共查询到20条相似文献，搜索用时 31 毫秒
1.
We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution. 相似文献
2.
Sissel Slettemark Mundal Eirik Keilegavlen Ivar Aavatsmark 《Computational Geosciences》2010,14(4):509-525
Control-volume multipoint flux approximations (MPFA) are discussed for the simulation of complex near-well flow using geometrically
flexible grids. Due to the strong non-linearity of the near-well flow, a linear model will, in general, be inefficient. Instead,
a model accounting for the logarithmic pressure behavior in the well vicinity is advocated. This involves a non-uniform refinement
of the grid in the radial direction. The model accounts for both near-well anisotropies and heterogeneities. For a full simulation
involving multiple wells, this single-well approach can easily be coupled with the reservoir model. Numerical simulations
demonstrate the convergence behavior of this model using various MPFA schemes under different near-well conditions for single-phase
flow regimes. Two-phase simulations support the results of the single-phase simulations. 相似文献
3.
M. Schneider B. Flemisch R. Helmig K. Terekhov H. Tchelepi 《Computational Geosciences》2018,22(2):565-586
This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids. 相似文献
4.
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
5.
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
6.
Benjamin Faigle Rainer Helmig Ivar Aavatsmark Bernd Flemisch 《Computational Geosciences》2014,18(5):625-636
A sequential solution procedure is used to simulate compositional two-phase flow in porous media. We employ a multiphysics concept that adapts the numerical complexity locally according to the underlying processes to increase efficiency. The framework is supplemented by a local refinement of the simulation grid. To calculate the fluxes on such grids, we employ a combination of the standard two-point flux approximation and a multipoint flux approximation where the grid is refined. This is then used to simulate a large-scale example related to underground CO2 storage. 相似文献
7.
R. Eymard T. Gallouët C. Guichard R. Herbin R. Masson 《Computational Geosciences》2014,18(3-4):285-296
We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, and mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two-point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain. 相似文献
8.
In this paper, we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper, we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regard to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular, we look at how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics. 相似文献
9.
We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method and the displacements are approximated by a continuous Galerkin finite element method. First-order convergence in space and time is established in appropriate norms for the pressure, velocity, and displacement. Numerical results are presented that illustrate the behavior of the method. 相似文献
10.
11.
Large-scale flow models constructed using standard coarsening procedures may not accurately resolve detailed near-well effects.
Such effects are often important to capture, however, as the interaction of the well with the formation can have a dominant
impact on process performance. In this work, a near-well upscaling procedure, which provides three-phase well-block properties,
is developed and tested. The overall approach represents an extension of a recently developed oil–gas upscaling procedure
and entails the use of local well computations (over a region referred to as the local well model (LWM)) along with a gradient-based
optimization procedure to minimize the mismatch between fine and coarse-scale well rates, for oil, gas, and water, over the
LWM. The gradients required for the minimization are computed efficiently through solution of adjoint equations. The LWM boundary
conditions are determined using an iterative local-global procedure. With this approach, pressures and saturations computed
during a global coarse-scale simulation are interpolated onto LWM boundaries and then used as boundary conditions for the
fine-scale LWM computations. In addition to extending the overall approach to the three-phase case, this work also introduces
new treatments that provide improved accuracy in cases with significant flux from the gas cap into the well block. The near-well
multiphase upscaling method is applied to heterogeneous reservoir models, with production from vertical and horizontal wells.
Simulation results illustrate that the method is able to accurately capture key near-well effects and to provide predictions
for component production rates that are in close agreement with reference fine-scale results. The level of accuracy of the
procedure is shown to be significantly higher than that of a standard approach which uses only upscaled single-phase flow
parameters. 相似文献
12.
Modeling semi-steady state near-well flow performance for horizontal wells in anisotropic reservoirs
This paper presents a novel methodology to model semi-steady state horizontal well flow performance in an anisotropic reservoir taking into account flow in the near-well region for an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. In order to apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical space transformation is used in concert with rotational transforms to obtain a virtual isotropic model. The transformation preserves the volumes and pressures. It is not a novel concept, but different from previous approaches in the sense that it is only applied in the near-well domain to formulate an equally isotropic media. As a result, the use of this virtual isotropic model requires the Dietz shape factor for an ellipse, transformed from the original cylindrical near-well domain. The Dietz shape factors are determined numerically in this research. The semi-steady state well/near-well model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented. Furthermore, the well/near-well model is applied in coupling with streamline reservoir model for a water flooding case. This appears to be the first coupling of a well hydraulics model and a streamline simulator. It presents the application of the well/near-well model in integrated reservoir simulation in an efficient and accurate manner. The results demonstrate that the coupling approach with a streamline reservoir model and the well/near-well is of great potential for advanced well simulation efficiently. 相似文献
13.
Subsurface flows are affected by geological variability over a range of length scales. The modeling of well singularity in
heterogeneous formations is important for simulating flow in aquifers and petroleum reservoirs. In this paper, two approaches
in calculating the upscaled well index to capture the effects of fine scale heterogeneity in near-well regions are presented
and applied. We first develop a flow-based near-well upscaling procedure for geometrically flexible grids. This approach entails
solving local well-driven flows and requires the treatment of geometric effects due to the nonalignment between fine and coarse
scale grids. An approximate coarse scale well model based on a well singularity analysis is also proposed. This model, referred
to as near-well arithmetic averaging, uses only the fine scale permeabilities at well locations to compute the coarse scale
well index; it does not require solving any flow problems. These two methods are systematically tested on three-dimensional
models with a variety of permeability distributions. It is shown that both approaches provide considerable improvement over
a simple (arithmetic) averaging approach to compute the coarse scale well index. The flow-based approach shows close agreement
to the fine scale reference model, and the near-well arithmetic averaging also offers accuracy for an appropriate range of
parameters. The interaction between global flow and near-well upscaling is also investigated through the use of global fine
scale solutions in near-well scale-up calculations. 相似文献
14.
Louis J. Durlofsky 《Mathematical Geology》2000,32(4):421-438
15.
Ivar Aavatsmark 《Computational Geosciences》2007,11(3):199-206
Control-volume formulations for elliptic equations often use two-point flux stencils, even for skew grids. Any two-point flux
stencil may be interpreted as a multipoint flux stencil. This yields a definition of the permeability (or conductivity) tensor.
Formulas for calculating the permeability tensor, based on the user-specified quantities in the two-point flux stencil, are
given. Numerical test examples demonstrate the validity of the derivation.
相似文献
16.
This paper extends the multipoint flux-approximation (MPFA) control-volume method to quadrilateral grids for which the adjacent cells do not necessarily share corners. Examples are grids with faults and locally refined grids. This paper gives a derivation of the method for such grids. The difference between two-point flux-approximation (TPFA) results and MPFA results for faults and local grid refinements is demonstrated for synthetic problems. Further, the results are compared with results from uniform fine-grid simulations. The effect of repeated fault patterns as well as anisotropy is investigated. Large errors may be found for the TPFA method for flow through a series of faults in an anisotropic medium. Finally, a comparison is done for a reservoir field application. 相似文献
17.
Ilya D. Mishev 《Computational Geosciences》2002,6(3-4):253-268
We extend and generalize recently proposed finite volume methods using the framework of mixed finite element methods. Proposed discretizations are defined for tensor permeabilities and naturally produce a generalization of harmonic averaging, and are therefore well suited for heterogeneous and anisotropic media. They are locally mass conservative and work on extremely flexible distorted meshes. Flux variables can be excluded locally and the resulting discretizations for the pressures has the same stencils for Voronoi/PEBI grids as 2-point finite volume discretizations currently used in many simulators. 相似文献
18.
Piyang Liu Gary Douglas Couples Jun Yao Zhaoqin Huang Wenhui Song Jingsheng Ma 《Computational Geosciences》2018,22(5):1187-1201
The two-scale continuum model is widely used in simulating the reactive dissolution process and predicting the optimum injection rate for carbonate reservoir acidizing treatment. The numerical methods of this model are currently based on structured grids, which are not applicable for complicated geometries. In this study, a general numerical scheme for simulating a reactive flow problem on both structured and unstructured grids is presented based on the finite volume method (FVM). The convection and diffusion terms involved in the reactive flow model are discretized by using the upwind scheme and two-point flux approximation (TPFA), respectively. The location of the centroid node inside each control volume is moved by using an optimization algorithm to make the connections with the surrounding elements as orthogonal as possible, which systematically improves the accuracy of the TPFA scheme. Additionally, in order to avoid the computational complexity resulting from the discretization of the non-linear term, the mass balance equation is only discretized in the spatial domain to get a set of ordinary differential equations (ODEs). These ODEs are coupled with the reaction equations and then solved using the numerical algorithm on ODEs. The accuracy and efficiency of the proposed method are studied by comparing the results obtained from the proposed numerical method with previous experimental and numerical results. This comparison indicates that, compared with the previous methods, the proposed method predicts the wormhole structure more accurately. Finally, the presented method is used to check the effect of the domain geometry, and it is found that the geometry of the flow domain has no effect on the optimum injection velocity, but the radial domain requires a larger breakthrough volume than the linear domain when other parameters are fixed. 相似文献
19.
One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the
saturation. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple
models. Hence, we consider in this paper two extensions to standard capillary pressure relationships. Firstly, to correct
the nonphysical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous
porous media, we apply a model with a capillary threshold pressure that controls the penetration process. Mathematically,
we rewrite this model as inequality constraint at the interfaces, which allows discontinuities in the saturation and pressure.
For the standard model, often finite-volume schemes resulting in a nonlinear system for the saturation are applied. To handle
the enhanced model at the interfaces correctly, we apply a mortar discretization method on nonmatching meshes. Introducing
the flux as a new variable allows us to solve the inequality constraint efficiently. This method can be applied to both the
standard and the enhanced capillary model. As nonlinear solver, we use an active set strategy combined with a Newton method.
Several numerical examples demonstrate the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence
of the retardation term.
This work was supported in part by IRTG NUPUS. 相似文献
20.
?ystein Pettersen 《Computational Geosciences》2012,16(2):211-230
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. 相似文献