Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.     

Nested gridding and streamline-based simulation for fast reservoir performance prediction

Detailed reservoir models routinely contain 10⁶–10⁸ grid blocks. These models often cannot be used directly in a reservoir simulation because of the time and memory required for solving the pressure grid on the fine grid. We propose a nested gridding technique that efficiently obtains an approximate solution for the pressure field. The domain is divided into a series of coarse blocks, each containing several fine cells. Effective mobilities are computed for each coarse grid block and the pressure is then found on the coarse scale. The pressure field within each coarse block is computed using flux boundary conditions obtained from the coarse pressure solution. Streamline-based simulation is used to move saturations forward in time. We test the method for a series of example waterflood problems and demonstrate that the method can give accurate estimates of oil production for large 3D models significantly faster than direct simulation using streamlines on the fine grid, making the method overall approximately up to 1,000 times faster than direct conventional simulation.     

Interface error analysis for numerical wave propagation

The numerical error associated with finite-difference simulation of wave propagation in discontinuous media consists of two components. The first component is a higher-order error that leads to grid dispersion; it can be controlled by higher-order methods. The second component results from misalignment between numerical grids and material interfaces. We provide an explicit estimate of the interface misalignment error for the second order in time and space staggered finite-difference scheme applied to the acoustic wave equation. Our analysis, confirmed by numerical experiments, demonstrates that the interface error results in a first-order time shift proportional to the distance between the interface and computational grids. A 2D experiment shows that the interface error cannot be suppressed by higher-order methods and indicates that our 1D analysis gives a good prediction about the behavior of the numerical solution in higher dimensions.      

Research Progress on Physical Parameterization Schemes in Numerical Weather Prediction Models

Atmospheric physics in numerical weather prediction model which predominantly determines the evolution of atmospheric processes is mainly described by physical parameterization. As a result, the development of physical parameterization has been a hot research issue in the area of numerical prediction for a long time. In this regard, the theoretical background and history of physical parameterization schemes for convection, microphysics, and planetary boundary layer, were reviewed in this study. It is suggested that the advance of physical parameterization for the model with high-resolution grid spaces should be considered as a principle issue for numerical model development in the future. Although the gird spaces in current operational numerical models generally decrease toward 10 km owing to the rapid development of high-performance computation, yet most of these schemes are designed for coarse grid spaces. Because of this kind of deficiency, the theoretical basis of these schemes inevitably faces controversy. Directions for development of physical parameterization were also suggested according to the trends of research in numerical prediction.     

Nearly perfectly matched layer boundary conditions for operator upscaling of the acoustic wave equation

Acoustic imaging and sensor modeling are processes that require repeated solution of the acoustic wave equation. Solution of the wave equation can be computationally expensive and memory intensive for large simulation domains. One scheme for speeding up solution of the wave equation is the operator-based upscaling method. The algorithm proceeds in two steps. First, the wave equation is solved for fine grid unknowns internal to coarse blocks assuming the coarse blocks do not need to communicate with neighboring blocks in parallel. Second, these fine grid solutions are used to form a new problem which is solved on the coarse grid. Accurate and efficient wave propagation schemes also must avoid artificial reflections off of the computational domain edges. One popular method for preventing artificial reflections is the nearly perfectly matched layer (NPML) method. In this paper, we discuss applying NPML to operator upscaling for the wave equation. We show that although we only apply NPML to the first step of this two step algorithm (directly affecting the fine grid unknowns only), we still see a significant reduction of reflections back into the domain. We describe three numerical experiments (one homogeneous medium experiment and two heterogeneous media examples) in which we validate that the solution of the wave equation exponentially decays in the NPML regions. Numerical experiments of acoustic wave propagation in two dimensions with a reasonable absorbing layer thickness resulted in a maximum pressure reflection of 3–8%. While the coarse grid acceleration is not explicitly damped in our algorithm, the tight coupling between the two steps of the algorithm results in only 0.1–1% of acceleration reflecting back into the computational domain.     

证据权模型中两种预测单元划分方式对比

证据权模型作为一种数据综合方法已被广泛应用于矿产资源定量预测与评价。在模糊证据权基础上,发展了基于地质单元思想的矢量证据图层构建和数据综合方法,并通过实例作具体阐述：它以矿点缓冲区图层作为训练图层,以各证据变量图层在空间上的叠置所形成的唯一地质单元作为评价对象,统一计算各个证据变量的证据权重,进而基于地质单元进行证据综合和后验概率成图。与基于栅格（或规则格网）的模型不同,基于矢量证据权模型以具有明确地质内涵的地质单元（而非规则网格单元）为预测单元,易于解释,并且消除了边界误差;相比基于规则格网划分所得到的成矿单元,以矿床（点）缓冲区作为训练对象,提高了已知矿点的代表性。实例表明：若预测单元大小为初始栅格大小整数倍,各缓冲等级平均面积计算误差为0.26%,否则面积平均误差达到6%;即使在预测单元大小为初始栅格大小整数倍情况下,矿点平均计算误差也达到4.78%。因此,基于地质单元思想的证据权预测单元划分方法在精度上优于基于栅格或规则格网方法。     

Machine-learning-based modeling of coarse-scale error,with application to uncertainty quantification

The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions.     

Development of a refinement criterion for adaptive mesh refinement in steam-assisted gravity drainage simulation

Steam-assisted gravity drainage (SAGD) is an enhanced oil recovery process for heavy oils and bitumens. Numerical simulations of this thermal process allow us to estimate the retrievable volume of oil and to evaluate the benefits of the project. As there exists a thin flow interface (compared to the reservoir dimensions), SAGD simulations are sensitive to the grid size. Thus, to obtain precise forecasts of oil production, very small-sized cells have to be used, which leads to prohibitive CPU times. To reduce these computation times, one can use an adaptive mesh refinement technique, which will only refine the grid in the interface area and use coarser cells outside. To this end, in this work, we introduce new refinement criteria, which are based on the work achieved in Kröner and Ohlberger (Math Comput 69(229):25–39, 2000) on a posteriori error estimators for finite volume schemes for hyperbolic equations. Through numerical experiments, we show that they enable us to decrease in a significant way the number of cells (and then CPU times) while maintaining a good accuracy in the results.     

Vertical equilibrium with sub-scale analytical methods for geological CO2 sequestration

Large-scale implementation of geological CO₂ sequestration requires quantification of risk and leakage potential. One potentially important leakage pathway for the injected CO₂ involves existing oil and gas wells. Wells are particularly important in North America, where more than a century of drilling has created millions of oil and gas wells. Models of CO₂ injection and leakage will involve large uncertainties in parameters associated with wells, and therefore a probabilistic framework is required. These models must be able to capture both the large-scale CO₂ plume associated with the injection and the small-scale leakage problem associated with localized flow along wells. Within a typical simulation domain, many hundreds of wells may exist. One effective modeling strategy combines both numerical and analytical models with a specific set of simplifying assumptions to produce an efficient numerical–analytical hybrid model. The model solves a set of governing equations derived by vertical averaging with assumptions of a macroscopic sharp interface and vertical equilibrium. These equations are solved numerically on a relatively coarse grid, with an analytical model embedded to solve for wellbore flow occurring at the sub-gridblock scale. This vertical equilibrium with sub-scale analytical method (VESA) combines the flexibility of a numerical method, allowing for heterogeneous and geologically complex systems, with the efficiency and accuracy of an analytical method, thereby eliminating expensive grid refinement for sub-scale features. Through a series of benchmark problems, we show that VESA compares well with traditional numerical simulations and to a semi-analytical model which applies to appropriately simple systems. We believe that the VESA model provides the necessary accuracy and efficiency for applications of risk analysis in many CO₂ sequestration problems.     

Multiscale mixed/mimetic methods on corner-point grids

Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation, approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately sized) coarse grid, one can retain the efficiency of an upscaling method and, at the same time, produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has been shown to be a particularly versatile approach. In this paper, we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. In principle, any stable and conservative method can be used. Here, we use a mimetic discretisation, which is a generalisation of mixed finite elements that gives a discrete inner product, allows for polyhedral elements, and can (easily) be extended to curved grid faces. The coarse grid can, in principle, be any partition of the subgrid, where each coarse block is a connected collection of subgrid cells. However, we argue that, when generating coarse grids, one should follow certain simple guidelines to achieve improved accuracy. We discuss partitioning in both index space and physical space and suggest simple processing techniques. The versatility and accuracy of the new multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. A variety of coarse grids, both violating and obeying the above mentioned guidelines, are employed. The MsMFEM solutions are compared with a reference solution obtained by direct simulation on the subgrid.     

Upscaling Hydraulic Conductivities in Cross-Bedded Formations

Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities.     

Upscaling facies models to preserve connectivity of designated facies

An upscaling algorithm has been developed that generates an irregular coarse grid that preserves flow connectivity by applying a rule-based upscaling algorithm to a fine-scale facies distribution. The algorithm is demonstrated using stochastically generated paleo-fluvial facies distributions. First, an irregular grid honoring the channel facies is created, followed by computation of effective anisotropic parameters for all coarse-grid cells. For the apparent layer-cake geometry of overbank deposits seen in outcrop, two local upscaling methods are compared: (1) the layered system approximation and (2) the mode. To assess upscaling performance, flow simulations for the original and upscaled grids are compared. The horizontal layered approximation (arithmetic mean) performs poorly, over-predicting lateral connectivity where even infrequent disconnection becomes important. Performance of the mode as an upscaling algorithm depends on the probability that a coarse-grid cell will be dominated by a single facies, and it performs surprisingly well because the upscaled grid-generation algorithm honors the channels, informing the upscaling process. Lastly, the irregular coarse grid was compared to a uniform coarse grid, showing superior performance with the irregular grid. The reduction in grid size achieved by irregular-grid generation will be a function of the geometrical complexity of the geologic objects to be honored.     

Folder: A numerical tool to simulate the development of structures in layered media

We present Folder, a numerical toolbox for modelling deformation in layered media subject to layer parallel shortening or extension in two dimensions. The toolbox includes a range of features that ensure maximum flexibility to configure model geometry, define material parameters, specify numerical parameters, and choose the plotting options. Folder builds on an efficient finite element method model and implements state of the art iterative and time integration schemes. We describe the basic Folder features and present several case studies of single and multilayer stacks subject to layer parallel shortening and extension. Folder additionally comprises an application that illustrates various analytical solutions of growth rates calculated for the cases of layer parallel shortening and extension of a single layer with interfaces perturbed with a single sinusoidal waveform. We further derive two novel analytical expressions for the growth rate in the cases of layer parallel shortening and extension of a linear viscous layer embedded in a linear viscous medium of a finite thickness. These solutions help understand mechanical instabilities in layered rocks and provide a unique opportunity for benchmarking of numerical codes. We demonstrate how Folder can be used for benchmarking of numerical codes. We test the accuracy of single-layer folding simulations using various 1) spatial and temporal resolutions, 2) iterative algorithms for non-linear materials, and 3) time integration schemes. The accuracy of the numerical results is quantified by: 1) comparing them to analytical solutions, if available, or 2) running convergence tests. As a result, we provide a map of the most optimal choice of grid size, time step, and number of iterations to keep the results of the numerical simulations below a given error for a given time integration scheme. Folder is an open source MATLAB application and comes with a user-friendly graphical interface. Folder is suitable for both educational and research purposes.     

Spatial grid correction for short-term numerical simulation results of carbon dioxide dissolution in saline aquifers

Numerical simulation is an essential component of many studies of geological storage of carbon dioxide, but care must be taken to ensure the accuracy of the results. Unlike several other possible sources of simulation errors, which have previously been considered in detail and have well-understood techniques for mitigating their effects, comparatively little discussion of the spatial grid dependence of the dissolution rate of carbon dioxide into the formation water has appeared in the literature, despite its importance to simulation studies of geological storage of carbon dioxide in saline aquifers. In many instances, sufficient refinement of the computational grid can be a practical solution. However, this approach is not always feasible, especially for large-scale simulations in three dimensions requiring multiple realisations, which commonly feature a coarse grid due to constraints on available computational capabilities. A measure of the error in the amount of dissolved carbon dioxide introduced by the use of a finite grid is therefore of great interest. In this study, the use of finite-sized grid blocks is shown to overestimate the amount of dissolved carbon dioxide in short-term results by a factor of 1?+?V _f/V _p, where V _f is the grid block volume at the saturation front and V _p is the total grid block volume of the plume. This result can be used in a number of ways to correct the calculated short-term dissolution in coarse-scale simulations so that the amount dissolved agrees better with that obtained from fine-scale simulations.     

煤层气藏数值模拟垂向网格优化

数值模拟是研究煤层气藏工程的一种常规方法,建立模型时人们常忽略垂向网格精细程度对模拟的影响,无法准确反映出垂向上流体流动规律、气水分异现象以及压降漏斗展布等,对模拟结果有很大影响。为了研究垂向网格划分精度对煤层气藏数值模拟过程的影响,采用不同模拟器对垂向网格的精细程度进行模拟计算,运用渗透率等效方法、局部网格加密方法和模拟器自带压裂方法模拟煤层压裂缝,总共模拟了3种方法15套方案。结果表明,垂向网格划分精度对煤层气生产影响较大,当网格步长达到1.5m时计算结果较为精确,可以满足模拟需求。网格数量及网格步长的合理划分,能够更好地呈现煤层中气水分异现象,有助于分析气水流动状态,便于历史拟合和产量预测。      

流体黏性对油井砂岩力学响应的影响

随着黏度较大的油藏陆续投入开发,油藏黏性对储层砂岩力学特性的影响研究意义重大。基于柱坐标系建立射孔试验的三维颗粒流数值模型,考虑不同黏性的流体运动对砂岩力学响应的影响,反映油井的出砂过程。砂岩的宏观应力曲线说明流速相同时,随着黏滞系数的增大,切向应力和偏应力均增大,使得砂岩剪切破坏的几率增大,砂岩更容易屈服破坏而出砂。另外,砂岩黏结应力图说明油井附近的应力较大,且随着黏滞系数增大,黏结张拉应力的增大是局部的,而剪应力的增大是全局的,且变化趋势更明显;颗粒的旋转也说明随着流体黏性的增大,颗粒旋转增大,砂岩形成离散颗粒而出砂的几率增大。上述结果与实际开采中的砂岩力学响应吻合,说明了在相同的外界条件下,黏性越大的流体运动对砂岩受力的影响越大,出砂越明显,该成果对不同黏性的油藏开采采用有效的防砂方法提供了重要的科学依据。     

Sensitivity of simulated cyclone Gonu intensity and track to variety of parameterizations: Advanced hurricane WRF model application

Domain configuration and several physical parameterization settings such as planetary boundary layer, cumulus convection, and ocean–atmosphere surface flux parameterizations can play significant roles in numerical prediction of tropical cyclones. The present study focuses to improve the prediction of the TC Gonu by investigating the sensitivity of simulations to mentioned configurations with the Advanced Hurricane WRF model. The experiments for domain design sensitivity with 27 km resolution has been shown moving the domains towards the east improve the results, due to better account for the large-scale process. The fixed and movable nests on a 9-km grid were considered separately within the coarse domain and their results showed that despite salient improvement in simulated intensity, an accuracy reduction in simulated track was observed. Increasing horizontal resolution to 3 km incredibly reduced the simulated intensity accuracy when compared to 27 km resolution. Thereafter, different initial conditions were experimented and the results have shown that the cyclone of 1000 hPa sea level pressure is the best simulation initial condition in predicting the track and intensity for cyclone Gonu. The sensitivity of simulations to ocean–atmosphere surface-flux parameterizations on a 9-km grid showed the combination of ‘Donelan scheme’ for momentum exchanges along with ‘Large and Pond scheme’ for heat and moisture exchanges provide the best prediction for cyclone Gonu intensity. The combination of YSU and MYJ PBL scheme with KF convection for prediction of track and the combination of YSU PBL scheme with KF convection for prediction of intensity are found to have better performance than the other combinations. These 22 sensitivity experiments also implicitly lead us to the conclusion that each particular forecast aspect of TC (e.g., track, intensity, etc.) will require its own special design.