Local Grid Refinement Method Research Articles

An adaptive local grid refinement method for 2D diffusion equation with variable coefficients based on block-centered finite differences

In this paper, an adaptive local grid refinement method based on block-centered finite differences is proposed for 2D diffusion equation with Neumann boundary condition. The method first identifies the regions with large local error, then these regions are divided until a prescribed tolerance is satisfied. The numerical solutions of unknown variable along with its first derivatives are obtained simultaneously. Theoretical analysis shows that the proposed method is second order accurate both on uniform and non-uniform meshes. Some high gradient problems are carried out to verify the efficiency and reliability of the adaptive local grid refinement method.

Evaluation of MODFLOW‐LGR in Connection with a Synthetic Regional‐Scale Model

This work studies costs and benefits of utilizing local-grid refinement (LGR) as implemented in MODFLOW-LGR to simulate groundwater flow in a buried tunnel valley interacting with a regional aquifer. Two alternative LGR methods were used: the shared-node (SN) method and the ghost-node (GN) method. To conserve flows the SN method requires correction of sources and sinks in cells at the refined/coarse-grid interface. We found that the optimal correction method is case dependent and difficult to identify in practice. However, the results showed little difference and suggest that identifying the optimal method was of minor importance in our case. The GN method does not require corrections at the models' interface, and it uses a simpler head interpolation scheme than the SN method. The simpler scheme is faster but less accurate so that more iterations may be necessary. However, the GN method solved our flow problem more efficiently than the SN method. The MODFLOW-LGR results were compared with the results obtained using a globally coarse (GC) grid. The LGR simulations required one to two orders of magnitude longer run times than the GC model. However, the improvements of the numerical resolution around the buried valley substantially increased the accuracy of simulated heads and flows compared with the GC simulation. Accuracy further increased locally around the valley flanks when improving the geological resolution using the refined grid. Finally, comparing MODFLOW-LGR simulation with a globally refined (GR) grid showed that the refinement proportion of the model should not exceed 10% to 15% in order to secure method efficiency.

Finite difference schemes for two-dimensional miscible displacement flow in porous media on composite triangular grids

Considering two-dimensional miscible displacement flow in porous media, the local grid refinement method of a coupled system on triangular cell-centered grids with local refinement in space is studied. Based on the balance equation, finite difference schemes of the coupled equations on composite grids are constructed. Studying their stability and convergence properties, the error estimate in the energy norm is obtained. Finally, a numerical example is given.

Foreword: Understanding through Modeling

Since 1998, five MODFLOW-themed conferences have been held, including ‘‘MODFLOW 98,’’ ‘‘MODFLOW 2001 and Other Modeling Odysseys,’’ ‘‘MODFLOW and More 2003: Understanding through Modeling,’’ ‘‘FiniteElement Models, MODFLOW and More 2004,’’ and ‘‘MODFLOW and More 2006: Managing Ground Water Systems.’’ This is the second theme issue of Ground Water to be derived from MODFLOW-themed conferences; the first was derived from the 2001 conference, as described by Hill et al. (2003). This theme issue is derived from the ‘‘MODFLOW and More 2003: Modeling through Understanding’’ conference held in September 2003 at the Colorado School of Mines in Golden, Colorado. Chosen contributions have evolved and expanded beyond their original scope and content as presented at the 2003 conference. In spite of using MODFLOW as a working theme, both the conferences and related journal issues involve many different modeling approaches and simulation programs. This theme issue begins with a Software Spotlight column by Barlow and Harbaugh. The column summarizes some of the current directions for MODFLOW development being pursued by the USGS. There include simulation of coupled ground water/surface water systems, variable-density flow and solute transport modeling, unsaturated-zone flow simulation, parameter estimation, and ground water management optimization modeling. The next two articles are issue papers by Hill and Gomez-Hernandez that debate the merits and pitfalls of simplicity vs. complexity in developing ground water models. Hill argues that all ground water models should start out simple and modelers need to add complexity carefully as warranted by the complexity (1) supported by the available data and (2) important to the predictions. On the other hand, Gomez-Hernandez argues that aquifers are inherently complex and their representation is uncertain. Thus, complexity in models, based on the stochastic, multiple-realization Monte-Carlo approach, is a must for proper evaluation and prediction. The exchange of the simplicity vs. complexity viewpoints is complemented by a following technical commentary by Haitjema. He advocates that training in basic hand calculations is important in ground water modeling curricula because analytic solutions to elementary ground water flow problems provide something that often is lost when using sophisticated computer models: insight. Nine research papers constitute the remainder of the theme issue. The first three deal with modeling techniques in flow and solute transport. Mehl et al. review several local grid refinement (LGR) methods and identify their advantages and drawbacks. They also demonstrate how a new LGR method can be successfully implemented to model surface water–ground water interactions. Romero and Silver take advantage of MODFLOW’s integrated finite-difference formulation to obtain more robust solutions for distorted grids. Herrera and Valocchi investigate a well-known but difficult problem in solute transport modeling: the presence of negative concentrations in solute transport solutions arising from advection and cross-dispersion terms. They propose corrective schemes aimed at preserving positive simulated concentrations. The next two papers address model calibration. Zyvoloski and Vesselinov demonstrate that grid properties and resolution, and the computational scheme, can have large effects on forward model predictions and on inverse parameter estimates. They quantify these effects using a series of oneand two-dimensional synthetic cases representing saturated and variably saturated flow problems. Edington and Poeter explore the connection between stratigraphic processes and the character of flow systems. They conclude that geologic processes controlling the ratio of accommodation space to sediment supply have a consistent effect on hydraulic properties. This suggests that joint inversion of geologic and flow models could be useful in characterizing the magnitude and distribution of hydraulic properties important to ground water flow and transport. 1Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487; czheng@ua.edu 2International Ground Water Modeling Center, Colorado School of Mines, Golden, CO 80401; epoeter@mines.edu 3U.S. Geological Survey, Boulder, CO 80303; mchill@usgs.gov 4Watermark Numerical Computing, Corinda 4075, Australia; johndoherty@ozemail.com.au Copyright a 2006 The Author(s). Journal compilationa2006National GroundWaterAssociation. doi: 10.1111/j.1745-6584.2006.00270.x

Comparison of Local Grid Refinement Methods for MODFLOW

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.

フィージビリティスタディにおけるガスコンデンセート貯留層評価―コンデンセートバンキングの簡便な表現方法について―

A reservoir modeling using simulation software on the market is popular to evaluate production profile and has specific requirement in an each phase through a project. Generally, a simpler model is preferred to examine many cases about production scenarios for a commercial evaluation in feasibility study phase. So, it is very important how to express subsurface uncertainty in a simple reservoir model.A condensate banking phenomenon which reduces productivity near the wellbore is a typical feature of gas reservoir, especially condensate rich gas reservoir. Evaluation of the condensate banking is an important issue from a viewpoint of production profile evaluation because of requirement from market to maintain plateau of production for the project. One of popular techniques to express the condensate baking is local grid refinement (LGR) method applied around each well in the compositional simulator.However, LGR application may not be a preferable option in the feasibility study phase, because more complicated reservoir model requires more calculation time. So, PI multiplier method was introduced without LGR option to keep simplicity of the reservoir model.An investigation of the PI multiplier applied in the coarse grided full field model without LGR to take into account productivity reduction by condensate baking has been presented in this paper.

An adaptive multigrid approach for the simulation of contaminant transport in the 3D subsurface

This paper presents an adaptive multigrid approach, combining adaptive local grid refinement and multigrid methods, in conjunction with the Lagrangian–Eulerian finite element method to simulate contaminant transport in the 3D subsurface. Adaptive local grid refinement can improve solution accuracy without sacrificing computational efficiency because computer efforts are focused on the rough regions (i.e., requiring high spatial resolution) of the problem domain. To implement adaptive grids, a backward/forward particle tracking technique is applied in the Lagrangian step, and the interpolation errors of the Lagrangian concentrations are compared with prescribed error tolerances to determine rough regions. A modular setting of the grid generation is then used to generate locally zooming grids and to prepare information for applying multigrid methods. The Lagrangian concentrations of the newly generated nodes at the finest grid level are also evaluated by performing a backward tracking. Multigrid strategies which can effectively eliminate the smooth component error through coarse grid correction are finally applied in the Eulerian step to solve the matrix equations for further saving of computer time. Example problems are used to demonstrate the success of this integrated approach.

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors

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.

Modeling fractured and faulted regions: Local grid refinement methods for implicit solvers

In order to give a full 3D description of reservoir fault zones, we use a non-overlapping, non-matching local grid refinement (LGR) technique in the multi-phase, multi-component hydrocarbon migration simulator Som (Secondary Oil Migration). The resulting small spatial scales severely restrict the time steps if the mass transport equations are solved explicitly. Thus, an implicit formulation is derived where the Jacobian terms are analytical expressions based on binary mixture thermodynamics. We will present details of the implementation of this method in the LGR framework using Newton-GMRES and a multigrid-like, two-level iterative solver.

Modeling fractured and faulted regions: Local grid refinement methods for implicit solvers

Modeling fractured and faulted regions: Local grid refinement methods for implicit solvers

Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes

A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback. Results indicate: (1) The new method exhibits quadratic convergence for homogenous systems and convergence equivalent to uniform-grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse-grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%.

Local grid refinement for improved description of leaks in industrial gas safety analysis

Reliable simulation of leaks, dispersion and dilution of flammable gas mixtures are important in order to estimate the risk in industrial gas safety analysis. Computational Fluid Dynamics (CFD) based tools are routinely used in these kinds of studies. There is, however, a constant conflict between the computational resources available in a typical engineering environment and the resolution required to capture all scales of the problem, and coarse grids are typically used. This paper describes the development of a local grid refinement method aimed at improving the simulation of the region close to the leak. Aspects of risk and safety analysis are discussed and descriptions of the model and the numerical algorithm are given. Results are presented for representative model problems.

A local grid refinement method for three-dimensional turbulent recirculating flows

A local grid refinement method is presented and applied to a three-dimensional turbulent recirculating flow. It is based on the staggered grid arrangement. The computational domain is covered by block-structured subgrids of different refinement levels. The exchange of information between the subgrids is fully conservative and all grids are treated implicitly. This allows for a simultaneous solution of one variable in all grids. All variables are stored in one-dimensional arrays. The solver selected for the solution of the discretised finite difference equations is the preconditioned bi-conjugate gradient (Bi-CG) method

Efficient simulation of single species and multispecies transport in groundwater with local adaptive grid refinement

The computational burden associated with multidimensional, multicomponent, numerical solute transport models can be prohibitive. To solve these CPU intensive problems efficiently and accurately, we have implemented and extended a local adaptive grid refinement method of Berger and Oliger (1984) to track error‐prone regions and supply high‐resolution subgrids where they are locally needed while maintaining relatively few nodes elsewhere on a coarse, base grid. Novel features include a unique method for detecting a priori where the numerical error is unacceptable, variable time step control which allows smaller time steps on subgrids than on the base grid, and a modular framework which allows easy exchange of partial differential equation solvers to accommodate different problem formulations. Three examples show how the subgrids track the error‐prone regions, how multiple subgrids are used for geometrically complex front shapes, and how increased resolution is provided for self‐sharpening fronts when multiple reacting species are considered. For all three examples, solutions with accuracies comparable to those achieved with a uniform fine grid are obtained at between 20 and 30% of the computational cost.

Computing brine transport in porous media with an adaptive‐grid method

AbstractAn adaptive‐grid finite‐difference method is applied to a model for non‐isothermal, coupled flow and transport of brine in porous media. In the vicinity of rock salt formations the salt concentration in the fluid becomes large, giving rise to disparate scales in the salt concentrations profiles. A typical situation one encounters is that of a sharp freshwater‐saltwater interface that moves in time. In such situations adaptive‐grid methods are more effective than standard fixed‐grid methods, since they refine the space grid locally and, hence, provide for substantial reduction in the number of grid points, memory use and CPU time. The adaptive‐grid method of this paper is a static, local uniform grid refinement method. Its main feature is that it integrates on nested sequences of locally uniformly refined Cartesian space grids, which are automatically adjusted in time to follow rapid spatial transitions. Variable time steps are used to cope with rapid temporal transitions, including a fast march to possible steady‐state solutions. For time stepping, the implicit, second‐order BDF scheme is used. Two specific example problems are numerically illustrated. The main physical properties involved here are advection and dispersion and in case of dominant advection sharp freshwater‐saltwater interfaces arise.

フレキシブル格子を用いた高精度油層シミュレーション

In reservoir simulation, finite difference method has long been used, which requires small grid size around a well to account for rapid pressure change. With finite difference grid for Cartesian coordinate, reducing grid size around a well results in large aspect ratio at cells far from the well and in decreasing accuracy. To deal with this problem, local grid refinement method has been developed and used.Another difficulty in Cartesian coordinate is the grid orientation, which greatly affects flow of fluids because pressure gradient can be calculated only in the direction of the grid. In the case of horizontal well simulation, gridding is dominated by the direction of the horizontal well, which makes existing finite difference approach nearly impossible to simulate.One solution to these problems is to use flexible grid. We have developed “VERDI/2D”, which is a prototype of two-dimensional, three-phase reservoir simulator using flexible grid. In VERDI/2D, control-volume finite element (CVFE) method is used to discretize flow equation. Voronoi grid method is used to generate grid by overlaying sub-grids (cylindrical or Cartesian) on background Cartesian grid at any location and any angle.The validity of VERDI/2D was proved by comparing calculation results with those of widely used commercial simulator “ECLIPSE” of Intera Information Technologies Ltd.

Simulation techniques for multiphase and multicomponent flows

AbstractThe simulation of multiphase and multicomponent fluid flows often requires large coupled systems of nonlinear partial differential equations. These equations are convection‐dominated, with important local diffusive effects. An operator splitting technique is used to address these different phenomena. Adaptive local grid refinement methods are then presented to resolve the moving internal boundary layers which arise and often govern the mass transfer between the phases. Owing to the large size of many application, efficiency in computation is critical. Concepts of domain decomposition for dynamic adaptive grid refinement are presented. Techniques for dividing the problems into pieces for parallel computation on separate processors are also discussed.