On divergence- and gradient-preserving coarse-graining for finite volume primitive equation ocean models

Remote sensing technology and the resulting high resolution geospatial data now allow for a detailed description of urban landscape, advancing the development of raster-based flood models. Previous studies have highlighted the critical role of finely resolved and accurate terrain data (5m or less) in capturing flow patterns in urban areas. However, using a uniform fine grid resolution over a rectangular domain generally results in dense grids and leads to large computational costs. The small cell size is often an overspecification for rural regions where the flow processes are changing much less rapidly. Unstructured grid models resolve this issue and trade off more complex programming and slower operation against being able to represent a given problem with fewer computational elements. An alternative solution has been recently proposed to apply the non-uniform structured grid, with fine grids covering only the regions where this detail is a necessity, for example to capture the preferential flow paths influenced by small-scale topographic features or man-made structures (river channels, buildings, roads, defences, etc.). Without this, the smoothing effect of mesh coarsening upon input topographical data in urban areas leads to a uncertain prediction of the inundation extent and the timing of inundation due to the simplified wetting process. Flow connectivity formed by the river channels and the road network, which has a strong control on urban floodplain hydraulics, is also better represented by mixing grid resolutions. Considering the large consequences in terms of economic losses caused by urban flooding, here we develop a GPU-accelerated non-uniform sub-/super-grid channel model (river channels with width below or above the fine grid resolution) for accurate and efficient urban flood modelling. Urban areas and the river channel network are forced to keep fine resolution, while a coarse representation, depending on the terrain gradient, is allowed for rural regions. This model allows the utilization of available sub-/super-grid scale bathymetric information for 1D in-channel flow representation, and a 2D model for floodplain with variable grid resolution, minimising the computational costs and below water line data requirements in the river channel. Three tests are set up to validate the model performance, and the results show that modelling the urban area with fine resolution improved the model reliability and accuracy, and reduces computational cost in rural areas where a coarse grid may be used.

Flow fields in a centrifugal fan for an indoor unit of an air-conditioner were calculated with finite element method-based large eddy simulation (LES) with the aim of predicting fan performance and aerodynamic noise in this study. The numerical simulation code employed throughout the LES was called FrontFlow/blue (FFB). We compared 10M grid [coarse grid] and 60M grid [fine grid] calculation results for investigation of influence of grid resolution. In the fine grid, the number of grid elements in blade-to-blade direction, and of region between the shroud and the bell mouth increased in particular. By calculating with the fine grid, calculated distributions of absolute velocities at blade exit reasonably agreed with experimental results. Because of this, maximum absolute velocity by fine grid near hub decreased as compared to those by coarse grid. Calculated sound pressure level by fine grid was therefore smaller than that by coarse grid, and the overestimation of sound pressure was suppressed by calculating with fine grid. This decrease of the absolute velocity was a first factor for the improvement of calculation accuracy. Moreover, number of captured streaks on the blade, hub, and shroud surfaces by fine grid increased as compared to those by coarse grid. As a result, size of streak by fine grid became smaller than that by coarse grid. Static pressure fluctuations by fine grid on the blade, hub, and shroud surfaces therefore reduced as compared to those by coarse grid. Aerodynamic noise was related to static pressure fluctuations according to Curle’s equation. This reduction of static pressure fluctuations was therefore a second factor for improvement of calculation accuracy.

Running a fine grid model with 107 - 109 of cells is possible using a supercomputer with 103 - 106 of CPUs but may not be always cost-effective. The most cost-effective way is to use a coarse grid model that is much smaller but with static/dynamic profiles very close to the fine grid model. This paper proposes a new layer optimization and upscaling method with the aim for creating a consistent coarse grid model. Unlike the industry's existing layer optimization and upscaling methods, the proposed method performs layer optimization and upscaling fully integrated with the Lorenz coefficient and curves (LCC). Coarse grid layers and their permeabilities are created by minimizing the difference between fine and coarse grid LCCs. The process consists of static and dynamic optimizations. The former is measured by LCC while the latter by pressure, GOR, and water-cut. A new LCC-based permeability upscaling method is developed to preserve the fine grid multiphase flow behaviors. A satisfactory coarse grid model is achieved when both static and dynamic criteria are met. The proposed method has been successfully applied to a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within and between the matrix and fracture/karst features. The reservoir drive mechanisms are fluid expansion, miscible gas injection and aquifer drive. The reservoir is undersaturated and has an abnormally high initial reservoir pressure. The fine-grid static model contains 104 million cells (370×225×625×2) and the optimized upscaled coarse-grid dynamic model has 8.3 million cells (370×225×50×2). The upscaled model can be run efficiently on the company's existing HPC infrastructure with a maximum of 64 CPUs. Excellent matches of the Lorenz coefficient maps for reservoir total/zones and Lorenz curves at all wells between the fine and coarse grid models have been achieved. Matches on the dynamic variables, e.g., pressure, gas breakthrough time, and GOR growth, in all producers are within the defined acceptable tolerances. The high quality of the static and dynamic matches between the coarse- and fine-grid models confirms that the reservoir properties of the coarse-grid model is very close to the fine-grid model and can be used a base model for history matching and uncertainty analysis.

Hybrid steam and in-situ combustion recovery processes have shown advantages over pure steam injection for recovery of oil sands resources, particularly with respect to reducing costs and lowering requirement for water and natural gas. However, it has been very challenging to predict field performance of hybrid steam and combustion processes with a reasonable degree of confidence. Usually, a combustion front has a thickness of only a few inches and high resolution grids are required to capture the steep temperature, saturation and fluid composition gradients in the vicinity of the combustion front. Using high resolution, fine grids to improve accuracy of simulation requires excessive computation time and, therefore, may be impractical for field scale modelling. It is important to have a robust simulation tool to accurately predict reservoir performance without compromising the computational efficiency. In this work, numerical modeling of a hybrid steam and combustion recovery process was performed in a typical Athabasca oil sands reservoir. A comprehensive new reaction kinetics model derived from laboratory results was incorporated to represent the complex chemical reactions in the combustion process. The hybrid recovery process utilized oxygen enriched air co-injection after several years of SAGD operation. In the numerical model, safe limits were set on producing well temperature and oxygen content of produced fluids. The initial grid size in the numerical model was at the centimeter scale resulting in large run time, and thus, in order to improve the computational efficiency, a dynamic gridding feature was applied. Parameters for controlling the creation of a dynamic grid and subsequently reverting back to the coarse grid have been examined in order to properly trigger the dynamic gridding feature in the model. Once the optimized dynamic gridding parameters were determined, several different well configurations were investigated. Comparisons were made between SAGD and hybrid steam/combustion processes in terms of cumulative water (steam) injection, cumulative oil production, and a steam-oil ratio. By comparing the simulation results from the fine grid model and the dynamic gridding model, it has been found that the temperature gradient is the best criterion to use for controlling dynamic gridding compared to fluid saturation and/or composition criteria. The threshold value for the temperature criterion was determined to be 35°C. The model locates the fine grids in close proximity to the combustion front where the temperature and fluid saturation gradients are the steepest and it places the coarse grid blocks elsewhere in the model. Comparisons are made between the computation time and the accuracy of the simulation and these demonstrate that dynamic grid amalgamation reduces the computation time significantly while maintaining reasonable computation accuracy of simulation. Compared with SAGD, the hybrid steam/in-situ combustion process reduced cumulative water usage (steam injection) by 20% to 27%, while the cumulative oil production remained the same. This paper provides a workflow for modelling of hybrid steam and combustion processes. Also, it is expected that this work will provide insights for field design of these hybrid thermal recovery processes.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

This paper presents a multidomain (MD) computational approach to multiscale (MS) compositional reservoir simulation amenable to parallel processing. The main objective is to minimize the upscaling process and preserve the pertinent flow characteristics of the fine-scale geological features of the reservoir. The preservation of the fine-scale reservoir characteristics yields higher-accuracy solution compared to an upscaled model. The computing algorithm involves breaking down the fine-grid (FG) computing mesh into an underlying coarse-grid (CG) subdomain. Unlike other MS techniques, the CG subdomain only serves as an assemblage of the FG cells and not as a computational CG overlaying the FG cells. The flow equations are then restructured into a global pressure equation which is solved on the FG using a large time step without solving for the compositions, saturations, etc. The convergence tolerance for the global pressure solution is soft (no iterations) to reduce computing time. The pressure solution is then used to calculate the molar flux of components only at the FG boundaries of the CG subdomain cells. Finally, each CG subdomain subject to its boundary conditions is solved independently using a smaller time step to compute cell pressures, compositions, and phase saturations. This procedure is amenable to parallel processing. This MS scheme was applied to time domain, in which the global pressure solution is obtained with a larger time step than in the local FG solution. This means that the molar flux boundary condition at the CG subdomain boundaries is not updated every time step, thus, buying additional computational advantage. This new technique was tested on several volatile oil and gas-condensate systems and excellent agreement was obtained with FG simulation with about 60% reduction in computing time using a single processor. Much greater time reduction is expected in a multi processing environment.

A non-linear two-dimensional vertically stratified cross-sectional model of a constant depth basin without rotation is used to investigate the influence of vertical and horizontal diffusion upon the wind-driven circulation in the basin and the associated temperature field. The influence of horizontal grid resolution, in particular the application of an irregular grid with high resolution in the coastal boundary layer is examined. The calculations show that the initial response to a wind impulse is downwelling at the downwind end of the basin with upwelling and convective mixing at the opposite end. Results from a two-layer analytical model show that the initial response is the excitation of an infinite number of internal seiche modes in order to represent the initial response which is confined to a narrow near coastal region. As time progresses, at the downwind end of the basin a density front propagates away from the boundary, with the intensity of its horizontal gradient and associated vertical velocity determined by both horizontal and vertical viscosity values. Calculations demonstrate the importance of high horizontal grid resolution in resolving this density gradient together with an accurate density advection scheme. The application of an irregular grid in the horizontal with high grid resolution in the nearshore region enables the initial response to be accurately reproduced although physically unrealistic short waves appear as the frontal region propagates onto the coarser grid. Parameterization of horizontal viscosity using a Smagorinsky-type formulation acts as a selective grid size-dependent filter, and removes the short-wave problem although enhanced smoothing can occur if the scaling coefficient in the formulation is too large. Calculations clearly show the advantages of using an irregular grid but also the importance of using a grid size-dependent filter to avoid numerical problems.

As large-scale, parallel computers have become more widely available and numerical models and algorithms have advanced, the range of physical phenomena that can be simulated has expanded dramatically. Many important science and engineering problems exhibit solutions with localized behavior where highly detailed salient features or large gradients appear in certain regions which are separated by much larger regions where the solution is smooth. Examples include chemically reacting flows with radiative heat transfer, high Reynolds-number flows interacting with solid objects, and combustion problems where the flame front is essentially a two-dimensional sheet occupying a small part of a three-dimensional domain. Modeling such problems numerically requires approximating the governing partial differential equations on a discrete domain, or grid. Grid spacing is an important factor in determining the accuracy and cost of a computation. A fine grid may be needed to resolve key local features, while a much coarser grid may suffice elsewhere. Employing a fine grid everywhere may be inefficient at best and, at worst, may make an adequately resolved simulation impractical. Moreover, the location and resolution of fine grid required for an accurate solution is a dynamic property of a problem's transient features and may not be known a priori. Adaptive mesh refinement (AMR) is a technique that can be used with both structured and unstructured meshes to adjust local grid spacing dynamically to capture solution features with an appropriate degree of resolution. Thus, computational resources can be focused where and when they are needed most to efficiently achieve an accurate solution without incurring the cost of a globally fine grid. Figure 8.1 shows two example computations using AMR. On the left is a structured mesh calculation of an impulsively sheared contact surface [3], and on the right is the fuselage and volume discretization of an RAH-66 Comanche helicopter [37]. Note the ability of both meshing methods to resolve simulation details by varying the local grid spacing.

Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

An application of the potential added value (PAV) concept suggested by Di Luca et al. (Clim Dyn 40:443–464, 2013a) is applied to ERA Interim driven runs of the regional climate model COSMO–CLM. They are performed for the time period 1979–2013 for the EURO–CORDEX domain at horizontal grid resolutions 0.11^{circ }, 0.22^{circ }, and 0.44^{circ } such that the higher resolved model grid fits into the next coarser grid. The concept of the potential added value is applied to annual, seasonal, and monthly means of the 2 m air temperature. Results show the highest potential added value at the run with the finest grid and generally increasing PAV with increasing resolution. The potential added value strongly depends on the season as well as the region of consideration. The gain of PAV is higher enhancing the resolution from 0.44^{circ } to 0.22^{circ } than from 0.22^{circ } to 0.11^{circ }. At grid aggregations to 0.88^{circ } and 1.76^{circ } the differences in PAV between the COSMO–CLM runs on the mentioned grid resolutions are maximal. They nearly vanish at aggregations to even coarser grids. In all cases the PAV is dominated by at least 80% by its stationary part.

: To illustrate Brandt's multigrid method this report considers the Helmholtz equation in cases where the frequency is close to an eigenfrequency. In a second part the eigenvalue problem is discussed. The basic idea for the treatment of the inhomogeneous problem is contained in Brandt's publications. Difficulties are encountered because the coarse grid eigenvalues differ from those of the finer grid. As a consequence, approximations to a solution in the coarse grid may differ widely from the corresponding approximation in a finer grid, if the frequency defined by the inhomogeneous problem is close to one of the eigenfrequencies. This may be detrimental to the aim of the coarse grid step; namely to remove from an approximation in a fine grid long waves which cause divergence of fine grid iterations. This is remedied by eliminating long wave contributions to the residual directly in the next finer grid. The success of this procedure still depends upon the choice of the subspace from which the long wave corrections in the finer grid are taken. Even this modified method fails if the frequency pertaining to the inhomogeneous term is too close to an eigenvalue, unless the subspace contains the pertinent eigenfunction.

A method for performing nested grid calculations with a large-eddy simulation code is described. A common numerical method is used for all meshes, and the grid architecture consists of a single outer or coarse grid, and nested or fine grids, which overlap in some common region. Inter-grid communication matches the velocity, pressure and potential temperature fields in the overlap region. Resolved and sub-grid scale (SGS) turbulent fluxes and kinetic energy on the fine grid are averaged to the coarse grid using a conservation rule equivalent to Germano's identity used to develop dynamic SGS models. Simulations of a slightly convective, strong shear planetary boundary layer were carried out with varying surface-layer resolutions. Grid refinements in the (x, y, z) directions of up to (5, 5, 2) times were employed. Two-way interaction solutions on the coarse and fine meshes are successfully matched in the overlap region on an instantaneous basis, and the turbulent motions on the fine grid blend smoothly into the coarse grid across the grid interface. With surface-layer grid nesting, significant increases in resolved eddy fluxes and variances are found. The energy-scale content of the vertical velocity, and hence vertical turbulent fluxes, appear to be most influenced by increased grid resolution. Vertical velocity spectra show that the dominant scale shifts towards higher wavenumbers (smaller scales) and the magnitude of the peak energy is increased by more than a factor of 3 with finer resolution. Outside of the nested region the average heat and momentum fluxes and spectra are slightly influenced by the fine resolution in the surface layer. From these results we conclude that fine resolution is required to resolve the details of the turbulent motions in the surface layer. At the same time, however, increased resolution in the surface layer does not appreciably alter the ensemble statistics of the resolved and SGS motions outside of the nested region.

In this paper, we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions. We approach this from the perspective of phase error, which allows us to exploit the hyperbolic nature of the underlying PDE. In particular, we consider three combinations of coarse grid operators and coarse grid solution approaches: (1) inexact coarse grid solve with direct discretization, (2) exact coarse grid solve with direct discretization, and (3) exact coarse grid solve with Galerkin coarse grid operator. For all these approaches, we show that the convergence behavior of multigrid can be precisely described by the phase error analysis ofthe coarse grid correction matrix, and we verify our results by numerical examples in one and two dimensions.

Modelling a pike (Esox lucius) population in a lowland river using a cellular automaton

A hybrid LES-RANS study on square cylinder unsteady heat transfer