‐2000 and MODFLOW‐SP were nearly negligible when the slope was less than 0.27, and they were noticeable during the transient flow stage and vanished in steady state when the slope increased above 0.27. We established a model considering the vertical flow using COMSOL Multiphysics to test the robustness of constrains used in MODFLOW‐SP. The results showed that streamlines quickly became parallel with the aquifer base except in the narrow regions near the boundaries when the initial flow was not parallel to the aquifer base. MODFLOW‐SP can be used to predict the hydraulic head of an unconfined aquifer along the profile perpendicular to the aquifer base when the slope was smaller than 0.50. The errors associated with constrains used in MODFLOW‐SP were small but noticeable when the slope increased to 0.75, and became significant for the slope of 1.0. 相似文献
7.
A Parallel PCG Solver for MODFLOW 总被引：2，自引：0，他引：2
In order to simulate large-scale ground water flow problems more efficiently with MODFLOW, the OpenMP programming paradigm was used to parallelize the preconditioned conjugate-gradient (PCG) solver with in this study. Incremental parallelization, the significant advantage supported by OpenMP on a shared-memory computer, made the solver transit to a parallel program smoothly one block of code at a time. The parallel PCG solver, suitable for both MODFLOW-2000 and MODFLOW-2005, is verified using an 8-processor computer. Both the impact of compilers and different model domain sizes were considered in the numerical experiments. Based on the timing results, execution times using the parallel PCG solver are typically about 1.40 to 5.31 times faster than those using the serial one. In addition, the simulation results are the exact same as the original PCG solver, because the majority of serial codes were not changed. It is worth noting that this parallelizing approach reduces cost in terms of software maintenance because only a single source PCG solver code needs to be maintained in the MODFLOW source tree. 相似文献
8.
Herman W. J. Kernkamp Henri A. H. Petit Herman Gerritsen Erik D. de Goede 《Ocean Dynamics》2005,55(3-4):351-369
In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems
and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal
co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same
for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting
in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest.
This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved
in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system
specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types.
Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical
orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying
a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution
based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon
surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear
grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with
regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach
combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very
large model domains. 相似文献
9.
Approaches for modeling lake-ground water interactions have evolved significantly from early simulations that used fixed lake stages specified as constant head to sophisticated LAK packages for MODFLOW. Although model input can be complex, the LAK package capabilities and output are superior to methods that rely on a fixed lake stage and compare well to other simple methods where lake stage can be calculated. Regardless of the approach, guidelines presented here for model grid size, location of three-dimensional flow, and extent of vertical capture can facilitate the construction of appropriately detailed models that simulate important lake-ground water interactions without adding unnecessary complexity. In addition to MODFLOW approaches, lake simulation has been formulated in terms of analytic elements. The analytic element lake package had acceptable agreement with a published LAKI problem, even though there were differences in the total lake conductance and number of layers used in the two models. The grid size used in the original LAKI problem, however, violated a grid size guideline presented in this paper. Grid sensitivity analyses demonstrated that an appreciable discrepancy in the distribution of stream and lake flux was related to the large grid size used in the original LAKI problem. This artifact is expected regardless of MODFLOW LAK package used. When the grid size was reduced, a finite-difference formulation approached the analytic element results. These insights and guidelines can help ensure that the proper lake simulation tool is being selected and applied. 相似文献
10.
Approaches to the simulation of unconfined flow and perched groundwater flow in MODFLOW 总被引：1，自引：0，他引：1
Various approaches have been proposed to manage the nonlinearities associated with the unconfined flow equation and to simulate perched groundwater conditions using the MODFLOW family of codes. The approaches comprise a variety of numerical techniques to prevent dry cells from becoming inactive and to achieve a stable solution focused on formulations of the unconfined, partially-saturated, groundwater flow equation. Keeping dry cells active avoids a discontinuous head solution which in turn improves the effectiveness of parameter estimation software that relies on continuous derivatives. Most approaches implement an upstream weighting of intercell conductance and Newton-Raphson linearization to obtain robust convergence. In this study, several published approaches were implemented in a stepwise manner into MODFLOW for comparative analysis. First, a comparative analysis of the methods is presented using synthetic examples that create convergence issues or difficulty in handling perched conditions with the more common dry-cell simulation capabilities of MODFLOW. Next, a field-scale three-dimensional simulation is presented to examine the stability and performance of the discussed approaches in larger, practical, simulation settings. 相似文献
11.
Regional finite-difference models tend to have large cell sizes, often on the order of 1–2 km on a side. Although the regional flow patterns in deeper formations may be adequately represented by such a model, the intricate surface water and groundwater interactions in the shallower layers are not. Several stream reaches and nearby wells may occur in a single cell, precluding any meaningful modeling of the surface water and groundwater interactions between the individual features. We propose to replace the upper MODFLOW layer or layers, in which the surface water and groundwater interactions occur, by an analytic element model (GFLOW) that does not employ a model grid; instead, it represents wells and surface waters directly by the use of point-sinks and line-sinks. For many practical cases it suffices to provide GFLOW with the vertical leakage rates calculated in the original coarse MODFLOW model in order to obtain a good representation of surface water and groundwater interactions. However, when the combined transmissivities in the deeper (MODFLOW) layers dominate, the accuracy of the GFLOW solution diminishes. For those cases, an iterative coupling procedure, whereby the leakages between the GFLOW and MODFLOW model are updated, appreciably improves the overall solution, albeit at considerable computational cost. The coupled GFLOW–MODFLOW model is applicable to relatively large areas, in many cases to the entire model domain, thus forming an attractive alternative to local grid refinement or inset models. 相似文献
12.
As a result of rock dissolution processes, karst aquifers exhibit highly conductive features such as caves and conduits. Within these structures, groundwater flow can become turbulent and therefore be described by nonlinear gradient functions. Some numerical groundwater flow models explicitly account for pipe hydraulics by coupling the continuum model with a pipe network that represents the conduit system. In contrast, the Conduit Flow Process Mode 2 (CFPM2) for MODFLOW-2005 approximates turbulent flow by reducing the hydraulic conductivity within the existing linear head gradient of the MODFLOW continuum model. This approach reduces the practical as well as numerical efforts for simulating turbulence. The original formulation was for large pore aquifers where the onset of turbulence is at low Reynolds numbers (1 to 100) and not for conduits or pipes. In addition, the existing code requires multiple time steps for convergence due to iterative adjustment of the hydraulic conductivity. Modifications to the existing CFPM2 were made by implementing a generalized power function with a user-defined exponent. This allows for matching turbulence in porous media or pipes and eliminates the time steps required for iterative adjustment of hydraulic conductivity. The modified CFPM2 successfully replicated simple benchmark test problems. 相似文献
13.
The solution to the 2-D time-dependent unsaturated flow equation is numerically approximated by a second-order accurate cell-centered finite-volume discretization on unstructured grids. The approximation method is based on a vertex-centered Least Squares linear reconstruction of the solution gradients at mesh edges.A Taylor series development in time of the water content dependent variable in a finite-difference framework guarantees that the proposed finite volume method is mass conservative. A Picard iterative scheme solves at each time step the resulting non-linear algebraic problem. The performance of the method is assessed on five different test cases and implementing four distinct soil constitutive relationships. The first test case deals with a column infiltration problem. It shows the capability of providing a mass-conservative behavior. The second test case verifies the numerical approximation by comparison with an analytical mixed saturated–unsaturated solution. In this case, the water drains from a fully saturated portion of a 1-D column. The third and fourth test cases illustrate the performance of the approximation scheme on sharp soil heterogeneities on 1-D and 2-D multi-layered infiltration problems. The 2-D case shows the passage of an abrupt infiltration front across a curved interface between two layers. Finally, the fifth test case compares the numerical results with an analytical solution that is developed for a 2-D heterogeneous soil with a source term representing plant roots. This last test case illustrates the formal second-order accuracy of the method in the numerical approximation of the pressure head. 相似文献
14.
《Advances in water resources》2004,27(9):899-912
This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497–511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, “cage-shell” interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size—a coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid. 相似文献
15.
基于应力、速度混合变量弹性波方程及任意四边形网格差分算子,给出了交错计算应力及速度的非规则网格弹性波应力一速度差分法该方法融合了有限元法能适应复杂形状边界及差分法无需计算刚度阵的特点,具有较高的计算精度,所需计算机存储空间较少,计算效率也很高．基于积分平衡方程引入了任意形状自由表面的边界条件,且通过局部滤波改善了自由表面边界条件的稳定性,使得该方法可应用于考虑地表形状影响的地震波数值模拟 相似文献
16.
M. Giudici F. Delay G. de Marsily G. Parravicini G. Ponzini A. Rosazza 《Stochastic Hydrology and Hydraulics》1998,12(3):191-204
The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow
models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice
for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data
are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques.
We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated
measurement errors and interpolation errors. The numerical results show that the DSM is stable. 相似文献
17.
The accuracy with which MODFLOW simulates surface water-groundwater interaction is examined for connected and disconnected losing streams. We compare the effect of different vertical and horizontal discretization within MODFLOW and also compare MODFLOW simulations with those produced by HydroGeoSphere. HydroGeoSphere is able to simulate both saturated and unsaturated flow, as well as surface water, groundwater and the full coupling between them in a physical way, and so is used as a reference code to quantify the influence of some of the simplifying assumptions of MODFLOW. In particular, we show that (1) the inability to simulate negative pressures beneath disconnected streams in MODFLOW results in an underestimation of the infiltration flux; (2) a river in MODFLOW is either fully connected or fully disconnected, while in reality transitional stages between the two flow regimes exist; (3) limitations in the horizontal discretization of the river can cause a mismatch between river width and cell width, resulting in an error in the water table position under the river; and (4) because coarse vertical discretization of the aquifer is often used to avoid the drying out of cells, this may result in an error in simulating the height of the groundwater mound. Conditions under which these errors are significant are investigated. 相似文献
18.
M. Giudici F. Delay G. de Marsily G. Parravicini G. Ponzini A. Rosazza 《Stochastic Environmental Research and Risk Assessment (SERRA)》1998,12(3):191-204
The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow
models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice
for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data
are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques.
We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated
measurement errors and interpolation errors. The numerical results show that the DSM is stable. 相似文献
19.
A numerical model of a riffle segment of a large river is applied to examine the kinematic flow structure at different phases of water regime. A finite-difference scheme on an irregular triangular grid is proposed for solving shallow-water equations. The results of field measurements and calculations are presented. 相似文献
20.
Vertical Discretization Impact in Numerical Modeling of Matrix Diffusion in Contaminated Groundwater
Shahla K. Farhat David T. Adamson Arun R. Gavaskar Sophia A. Lee Ronald W. Falta Charles J. Newell 《Ground Water Monitoring & Remediation》2020,40(2):52-64
Understanding the effects of contaminants that can diffuse into low-permeability (“low-k”) zones is crucial for effective groundwater remedial decision-making. Because low-k zones can serve as low-level sources of contamination to more transmissive zones over time, an accurate evaluation of the impacts of matrix diffusion at contaminated sites is vital. This study compared numerical groundwater flow and transport simulations using MODFLOW/RT3D at a hypothetical site using three cases, each with increasing discretization of the vertical 10-m thick domain: (1) a coarse multilayer heterogeneous grid based on one layer for each of four different hydrogeological units, (2) a “low-resolution” discretization approach where the low-k units were divided into several sublayers giving the model 10 layers, and (3) a “high-resolution” numerical model with 199 layers that are a few centimeters thick. When comparing the results of each case, significant differences were observed between the discretizations used, even though all other model input data were identical. The conventional grid models (Cases 1 and 2) appeared to underestimate groundwater plume concentrations by a factor ranging from 1.1 to 36 when compared to the high-resolution grid model (Case 3), and underestimated predicted cleanup times by more than a factor of 10 for some of the hypothetical sampling points in the modeling domain. These results validate the implication of Chapman et al. (2012), that conventional vertical discretization of numerical groundwater flow and transport models at contaminated sites (with layers that are greater than 1 m thick) can lead to significant errors when compared to more accurate high-resolution vertical discretization schemes (layers that are centimeters thick). 相似文献