A novel localization scheme for scalar uncertainties in ensemble-based data assimilation methods

History matching is a crucial process during petroleum field management. Historical production and injection data are used to update uncertainties in the reservoir simulation models and generate more consistent models. Reservoir behavior must be predicted with a certain level of confidence as it is used to optimize key factors of oil and gas projects, such as oil production and/or net present value. History matching is a challenging inverse problem, in which the ultimate goal is to determine the uncertain parameters that lead to the known answer (historical production and injection data, for instance).

According to Oliver and Chen (2011), history matching is an ill-posed process, i.e., multiple combinations of the uncertain parameters can match historical data. Therefore, as the exact characteristics of the real reservoir are unknown, a single model is insufficient to represent the reservoir. As several combinations of uncertainties can match historical data, all possible combinations within a defined search space should be considered, taking into account the main geological and operational features.

Several methods, such as Markov Chain Monte Carlo (MCMC) and gradient-based methods are proven to be capable of matching simulated to observed data. However, these either are computationally demanding or concentrate the response in specific regions (local minimum), failing to find some of the possible models that can represent the reservoir in a proper manner (Yustres et al. 2011; Oliver and Chen 2011).

Ensemble-based methods derived from Ensemble Kalman Filter (EnKF) are a good alternative for history matching. EnKF is a sequential data assimilation algorithm proposed by Evensen (1994) that represents the uncertainties in the models and in the state vector through a finite number of models by a covariance matrix. Furthermore, the ability of storing and propagating this matrix made possible to work with models containing a great amount of data with low computational effort. Usually, EnKF and other ensemble-based methods are used in conjunction with localization techniques. This type of technique was first used in data assimilation problems by Houtekamer and Mitchell (1998), where they utilized distance-dependent localization to constrain the updates of the uncertainties into a defined region based on the observed data. Its use allowed to generate more consistent models, avoiding spurious correlations and underestimation of uncertainties (ensemble collapse), as also mentioned by Aanonsen et al. (2009), Arroyo-Negrete et al. (2008) and Emerick and Reynolds (2011). Note that there are other types of localization techniques, such as non-distance-dependent localization (Furrer and Bengtsson 2007) and streamline-based localization (Arroyo-Negrete et al. 2008).

Alternatively, Emerick and Reynolds (2013) developed the Ensemble Smoother with Multiple Data Assimilation (ES-MDA) which is an iterative ensemble-based method similar to EnKF but it assimilates all historical data in one single update multiple times. Emerick (2016) and Soares et al. (2018) used ES-MDA with distance-dependent localization, and pointed out that the distance-dependent localization technique is used to update only petrophysical uncertainties, which has single values for each grid cell, such as porosity and permeability images. Scalar uncertainties are represented by a single value for the whole model or part of it (rock compressibility, water relative permeability, etc.) and are not updated with the distance-dependent localization technique. Consequently, the variability is strongly reduced, suggesting the potential exclusion of models that can represent the reservoir.

Furthermore, previous knowledge about the ES-MDA methodology demonstrated that the method achieved different values for scalar uncertainties when changing some key parameters, such as inflation factor. Therefore, the method neglected some models with a good representation of the real reservoir. Clearly, the search space of uncertain scalar parameters requires better scanning to improve the representativeness of the reservoir. Consequently, more different scenarios must be generated to reliably forecast production for future field management decisions.

Ne is the number of models per ensemble, Nd the number of data, σ² the variance and \(\bar{m}^{\text{f}}\) the mean of certain attribute. It is important to point out that the matrix shown in Eq. 5 is diagonal because of the difficulty to measure the correlation between observed data error. More information about the method can be found in Emerick and Reynolds (2013).

The ρ matrix comprises the elements ρ_ab, where ‘a’ is the number of rows and ‘b,’ the number of columns. The rows represent the uncertainties in each grid block and the column represents historical data at specific historical times. Therefore, depending on the location of the grid block in relation to a specific well, ρ_ab will be equal to the value calculated by Eq. 7, depending on h and L.

Chen and Oliver (2014) used an iterative ensemble smoother method, similar to ES-MDA, in the Norne field (real case). They highlighted the improvement in the history-matching process over a manual process. Nevertheless, because of the high complexity of the case, they believe that the localization approach they used could be improved.

Morosov and Schiozer (2016) carried out a closed-loop reservoir management and development process using the UNISIM-I-D benchmark case (Avansi and Schiozer 2015) and, during history-matching phases, they used ES-MDA without a localization technique. They reported a strong reduction in the variability of the uncertainties, resulting in the exclusion of several models that could have a good representation of the reservoir. In fact, they observed that simulated responses did not capture the reference model, which represented the real reservoir.

Silva et al. (2017) also presented a closed-loop process for the UNISIM-I-D benchmark with ES-MDA. The main difference from the work of Morosov and Schiozer (2016) was the use of distance-dependent localization. They reported a better response when compared with the reference model. However, localization range was constant for all data and all wells, and it was used to update only petrophysical images.

Soares et al. (2018) proposed a methodology using ES-MDA in history matching with distance-dependent localization based on streamlines. They observed that localization improved the petrophysical images generated after the application of the method. On the other hand, they pointed out that uncertainties such as water relative permeability produced a significant reduction in variability since distance-dependent localization is not used to update scalar uncertainties.

Additionally, Luo et al. (2016) highlighted that some data, especially seismic, may not be associated with specific physical locations (such as scalar uncertainties). Therefore, the use of distance-dependent localization becomes an issue. Because of this, they proposed a correlation-based adaptive localization to be used with seismic data in a history-matching process, where they considered a threshold value to update the parameters, following the same logic of Furrer and Bengtsson (2007). Finally, they noted that the proposed methodology should also work for production data and values ranging from 0 to 1 could be used for ρ, as in the distance-dependent localization.

Lacerda et al. (2019) compared several methods (NDDL—Furrer and Bengtsson (2007), correlation-based—Luo et al. (2016), among others) with the same objective, and because most of these methods need some internal parameters that can have a large effect on the final response, their use might require the calibration of these parameters.

In this work, we propose and compare three different approaches to increase the variability of scalar uncertainties without impairing data match, model consistency or production forecast. Figure 1 shows the different approaches used.

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Each approach is explained below.

We applied the methodology to a benchmark case with a known response, UNISIM-I-H. Avansi and Schiozer (2015) developed a reference model (UNISIM-I-R) based on data from the Namorado field, an offshore field located in Campos Basin (Brazil). UNISIM-I-R is a very fine grid model containing 326 × 234 × 157 cells (3,408,633 active cells) with a resolution of 25 × 25 × 1 m. Therefore, UNISIM-I-R is assumed to be the reference case, which corresponds to the true model of the benchmark.

In order to test and validate new methodologies, Avansi and Schiozer developed the UNISIM-I-H, which is a coarser model derived from the reference model (UNISIM-I-R), and despite being a synthetic case, it has the characteristics and complexity of a real field.

UNISIM-I-H represents a reservoir composed by consolidated sandstones with a sealing fault, which divides the field into two production areas: main block and East block, as shown in Fig. 2. Avansi and Schiozer (2015) discretized the model in 81 × 58 × 20 cells and each cell measures, on average, 100 × 100 × 8 m. The model has 14 producers and 11 injectors (only water injection), and a black-oil simulator is used. Among the 14 producers, 4 are verticals and 10 are horizontal located at the top of the reservoir. Injectors are located at the basis of the field. The historical period is 11 years and forecast time is from 11 to 30 years. Figure 2 also shows the distribution of the wells.

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Avansi and Schiozer, (2015) selected the range of the uncertain parameters based on the knowledge of the Namorado field while avoiding values that are not compatible with the case. Finally, because UNISIM-I-H is a benchmark case and different authors have been using it (Morosov and Schiozer 2016; Silva et al. 2017; Soares et al. 2018), it is important to keep some basic features of the case in order to validate and compare different methodologies.

NQDS values were similar for all variables. Figure 3 shows NQDS for q_w, which presented a symmetrical response for most wells, with values close to zero. The exception, PROD023A, tended to overestimate water production for all approaches. Figure 4 illustrates water production for wells PROD012A (Fig. 4a) and PROD023A (Fig. 4b) under the FCC approach. PROD012A presented water rate curves close to the historical points, as NQDS values suggested, while PROD023A tended to overestimate values, especially at the beginning of water production.

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1. B.

   Data match

    

In Fig. 5, it is possible to see the distribution of the scalar uncertainties, where the y-axis represents their values. As suggested by Soares et al. (2018) and as we can see in Fig. 5, both BC and BCC-C_MD caused a large reduction in the variability in scalar uncertainties. The only exception was DWOC because water–oil contact is uncertain only in the East block, thus, the only data used for updates were from wells located in that part of the field (PROD023A, PROD024A, PROD025A, INJ007 and INJ010). Furthermore, the final convergence for different responses, as CPOR and k_rwiro values demonstrated, indicated the presence of other possible responses for this specific problem.

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BCC–R (λ = 0.1) showed that there are other possible responses when analyzing only BC and BCC-C_MD, as the convergence of MOD (Fig. 5c) illustrates. Moreover, because fewer data points were used to update the uncertainties, BCC–R (λ = 0.1) presented higher variability in the final answer. This higher variability, in conjunction with lower data mismatch, is a key point for the proposed methodology, as we aim to represent all possible models that might have a good representation of the real reservoir and, therefore, generate a more reliable forecast. However, the final response for MOD (Fig. 5c), for example, depicted that BCC–R (λ = 0.1) also found different answers to previous applications.

For BCC–R (λ = 0.2), the most influential data were used to update scalar uncertainties, generating higher variability for this approach compared with BCC–R (λ = 0.1), as CPOR (Fig. 5a) and P_krw (Fig. 5e) values show. Consequently, BCC–R (λ = 0.2) generated even more variability while respecting historical data. Nevertheless, as observed for BCC–R (λ = 0.1), when using λ = 0.2, MOD values concentrated in different values, demonstrating that the methodology can be further improved.

Note that for DWOC, BCC–R (λ = 0.1) and BCC–R (λ = 0.2) presented the same distribution as the initial (prior) ensemble of models. This was because the correlation between the objective function and this variable was weak, i.e., R_ab values were < 0.1 and 0.2, respectively. Consequently, ρ_ab = 0, resulted in m^a = m^f, as shown in Eq. 6.

Finally, FCC generated more likely levels of influence for each data point when updating scalar uncertainties and, because of this, FCC achieved better variability for almost all attributes. With the exception of MOD, FCC encompassed most of the responses from other approaches. Thus, the higher variability of the scalar uncertainties contributed to generating more different scenarios and the lower data mismatch shown in part A demonstrated that these models represent more adequate the reference model compared to BCC.

Figure 6 illustrates the final distribution of the scalar parameters as in Fig. 5, but using histograms (relative frequency) for k_rwiro, where we can see the higher variability for the FCC approach (Fig. 6f).

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In addition, despite different responses for the scalar uncertainties, petrophysical uncertain attributes showed very similar patterns for the mean of ln (k_x) (Fig. 7).

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1. C.

   Filter

    

Using the acceptable models defined in the previous step (C), we forecasted field behavior and noticed that all approaches were capable of encompassing the reference response for cumulative oil production (N_p) and cumulative water production (W_p). Note that the forecast reference corresponds to the UNISIM-I-R, which is the reference field treated as our true answer. Figure 8 depicts N_p for all approaches and, because of different combinations of scalar uncertainties, the approaches presented different responses. For BCC–C_MD and BCC–R (λ = 0.1), the reference answer was at the lower limit of the final ensemble while other approaches presented more symmetrical responses compared with the reference response. Figure 9 illustrates W_p and the similar responses for all approaches. Furthermore, FCC showed a slightly higher variability in the forecast when compared with all other approaches. Note that we filtered most of the models, therefore the curves in gray are hidden by the colorful lines in Figs. 8 and 9.

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In conclusion, FCC affected the scalar parameters just as the distance-dependent localization affected the petrophysical images. Therefore, values of ρ ranging from 0 to 1 were assigned, which helped to increase the variability of the scalar uncertainties in a more plausible way, performing a better exploration in the defined search space. However, as all approaches assessed in this work used correlation between production/injection data and uncertain attributes, care must be taken because if there is any spurious correlation among those variables, once it will be carried during the history-matching process. As a result, the process might generate models that poorly represent the real reservoir. To address this, we recommend using a large number of models (around 500), as already suggested by Soares et al. (2018) and Aanonsen (2009).

Finally, ensemble-based methods have been attracting attention within the oil industry due to its ability to handle large models and its relative easiness of implementation compared to other data assimilation methods, such as gradient-based and MCMC. Several works showed the effectiveness of the method when performing history matching. As already mentioned by authors such as Aanonsen et al. (2009) and Emerick and Reynolds (2011), for instance, the use of ensemble-based methods with distance-dependent localization helps to generate more geological consistent models. Nevertheless, one drawback of this method is the possible collapse of the response on the final ensemble. Therefore, the methods described in this work (BCC and FCC, for example) can help dealing with this issue, making ensemble-based methods more suitable for data assimilation. In addition, compared to methods as MCMC, ensemble-based methods demands less computational time and the methodology proposed in this work did not result in any extra significant computational time compared to other ensemble-based method methodologies.

This work presented new approaches based on localization techniques to increase the variability of scalar uncertainties with the application of the ES-MDA method.

The approach BCC-C_MD presented a similar response to the Base Case (BC), where no localization technique was used to update scalar uncertainties, with a strong reduction in scalar uncertainties.

BCC–R generated better results when compared with BCC-C_MD, and for higher values of λ, only a few data points were used to update scalar uncertainties and this explains the higher variability of BCC–R (λ = 0.2) compared with BCC–R (λ = 0.1).

FCC identified more plausible levels of influence of each data point for uncertain scalar attributes and used this knowledge to update them. This generated higher variability for the scalar uncertainties without losing quality in data matches and production forecasts.

Note that if too few models are used, spurious correlations may be generated. Consequently, this may affect the updates of petrophysical and scalar uncertainties and, therefore, the data matches and production forecasts.