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1.
The adaptive Gaussian mixture filter (AGM) was introduced as a robust filter technique for large-scale applications and an alternative to the well-known ensemble Kalman filter (EnKF). It consists of two analysis steps, one linear update and one weighting/resampling step. The bias of AGM is determined by two parameters, one adaptive weight parameter (forcing the weights to be more uniform to avoid filter collapse) and one predetermined bandwidth parameter which decides the size of the linear update. It has been shown that if the adaptive parameter approaches one and the bandwidth parameter decreases, as an increasing function of the sample size, the filter can achieve asymptotic optimality. For large-scale applications with a limited sample size, the filter solution may be far from optimal as the adaptive parameter gets close to zero depending on how well the samples from the prior distribution match the data. The bandwidth parameter must often be selected significantly different from zero in order to make large enough linear updates to match the data, at the expense of bias in the estimates. In the iterative AGM we introduce here, we take advantage of the fact that the history matching problem is usually estimation of parameters and initial conditions. If the prior distribution of initial conditions and parameters is close to the posterior distribution, it is possible to match the historical data with a small bandwidth parameter and an adaptive weight parameter that gets close to one. Hence, the bias of the filter solution is small. In order to obtain this scenario, we iteratively run the AGM throughout the data history with a very small bandwidth to create a new prior distribution from the updated samples after each iteration. After a few iterations, nearly all samples from the previous iteration match the data, and the above scenario is achieved. A simple toy problem shows that it is possible to reconstruct the true posterior distribution using the iterative version of the AGM. Then a 2D synthetic reservoir is revisited to demonstrate the potential of the new method on large-scale problems. 相似文献
2.
Andreas S. Stordal Hans A. Karlsen Geir Nævdal Hans J. Skaug Brice Vallès 《Computational Geosciences》2011,15(2):293-305
The nonlinear filtering problem occurs in many scientific areas. Sequential Monte Carlo solutions with the correct asymptotic
behavior such as particle filters exist, but they are computationally too expensive when working with high-dimensional systems.
The ensemble Kalman filter (EnKF) is a more robust method that has shown promising results with a small sample size, but the
samples are not guaranteed to come from the true posterior distribution. By approximating the model error with a Gaussian
distribution, one may represent the posterior distribution as a sum of Gaussian kernels. The resulting Gaussian mixture filter
has the advantage of both a local Kalman type correction and the weighting/resampling step of a particle filter. The Gaussian
mixture approximation relies on a bandwidth parameter which often has to be kept quite large in order to avoid a weight collapse
in high dimensions. As a result, the Kalman correction is too large to capture highly non-Gaussian posterior distributions.
In this paper, we have extended the Gaussian mixture filter (Hoteit et al., Mon Weather Rev 136:317–334, 2008) and also made the connection to particle filters more transparent. In particular, we introduce a tuning parameter for the
importance weights. In the last part of the paper, we have performed a simulation experiment with the Lorenz40 model where
our method has been compared to the EnKF and a full implementation of a particle filter. The results clearly indicate that
the new method has advantages compared to the standard EnKF. 相似文献
3.
A probabilistic parametrization for geological uncertainty estimation using the ensemble Kalman filter (EnKF) 总被引：2，自引：0，他引：2
In the past years, many applications of history-matching methods in general and ensemble Kalman filter in particular have been proposed, especially in order to estimate fields that provide uncertainty in the stochastic process defined by the dynamical system of hydrocarbon recovery. Such fields can be permeability fields or porosity fields, but can also fields defined by the rock type (facies fields). The estimation of the boundaries of the geologic facies with ensemble Kalman filter (EnKF) was made, in different papers, with the aid of Gaussian random fields, which were truncated using various schemes and introduced in a history-matching process. In this paper, we estimate, in the frame of the EnKF process, the locations of three facies types that occur into a reservoir domain, with the property that each two could have a contact. The geological simulation model is a form of the general truncated plurigaussian method. The difference with other approaches consists in how the truncation scheme is introduced and in the observation operator of the facies types at the well locations. The projection from the continuous space of the Gaussian fields into the discrete space of the facies fields is realized through in an intermediary space (space with probabilities). This space connects the observation operator of the facies types at the well locations with the geological simulation model. We will test the model using a 2D reservoir which is connected with the EnKF method as a data assimilation technique. We will use different geostatistical properties for the Gaussian fields and different levels of the uncertainty introduced in the model parameters and also in the construction of the Gaussian fields. 相似文献
4.
The study has been focused on examining the usage and the applicability of ensemble Kalman filtering techniques to the history
matching procedures. The ensemble Kalman filter (EnKF) is often applied nowadays to solving such a problem. Meanwhile, traditional
EnKF requires assumption of the distribution’s normality. Besides, it is based on the linear update of the analysis equations.
These facts may cause problems when filter is used in reservoir applications and result in sampling error. The situation becomes
more problematic if the a priori information on the reservoir structure is poor and initial guess about the, e.g., permeability
field is far from the actual one. The above circumstance explains a reason to perform some further research concerned with
analyzing specific modification of the EnKF-based approach, namely, the iterative EnKF (IEnKF) scheme, which allows restarting
the procedure with a new initial guess that is closer to the actual solution and, hence, requires less improvement by the
algorithm while providing better estimation of the parameters. The paper presents some examples for which the IEnKF algorithm
works better than traditional EnKF. The algorithms are compared while estimating the permeability field in relation to the
two-phase, two-dimensional fluid flow model. 相似文献
5.
6.
Continuous Facies Updating Using the Ensemble Kalman Filter and the Level Set Method 总被引：1，自引：1，他引：0
We present a methodology based on the ensemble Kalman filter (EnKF) and the level set method for the continuous model updating of geological facies with respect to production data. Geological facies are modeled using an implicit surface representation and conditioned to production data using the ensemble Kalman filter. The methodology is based on Gaussian random fields used to deform the facies boundaries. The Gaussian random fields are used as the model parameter vector to be updated sequentially within the EnKF when new measurements are available. We show the successful application of the methodology to two synthetic reservoir models. 相似文献
7.
In this paper we present an extension of the ensemble Kalman filter (EnKF) specifically designed for multimodal systems. EnKF
data assimilation scheme is less accurate when it is used to approximate systems with multimodal distribution such as reservoir
facies models. The algorithm is based on the assumption that both prior and posterior distribution can be approximated by
Gaussian mixture and it is validated by the introduction of the concept of finite ensemble representation. The effectiveness
of the approach is shown with two applications. The first example is based on Lorenz model. In the second example, the proposed
methodology combined with a localization technique is used to update a 2D reservoir facies models. Both applications give
evidence of an improved performance of the proposed method respect to the EnKF. 相似文献
8.
Elisabeth Peters 《Computational Geosciences》2011,15(2):345-358
The ensemble Kalman filter (EnKF) appears to give good results for matching production data at existing wells. However, the
predictive power of these models outside of the existing wells is much more uncertain. In this paper, for a channelized reservoir
for five different cases with different levels of information the production history is matched using the EnKF. The predictive
power of the resulting model is tested for the existing wells and for new wells. The results show a consistent improvement
for the predictions at the existing wells after assimilation of the production data, but not for prediction of production
at new well locations. The latter depended on the settings of the problem and prior information used. The results also showed
that the fit during the history match was not always a good predictor for predictive capabilities of the history match model.
This suggests that some form of validation outside of observed wells is essential. 相似文献
9.
The ensemble Kalman filter (EnKF) has become a popular method for history matching production and seismic data in petroleum
reservoir models. However, it is known that EnKF may fail to give acceptable data matches especially for highly nonlinear
problems. In this paper, we introduce a procedure to improve EnKF data matches based on assimilating the same data multiple
times with the covariance matrix of the measurement errors multiplied by the number of data assimilations. We prove the equivalence
between single and multiple data assimilations for the linear-Gaussian case and present computational evidence that multiple
data assimilations can improve EnKF estimates for the nonlinear case. The proposed procedure was tested by assimilating time-lapse
seismic data in two synthetic reservoir problems, and the results show significant improvements compared to the standard EnKF.
In addition, we review the inversion schemes used in the EnKF analysis and present a rescaling procedure to avoid loss of
information during the truncation of small singular values. 相似文献
10.
In recent years, data assimilation techniques have been applied to an increasingly wider specter of problems. Monte Carlo
variants of the Kalman filter, in particular, the ensemble Kalman filter (EnKF), have gained significant popularity. EnKF
is used for a wide variety of applications, among them for updating reservoir simulation models. EnKF is a Monte Carlo method,
and its reliability depends on the actual size of the sample. In applications, a moderately sized sample (40–100 members)
is used for computational convenience. Problems due to the resulting Monte Carlo effects require a more thorough analysis
of the EnKF. Earlier we presented a method for the assessment of the error emerging at the EnKF update step (Kovalenko et
al., SIAM J Matrix Anal Appl, in press). A particular energy norm of the EnKF error after a single update step was studied.
The energy norm used to assess the error is hard to interpret. In this paper, we derive the distribution of the Euclidean
norm of the sampling error under the same assumptions as before, namely normality of the forecast distribution and negligibility
of the observation error. The distribution depends on the ensemble size, the number and spatial arrangement of the observations,
and the prior covariance. The distribution is used to study the error propagation in a single update step on several synthetic
examples. The examples illustrate the changes in reliability of the EnKF, when the parameters governing the error distribution
vary. 相似文献
11.
Wiktoria Lawniczak Remus Hanea Arnold Heemink Dennis McLaughlin 《Computational Geosciences》2009,13(2):245-254
Reservoir management requires periodic updates of the simulation models using the production data available over time. Traditionally,
validation of reservoir models with production data is done using a history matching process. Uncertainties in the data, as
well as in the model, lead to a nonunique history matching inverse problem. It has been shown that the ensemble Kalman filter
(EnKF) is an adequate method for predicting the dynamics of the reservoir. The EnKF is a sequential Monte-Carlo approach that
uses an ensemble of reservoir models. For realistic, large-scale applications, the ensemble size needs to be kept small due
to computational inefficiency. Consequently, the error space is not well covered (poor cross-correlation matrix approximations)
and the updated parameter field becomes scattered and loses important geological features (for example, the contact between
high- and low-permeability values). The prior geological knowledge present in the initial time is not found anymore in the
final updated parameter. We propose a new approach to overcome some of the EnKF limitations. This paper shows the specifications
and results of the ensemble multiscale filter (EnMSF) for automatic history matching. EnMSF replaces, at each update time,
the prior sample covariance with a multiscale tree. The global dependence is preserved via the parent–child relation in the
tree (nodes at the adjacent scales). After constructing the tree, the Kalman update is performed. The properties of the EnMSF
are presented here with a 2D, two-phase (oil and water) small twin experiment, and the results are compared to the EnKF. The
advantages of using EnMSF are localization in space and scale, adaptability to prior information, and efficiency in case many
measurements are available. These advantages make the EnMSF a practical tool for many data assimilation problems. 相似文献
12.
Traditional ensemble-based history matching method, such as the ensemble Kalman filter and iterative ensemble filters, usually update reservoir parameter fields using numerical grid-based parameterization. Although a parameter constraint term in the objective function for deriving these methods exists, it is difficult to preserve the geological continuity of the parameter field in the updating process of these methods; this is especially the case in the estimation of statistically anisotropic fields (such as a statistically anisotropic Gaussian field and facies field with elongated facies) with uncertainties about the anisotropy direction. In this work, we propose a Karhunen-Loeve expansion-based global parameterization technique that is combined with the ensemble-based history matching method for inverse modeling of statistically anisotropic fields. By using the Karhunen-Loeve expansion, a Gaussian random field can be parameterized by a group of independent Gaussian random variables. For a facies field, we combine the Karhunen-Loeve expansion and the level set technique to perform the parameterization; that is, for each facies, we use a Gaussian random field and a level set algorithm to parameterize it, and the Gaussian random field is further parameterized by the Karhunen-Loeve expansion. We treat the independent Gaussian random variables in the Karhunen-Loeve expansion as the model parameters. When the anisotropy direction of the statistically anisotropic field is uncertain, we also treat it as a model parameter for updating. After model parameterization, we use the ensemble randomized maximum likelihood filter to perform history matching. Because of the nature of the Karhunen-Loeve expansion, the geostatistical characteristics of the parameter field can be preserved in the updating process. Synthetic cases are set up to test the performance of the proposed method. Numerical results show that the proposed method is suitable for estimating statistically anisotropic fields. 相似文献
13.
Anca Maria Hanea Maria Gheorghe Remus Hanea Dan Ababei 《Computational Geosciences》2013,17(6):929-949
Reservoir simulation models are used both in the development of new fields and in developed fields where production forecasts are needed for investment decisions. When simulating a reservoir, one must account for the physical and chemical processes taking place in the subsurface. Rock and fluid properties are crucial when describing the flow in porous media. In this paper, the authors are concerned with estimating the permeability field of a reservoir. The problem of estimating model parameters such as permeability is often referred to as a history-matching problem in reservoir engineering. Currently, one of the most widely used methodologies which address the history-matching problem is the ensemble Kalman filter (EnKF). EnKF is a Monte Carlo implementation of the Bayesian update problem. Nevertheless, the EnKF methodology has certain limitations that encourage the search for an alternative method.For this reason, a new approach based on graphical models is proposed and studied. In particular, the graphical model chosen for this purpose is a dynamic non-parametric Bayesian network (NPBN). This is the first attempt to approach a history-matching problem in reservoir simulation using a NPBN-based method. A two-phase, two-dimensional flow model was implemented for a synthetic reservoir simulation exercise, and initial results are shown. The methods’ performances are evaluated and compared. This paper features a completely novel approach to history matching and constitutes only the first part (part I) of a more detailed investigation. For these reasons (novelty and incompleteness), many questions are left open and a number of recommendations are formulated, to be investigated in part II of the same paper. 相似文献
14.
Using a small ensemble size in the ensemble Kalman filter methodology is efficient for updating numerical reservoir models
but can result in poor updates following spurious correlations between observations and model variables. The most common approach
for reducing the effect of spurious correlations on model updates is multiplication of the estimated covariance by a tapering
function that eliminates all correlations beyond a prespecified distance. Distance-dependent tapering is not always appropriate,
however. In this paper, we describe efficient methods for discriminating between the real and the spurious correlations in
the Kalman gain matrix by using the bootstrap method to assess the confidence level of each element from the Kalman gain matrix.
The new method is tested on a small linear problem, and on a water flooding reservoir history matching problem. For the water
flooding example, a small ensemble size of 30 was used to compute the Kalman gain in both the screened EnKF and standard EnKF
methods. The new method resulted in significantly smaller root mean squared errors of the estimated model parameters and greater
variability in the final updated ensemble. 相似文献
15.
集合卡曼滤波由于易于使用而被广泛地应用到陆面数据同化研究中,它是建立在模型为线性、误差为正态分布的假设上,而实际土壤湿度方程是高度非线性的,并且当土壤过干或过湿时会发生样本偏斜.为了全面评估它在同化表层土壤湿度观测来反演土壤湿度廓线的性能,特引入不需要上述假设的采样重要性重采样粒子滤波,比较非线性和偏斜性对同化算法的影响.结果显示:不管是小样本还是大样本,集合卡曼滤波都能快速、准确地逼近样本均值,而粒子滤波只有在大样本时才能缓慢地趋近;此外,集合卡曼滤波的粒子边缘概率密度及其偏度和峰度与粒子滤波完全不同,前者粒子虽不完全满足正态分布,但始终为单峰状态,而后者粒子随同化推进经历了单峰到双峰再到单峰的变化. 相似文献
16.
In this work, we present an efficient matrix-free ensemble Kalman filter (EnKF) algorithm for the assimilation of large data
sets. The EnKF has increasingly become an essential tool for data assimilation of numerical models. It is an attractive assimilation
method because it can evolve the model covariance matrix for a non-linear model, through the use of an ensemble of model states,
and it is easy to implement for any numerical model. Nevertheless, the computational cost of the EnKF can increase significantly
for cases involving the assimilation of large data sets. As more data become available for assimilation, a potential bottleneck
in most EnKF algorithms involves the operation of the Kalman gain matrix. To reduce the complexity and cost of assimilating
large data sets, a matrix-free EnKF algorithm is proposed. The algorithm uses an efficient matrix-free linear solver, based
on the Sherman–Morrison formulas, to solve the implicit linear system within the Kalman gain matrix and compute the analysis.
Numerical experiments with a two-dimensional shallow water model on the sphere are presented, where results show the matrix-free
implementation outperforming an singular value decomposition-based implementation in computational time. 相似文献
17.
To more correctly estimate the error covariance of an evolved state of a nonlinear dynamical system, the second and higher-order
moments of the prior error need to be known. Retrospective optimal interpolation (ROI) may require relatively less information
on the higher-order moments of the prior errors than an ensemble Kalman filter (EnKF) because it uses the initial conditions
as the background states instead of forecasts. Analogous to the extension of a Kalman filter into an EnKF, an ensemble retrospective
optimal interpolation (EnROI) technique was derived using the Monte Carlo method from ROI. In contrast to the deterministic
version of ROI, the background error covariance is represented by a background ensemble in EnROI. By sequentially applying
EnROI to a moving limited analysis window and exploiting the forecast from the average of the background ensemble of EnROI
as a guess field, the computation costs for EnROI can be reduced. In the numerical experiment using a Lorenz-96 model and
a Model-III of Lorenz with a perfect-model assumption, the cost-effectiveness of the suboptimal version of EnROI is demonstrated
to be superior to that of EnKF using perturbed observations. 相似文献
18.
Reza Tavakoli Gergina Pencheva Mary F. Wheeler Benjamin Ganis 《Computational Geosciences》2013,17(1):83-97
We present a parallel framework for history matching and uncertainty characterization based on the Kalman filter update equation for the application of reservoir simulation. The main advantages of ensemble-based data assimilation methods are that they can handle large-scale numerical models with a high degree of nonlinearity and large amount of data, making them perfectly suited for coupling with a reservoir simulator. However, the sequential implementation is computationally expensive as the methods require relatively high number of reservoir simulation runs. Therefore, the main focus of this work is to develop a parallel data assimilation framework with minimum changes into the reservoir simulator source code. In this framework, multiple concurrent realizations are computed on several partitions of a parallel machine. These realizations are further subdivided among different processors, and communication is performed at data assimilation times. Although this parallel framework is general and can be used for different ensemble techniques, we discuss the methodology and compare results of two algorithms, the ensemble Kalman filter (EnKF) and the ensemble smoother (ES). Computational results show that the absolute runtime is greatly reduced using a parallel implementation versus a serial one. In particular, a parallel efficiency of about 35 % is obtained for the EnKF, and an efficiency of more than 50 % is obtained for the ES. 相似文献
19.
The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale
problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly
difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g.,
water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a
limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the
model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application
of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares
data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the
data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state
space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance
with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with
observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with
second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root
filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem
is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range
by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model
dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem
in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving
ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the
ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance
from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble
methods when applied to nonlinear problems. 相似文献
20.
Investigation of the sampling performance of ensemble-based methods with a simple reservoir model 总被引：1，自引：0，他引：1
The application of the ensemble Kalman filter (EnKF) for history matching petroleum reservoir models has been the subject of intense investigation during the past 10 years. Unfortunately, EnKF often fails to provide reasonable data matches for highly nonlinear problems. This fact motivated the development of several iterative ensemble-based methods in the last few years. However, there exists no study comparing the performance of these methods in the literature, especially in terms of their ability to quantify uncertainty correctly. In this paper, we compare the performance of nine ensemble-based methods in terms of the quality of the data matches, quantification of uncertainty, and computational cost. For this purpose, we use a small but highly nonlinear reservoir model so that we can generate the reference posterior distribution of reservoir properties using a very long chain generated by a Markov chain Monte Carlo sampling algorithm. We also consider one adjoint-based implementation of the randomized maximum likelihood method in the comparisons. 相似文献