Integration of Markov mesh models and data assimilation techniques in complex reservoirs

We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics.     

An efficient matrix-free algorithm for the ensemble Kalman filter

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.     

Cross-covariances and localization for EnKF in multiphase flow data assimilation

The ensemble Kalman filter has been successfully applied for data assimilation in very large models, including those in reservoir simulation and weather. Two problems become critical in a standard implementation of the ensemble Kalman filter, however, when the ensemble size is small. The first is that the ensemble approximation to cross-covariances of model and state variables to data can indicate the presence of correlations that are not real. These spurious correlations give rise to model or state variable updates in regions that should not be updated. The second problem is that the number of degrees of freedom in the ensemble is only as large as the size of the ensemble, so the assimilation of large amounts of precise, independent data is impossible. Localization of the Kalman gain is almost universal in the weather community, but applications of localization for the ensemble Kalman filter in porous media flow have been somewhat rare. It has been shown, however, that localization of updates to regions of non-zero sensitivity or regions of non-zero cross-covariance improves the performance of the EnKF when the ensemble size is small. Localization is necessary for assimilation of large amounts of independent data. The problem is to define appropriate localization functions for different types of data and different types of variables. We show that the knowledge of sensitivity alone is not sufficient for determination of the region of localization. The region depends also on the prior covariance for model variables and on the past history of data assimilation. Although the goal is to choose localization functions that are large enough to include the true region of non-zero cross-covariance, for EnKF applications, the choice of localization function needs to balance the harm done by spurious covariance resulting from small ensembles and the harm done by excluding real correlations. In this paper, we focus on the distance-based localization and provide insights for choosing suitable localization functions for data assimilation in multiphase flow problems. In practice, we conclude that it is reasonable to choose localization functions based on well patterns, that localization function should be larger than regions of non-zero sensitivity and should extend beyond a single well pattern.     

Fast linearized forecasts for subsurface flow data assimilation with ensemble Kalman filter

Ensemble methods present a practical framework for parameter estimation, performance prediction, and uncertainty quantification in subsurface flow and transport modeling. In particular, the ensemble Kalman filter (EnKF) has received significant attention for its promising performance in calibrating heterogeneous subsurface flow models. Since an ensemble of model realizations is used to compute the statistical moments needed to perform the EnKF updates, large ensemble sizes are needed to provide accurate updates and uncertainty assessment. However, for realistic problems that involve large-scale models with computationally demanding flow simulation runs, the EnKF implementation is limited to small-sized ensembles. As a result, spurious numerical correlations can develop and lead to inaccurate EnKF updates, which tend to underestimate or even eliminate the ensemble spread. Ad hoc practical remedies, such as localization, local analysis, and covariance inflation schemes, have been developed and applied to reduce the effect of sampling errors due to small ensemble sizes. In this paper, a fast linear approximate forecast method is proposed as an alternative approach to enable the use of large ensemble sizes in operational settings to obtain more improved sample statistics and EnKF updates. The proposed method first clusters a large number of initial geologic model realizations into a small number of groups. A representative member from each group is used to run a full forward flow simulation. The flow predictions for the remaining realizations in each group are approximated by a linearization around the full simulation results of the representative model (centroid) of the respective cluster. The linearization can be performed using either adjoint-based or ensemble-based gradients. Results from several numerical experiments with two-phase and three-phase flow systems in this paper suggest that the proposed method can be applied to improve the EnKF performance in large-scale problems where the number of full simulation is constrained.     

Ensemble filter methods with perturbed observations applied to nonlinear problems

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.     

Multiscale ensemble filtering for reservoir engineering applications

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.     

Multimodal ensemble Kalman filtering using Gaussian mixture models

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.     

Sampling error distribution for the ensemble Kalman filter update step

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.     

Comparing the adaptive Gaussian mixture filter with the ensemble Kalman filter on synthetic reservoir models

Over the last years, the ensemble Kalman filter (EnKF) has become a very popular tool for history matching petroleum reservoirs. EnKF is an alternative to more traditional history matching techniques as it is computationally fast and easy to implement. Instead of seeking one best model estimate, EnKF is a Monte Carlo method that represents the solution with an ensemble of state vectors. Lately, several ensemble-based methods have been proposed to improve upon the solution produced by EnKF. In this paper, we compare EnKF with one of the most recently proposed methods, the adaptive Gaussian mixture filter (AGM), on a 2D synthetic reservoir and the Punq-S3 test case. AGM was introduced to loosen up the requirement of a Gaussian prior distribution as implicitly formulated in EnKF. By combining ideas from particle filters with EnKF, AGM extends the low-rank kernel particle Kalman filter. The simulation study shows that while both methods match the historical data well, AGM is better at preserving the geostatistics of the prior distribution. Further, AGM also produces estimated fields that have a higher empirical correlation with the reference field than the corresponding fields obtained with EnKF.     

An enhanced ensemble Kalman filter scheme incorporating model error in sequential coupling between flow and geomechanics

In this work, we construct a new methodology for enhancing the predictive accuracy of sequential methods for coupling flow and geomechanics while preserving low computational cost. The new computational approach is developed within the framework of the fixed-stress split algorithm procedure in conjunction with data assimilation based on the ensemble Kalman filter (EnKF). In this context, we identify the high-fidelity model with the two-way formulation where additional source term appears in the flow equation containing the time derivative of total mean stress. The iterative scheme is then interlaced with data assimilation steps, which also incorporate the modeling error inherent to the EnKF framework. Such a procedure gives rise to an “enhanced one-way formulation,” exhibiting substantial improvement in accuracy compared with the classical one-way method. The governing equations are discretized by mixed finite elements, and numerical simulation of a 2D slab problem between injection and production wells illustrate the tremendous achievement of the method proposed herein.     

评估集合卡曼滤波反演土壤湿度廓线的性能

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.     

基于集合卡尔曼滤波的多相流模型参数估计——以室内二维砂箱中重质非水相污染物入渗为例

重质非水相有机污染物(DNAPL)泄漏到地下后,其运移与分布特征受渗透率非均质性影响显著。为刻画DNAPL污染源区结构特征,需进行参数估计以描述水文地质参数的非均质性。本研究构建了基于集合卡尔曼滤波方法(EnKF)与多相流运移模型的同化方案,通过融合DNAPL饱和度观测数据推估非均质介质渗透率空间分布。通过二维砂箱实际与理想算例,验证了同化方法的推估效果,并探讨了不同因素对同化的影响。研究结果表明:基于EnKF方法同化饱和度观测资料可有效地推估非均质渗透率场;参数推估精度随观测时空密度的增大而提高;观测点位置分布对同化效果有所影响,布置在污染集中区域的观测数据对于参数估计具有较高的数据价值。     

Groundwater parameter estimation using the ensemble Kalman filter with localization

The ensemble Kalman filter (EnKF), an efficient data assimilation method showing advantages in many numerical experiments, is deficient when used in approximating covariance from an ensemble of small size. Implicit localization is used to add distance-related weight to covariance and filter spurious correlations which weaken the EnKF??s capability to estimate uncertainty correctly. The effect of this kind of localization is studied in two-dimensional (2D) and three-dimensional (3D) synthetic cases. It is found that EnKF with localization can capture reliably both the mean and variance of the hydraulic conductivity field with higher efficiency; it can also greatly stabilize the assimilation process as a small-size ensemble is used. Sensitivity experiments are conducted to explore the effect of localization function format and filter lengths. It is suggested that too long or too short filter lengths will prevent implicit localization from modifying the covariance appropriately. Steep localization functions will greatly disturb local dynamics like the 0-1 function even if the function is continuous; four relatively gentle localization functions succeed in avoiding obvious disturbance to the system and improve estimation. As the degree of localization of the L function increases, the parameter sensitivity becomes weak, making parameter selection easier, but more information may be lost in the assimilation process.     

不同滤波算法在土壤湿度同化中的应用

为研究不同滤波算法在土壤湿度同化中的有效性,以及土壤湿度模拟结果对模型参数的敏感性,结合简单生物圈模型SiB2,设置敏感性实验,探求土壤饱和水力传导度对土壤湿度模拟结果的影响;并在此基础上,采用集合卡尔曼滤波（EnKF）、无迹卡尔曼滤波（UKF）和无迹粒子滤波（UPF）开展土壤湿度实时同化实验。结果表明：土壤饱和水力传导度能显著影响土壤湿度模拟精度;利用EnKF、UKF、UPF同化站点观测数据,均能改善土壤湿度模拟结果;3种同化方法在不同土壤层的同化效果不同,在土壤表层,EnKF的有效性优于UKF和UPF,在根域层和土壤深层,3种滤波方法有效性在降雨前后相差较大。因此,针对性地选择同化方法,是提高土壤湿度模拟精度的有效手段。     

A probabilistic parametrization for geological uncertainty estimation using the ensemble Kalman filter (EnKF)

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.     

Continuous Facies Updating Using the Ensemble Kalman Filter and the Level Set Method

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.     

基于集合卡尔曼平滑算法的土壤水分同化

为研究观测资料稀少情况下土壤质地及有机质对土壤水分同化的影响,发展了集合卡尔曼平滑(Ensemble Kalman Smooth, EnKS)的土壤水分同化方案。利用黑河上游阿柔冻融观测站2008年6月1日至10月29日的观测数据,使用EnKS算法将表层土壤水分观测数据同化到简单生物圈模型(Simple Biosphere Model 2, SiB2)中,分析不同方案对土壤水分估计的影响,并与集合卡尔曼滤波算法(EnKF)的结果进行比较。研究结果表明,土壤质地和有机质对表层土壤水分模拟结果影响最大而对深层的影响相对较小;利用EnKF和EnKS算法同化表层土壤水分观测数据,均能够显著提高表层和根区土壤水分估计的精度,EnKS算法的精度略高于EnKF且所受土壤质地和有机质的影响小于EnKF;当观测数据稀少时,EnKS算法仍然可以得到较高精度的土壤水分估计。     

Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications

Sampling errors can severely degrade the reliability of estimates of conditional means and uncertainty quantification obtained by the application of the ensemble Kalman filter (EnKF) for data assimilation. A standard recommendation for reducing the spurious correlations and loss of variance due to sampling errors is to use covariance localization. In distance-based localization, the prior (forecast) covariance matrix at each data assimilation step is replaced with the Schur product of a correlation matrix with compact support and the forecast covariance matrix. The most important decision to be made in this localization procedure is the choice of the critical length(s) used to generate this correlation matrix. Here, we give a simple argument that the appropriate choice of critical length(s) should be based both on the underlying principal correlation length(s) of the geological model and the range of the sensitivity matrices. Based on this result, we implement a procedure for covariance localization and demonstrate with a set of distinctive reservoir history-matching examples that this procedure yields improved results over the standard EnKF implementation and over covariance localization with other choices of critical length.