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1.
In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system.
A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic
transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase
flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase
flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system. 相似文献
2.
3.
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
4.
Use of Border Regions for Improved Permeability Upscaling 总被引：1，自引：0，他引：1
A procedure for the improved calculation of upscaled grid block permeability tensors on Cartesian grids is described and applied. The method entails the use of a border region of fine-scale cells surrounding the coarse block for which the upscaled permeability is to be computed. The implementation allows for the use of full-tensor permeability fields on the fine and coarse scales. Either periodic or pressure–no flow boundary conditions are imposed over the extended local domain (target block plus border regions) though averaged quantities, used to compute the upscaled permeability tensor, are computed only over the target block region. Flow and transport results using this procedure are compared to those from standard methods for different types of geological and simulation models. Improvement using the new approach is consistently observed for the cases considered, though the degree of improvement varies for different models and flow quantities. 相似文献
5.
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
6.
Odd Kolbjørnsen Marita Stien Heidi Kjønsberg Bjørn Fjellvoll Petter Abrahamsen 《Mathematical Geosciences》2014,46(2):205-225
A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation. 相似文献
7.
8.
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
9.
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
10.
The spatial distribution of rock properties in porous media, such as permeability and porosity, often is strongly variable. Therefore, these properties usefully may be considered as a random field. However, this variability is correlated frequently on length scales comparable to geological lengths (for example, scales of sand bodies or facies). To solve various engineering problems (for example, in the oil recovery process) numerical models of a porous medium often are used. A need exists then to understand correlated random fields and to generate them over discretized numerical grids. The paper describes the general mathematical methods required to do this, with one particular method (the nearest neighbor model) described in detail. How parameters of the mathematical model may be related to rock property statistics for the nearest neighbor model is shown. The method is described in detail in one, two, and three dimensions. Examples are given of how model parameters may be determined from real data. 相似文献
11.
Regional scale models of groundwater flow and transport often employ domain discretizations with grid blocks larger than typical scales of field data. For heterogeneous formations, this difference in scales is often handled by using effective (upscaled) parameters. We investigate the problem of upscaling hydraulic conductivity and transmissivity from a small scale of measurement to a larger scale of grid blocks. Transmissivity statistics is expressed in terms of statistics of hydraulic conductivity, and expressions for the effective (upscaled) hydraulic conductivity K
eff and transmissivity T
eff for steady state flow in confined heterogeneous aquifers are derived by means of stochastic averaging and perturbation analysis. These expressions reveal that the commonly used relation T
eff = BK
eff, where B is the confined aquifer thickness, is not generally valid. 相似文献
12.
Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be “upscaled” to one “equivalent homogeneous” coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented. 相似文献
13.
In this paper, a multiscale homogenization approach is developed for fully coupled saturated porous media to represent the idealized sugar cube model, which is generally employed in fractured porous media on the basis of dual porosity models. In this manner, an extended version of the Hill-Mandel theory that incorporates the microdynamic effects into the multiscale analysis is presented, and the concept of the deformable dual porosity model is demonstrated. Numerical simulations are performed employing the multiscale analysis and dual porosity model, and the results are compared with the direct numerical simulation through 2 numerical examples. Finally, a combined multiscale-dual porosity technique is introduced by employing a bridge between these 2 techniques as an alternative approach that reduces the computational cost of numerical simulation in modeling of heterogeneous deformable porous media. 相似文献
14.
This paper addresses the problem of explicit fractured media modelling in an operational case. On one side, realistic fracture models are mainly used for research purposes in order to investigate better the flow behaviour impacted by the complex multi-scale fracture network. Often, a very fine grid and hence an increased computation time are needed. On the other hand, an operational fractured reservoir is still generally modelled using an implicit fracture media representation. The upscaled petrophysical properties and dual media are defined on a coarse grid to limit the computational time of dynamic simulation. The challenge of this work is to demonstrate that an explicit fracture modelling is not reserved only for the research domain, but can be applied to an operational case study. The static model is constructed using a multiple point statistics approach in order to represent complex interaction patterns of fractures and faults observed at the analogue outcrop. The dynamic behaviour is simulated based on this spatial fracture network representation. 相似文献
15.
Statistical assignment of upscaled flow functions for an ensemble of geological models 总被引：1，自引：0，他引：1
Upscaled flow functions are often needed to account for the effects of fine-scale permeability heterogeneity in coarse-scale
simulation models. We present procedures in which the required coarse-scale flow functions are statistically assigned to an
ensemble of upscaled geological models. This can be viewed as an extension and further development of a recently developed
ensemble level upscaling (EnLU) approach. The method aims to efficiently generate coarse-scale flow models capable of reproducing
the ensemble statistics (e.g., cumulative distribution function) of fine-scale flow predictions for multiple reservoir models.
The most expensive part of standard coarsening procedures is typically the generation of upscaled two-phase flow functions
(e.g., relative permeabilities). EnLU provides a means for efficiently generating these upscaled functions using stochastic
simulation. This involves the use of coarse-block attributes that are both fast to compute and correlate closely with the
upscaled two-phase functions. In this paper, improved attributes for use in EnLU, namely the coefficient of variation of the
fine-scale single-phase velocity field (computed during computation of upscaled absolute permeability) and the integral range
of the fine-scale permeability variogram, are identified. Geostatistical simulation methods, which account for spatial correlations
of the statistically generated upscaled functions, are also applied. The overall methodology thus enables the efficient generation
of coarse-scale flow models. The procedure is tested on 3D well-driven flow problems with different permeability distributions
and variable fluid mobility ratios. EnLU is shown to capture the ensemble statistics of fine-scale flow results (water and
oil flow rates as a function of time) with similar accuracy to full flow-based upscaling methods but with computational speedups
of more than an order of magnitude. 相似文献
16.
The simulation of flow in porous and fibrous permeable media is of high importance in many scientific and industrial applications. Although the finite element models at the representative elementary volume scales are used to solve a huge amount of scientific and engineering problems, they are hardly used to efficiently simulate pore-fluid flow problems at the particle scales. This encourages the development of numerical models to match the needs of such studies. In this paper, we propose a new Gray Lattice Boltzmann numerical model for simulating fluid flow in permeable media. Unlike most previous models, our proposed model has the ability to simulate multi-layers and space-variable permeability while preserving the continuity of the macroscopic velocity field. The model is verified with the available analytical solutions and a derived analytical expression for the case of variable porosity. In addition, we examine the importance of introducing a transition layer with a defined porosity function near the boundaries and interfaces. If this layer exists in practice, then the numerical results reveal that it cannot be neglected, and its impact is significant on the obtained velocity distribution. Finally, in the light of the obtained results, we can state that the proposed model has great potential to simulate complex and heterogeneous media with smoothness and accuracy, so that it may enrich the research content of the emerging computational geosciences. 相似文献
17.
Sabine Attinger 《Computational Geosciences》2003,7(4):253-273
This paper focuses on heterogeneous soil conductivities and on the impact their resolution has on a solution of the piezometric
head equation: owing to spatial variations of the conductivity, the flow properties at larger scales differ from those found
for experiments performed at smaller scales. The method of coarse graining is proposed in order to upscale the piezometric
head equation on arbitrary intermediate scales. At intermediate scales large scale fluctuations of the conductivities are
resolved, whereas small scale fluctuations are smoothed by a partialy spatial filtering procedure. The filtering procedure
is performed in Fourier space with the aid of a low-frequency cut-off function. We derive the partially upscaled head equations.
In these equations, the impact of the small scale variability is modeled by scale dependent effective conductivities which
are determined by additional differential equations. Explicit results for the scale dependent conductivity values are presented
in lowest order perturbation theory. The perturbation theory contributions are summed up with using a renormalisation group
analysis yielding explicit results for the effective conductivity in isotropic media. Therefore, the results are also valid
for highly heterogeneous media. The results are compared with numerical simulations performed by Dykaar and Kitanidis (1992).
The method of coarse graining combined by a renormalisation group analysis offers a tool to derive exact and explicit expressions
for resolution dependent conductivity values. It is, e.g., relevant for the interpretation of measurement data on different
scales and for reduction of grid-block resolution in numerical modeling.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
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
19.
Coupled hydro-mechanical (HM) processes are significant in geological engineering such as oil and gas extraction, geothermal energy, nuclear waste disposal and for the safety assessment of dam foundations and rock slopes, where the geological media usually consist of fractured rock masses. In this study, we developed a model for the analysis of coupled hydro-mechanical processes in porous rock containing dominant fractures, by using the numerical manifold method (NMM). In the current model, the fractures are regarded as different material domains from surrounding rock, i.e., finite-thickness fracture zones as porous media. Compared with the rock matrix, these fractured porous media are characterized with nonlinear behavior of hydraulic and mechanical properties, involving not only direct (poroelastic) coupling but also indirect (property change) coupling. By combining the potential energy associated with mechanical responses, fluid flow and solid–fluid interactions, a new formulation for direct HM coupling in porous media is established. For indirect coupling associated with fracture opening/closure, we developed a new approach implicitly considering the nonlinear properties by directly assembling the corresponding strain energy. Compared with traditional methods with approximation of the nonlinear constitutive equations, this new formulation achieves a more accurate representation of the nonlinear behavior. We implemented the new model for coupled HM analysis in NMM, which has fixed mathematical grid and accurate integration, and developed a new computer code. We tested the code for direct coupling on two classical poroelastic problems with coarse mesh and compared the results with the analytical solutions, achieving excellent agreement, respectively. Finally, we tested for indirect coupling on models with a single dominant fracture and obtained reasonable results. The current poroelastic NNM model with a continuous finite-thickness fracture zone will be further developed considering thin fractures in a discontinuous approach for a comprehensive model for HM analysis in fractured porous rock masses. 相似文献