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1.
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
2.
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
3.
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
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.
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
6.
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
7.
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
8.
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
9.
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
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.
In this paper, we develop a multiscale model reduction technique that describes shale gas transport in fractured media. Due to the pore-scale heterogeneities and processes, we use upscaled models to describe the matrix. We follow our previous work (Akkutlu et al. Transp. Porous Media 107(1), 235–260, 2015), where we derived an upscaled model in the form of generalized nonlinear diffusion model to describe the effects of kerogen. To model the interaction between the matrix and the fractures, we use Generalized Multiscale Finite Element Method (Efendiev et al. J. Comput. Phys. 251, 116–135, 2013, 2015). In this approach, the matrix and the fracture interaction is modeled via local multiscale basis functions. In Efendiev et al. (2015), we developed the GMsFEM and applied for linear flows with horizontal or vertical fracture orientations aligned with a Cartesian fine grid. The approach in Efendiev et al. (2015) does not allow handling arbitrary fracture distributions. In this paper, we (1) consider arbitrary fracture distributions on an unstructured grid; (2) develop GMsFEM for nonlinear flows; and (3) develop online basis function strategies to adaptively improve the convergence. The number of multiscale basis functions in each coarse region represents the degrees of freedom needed to achieve a certain error threshold. Our approach is adaptive in a sense that the multiscale basis functions can be added in the regions of interest. Numerical results for two-dimensional problem are presented to demonstrate the efficiency of proposed approach. 相似文献
12.
Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior
is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal
with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model).
Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit
algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation).
The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homo geneous
elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities.
The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/
sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary
pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local
problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute
the fine-scale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz
iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of
nonlinear compressible multiphase flow in strongly heterogeneous formations. 相似文献
13.
In this paper, we propose multilevel Monte Carlo (MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multiphase flow and transport. The contribution of this paper is twofold: (1) a design of ensemble level mixed multiscale finite element methods and (2) a novel use of mixed multiscale finite element methods within multilevel Monte Carlo techniques to speed up the computations. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. In this paper, we consider two ensemble level mixed multiscale finite element methods: (1) the no-local-solve-online ensemble level method (NLSO); and (2) the local-solve-online ensemble level method (LSO). The first approach was proposed in Aarnes and Efendiev (SIAM J. Sci. Comput. 30(5):2319-2339, 2008) while the second approach is new. Both mixed multiscale methods use a number of snapshots of the permeability media in generating multiscale basis functions. As a result, in the off-line stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method, one uses the whole set of precomputed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method, one uses the precomputed functions to construct a multiscale basis for a particular realization. With this basis, the solution corresponding to this particular realization is approximated in LSO mixed multiscale finite element method (MsFEM). In both approaches, the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to precompute a multiscale basis. In this paper, ensemble level multiscale methods are used in multilevel Monte Carlo methods (Giles 2008a, Oper.Res. 56(3):607-617, b). In multilevel Monte Carlo methods, more accurate (and expensive) forward simulations are run with fewer samples, while less accurate (and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations based on the number of coarse degrees of freedom, one can show that MLMC methods can provide better accuracy at the same cost as Monte Carlo (MC) methods. The main objective of the paper is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. Further, we use both approaches in the context of MLMC to speedup MC calculations. 相似文献
14.
A multiscale method for the dynamic analysis of underground structures is proposed, which involves the concurrent discretization of the entire domain with both coarse‐scale and fine‐scale finite element meshes. The coarse‐scale mesh is employed to capture seismic response characteristics of the integral system, whereas the fine‐scale mesh describes in detail the dynamic response in positions of potential damage or interest. For both the coarse‐scale and fine‐scale meshes to overlap, a bridging scale term is introduced so that compatibility of dynamic behavior between the coarse‐ and fine‐scale models is enforced. Both material and contact nonlinearities are considered in the multiscale model. As an application, the model is used for large‐scale seismic response of a newly built long‐distance shield tunnel. Results show that this multiscale method does not have spurious wave reflections at the fine/coarse interface and does not need filtering procedures, which is an advantage compared with the displacement coupling method. Stress and deformation response in lining segments and their connecting bolts are investigated and analyzed within the fine‐scale model, and the capacity of critical structural components, such as bolts and joints is evaluated. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
15.
16.
Strain‐softening in geomaterials often leads to ill‐posed boundary‐valued problems (BVP), which cannot be solved with finite element methods without introducing some kind of regularization such as nonlocal plasticity. Hereafter we propose to apply spectral analysis for testing the performance of nonlocal plasticity in regularizing ill‐posed BVP and producing mesh‐independent solutions when local plasticity usually fails. The spectral analysis consists of examining the eigenvalues and eigenvectors of the global tangential stiffness matrix of the incremental equilibrium equations. Based on spectral analysis, we propose a criterion for passing or failing the test of constitutive regularization in the context of BVP. If the eigenvalues of the tangential operator are all positive then the regularization succeeds, otherwise it fails and may not prevent artificial mesh‐dependent solutions from appearing. The approach is illustrated in the particular case of a biaxial compression with strain‐softening plasticity. In this particular case, local softening plasticity is found to produce negative eigenvalues in the tangential stiffness matrix, which indicates ill‐posed BVP. In contrast, nonlocal softening plasticity always produces positive eigenvalues, which regularizes ill‐posed BVP. The dominant eigenvectors, which generate localized deformation patterns, have a bandwidth independent of mesh size, provided that the mesh is fine enough to capture localization. These mesh‐independent eigenmodes explain why nonlocal plasticity produces numerical solutions that are mesh‐independent. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
17.
18.
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
19.
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
20.
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