A Comparison Study Between an Adaptive Quadtree Grid and Uniform Grid Upscaling for Reservoir Simulation

An adaptive quadtree grid generation algorithm is developed and applied for tracer and multiphase flow in channelized heterogeneous porous media. Adaptivity was guided using two different approaches. In the first approach, wavelet transformation was used to generate a refinement field based on permeability variations. The second approach uses flow information based on the solution of an initial-time fine-scale problem. The resulting grids were compared with uniform grid upscaling. For uniform upscaling, two commonly applied methods were used: renormalization upscaling and local-global upscaling. The velocities obtained by adaptive grid and uniformly upscaled grids, were downscaled. This procedure allows us to separate the upscaling errors, on adaptive and uniform grids, from the numerical dispersion errors resulting from solving the saturation equation on a coarse grid. The simulation results obtained by solving on flow-based adaptive quadtree grids for the case of a single phase flow show reasonable agreement with more computationally demanding fine-scale models and local-global upscaled models. For the multiphase case, the agreement is less evident, especially in piston-like displacement cases with sharp frontal movement. In such cases a non-iterative transmissibility upscaling procedure for adaptive grid is shown to significantly reduce the errors and make the adaptive grid comparable to iterative local-global upscaling. Furthermore, existence of barriers in a porous medium complicates both upscaling and grid adaptivity. This issue is addressed by adapting the grid using a combination of flow information and a permeability based heuristic criterion.     

Upscaling of Channel Systems in Two Dimensions Using Flow-Based Grids

A methodology for the gridding and upscaling of geological systems characterized by channeling is presented. The overall approach entails the use of a flow-based gridding procedure for the generation of variably refined grids capable of resolving the channel geometry, a specialized full-tensor upscaling method to capture the effects of permeability connectivity, and the use of a flux-continuous finite volume method applicable to full tensor permeability fields and non-orthogonal grids. The gridding and upscaling procedures are described in detail and then applied to several two-dimensional systems. Significant improvement in the accuracy of the coarse scale models, relative to that obtained using uniform Cartesian coarse scale models, is achieved in all cases. It is shown that, for some systems, improvement results from the use of the flow-based grid, while in other cases the improvement is mainly due to the new upscaling method.     

Numerical simulation of the interaction of a transverse jet with a supersonic flow using different turbulence models

This paper presents a numerical simulation of the flow resulting from transverse jet injection into a supersonic flow through a slot nozzle at different pressures in the injected jet and the crossflow. Calculations on grids with different resolutions use the Spalart–Allmaras turbulence model, the k–ε model, the k–ω model, and the SST model. Based on a comparison of the calculated and experimental data on the wall pressure distribution, the length of the recirculation area, and the depth of jet penetration into the supersonic flow, conclusions are made on the accuracy of the calculation results for the different turbulence models and the applicability of these models to similar problems.     

Study of the Spectral Difference Numerical Dissipation for Turbulent Flows Using Unstructured Grids

In this paper, the numerical dissipation properties of the Spectral Difference (SD) method are studied in the context of vortex dominated flows and wall-bounded turbulence, using uniform and distorted grids. First, the validity of using the SD numerical dissipation as the only source of subgrid dissipation (the so-called Implicit-LES approach) is assessed on regular grids using various polynomial degrees (namely, p = 3, p = 4, p = 5) for the Taylor-Green vortex flow configuration at R e = 5 000. It is shown that the levels of numerical dissipation greatly depend on the order of accuracy chosen and, in turn, lead to an incorrect estimation of the viscous dissipation levels. The influence of grid distortion on the numerical dissipation is then assessed in the context of finite Reynolds number freely-decaying and wall-bounded turbulence. Tests involving different amplitudes of distortion show that highly skewed grids lead to the presence of small-scale, noisy structures, emphasizing the need of explicit subgrid modeling or regularization procedures when considering coarse, high-order SD computations on unstructured grids. Under-resolved, high-order computations of the turbulent channel flow at R e _τ = 1000 using highly-skewed grids are considered as well and present a qualitatively similar agreement to results obtained on a regular grid.     

Dual-Porosity Coarse-Scale Modeling and Simulation of Highly Heterogeneous Geomodels

Accurate upscaling of highly heterogeneous subsurface reservoirs remains a challenge in the context of modeling of flow and transport. In this work, we address this challenge with emphasis on the representation of the displacement efficiency in coarse-scale modeling. We propose a dual-porosity upscaling approach to handle displacement calculations in high resolution and highly heterogeneous formations. In this approach, the pore space is arranged into two levels of porosity based on flow contribution, and a dual-porosity dual-permeability flow model is adapted for coarse-scale flow simulation. The approach uses fine-scale streamline information to transform a heterogeneous geomodel into a coarse dual-continuum model that preserves the global flow pathways adequately. The performance of the proposed technique is demonstrated for two heterogeneous reservoirs using both black oil (waterflooding) and compositional (gas injection) modeling approaches. We demonstrate that the coarse dual-porosity models predict the breakthrough times accurately and reproduce the post-breakthrough responses adequately. This is in contrast to conventional single-porosity upscaling techniques that overestimate breakthrough times and displacement efficiencies (sweep). By preserving large-scale heterogeneities, coarse dual-porosity models are demonstrated to be significantly less sensitive to the level of upscaling, when compared to conventional single-porosity upscaling. Accordingly, the proposed upscaling approach is a relevant and suitable technique for upscaling of highly heterogeneous geomodels.     

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks’ centers is then used with an optimized Delauney triangulation method and the advancing-front technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.     

Modeling of Two-Phase Flow in Fractured Porous Media on Unstructured Non-Uniformly Coarsened Grids

In this article, we investigate two strategies for coarsening fractured geological models. The first approach, which generates grids that resolve the fractures, is referred to as explicit fracture-matrix separation (EFMS). The second approach is based on a non-uniform coarsening strategy introduced in Aarnes et al. (Adv Water Resour 30(11):2177–2193, 2007a). A series of two-phase flow simulations where the saturation is modeled on the respective coarse grids are performed. The accuracy of the resulting solutions is examined, and the robustness of the two strategies is assessed with respect to number of fractures, degree of coarsening, well locations, phase viscosities, and fracture permeability. The numerical results show that saturation solutions obtained on the non-uniform coarse grids are consistently more accurate than the corresponding saturation solutions obtained on the EFMS grids. The numerical results also reveal that it is much easier to tune the upscaling factor with the non-uniform coarsening approach.     

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.     

An Upscaling–Static-Downscaling Scheme for Simulation of Enhanced Oil Recovery Processes

We investigate whether upscaling errors for EOR simulation can be reduced by an upscaling–static-downscaling method where the scales of simulation for the pressure and saturation/concentration switch between coarse simulation model and fine geological model. We apply a static downscaling that has been previously shown to be reliable for water flooding. We use the same algorithm of static downscaling for EOR processes that have been used for water flooding. Different EOR processes are considered: polymer, surfactant and thermal. This range of flooding processes ensures that we are examining more physically complicated systems than water flooding. For these processes, one major difference from water flooding is existence of a secondary front. The effective capturing of this front is a criterion of accuracy for upscaling because, for this front, the coupling of dispersion with the fractional flow creates excessive smearing. A scheme for numerical dispersion control is implemented to both upscaled and downscaled models to determine and reduce the sensitivity to dispersion errors.     

Multiblock Pore-Scale Modeling and Upscaling of Reactive Transport: Application to Carbon Sequestration

In order to safely store CO₂ in depleted reservoirs and deep saline aquifers, a better understanding of the storage mechanisms of CO₂ is needed. Reaction of CO₂ with minerals to form precipitate in the subsurface helps to securely store CO₂ over geologic time periods, but a concern is the formation of localized channels through which CO₂ could travel at large, localized rates. Pore-scale network modeling is an attractive option for modeling and understanding this inherently pore-level process, but the relatively small domains of pore-scale network models may prevent accurate upscaling. Here, we develop a transient, single-phase, reactive pore-network model that includes reduction of throat conductivity as a result of precipitation. The novelty of this study is the implementation of a new mortar/transport method for coupling pore networks together at model interfaces that ensure continuity of pressures, species concentrations, and fluxes. The coupling allows for modeling at larger scales which may lead to more accurate upscaling approaches. Here, we couple pore-scale models with large variation in permeability and porosity which result in initial preferential pathways for flow. Our simulation results suggest that the preferential pathways close due to precipitation, but are not redirected at late times.     

Accurate Representation of Near-well Effects in Coarse-Scale Models of Primary Oil Production

Near-well effects can have a strong impact on many subsurface flow processes. In oil production, because dissolved gas is released from the oil phase when the pressure falls below the bubble point, the detailed pressure field in the immediate vicinity of a production well strongly impacts gas (and thus oil) production. This effect is complicated by the interplay of fine-scale heterogeneity and two-phase flow physics, and can be difficult to capture in coarse-grid simulations. In this article, we develop and apply a new upscaling (coarse-graining) procedure to capture such near-well subgrid effects in coarse-scale flow simulation models. The method entails the use of preprocessing computations over near-well domains [referred to as local well models (LWM)] for the determination of upscaled single-phase and two-phase near-well parameters. These parameters are computed by minimizing the mismatch between fine and coarse-scale flows over the LWM. Minimization is accomplished using a gradient-based optimization procedure, with gradients calculated through solution of adjoint equations. The boundary conditions applied on the LWM can impact the upscaled parameters, but these boundary conditions depend on the global flow and are not, therefore, known a priori. In order to circumvent this difficulty, an adaptive local–global procedure is applied. This entails performing a global coarse-scale simulation with initial estimates for well-block parameters. The resulting pressure and saturation fields are then used to define local boundary conditions for the near-well computations. The overall procedure is applied to several example problems and is shown to provide results in close agreement with reference fine-scale computations. Significant improvement in accuracy over existing near-well upscaling treatments is demonstrated, particularly for a heavy oil case with oil viscosity of ~10⁴ cp.     

A GPU-based incompressible Navier–Stokes solver on moving overset grids

In pursuit of obtaining high fidelity solutions to the fluid flow equations in a short span of time, graphics processing units (GPUs) which were originally intended for gaming applications are currently being used to accelerate computational fluid dynamics (CFD) codes. With a high peak throughput of about 1 TFLOPS on a PC, GPUs seem to be favourable for many high-resolution computations. One such computation that involves a lot of number crunching is computing time accurate flow solutions past moving bodies. The aim of the present paper is thus to discuss the development of a flow solver on unstructured and overset grids and its implementation on GPUs. In its present form, the flow solver solves the incompressible fluid flow equations on unstructured/hybrid/overset grids using a fully implicit projection method. The resulting discretised equations are solved using a matrix-free Krylov solver using several GPU kernels such as gradient, Laplacian and reduction. Some of the simple arithmetic vector calculations are implemented using the CU++: An Object Oriented Framework for Computational Fluid Dynamics Applications using Graphics Processing Units, Journal of Supercomputing, 2013, doi:10.1007/s11227-013-0985-9 approach where GPU kernels are automatically generated at compile time. Results are presented for two- and three-dimensional computations on static and moving grids.     

An efficient semi‐implicit subgrid method for free‐surface flows on hierarchical grids

We present a new modelling strategy for improving the efficiency of computationally intensive flow problems in environmental free‐surface flows. The approach combines a recently developed semi‐implicit subgrid method with a hierarchical grid solution strategy. The method allows the incorporation of high‐resolution data on subgrid scale to obtain a more accurate and efficient hydrodynamic model. The subgrid method improves the efficiency of the hierarchical grid method by providing better solutions on coarse grids. The method is applicable to both steady and unsteady flows, but we particularly focus on river flows with steady boundary conditions. There, the combined hierarchical grid–subgrid method reduces the computational effort to obtain a steady state with factors up to 43. For unsteady models, the method can be used for efficiently generating accurate initial conditions on high‐resolution grids. Additionally, the method provides automatic insight in grid convergence. We demonstrate the efficiency and applicability of the method using a schematic test for the vortex shedding around a circular cylinder and a real‐world river case study. Copyright © 2015 John Wiley & Sons, Ltd.     

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions.

     

The Development of Appropriate Upscaling Procedures

Permeability upscaling should be carried out with careful attention to the nature of rock heterogeneities. While there are many large-scale features which must be taken into account, there are also important heterogeneities at the small-scale. Many sedimentary structures contain laminae at the mm–cm scale, and beds at the m-scale, which give rise to strong contrasts in permeability. We use a 2D model derived from a photo-panel of an aeolian outcrop, along with permeability measurements from a North Sea oil field, to demonstrate the effects of small-scale heterogeneity. This model is similar in size to a typical cell of a reservoir geological model. We take imaginary probe and core plug measurements from the model, average them, and compare these with the effective permeability for the model computed from a finite difference flow calculation. Although this procedure is standard practice, we show that it can lead to biased estimates of the permeabilities used in flow simulation. As an alternative we suggest using models of representative beds, and performing flow simulation to calculate effective permeabilities for both single-phase and two-phase flow.     

Combining the SSG/LRR-<Emphasis Type="Italic">ω</Emphasis> differential reynolds stress model with the detached eddy and laminar-turbulent transition models

The procedure of incorporating the detached eddy method and a model of laminar-turbulent transition into the SSG/LRR-ω turbulence model is presented. The approach proposed can be regarded as the generalization of the existing models intended to perform calculations with the SST turbulence model to the case of their use with the SSG/LRR-ω model. The advantage of the approach developed over the RANS turbulence models based on the Boussinesq hypothesis is demonstrated with respect to the problems of flow past an airfoil and cold jet outflow.     

多尺度嵌入式离散裂缝模型模拟方法

天然裂缝性油藏和人工压裂油藏内裂缝形态多样,分布复杂,传统的离散裂缝模型将裂缝作为基岩网格的边界,采用非结构化网格进行网格划分,其划分过程复杂,计算量大。嵌入式离散裂缝模型划分网格时不需要考虑油藏内的裂缝形态,只需对基岩系统进行简单的网格剖分,可以大大降低网格划分的复杂度,从而提高计算效率。然而,在油藏级别的数值模拟和人工压裂裂缝下的产能分析中,仍然存在计算量巨大、模拟时间过长的问题。本文提出嵌入式离散裂缝模型的多尺度数值计算格式,使用多尺度模拟有限差分法研究嵌入式离散裂缝模型渗流问题。通过在粗网格上求解局部流动问题计算多尺度基函数,多尺度基函数可以捕捉裂缝与基岩间的相互关系,反映单元内的非均质性,因此该方法既有传统尺度升级法的计算效率,又可以保证计算精度,数值结果表明这是一种有效的裂缝性油藏数值模拟方法。     

Reynolds-Averaged,Scale-Adaptive and Large-Eddy Simulations of Premixed Bluff-Body Combustion Using the Eddy Dissipation Concept

A lean premixed propane/air bluff-body stabilized flame (Volvo test rig) is calculated using the Scale-Adaptive Simulation turbulence model (SAS) and Large-Eddy simulations (LES) as well as the conventional Reynolds-averaged approach (RAS). RAS and SAS are closed by the standard k-?? and the k-ω Shear Stress Transport (SST) turbulence models, respectively. The conventional Smagorinsky and the k-equation sub-grid scales models are used for the LES closure. Effects of the sub-grid scalar flux modeling using the classical gradient hypothesis and Clark’s tensor diffusivity closures both for the inert and reactive LES flows are discussed. The Eddy Dissipation Concept (EDC) is used for the turbulence-chemistry interaction. It assumes that molecular mixing and the subsequent combustion occur in the ’fine structures’ (smaller dissipative eddies, which are close to the Kolmogorov scales). Assuming the full turbulence energy cascade, the characteristic length and velocity scales of the ’fine structures’ are evaluated using different turbulence models (RAS, SAS and LES). The finite-rate chemical kinetics is taken into account by treating the ’fine structures’ as constant pressure and adiabatic homogeneous reactors, calculated as a system of ordinary-differential equations (ODEs) described by a Perfectly Stirred Reactor (PSR) concept. Several further enhancements to model the PSRs are proposed, including a new Livermore Solver (LSODA) for integrating stiff ODEs and a new correction to calculate the PSR time scales. All models have been implemented as a stand-alone application \(\text {edcPisoFoam}\) based on the OpenFOAM technology. Additionally, several RAS calculations were performed using the Turbulence Flame Speed Closure model in Ansys Fluent to assess effects of the heat losses by modeling the conjugate heat transfer between the bluff-body and the reactive flow. Effects of the turbulence Schmidt number on RAS results are discussed as well. Numerical results are compared with available experimental data. Reasonable consistency between experimental data and numerical results provided by RAS, SAS and LES is observed. In general, there is satisfactory agreement between present LES-EDC simulations, numerical results by other authors and measurements without any major modification to the EDC closure constants, which gives a quite reasonable indication on the adequacy and accuracy of the method and its further application for turbulent premixed combustion simulations.