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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.
Multiscale mixed/mimetic methods on corner-point grids 总被引：1，自引：0，他引：1
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. 相似文献
3.
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
4.
The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308–317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods. 相似文献
5.
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
6.
The present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing defining accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called multiscale hybrid-mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach (Harder et al. 2013). We extend here the MH-mixed method to a finite volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively w.r.t. a posteriori error estimators. The method is applied to single phase (well-testing) and multiphase flow in heterogeneous porous media. 相似文献
7.
A new method for upscaling fine scale permeability fields to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, applies a triangle-based finite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local fine scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of flow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a flow-based gridding technique. In this case, the approach is shown to provide results that are significantly more accurate than those obtained using standard techniques. 相似文献
8.
Luz Angélica Caudillo-Mata Eldad Haber Christoph Schwarzbach 《Computational Geosciences》2017,21(5-6):963-980
In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique. 相似文献
9.
The problem of multiphase phase flow in heterogeneous subsurface porous media is one involving many uncertainties. In particular, the permeability of the medium is an important aspect of the model that is inherently uncertain. Properly quantifying these uncertainties is essential in order to make reliable probabilistic-based predictions and future decisions. In this work, a measure-theoretic framework is employed to quantify uncertainties in a two-phase subsurface flow model in high-contrast media. Given uncertain saturation data from observation wells, the stochastic inverse problem is solved numerically in order to obtain a probability measure on the space of unknown permeability parameters characterizing the two-phase flow. As solving the stochastic inverse problem requires a number of forward model solves, we also incorporate the use of a conservative version of the generalized multiscale finite element method for added efficiency. The parameter-space probability measure is used in order to make predictions of saturation values where measurements are not available, and to validate the effectiveness of the proposed approach in the context of fine and coarse model solves. A number of numerical examples are offered to illustrate the measure-theoretic methodology for solving the stochastic inverse problem using both fine and coarse solution schemes. 相似文献
10.
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
11.
Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code. 相似文献
12.
多尺度有限单元法求解非均质多孔介质中的三维地下水流问题 总被引：7，自引：0，他引：7
应用多尺度有限单元法模拟非均质多孔介质中的三维地下水流问题。与传统有限单元法相比，多尺度有限单元法的基函数具有能反映单元内参数变化的优点，所以这种方法能在大尺度上抓住解的小尺度特征获得较精确的解。在介绍多尺度有限单元法求解非均质多孔介质中三维地下水流问题的基本原理之后，对参数水平方向渐变垂直方向突变的非均质多孔介质中的三维地下水流和Borden实验场的三维地下水流分别用多尺度有限单元法和传统等参有限单元法进行了计算，结果表明在模拟高度非均质多孔介质中的三维地下水流问题时，多尺度有限单元法比传统有限单元法有效，既节省计算量又有较高的精度；在模拟非均质性弱的多孔介质中的三维地下水流问题时，多尺度有限单元法虽然也能在大尺度上获得较为精确的解，但效果不明显。 相似文献
13.
A method for multiscale parameter estimation with application to reservoir history matching is presented. Starting from a
given fine-scale model, coarser models are generated using a global upscaling technique where the coarse models are tuned
to match the solution of the fine model. Conditioning to dynamic data is done by history-matching the coarse model. Using
consistently the same resolution both for the forward and inverse problems, this model is successively refined using a combination
of downscaling and history matching until model-matching dynamic data are obtained at the finest scale. Large-scale corrections
are obtained using fast models, which, combined with a downscaling procedure, provide a better initial model for the final
adjustment on the fine scale. The result is thus a series of models with different resolution, all matching history as good
as possible with this grid. Numerical examples show that this method may significantly reduce the computational effort and/or
improve the quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine
scale. 相似文献
14.
The lattice Boltzmann (LB) method is an efficient technique for simulating fluid flow through individual pores of complex porous media. The ease with which the LB method handles complex boundary conditions, combined with the algorithm’s inherent parallelism, makes it an elegant approach to solving flow problems at the sub-continuum scale. However, the realities of current computational resources can limit the size and resolution of these simulations. A major research focus is developing methodologies for upscaling microscale techniques for use in macroscale problems of engineering interest. In this paper, we propose a hybrid, multiscale framework for simulating diffusion through porous media. We use the finite element (FE) method to solve the continuum boundary-value problem at the macroscale. Each finite element is treated as a sub-cell and assigned permeabilities calculated from subcontinuum simulations using the LB method. This framework allows us to efficiently find a macroscale solution while still maintaining information about microscale heterogeneities. As input to these simulations, we use synchrotron-computed 3D microtomographic images of a sandstone, with sample resolution of 3.34 μm. We discuss the predictive ability of these simulations, as well as implementation issues. We also quantify the lower limit of the continuum (Darcy) scale, as well as identify the optimal representative elementary volume for the hybrid LB–FE simulations. 相似文献
15.
General coupling extended multiscale FEM for elasto‐plastic consolidation analysis of heterogeneous saturated porous media 
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下载免费PDF全文 This paper presents a general coupling extended multiscale FEM (GCEMs) for solving the coupling problem of elasto‐plastic consolidation of heterogeneous saturated porous media. In the GCEMs, the numerical multiscale base functions for the solid skeleton and fluid phase of the coupling system are all constructed on the basis of the equivalent stiffness matrix of the unit cell, which not only contain the interaction between the solid and fluid phases but also consider the time effect. Furthermore, in order to improve the computational accuracy for two‐dimensional problems, a multi‐node coarse element strategy for the GCEMs is proposed, and a two‐scale iteration algorithm for the elasto‐plastic consolidation analysis is developed. Some one‐dimensional and two‐dimensional homogeneous and heterogeneous numerical examples are carried out to validate the proposed method through the comparison with the coupling multiscale FEM and standard FEM. Numerical results show that the newly developed GCEMs can almost preserve the same convergent property as the standard FEM and also possesses the advantages of high computational efficiency. In addition, the GCEMs can be easily applied to other coupling multifield and multiphase transient problems. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
16.
扩展有限元法模拟裂纹时独立于网格,因此该方法是目前求解裂纹问题最有效的数值方法。为了在计算代价不大的情况,实现大型结构分析中考虑小裂纹或提高裂纹附近精度,在裂纹附近一般采用小尺度单元,其他区域采用大尺度单元。提出了分析三维裂纹问题的多尺度扩展有限元法,在需要的地方采用小尺度单元。基于点插值构造了六面体任意节点单元。所有尺度单元都采用8节点六面体单元,这样六面体任意节点单元可方便有效地连接不同尺度单元。采用互作用积分法计算三维应力强度因子。边裂纹和中心圆裂纹算例分析结果表明,该方法是正确和有效的。 相似文献
17.
This paper presents a second-order work analysis in application to geotechnical problems by using a novel effective multiscale approach. To abandon complicated equations involved in conventional phenomenological models, this multiscale approach employs a micromechanically-based formulation, in which only four parameters are involved. The multiscale approach makes it possible a coupling of the finite element method (FEM) and the micromechanically-based model. The FEM is used to solve the boundary value problem (BVP) while the micromechanically-based model is utilized at the Gauss point of the FEM. Then, the multiscale approach is used to simulate a three-dimensional triaxial test and a plain-strain footing. On the basis of the simulations, material instabilities are analyzed at both mesoscale and global scale. The second-order work criterion is then used to analyze the numerical results. It opens a road to interpret and understand the micromechanisms hiding behind the occurrence of failure in geotechnical issues. 相似文献
18.
The multiscale finite-volume (MSFV) method has been developed to solve multiphase flow problems on large and highly heterogeneous
domains efficiently. It employs an auxiliary coarse grid, together with its dual, to define and solve a coarse-scale pressure
problem. A set of basis functions, which are local solutions on dual cells, is used to interpolate the coarse-grid pressure
and obtain an approximate fine-scale pressure distribution. However, if flow takes place in presence of gravity (or capillarity),
the basis functions are not good interpolators. To treat this case correctly, a correction function is added to the basis
function interpolated pressure. This function, which is similar to a supplementary basis function independent of the coarse-scale
pressure, allows for a very accurate fine-scale approximation. In the coarse-scale pressure equation, it appears as an additional
source term and can be regarded as a local correction to the coarse-scale operator: It modifies the fluxes across the coarse-cell
interfaces defined by the basis functions. Given the closure assumption that localizes the pressure problem in a dual cell,
the derivation of the local problem that defines the correction function is exact, and no additional hypothesis is needed.
Therefore, as in the original MSFV method, the only closure approximation is the localization assumption. The numerical experiments
performed for density-driven flow problems (counter-current flow and lock exchange) demonstrate excellent agreement between
the MSFV solutions and the corresponding fine-scale reference solutions. 相似文献
19.
Similar to plane strain, axisymmetric stress problem is also highly kinematics constrained. Standard displacement‐based finite element exhibits volumetric locking issue in simulating nearly/fully incompressible material or isochoric plasticity under axisymmetric loading conditions, which severely underestimates the deformation and overestimates the bearing capacity for structural/geotechnical engineering problems. The aim of this paper is to apply variational multiscale method to produce a stabilized mixed displacement–pressure formulation, which can effectively alleviate the volumetric locking issue for axisymmetric stress problem. Both nearly incompressible elasticity and isochoric J2 elastoplasticity are investigated. First‐order 3‐node triangular and 4‐node quadrilateral elements are tested for locking issues. Several representative simulations are provided to demonstrate the performance of the linear elements, which include the convergence study and comparison with closed‐form solutions. A comparative study with pressure Laplacian stabilized formulation is also presented. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献