A Robust Iterative Ensemble Smoother Method for Efficient History Matching and Uncertainty Quantification

Channelized reservoirs consist of sinusoid patterns for sandstone. It is the most important for decision making to characterize connectivities because they significantly affect reservoir performances. Ensemble Kalman filter (EnKF) and ensemble smoother (ES) modify reservoir models using dynamic data. However, it is difficult for them to apply to channelized reservoirs because they assume that model parameters follow Gaussian distribution. The purpose of this research is to characterize 3D channel connectivities using ensemble-based methods. We use the concept of multiple Kalman gains for improved inverse modeling and it is applied to ES for fast history matching. Multiple Kalman gains are calculated by distance-based method such as hausdorff distance and kernel kmeans clustering. The proposed method, ES with multiple Kalman gains, is compared to EnKF and ES for 3D synthetic case. It solves overshooting problem in ES and describes better channel connectivities and bimodal distribution than EnKF. Furthermore, it requires only 6.4% simulation time of EnKF. When oil production rates and water cuts are predicted by updated ensembles, only the proposed method gives reliable uncertainty quantifications while EnKF and ES deviate from the true productions. Therefore, the proposed method can be utilized for fast decision making tool for 3D channelized reservoirs.

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).

Time-lapse seismic data has been widely used for the detailed reservoir characterization, and data assimilation algorithms are commonly used for petroleum reservoir history matching of production data. However, hardly any seismic data has been integrated into the reservoir inverse modeling workflow, due to the large data size, finer gridding, and especially the scarcity in time of the seismic data. Popular ensemble-based reservoir inverse modeling methods such as ensemble Kalman filter (EnKF) also face the problems of high computational cost due to the storage of the intermediate variables, restarting the reservoir simulating process and the inconsistency of the full-step and step-wise simulations. The intrinsic sequential data assimilations characteristics of the EnKF set obstacles to assimilate 4D seismic data. The alternative method can be the ensemble smoother (ES), which is an ensemble based method for data assimilation. The ES is based on a Bayesian updating scheme of the reservoir model to match the production history and improve the production forecast. To improve the algorithm convergence, a multiple data assimilation (MDA) method was proposed by Emerick and Reynolds (2012a). The algorithm we used is called ensemble smoother with multiple data assimilation (ESMDA), we modified the ESMDA to integrate the geophysical data, and created the new workflow to history match both the production and geophysical data. The available production data is well measurements, including oil production rate, well water cut and bottom-hole pressure, as well as time-lapse geophysical data, including P-wave impedance. In this paper, we first formulated the mathematical descriptions of ESMDA, we then down-scaled the seismic data, and modified the data assimilation algorithms, illustrated the history matching workflow for both production and geophysical data, finally, we showed how it works with a case study on a water flooding operation in a synthetic reservoir. In comparison, we also showed the history matching only on the production data, which yields an inferior results than the one matches both production and seismic data.

This work focuses on the improvement of an integrated methodology for the automatic history matching of compartmentalised reservoirs using 4D seismic results, stochastic initialization and the Ensemble Kalman Filter method. We show the comparison of two different history matching approaches using the Ensemble Kalman Filter (EnKF) to update the Fault Transmissibility Multipliers (FTM) initially estimated with and without considering the 4D seismic results. In this study, the parameters updated during the history matching are two-phase fault transmissibility multipliers (FTM), absolute permeability and effective porosity of a synthetic realistic 3D reservoir. The true impedance map and the changes in reservoir pressure and saturation were previously computed from 4D seismic results. The systematic estimation of two-phase fault transmissibility multipliers is based on the integration of the collected 4D seismic results and an established method validated in our previous work based on a deterministic model, using the gradient-based History Matching, Levenberg Marquardt method (LM). We present the history matching of a synthetic reservoir using the Ensemble Kalman Filter (EnKF) considering the 4D seismic results to update the models and geostatistical techniques to produce the initial geological models. The stochastic method used is the Sequential Gaussian Simulation (SGS) technique to generate 100 initial models. During history matching using the EnKF, the saturation distributions are computed from the forward modelling of a two-phase system (oil-water). The impedance maps are then estimated using the Gassmann equation and compared with the true impedance map as part of the History Matching process. To validate the results, the cost function consisting of two components is calculated, the first is the structural similarity index of the two reconstructed impedance images to the real impedance image and the second is the RMS cost function value, l2 – norm of the difference between the true (real) and the simulated pressure-production data. The EnKF history matching using the two-phase FTM values considering 4D seismic results produced lower cost function values compared with the model using the initial FTM multiplier without considering 4D seismic results. The EnKF history matching algorithm using 4D seismic presented in this work produced results closer to the true reservoir impedance map compared to our previous 4D gradient based history matching method.

As the role of reservoir flow simulation increasingly impacts existing operations and field development decisions, it follows that rigor, fitness and consistency should be imposed on the calibration of reservoir flow models to dynamic data through history matching. Although a wealth of history matching techniques exist in the petroleum literature that propose novel algorithms or share case studies, seldom does the content guide the modeler in fit-for-purpose reservoir model calibration for an operating asset. To evaluate the applicability of these diverse techniques against standards required for reservoir management, an internal study was performed to benchmark four assisted history matching (AHM) techniques commonly promoted in the oil and gas industry. The techniques were vetted against a comprehensive suite of modeling requirements for multiple asset classes, integrating a variety of historical dynamic data types through the calibration of reservoir properties that control flow behavior from the field to inter-well scale. The methods benchmarked were: (1) Design of Experiments (DoE)-based, (2) Ensemble Kalman Filter and Ensemble Smoother, (3) Genetic Algorithm and (4) Generalized Travel Time Inversion. This manuscript focuses solely on the DoE-based technique.In order to consistently benchmark the techniques, a set of standards was defined against which each was evaluated to determine its suitability for widespread history matching applications. The standards involve: the capacity to parameterize (and therefore calibrate) a diversity of reservoir flow model attributes, the capacity to integrate different types of dynamic data, the level of independence from the flow simulator and the capability to provide probabilistic outcomes for predictive uncertainty assessment. Of the four techniques, the DoE-based approach uniquely satisfied all requirements. Its history matching workflow has the flexibility to incorporate any form of reservoir model parameter and to assimilate a history matching error metric for any individual or group of historical data types; therefore, benchmarking established DoE-based techniques as unambiguously the most compliant with generic asset modeling requirements. The approach was also identified as the most straightforward, both theoretically and in practical computation, and therefore applicable to the broadest range of practitioners. Perhaps most importantly, the approach demonstrated the capacity for accurate quantification of uncertainty (or non-uniqueness) in reservoir quality resulting from an exhaustive, although approximate, exploration of model parameter space and the associated history matching error metric(s).This manuscript compiles the results and insights gained from benchmarking of the DoE-based techniques through the proposal of a comprehensive assisted history matching workflow. The workflow is designed for generality while providing best practices that guide the modeler in fit-for-purpose application. Limitations of the workflow are also recognized. Key components include: selection and screening of calibration parameters based on statistical significance, development of surrogate models to characterize the relationship between parameters and the simulated historical data being integrated into the reservoir model, use of the surrogate models for exhaustive yet efficient exploration of parameter space to identify (non-unique) history matched models, and (deterministic or probabilistic) discrete reservoir model selection for use in forecast-based decision making. Each step of the AHM workflow is presented from a conceptual and applied perspective, and field applications are provided to demonstrate key concepts. Although the applications presented include two deepwater Gulf of Mexico assets, the workflow and insights provided are developed from benchmarking across a diverse suite of asset types.

Technology Focus Many reservoir engineers dislike the very idea of automatic history matching applied to real full-field studies. They believe there is no artificial substitute for experienced reasoning, deep understanding of the reservoir mechanisms, and attention to real-life practical aspects of the problem. Some use terms such as art and intuition. For them, even if computers long ago learned to play chess, computers will never be able to perform real-case history matching on their own or at least they are still too far from this achievement. Very often, during technical sessions, immediately following an advanced mathematical presentation on history matching, someone in the audience makes his or her point about the limits of automatic approaches. To avoid disputes, experienced speakers prefer less pretentious expressions such as assisted or semiautomatic history matching. Indeed, history matching can be seen as a two-step iterative process, normally requiring many cycles to be completed. Broadly speaking, the first step is about analysis and setting the problem parameters, and the second step is about searching for and computing solutions. We start our discussion with the second part, which has a more obvious algorithmic nature. There has being a great deal of research and progress in this area. The ensemble Kalman filter is dominating the scene, but gradient-based methods and global-optimization stochastic methods are attracting merited attention. Most published contributions come from universities, and, typically, papers include examples to demonstrate successful algorithm application. These examples can be simple synthetic or somewhat-more-realistic cases, but the discussion is naturally focused on the solution method and not on the entire problem as found in the field. The first part of the problem is less mathematized, for now, and involves essential tasks such as to be clear about the practical purposes and requirements in the particular context; to have a full understanding of the quality of the reservoir model and the production data; to design or redesign well-justified objective functions; to set adequate parameterization, considering the main uncertainties and their effect on the simulation results; to represent properly and sample the uncertainty space; and to evaluate results from the previous steps of the history-matching process judiciously. Unfortunately, the strategies used to consider this part of the problem are much less discussed and documented. In fact, many of these tasks are open to further formalization and, ultimately, can be automated also. We definitely need more papers illuminating these other aspects of the reservoir-engineering problem, instead of relying on intuition. Recommended additional reading at OnePetro: www.onepetro.org. SPE 146292 Practical Assisted History Matching and Probabilistic Forecasting Procedure: A West Africa Case Study by B.O. Dujardin, SPE, Chevron, et al. SPE 141929 An Ensemble Smoother for Assisted History Matching by J.-A. Skjervheim, Statoil, et al. SPE 142497 Advanced History-Matching Techniques Reviewed by R. Rwechungura, Norwegian University of Science and Technology, et al. SPE 143067 History Matching and Uncertainty Quantification: Multiobjective Particle-Swarm-Optimization Approach by Linah Mohamed, SPE, Heriot-Watt University, et al.

History matching is a challenging reverse problem due to the inherited uncertainty from Earth modeling. History matching process mandates building one or more sets of representative reservoir models with minimum misfit between the reservoir models and observed field data. There are several commercial assisted history matching tools and workflows to facilitate and orchestrate this process. This paper provides performance assessment and process evaluation between two of the commonly used optimization algorithms: the Evolutionary Algorithm (EA) and the Ensemble Smoother with Multiple Data Assimilation (ES-MDA) in an automated and closed loop workflows. The workflow starts with a parametrization step to identify and quantify the static and dynamic uncertainties in the reservoir model. The objective function is then constructed to guide and measure the convergence of the optimization process. The workflow is then executed to start the model calibration and history matching process. The algorithms, EA and ES-MDA, take different routes during the optimization process. EA works by selecting the best cases (i.e. most fitting genes that resulted in a minimum misfit) in each cycle, and generates a number of simulation cases (new generation) based on the predefined gene selection criteria. The iterative ES-MDA algorithm maintains the same number of simulation cases in each iteration (as many as in the initial Ensemble) throughout the assimilation process. The EA and ES-MDA optimization algorithms have their advantages and disadvantages to the history matching process. For instance, both workflows preserve the geological consistency with the dynamic reservoir models. The automation that both workflows provide incorporates G&G (Geological and Geophysical) workflows to capture, and carryover the static uncertainty to ensure consistency throughout the history matching process. In addition, in both cases, multiple simulation models are generated to cover reservoir static and dynamic uncertainties, which are crucial to the predictive capabilities of the history matched models. Both algorithms, as known, are stochastics in nature (i.e. cannot guarantee one deterministic solution) and, therefore. the results are highly dependent on the initial ensemble. EA is highly effective in searching the solution space, it requires a large number of simulation runs to converge a solution. On the other hand, ES-MDA does not exceed the number of initial ensembles in each iteration regardless of the number of iterations. This paper presents an automated workflow that leads to multiple reliable history matched simulation models. All of the generated models are geologically consistent and capture all model uncertainties that are needed to be preserved and carried forward in prediction. Moreover, this helps in updating the models seamlessly whenever new data is available by simply re-running the workflow with the model updates.

Full-physics models in history matching and optimization can be computationally expensive since these problems usually require hundreds of simulations or more. We have previously implemented a physics-based data-driven network model with a commercial simulator that serves as a surrogate without the need to build the 3-D geological model. In this paper, we reconstruct the network model to account for complex reservoir conditions of mature fields and successfully apply it to a diatomite reservoir in the San Joaquin Valley (SJV) for rapid history matching and optimization. The reservoir is simplified into a network of 1-D connections between well perforations. These connections are discretized into grid blocks and the grid properties are calibrated to historical production data. Elevation change, saturation distribution, capillary pressure, and relative permeability are accounted for to best represent the mature field conditions. To simulate this physics-based network model through a commercial simulator, an equivalent 2-D Cartesian model is designed where rows correspond to the above-mentioned connections. Thereafter, the history matching can be performed with the Ensemble Smoother with Multiple Data Assimilation (ESMDA) algorithm under a sequential iterative process. A representative model after history matching is then employed for well control optimization. The network model methodology has been successfully applied to the waterflood optimization for a 56-well sector model of a diatomite reservoir in the SJV. History matching result shows that the network model honors field-level production history and gives reasonable matches for most of the wells, including pressure and flow rate. The calibrated ensemble from the last iteration of history matching yields a satisfactory production prediction, which is verified by the remaining historical data. For well control optimization, we select the P50 model to maximize the Net Present Value (NPV) in 5 years under provided well/field constraints. This confirms that the calibrated network model is accurate enough for production forecasts and optimization. The use of a commercial simulator in the network model provided flexibility to account for complex physics, such as elevation difference between wells, saturation non-equilibrium, and strong capillary pressure. Unlike traditional big-loop workflow that relies on a detailed characterization of geological models, the proposed network model only requires production data and can be built and updated rapidly. The model also runs much faster (tens of seconds) than a full-physics model due to the employment of much fewer grid blocks. To our knowledge, this is the first time this physics-based data-driven network model is applied with a commercial simulator on a field waterflood case. Unlike approaches developed with analytic solutions, the use of commercial simulator makes it feasible to be further extended for complex processes, e.g., thermal or compositional flow. It serves as an useful surrogate model for both fast and reliable decision-making in reservoir management.

Reservoir history matching by ensemble smoother with principle component and sensitivity analysis for heterogeneous formations

Summary In general, a probabilistic framework for a modeling process involves two uncertainty spaces: model parameters and state variables (or predictions). The two uncertainty spaces in reservoir simulation are connected by the governing equations of flow and transport in porous media in the form of a reservoir simulator. In a forward problem (or a predictive run), the reservoir simulator directly maps the uncertainty space of the model parameters to the uncertainty space of the state variables. Conversely, an inverse problem (or history matching) aims to improve the descriptions of the model parameters by using the measurements of state variables. However, we cannot solve the inverse problem directly in practice. Numerous algorithms, including Kriging-based inversion and the ensemble Kalman filter (EnKF) and its many variants, simplify the system by using a linear assumption. The purpose of this paper is to improve the integration of measurement errors in the history-matching algorithms that rely on the linear assumption. The statistical moment equation (SME) approach with the Kriging-based inversion algorithm is used to illustrate several practical examples. In the Motivation section, an example of pressure conditioning has a measurement that contains no additional information because of its significant measurement error. This example highlights the inadequacy of the current method that underestimates the conditional uncertainty for both model parameters and predictions. Accordingly, we derive a new formula that recognizes the absence of additional information and preserves the unconditional uncertainty. We believe this to be the consistent behavior to integrate measurement errors. Other examples are used to validate the new formula with both linear and nonlinear (i.e., the saturation equation) problems, with single and multiple measurements, and with different configurations of measurement errors. For broader applications, we also develop an equivalent formula for algorithms in the Monte Carlo simulation (MCS) approach, such as EnKF and ensemble smoother (ES).

Multiple assisted history matching (AHM) techniques commonly applied in the oil and gas industry were benchmarked for their applicability to field studies. The benchmarking effort was aimed at exploration of the theoretical as well as practical benefits, challenges and tradeoffs of the AHM techniques as they are utilized for asset development and reservoir management. This paper specifically discusses benchmarking of ensemble-based methods as applied for history matching of a key deepwater asset for the corporation. Two variations of ensemble-based methods are examined: the ensemble Kalman filter (EnKF) (Evensen, 2009) and the ensemble smoother with multiple data assimilation (ES-MDA) (Emerick and Reynolds, 2013). In both techniques, the ensemble of parameter estimates evolves as field data are assimilated into the reservoir characterization, either sequentially using the EnKF or iteratively using the ES-MDA. Both of these variations have been deployed with adaptive bootstrap-based screening (Zhang and Oliver, 2011) to reduce the sampling error of ensemble estimates. One significant strength of ensemble-based methods is their ability to estimate high-resolution model parameters. In this benchmark study, however, the ensemble-based methods were applied for estimation of a group of low-resolution model parameters including regional property multipliers and pilot points. The insights gained from the inverse problem of constraining reservoir simulation models to historical production data can be particularly valuable in guiding the practitioner in re-definition of the prior model uncertainty and towards the model solution. The study shows that a major challenge in the application of ensemble-based methods is parameter over-adjustment resulting from the improper addition of observation errors associated with the actual observations. In practice, multiple experiments are required for the user to iteratively improve the optimal level of the added errors. As the number of observations grows large, the increase in computational cost and decrease in ensemble variability are additional challenges in the application of ensemble-based methods. While assimilating different types of data, a balanced proportion of these data types should be maintained. These points are illustrated and discussed in the field application section of the paper.

SummaryFull-physics models in history matching (HM) and optimization can be computationally expensive because these problems usually require hundreds of simulations or more. In a previous study, a physics-based data-driven network model was implemented with a commercial simulator that served as a surrogate without the need to build a 3D geological model. In this paper, the network model is reconstructed to account for complex reservoir conditions of mature fields and successfully apply it to a diatomite reservoir in the San Joaquin Valley, California, for rapid HM and optimization.The reservoir is simplified into a network of 1D connections between well perforations. These connections are discretized into gridblocks, and the grid properties are calibrated to historical production data. Elevation change, saturation distribution, capillary pressure, and relative permeability are accounted for to best represent the mature field conditions. To simulate this physics-based network model through a commercial simulator, an equivalent Cartesian model is designed where rows correspond to the previously mentioned connections. Thereafter, the HM can be performed with the ensemble smoother with multiple data assimilation (ESMDA) algorithm under a sequential iterative process. A representative model after HM is then used for well control optimization.The network model methodology has been successfully applied to the waterflood optimization for a 56-well sector model of a diatomite reservoir in the San Joaquin Valley. HM results show that the network model matches with field level production history and gives reasonable matches for most of the wells, including pressure and volumetric data. The calibrated posterior ensemble of HM yields a satisfactory production prediction that is verified by the remaining historical data. For well control optimization, the P50 model is selected to maximize the net present value (NPV) in 5 years under provided well/field constraints. This confirms that the calibrated network model is accurate enough for production forecasts and optimization. The use of a commercial simulator in the network model provided flexibility to account for complex physics, such as elevation difference between wells, saturation nonequilibrium, and strong capillary pressure. Unlike the traditional big-loop workflow that relies on a detailed characterization of geological models, the proposed network model only requires production data and can be built and updated rapidly. The model also runs much faster (tens of seconds) than a full-physics model because of the use of much fewer gridblocks.To our knowledge, this is the first time this physics-based data-driven network model is applied with a commercial simulator on a field waterflood case. Unlike approaches developed with analytic solutions, the use of a commercial simulator makes it feasible to be further extended for complex processes (e.g., thermal or compositional flow). It serves as a useful surrogate model for both fast and reliable decision-making in reservoir management.

Recently, the Ensemble Kalman Filter (EnKF) has emerged as an effective tool for performing continuous updating of petroleum reservoir simulation models. The method is firmly grounded on the theory of Kalman filters and sequential Monte Carlo techniques. The ability of the method to sequentially update the spatial properties in petroleum reservoir models, such as permeability and porosity, by integrating the dynamic production data makes it a very attractive approach. Moreover, the method takes into account the production uncertainty in the reservoir models by using error covariance matrices for the measurement vector (Production and injection rates, Gas-Oil ratio, Steam-Oil ratio, etc.) and the state vector (pressure, saturation, permeability, porosity). Similar to the traditional Kalman filter, the covariance matrices have to be tuned to reflect the uncertainty in the model and the measurements. We consider two unconventional oil reservoir models: 1) highly heterogeneous black-oil reservoir model, and 2) heterogeneous SAGD reservoir model. The results will demonstrate the advantage of using the localized EnKF for effective history matching using ensemble sizes relatively lower than what otherwise would be required with the ordinary EnKF. The results will also show the advantages of using prior knowledge available from the wells (permeability and porosity measurements) to generate initial realizations. One of the main practical advantages of history matching using the EnKF over traditional optimization based approaches is its low computational effort. The computational cost is dominated by Monte Carlo simulation of the ensemble of models only. Thus, significant computational time saving is possible by running each of the ensemble simulations on independent processors in a parallel mode. Moreover, the method can be easily integrated with any commercial reservoir simulation software.

Saturation-based localization within EnKF approach for history matching in SAGD processes

History matching geological facies models based on ensemble smoother and deep generative models