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Efficient Selection of Reservoir Model Outputs within an Emulation-Based Bayesian History-Matching Uncertainty Analysis
Abstract
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.
... Bayesian artificial neural networks (BANNs) (Neal, 1996) can also be used (e.g., Hauser et al., 2011;Tarasov et al., 2012). Given their greater familiarity and lower im-1410 plementation costs, pure linear regression models (such as the lm() function in R) can be efficiently used for initial waves (or depending on the desired reduction in the non-implausible space, can be used for a full history matching exercise, c.f. Ferreira et al., 2020). ...
... The use of standard linear regression emulators has the major advantage of much wider familiarity within and outside of the statistical community. For initial waves in history matching, Ferreira et al. (2020) have demonstrated for a high dimensional 1415 (2136 outputs) synthetic case study of a geological water reservoir model that standard linear regression for second-order polynomial emulators can be quite effective. They achieved a 99.5% reduction in the parameter space after two waves. ...
In the geosciences, complex computational models have become a common tool for making statements about past earth system evolution. However, the relationship between model output and the actual earth system (or component thereof) is generally poorly specified and even more poorly assessed. This is especially challenging for the paleo sciences for which data constraints are sparse and have large uncertainties. Bayesian inference offers, in principle, a self-consistent and rigorous framework for assessing this relationship as well as a coherent approach to combining data constraints with computational modelling. Though “Bayesian” is becoming more common in paleoclimate and paleo ice sheet publications, our impression is that most scientists in these fields have little understanding of what this actually means nor are they able to evaluate the quality of such inference. This is especially unfortunate given the correspondence between Bayesian inference and the classical concept of the scientific method. Herein, we examine the relationship between a complex model and a system of interest, or in equivalent words (from a statistical perspective), how uncertainties describing this relationship can be assessed and accounted for in a principled and coherent manner. By way of a simple example, we show how inference, whether Bayesian or not, can be severely broken if uncertainties are erroneously assessed. We explain and decompose Bayes Rule (more commonly known as Bayes Theorem), examine key components of Bayesian inference, offer some more robust and easier to attain stepping stones, and provide suggestions on implementation and how the community can move forward. This overview is intended for all interested in making and/or evaluating inferences about the past evolution of the Earth system (or any of its components), with a focus on past ice sheet and climate evolution during the Quaternary.
... can be used for initial waves (or depending on the desired reduction in the non-implausible space, can be used for a full history matching exercise, c.f. Ferreira et al., 2020). ...
... The use of standard linear regression emulators has the major advantage of much wider familiarity within and outside of the statistical community. For initial waves in history matching, Ferreira et al. (2020) have demonstrated for a high dimensional (2136 outputs) synthetic case study of a geological water reservoir model that standard linear regression for second-order polynomial emulators can be quite effective. They achieved a 99.5% reduction in the parameter space after two waves. ...
In the geosciences, complex computational models have become a common tool for making statements about past earth system evolution. However, the relationship between model output and the actual earth system (or component thereof) is generally poorly specified and even more poorly assessed. This is especially challenging for the paleo sciences for which data constraints are sparse and have large uncertainties. Bayesian inference offers a self-consistent and rigorous framework for assessing this relationship as well as a coherent approach to combining data constraints with computational modelling. Though “Bayesian” is becoming more common in paleoclimate and paleo ice sheet publications, our impression is that most scientists in these fields have little understanding of what this actually means nor are they able to evaluate the quality of such inference. This is especially unfortunate given the correspondence between Bayesian inference and the classical concept of the scientific method. Herein, we examine the relationship between a complex model and a system of interest, or in equivalent words (from a statistical perspective), how uncertainties describing this relationship can be assessed and accounted for in a principled and coherent manner. By way of a simple example, we show how inference can be severely broken if uncertainties are erroneously assessed. We explain and decompose Bayes Rule (more commonly known as Bayes Theorem), examine key components of Bayesian inference, offer some more robust and easier to attain stepping stones, and provide suggestions on implementation and how the community can move forward. This overview is intended for all interested in making and/or evaluating inferences about the past evolution of the Earth system (or any of its components), with a nominal focus on past ice sheet and climate evolution during the Quaternary.
... Another challenge is the variety of observed data sources for data conditioning. Methods such as multiobjective optimization in a Bayesian framework were applied to address this topic (Olalotiti-Lawal and Datta-Gupta 2015; Hutahaean et al. 2017;Ferreira et al. 2020). Recently, a number of additional ML algorithms, such as support-vector regression and convolutional and recurrent neural networks, were applied to various aspects of data conditioning (Ersahin et al. 2019; Alakeely and Horne 2020; da Silva et al. 2020). ...
Forecasting production from hydrocarbon fields is challenging because of the large number of uncertain model parameters and the multitude of observed data that are measured. The large number of model parameters leads to uncertainty in the production forecast from hydrocarbon fields. Changing operating conditions [e.g., implementation of improved oil recovery or enhanced oil recovery (EOR)] results in model parameters becoming sensitive in the forecast that were not sensitive during the production history. Hence, simulation approaches need to be able to address uncertainty in model parameters as well as conditioning numerical models to a multitude of different observed data.
Sampling from distributions of various geological and dynamic parameters allows for the generation of an ensemble of numerical models that could be falsified using principal-component analysis (PCA) for different observed data. If the numerical models are not falsified, machine-learning (ML) approaches can be used to generate a large set of parameter combinations that can be conditioned to the different observed data. The data conditioning is followed by a final step ensuring that parameter interactions are covered. The methodology was applied to a sandstone oil reservoir with more than 70 years of production history containing dozens of wells.
The resulting ensemble of numerical models is conditioned to all observed data. Furthermore, the resulting posterior-model parameter distributions are only modified from the prior-model parameter distributions if the observed data are informative for the model parameters. Hence, changes in operating conditions can be forecast under uncertainty, which is essential if nonsensitive parameters in the history are sensitive in the forecast.
... [1] Bayesian approaches to differentiate between models based on the likelihood of the posterior model are considered in this study and it take into account that the uncertainties in the parameters are usually believed to be univariate Gaussian random areas by applying the principles of model probability and model probability. [16] For the uncertainty reduction of complicated calculations, this research used the emulation-based Bayesian historymatching uncertainty analysis, which is structured to resolve issues with a massive amount of inputs and outputs also explain how to select collections of outputs that are appropriate for simulation effectively and that are highly descriptive in order to minimize the space of the input parameter and evaluate various types of outputs and objective functions. ...
History matching is one of the most complicated problems which requires long time period and many attempts to get an acceptable match because it is ill-post invers problem. For this purpose, many theories have been developed to solve this complexity.
In this paper we show the automatic history matching methods and the different studies that use those methods over the years.
History matching is one of the most complicated problems which requires long time period and many attempts to get an acceptable match because it is ill-post invers problem. For this purpose, many theories have been developed to solve this complexity. In this paper we show the automatic history matching methods and the different studies that use those methods over the years. Abstract History matching is one of the most complicated problems which requires long time period and many attempts to get an acceptable match because it is ill-post invers problem. For this purpose, many theories have been developed to solve this complexity. In this paper we show the automatic history matching methods and the different studies that use those methods over the years.
Data assimilation (DA) for uncertainty reduction based on reservoir simulation models is generally an intense and high time-consuming process due to the number of uncertain parameters involved and the computational time required to run the flow simulation models. Depending on the degree of model description (fidelity), DA can take days or even weeks to be executed because it can take several hundred (or thousands) of reservoir simulations, even applying efficient DA methods. Since quick decisions on reservoir development and management phases have to be made, the process must be carried out in a suitable and affordable computational time. There are two complimentary ways of dealing with this problem: the first is reducing the model simulation time (trying to maintain the quality of results), and the second is the application of efficient and effective DA methods. In this work, we propose a methodology that consists of generating and using efficient and effective lower-fidelity models (LFM), combined with the application of the Ensemble Smoother with Multiple Data Assimilation (ES-MDA) method which is the state-of-art on data assimilation combining efficiency and efficacy. The focus is to present a comprehensive and robust analysis of the DA process by assessing different model fidelities to achieve the best trade-off between computational time and quality of results. We applied the methodology in a real field using six different fidelities: an initial (fit-for-purpose) model named the medium-fidelity model (MFM) and five LFM. Their performances are benchmarked against the MFM results. For each fidelity model, we ran a DA process and compared the results, including (1) quality of data match and computational time, (2) prior and posterior attribute distributions, and (3) predictive capacity of the models. To conclude, we showed that the DA process using LFM reduced the computational time from days to hours, being up to about 11 times faster than the process using the original model with similar or even better results, making it clear the advantage of building, evaluating and selecting a lower-fidelity model based on the purpose of the study. This is important because, after the data assimilation, high number of reservoir simulations (typically thousands) is necessary for reliable decisions in the context of reservoir development and management.
Simulator imperfection, often known as model error, is ubiquitous in practical data assimilation problems. Despite the enormous efforts dedicated to addressing this problem, properly handling simulator imperfection in data assimilation remains to be a challenging task. In this work, we propose an approach to dealing with simulator imperfection from a point of view of functional approximation that can be implemented through a certain machine learning method, such as kernel-based learning adopted in the current work. To this end, we start from considering a class of supervised learning problems, and then identify similarities between supervised learning and variational data assimilation. These similarities found the basis for us to develop an ensemble-based learning framework to tackle supervised learning problems, while achieving various advantages of ensemble-based methods over the variational ones. After establishing the ensemble-based learning framework, we proceed to investigate the integration of ensemble-based learning into an ensemble-based data assimilation framework to handle simulator imperfection. In the course of our investigations, we also develop a strategy to tackle the issue of multi-modality in supervised-learning problems, and transfer this strategy to data assimilation problems to help improve assimilation performance. For demonstration, we apply the ensemble-based learning framework and the integrated, ensemble-based data assimilation framework to a supervised learning problem and a data assimilation problem with an imperfect forward simulator, respectively. The experiment results indicate that both frameworks achieve good performance in relevant case studies, and that functional approximation through machine learning may serve as a viable way to account for simulator imperfection in data assimilation problems.
Time-consuming numerical simulators for solving groundwater flow and dissolution models of physico-chemical processes in deep aquifers normally require some of the model inputs to be defined in high-dimensional spaces in order to return realistic results. Sometimes, the outputs of interest are spatial fields leading to high-dimensional output spaces. Although Gaussian process emulation has been satisfactorily used for computing faithful and inexpensive approximations of complex simulators, these have been mostly applied to problems defined in low-dimensional input spaces. In this paper, we propose a method for simultaneously reducing the dimensionality of very high-dimensional input and output spaces in Gaussian process emulators for stochastic partial differential equation models while retaining the qualitative features of the original models. This allows us to build a surrogate model for the prediction of spatial fields in such time-consuming simulators. We apply the methodology to a model of convection and dissolution processes occurring during carbon capture and storage.
Computer models are now widely used across a range of scientific disciplines to describe various complex physical systems, however to perform full uncertainty quantification we often need to employ emulators. An emulator is a fast statistical construct that mimics the complex computer model, and greatly aids the vastly more computationally intensive uncertainty quantification calculations that a serious scientific analysis often requires. In some cases, the complex model can be solved far more efficiently for certain parameter settings, leading to boundaries or hyperplanes in the input parameter space where the model is essentially known. We show that for a large class of Gaussian process style emulators, multiple boundaries can be formally incorporated into the emulation process, by Bayesian updating of the emulators with respect to the boundaries, for trivial computational cost. The resulting updated emulator equations are given analytically. This leads to emulators that possess increased accuracy across large portions of the input parameter space. We also describe how a user can incorporate such boundaries within standard black box GP emulation packages that are currently available, without altering the core code. Appropriate designs of model runs in the presence of known boundaries are then analysed, with two kinds of general purpose designs proposed. We then apply the improved emulation and design methodology to an important systems biology model of hormonal crosstalk in Arabidopsis Thaliana.
Complex stochastic models are commonplace in epidemiology, but their utility depends on their calibration to empirical data. History matching is a (pre)calibration method that has been applied successfully to complex deterministic models. In this work, we adapt history matching to stochastic models, by emulating the variance in the model outputs, and therefore accounting for its dependence on the model's input values. The method proposed is applied to a real complex epidemiological model of human immunodeficiency virus in Uganda with 22 inputs and 18 outputs, and is found to increase the efficiency of history matching, requiring 70% of the time and 43% fewer simulator evaluations compared with a previous variant of the method. The insight gained into the structure of the human immunodeficiency virus model, and the constraints placed on it, are then discussed.
Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulator's input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulator's behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was 10(11) times smaller than the original input space. Simulator evaluations made within this region were found to have a 65% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.
Background
Many mathematical models have now been employed across every area of systems biology. These models increasingly involve large numbers of unknown parameters, have complex structure which can result in substantial evaluation time relative to the needs of the analysis, and need to be compared to observed data of various forms. The correct analysis of such models usually requires a global parameter search, over a high dimensional parameter space, that incorporates and respects the most important sources of uncertainty. This can be an extremely difficult task, but it is essential for any meaningful inference or prediction to be made about any biological system. It hence represents a fundamental challenge for the whole of systems biology. Methods
Bayesian statistical methodology for the uncertainty analysis of complex models is introduced, which is designed to address the high dimensional global parameter search problem. Bayesian emulators that mimic the systems biology model but which are extremely fast to evaluate are embeded within an iterative history match: an efficient method to search high dimensional spaces within a more formal statistical setting, while incorporating major sources of uncertainty. ResultsThe approach is demonstrated via application to a model of hormonal crosstalk in Arabidopsis root development, which has 32 rate parameters, for which we identify the sets of rate parameter values that lead to acceptable matches between model output and observed trend data. The multiple insights into the model’s structure that this analysis provides are discussed. The methodology is applied to a second related model, and the biological consequences of the resulting comparison, including the evaluation of gene functions, are described. Conclusions
Bayesian uncertainty analysis for complex models using both emulators and history matching is shown to be a powerful technique that can greatly aid the study of a large class of systems biology models. It both provides insight into model behaviour and identifies the sets of rate parameters of interest.
Ensemble-based methods have gained popularity as reservoir history-matching techniques. The advantages typically attributed to these methods include the possibility of adjusting a large number of model parameters at a reasonable computational cost, the generation of several alternative models conditioned to data and the ease of implementation. In fact, it is straightforward to adapt these methods to handle different types of data and model variables. Moreover, they are easily coupled with commercial reservoir simulators. Among these methods, the ensemble Kalman filter (EnKF) is by far the most investigated. Iterative forms of the ensemble smoother (ES), on the other hand, are less widespread in the literature. However, ensemble smoothers are much better suited to practical history-matching applications, because they do not require updating dynamical (state) variables and consequently avoid the frequent simulation restarts required by EnKF. This paper presents the results of an investigation on the performance of a variant of ES, namely, ensemble smoother with multiple data assimilation (ES-MDA), to history match production and seismic data of a real field. The paper discusses the quality of the data matches, the plausibility of the history matched models, the ability of the posterior ensemble to assess the uncertainty in the forecasted water production, the effect of the number of iterations and localization. The paper also includes two appendix sections. The first one presents two alternative implementations of the ES-MDA method. The second appendix presents the matrix operations for an efficient implementation of the analysis.
The focus of this work is on an alternative implementation of the iterative
ensemble smoother (iES). We show that iteration formulae similar to those used
in \cite{chen2013-levenberg,emerick2012ensemble} can be derived by adopting a
regularized Levenberg-Marquardt (RLM) algorithm \cite{jin2010regularized} to
approximately solve a minimum-average-cost (MAC) problem. This not only leads
to an alternative theoretical tool in understanding and analyzing the behaviour
of the aforementioned iES, but also provides insights and guidelines for
further developments of the smoothing algorithms. For illustration, we compare
the performance of an implementation of the RLM-MAC algorithm to that of the
approximate iES used in \cite{chen2013-levenberg} in three numerical examples:
an initial condition estimation problem in a strongly nonlinear system, a
facies estimation problem in a 2D reservoir and the history matching problem in
the Brugge field case. In these three specific cases, the RLM-MAC algorithm
exhibits comparable or even better performance, especially in the strongly
nonlinear system.
Cosmologists at the Institute of Computational Cosmology, Durham University,
have developed a state of the art model of galaxy formation known as Galform,
intended to contribute to our understanding of the formation, growth and
subsequent evolution of galaxies in the presence of dark matter. Galform
requires the specification of many input parameters and takes a significant
time to complete one simulation, making comparison between the model's output
and real observations of the Universe extremely challenging. This paper
concerns the analysis of this problem using Bayesian emulation within an
iterative history matching strategy, and represents the most detailed
uncertainty analysis of a galaxy formation simulation yet performed.
We present a new method for solving the history matching problem by gradient-based optimization within a probabilistic framework. The focus is on minimizing the number of forward simulations and conserving geological realism of the solutions. Geological a priori information is taken into account by means of multipoint statistics borrowed from training images. Then production data and prior information are integrated into a single differentiable objective function, minimizer of which has a high posterior value. Solving the proposed optimization problem for an ensemble of different starting models, we obtain a set of solutions honouring both data and prior information.
The process of conditioning the geological or the static model to production data is typically known as history matching (HM). The economic viability of a petroleum recovery project is greatly influenced by the reservoir production performance under the current and future operating conditions. Therefore evaluation of the past and present reservoir performance and forecast of its future are essential in reservoir management process. At this point history matching plays a very important role in model updating and hence optimum forecasting, researchers are looking for new techniques, methods and algorithms to improve it.
This paper therefore reviews HM and its advancements to date including time-lapse seismic data integration. The paper covers manual and automatic HM, minimization algorithms including gradient and non gradient methods. It reviews the advantages and disadvantages of using one method over the other. Gradient methods covered include conjugate gradient, steepest descent, Gauss-Newton and Quasi-Newton. Non-gradient methods covered includes evolutionary strategies, genetic algorithm and Kalman filter (ensemble Kalman filter).It also addresses re-parameterization techniques including principal component analysis (PCA) and discrete cosine transforms (DCT). The methods are evaluated using a data set based on data from the Norne Field in the Norwegian Sea provided by Statoil and its partners to the Center of Integrated in Petroleum Industry (IO Center) at the Norwegian University of Science and Technology (NTNU).
Mathematical models, usually implemented in computer programs known as simulators, are widely used in all areas of science and technology to represent complex real-world phenomena. Simulators are often suciently complex that they take appreciable amounts of computer time or other resources to run. In this context, a methodology has been developed based on building a statistical representation of the simulator, known as an emulator. The principal approach to building emulators uses Gaussian processes. This work presents some diagnostics to validate and assess the adequacy of a Gaussian process emulator as surrogate for the simulator. These diagnostics are based on comparisons between simulator outputs and Gaussian process emulator outputs for some test data, known as validation data, dened by a sample of simulator runs not used to build the emulator. Our diagnostics take care to account for correlation between the validation data. In order to illustrate a validation procedure, these diagnostics are applied to two dierent data sets.
We thank the discussants David Poole, Pritam Ranjan, Earl Lawrence, David Higdon, and David van Dyk for their commentaries on our paper, and for raising many interesting points for discussion, ranging from practical questions related to the imple- mentation of our methodology to fundamental issues of the role and purpose of Bayesian analysis in science.
In many scientific disciplines complex computer models are used to understand the behaviour of large scale physical systems. An uncertainty anal- ysis of such a computer model known as Galform is presented. Galform models the creation and evolution of approximately one million galaxies from the begin- ning of the Universe until the current day, and is regarded as a state-of-the-art model within the cosmology community. It requires the specification of many in- put parameters in order to run the simulation, takes significant time to run, and provides various outputs that can be compared with real world data. A Bayes Linear approach is presented in order to identify the subset of the input space that could give rise to acceptable matches between model output and measured data. This approach takes account of the major sources of uncertainty in a consistent and unified manner, including input parameter uncertainty, function uncertainty, observational error, forcing function uncertainty and structural uncertainty. The approach is known as History Matching, and involves the use of an iterative suc- cession of emulators (stochastic belief specifications detailing beliefs about the Galform function), which are used to cut down the input parameter space. The analysis was successful in producing a large collection of model evaluations that exhibit good fits to the observed data.
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 introduces a new method for history matching, combining the Iterative Discrete Latin Hypercube (IDLHC) with multi-start Simulated Annealing methods. The proposed method, named IDLHCSA, combines the potential of the IDLHC in finding good matched models while preserving the diversity of solutions and the potential of the SA with local control in finding local (refined) solutions. The IDLHCSA was applied in two cases. The first is a simple reservoir model used as proof of concept. The second is a complex benchmark case (UNISIM-I-H) based on the Namorado field, located at the Campos Basin, Brazil. The robustness and effectiveness of the proposed method are demonstrated by comparison with other consolidated methods. It is demonstrated here that, when compared to other methodologies, the proposed method is more effective in finding multiple solutions for the history matching problem while maintaining solution diversity. The production forecast is analyzed and the predictive capacity of the matched models is assessed. The paper reveals that obtaining good matched models does not ensure reliable forecasts. The proposed method was able to find matched models which provided more reliable forecasts when compared to other methods.
History matching is an important process for uncertainty quantification of reservoir performance. Hundreds of reservoir simulation cases, or more, are typically required to sample the posterior probability density function of the uncertain parameters. The process is extremely challenging for shale reservoirs due to large uncertainties including reservoir heterogeneity, wide range of permeability, gas desorption, and fracture properties. Direct application of stimulated rock volume or dual continuum model fails to capture fracture connectivity and fracture conductivity that dominate the fluid transport in most shale developments. Embedding discrete fractures in reservoir simulation model is thus a preferable method to attain more realistic reservoir behavior. However, the option of using local grid refinement to embed the fractures is computationally expensive as well as intrusive to a reservoir model. Even more challenging is generating a huge set of fracture-embedded reservoir cases during the history matching process. Nevertheless, recent developments in a methodology called Embedded Discrete Fracture Model (EDFM) have successfully overcome the computational complexity to include discrete fractures in reservoir simulations. We demonstrate the implementation of the assisted history matching workflow coupled with EDFM to history match a real field case of shale gas condensate reservoir in Duvernay Shale, Canada. In the workflow, the EDFM preprocessor is fully coupled with a commercial reservoir simulator and proxy-based Markov chain Monte Carlo (MCMC) algorithm. Moreover, to further reduce the computational cost, the single-fractured submodel is used to represent the full reservoir model for this case study. Using the proposed assisted history matching workflow and the appropriate submodel enables efficient and successful uncertainty quantification of this shale play.
Computer simulators are a useful tool for understanding complicated systems. However, any inferences made from them should recognize the inherent limitations and approximations in the simulator’s predictions for reality, the data used to run and calibrate the simulator, and the lack of knowledge about the best inputs to use for the simulator. This article describes the methods of emulation and history matching, where fast statistical approximations to the computer simulator (emulators) are constructed and used to reject implausible choices of input (history matching). Also described is a simple and tractable approach to estimating the discrepancy between simulator and reality induced by certain intrinsic limitations and uncertainties in the simulator and input data. Finally, a method for forecasting based on this approach is presented. The analysis is based on the Bayes linear approach to uncertainty quantification, which is similar in spirit to the standard Bayesian approach but takes expectation, rather than probability, as the primitive for the theory, with consequent simplifications in the prior uncertainty specification and analysis.
Approximate Bayesian Computation (ABC) and other simulation-based inference methods are becoming increasingly used for inference in complex systems, due to their relative ease-of-implementation. We briefly review some of the more popular variants of ABC and their application in epidemiology, before using a real-world model of HIV transmission to illustrate some of challenges when applying ABC methods to high dimensional, computationally intensive models. We then discuss an alternative approach—history matching—that aims to address some of these issues, and conclude with a comparison between these different methodologies.
Assisted history matching approach has been arisen in the last few decades in an attempt to make the process of history matching faster and easier. Assisted history matching simply involves converting the history matching problem to an optimization problem. One main aspect of the assisted history matching is building proxy model that interpolates the relationship between the objective function and the history matching parameters. Several proxy modeling techniques are introduced in the literature, some are useful and some are not.
This paper provides a comparative study between four powerful proxy modeling techniques in assisted history matching; Thin Plate Spline, Radial Basis Function, Kriging, and Artificial Neural Network. Two test problems of different reservoir engineering approaches (material balance and reservoir simulation) are used to test and compare the performance of the studied proxy methods on solving assisted history matching problems. To make the comparison reliable, a performance indicator is developed to compare between the four studied techniques. The performance indicator represents the relative error between the estimated values of history matching parameters calculated using the studied proxy modeling methods and their exact solutions.
The results of this work indicate that the Kriging and artificial neural network proxy techniques are more efficient and superior to thin plate spline and radial basis function.
This paper presents an innovative application of a new class of parallel interacting Markov chains Monte Carlo to solve the Bayesian history matching (BHM) problem. BHM consists of sampling a posterior distribution given by the Bayesian theorem. Markov chain Monte Carlo (MCMC) is well suited for sampling, in principle, any type of distribution; however the number of iteration required by the traditional single-chain MCMC can be prohibitive in BHM applications. Furthermore, history matching is typically a highly nonlinear inverse problem, which leads in very complex posterior distributions, characterized by many separated modes. Therefore, single chain can be trapped into a local mode. Parallel interacting chains is an interesting way to overcome this problem, as shown in this paper. In addition, we presented new approaches to define starting points for the parallel chains. For validation purposes, the proposed methodology is firstly applied in a simple but challenging cross section reservoir model with many modes in the posterior distribution. Afterwards, the application to a realistic case integrated to geostatistical modelling is also presented. The results showed that the combination of parallel interacting chain with the capabilities of distributed computing commonly available nowadays is very promising to solve the BHM problem.
History matching is a model (pre-)calibration method that has been applied to computer models from a wide range of scientific disciplines. In this work we apply history matching to an individual-based epidemiological model of HIV that has 96 input and 50 output parameters, a model of much larger scale than others that have been calibrated before using this or similar methods. Apart from demonstrating that history matching can analyze models of this complexity, a central contribution of this work is that the history match is carried out using linear regression, a statistical tool that is elementary and easier to implement than the Gaussian process-based emulators that have previously been used. Furthermore, we address a practical difficulty with history matching, namely, the sampling of tiny, nonimplausible spaces, by introducing a sampling algorithm adjusted to the specific needs of this method. The effectiveness and simplicity of the history matching method presented here shows that it is a useful tool for the calibration of computationally expensive, high dimensional, individual-based models.
Reservoir characterization is the key to success in history matching and production forecasting. Thus, numerical simulation becomes a powerful tool to achieve a reliable model by quantifying the effect of uncertainties in field development and management planning, calibrating a model with history data, and forecasting field production. History matching is integrated into several areas, such as geology (geological characterization and petrophysical attributes), geophysics (4D-seismic data), statistical approaches (Bayesian theory and Markov field), and computer science (evolutionary algorithms). Although most integrated-history-matching studies use a unique objective function (OF), this is not enough. History matching by simultaneous calibrations of different OFs is necessary because all OFs must be within the acceptance range as well as maintain the consistency of generated geological models during reservoir characterization. The main goal of this work is to integrate history matching and reservoir characterization, applying a simultaneous calibration of different OFs in a history-matching procedure, and keeping the geological consistency in an adjustment approach to reliably forecast production. We also integrate virtual wells and geostatistical methods into the reservoir characterization to ensure realistic geomodels, avoiding the geological discontinuities, to match the reservoir numerical model. The proposed methodology comprises a geostatistical method to model the spatial reservoir-property distribution on the basis of the well-log data; numerical simulation; and adjusting conditional realizations (models) on the basis of geological modeling (variogram model, vertical-proportion curve, and regularized well-log data). In addition, reservoir uncertainties are included, simultaneously adjusting different OFs to evaluate the history-matching process and virtual wells to perturb geological continuities. This methodology effectively preserves the consistency of geological models during the history-matching process. We also simultaneously combine different OFs to calibrate and validate the models with well-production data. Reliable numerical and geological models are used in forecasting production under uncertainties to validate the integrated procedure.
Response surface methods are commonly used in history matching process to approximate the functional relationship between the input parameters and the aggregated mismatch. The quality of the proxy (accuracy in prediction) degrades as the nonlinearity of the response surface increases. However, commonly-used definitions of aggregated mismatch, such as root mean squared error (RMSE) or mean absolute error (MAE), are highly nonlinear. As a result, the quality of the proxy for aggregated mismatch can be unsatisfying in many cases.
In this work, we propose the proxy-for-data (PFD) approach, in which one proxy is built for each observation data point and then the data values predicted by those proxies are used to calculate the aggregated mismatch. Because proxies are constructed for the data themselves rather than for the aggregated mismatch, the nonlinearity of the aggregated mismatch definition will not affect the quality of the proxy. It is shown in multiple test cases that the new approach could potentially improve proxy quality for different types of proxies and different aggregated mismatch definitions. For cases with a large amount of observation data points, we also show that the use of reduced-order modeling can efficiently reduce the number of proxies needed and achieve similar improvement. The new approach is successfully applied to both synthetic and field examples and both examples show improved proxy quality.
This paper describes a new iterative procedure for probabilistic history matching using a discrete Latin Hypercube (DLHC) sampling method and nonparametric density estimation. The iterative procedure consists of selecting a set of models based on the history matching quality (normalized misfit) to generate histograms. The histograms are smoothed and used to estimate marginal probability densities of reservoir attributes. Three selection methods are evaluated. One of them is based on a global objective function (GOF) and the others are based on a local objective function (LOF), which is composed of influenced reservoir responses identified with the aim of a correlation matrix. The methodology was successfully applied to the UNISIM-I-H benchmark case, which is a reservoir model based on Namorado field, Campos basin, Brazil. Eight iterations with 450 combinations for each one were adequate to address the problem studied in this paper. To demonstrate the robustness of the proposed method and the consistency of the results, the iterative process was repeated 10 times and the discrepancy among the results was very small. The proposed method exhibited good convergence along the iterations, reducing the variability of the objective function (average normalized misfit) by approximately 90% from the first to the last iteration. Robustness, efficiency and facility of implementation are the key features of the proposed methodology.
In this work, we propose an ensemble 4D-seismic history-matching framework for reservoir characterization. Compared with similar existing frameworks in the reservoir-engineering community, the proposed one consists of some relatively new ingredients, in terms of the type of seismic data in choice, wavelet multiresolution analysis for the chosen seismic-data and related-data noise estimation, and the use of recently developed iterative ensemble history-matching algorithms.
Typical seismic data used for history matching, such as acoustic impedance, are inverted quantities, whereas extra uncertainties may arise during the inversion processes. In the proposed framework, we avoid such intermediate inversion processes. In addition, we also adopt wavelet-based sparse representation to reduce data size. Concretely, we use intercept and gradient attributes derived from amplitude vs. angle (AVA) data, apply multilevel discrete wavelet transforms (DWTs) to attribute data, and estimate noise level of resulting wavelet coefficients. We then select the wavelet coefficients above a certain threshold value, and history match these leading wavelet coefficients with an iterative ensemble smoother (iES).
As a proof-of-concept study, we apply the proposed framework to a 2D synthetic case originated from a 3D Norne field model. The reservoir-model variables to be estimated are permeability (PERMX) and porosity (PORO) at each active gridblock. A rock-physics model is used to calculate seismic parameters (velocity and density) from reservoir properties (porosity, fluid saturation, and pressure); then, reflection coefficients are generated with a linearized AVA equation that involves velocity and density. AVA data are obtained by computing the convolution between reflection coefficients and a Ricker wavelet function. The multiresolution analysis applied to the AVA attributes helps to obtain a good estimation of noise level and substantially reduce the data size. We compare history-matching performance in three scenarios: (S1) with production data only, (S2) with seismic data only, and (S3) with both production and seismic data. In either Scenario S2 or Scenario S3, we also inspect two sets of experiments, one with the original seismic data (full-data experiment) and the other adopting sparse representation (sparse-data experiment). Our numerical results suggest that, in this particular case study, the use of production data largely improves the estimation of permeability, but has little effect on the estimation of porosity. Using seismic data only improves the estimation of porosity, but not that of permeability. In contrast, using both production and 4D-seismic data improves the estimation accuracies of both porosity and permeability. Moreover, in either Scenario S2 or Scenario S3, provided that a certain stopping criterion is equipped in the iES, adopting sparse representation results in better history-matching performance than using the original data set.
Reservoir characterization is very important to the success of a history matching and production forecasting. Thus, numerical simulation becomes a powerful tool for the reservoir engineer in quantifying the impact of uncertainties in field development and management planning, calibrating a model with history data and forecasting field production, resulting in a reliable numerical model. History matching has been integrated into several areas, such as geology (geological characterization and petrophysical attributes), geophysics (4D seismic data), statistical approaches (Bayesian theory and Markov field), and computer science (evolutionary algorithms). Although most integrated history-matching studies use a unique objective function (OF), this is not enough. A history matching by simultaneous calibrations of different OF is necessary because all wells must have their OF near the acceptance range as well as maintain the consistency of generated geological models during reservoir characterization. The main goal of this work is to integrate history matching and reservoir characterization; applying a simultaneous calibration of different OF in a history matching procedure and keeping the geological consistency in an adjustment approach to reliably forecast production. We also integrate virtual wells and geostatistical methods into the reservoir characterization to ensure realistic geomodels without creating the geological discontinuities to match the reservoir numerical model. The proposed integrated calibration methodology consists of using a geostatistical method for modelling the spatial reservoir property distribution based on the well log data, running a numerical simulator and adjusting conditional realizations (models) based on geological modeling (variogram model, vertical proportion curve and regularized well log data) and reservoir uncertainties, using a simultaneous adjustment of different OF to evaluate the history matching process and virtual wells to perturb geological continuities such as channels and barriers. In conclusion, we present an effective methodology to preserve the consistency of geological models during history matching process. In addition, we simultaneously combine different OF to calibrate and validate the models with well production data. Reliable numerical and geological models are used in the forecasting production under uncertainties to validate the integrated procedure. Copyright © (2015) by the Offshore Technology Conference All rights reserved.
Assisted history matching procedures are usually implemented within an optimization procedure, where the goal is to minimize the object function, which is written in terms of the error between observed and simulated data. One of the problems of the process is that the objective function represented by a single value is not sufficient to express the complexity of the problem. The errors that are normally measured for each well and for all production rates and pressure are converted to a single value; usually the norm of all error vectors. If the problem is well behaved and the objective function quickly converges within a desired tolerance, this would not be a problem, but this is usually not the case for complex history matching processes, since it is very difficult to find a model (or a set of them) that matches with a reasonable tolerance the production profiles for every wells. This work proposes a new approach to deal simultaneously with several objective functions (for instance, well rates and pressure). The methodology follows a probabilistic approach where several simulation models are handled during the entire procedure. The initial step is to generate several simulation models by combining the most important uncertainties of the reservoir. Then, an iterative procedure is performed to iteratively change the reservoir attributes and filter the models that are closer to history data. This procedure encompass a re-characterization step, where the probability of the attributes are updated, global multipliers are applied and local changes are made around problematic wells in order to provide a set of model that yield production and pressure curves with better dispersion compared to history data for every well. The key point of the methodology is that for every iteration the errors of each model, well and objective function can be visualized in a very concise plot that is based on the normalized quadratic error of each curve. The plot clearly shows global and local problems of the set of simulation models, so it is a good indicator of the changes to be made in the next iteration. Another point to highlight is that the same type of iterative procedure is performed to integrate 4D seismic into the process. Thus, after selecting a set of models with a good representation of well history data the same iterative process is repeated to generate a new set of models that match 4D seismic data as well. Thus, the proposed methodology integrates 4D seismic and well history data to reduce uncertainties with a probabilistic approach. To validate the methodology an application is shown where it can be seen the gradual improvement achieved for the models along the iterations.
In this paper we present and illustrate basic Bayesian techniques for the uncertainty analysis of complex physical systems modelled by computer simulators. We focus on emulation and history matching and also discuss the treatment of observational errors and structural discrepancies in time series. We exemplify such methods using a four-box model for the termohaline circulation. We show how these methods may be applied to systems containing tipping points and how to treat possible discontinuities using multiple emulators.
History matching is a challenging and time-consuming task related to reservoir simulation. Probabilistic approaches using dynamic data are often used to reduce reservoir uncertainties and improve matching. This work presents a new process to evaluate and reduce reservoir uncertainties using multivariate analysis incorporating the interaction between reservoir properties.
The proposed uncertainty reduction workflow provides a multivariate approach without the use of proxy models, allowing understanding of the reservoir response through the R² matrix as well as more reliable reservoir predictions. The methodology offers a quantitative analysis and a new tool to evaluate and reduce uncertainties. The process uses a Latin Hypercube (LHC) to sample the reservoir attribute range and a smoothed mismatch data set from the LHC selected objective functions. The attribute interval, which minimizes the mismatch, is identified through polynomial fitting. The main objective is to reduce uncertainties considering the reservoir attributes range and a multivariate sensitivity matrix.
The methodology was firstly applied to a simple synthetic reservoir simulation model with 20 uncertainty attributes and we drew the following conclusions: (1) R² sensitivity matrix clearly showed the key physical features of the reservoir model; (2) all reservoir attributes ranges were reduced, providing a set of simulation models with improved history matching. We successfully applied to the UNISIM-I-H reservoir model based on Namorado field, Campos basin, Brazil.
Calibration of stochastic traffic microsimulation models is a challenging task. This paper proposes a fast iterative probabilistic precalibration framework and demonstrates how it can be successfully applied to a real-world traffic simulation model of a section of the M40 motorway and its surrounding area in the U.K. The efficiency of the method stems from the use of emulators of the stochastic microsimulator, which provides fast surrogates of the traffic model. The use of emulators minimizes the number of microsimulator runs required, and the emulators' probabilistic construction allows for the consideration of the extra uncertainty introduced by the approximation. It is shown that automatic precalibration of this real-world microsimulator, using turn-count observational data, is possible, considering all parameters at once, and that this precalibrated microsimulator improves on the fit to observations compared with the traditional expertly tuned microsimulation.
We apply an established statistical methodology called history matching
to constrain the parameter space of a coupled non-flux-adjusted climate
model (the third Hadley Centre Climate Model; HadCM3) by using a
10,000-member perturbed physics ensemble and observational metrics.
History matching uses emulators (fast statistical representations of
climate models that include a measure of uncertainty in the prediction
of climate model output) to rule out regions of the parameter space of
the climate model that are inconsistent with physical observations given
the relevant uncertainties. Our methods rule out about half of the
parameter space of the climate model even though we only use a small
number of historical observations. We explore 2 dimensional projections
of the remaining space and observe a region whose shape mainly depends
on parameters controlling cloud processes and one ocean mixing
parameter. We find that global mean surface air temperature (SAT) is the
dominant constraint of those used, and that the others provide little
further constraint after matching to SAT. The Atlantic meridional
overturning circulation (AMOC) has a non linear relationship with SAT
and is not a good proxy for the meridional heat transport in the
unconstrained parameter space, but these relationships are linear in our
reduced space. We find that the transient response of the AMOC to
idealised CO2 forcing at 1 and 2 % per year shows a greater
average reduction in strength in the constrained parameter space than in
the unconstrained space. We test extended ranges of a number of
parameters of HadCM3 and discover that no part of the extended ranges
can by ruled out using any of our constraints. Constraining parameter
space using easy to emulate observational metrics prior to analysis of
more complex processes is an important and powerful tool. It can remove
complex and irrelevant behaviour in unrealistic parts of parameter
space, allowing the processes in question to be more easily studied or
emulated, perhaps as a precursor to the application of further relevant
constraints.
This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties.
The use of the ensemble smoother (ES) instead of the ensemble Kalman filter increases the nonlinearity of the update step during data assimilation and the need for iterative assimilation methods. A previous version of the iterative ensemble smoother based on Gauss–Newton formulation was able to match data relatively well but only after a large number of iterations. A multiple data assimilation method (MDA) was generally more efficient for large problems but lacked ability to continue “iterating” if the data mismatch was too large. In this paper, we develop an efficient, iterative ensemble smoother algorithm based on the Levenberg–Marquardt (LM) method of regularizing the update direction and choosing the step length. The incorporation of the LM damping parameter reduces the tendency to add model roughness at early iterations when the update step is highly nonlinear, as it often is when all data are assimilated simultaneously. In addition, the ensemble approximation of the Hessian is modified in a way that simplifies computation and increases stability. We also report on a simplified algorithm in which the model mismatch term in the updating equation is neglected. We thoroughly evaluated the new algorithm based on the modified LM method, LM-ensemble randomized maximum likelihood (LM-EnRML), and the simplified version of the algorithm, LM-EnRML (approx), on three test cases. The first is a highly nonlinear single-variable problem for which results can be compared against the true conditional pdf. The second test case is a one-dimensional two-phase flow problem in which the permeability of 31 grid cells is uncertain. In this case, Markov chain Monte Carlo results are available for comparison with ensemble-based results. The third test case is the Brugge benchmark case with both 10 and 20 years of history. The efficiency and quality of results of the new algorithms were compared with the standard ES (without iteration), the ensemble-based Gauss–Newton formulation, the standard ensemble-based LM formulation, and the MDA. Because of the high level of nonlinearity, the standard ES performed poorly on all test cases. The MDA often performed well, especially at early iterations where the reduction in data mismatch was quite rapid. The best results, however, were always achieved with the new iterative ensemble smoother algorithms, LM-EnRML and LM-EnRML (approx).
Computer simulators are commonly used in many areas of science to investigate complex physical systems of interest. The output of such sim- ulators is often expensive and time-consuming to obtain, thus limiting the amount of information we can gather about their behaviour. In some cases, an approximate simulator is also available, which can be evaluated in a fraction of the time and which shares many qualitative features with the original simulator. Since the approximate simulator is cheap to evalu- ate, large numbers of evaluations can be performed and its behaviour can be investigated thoroughly. We propose a technique to construct efficient small sample designs for runs of the full simulator by exploiting the infor- mation gained from the approximate simulator via Bayes linear methods. The methodology is illustrated with an example concerning a computer simulation of a hydrocarbon reservoir.
For random variables with a unimodal Legesgue density, the 3[sgrave] rule is proved by elementary calculus. It emerges as a special case of the Vysochanskiĭ-Petunin inequality, which in turn is based on the Gauss inequality.
The ensemble Kalman filter (EnKF) has attracted attention as a useful method for solving the history matching problem. The EnKF is based on the simpler Kalman filter, which is an efficient recursive filter that estimates the state of a linear dynamical system from a series of noisy measurements. Model parameters for a dynamic two-phase model of fluid flow in a well were tuned and the method was applied to a set of full-scale experimental data. Improved predictions of the pressure behavior of the well were obtained. The EnKF was employed to update permeability fields for near-well reservoir models. The reservoir model is updated using EnKF and an optimal flooding strategy is found using adjoint-based optimization. Closed-loop optimization methods that do not require adjoints have been developed.
We consider the problem of designing for complex high-dimensional computer models that can be evaluated at different levels of accuracy. Ordinarily, this requires performing many expensive evaluations of the most accurate version of the computer model to obtain a reasonable coverage of the design space. In some cases, it is possible to supplement the information from the accurate model evaluations with a large number of evaluations of a cheap, approximate version of the computer model to enable a more informed design choice. We describe an approach that combines the information from both the approximate model and the accurate model into a single multiscale emulator for the computer model. We then propose a design strategy for selecting a small number of expensive evaluations of the accurate computer model based on our multiscale emulator and a decomposition of the input parameter space. We illustrate our methodology with an example concerning a computer simulation of a hydrocarbon reservoir.
The Norne field is the largest discovery on the Norwegian continental shelf in more than a decade with recoverable oil reserves of 450 Million bbl. Reservoir drive mechanism will be a combination of gas and water reinjection. The largest monohull production ship in the world will be built to develop the field. To improve theproject economics and company performance a clearobjective was established to reduce investments costs by 25-30% and a time from find to first oil of 5-6 years. The paper will describe both the concept development chosen to achieve these goals and the method of working used to find a simple robust solution to meet uncertainties in project development, reservoir nperformance and regional potential while maintaining a fast track approach.
INTRODUCTION
The block 6608/1O Norne field is the largest discovery on the Norwegian continental shelf in more than a decade. The field extends for 10 km, is 2 km wide and is located 200 km west of the mid-Norway coast in 380 m waters.
The Norne field is owned by a partnership of Statoil (50%), Saga (15%), Norsk Hydro (15%), Enterprise (1O%) and Agip (10%). Statoil is the operator.
Well 6608/1 O-2 first penetrated the Norne reservoir in December 1991. Appraisal well 6608/1 O-3 was drilled in 1993 and proved the field?s northerly extension.
Based on results from those two wells, a development project began in 1993.
Exploration well 6608/1 O-4 was drilled in a separate smaller structure north-east of Nome and proved some additional reserves.
To improve project economics and company performance, a clear objective was established to reduce investment costs by 25-30% to the established levelcurrent in 1993. The Nome organization in Sept. ?94 submitted a Plan for Development and Operation (PDO) to the Norwegian authorities for final approval firstquarter of 1995.
NORNE RESERVOIR
Norne hydrocarbons are located in sandstone rocks of the Lower and Middle Jurassic age. The column consists of a 100 m thick oil zone with an overlying gas cap. The reservoir is a flat structure and the top reservoir is at a depth of about 2525 m below mean sea level. The reservoir pressure is close to hydrostatic. At reference depth 2639 m below mean sea level the formation pressure is 273 bar and the temperature 98 degrees C. The reservoir quality is generally good with 100-to 2,500 md horizontal permeabilities.
Total field reserves are estimated at 160 mill. Sm3 of oiland 29 mrd. Sm3 of gas (free gas and associated gas). Based on reservoir simulation and risk analysis the most likely recoverable oil reserves is about 72 mill. Sm3 for a recovery rate Of roughly 45%.
The reservoir drive mechanism, will be a combination of water and gas injection. Produced gas will be reinfected into the gas cap and water will be injected into the water zone. The concept will be designed for alternating water and gas injection in the southern part of the field. Production wells will be located to delay gas and water breakthrough and to minimize gas and water production.
History matching is a type of inverse problem in which observed reservoir behavior is used to estimate reservoir model variables
that caused the behavior. Obtaining even a single history-matched reservoir model requires a substantial amount of effort,
but the past decade has seen remarkable progress in the ability to generate reservoir simulation models that match large amounts
of production data. Progress can be partially attributed to an increase in computational power, but the widespread adoption
of geostatistics and Monte Carlo methods has also contributed indirectly. In this review paper, we will summarize key developments
in history matching and then review many of the accomplishments of the past decade, including developments in reparameterization
of the model variables, methods for computation of the sensitivity coefficients, and methods for quantifying uncertainty.
An attempt has been made to compare representative procedures and to identify possible limitations of each.
KeywordsHistory matching–Review–Ensemble Kalman filter–Parameterization
For many large-scale datasets it is necessary to reduce dimensionality to the point where further exploration and analysis can take place. Principal variables are a subset of the original variables and preserve, to some extent, the structure and information carried by the original variables. Dimension reduction using principal variables is considered and a novel algorithm for determining such principal variables is proposed. This method is tested and compared with 11 other variable selection methods from the literature in a simulation study and is shown to be highly effective. Extensions to this procedure are also developed, including a method to determine longitudinal principal variables for repeated measures data, and a technique for incorporating utilities in order to modify the selection process. The method is further illustrated with real datasets, including some larger UK data relating to patient outcome after total knee replacement.
We describe an approach, termed reified analysis, for linking the behaviour of mathematical models with inferences about the physical systems which the models represent. We describe the logical basis for the approach, based on coherent assessment of the implications of deficiencies in the mathematical model. We show how the statistical analysis may be carried out by specifying stochastic relationships between the model that we have, improved versions of the model that we might construct, and the system itself. We illustrate our approach with an example concerning the potential shutdown of the Thermohaline circulation in the Atlantic Ocean.
The Bayesian approach to quantifying, analysing and reducing uncertainty in the application of complex process models is attracting increasing attention amongst users of such models. The range and power of the Bayesian methods is growing and there is already a sizeable literature on these methods. However, most of it is in specialist statistical journals. The purpose of this tutorial is to introduce the more general reader to the Bayesian approach.
Semi-analytic models are a powerful tool for studying the formation of galaxies. However, these models inevitably involve a significant number of poorly constrained parameters that must be adjusted to provide an acceptable match to the observed universe. In this paper, we set out to quantify the degree to which observational data-sets can constrain the model parameters. By revealing degeneracies in the parameter space we can hope to better understand the key physical processes probed by the data. We use novel mathematical techniques to explore the parameter space of the GALFORM semi-analytic model. We base our investigation on the Bower et al. 2006 version of GALFORM, adopting the same methodology of selecting model parameters based on an acceptable match to the local bJ and K luminosity functions. The model contains 16 parameters that are poorly constrained, and we investigate this parameter space using the Model Emulator technique, constructing a Bayesian approximation to the GALFORM model that can be rapidly evaluated at any point in parameter space. By combining successive waves of emulation, we show that only 0.26% of the initial volume is of interest for further exploration. However, within this region we show that the Bower et al. 2006 model is only one choice from an extended sub-space of model parameters that can provide equally acceptable fits. We explore the geometry of this region and begin to explore the physical connections between parameters that are exposed by this analysis. We also consider the impact of adding additional observational data to further constrain the parameter space. Comment: 33 pages, 15 figures. Accepted by MNRAS
The text provides a thorough coverage of Bayes linear analysis, from the development of the basic language to the collection of algebraic results needed for efficient implementation, with detailed practical examples.
The book covers:
The importance of partial prior specifications for complex problems where it is difficult to supply a meaningful full prior probability specification.
Simple ways to use partial prior specifications to adjust beliefs, given observations.
Interpretative and diagnostic tools to display the implications of collections of belief statements, and to make stringent comparisons between expected and actual observations.
General approaches to statistical modelling based upon partial exchangeability judgements.
Bayes linear graphical models to represent and display partial belief specifications, organize computations, and display the results of analyses.
We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the fitted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-fitting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.