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Evazi, M., and H. Mahani. "Unstructured-Coarse-Grid Generation Using Background-Grid Approach." SPE Journal 15, no. 02 (March 3, 2010): 326–40. http://dx.doi.org/10.2118/120170-pa.
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Summary Reservoir flow simulation involves subdivision of the physical domain into a number of gridblocks. This is best accomplished with optimized gridpoint density and a minimized number of gridblocks, especially for coarse-grid generation from a fine-grid geological model. In any coarse-grid generation, proper distribution of gridpoints, which form the basis of numerical gridblocks, is a challenging task. We show that this can be achieved effectively by a novel grid-generation approach based on a background grid that stores gridpoint spacing parameters. Spacing parameter (L) can be described by Poisson's equation (∇2L = G), where the local density of gridpoints is controlled by a variable source term (G); see Eq. 1. This source term can be based on different gridpoint density indicators, such as permeability variations, fluid velocity, or their combination (e.g., vorticity) where they can be extracted from the reference fine grid. Once a background grid is generated, advancing-front triangulation (AFT) and then Delaunay tessellation are invoked to form the final (coarse) gridblocks. The algorithm produces grids varying smoothly from high- to low-density gridpoints, thus minimizing use of grid-smoothing and -optimization techniques. This algorithm is quite flexible, allowing choice of the gridding indicator, hence providing the possibility of comparing the grids generated with different indicators and selecting the best. In this paper, the capabilities of approach in generation of unstructured coarse grids from fine geological models are illustrated using 2D highly heterogeneous test cases. Flexibility of algorithm to gridding indicator is demonstrated using vorticity, permeability variation, and velocity. Quality of the coarse grids is evaluated by comparing their two-phase-flow simulation results to those of fine grid and uniform coarse grid. Results demonstrate the robustness and attractiveness of the approach, as well as relative quality/performance of grids generated by using different indicators.
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Krogstad, S., V. L. L. Hauge, and A. F. F. Gulbransen. "Adjoint Multiscale Mixed Finite Elements." SPE Journal 16, no. 01 (August 23, 2010): 162–71. http://dx.doi.org/10.2118/119112-pa.
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Summary We develop an adjoint model for a simulator consisting of a multiscale pressure solver and a saturation solver that works on flow-adapted grids. The multiscale method solves the pressure on a coarse grid that is close to uniform in index space and incorporates fine-grid effects through numerically computed basis functions. The transport solver works on a coarse grid adapted by a fine-grid velocity field obtained by the multiscale solver. Both the multiscale solver for pressure and the flow-based coarsening approach for transport have shown earlier the ability to produce accurate results for a high degree of coarsening. We present results for a complex realistic model to demonstrate that control settings based on optimization of our multiscale flow-based model closely match or even outperform those found by using a fine-grid model. For additional speed-up, we develop mappings used for rapid system updates during the timestepping procedure. As a result, no fine-grid quantities are required during simulations and all fine-grid computations (multiscale basis functions, generation of coarse transport grid, and coarse mappings) become a preprocessing step. The combined methodology enables optimization of waterflooding on a complex model with 45,000 grid cells in a few minutes.
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Li, Jun, and Donald Brown. "Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media." Geofluids 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/1740693.
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An upscaled Lattice Boltzmann Method (LBM) for flow simulations in heterogeneous porous media at the Darcy scale is proposed in this paper. In the Darcy-scale simulations, the Shan-Chen force model is used to simplify the algorithm. The proposed upscaled LBM uses coarser grids to represent the average effects of the fine-grid simulations. In the upscaled LBM, each coarse grid represents a subdomain of the fine-grid discretization and the effective permeability with the reduced-order models is proposed as we coarsen the grid. The effective permeability is computed using solutions of local problems (e.g., by performing local LBM simulations on the fine grids using the original permeability distribution) and used on the coarse grids in the upscaled simulations. The upscaled LBM that can reduce the computational cost of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with averaged results obtained using a fine grid.
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Du, Shouhong, Larry S. Fung, and Ali H. Dogru. "Aquifer Acceleration in Parallel Implicit Field-Scale Reservoir Simulation." SPE Journal 23, no. 02 (February 12, 2018): 614–24. http://dx.doi.org/10.2118/182686-pa.
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Summary Grid coarsening outside of the areas of interest is a common method to reduce computational cost in reservoir simulation. Aquifer regions are candidates for grid coarsening. In this situation, upscaling is applied to the fine grid to generate coarse-grid flow properties. The efficacy of the approach can be judged easily by comparing the simulation results between the coarse-grid model and the fine-grid model. For many reservoirs in the Middle East bordered by active aquifers, transient water influx is an important recovery mechanism that needs to be modeled correctly. Our experience has shown that the standard grid coarsening and upscaling method do not produce correct results in this situation. Therefore, the objective of this work is to build a method that retains the fine-scale heterogeneities to accurately represent the water movement, but to significantly reduce the computational cost of the aquifer grids in the model. The new method can be viewed as a modified two-level multigrid (MTL-MG) or a specialized adaptation of the multiscale method. It makes use of the vertical-equilibrium (VE) concept in the fine-scale pressure reconstruction in which it is applicable. The method differs from the standard grid coarsening and upscaling method in which the coarse-grid properties are computed a priori. Instead, the fine-scale information is restricted to the coarse grid during Newton's iteration to represent the fine-scale flow behavior. Within the aquifer regions, each column of fine cells is coarsened vertically based on fine-scale z-transmissibility. A coarsened column may consist of a single amalgamated aquifer cell or multiple vertically disconnected aquifer cells separated by flow barriers. The pore volume (PV), compressibility, and lateral flow terms of the coarse cell are restricted from the fine-grid cells. The lateral connectivity within the aquifer regions and the one between the aquifer and the reservoir are honored, inclusive of the fine-scale description of faults, pinchouts, and null cells. Reservoir regions are not coarsened. Two alternatives exist for the fine-scale pressure reconstruction from the coarse-grid solution. The first method uses the VE concept. When VE applies, pressure variation can be analytically computed in the solution update step. Otherwise, the second method is to apply a 1D z-line solve for the fine-scale aquifer pressure from the coarse-grid solution. Simulation results for several examples are included to demonstrate the efficacy and efficiency of the method. We have applied the method to several Saudi Arabian complex full-field simulation models in which the transient aquifer water influx has been identified as a key factor. These models include dual-porosity/dual-permeability (DPDP) models, as well as models with faults and pinchouts in corner-point-geometry grids, for both history match and prediction period. The method is flexible and allows for the optional selection of aquifer regions to be coarsened, either only peripheral aquifers or both the peripheral and bottom aquifers. The new method gives nearly identical results compared with the original runs without coarsening, but with significant reduction in computer time or hardware cost. These results will be detailed in the paper.
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Gibson, Richard L., Kai Gao, Eric Chung, and Yalchin Efendiev. "Multiscale modeling of acoustic wave propagation in 2D media." GEOPHYSICS 79, no. 2 (March 1, 2014): T61—T75. http://dx.doi.org/10.1190/geo2012-0208.1.
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Conventional finite-difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finite-element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine- and coarse-scale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the fine-scale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the fine-scale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, fine-scale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.
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Wu, Gruorong, and Jie Ouyang. "Use of Precise Area Fraction Model for Fine Grid DEM Simulation of ICFB with Large Particles." Symmetry 12, no. 3 (March 4, 2020): 399. http://dx.doi.org/10.3390/sym12030399.
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The heterogeneous structures in a gas–solid fluidized bed can be resolved in discrete element simulation so long as the grid is fine enough. In order to conveniently calculate mean porosity in fine grid simulations, a precise area fraction model is given for two-dimensional simulations. The proposed area fraction model is validated by the discrete element simulation test on a small-scale internal circulation fluidized system of large particles, using a fine grid size of two particle diameters. Simulations show that the discrete element method can perform well in modelling time-varying waveforms for the physical quantities in an internal circulating fluidized bed, employing the precise gas area fraction model. This thought of precise calculation can be generalized to construct a volume fraction porosity model for three-dimensional simulation by use of the similar symmetry of a rectangular grid. Moreover, to construct these area and volume fraction models is to enrich and perfect the underlying model of fine grid simulation.
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Chen, Chuanjun, Wei Liu, and Xin Zhao. "A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/803615.
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We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).
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KAWASAKI, Akira, Mitsuo OBATA, and Heihachi SHIMADA. "AN IMAGE-PROCESSING SYSTEM FOR FINE-GRID METHOD." Nondestructive Testing Communications 4, no. 2-3 (January 1988): 49. http://dx.doi.org/10.1080/02780898808962106.
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Tobias, M. E. "Grid for CT-guided percutaneous fine-needle aspiration." American Journal of Roentgenology 158, no. 2 (February 1992): 459–60. http://dx.doi.org/10.2214/ajr.158.2.1729811.
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Erturk, E., and O. Gokcol. "Fine grid numerical solutions of triangular cavity flow." European Physical Journal Applied Physics 38, no. 1 (March 21, 2007): 97–105. http://dx.doi.org/10.1051/epjap:2007057.
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Bischoff‐Kim, A., M. H. Montgomery, and D. E. Winget. "Fine Grid Asteroseismology of G117‐B15A and R548." Astrophysical Journal 675, no. 2 (March 10, 2008): 1505–11. http://dx.doi.org/10.1086/527287.
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Chen, Chuanjun, and Wei Liu. "A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/391918.
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A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.
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Wang, Chen, Hong Ai, Lie Wu, and Yun Yang. "A Fine-Grained Access Control Model for Smart Grid." Applied Mechanics and Materials 513-517 (February 2014): 772–76. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.772.
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The smart grid that the next-generation electric power system is studied intensively as a promising solution for energy crisis. One important feature of the smart grid is the integration of high-speed, reliable and secure data communication networks to manage the complex power systems effectively and intelligently. The goal of smart grid is to achieve the security of operation, economic efficient and environmental friendly. To achieve this goal, we proposed a fine-grained access control model for smart grid. In order to improve the security of smart grid, an access-trust-degree algorithm is proposed to evaluate the reliability of the user who want to access to the smart grid.
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Ruangtrakoon, Natthawut, and Eakarach Bumrungthaichaichan. "Influence of grid distribution on CFD model of compressible flow inside the primary nozzle and mixing chamber used in refrigeration application." MATEC Web of Conferences 192 (2018): 02045. http://dx.doi.org/10.1051/matecconf/201819202045.
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In this study, the influence of grid distribution on CFD model of the primary nozzle and mixing chamber used in refrigeration application was primarily investigated. The only one geometry of primary nozzle and mixing chamber was modeled. The two different grid distributions, fine near-wall grid and regular grid with the identical total grid number, were simulated to investigate the flow phenomena inside the considered system. The appropriate boundary conditions and numerical methods were carefully employed. The simulated entrainment ratios obtained by two different grid arrangements were validated by comparing with the reliable experimental data. The results revealed that the Mach number distributions of these models were different. Further, the outlet total pressure predicted by fine near-wall grid was about 1.3% higher than that obtained by regular grid.
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Li, Hangyu, Jeroen C. Vink, and Faruk O. Alpak. "A Dual-Grid Method for the Upscaling of Solid-Based Thermal Reactive Flow, With Application to the In-Situ Conversion Process." SPE Journal 21, no. 06 (June 2, 2016): 2097–111. http://dx.doi.org/10.2118/173248-pa.
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Summary Thermal-reactive compositional-flow simulation in porous media is essential to model thermal-oil-recovery processes for extraheavy-hydrocarbon resources, and an example is the in-situ conversion process (ICP) developed by Shell for oil-shale production. Computational costs can be very high for such a complex system, which makes direct fine-scale simulations prohibitively time-consuming for large field-scale applications. This motivates the use of coarse grids for thermal-reactive compositional-flow simulation. However, significant errors are introduced by use of coarse-scale models without carefully computing the appropriate coarse parameters. In this paper, we develop an innovative dual-grid method to effectively capture the fine-scale reaction rates in coarse-scale ICP-simulation models. In our dual-grid method, coupled thermal-reactive compositional-flow equations are solved only on the coarse scale, with the kinetic parameters (frequency factors) calculated on the basis of fine-scale computations, such as temperature downscaling and fine-scale reaction-rate calculation. A dual-grid treatment for the heater-well model is also developed with coarse-scale heater-well indices calculated on the basis of fine-scale well results. The dual-grid heater-well treatment is able to provide accurate heater temperatures. The newly developed dual-grid method is applied to realistic cross-sectional ICP-pattern models with a vertical production well and multiple horizontal heater wells operated subject to fixed and time-varying heater powers. It is shown that the dual-grid model delivers results that are in close agreement with the fine-scale reference results for all quantities of interest. Despite the fact that the dual-grid method is implemented at the simulation-deck level, by use of the flexible scripting and monitor functionalities of our proprietary simulation package, significant computational improvements are achieved for all cases considered.
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Karim, Md Masud, Navin Bindra, and Mohammed Mukteruzzaman. "Effects of Fine Grid NO2 Modeling around a Thermal Power Station: A Case Study." Current Environmental Management 7, no. 1 (January 11, 2021): 67–79. http://dx.doi.org/10.2174/2666214007999200826110202.
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Objective: The objective of this study was to assess the outcome of fine grid modeling on top of a courser grid to predict the NO2 concentration more accurately in an airshed and at sensitive receptors. Methods: This study assessed the cumulative NO2 impact of all major emission sources in an airshed using USEPA regulatory model AERMOD. A 50 km by 50 km airshed is considered in the study. Micro-environmental pollution was modeled using a refined grid analysis. An area (4 km x 4 km) close to GTPS, with receptors at every 150 m apart and a further fine grid (500 m x 500 m) with receptors at every 50 m apart, was modeled. Results: Coarse grid modeling showed annual average NO2 concentration levels within applicable standards. However, when fine grid modeling was conducted, the predicted annual average NO2 concentration levels were found to exceed World Bank Guidelines and Bangladesh Standards. A source contribution analysis showed that a quick rental power plant (natural gas generators) without proper stacks contributed a significant portion of the maximum 1-hr and annual average NO2 concentration (76% and 86%, respectively). Conclusion: The findings of fine grid modeling can be used at the policy level of the government to enforce environmental regulations on the minimum height requirements of stacks and city planning, avoiding downwind directions and the close proximity of powerplants to safeguard human health.
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Kornev, Nikolai, Jordan Denev, and Sina Samarbakhsh. "Theoretical Background of the Hybrid VπLES Method for Flows with Variable Transport Properties." Fluids 5, no. 2 (April 10, 2020): 45. http://dx.doi.org/10.3390/fluids5020045.
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The paper presents the theoretical basis for the extension of the V π LES method, originally developed in recent works of the authors for incompressible flows, to flows with variable density and transport properties but without chemical reactions. The method is based on the combination of grid based and grid free computational particle techniques. Large scale motions are modelled on the grid whereas the fine scale ones are modelled by particles. The particles represent the fine scale vorticity, and scalar quantities like e.g., temperature, mass fractions of species, density and mixture fraction. Coupled system of equations is derived for large and fine scales transport.
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Tsanis, Ioannis K., and Jian Wu. "A Nested-Grid Hydrodynamic/Pollutant Transport Model for Nearshore Areas in Hamilton Harbour." Water Quality Research Journal 30, no. 2 (May 1, 1995): 205–30. http://dx.doi.org/10.2166/wqrj.1995.022.
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Abstract A nested-grid depth-averaged circulation model was developed and applied to three nearshore areas in Hamilton Harbour: the western basin, LaSalle Park waterfront and the northeastern shoreline. The grid sizes used were 100 m for the whole harbour, and 25 m for the three nearshore areas. General features of current circulation and horizontal mixing times under various wind directions and speeds were obtained for the whole harbour using the coarse-grid model. The fine-grid model (water elevations and current information on the open boundaries were obtained from the whole harbour model) then provided current patterns which were used to drive the pollutant transport model. Simulation results reveal that the current in the fine-grid model is close to the current from the coarse-grid model, while more detailed current structures are explored. The water elevations from the fine-grid model agree well with the elevations from the coarse-grid one. The impact of artificial islands was examined by studying changes in current patterns, pollutant peaks, exposure and flushing time in different locations of concern. The design proposed provides: (i) minimum change in the existing current patterns; (ii) avoidance of pollutant hot spots; and (iii) minimum changes in the flushing time of pollutants.
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von Clarmann, T. "Smoothing error pitfalls." Atmospheric Measurement Techniques 7, no. 9 (September 18, 2014): 3023–34. http://dx.doi.org/10.5194/amt-7-3023-2014.
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Abstract. The difference due to the content of a priori information between a constrained retrieval and the true atmospheric state is usually represented by a diagnostic quantity called smoothing error. In this paper it is shown that, regardless of the usefulness of the smoothing error as a diagnostic tool in its own right, the concept of the smoothing error as a component of the retrieval error budget is questionable because it is not compliant with Gaussian error propagation. The reason for this is that the smoothing error does not represent the expected deviation of the retrieval from the true state but the expected deviation of the retrieval from the atmospheric state sampled on an arbitrary grid, which is itself a smoothed representation of the true state; in other words, to characterize the full loss of information with respect to the true atmosphere, the effect of the representation of the atmospheric state on a finite grid also needs to be considered. The idea of a sufficiently fine sampling of this reference atmospheric state is problematic because atmospheric variability occurs on all scales, implying that there is no limit beyond which the sampling is fine enough. Even the idealization of infinitesimally fine sampling of the reference state does not help, because the smoothing error is applied to quantities which are only defined in a statistical sense, which implies that a finite volume of sufficient spatial extent is needed to meaningfully discuss temperature or concentration. Smoothing differences, however, which play a role when measurements are compared, are still a useful quantity if the covariance matrix involved has been evaluated on the comparison grid rather than resulting from interpolation and if the averaging kernel matrices have been evaluated on a grid fine enough to capture all atmospheric variations that the instruments are sensitive to. This is, under the assumptions stated, because the undefined component of the smoothing error, which is the effect of smoothing implied by the finite grid on which the measurements are compared, cancels out when the difference is calculated. If the effect of a retrieval constraint is to be diagnosed on a grid finer than the native grid of the retrieval by means of the smoothing error, the latter must be evaluated directly on the fine grid, using an ensemble covariance matrix which includes all variability on the fine grid. Ideally, the averaging kernels needed should be calculated directly on the finer grid, but if the grid of the original averaging kernels allows for representation of all the structures the instrument is sensitive to, then their interpolation can be an adequate approximation.
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Liao, Yeqing, and Quan Liu. "Research on Fine-grained Job scheduling in Grid Computing." International Journal of Information Engineering and Electronic Business 1, no. 1 (October 18, 2009): 9–16. http://dx.doi.org/10.5815/ijieeb.2009.01.02.
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Schoch, K. F. "BALL GRID ARRAY AND FINE PITCH PERIPHERAL INTERCONNECTS [Books]." IEEE Electrical Insulation Magazine 12, no. 4 (July 1996): 43. http://dx.doi.org/10.1109/mei.1996.526951.
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Kuroda, K., S. Sillou, and F. Takeutchi. "Comments on some publications concerning fine grid dynode PMTs." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 274, no. 1-2 (January 1989): 409–10. http://dx.doi.org/10.1016/0168-9002(89)90410-5.
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Alekseev, Valentin, Qili Tang, Maria Vasilyeva, Eric T. Chung, and Yalchin Efendiev. "Mixed Generalized Multiscale Finite Element Method for a Simplified Magnetohydrodynamics Problem in Perforated Domains." Computation 8, no. 2 (June 23, 2020): 58. http://dx.doi.org/10.3390/computation8020058.
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In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions.
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Mlacnik, Martin, Louis J. Durlofsky, and Zoltan E. Heinemann. "Sequentially Adapted Flow-Based PEBI Grids for Reservoir Simulation." SPE Journal 11, no. 03 (September 1, 2006): 317–27. http://dx.doi.org/10.2118/90009-pa.
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Summary A technique for the sequential generation of perpendicular-bisectional (PEBI) grids adapted to flow information is presented and applied. The procedure includes a fine-scale flow solution, the generation of an initial streamline-isopotential grid, grid optimization, and upscaling. The grid optimization is accomplished through application of a hybrid procedure with gradient and Laplacian smoothing steps, while the upscaling is based on a global-local procedure that makes use of the global solution used in the grid-determination step. The overall procedure is successfully applied to a complex channelized reservoir model involving changing well conditions. The gridding and upscaling procedures presented here may also be suitable for use with other types of structured or unstructured grid systems. Introduction Modern geological and geostatistical tools provide highly detailed descriptions of the spatial variation of reservoir properties, resulting in fine-grid models consisting of 107 to 108 gridblocks. As a consequence of this high level of detail, these models cannot be used directly in numerical reservoir simulators, but need to be coarsened significantly. Coarsening requires the averaging of rock parameters from the fine scale to the coarse scale. This process is referred to as upscaling. For simulation of flow in porous media, the upscaling of permeability is of particular interest. A large body of literature exists on this topic; for a comprehensive review of existing techniques, see Durlofsky (2005). To preserve as much of the geological information of the fine grid as possible, the grid coarsening should not be performed uniformly, but with more refinement in areas that are expected to have large impact on the flow, including structural features, such as faults. Although grid-generation techniques based on purely static, nonflow-based considerations have been shown to produce reasonable results(Garcia et al. 1992), the application of flow-based grids is often preferable. Flow-based grids require the solution of some type of fine-scale problem. They are then constructed by exploiting the information obtained from streamlines (and possibly isopotentials) either directly or indirectly. Depending on the type of grid used, points will be defined as cell vertices or nodes, resulting in either a corner-point geometry or point-distributed grid. Several gridding techniques for reservoir simulation have been introduced along these lines, as we now discuss.
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Darman, N. H., L. J. Durlofsky, K. S. Sorbie, and G. E. Pickup. "Upscaling Immiscible Gas Displacements: Quantitative Use of Fine-Grid Flow Data in Grid-Coarsening Schemes." SPE Journal 6, no. 01 (March 1, 2001): 47–56. http://dx.doi.org/10.2118/69674-pa.
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Xia, Chao, Yi Du Yang, and Hai Bi. "A Two-Grid Discretization Scheme for a Sort of Steklov Eigenvalue Problem." Advanced Materials Research 557-559 (July 2012): 2087–91. http://dx.doi.org/10.4028/www.scientific.net/amr.557-559.2087.
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On the basis of Yang and Bi’s work (SIAM J Numer Anal 49, p.1602-1624), this paper discusses a discretization scheme for a sort of Steklov eigenvalue problem and proves the high effiency of the scheme. With the scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. And the resulting solution can maintain an asymptotically optimal accuracy. Finally, the numerical results are provided to support the theoretical analysis..
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Mozgaleva, Marina L. "TWO-STAGE GRID METHOD OF SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF DISCRETE HAAR BASIS." International Journal for Computational Civil and Structural Engineering 13, no. 1 (March 22, 2017): 69–85. http://dx.doi.org/10.22337/2587-9618-2017-13-1-69-85.
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The distinctive paper is devoted to development of two-stage numerical method. At the first stage, the discrete problem is solved on a coarse grid, where the number of nodes in each direction is the same and is a pow-er of 2. Then the number of nodes in each direction is doubled and the resulting solution on a coarse grid using a discrete Haar basis is defined at the nodes of the fine grid as the initial approximation. At the second stage, we ob-tain a solution in the nodes of the fine grid using the most appropriate iterative method,. Test examples of the solu-tion of one-dimensional, two-dimensional and three-dimensional boundary problems are under consideration
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Cho, Yongchae, and Richard L. Gibson, Jr. "Reverse time migration via frequency-adaptive multiscale spatial grids." GEOPHYSICS 84, no. 2 (March 1, 2019): S41—S55. http://dx.doi.org/10.1190/geo2018-0292.1.
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Reverse time migration (RTM) is widely used because of its ability to recover complex geologic structures. However, RTM also has a drawback in that it requires significant computational cost. In RTM, wave modeling accounts for the largest part of the computing cost for calculating forward- and backward-propagated wavefields before applying an imaging condition. For this reason, we have applied a frequency-adaptive multiscale spatial grid to enhance the efficiency of the wave simulations. To implement wave modeling for different values of the spatial grid interval, we apply a model reduction technique, the generalized multiscale finite-element method (GMsFEM), which solves local spectral problems on a fine grid to simulate wave propagation on a coarser grid. We can enhance the speed of computation without sacrificing accuracy by using coarser grids for lower frequency waves, while applying a finer grid for higher frequency waves. In the proposed method, we can control the size of the coarse grid and level of heterogeneity of the wave solutions to tune the trade-off between speedup and accuracy. As we increase the expected level of complexity of the wave solutions, the GMsFEM wave modeling can capture more detailed features of waves. After computing the forward and backward wavefield on the coarse grid, we reproject the coarse wave solutions to the fine grid to construct the RTM gradient image. Although wave solutions are computed on a coarse grid, we still obtain the RTM images without reducing the image resolution by projecting coarse wave solutions to the fine grid. We determine the efficiency of the proposed imaging method using the Marmousi-2 model. We compare the RTM images using GMsFEM with a fixed coarse mesh and a multiple frequency-adaptive coarse meshes to indicate the image quality and computational speed of the new approach.
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Li, Dongze, Xiang Li, Yongqiang Cheng, Yuliang Qin, and Hongqiang Wang. "Radar Coincidence Imaging under Grid Mismatch." ISRN Signal Processing 2014 (April 22, 2014): 1–8. http://dx.doi.org/10.1155/2014/987803.
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Radar coincidence imaging is an instantaneous imaging technique which does not depend on the relative motion between targets and radars. High-resolution, fine-quality images can be obtained using a single pulse either for stationary targets or for complexly maneuvering ones. There are two image-reconstruction algorithms used for radar coincidence imaging, that is, the correlation method and the parameterized method. In comparison with the former, the parameterized method can achieve much higher resolution but is seriously sensitive to grid mismatch. In the presence of grid mismatch, neither of the two algorithms can obtain recognizable high-resolution images. The above problem largely limits the applicability of radar coincidence imaging in actual imaging scenes where grid mismatch generally exists. This paper proposes a joint correlation-parameterization algorithm, which uses the correlation method to estimate the grid-mismatch error and then iteratively modifies the results of the parameterized method. The proposed algorithm can achieve high resolution with fine imagery quality under the grid mismatch. Examples are provided to illustrate the improvement of the proposed method.
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Lauritzen, Peter H., and Ramachandran D. Nair. "Monotone and Conservative Cascade Remapping between Spherical Grids (CaRS): Regular Latitude–Longitude and Cubed-Sphere Grids." Monthly Weather Review 136, no. 4 (April 1, 2008): 1416–32. http://dx.doi.org/10.1175/2007mwr2181.1.
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Abstract A high-order monotone and conservative cascade remapping algorithm between spherical grids (CaRS) is developed. This algorithm is specifically designed to remap between the cubed-sphere and regular latitude–longitude grids. The remapping approach is based on the conservative cascade method in which a two-dimensional remapping problem is split into two one-dimensional problems. This allows for easy implementation of high-order subgrid-cell reconstructions as well as the application of advanced monotone filters. The accuracy of CaRS is assessed by remapping analytic fields from the regular latitude–longitude grid to the cubed-sphere grid. In terms of standard error measures, CaRS is found to be competitive relative to an existing algorithm when regridding from a fine to a coarse grid and more accurate when regridding from a coarse to a fine grid.
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Marchand, Roger, and Thomas Ackerman. "A Cloud-Resolving Model with an Adaptive Vertical Grid for Boundary Layer Clouds." Journal of the Atmospheric Sciences 68, no. 5 (May 1, 2011): 1058–74. http://dx.doi.org/10.1175/2010jas3638.1.
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Abstract Accurate cloud-resolving model simulations of cloud cover and cloud water content for boundary layer clouds are difficult to achieve without vertical grid spacing well below 100 m, especially for inversion-topped stratocumulus. The need for fine vertical grid spacing presents a significant impediment to global or large regional simulations using cloud-resolving models, including the Multiscale Modeling Framework (MMF), in which a two-dimensional or small three-dimensional cloud-resolving model is embedded into each grid cell of a global climate model in place of more traditional cloud parameterizations. One potential solution to this problem is to use a model with an adaptive vertical grid (i.e., a model that is able to add vertical layers where and when needed) rather than trying to use a fixed grid with fine vertical spacing throughout the boundary layer. This article examines simulations with an adaptive vertical grid for three well-studied stratocumulus cases based on observations from the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) experiment, the Atlantic Stratocumulus Transition Experiment (ASTEX), and the Atlantic Trade Cumulus Experiment (ATEX). For each case, three criteria are examined for determining where to add or remove vertical layers. One criterion is based on the domain-averaged potential temperature profile; the other two are based on the ratio of the estimated subgrid-scale to total water flux and turbulent kinetic energy. The results of the adaptive vertical grid simulations are encouraging in that these simulations are able to produce results similar to simulations using fine vertical grid spacing throughout the boundary layer, while using many fewer vertical layers.
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Mittet, Rune. "On the internal interfaces in finite-difference schemes." GEOPHYSICS 82, no. 4 (July 1, 2017): T159—T182. http://dx.doi.org/10.1190/geo2016-0477.1.
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Implementing sharp internal interfaces in finite-difference schemes with high spatial accuracy is challenging. The propagation of fields in a locally homogeneous part of a model can be performed with spectral accuracy. The implementations of interfaces are generally considered accurate to, at best, second order. This situation can be improved by a proper band limitation of the simulation grid. Interfaces can be located anywhere on the grid; however, the fine detail information regarding the interface location must be imprinted correctly in the coarse simulation grid. This can be done by starting out with a representation of a sharp material jump in the wavenumber domain and limiting the highest wavenumber to the maximum wavenumber allowed for the simulation grid. The resulting wavenumber representation is then transformed to the space domain. An alternative procedure is to create a fine grid model that is low-pass filtered to remove wavenumbers above the maximum wavenumber allowed for the coarse simulation grid. The fine grid is thereafter sampled at the required coordinates for the coarse simulation grid. An accurate and flexible interface implementation is a requisite for reducing staircase diffractions in higher dimensional finite-difference simulations. Our strategy achieves this. The frequency content of the source must be constrained to a level in which the spatial sampling is at approximately four to five grid points per shortest wavelength. Simulation results indicate that the implementation of the interface is accurate to at least the sixth order for large contrasts. Our method can be used for all systems of partial differential equations that formally can be expressed as a material parameter times a dynamic field on one side of the equal sign and with spatial derivatives on the other side of the equal sign. For geophysical simulations, the most important cases will be the Maxwell equations and the acoustic and elastic wave equations.
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Chen, Yanping, and Hanzhang Hu. "Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics." Communications in Computational Physics 19, no. 5 (May 2016): 1503–28. http://dx.doi.org/10.4208/cicp.scpde14.46s.
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AbstractThe miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.
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Ibrahim, Maizura, Hamidah Ibrahim, Azizol Abdullah, and Rohaya Latip. "ENHANCING THE PERFORMANCE OF ADVANCED FINE-GRAINED GRID AUTHORIZATION SYSTEM." Journal of Computer Science 10, no. 12 (December 1, 2014): 2576–83. http://dx.doi.org/10.3844/jcssp.2014.2576.2583.
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Ryu, Ji Song, Se Hoon Oh, Kyung Ju Min, and Jong Wook Ahn. "Fine Dust Information Visualize Technique Utilized 3D Geospatial Grid System." Journal of Korean Society for Geospatial Information Science 27, no. 6 (November 30, 2019): 35–42. http://dx.doi.org/10.7319/kogsis.2019.27.6.035.
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Jung, Hyunjoon, Hyuck Han, Hyungsoo Jung, and Heon Y. Yeom. "Dynamic and fine-grained authentication and authorisation for grid computing." International Journal of Critical Infrastructures 4, no. 3 (2008): 308. http://dx.doi.org/10.1504/ijcis.2008.017443.
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Wrzesińska, Gosia, Rob V. van Nieuwpoort, Jason Maassen, Thilo Kielmann, and Henri E. Bal. "Fault-Tolerant Scheduling of Fine-Grained Tasks in Grid Environments." International Journal of High Performance Computing Applications 20, no. 1 (February 2006): 103–14. http://dx.doi.org/10.1177/1094342006062528.
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Fan, Na, Lian-Feng Zhao, Xiao-Bi Xie, and Zhen-Xing Yao. "A discontinuous-grid finite-difference scheme for frequency-domain 2D scalar wave modeling." GEOPHYSICS 83, no. 4 (July 1, 2018): T235—T244. http://dx.doi.org/10.1190/geo2017-0535.1.
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The discontinuous-grid method can greatly reduce the storage requirement and computational cost in finite-difference modeling, especially for models with large velocity contrasts. However, this technique is mostly applied to time-domain methods. We have developed a discontinuous-grid finite-difference scheme for frequency-domain 2D scalar wave modeling. Special frequency-domain finite-difference stencils are designed in the fine-coarse grid transition zone. The coarse-to-fine-grid spacing ratio is restricted to [Formula: see text], where [Formula: see text] is a positive integer. Optimization equations are formulated to obtain expansion coefficients for irregular stencils in the transition zone. The proposed method works well when teamed with commonly used 9- and 25-point schemes. Compared with the conventional frequency-domain finite-difference method, the proposed discontinuous-grid method can largely reduce the size of the impedance matrix and number of nonzero elements. Numerical experiments demonstrated that the proposed discontinuous-grid scheme can significantly reduce memory and computational costs, while still yielding almost identical results compared with those from conventional uniform-grid simulations. When tested for a very long elapsed time, the frequency-domain discontinuous-grid method does not show instability problems as its time-domain counterpart usually does.
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Karimi-Fard, M., and L. J. J. Durlofsky. "Accurate Resolution of Near-Well Effects in Upscaled Models Using Flow-Based Unstructured Local Grid Refinement." SPE Journal 17, no. 04 (November 29, 2012): 1084–95. http://dx.doi.org/10.2118/141675-pa.
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Summary We present a new approach for representing wells in coarse-scale reservoir simulation models. The technique is based on an expanded well model concept which provides a systematic procedure for the construction of the near-well grid. The method proceeds by first defining an underlying fine-scale model, in which the well and any key near-well features such as hydraulic fractures are fully resolved using an unstructured grid. In the (coarse) simulation model, the geometry of the grid in the expanded well region, and the associated "radial" transmissibilities, are determined from the solution of a fine-scale, single-phase, well-driven flow problem. The coarse-scale transmissibilities outside of the well region are computed using existing local upscaling techniques or by applying a new global upscaling procedure. Thus, through use of near-well flow-based gridding and generalized local grid refinement, this methodology efficiently incorporates the advantages of highly-resolved unstructured grid representations of wells into coarse models. The overall model provided by this technique is compatible with any reservoir simulator that allows general unstructured cell-to-cell connections (model capabilities, in terms of flow physics, are defined by the simulator). The expanded well modeling approach is applied to challenging 3D problems involving injection and production in a low-permeability heterogeneous reservoir, tight-gas production by a hydraulically-fractured well, and production in a gas-condensate reservoir. In the first two cases, where it is possible to simulate the fine-grid unstructured model, results using the expanded well model closely match the reference solutions, while standard approaches lead to significant error. In the gas-condensate example, which involves a nine-component compositional model, the reference solution is not computed, but the solution using the expanded well model is shown to be physically reasonable while standard coarse-grid solutions show large variation under grid refinement.
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Huq, Sadiq, Frederik De Roo, Siegfried Raasch, and Matthias Mauder. "Vertically nested LES for high-resolution simulation of the surface layer in PALM (version 5.0)." Geoscientific Model Development 12, no. 6 (June 28, 2019): 2523–38. http://dx.doi.org/10.5194/gmd-12-2523-2019.
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Abstract. Large-eddy simulation (LES) has become a well-established tool in the atmospheric boundary layer research community to study turbulence. It allows three-dimensional realizations of the turbulent fields, which large-scale models and most experimental studies cannot yield. To resolve the largest eddies in the mixed layer, a moderate grid resolution in the range of 10 to 100 m is often sufficient, and these simulations can be run on a computing cluster with a few hundred processors or even on a workstation for simple configurations. The desired resolution is usually limited by the computational resources. However, to compare with tower measurements of turbulence and exchange fluxes in the surface layer, a much higher resolution is required. In spite of the growth in computational power, a high-resolution LES of the surface layer is often not feasible: to fully resolve the energy-containing eddies near the surface, a grid spacing of O(1 m) is required. One way to tackle this problem is to employ a vertical grid nesting technique, in which the surface is simulated at the necessary fine grid resolution, and it is coupled with a standard, coarse, LES that resolves the turbulence in the whole boundary layer. We modified the LES model PALM (Parallelized Large-eddy simulation Model) and implemented a two-way nesting technique, with coupling in both directions between the coarse and the fine grid. The coupling algorithm has to ensure correct boundary conditions for the fine grid. Our nesting algorithm is realized by modifying the standard third-order Runge–Kutta time stepping to allow communication of data between the two grids. The two grids are concurrently advanced in time while ensuring that the sum of resolved and sub-grid-scale kinetic energy is conserved. We design a validation test and show that the temporally averaged profiles from the fine grid agree well compared to the reference simulation with high resolution in the entire domain. The overall performance and scalability of the nesting algorithm is found to be satisfactory. Our nesting results in more than 80 % savings in computational power for 5 times higher resolution in each direction in the surface layer.
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Shutts, Glenn, Thomas Allen, and Judith Berner. "Stochastic parametrization of multiscale processes using a dual-grid approach." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1875 (April 29, 2008): 2623–39. http://dx.doi.org/10.1098/rsta.2008.0035.
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Some speculative proposals are made for extending current stochastic sub-gridscale parametrization methods using the techniques adopted from the field of computer graphics and flow visualization. The idea is to emulate sub-filter-scale physical process organization and time evolution on a fine grid and couple the implied coarse-grained tendencies with a forecast model. A two-way interaction is envisaged so that fine-grid physics (e.g. deep convective clouds) responds to forecast model fields. The fine-grid model may be as simple as a two-dimensional cellular automaton or as computationally demanding as a cloud-resolving model similar to the coupling strategy envisaged in ‘super-parametrization’. Computer codes used in computer games and visualization software illustrate the potential for cheap but realistic simulation where emphasis is placed on algorithmic stability and visual realism rather than pointwise accuracy in a predictive sense. In an ensemble prediction context, a computationally cheap technique would be essential and some possibilities are outlined. An idealized proof-of-concept simulation is described, which highlights technical problems such as the nature of the coupling.
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Zhou, Jie, Long Chen, Yunqing Huang, and Wansheng Wang. "An Efficient Two-Grid Scheme for the Cahn-Hilliard Equation." Communications in Computational Physics 17, no. 1 (November 28, 2014): 127–45. http://dx.doi.org/10.4208/cicp.231213.100714a.
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AbstractA two-grid method for solving the Cahn-Hilliard equation is proposed in this paper. This two-grid method consists of two steps. First, solve the Cahn-Hilliard equation with an implicit mixed finite element method on a coarse grid. Second, solve two Poisson equations using multigrid methods on a fine grid. This two-grid method can also be combined with local mesh refinement to further improve the efficiency. Numerical results including two and three dimensional cases with linear or quadratic elements show that this two-grid method can speed up the existing mixed finite method while keeping the same convergence rate.
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Christie, M. A., and M. J. Blunt. "Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques." SPE Reservoir Evaluation & Engineering 4, no. 04 (August 1, 2001): 308–17. http://dx.doi.org/10.2118/72469-pa.
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Summary This paper presents the results of the 10th SPE Comparative Solution Project on Upscaling. Two problems were chosen. The first problem was a small 2D gas-injection problem, chosen so that the fine grid could be computed easily and both upscaling and pseudoization methods could be used. The second problem was a waterflood of a large geostatistical model, chosen so that it was hard (though not impossible) to compute the true fine-grid solution. Nine participants provided results for one or both problems. Introduction The SPE Comparative Solution Projects provide a vehicle for independent comparison of methods and a recognized suite of test data sets for specific problems. The previous nine comparative solution projects1–9 have focused on black-oil, compositional, dual-porosity, thermal, or miscible simulations, as well as horizontal wells and gridding techniques. The aim of the 10th Comparative Solution Project was to compare upgridding and upscaling approaches for two problems. Full details of the project, and data files available for downloading, can be found on the project's Web site.10 The first problem was a simple, 2,000-cell 2D vertical cross section. The specified tasks were to apply upscaling or pseudoization methods and to obtain solutions for a specified coarse grid and a coarse grid selected by the participant. The second problem was a 3D waterflood of a 1.1-million-cell geostatistical model. This model was chosen to be sufficiently detailed so that it would be hard, though not impossible, to run the fine-grid solution and use classical pseudoization methods. We will not review the large number of upscaling approaches here. For a detailed description of these methods, see any of the reviews of upscaling and pseudoization techniques, such as Refs. 11 through 14. Description of Problems Model 1. The model is a two-phase (oil and gas) model that has a simple 2D vertical cross-sectional geometry with no dipping or faults. The dimensions of the model are 2,500 ft long×25 ft wide×50 ft thick. The fine-scale grid is 100×1×20, with uniform size for each of the gridblocks. The top of the model is at 0.0 ft, with initial pressure at this point of 100 psia. Initially, the model is fully saturated with oil (no connate water). Full details are provided in Appendix A. The permeability distribution is a correlated, geostatistically generated field, shown in Fig. 1. The fluids are assumed to be incompressible and immiscible. The fine-grid relative permeabilities are shown in Fig. 2. Residual oil saturation was 0.2, and critical gas saturation was 0. Capillary pressure was assumed to be negligible in this case. Gas was injected from an injector located at the left of the model, and dead oil was produced from a well to the right of the model. Both wells have a well internal diameter of 1.0 ft and are completed vertically throughout the model. The injection rate was set to give a frontal velocity of 1 ft/D (about 0.3 m/d or 6.97 m3/d), and the producer is set to produce at a constant bottomhole pressure limit of 95 psia. The reference depth for the bottomhole pressure is at 0.0 ft (top of the model). The specified tasks were to apply an upscaling or pseudoization method in the following scenarios.2D: 2D uniform 5×1×5 coarse-grid model.2D: 2D nonuniform coarsening, maximum 100 cells. Directional pseudorelative permeabilities were allowed if necessary. Model 2. This model has a sufficiently fine grid to make the use of any method that relies on having the full fine-grid solution almost impossible. The model has a simple geometry, with no top structure or faults. The reason for this choice is to provide maximum flexibility in the selection of upscaled grids. At the fine geological model scale, the model is described on a regular Cartesian grid. The model dimensions are 1,200×2,200×170 ft. The top 70 ft (35 layers) represent the Tarbert formation, and the bottom 100 ft (50 layers) represent Upper Ness. The fine-scale cell size is 20×10×2 ft. The fine-scale model has 60×220×85 cells (1.122×106 cells). The porosity distribution is shown in Fig 3. The model consists of part of a Brent sequence. The model was originally generated for use in the PUNQ project.15 The vertical permeability of the model was altered from the original; originally, the model had a uniform kV/kH across the whole domain. The model used here has a kV/kH of 0.3 in the channels and a kV/kH of 10–3 in the background. The top part of the model is a Tarbert formation and is a representation of a prograding near-shore environment. The lower part (Upper Ness) is fluvial. Full details are provided in Appendix B. Participants and Methods Chevron. Results were submitted for Model 2 using CHEARS, Chevron's in-house reservoir simulator. They used the parallel version and the serial version for the fine-grid model and the serial version for the scaled-up model. Coats Engineering Inc. Runs were submitted for both Model 1 and Model 2. The simulation results were generated with SENSOR. GeoQuest. A solution was submitted for Model 2 only, with coarse-grid runs performed using ECLIPSE 100. The full fine-grid model was run using FRONTSIM, a streamline simulator,16 to check the accuracy of the upscaling. The coarse-grid models were constructed with FloGrid, a gridding and upscaling application. Landmark. Landmark submitted entries for both Model 1 and Model 2 using the VIP simulator. The fine grid for Model 2 was run with parallel VIP. Phillips Petroleum. Solutions were submitted for both Model 1 and Model 2. The simulator used was SENSOR. Roxar. Entries were submitted for both Model 1 and Model 2. The simulation results presented were generated with the black-oil implicit simulator Nextwell. The upscaled grid properties were generated using RMS, specifically the RMSsimgrid option. Streamsim. Streamsim submitted an entry for Model 2 only. Simulations were run with 3DSL, a streamline-based simulator.17 TotalFinaElf. TotalFinaElf submitted a solution for Model 2 only. The simulator used for the results presented was ECLIPSE; results were checked with the streamline code 3DSL. U. of New South Wales. The U. of New South Wales submitted results for Model 1 only, using CMG's IMEX simulator.
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Mohd Sakri, Fadhilah, Mohamed Sukri Mat Ali, and Sheikh Ahmad Zaki. "Benchmark on the Dynamics of Liquid Draining Inside a Tank." E3S Web of Conferences 95 (2019): 02009. http://dx.doi.org/10.1051/e3sconf/20199502009.
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Immense information and details observation of flow physics inside a draining tank can be achieved by adopting reliable numerical simulations. Yet the accuracy of numerical results has been always debatable and it is mainly affected by the grid convergence error and computational modeling approaches. Hence, this study is divided into two stages. In the first stage, this paper determines a systematic method of refining a computational grid for a liquid draining inside a tank using OpenFOAM software. The sensitivity of the computed flow field on different mesh resolutions is also examined. In order to study the effect of grid dependency, three different grid refinements are investigated: fine, medium and coarse grids. By using a form of Richardson extrapolation and Grid Convergence Index (GCI), the level of grid independence is attained. In this paper, a monotonic convergence criteria is reached when the fine grid has the GCI value below 10% for each parameter. In the second stage, different computational modeling approaches (DNS, RANS k-ε, RANS k-ω and LES turbulence models) are investigated using the finer grid from the first stage. The results for the draining time and flow visualization of the generation of an air-core are in a good agreement with the available published data. The Direct Numerical Simulation (DNS) seems most reasonably satisfactory for VOF studies relating air-core compared to other different turbulence modeling approaches.
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Goodfriend, Elijah, Fotini Katopodes Chow, Marcos Vanella, and Elias Balaras. "Improving Large-Eddy Simulation of Neutral Boundary Layer Flow across Grid Interfaces." Monthly Weather Review 143, no. 8 (August 1, 2015): 3310–26. http://dx.doi.org/10.1175/mwr-d-14-00392.1.
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Abstract Increasing computational power has enabled grid resolutions that support large-eddy simulation (LES) of the atmospheric boundary layer. These simulations often use grid nesting or adaptive mesh refinement to refine the grid in regions of interest. LES generates errors at grid refinement interfaces, such as resolved energy accumulation, that may compromise solution accuracy. In this paper, the authors test the ability of two LES formulations and turbulence closures to mitigate errors associated with the use of LES on nonuniform grids for a half-channel approximation to a neutral atmospheric boundary layer simulation. Idealized simulations are used to examine flow across coarse–fine and fine–coarse interfaces, as would occur in a two-way nested configuration or with block structured adaptive mesh refinement. Specifically, explicit filtering of the advection term and the mixed model are compared to a standard LES formulation with an eddy viscosity model. Errors due to grid interfaces are evaluated by comparison to uniform grid solutions. It is found that explicitly filtering the advection term provides significant benefits, in that it allows both mass and momentum to be conserved across grid refinement interfaces. The mixed model reduces unphysical perturbations generated by wave reflection at the interfaces. These results suggest that the choice of LES formulation and turbulence closure can be used to help control grid refinement interface errors in atmospheric boundary layer simulations.
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Spiridonov, Denis, Maria Vasilyeva, Eric T. Chung, Yalchin Efendiev, and Raghavendra Jana. "Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography." Mathematics 8, no. 6 (June 3, 2020): 904. http://dx.doi.org/10.3390/math8060904.
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In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.
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Ruan, Chunlei. "Multiscale Numerical Study of 3D Polymer Crystallization during Cooling Stage." Mathematical Problems in Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/802420.
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We aim to study the behavior of polymer crystallization during cooling stage in injection molding more accurately, the multiscale model and multiscale algorithm proposed in our previous work (Ruan et al., 2012) have been extended to the 3D polymer crystallization case. Our multiscale model incorporates two distinct length scales: a coarse grid for the heat diffusion and a fine grid for the crystal morphology evolution (nucleation, growth, and impingement). Our multiscale algorithm couples the different methods on different length scales, namely, the finite volume method (FVM) on the coarse grid and the pixel coloring method on the fine grid. By using these multiscale model and multiscale algorithm, simulations for 3D polymer crystallization are carried out. Macroscopic variables, for example, temperature, relative crystallinity, as well as the microscopic structural characters, for example, crystal morphology development, and mean size of spherulites, are investigated at various cooling conditions. We also show the importance of coupling heat transfer with crystallization as well as 3D numerical studies.
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Hui, Mun-Hong (Robin), Mohammad Karimi-Fard, Bradley Mallison, and Louis J. Durlofsky. "A General Modeling Framework for Simulating Complex Recovery Processes in Fractured Reservoirs at Different Resolutions." SPE Journal 23, no. 02 (January 8, 2018): 598–613. http://dx.doi.org/10.2118/182621-pa.
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Summary A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, either at the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral-gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed with a general flow-based procedure. Numerical results are presented for waterflood, sour-gas injection, and gas-condensate primary production for fracture models with matrix and fracture heterogeneities. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that with our methodology, two orders of magnitude of speedup can typically be achieved with models that introduce less than approximately 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.
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Si, Weijian, Fuhong Zeng, Changbo Hou, and Zhanli Peng. "A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays." Sensors 18, no. 9 (September 10, 2018): 3025. http://dx.doi.org/10.3390/s18093025.
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Recently, many sparse-based direction-of-arrival (DOA) estimation methods for coprime arrays have become popular for their excellent detection performance. However, these methods often suffer from grid mismatch problem due to the discretization of the potential angle space, which will cause DOA estimation performance degradation when the target is off-grid. To this end, we proposed a sparse-based off-grid DOA estimation method for coprime arrays in this paper, which includes two parts: coarse estimation process and fine estimation process. In the coarse estimation process, the grid points closest to the true DOAs, named coarse DOAs, are derived by solving an optimization problem, which is constructed according to the statistical property of the vectorized covariance matrix estimation error. Meanwhile, we eliminate the unknown noise variance effectively through a linear transformation. Due to finite snapshots effect, some undesirable correlation terms between signal and noise vectors exist in the sample covariance matrix. In the fine estimation process, we therefore remove the undesirable correlation terms from the sample covariance matrix first, and then utilize a two-step iterative method to update the grid biases. Combining the coarse DOAs with the grid biases, the final DOAs can be obtained. In the end, simulation results verify the effectiveness of the proposed method.
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Zhang, Xuejuan, Lei Zhang, Dandan Wang, Kuo Lan, Xuesong Zhou, Hongyu Yu, Ruhao Liu, and Xueying Lv. "Nonuniform grid upscaling method for geologic model of oil reservoir: A case study of the W block in the northern part of the Songliao Basin." Interpretation 9, no. 2 (April 7, 2021): T443—T452. http://dx.doi.org/10.1190/int-2020-0112.1.
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At present, uniform upscaling division methods are routinely used to upscale geologic model grids, resulting in overly fine grids in some areas of the model. To improve computational efficiency, we have examined the effect of model upscaling with different upscaling parameters with the goal of producing a nonuniform grid with uniform accuracy. We based our nonuniform upscaling grid method on geologic characteristics including reservoir thickness, physical properties, reservoir spacing, and water flooding. Most of the logging curves of thin reservoirs are finger-like, allowing us to define the grid size according to the reservoir thickness. We use two different strategies to discretize uniform and composite reservoirs and represent reservoir thickness that exhibit bell- and funnel-shaped logging curves. Although one grid point accurately represents a uniform reservoir, we find that composite reservoirs require four or five points to accurately represent the physical properties of a composite reservoir. For the thick reservoirs (>5 m) with box- or composite-type logging curves, the physical properties inside the reservoir do not change much; therefore, we use a grid point to represent the reservoir thickness information. Using these metrics, we constructed alternative “moderate” and “efficient” vertical grid upscaling strategies. Taking the 15 sedimentary units with a total thickness of 72 m as an example, the statistical results show that the computational efficiency using our data-adaptive grid can be increased more than five times compared to the traditional uniform fine-grid method while retaining the same accuracy.