Correlation-Based Localization Effective in Ensemble-Based History Matching
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Ensemble-based methods are considered to be state-of-the-art history-matching algorithms. However, in practice, they often suffer from ensemble collapse, a phenomenon that deteriorates history-matching performance. An ensemble history-matching algorithm is equipped customarily with a localization scheme to prevent ensemble collapse. To enhance the applicability of localization to various history-matching problems, the authors adopt an adaptive localization scheme that exploits the correlations between model variables and observations.
In the current work, the authors focus on adopting an efficient adaptive localization scheme, previously established in the literature, for the full Norne Field case study. The adaptive localization scheme exploits the information of sample correlation coefficients between an ensemble of model variables and the corresponding realizations of simulated observations. The adaptive localization scheme uses a data-selection procedure; however, instead of physical distances between the locations of model variables and observations being used for data selection, the magnitudes of the sample correlation coefficients are used for data selection through a hard-thresholding strategy (i.e., keep or kill).
To conduct data selection in the adaptive localization scheme, one specifies a positive correlation-threshold value. For a given observation, if the magnitude of the sample correlation coefficient between a model variable and the simulated observation is greater than the threshold value, then the observation will be used to update that model variable. Otherwise, one discards the observation in the update of that model variable.
This described data-selection procedure essentially means that a given model variable is updated using only the observations that have significant correlations with the model variable. The rationale behind this hard-thresholding strategy is the interpretation of the magnitude of the correlation coefficient as a measure to detect the causal relation between a model variable and an observation, and the effect is the suppression of spurious correlations caused by a finite sample size.
Correlation-based localization can overcome or alleviate the issues arising in distance-based localization, such as the use of nonlocal and time-lapse observations, the need of physical locations for model variables and measurements, and the different degrees of correlations or sensitivities of model variables to observations. Because data selection depends on the magnitudes of sample correlation coefficients between model variables and the corresponding simulated observations, the measurements need not have associated physical locations.
Model variables are thus selected by using those observations that exhibit strong-enough correlations, regardless of the physical distances between observations and model variables. As a result, correlation-based localization can be used to localize nonlocal observations. The changes of correlations caused by the effect of time-lapse observations or different types of model variables will be taken into account automatically in correlation-based localization, and this makes the proposed localization scheme more-adaptive and more-flexible in various situations.
To the authors’ knowledge, this work represents the first time that a correlation-based localization scheme is applied to ensemble-based history matching in real field case studies. The paper aims to show that, in one particular case study, correlation-based localization leads to history-matching performance (in terms of data mismatch) that is close to or better than that of distance-based localization. Moreover, in practical applications, correlation-based localization is more straightforward to implement and use and can be transferred between different case studies. As a result, correlation-based localization can serve as a viable alternative to distance-based localization for ensemble-based history-matching problems in real field case studies.
Distance- and Correlation-Based Localization Applied to History Matching in the Norne Field Case Study
Production Data. The production data used in history matching include gas-, oil-, and water-production rates (WGRPH, WOPRH, and WWPRH, respectively) in the Norne database from November 1997 through December 2006, with a total of 5,038 data elements. A data-screening procedure is applied to select a part of the whole production data for history matching in such a way that the time intervals between two consecutive time instances of the assimilated data are approximately 1 month (so that the temporal correlations of the observation errors of production data might be negligible), whereas the production data not selected for history matching will be used for cross validation instead. After data screening, the number of production data used for history matching is 2,358, and the number of production data for cross validation is 2,680.
The authors have conducted history-matching experiments using all available WGPRH, WOPRH, and WWPRH data without data screening and found similar results in terms of the performance comparison of distance- and correlation-based localization schemes. The authors have also tried to apply an iterative ensemble smoother (IES) without any localization and observed ensemble collapse in history-matched reservoir models. For conciseness, this paper focuses on reporting history-matching results with data screening, for which the IES is equipped with either the distance- or the correlation-based localization scheme.
Implementation of Correlation-Based Localization. An implementation of correlation-based localization requires the specification of the correlation threshold values for different pairs of model variables and production data. In the current case study, computation of the threshold values in which the noise levels of the correlation fields are estimated is performed using a wavelet-based denoising method by exploiting the spatial correlations in the correlation fields. Roughly speaking, the rationale behind wavelet-based denoising is dependent on the assumption that the relatively low-frequency components in the frequency domain are dominated by signals, whereas the relatively high-frequency components mainly contain noise instead. Because a wavelet transform involves passing noisy signals through a filter bank that is designed to separate high- and low-frequency components, one can simply use the wavelet coefficients resulting from the high-pass filters to estimate the noise level. Concretely, in the current study, three-level discrete wavelet transforms are applied to all correlation fields, using 3D Daubechies wavelets with two vanishing moments.
For the free parameters, in general, the authors do not assume that they have associated spatial distributions. (By “free” parameters, they mean parameters that do not have associated reservoir gridblocks.) As a result, one might not apply an image-denoising-based method directly to estimate the threshold values. To tackle this issue, one might choose to introduce hard-coded empirical threshold values to the free parameters.
In terms of computational cost, correlation-based localization is more expensive than distance-based localization, although the cost in the former is still acceptable for practical history-matching problems. Because the computational cost of correlation-based localization depends on the number of observation elements, it is expected that one can cut down the computational cost by reducing the effective number of observation elements, such as by projecting observational data onto the ensemble subspace.
For the IES equipped with either localization scheme, the data-mismatch values of the estimated ensembles of reservoir models tend to decrease as the iteration progresses and they are all lower than the data-mismatch value of the manually history-matched reservoir model from the Norne database after the first four iteration steps. On average, the IES with correlation-based localization results in 14% lower final data mismatch than that seen with distance-based localization. Both localization schemes manage to maintain substantial spreads in the ensembles of data-mismatch values, which serves as a sign to indicate that ensemble collapse is avoided.
Compared with the production-data profiles of the initial ensemble, those of the final ensembles for the IES with distance- and correlation-based localization tend to have lower spread and match the real production data used in history matching better. In addition, the production-data profiles of the final ensembles also exhibit better matching to the real production data than those of the manually history-matched reservoir model provided in the Norne database.
Compared with the mean map of the initial ensemble, there are significant changes in the mean maps of the final history-matched ensembles. In addition, the mean maps of the final ensembles also appear different from each other because of different schemes used for localization. For correlation-based localization, the spatial distributions of the tapering coefficients are not confined to be unimodal, whereas relatively large tapering coefficients can be found along the edges of the area with active gridblocks. As a result, the petrophysical parameters on the active gridblocks along the edges have more changes (and, hence, smaller ensemble spreads) compared with the same petrophysical parameters with distance-based localization.
In the featured case study, 2,680 production data were reserved for cross validation. Here, cross validation is performed by comparing the corresponding simulated production rates of the final history-matched ensembles to these 2,680 observed data. Fig. 1 shows the box plots of data mismatch between the production data for cross validation and the corresponding simulated data from the final ensembles with distance- and correlation-based localization, respectively. In this case, on average, correlation-based localization also leads to lower data mismatch than distance-based localization.
As the focus of the current work, the efficacy of correlation-based localization is demonstrated through a case study in the full Norne Field. The authors document how distance- and correlation-based localization schemes are implemented in this real field case study and inspect the distinctions in the history-matching results that are caused by the different characteristics of the localization schemes. In this particular case study, correlation-based localization leads to lower data mismatch for production data used for both history matching and cross validation. In addition, experience indicates that correlation-based localization is more straightforward to use in practical history-matching problems (especially for reservoir models that have a relatively large number of production/injection wells) and has better reusability for different case studies. These attractive features make correlation-based localization a viable alternative to conventional distance-based localization in practical history-matching problems.
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Correlation-Based Localization Effective in Ensemble-Based History Matching
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