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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 linearGaussian 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.

Summary Recently, Emerick and Reynolds (2012) introduced the ensemble smoother with multiple data assimilations (ES-MDA) for assisted history matching. With computational examples, they demonstrated that ES-MDA provides both a better data match and a better quantification of uncertainty than is obtained with the ensemble Kalman filter (EnKF). However, similar to EnKF, ES-MDA can experience near ensemble collapse and results in too many extreme values of rock-property fields for complex problems. These negative effects can be avoided by a judicious choice of the ES-MDA inflation factors, but, before this work, the optimal inflation factors could only be determined by trial and error. Here, we provide two automatic procedures for choosing the inflation factor for the next data-assimilation step adaptively as the history match proceeds. Both methods are motivated by knowledge of regularization procedures—the first is intuitive and heuristical; the second is motivated by existing theory on the regularization of least-squares inverse problems. We illustrate that the adaptive ES-MDA algorithms are superior to the original ES-MDA algorithm by history matching three-phase-flow production data for a complicated synthetic problem in which the reservoir-model parameters include the porosity, horizontal and vertical permeability fields, depths of the initial fluid contacts, and the parameters of power-law permeability curves.

History matching is a jargon used to refer to the data assimilation problem in oil and gas reservoirs. The literature about history matching is vast and despite the impressive number of methods proposed and the significant progresses reported in the last decade, conditioning reservoir models to dynamic data is still a challenging task. Ensemble-based methods are among the most successful and efficient techniques currently available for history matching. These methods are usually able to achieve reasonable data matches, especially if an iterative formulation is employed. However, they sometimes fail to preserve the geological realism of the model, which is particularly evident in reservoir with complex facies distributions. This occurs mainly because of the Gaussian assumptions inherent in these methods. This fact has encouraged an intense research activity to develop parameterizations for facies history matching. Despite the large number of publications, the development of robust parameterizations for facies remains an open problem.Deep learning techniques have been delivering impressive results in a number of different areas and the first applications in data assimilation in geoscience have started to appear in literature. The present paper reports the current results of our investigations on the use of deep neural networks towards the construction of a continuous parameterization of facies which can be used for data assimilation with ensemble methods. Specifically, we use a convolutional variational autoencoder and the ensemble smoother with multiple data assimilation. We tested the parameterization in three synthetic history-matching problems with channelized facies. We focus on this type of facies because they are among the most challenging to preserve after the assimilation of data. The parameterization showed promising results outperforming previous methods and generating well-defined channelized facies. However, more research is still required before deploying these methods for operational use.

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 historymatching examples that this procedure yields improved results over the standard EnKF implementation and over covariance localization with other choices of critical length.

The ensemble smoother with multiple data assimilation (ES-MDA) is becoming a popular assisted history matching method. In its standard form, the method requires the specification of the number of iterations in advance. If the selected number of iterations is not enough, the entire data assimilation must be restarted. Moreover, ES-MDA also requires the selection of data-error covariance inflations. The typical choice is to select constant values. However, previous works indicate that starting with large inflation and gradually decreasing during the data assimilation steps may improve the quality of the final models.This paper presents an analysis of the use of geometrically decreasing sequences of the data-error covariance inflations. In particular, the paper investigates a recently introduced procedure based on the singular values of a sensitivity matrix computed from the prior ensemble. The paper also introduces a novel procedure to select the inflation factors. The performance of the data assimilation schemes is evaluated in three reservoir history-matching problems with increasing level of complexity. The first problem is a small synthetic case which illustrates that the standard ES-MDA scheme with constant inflation may result in overcorrection of the permeability field and that a geometric sequence can alleviate this problem. The second problem is a recently published benchmark and the third one is a field case with real production data. The data assimilation schemes are compared in terms of a data-mismatch and a model-change norm. The first norm evaluates the ability of the models to reproduce the observed data. The second norm evaluates the amount of changes in the prior model. The results indicate that geometric inflations can generate solutions with good balance between the two norms.