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

A local grid refinement method is presented and applied to a three‐dimensional turbulent recirculating flow. It is based on the staggered grid arrangement. The computational domain is covered by block‐structured subgrids of different refinement levels. The exchange of information between the subgrids is fully conservative and all grids are treated implicitly. This allows for a simultaneous solution of one variable in all grids. All variables are stored in one‐dimensional arrays. The solver selected for the solution of the discretised finite difference equations is the preconditioned bi‐conjugate gradient (Bi‐CG) method. For the case examined (turbulent flow around a surface‐mounted cube), it was found that the latter method converges faster than the line solver. The locally refined mesh improved the accuracy of the pressure distribution on cube faces compared with a coarse mesh and yielded the same results as a fine single mesh, with a 62% gain in computer time. Copyright © 1999 John Wiley & Sons, Ltd.     

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

A family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2 :259–290; Comput. Geo. 2002; 6 :433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51 :1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve:

• (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice.
• (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution.
• (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution.

The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.     

含泥质夹层油藏蒸汽注采过程的自适应网格算法研究

根据泥质夹层的低渗特性及空间分布，本文提出了一种含泥质夹层油藏网格渗透率的粗化计算方法，并在此基础上，将自适应网格算法应用于含泥质夹层油藏的数值模拟，提升其计算效率．在计算过程中，网格的动态划分仅依据流体物理量的变化，泥质夹层区域不全部采用细网格，仅针对流动锋面处的泥质夹层采用细网格，其余泥质夹层处采用不同程度的粗网格．相较于传统算法，网格数大幅下降．数值算例表明，自适应网格算法的计算结果精度与全精细网格一致，能够准确模拟出泥质夹层对于流体的阻碍作用，同时计算效率得到大幅提升，约为全精细网格算法的3~7 倍．     

Flexible Unstructured Gridding for 2D Reservoir Coarse Scale Modeling using Fine Scale Permeability Filtering

Accurate up-scaling is an essential part of creating a valid reservoir coarse scale dynamic model. In this article, unstructured discretization of spatial domain is accompanied by numerical permeability up-scaling in order to construct an accurate coarse scale model. A new technique for generating a course scale triangular mesh is presented in which the density of elements in key flow regions is kept high to capture accuracy. The fine scale permeability map is investigated using image processing techniques, especially steerable filters, and the results are converted into a high-resolution element size map. This element size map will be refined by the integration of other important factors such as well-position effects and used to construct a coarse triangular mesh. The combination of flux-continuous pressure approximation and mass conservative, total variation diminishing finite volume schemes have been considered to solve two phase flow equations on the control volume finite element mesh. Fine scale simulations results are compared with the coarse scale ones for a series of water flooding examples to investigate the efficiency and accuracy of the presented gridding methodology. This method is developed for 2D cases, but can be easily extended to 3D problems.     

Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations

We present a novel unstructured coarse grid generation technique based on vorticity for upscaling two-phase flow in permeable media. In the technique, the fineness of the gridblocks throughout the domain is determined by vorticity distribution such that where the larger is the vorticity at a region, the finer are the gridblocks at that region. Vorticity is obtained from single-phase flow on original fine grid, and is utilized to generate a background grid which stores spacing parameter, and is used to steer generation of triangular and finally Voronoi grids. This technique is applied to two channelized and heterogeneous models and two-phase flow simulations are performed on the generated coarse grids and, the results are compared with the ones of fine scale grid and uniformly gridded coarse models. The results show a close match of unstructured coarse grid flow results with those of fine grid, and substantial accuracy compared to uniformly gridded coarse grid model.     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

Upscaling and Simulation of Waterflooding in Heterogeneous Reservoirs Using Wavelet Transformations: Application to the SPE-10 Model

We have developed an accurate and highly efficient method for upscaling and simulation of immiscible displacements in three-dimensional (3D) heterogeneous reservoirs, which is an extension of the technique that we developed previously for 2D systems. The method utilizes wavelet transformations (WTs) to upscale the geological model of a reservoir, based on the spatial distribution of the single-phase permeabilities and the locations of the wells in the reservoir. It generates a non-uniform grid in which the resolved structure of the fine grid around the wells, as well as in the high-permeability sectors, are preserved, but the rest of the grid is upscaled. A robust uplayering procedure is used to reduce the number of the layers, and the WTs are used to upscale each layer areally. To demonstrate the method’s accuracy and efficiency, we have applied it to the geological model of a highly heterogeneous reservoir put forward in the tenth Society of Petroleum Engineers comparative solution project (the SPE-10 model), and carried out simulation of waterflooding in the upscaled model. Various upscaling scenarios were examined, and although some of them resulted in efficient simulations and accurate predictions, the results when non-uniform upscaling is used based on the WT technique are in excellent agreement with the solution of the same problem in the fine grid of the SPE-10 model. Most importantly, the speed-up factors that we obtain are several orders of magnitude. Hence, the method renders it unnecessary to use massively parallel computations for such problems.     

Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution

An improved vorticity-based gridding technique is presented and applied to create optimal non-uniform Cartesian coarse grid for numerical simulation of two-phase flow. The optimal coarse grid distribution (OCGD) is obtained in a manner to capture variations in both permeability and fluid velocity of the fine grid using a single physical quantity called “vorticity”. Only single-phase flow simulation on the fine grid is required to extract the vorticity. Based on the fine-scale vorticity information, several coarse grid models are generated for a given fine grid model. Then the vorticity map preservation error is used to predict how well each coarse grid model reproduces the fine-scale simulation results. The coarse grid model which best preserves the fine-scale vorticity, i.e. has the minimum vorticity map preservation error is recognized as an OCGD. The performance of vorticity-based optimal coarse grid is evaluated for two highly heterogeneous 2D formations. It is also shown that two-phase flow parameters such as mobility ratio have only minor impact on the performance of the predicted OCGD.     

Application of Effective Flux Boundary Conditions to Two-Phase Upscaling in Porous Media

A new algorithm is introduced for upscaling relative permeabilities, and tested in simulations of two-dimensional reservoir displacement processes. The algorithm is similar to existing algorithms for computing upscaled relative permeabilities from subgrid simulations, but uses new boundary conditions for the pressure field. The new 'effective flux boundary conditions' were introduced in a previous paper and provide a more accurate estimate of flux through high permeability channels. The algorithm was tested in conjunction with uniform grid coarsening and upscaled absolute permeabilities for a broad range of coarsenings. The permeability fields were highly heteroge-neous and layered, and were obtained from synthetic data and from conditioned realizations of actual oil reservoirs. The algorithm was tested for a wide variety of grid aspect ratios, and for both viscous-and gravity-dominated flow. Typical fine grids were of the order of 100×100 cells; the coarsest scaled-up grids were on the order of 5×5 cells. The quality of scale up was evaluated by comparing oil cut curves for the fine and coarse grid simulations. We consistently obtained excellent agreement, even at the coarsest levels of scale up.     

Parallel solution of lifting rotors in hover and forward flight

An implicit unsteady, multiblock, multigrid, upwind solver including mesh deformation capability, and structured multiblock grid generator, are presented and applied to lifting rotors in both hover and forward flight. To allow the use of very fine meshes and, hence, better representation of the flow physics, a parallel version of the code has been developed. It is demonstrated that once the grid density is sufficient to capture enough turns of the tip vortices, hover exhibits oscillatory behaviour of the wake, even using a steady formulation. An unsteady simulation is then presented, and detailed analysis of the time‐accurate wake history is performed and compared to theoretical predictions. Forward flight simulations are also presented and, again, grid density effects on the wake formation investigated. Parallel performance of the code using up to 1024 CPU's is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

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Solution of shallow water equations for complex flow domains via boundary-fitted co-ordinates

Many problems of applied oceanography and environmental science demand the solution of the momentum, mass and energy equations on physical domains having curving coastlines. Finite-difference calculations representing the boundary as a step function may give inaccurate results near the coastline where simulation results are of greatest interest for numerous applications. This suggests the use of methods which are capable of handling the problem of boundary curvature. This paper presents computational results for the shallow water equations on a circular ring of constant depth, employing the concept of boundary fitted grids (BFG) for an accurate representation of the boundary. All calculations are performed on a rectangle in the transformed plane using a mesh with square grid spacing. Comparisons of the simulations of transient normal mode oscillations and analytic solutions are shown, demonstrating that this technique yields accurate results in situations (provided that there is a reasonable choice of grid) involving a curved boundary. The software developed allows application to any two-dimensional area, regardless of the complexity of the geometry. Simulation runs were made with two co-ordinate systems. For the first system, the grid point distribution was obtained from polar co-ordinates. For the second one, grid point positions were calculated numerically, solving Poisson's equation. It was found that small variations in the metric coefficients do not deteriorate the accuracy of the simulation results. Moreover, comparisons of surface elevation and velocity components at grid points near the inner and outer radii obtained from an x?y Cartesian grid model with the BFG simulation were made. The former model produced inacccuracies at grid points near boundaries, and, owing to the large number of mesh points used to yield the necessary fine resolution, the computation time was found to be a factor of three higher.     

Seismic propagation simulation in complex media with non-rectangular irregular-grid finite-difference

This paper presents a finite-difference (FD) method with spatially non-rectangular irregular grids to simulate the elastic wave propagation. Staggered irregular grid finite difference operators with a second-order time and spatial accuracy are used to approximate the velocity-stress elastic wave equations. This method is very simple and the cost of computing time is not much. Complicated geometries like curved thin layers, cased borehole and nonplanar interfaces may be treated with nonrectangular irregular grids in a more flexible way. Unlike the multi-grid scheme, this method requires no interpolation between the fine and coarse grids and all grids are computed at the same spatial iteration. Compared with the rectangular irregular grid FD, the spurious diffractions from “staircase” interfaces can easily be eliminated without using finer grids. Dispersion and stability conditions of the proposed method can be established in a similar form as for the rectangular irregular grid scheme. The Higdon‘s absorbing boundary condition is adopted to eliminate boundary reflections. Numerical simulations show that this method has satisfactory stability and accuracy in simulating wave propagation near rough solid-fluid interfaces. The computation costs are less than those using a regular grid and rectangular grid FD method.     

Large-eddy Simulation of Triangular-stabilized Lean Premixed Turbulent Flames: Quality and Error Assessment

In this numerical study, an algebraic flame surface wrinkling (AFSW) reaction submodel based on the progress variable approach is implemented in the large-eddy simulation (LES) context and validated against the triangular stabilized bluff body flame configuration measurements i.e. in VOLVO test rig. The quantitative predictability of the AFSW model is analyzed in comparison with another well validated turbulent flame speed closure (TFC) combustion model in order to help assess the behaviour of the present model and to further help improve the understanding of the flow and flame dynamics. Characterization of non-reacting (or cold) and reacting flows are performed using various subgrid scale models for consistent grid size variation with 300,000 (coarse), 1.2 million (intermediate) and 2.4 million (fine) grid cells. For non-reacting flows at inlet velocity of 17?m/s and inlet temperature 288?K, coarse grid leads to over prediction of turbulence quantities due to low dissipation at the early stage of flow development behind the bluff body that convects downstream eventually polluting the resulting solution. The simulated results with the intermediate (and fine) grid for mean flow and turbulence quantities, and the vortex shedding frequency (f_s) closely match experimental data. For combusting flows for lean propane/air mixtures at 35?m/s and 600?K, the vortex shedding frequency increase threefold compared with cold scenario. The predicted results of mean, rms velocities and reaction progress variable are generally in good agreement with experimental data. For the coarse grid the combustion predictions show a shorter recirculation region due to higher turbulent burning rate. Finally, both cold and reacting LES data are analyzed for uncertainty in the solution using two quality assessment techniques: two-grid estimator by Celik, and model and grid variation by Klein. For both approaches, the resolved turbulent kinetic energy is used to estimate the grid quality and error assessment. The quality assessment reveals that the cold flows are well resolved even on the intermediate mesh, while for the reacting flows even the fine mesh is locally not sufficient in the flamelet region. The Klein approach estimates that depending on the recirculation region in cold scenario both numerical and model errors rise near the bluff-body region, while in combusting flows these errors are significant behind the stabilizing point due to preheating of unburned mixture and reaction heat release. The total error mainly depends on the numerical error and the influence of model error is low for this configuration.     

An unstructured grid Reynolds stress model for separated turbulent flow simulations

Most of the turbulence models in the literature contain simplified assumptions which make them computationally inexpensive but of limited accuracy for the solution of separated turbulent flows. Dramatic improvements in computer processing speed and parallel processing make it possible to use more complete models, such as Reynolds Stress Models, for separated turbulent flow simulations, which is the focus of this work. The Reynolds Stress Model consists of coupling the Reynolds transport equations with the Favre–Reynolds averaged Navier–Stokes equations, which results in a system of 12 coupled non-linear partial differential equations. The solutions are obtained by running the PUMA_RSM computational fluid dynamics code on unstructured meshes. The equations are solved all the way to the wall without using any wall functions. Results for high Reynolds number flow around a 6:1 prolate spheroid and a Bell 214ST fuselage are presented. For the prolate spheroid basic flow features such as cross-flow separation are simulated. Predictions of circumferential locations of cross flow separation points are in good agreement with the experiment. A grid refinement study is performed to improve the computations. The fine mesh solution predicted locations of primary and secondary separation points with errors of roughly 2° and 0°, respectively. Flow simulations around an isolated Bell 214ST helicopter fuselage were also performed. Predicted pressure and drag force correlate well with the wind tunnel data, with a less than 10% deviation from the experiment. Drag predictions also show relative speed of Reynolds Stress Model compared to Large Eddy Simulation to compute time averaged quantities. For numerical solutions parallel processing is applied with the MPI communication standard. The code used in this study is run on Beowulf clusters. The parallel performance of the code PUMA_RSM is analysed and presented.     

The two‐grid stabilization of equal‐order finite elements for the stokes equations

This work presents a two‐grid stabilized method of equal‐order finite elements for the Stokes problems. This method only offsets the discrete pressure space by the residual of pressure on two grids to circumvent the discrete Babu?ka–Brezzi condition. The method can be done locally in a two‐grid approach without stabilization parameter by projecting the pressure onto a finite element space based on coarse mesh. Also, it leads to a linear system with minimal additional cost in implement. Optimal error estimates are obtained. Finally, some numerical simulations are presented to show stability and accuracy properties of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.