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1.
Downhole electrical heating can be used to achieve the high temperatures required for in situ upgrading of oil shale or oil
sands. Heater-well models are needed if this process is to be simulated accurately. The traditional Peaceman approach used
for fluid injection and production wells may not be applicable because it does not capture transient effects, which can be
important in downhole heater models. Standard models also neglect the effects of heterogeneity and temperature dependence
in the rock properties. Here, we develop two new models for representing heater wells in reservoir simulators. The first model
is applicable for homogeneous systems with properties that are not temperature dependent. For such cases, we develop a semi-analytical
procedure based on Green’s functions to construct time-dependent heater-well indexes and heater-block thermal transmissibilities.
For the general case, which can include both fine-scale heterogeneity and nonlinearity due to the temperature dependence of
rock properties, we present a numerical procedure for constructing the heater-well model. This technique is essentially a
near-well upscaling method and requires a local fine-scale solution in the near-well region. The boundary conditions are determined
using a local-global treatment. The accuracy of the new heater-well models is demonstrated through comparison to reference
solutions for example problems. The approach is then applied for the coarse-scale modeling of the in situ upgrading of oil
shale, which entails a thermal-compositional simulation with chemical reactions. The model is shown to provide an accurate
and efficient solution for this challenging problem. 相似文献
2.
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
3.
Uncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. We apply a newly developed global compositional upscaling method to provide coarse-scale parameters and functions for selected realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of a-factors to capture component fluxes. Additional features include near-well upscaling for all coarse parameters and functions, and iteration on the a-factors, which is shown to improve accuracy. In the EnLU framework, this global upscaling is applied for only a few selected realizations. For 90 % or more of the realizations, upscaled functions are assigned statistically based on quickly computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of two-dimensional models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50, and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures. 相似文献
4.
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.
Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities. 相似文献
7.
A multi-resolution workflow to generate high-resolution models constrained to dynamic data 总被引：2，自引：0，他引：2
Céline Scheidt Jef Caers Yuguang Chen Louis J. Durlofsky 《Computational Geosciences》2011,15(3):545-563
Distance-based stochastic techniques have recently emerged in the context of ensemble modeling, in particular for history
matching, model selection and uncertainty quantification. Starting with an initial ensemble of realizations, a distance between
any two models is defined. This distance is defined such that the objective of the study is incorporated into the geological
modeling process, thereby potentially enhancing the efficacy of the overall workflow. If the intent is to create new models
that are constrained to dynamic data (history matching), the calculation of the distance requires flow simulation for each
model in the initial ensemble. This can be very time consuming, especially for high-resolution models. In this paper, we present
a multi-resolution framework for ensemble modeling. A distance-based procedure is employed, with emphasis on the rapid construction
of multiple models that have improved dynamic data conditioning. Our intent is to construct new high-resolution models constrained
to dynamic data, while performing most of the flow simulations only on upscaled models. An error modeling procedure is introduced
into the distance calculations to account for potential errors in the upscaling. Based on a few fine-scale flow simulations,
the upscaling error is estimated for each model using a clustering technique. We demonstrate the efficiency of the method
on two examples, one where the upscaling error is small, and another where the upscaling error is significant. Results show
that the error modeling procedure can accurately capture the error in upscaling, and can thus reproduce the fine-scale flow
behavior from coarse-scale simulations with sufficient accuracy (in terms of uncertainty predictions). As a consequence, an
ensemble of high-resolution models, which are constrained to dynamic data, can be obtained, but with a minimum of flow simulations
at the fine scale. 相似文献
8.
The aim of upscaling is to determine equivalent homogeneous parameters at a coarse-scale from a spatially oscillating fine-scale parameter distribution. To be able to use a limited number of relatively large grid-blocks in numerical oil reservoir simulators or groundwater models, upscaling of the permeability is frequently applied. The spatial fine-scale permeability distribution is generally obtained from geological and geostatistical models. After upscaling, the coarse-scale permeabilities are incorporated in the relatively large grid-blocks of the numerical model. If the porous rock may be approximated as a periodic medium, upscaling can be performed by the method of homogenization. In this paper the homogenization is performed numerically, which gives rise to an approximation error. The complementarity between two different numerical methods – the conformal-nodal finite element method and the mixed-hybrid finite element method – has been used to quantify this error. These two methods yield respectively upper and lower bounds for the eigenvalues of the coarse-scale permeability tensor. Results of 3D numerical experiments are shown, both for the far field and around wells. 相似文献
9.
In subsurface flow modeling, compositional simulation is often required to model complex recovery processes, such as gas/CO 2 injection. However, compositional simulation on fine-scale geological models is still computationally expensive and even prohibitive. Most existing upscaling techniques focus on black-oil models. In this paper, we present a general framework to upscale two-phase multicomponent flow in compositional simulation. Unlike previous studies, our approach explicitly considers the upscaling of flow and thermodynamics. In the flow part, we introduce a new set of upscaled flow functions that account for the effects of compressibility. This is often ignored in the upscaling of black-oil models. In the upscaling of thermodynamics, we show that the oil and gas phases within a coarse block are not at chemical equilibrium. This non-equilibrium behavior is modeled by upscaled thermodynamic functions, which measure the difference between component fugacities among the oil and gas phases. We apply the approach to various gas injection problems with different compositional features, permeability heterogeneity, and coarsening ratios. It is shown that the proposed method accurately reproduces the averaged fine-scale solutions, such as component overall compositions, gas saturation, and density solutions in the compositional flow. 相似文献
10.
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
11.
Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds. 相似文献
12.
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
13.
Machine-learning-based modeling of coarse-scale error,with application to uncertainty quantification
The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions. 相似文献
14.
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
15.
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
17.
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
18.
The paper is devoted to the upscaling method appropriate for single-phase flow in media with discontinuous permeability distribution.
The suggested algorithm is a modification of the iterative adaptive local–global upscaling developed by Chen and coauthors.
The key feature of this method is a consistency between local and coarse global calculated characteristics. In this work,
we apply a modified procedure to determine the boundary conditions used in the local fine-scale computation. To increase the
accuracy of these boundary conditions on each iteration, we involve an additional preliminary step based on the results of
coarse scale calculations from the previous iteration. Numerical tests show an essential improvement of the accuracy of upscaled
flow rates for most of the realizations of statistical permeability distribution. Although the developed method is universal,
its efficiency increases with increasing of permeability contrast. 相似文献
19.
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
20.
We describe a new approach for simulation of multiphase flows through heterogeneous porous media, such as oil reservoirs.
The method, which is based on the wavelet transformation of the spatial distribution of the single-phase permeabilities, incorporates
in the upscaled computational grid all the relevant data on the permeability, porosity, and other important properties of
a porous medium at all the length scales. The upscaling method generates a nonuniform computational grid which preserves the resolved structure
of the geological model in the near-well zones as well as in the high-permeability sectors and upscales the rest of the geological
model. As such, the method is a multiscale one that preserves all the important information across all the relevant length
scales. Using a robust front-detection method which eliminates the numerical dispersion by a high-order total variation diminishing
method (suitable for the type of nonuniform upscaled grid that we generate), we obtain highly accurate results with a greatly
reduced computational cost. The speed-up in the computations is up to over three orders of magnitude, depending on the degree
of heterogeneity of the model. To demonstrate the accuracy and efficiency of our methods, five distinct models (including
one with fractures) of heterogeneous porous media are considered, and two-phase flows in the models are studied, with and
without the capillary pressure. 相似文献