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Summary When performing classic uncertainty reduction according to dynamic data, a large number of reservoir simulations need to be evaluated at high computational cost. As an alternative, we construct Bayesian emulators that mimic the dominant behavior of the reservoir simulator, and which are several orders of magnitude faster to evaluate. We combine these emulators within an iterative procedure that involves substantial but appropriate dimensional reduction of the output space (which represents the reservoir physical behavior, such as production data), enabling a more effective and efficient uncertainty reduction on the input space (representing uncertain reservoir parameters) than traditional methods, and with a more comprehensive understanding of the associated uncertainties. This study uses the emulation-based Bayesian history-matching (BHM) uncertainty analysis for the uncertainty reduction of complex models, which is designed to address problems with a high number of both input and output parameters. We detail how to efficiently choose sets of outputs that are suitable for emulation and that are highly informative to reduce the input-parameter space and investigate different classes of outputs and objective functions. We use output emulators and implausibility analysis iteratively to perform uncertainty reduction in the input-parameter space, and we discuss the strengths and weaknesses of certain popular classes of objective functions in this context. We demonstrate our approach through an application to a benchmark synthetic model (built using public data from a Brazilian offshore field) in an early stage of development using 4 years of historical data and four producers. This study investigates traditional simulation outputs (e.g., production data) and also novel classes of outputs, such as misfit indices and summaries of outputs. We show that despite there being a large number (2,136) of possible outputs, only very few (16) were sufficient to represent the available information; these informative outputs were used using fast and efficient emulators at each iteration (or wave) of the history match to perform the uncertainty-reduction procedure successfully. Using this small set of outputs, we were able to substantially reduce the input space by removing 99.8% of the original volume. We found that a small set of physically meaningful individual production outputs were the most informative at early waves, which once emulated, resulted in the highest uncertainty reduction in the input-parameter space, while more complex but popular objective functions that combine several outputs were only modestly useful at later waves. The latter point is because objective functions such as misfit indices have complex surfaces that can lead to low-quality emulators and hence result in noninformative outputs. We present an iterative emulator-based Bayesian uncertainty-reduction process in which all possible input-parameter configurations that lead to statistically acceptable matches between the simulated and observed data are identified. This methodology presents four central characteristics: incorporation of a powerful dimension reduction on the output space, resulting in significantly increased efficiency; effective reduction of the input space; computational efficiency, and provision of a better understanding of the complex geometry of the input and output spaces.

Parameterization is a crucial step during uncertainty reduction of reservoir properties using dynamic data. It establishes the search space based on prior knowledge of the model and can have a significant influence on the final response. A less-appropriate parameterization might fail to have a reasonable representation of the reservoir and lead to models unable to predict the correct reservoir characteristics. Parameterization of petrophysical images (as facies, porosities, and permeabilities) plays an essential role during data assimilation processes due to the strong influence in fluid flow in the porous media. This work shows how important the parameterization of petrophysical images is and how a less-appropriate parameterization can affect history-matching and uncertainty reduction process. Using a benchmark case, we compare two parameterization techniques, one capable of treating all blocks in the model (distance-dependent covariance localization), which is considered more appropriate, and one that considers a group of blocks under the same update rule (zonation) (less-appropriate). Results show that parameterization of petrophysical images has a high impact on the final response, and a less-appropriate parameterization, as the zonation, can generate higher data mismatches and fail to represent the real reservoir response. The analysis carried in this work quantifies and qualifies the impact of the parameterization of the petrophysical images in the data assimilation for the uncertainty reduction process.

History matching (HM) is an inverse problem where uncertainties in attributes are reduced by comparison with observed dynamic data. Typically, normalized misfit summarizes dissimilarities between observed and simulation data. Especially for long-time series, objective functions (OFs) aggregate multiple events and tendencies relevant to field performance in a single indicator (e.g. water rate and breakthrough time). To capture the attributes influencing the reservoir behavior, we evaluate the assimilation of data series through additional OFs, obtained from splitting time-series data. In this study, two additional OF groups supplement the time-series misfits: Breakthrough Deviation (BD) indicating dissimilarities in water breakthrough time; Productivity Deviation (PD), representing mismatches of the well potential, mainly impacting the transition from history to forecast conditions. The Productivity Deviation (PD) is adapted from previous studies. Instead of simulating the last time of the historical period under forecast conditions, we propose keeping it under historical data. The change is the historical data used as target condition to the simulator: Bottom Hole Pressure (BHP) in place of liquid production and water injection rates; with this, we estimate a mismatch in well productivity, while avoiding the influence of other boundary conditions in the evaluation. Two applications (1 & 2), assimilating different OF quantities, highlight the influence of the additional groups. Application 1 only computes time-series misfit (64 OFs) whereas Application 2 includes the BD and PD (counting 128 OFs). The iterative HM method presents flexibility regarding OFs assimilated and incorporation of uncertain attributes. UNISIM-I-H case allows us to evaluate the HM considering history and forecast data. We examine differences between the 450 scenarios resulting of data assimilation for each application through four perspectives. Application 2 resulted in scenarios with better predictability of the field behavior and smoother transitions between field history and forecast periods. Field cumulative oil production of Application 2 is also forecasted closer to the reference data when compared to Application 1; all forecast periods (1, 5 and 19 years) emphasize this impact. Some wells presented breakthrough time closer to the reference for Application 2. The challenging achievement of exact BD matches leads to the third advantage of the additional indicators. These OFs supply supplementary information to the diagnosis of scenarios, identifying unnoticed problems in the traditional approach. Finally, even with an overall better performance, some of the well OFs presented poorer matches for Application 2. To explain this, we analyzed the relationship between attributes and the OFs used to update the attributes. In conclusion, the improved forecast of the simulation scenarios indicates that superior performance of the HM process is possible by splitting the available dynamic data in relevant additional OFs. This study presents a case application with 11 years of field history, in which additional OFs, derived from dynamic data, add value to the reservoir characterization. They allow the influence of uncertain attributes to be captured for relevant events in reservoir performance. We also show how the increased quantity of OFs assimilated makes the HM harder for some OFs.

When performing classic uncertainty reduction based on dynamic data, a large number of reservoir simulations need to be evaluated at high computational cost. As an alternative, we construct Bayesian emulators that mimic the dominant behaviour of the reservoir simulator, and which are several orders of magnitude faster to evaluate. We combine these emulators within an iterative procedure that involves substantial but appropriate dimensional reduction of the output space, enabling a more effective and efficient uncertainty reduction on the input space than traditional methods, and with a more comprehensive understanding of the associated uncertainties. This study uses a Bayesian statistical approach for uncertainty reduction of complex models which is designed to address problems with high number of both input and output parameters. We detail how to efficiently choose sets of outputs that are suitable for emulation and that are highly informative to reduce the input parameter space and investigate different classes of outputs and objective functions. We use output emulators and implausibility analysis iteratively to perform input space reduction, and we discuss the strengths and weaknesses of certain popular classes of objective function in this context. We demonstrate our approach via an application to a benchmark synthetic model (built using public data from a Brazilian offshore field) in an early stage of development using four years of historical data and four producers. This study investigates traditional simulation outputs (e.g. production data) and also novel classes of outputs, such as misfit indexes and summaries of outputs. We show that despite there being a large number (2,136) of possible outputs, only a very small number (16) was sufficient to represent the available information; these informative outputs were utilized using fast and efficient emulators at each iteration (or wave) of the history match to perform the uncertainty reduction procedure successfully. Using this small set of outputs, we were able to substantially reduce the input space by removing 99.8% of the original volume. We found that a small set of physically meaningful individual production outputs were the most informative at early waves, which once emulated, resulted in the highest space reduction, while more complex but popular objective functions that combine several outputs were only modestly useful at later waves. The latter point is due to objective functions such as misfit indices having complex surfaces that can lead to low-quality emulators and hence result in non-informative outputs. We present an iterative emulator-based Bayesian uncertainty reduction process in which all possible input parameter configurations that lead to statistically acceptable matches between the simulated and observed data are identified. This methodology presents four central characteristics: (1) incorporation of a powerful dimension reduction on the output space, resulting in significantly increased efficiency, (2) effective reduction of the input space, (3) computational efficiency, and (4) provision of a better understanding of the complex geometry of the input and output spaces.

Reservoir simulation models incorporate physical laws and reservoir characteristics. They represent our understanding of sub-surface structures based on the available information. Emulators are statistical representations of simulation models, offering fast evaluations of a sufficiently large number of reservoir scenarios, to enable a full uncertainty analysis. Bayesian History Matching (BHM) aims to find the range of reservoir scenarios that are consistent with the historical data, in order to provide comprehensive evaluation of reservoir performance and consistent, unbiased predictions incorporating realistic levels of uncertainty, required for full asset management. We describe a systematic approach for uncertainty quantification that combines reservoir simulation and emulation techniques within a coherent Bayesian framework for uncertainty quantification.Our systematic procedure is an alternative and more rigorous tool for reservoir studies dealing with probabilistic uncertainty reduction. It comprises the design of sets of simulation scenarios to facilitate the construction of emulators, capable of accurately mimicking the simulator with known levels of uncertainty. Emulators can be used to accelerate the steps requiring large numbers of evaluations of the input space in order to be valid from a statistical perspective. Via implausibility measures, we compare emulated outputs with historical data incorporating major process uncertainties. Then, we iteratively identify regions of input parameter space unlikely to provide acceptable matches, performing more runs and reconstructing more accurate emulators at each wave, an approach that benefits from several efficiency improvements. We provide a workflow covering each stage of this procedure.The procedure was applied to reduce uncertainty in a complex reservoir case study with 25 injection and production wells. The case study contains 26 uncertain attributes representing petrophysical, rock-fluid and fluid properties. We selected phases of evaluation considering specific events during the reservoir management, improving the efficiency of simulation resources use. We identified and addressed data patterns untracked in previous studies: simulator targets, e.g. liquid production, and water breakthrough lead to discontinuities in relationships between outputs and inputs. With 15 waves and 115 valid emulators, we ruled out regions of the searching space identified as implausible, and what remained was only a small proportion of the initial space judged as non-implausible (~10−11%). The systematic procedure showed that uncertainty reduction using iterative Bayesian History Matching has the potential to be used in a large class of reservoir studies with a high number of uncertain parameters.We advance the applicability of Bayesian History Matching for reservoir studies with four deliveries: (a) a general workflow for systematic BHM, (b) the use of phases to progressively evaluate the historical data; and (c) the integration of two-class emulators in the BHM formulation. Finally, we demonstrate the internal discrepancy as a source of error in the reservoir model.

This paper proposes new objective functions to assimilate dynamic data for history matching, and evaluates their influence on the uncertainty conditioning. Representative events are observed and evaluated separately for the available dynamic data. The proposed objective functions evaluate two specific events: (1) the production transition behavior between the historical and forecasting period, and (2) the water breakthrough time. To assess production transition behavior, the deviation between the latest available historical data is compared with the forecast value, at a specific moment, under forecasting conditions. To assess water breakthrough, the irruption time error is measured in addition to the water-rate objective function. The new objective functions are normalized using the Normalized Quadratic Deviation with Sign, for comparison with conventional objective functions (i.e. NQDS-oil production rate). These additional objective functions are included in a probabilistic and multi-objective history matching and applied to the UNISIM-I-M benchmark for validation. Two history-matching procedures evaluate the impact of the additional objective functions, based on the same parameterization, boundary conditions and number of iterations. The first procedure (Procedure A) includes objective functions traditionally used such as fluid rates and bottom-hole pressure, computed using all the historical data points. The second procedure (Procedure B) considers the same as for A as well as the two additional objective functions. The advantages of including the additional objective functions was the supplementary data used to constrain the uncertainties, improving attribute updates. Consequently, Procedure B generated better-matched models considering the historical period and more consistent forecasts for both field and well behavior when compared to available reference data. The addition of the breakthrough deviation improved the quality of the match for water rates because breakthrough deviation is sensitive to reservoir attributes different to those objective functions related to water rate. The production transition error assisted the identification of scenarios that under or overestimated well capacity. Production transition error also improved the transition of the models from the historical to the forecasting period, reducing fluctuations due to the changes in boundary conditions. Despite the increased number of objective functions to be matched, the improved reliability for forecasting is an incentive for further study. Other representative events, such as oil rate before and after the start of water production could be separated and evaluated, for example. The improved reliability for forecasting supports the inclusion of the proposed objective functions in history-matching procedures.