On the relationship between the multiscale finite-volume method and domain decomposition preconditioners

In this paper, we review the classical nonoverlapping domain decomposition (NODD) preconditioners, together with the newly developed multiscale control volume (MSCV) method. By comparing the formulations, we observe that the MSCV method is a special case of a NODD preconditioner. We go on to suggest how the more general framework of NODD can be applied in the multiscale context to obtain improved multiscale estimates.     

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.     

非规则网格下二维浅水流动的数值模拟

建立了一种在非规则结构化网格上求解平面二维浅水流动的有限体积方法。通过采用地形在离散网格内双线性变化及离散网格界面间地形连续的地形逼近方法和应用可以有效处理间断问题的Roe格式来离散浅水方程中的对流项,并通过VanLeer提出的状态插值法提高格式精度。在计算原始变量在网格内的插值梯度时,采用最小二乘方法求变量的最优梯度代替差分计算梯度,从而可采用任意形状的不规则四边形网格离散计算域。计算实例表明,该方法能够计算间断问题并能够处理各种复杂流态的过渡,具有较好适应性和计算精度,能够满足不同实际问题的计算要求。     

Horizontal simulation grids as alternative to structure-based grids for thin oil-zone problems: a comparison study on a Troll segment

The layering in reservoir simulation grids is often based on the geology, e.g., structure tops. In this paper we investigate the alternative of using horizontal layers, where the link to the geology model is by the representation of the petrophysics alone. The obvious drawback is the failure to honor the structure in the grid geometry. On the other hand, a horizontal grid will honor the initial fluid contacts perfectly, and horizontal wells can also be accurately represented. Both these issues are vital in thin oil-zone problems, where horizontal grids may hence be a viable alternative. To investigate this question, a number of equivalent simulation models were built for a segment of the Troll Field, both geology-based and horizontal, and various combinations of these. In the paper, it is demonstrated that the horizontal grid was able to capture the essentials of fluid flow with the same degree of accuracy as the geology-based grid, and near-well flow was considerably more accurate. For grids of comparable resolution, more reliable results were obtained by a horizontal grid than a geo-grid. A geo-grid with local grid refinement and a horizontal grid produced almost identical results, but the ratio of computing times was almost 20 in favor of the horizontal grid. In the one-phase regions of the reservoir, relatively coarse cells can be used without significant loss of accuracy.     

A domain decomposition method for elastic wave propagation based on Huygens' principle

Classical domain decomposition techniques allow the modelling of wave propagation in large structures with multiprocessor resources. Indeed, domain decomposition can be viewed as a natural way for parallelism. In the first part of our work we analyze a well-known technique, the Schur complement method, and discuss its viability in terms of efficiency and complexity. Later, we introduce a novel domain decomposition algorithm, which is aimed at message-passing environments and reduces considerably the communication between processes. The result is a technique very competitive in the framework of parallel computing, which gives good results from the modelling viewpoint.     

A RAM-economizing domain decomposition technique for regional high-resolution groundwater simulation

A major limitation of the use of high-resolution groundwater models on a regional scale for resource management by water companies is the excessive RAM requirements of such models which surpass the capacity of today's PCs. A strategy is presented to overcome this problem by using overlapping domain decomposition techniques. Furthermore, because very long computing time is the bottleneck for the practical use of this technique for large groundwater models, an analysis is also presented of a number of methods implemented to increase calculation performance. The approach presented here is characterized by a fairly simple structure that represents a generalized relaxation algorithm. It can be adapted for use with finite element as well as with finite difference methods. Electronic Publication     

Lagrange multiplier-based domain decomposition methods for a non-linear sedimentary basin problem

We present two Lagrange multiplier-based domain decomposition methods for solving iteratively a non-linear sedimentary basin problem. Both methods are Uzawa-type algorithms derived from saddle point equations of Lagrangian and augmented Lagrangian functionals. Numerical experiments show that both methods are scalable with respect to the mesh size.     

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

Probabilistic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell’s equations as used in the magnetotelluric method. The domain is split into non-overlapping sub-domains and the solution on the sub-domain boundaries is obtained by evaluating the stochastic form of the exact solution of Maxwell’s equations by a Monte-Carlo approach. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell’s equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function-based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.     

A comparison of multiscale methods for elliptic problems in porous media flow

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation. The research is funded by the Research Council of Norway under grant nos. 152732 and 158908.     

Monotonicity for MPFA methods on triangular grids

Flow in a porous medium can be described by a set of non-linear partial differential equations. The pressure variable satisfies a maximum principle, which guarantees that the solution will have no oscillations. A discretisation of the pressure equation should preserve this monotonicity property. Whether a numerical method is monotone will depend both on the medium and on the grid. We study monotonicity of Multi-point Flux Approximation methods on triangular grids. We derive necessary conditions for monotonicity on uniform grids. Further, we study the robustness of the methods on rough grids, and quantify the violations of the maximum principle. These investigations are done for single phase flow, however, they are supported by two-phase simulations.     

基于区域分解法的水文地质参数寻优研究

在建立地下水流模型的过程中,水文地质参数寻优一直是较为复杂的步骤之一,具体难点包括寻优方法的取用,为保持总体平衡所引起的参数峰值异常以及总体寻优需要大量的计算机时等问题.本文运用区域分解法(Domain Decomposition method,DDM)的基本思想,将整个区域的参数寻优问题分解为各参数分区内的子域问题求解,通过寻找整个区域上的Nash均衡最终获得各子域上的最优参数.实验算例及其结果证明应用该方法实现水文地质参数自动寻优,不但具有高度的可靠性,同时优化问题的规模减小.此举不但减少了求解过程所需要的CPU时间,而且提高了参数拟合度.     

岩体通信技术研究中应注意的几个问题

岩体通信是通信信号在地层中的传播与接收,属无线电通信的一个分支。阐述了电流场通信的概念;指出电流场通信在煤矿应用的安全性不仅与信号功率有关,而且与信号频率相关;提出了岩体通信设备研制中频率确定的基本原则和岩体通信领域需要进一步研究的问题。     

Hyperion高光谱影像的经验模态分解及其几个问题的探讨

通过经验模态分解(EMD)方法,可以把高光谱曲线分解成一系列由高频到低频的本征模态函数(IMF),从而可以从不同层次分析高光谱数据特征,并进行特征提取.把EMD方法用于Hyperion高光谱数据处理,通过对每一个像元的高光谱曲线进行经验模态分解,构成IMF系列影像.通过分析IMF影像表明:对于不同的地物,其IMF也不同;不同波段,不但其噪声水平不同,而且噪声性质也不一样;原始影像的Smile效应在IMF中可以明显表现出来.由于IMF影像与原始影像的波段有一一对应的关系,因此与其它信号分析方法(如小波分析及FFT方法)相比,EMD方法得到的结果更直观,更易于数据分析.但是EMD结果受极值、插值方法、IMF判别准则以及端点效应的影响较大.     

Multidimensional upstream weighting for multiphase transport on general grids

The governing equations for multiphase flow in porous media have a mixed character, with both nearly elliptic and nearly hyperbolic variables. The flux for each phase can be decomposed into two parts: (1) a geometry- and rock-dependent term that resembles a single-phase flux; and (2) a mobility term representing fluid properties and rock–fluid interactions. The first term is commonly discretized by two- or multipoint flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single-point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, SPU suffers from grid orientation effects. An important example of this, which will be considered in this work, is the displacement of a heavy oil by water. For these adverse mobility ratio flows, the governing equations are unstable at the modeling scale, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results. In this work, we present a framework for multidimensional upstream weighting for multiphase flow with buoyancy on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multidimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for time integration of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multidimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multidimensional upstream weighting is used.     

Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I

Three Galerkin methods using discontinuous approximation spaces are introduced to solve elliptic problems. The underlying bilinear form for all three methods is the same and is nonsymmetric. In one case, a penalty is added to the form and in another, a constraint on jumps on each face of the triangulation. All three methods are locally conservative and the third one is not restricted. Optimal a priori hp error estimates are derived for all three procedures.     

Finite-difference strategy for elastic wave modelling on curved staggered grids

Waveform modelling is essential for seismic imaging and inversion. Because including more physical characteristics can potentially yield more accurate Earth models, we analyse strategies for elastic seismic wave propagation modelling including topography. We focus on using finite differences on modified staggered grids. Computational grids can be curved to fit the topography using distribution functions. With the chain rule, the elasto-dynamic formulation is adapted to be solved directly on curved staggered grids. The chain-rule approach is computationally less expensive than the tensorial approach for finite differences below the 6th order, but more expensive than the classical approach for flat topography (i.e. rectangular staggered grids). Free-surface conditions are evaluated and implemented according to the stress image method. Non-reflective boundary conditions are simulated via a Convolutional Perfect Matching Layer. This implementation does not generate spurious diffractions when the free-surface topography is not horizontal, as long as the topography is smoothly curved. Optimal results are obtained when the angle between grid lines at the free surface is orthogonal. The chain-rule implementation shows high accuracy when compared to the analytical solution in the case of the Lamb’s problem, Garvin’s problem and elastic interface.     

非结构网格上的三维浅水流动数值模型

针对当前复杂环境水流模拟的需求,建立了新型的基于特征型高分辨率数值算法的三维非结构网格浅水动力模型。模型采用有限体积法离散sigma坐标下的三维浅水方程,运用Roe黎曼近似解评估水平界面通量。模型网格拟合边界能力强,可根据需要局部加密;格式数值性能优良,具有守恒性、单调迎风性、高数值分辨率等特性。同时,应用干湿判别法处理动边界,以适应浅滩地形漫/露过程模拟的需要。封闭水池内部风生环流、干河床上溃坝过程和长江口实际潮流场的模拟从不同侧面展示了模型的特点,结果表明它能够准确地预测水流的三维流动结构,而且计算简单高效,具有良好的数值稳定性。     

裂隙岩体块体化程度评价方法的若干问题修正

块体化程度是评价岩体完整性的一种新指标,能从三维角度表征岩体破碎程度,但目前该方法仍存在未充分考虑岩体切割程度及块体规模、未限定基础应用条件、块体化程度等级划分不合理等缺陷。针对上述缺陷,深入分析了其产生原因,并借鉴岩体切割程度、三维块度模数、体积RQD等计算原理,限定了块体体积百分比相关概念,提出了考虑岩体完整性块度尺寸效应的块体体积综合百分比计算方法,确立了块体化程度等级及分级指标取值依据,建立了修正的块体化程度评价方法。通过对比块体化程度评价修正方法、岩土工程规范对岩体完整性的划分结果,分析了修正方法的合理性。分别以广西铜坑矿锌多金属矿体、乌东德水电工程的块体研究数据为基础,开展了修正块体化程度评价方法的实例验证,结果表明：利用修正块体化程度评价方法计算的铜坑矿+255 m中段4#试验区岩体块体体积百分比为11.18%,属轻度块状化岩体;乌东德水电站PD49-1平硐、PD4支硐块体体积百分比分别为12.847%、10.168%,均属于轻度块状化岩体,岩体完整性程度为较完整级别。与现有块体化程度评价方法比较,修正方法计算的块体体积百分比能够更准确地从三维角度表征真实岩体的完整性。研究成果对精确刻画岩体三维完整性具有重要意义。     

Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids

Improved and enhanced oil recovery methods require sophisticated simulation tools to predict the injected flow pass together with the chemical reactions inside it. One approach is application of higher-order numerical schemes to avoid excessive numerical diffusion that is very typical for transport processes. In this work, we provide a first step towards higher-order schemes applicable on general polyhedral and corner-point grids typically used in reservoir simulation. We compare three possible approaches of linear reconstruction and slope limiting techniques on a variety of different meshes in two and three spatial dimensions and discuss advantages and disadvantages.