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1.
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. 相似文献
2.
K.-A. Lie O. Møyner J. R. Natvig A. Kozlova K. Bratvedt S. Watanabe Z. Li 《Computational Geosciences》2017,21(5-6):981-998
For the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs. 相似文献
3.
We present a variational multiscale mixed finite element method for the solution of Darcy flow in porous media, in which both
the permeability field and the source term display a multiscale character. The formulation is based on a multiscale split
of the solution into coarse and subgrid scales. This decomposition is invoked in a variational setting that leads to a rigorous
definition of a (global) coarse problem and a set of (local) subgrid problems. One of the key issues for the success of the
method is the proper definition of the boundary conditions for the localization of the subgrid problems. We identify a weak
compatibility condition that allows for subgrid communication across element interfaces, a feature that turns out to be essential
for obtaining high-quality solutions. We also remove the singularities due to concentrated sources from the coarse-scale problem
by introducing additional multiscale basis functions, based on a decomposition of fine-scale source terms into coarse and
deviatoric components. The method is locally conservative and employs a low-order approximation of pressure and velocity at
both scales. We illustrate the performance of the method on several synthetic cases and conclude that the method is able to
capture the global and local flow patterns accurately. 相似文献
4.
Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow 总被引：1，自引：0，他引：1
Todd Arbogast 《Computational Geosciences》2002,6(3-4):453-481
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. 相似文献
5.
6.
Knut-Andreas Lie Jostein R. Natvig Stein Krogstad Yahan Yang Xiao-Hui Wu 《Computational Geosciences》2014,18(3-4):357-372
A Dirichlet–Neumann representation method was recently proposed for upscaling and simulating flow in reservoirs. The DNR method expresses coarse fluxes as linear functions of multiple pressure values along the boundary and at the center of each coarse block. The number of flux and pressure values at the boundary can be adjusted to improve the accuracy of simulation results and, in particular, to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high-permeability streaks or channels. As an alternative, the multiscale mixed finite-element (MsMFE) method was designed to obtain fine-scale fluxes at the cost of solving a coarsened problem, but can also be used as upscaling methods that are flexible with respect to geometry and topology of the coarsened grid. Both methods can be expressed in mixed-hybrid form, with local stiffness matrices obtained as “inner products” of numerically computed basis functions with fine-scale sub-resolution. These basis functions are determined by solving local flow problems with piecewise linear Dirichlet boundary conditions for the DNR method and piecewise constant Neumann conditions for MsMFE. Adding discrete pressure points in the DNR method corresponds to subdividing faces in the coarse grid and hence increasing the number of basis functions in the MsMFE method. The methods show similar accuracy for 2D Cartesian cases, but the MsMFE method is more straightforward to formulate in 3D and implement for general grids. 相似文献
7.
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. 相似文献
8.
Knut–Andreas Lie Stein Krogstad Ingeborg Skjelkv?le Ligaarden Jostein Roald Natvig Halvor M?ll Nilsen B?rd Skaflestad 《Computational Geosciences》2012,16(2):297-322
Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect
to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for
polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is
well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research
into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called
multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose
novel mathematical methods but instead presents an open-source Matlab? toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation.
The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing
and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point
and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising
petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well
as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a
generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method
as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the
toolkit contains add-on modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale
methods and adjoint methods for use in optimisation of rates and placement of wells. 相似文献
9.
Mortar Upscaling for Multiphase Flow in Porous Media 总被引：1，自引：0，他引：1
In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly. 相似文献
10.
11.
I. Aavatsmark G. T. Eigestad R. A. Klausen M. F. Wheeler I. Yotov 《Computational Geosciences》2007,11(4):333-345
This paper investigates different variants of the multipoint flux approximation (MPFA) O-method in 2D, which rely on a transformation
to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different
methods appear, depending on where the transformed permeability is evaluated. Midpoint and corner-point evaluations are considered.
Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence
of the MPFA methods is investigated numerically. For corner-point evaluation of the reference permeability, the same convergence
behavior as the O-method in the physical space is achieved when the grids are refined uniformly or when grid perturbations
of order h
2 are allowed. For h
2-perturbed grids, the convergence of the normal velocities is slower for the midpoint evaluation than for the corner-point
evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods.
The relations to the MFE methods are used to explain the loss of convergence.
Wheeler was partially supported by NSF grant DMS 0411413 and the DOE grant DE-FGO2-04ER25617. Yotov was supported in part
by the DOE grant DE-FG02-04ER25618, the NSF grant DMS 0411694 and the J. Tinsley Oden Faculty Fellowship, The University of
Texas at Austin. 相似文献
12.
James V. Lambers Margot G. Gerritsen Bradley T. Mallison 《Computational Geosciences》2008,12(3):399-416
We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations.
The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled
grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint
stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are
required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to
ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations
are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without
full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly
used local methods and also compares favorably with a local–global upscaling method. 相似文献
13.
在煤层气排采分析及数值模拟研究中,煤储层结构的三维建模是数值模拟和结果分析的起点,具有关键作用。三维长方体等传统网格,难以表达地层的起伏变化,角点网格具有表达起伏变化地层的优势。从煤层气动态可视化的角度出发,基于角点网格建立了煤储层三维地质模型。研究了角点网格模型的数据特点及生成角点网格的方法,采用C#编程语言并结合OpenGL图形接口,开发了煤储层三维可视化软件模型,利用该模型表达了山西省沁水盆地潘庄区块的煤储层地质构造。结果表明,角点网格适用于煤储层三维模型的构建,能较好的表达煤层的结构特征。 相似文献
14.
Erick R. Burns Larry R. Bentley Rene Therrien Clayton V. Deutsch 《Hydrogeology Journal》2010,18(6):1357-1373
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. 相似文献
15.
We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities
or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an
initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global
(LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce
flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued
until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of
LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore,
the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a
new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG
over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration
of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative
to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another
attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is,
the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic
boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic
refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured
grids. 相似文献
16.
Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation
approximations introduced within the multiscale framework are dependent upon both the heterogeneity of the reservoir and the
structure of the computational grid. While previous works on multiscale control volume methods have focused on heterogeneous
elliptic problems on regular Cartesian grids, we have tested the multiscale control volume formulations on two-dimensional
elliptic problems involving heterogeneous media and irregular grid structures. Our study shows that the tangential flow approximation
commonly used within multiscale methods is not suited for problems involving rough grids. We present a more robust mass conservative
domain decomposition preconditioner for simulating flow in heterogeneous porous media on general grids. 相似文献
17.
Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds. 相似文献
18.
油藏精细地质模型网格粗化算法及其效果 总被引：1，自引：0，他引：1
在前人研究基础上, 根据DP(Dykstra-Parsons)系数能定量评价储层非均质性, 微网格块的渗透率值粗化后, 其等效渗透率的上、下限(Cmin、Cmax)能反映渗透率的各向异性的特点, 提出了一种运算速度快和相对有效的网格粗化算法。该算法能考虑到储层非均质性对不同方向渗透率值的影响, 且求解过程相对简单。应用该方法对鄂尔多斯盆地中部某油藏的陆相储层的精细地质模型进行了网格粗化计算, 然后在粗化后的模型上进行油藏数值模拟研究, 同时针对研究区地质背景和产出流体微可压缩的物性特征, 首次利用流线模拟器对精细地质模型进行了油藏数值模拟研究, 并以此结果为标准, 对该网格粗化算法时效性进行了系统评价。分析表明, 该算法具有较快的计算速度和较高的可靠性, 是解决储层非均质强、物性差的陆相成因油藏精细油藏数值模拟的一种行之有效的手段。 相似文献
19.
This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast. 相似文献
20.
Wells are seldom modeled explicitly in large scale finite difference reservoir simulations. Instead, the well is coupled to the reservoir through the use of a well index, which relates wellbore flow rate and pressure to grid block quantities. The use of an accurate well index is essential for the detailed modeling of nonconventional wells; i.e., wells with an arbitrary trajectory or multiple branches. The determination of a well index for such problems is complicated, particularly when the simulation grid is irregular or unstructured. In this work, a general framework for the calculation of accurate well indices for general nonconventional wells on arbitrary grids is presented and applied. The method entails the use of an accurate semianalytical well model based on Green's functions as a reference single phase flow solution. This result is coupled with a finite difference calculation to provide an accurate well index for each grid block containing a well segment. The method is demonstrated on a number of homogeneous example cases involving deviated, horizontal and multilateral wells oriented skew to the grid. Both Cartesian and globally unstructured multiblock grids are considered. In all these cases, the method is shown to provide results that are considerably more accurate compared to results using standard procedures. The method is also applied to heterogeneous problems involving horizontal wells, where it is shown to be capable of approximating the effects of subgrid heterogeneity in coarse finite difference models. 相似文献