\n In applications of ensemble-based history matching, it is common to conduct Kalman gain or covariance localization to mitigate spurious correlations and excessive variability reduction resulting from the use of relatively small ensembles. Another alternative strategy not very well explored in reservoir applications is to apply a local analysis scheme, which consists of defining a smaller group of local model variables and observed data (observations), and perform history matching within each group individually. This work aims to demonstrate the practical advantages of a new local analysis scheme over the Kalman gain localization in a 4D seismic history-matching problem that involves big seismic data sets. In the proposed local analysis scheme, we use a correlation-based adaptive data-selection strategy to choose observations for the update of each group of local model variables. Compared to the Kalman gain localization scheme, the proposed local analysis scheme has an improved capacity in handling big models and big data sets, especially in terms of computer memory required to store relevant matrices involved in ensemble-based history-matching algorithms. In addition, we show that despite the need for a higher computational cost to perform model update per iteration step, the proposed local analysis scheme makes the ensemble-based history-matching algorithm converge faster, rendering the same level of data mismatch values at a faster pace. Meanwhile, with the same numbers of iteration steps, the ensemble-based history-matching algorithm equipped with the proposed local analysis scheme tends to yield better qualities for the estimated reservoir models than that with a Kalman gain localization scheme. As such, the proposed adaptive local analysis scheme has the potential of facilitating wider applications of ensemble-based algorithms to practical large-scale history-matching problems.

Handling Big Models and Big Data Sets in History-Matching Problems through an Adaptive Local Analysis Scheme

The joint identification of the parameters defining a contaminant source and the heterogeneous distribution of the hydraulic conductivities of the aquifer where the contamination took place is a difficult task. Previous studies have demonstrated the applicability of the restart normal-score ensemble Kalman filter (rNS-EnKF) in synthetic cases making use of sufficient hydraulic head and concentration data. This study shows an application of the same technique to a non-synthetic case under laboratory conditions and discusses the difficulties found on its application and the avenues taken to solve them. The method is first tested using a synthetic case that mimics the sandbox experiment to establish the minimum number of ensemble members and the best technique to prevent the filter collapsing. The synthetic case shows that among different techniques based on update damping and covariance inflation, the Bauser’s covariance inflation method works best in preventing filter collapse. Its application to the sandbox data shows that the rNS-EnKF can benefit from Bauser’s inflation to reduce the number of ensemble realizations substantially in comparison with a filter without inflation, arriving at a good joint identification of both the contaminant source and the spatial heterogeneity of the conductivities.

Contaminant Spill in a Sandbox with Non-Gaussian Conductivities: Simultaneous Identification by the Restart Normal-Score Ensemble Kalman Filter

An original graph clustering approach to efficient localization of error covariances is proposed within an ensemble-variational data assimilation framework. Here the localization term is very generic and refers to the idea of breaking up a global assimilation into subproblems. This unsupervised localization technique based on a linearized
state-observation measure is general and does not rely on any prior information such as relevant spatial scales, empirical cut-off radius or homogeneity assumptions. It automatically segregates the state and observation variables in an optimal number of clusters (otherwise named as subspaces or communities), more amenable to scalable data assimilation.
The application of this method does not require underlying block-diagonal structures of prior covariance matrices. In order to deal with inter-cluster connectivity, two alternative data adaptations are proposed. 
Once the localization is completed, an adaptive covariance diagnosis and tuning is performed within each cluster. 
Numerical tests show that this approach is less costly and more flexible than a global covariance tuning, and most often results in more accurate background and observations error covariances.

/pdf/a-graph-clustering-approach-to-localization-for-adaptive-exabh6x1ty.pdf

A graph clustering approach to localization for adaptive covariance tuning in data assimilation based on state-observation mapping

In the field of mineral resources extraction, one main challenge is to meet production targets in terms of geometallurgical properties. These properties influence the processing of the ore and are often represented in resource modeling by coregionalized variables with a complex relationship between them. Valuable data are available about geometalurgical properties and their interaction with the beneficiation process given sensor technologies during production monitoring. The aim of this research is to update resource models as new observations become available. A popular method for updating is the ensemble Kalman filter. This method relies on Gaussian assumptions and uses a set of realizations of the simulated models to derive sample covariances that can propagate the uncertainty between real observations and simulated ones. Hence, the relationship among variables has a compositional nature, such that updating these models while keeping the compositional constraints is a practical requirement in order to improve the accuracy of the updated models. This paper presents an updating framework for compositional data based on ensemble Kalman filter which allows us to work with compositions that are transformed into a multivariate Gaussian space by log-ratio transformation and flow anamorphosis. This flow anamorphosis, transforms the distribution of the variables to joint normality while reasonably keeping the dependencies between components. Furthermore, the positiveness of those variables, after updating the simulated models, is satisfied. The method is implemented in a bauxite deposit, demonstrating the performance of the proposed approach.

/pdf/resource-model-updating-for-compositional-geometallurgical-2vp9aqwdgh.pdf

Resource Model Updating For Compositional Geometallurgical Variables

Contaminant source and aquifer characterization (CSAC) is critical in groundwater pollution evaluation and remediation. The ensemble smoother with multiple data assimilation (ES-MDA) is utilized to jointly identify contaminant source information and hydraulic conductivities by assimilating time-lapse electrical resistivity tomography (ERT) data. In a synthetic profile with a time-varying release history in a heterogeneous aquifer, we verify the performance of the proposed data assimilation framework. The results show that the CSAC problem could be handled by the proposed approach. The time-varying release history and the high permeability area can be identified with adequate time-lapse ERT measurements. Further reproduction of the evolution of the plume after CSAC also shows consistency with the reference plume. The poor conditioning inversion caused by the filter inbreeding is analyzed by comparing four scenarios with different apparent resistivity measurements. Furthermore, we also evaluate the impact of uncertainties in the petrophysical properties and geophysical observations on our data assimilation framework. The results show that the proposed ES-MDA data assimilation framework could provide a convincing inversion of the time-varying release history and hydraulic conductivities. 

Contaminant source and aquifer characterization: An application of ES-MDA demonstrating the assimilation of geophysical data

Inverse modeling techniques for estimating reservoir parameters (e.g., transmissivity, permeability) utilize some dynamic (secondary) information (e.g., hydraulic head or production data at well locations). Ensemble based data assimilation methods are one such class of inverse modeling techniques. Ensemble Kalman filter (EnKF) in particular is built around the basic framework of linearly updating a state vector $${\psi ^f}$$
 to $${\psi ^a}$$
 , based on observed dynamic data. Components of the state vector are reservoir model parameters such as transmissivity, permeability, storativity, porosity, hydraulic head, and phase-saturation. Although EnKF is able to update a large number of parameters successively as data become available, it has some shortcomings. It is optimal only in the cases where multivariate joint distribution describing the relationship between components in the state vector is Gaussian. The relationship between the model parameters and the observed data space is quantified in terms of a covariance function, which is adequate to describe a multi-Gaussian distribution. These assumptions and simplifications may result in updated models that are inconsistent with the prior geology (exhibiting non-Gaussian characteristics) and that, in turn, may yield inaccurate predictions of reservoir performance. The aim of this research work is to propose a novel method for data assimilation which is free from the Gaussian assumptions. This new method can be used to assimilate dynamic data into reservoir models using an ensemble-based approach. Model updates are performed after transforming the model parameters into indicator variables. The indicator transform is insensitive to non-linear operations, and it thus allows us to overcome some of the limitations of the traditional EnKF.

Ensemble-Based Assimilation of Nonlinearly Related Dynamic Data in Reservoir Models Exhibiting Non-Gaussian Characteristics


 History matching is an integral part of field development planning for oil and gas reservoirs. It can be viewed as an inverse modeling technique which utilizes information contained in the observed flow response variables like flow rate, well bottom hole pressure etc. to better quantify the spatial distribution of reservoir model parameters like permeability or porosity. As the reservoir parameters and flow response variables are often related by non-linear relationships, the solution to the inverse problem of history matching is often non-unique. This makes it a problem that can be better handled by stochastic approaches than deterministic approaches. The proposed approach called ‘Indicator-based Data Assimilation’ (InDA) is suitable for such problems. To initiate the process, multiple realizations of the reservoir model parameters are generated based on the available information related to the reservoir description. The ensemble of realizations serves as a measure of initial uncertainty in the spatial distribution of model parameters. Next, flow simulations are run on the ensemble of models and the difference between the observed and simulated flow response variables are used to update the reservoir parameters using InDA. With successive updates, the uncertainty in the model parameters is reduced and the spatial distribution approaches the "true" distribution. Considering the residual uncertainty of the final updated models, reliable field development planning decisions can be made. The proposed method is validated using a realistic reservoir with complex channel-like features emulating a reservoir formed in a fluvial depositional environment. Liquid rate data from existing wells are used for updating reservoir parameters for several time steps using InDA. A comparison of the spatial distribution of final model parameters with the reference model used for validation shows a good match. After the history matching period, existing and new infill wells are run in a forecast mode where the observed and simulated flow responses show a good match. As majority of oil reservoirs comprise of high permeability oil-bearing zones in form of channels passing across low permeability zones, the statistical permeability distribution is bimodal making it non-Gaussian. It is shown that ‘Indicator Transformation’ of variables used in InDA preserves the non-Gaussian structure of the permeability field in comparison to methods like ‘Ensemble Kalman Filter’ (EnKF) that are sub-optimal in such cases.

Ensemble-based history matching and uncertainty quantification for reservoirs exhibiting complex non-Gaussian characteristics

Indicator-based data assimilation with multiple-point statistics for updating an ensemble of models with non-Gaussian parameter distributions

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Ensemble-Based Assimilation of Nonlinearly Related Dynamic Data in Reservoir Models Exhibiting Non-Gaussian Characteristics

Ensemble-based history matching and uncertainty quantification for reservoirs exhibiting complex non-Gaussian characteristics

Indicator-based data assimilation with multiple-point statistics for updating an ensemble of models with non-Gaussian parameter distributions