Multi-Fidelity Bayesian Approach for History Matching in Reservoir Simulation

The scope of this paper is to apply a model-correction-based strategy for efficient reliability analysis of uncertain dynamical systems based on a low-fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of a high-fidelity (HF) model. In the model-correction approach utilized, the LF model is calibrated to the HF model close to the so-called most probable point in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single-span railway bridge modelled as simply supported Euler–Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge–train interaction by modelling the passing train as mass-spring-damper system, however increasing the computational effort of the limit state function evaluations. Failure probabilities evaluated with the model-correction approach are contrasted and discussed with failure probabilities of the sophisticated bridge–train interaction model evaluated with the first-order reliability method (FORM). It is demonstrated that the efficiency of the method depends on the correlation between the LF and the HF model. A comparison of the results of FORM and the model-correction-based approach shows that the latter provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

• Differentiable multi-fidelity inference: BMFIA enables gradient-based Bayesian inference for high-dimensional, non-differentiable multi-physics problems using low-fidelity model gradients. • Efficiency with minimal HF data: Accurate posterior estimates are achieved from only 100–300 HF–LF simulations with orders-of-magnitude speed-ups over HF-only inference. • Accuracy despite crude LF models: Robust posterior reconstruction is maintained even with weakly correlated or simplified low-fidelity models. High-dimensional Bayesian inverse analysis ( dim ≫ 100 ) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or nonexistent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w. r. t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA) , which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity (MF) conditional density (discriminative model). We demonstrate how this MF conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved and hence high-dimensional spatial reconstruction problems in two-dimensional Euclidean domains with static posteriors, given nonlinear and transient coupled poro-elastic media physics. We show that the resulting static MF posteriors are in excellent agreement with the (usually inaccessible) HF posteriors or ground-truth data and note that extending the framework to arbitrary three-dimensional domains is a natural and important direction for future work.

What is Transferred and How Much is Retained? A Simulation Study of Complex Surgical Skills

An increasing number of field development projects include rigorous uncertainty quantification workflows based on parameterized subsurface uncertainties. Reservoir model calibration workflows for reservoir simulation models including historical production data, also called history matching, deliver non-unique solutions and remain technically challenging. In addition, the validation process of the reservoir simulation model often introduces a break of the conceptual connection to the geological model. This raises questions on how to quantify the deviation between the calibrated simulation model and the original geological model. Workflow designs for history matching require scalable and efficient optimization techniques to address project needs. Derivative-free techniques like Markov Chain Monte Carlo (MCMC) are used for optimization and uncertainty quantification. Adjoint techniques derive analytical sensitivities directly from the flow equations. For history matching those sensitivities are efficiently used for property updates on grid block level. Both techniques have different characteristics and support alternative history matching strategies like global vs. local, stochastic vs. deterministic. In this work both techniques will be applied in an integrated workflow design to the Norne field. The Norne field is a North Sea oil-and-gas reservoir with approximately 30 wells, with one third being used for WAG injection for pressure support. Field data was previously released by Statoil and made available for a public benchmark study (NTNU Norway) testing history matching techniques including production and time-lapsed seismic data. We focus on well production data for history matching. MCMC is used for global parameter updates and uncertainty quantification in a Bayesian context. An implementation of an adjoint technique is applied for analytical sensitivity calculations and local parameter adjustments of rock properties. History matching results are presented for field wide and well-by-well production data. Consistency checks between updated and original geological model are presented for rock property distribution maps. Geostatistical measures including spatial correlations are used to quantify deviations between updated and original geological model. In conclusion scalability and performance efficiency of the practical workflow implementation is discussed with a perspective of a consistent feedback loop from history matching to geological modeling.

The purpose of history matching is to achieve geological realizations calibrated to the historical performance of the reservoir. For complex geological structures it is usually intractable to run tens of thousands of full reservoir simulation to trace the most probable geological model. Hence the inadequacy of the history-matching results frequently leads to poor estimation of the true model and high uncertainty in production forecasting. Reduced-order modeling procedures, which have been applied in many application areas including reservoir simulation, represent a promising means for constructing efficient surrogate models. Nonlinear dimensionality reduction techniques allow for encapsulating the high-resolution complex geological description of reservoir into a low-dimensional subspace, which significantly reduces number of unknowns and provides an efficient way to construct a proxy model based on the the reduced-dimension parameters. Polynomial Chaos Expansions (PCE) is a powerful tool to quantify uncertainty in dynamical system when there is probabilistic uncertainty in the system parameters. In reservoir simulation it has been shown to be more accurate and efficient compared to traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution is proved when the order of the PCE is increased. Accordingly PCE proxy can be used as the pseudo-simulator to represent the surface responses of the uncertain variables. When the objective and constraints of a reservoir model is described by multivariate polynomial functions, there are very efficient algorithms to compute the global solutions. We have developed a workflow at which incorporates PCE to find the global minimum of the misfit surface and assess the uncertainty associated with. The accuracy of the PCE proxy increases with the additional trial runs of the reservoir simulator. We conduct a two dimensional synthetic case study of a fluvial channel as well as a real field example to demonstrate the effectiveness of this approach. Kernel Principal Component Analysis (KPCA) is used to parameterize the complex geological structure. The study has revealed useful reservoir information and delivered more reliable production forecasts. PCE-based history match enhances the quality and efficiency of the estimation of the most probable geological model and improve the confidence interval of production forecasts.

High resolution sequence stratigraphical methods have been used in the detailed geological modeling of an element of the Ness formation of a North Sea field. This updated geological description has been used as input to the stochastic geological model MOHERES. 14 realizations of facies architecture and petrophysical properties have been generated. The stochastically generated realizations have been scaled up to a refined element of an existing deterministic, history matched reservoir simulation model for the full Upper Brent reservoir. The upscaled versions of the realizations have been connected to the full field model, and reservoir simulations have been performed to compare twelve years of production history with simulated results for well production, RFT- and PLT-data. Six of the realizations gave good to very good results when compared to measured production data. These realizations were used for simulation of the future production performance to the year 2010, making it possible to estimate its uncertainty.

Using commercial numerical reservoir simulators to build a full field reservoir model and simultaneously history match multiple dynamic variables for a highly complex, offshore mature field in Malaysia, had proven to be challenging, manpower intensive, highly expensive, and not very successful. This field includes almost two hundred wells that have been completed in more than 60 different, non-continuous reservoir layers. The field has been producing oil, gas and water for decades. The objective of this article is to demonstrate how Artificial Intelligence (AI) and Machine Learning is used to build a purely data-driven reservoir simulation model that successfully history match all the dynamic variables for all the wells in this field and subsequently used for production forecast. The model has been validated in space and time. The AI and Machine Learning technology that was used to build the dynamic reservoir simulation and modeling is called spatio-temporal learning. Spatio-temporal learning is a machine-learning algorithm specifically developed for data-driven modeling of the physics of fluid flow through porous media. Spatio-temporal learning is used in the context of Deconvolutional Neural Networks. In this article Spatio-temporal Learning and Deconvolutional Neural Networks will be explained. This new technology is the result of more than 20 years of research and development in the application of AI and Machine Learning in reservoir modeling. This technology develops a coupled reservoir and wellbore model that for this particular oil & gas field in Malaysia uses choke setting, well-head pressure and well-head temperature as input and simultaneously history matches Oil production, GOR, WC, reservoir pressure, and water saturation for more than a hundred wells through a completely automated process. Once the data-driven reservoir model is developed and history matched, it is blind validated in space and time in order to establish a reliable and robust reservoir model to be used for decision making purposes and opportunity generation to maximise the field value. The concepts and the methodology of history match of multiple wells, individual offshore platforms, and the entire field will be presented in this article along with the results of blind validation and production forecasting. Results of using this model to perform uncertainty quantification will also be presented. A case study of a highly complex mature field with large number of wells and years of production has been used to be studied and simulated by this data-driven approach. Time, efforts, and resources required for the development of the dynamic reservoir simulation models using AI and Machine Learning is considerably less than time and resources required using the commercial numerical simulators. It is validated that the TDM developed model can make very reasonable prediction of field behavior and production from the existing wells based on modification of operational constraints and can be a prudent complementary tool to conventional numerical simulators for such complex assets.

The interaction of trailing vortices with lifting surfaces is investigated using two levels of modeling fidelity. An overset mesh-based Reynolds-averaged Navier–Stokes solver is considered as the high-fidelity computational model. A lower-fidelity model is developed by combining a vortex panel method with a propeller aerodynamic model and slipstream theory. The high-fidelity model is first validated against available experimental data obtained from the interaction of a trailing vortex generated by an upstream wing with a downstream wing. The ability of both models to represent the development of the vortex wake and integrated loads is assessed for a number of parametric configurations, including a case in which the vortex core directly impinges on the wing surface. Following this, isolated propeller and wing-mounted propeller configurations are studied. In all of these cases, the high-fidelity model is effective in predicting the details of the flow and integrated airloads. The low-fidelity model, although less accurate, is shown to accurately predict interactional air loads and performance at orders of magnitude less cost than the high-fidelity model, justifying its role as a viable tool in design and trajectory-planning applications.

The project objective was to detail better ways to assess and exploit intelligent oil and gas field information through improved modeling, sensor technology, and process control to increase ultimate recovery of domestic hydrocarbons. To meet this objective we investigated the use of permanent downhole sensors systems (Smart Wells) whose data is fed real-time into computational reservoir models that are integrated with optimized production control systems. The project utilized a three-pronged approach (1) a value of information analysis to address the economic advantages, (2) reservoir simulation modeling and control optimization to prove the capability, and (3) evaluation of new generation sensor packaging to survive the borehole environment for long periods of time. The Value of Information (VOI) decision tree method was developed and used to assess the economic advantage of using the proposed technology; the VOI demonstrated the increased subsurface resolution through additional sensor data. Our findings show that the VOI studies are a practical means of ascertaining the value associated with a technology, in this case application of sensors to production. The procedure acknowledges the uncertainty in predictions but nevertheless assigns monetary value to the predictions. The best aspect of the procedure is that it builds consensus within interdisciplinary teams The reservoir simulation and modeling aspect of the project was developed to show the capability of exploiting sensor information both for reservoir characterization and to optimize control of the production system. Our findings indicate history matching is improved as more information is added to the objective function, clearly indicating that sensor information can help in reducing the uncertainty associated with reservoir characterization. Additional findings and approaches used are described in detail within the report. The next generation sensors aspect of the project evaluated sensors and packaging survivability issues. Our findings indicate that packaging represents the most significant technical challenge associated with application of sensors in the downhole environment for long periods (5+ years) of time. These issues are described in detail within the report. The impact of successful reservoir monitoring programs and coincident improved reservoir management is measured by the production of additional oil and gas volumes from existing reservoirs, revitalization of nearly depleted reservoirs, possible re-establishment of already abandoned reservoirs, and improved economics for all cases. Smart Well monitoring provides the means to understand how a reservoir process is developing and to provide active reservoir management. At the same time it also provides data for developing high-fidelity simulation models. This work has been a joint effort with Sandia National Laboratories and UT-Austin's Bureau of Economic Geology, Department of Petroleum and Geosystems Engineering, and the Institute of Computational and Engineering Mathematics.

The map of permeability distribution in the reservoirs is considered one of the most essential steps of the geologic model building due to its governing the fluid flow through the reservoir which makes it the most influential parameter on the history matching than other parameters. For that, it is the most petrophysical properties that are tuned during the history matching. Unfortunately, the prediction of the relationship between static petrophysics (porosity) and dynamic petrophysics (permeability) from conventional wells logs has a sophisticated problem to solve by conventional statistical methods for heterogeneous formations. For that, this paper examines the ability and performance of the artificial intelligence method in permeability prediction and compared its results with the flow zone indicator methods for a carbonate heterogeneous Iraqi formation. The methodology of the research can be Summarized by permeability was estimated by using two methods: Flow zone indicator and Artificial intelligence, two reservoir models are built, where the difference between them is in permeability method estimation, and the simulation run will be conducted on both of the models, and the permeability estimation methods will be examined by comparing their effect on the model history matching. The results showed that the model with permeability predicted by using artificial intelligence matched the observed data for different reservoir responses more accurately than the model with permeability predicted by the flow zone indicator method. That conclusion is represented by good matching between observed data and simulated results for all reservoir responses such for the artificial intelligence model than the flow zone indicator model.

The essence of reservoir simulation is to forecast field performance, ultimate recovery and to evaluate the effects of different operational conditions on recovery while keeping the economics of each scenario in view. It is important to note that in simulation one of the characteristics of a history match is that the outcome is a non-unique solution and it is influenced by many factors including porosity, permeability, thickness, saturation, PVT, relative permeability etc. A combination of these parameters result in a match and is not unique to that reservoir, thus this may not perfectly represent the condition of that reservoir. A common area of struggle is the transition of a model from pre-production vintage to a history matched solution where the only additional data available is production without having to build the model from scratch. It is therefore important to have a rule of thumb to validate the robustness of a simulation pre-production forecast prior to the onset of production. The effect of this may result in a pessimistic or overly optimistic model which translates to a non-representative reserves addition. The Fotis reservoir has been used as a case study to highlight the importance of a robust pre-production forecast prior to the onset of production and history match. Fotis is a NAG reservoir with a fault assisted dip closure trap. The reservoir sand is interpreted to be deposited in a shallow marine setting (consisting of shoreface and channels deposits). Reservoir properties are good (Avg Por. 17%; perm ~700 mD). Base case In-Place volume is about half a Tscf. Some 27% of the inplace volumes have so far been produced. The reservoir simulation model went live in 2010 without a history match because there was no production and pressure data. Production commenced in 2011. Well and Reservoir Management Philosophy for the reservoir was to produce the well at 100MMscf/d thus this was captured in the model though model forecast plateau seemed conservative for the low case model. The model was history matched in 2015 post the acquisition of pressure data. Inability to achieve history match within acceptable limits on the low case, even as the current well production rate of 100MMscf/d was sustained confirm the conservative nature of the low case model. This paper analyzes and highlights pitfalls to be avoided in the pre-production models and how multi-disciplinary solutions could drive a more representative output that supports robust life cycle predictions from the model.

In this work, a method for the application of multi-fidelity modeling to the reliability analysis of 2D elastostatic structures using the boundary element method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a center circular hole subjected to uniaxial tension using Monte Carlo simulations (MCS), the first-order reliability method (FORM), and the second-order reliability method (SORM). Two BEM models were investigated, a low-fidelity model (LFM) of 20 elements and a high-fidelity model (HFM) of 100 elements. The response of these models at several design points was used to create multi-fidelity models (MFMs) utilizing second-order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were significantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM.

The objective of this paper is to efficiently perform the reliability assessment of shell and tube heat exchangers. Although inexpensive empirical/analytical heat exchanger models exist and could be used for brute force Monte-Carlo simulations (MCS) based reliability analysis, they typically do not characterize the shell side flow accurately. Hence, higher fidelity models are often needed for predicting the shell and tube heat exchanger performance with a desired level of accuracy. These higher fidelity models are generally associated with higher computational costs, making them impractical for MCS-based reliability analysis. To circumvent this problem, a multi-fidelity technique leveraging the lower and higher fidelity models is proposed to locally construct the failure boundaries using support vector machines (SVM). For this purpose, an adaptive sampling scheme, which explores the regions of inconsistency between failure boundaries from lower and higher fidelity models, is developed.

In the community of petroleum engineering, the use of surrogate modelling techniques have recently gained more and more popularity to improve the efficiency of history matching. However, it is still not possible to fully utilize their potential in realistic applications. One of the challenges is to retain high accuracy while increasing the computational efficiency using a surrogate model. This dissertation proposed a projection-based reduced-order model and a data-driven deep convolutional neural network. In the first part of the thesis, a non-intrusive subdomain POD-TPWL method for solving gradient-based reservoir history matching problems is presented. It is a projection-based reduced-order modelling approach wherein the adjoint model of the original high-dimensional non-linear model is approximated by a subdomain reduced-order linear model. Furthermore, by introducing domain decomposition for the reduced-order model and by restricting the number of uncertain parameter patterns to the subdomains, the number of full order simulations required for the derivation of this surrogate model is reduced drastically. In the second part of the thesis, we propose two kinds of deep-learning inversion frameworks for efficiently solving large-scale history matching problems. The first deep-learning deterministic inversion framework primarily explores the possibility of applying a DNN surrogate to approximate the gradient of the objective function by making use of auto-differentiation (AD). In combination with the DNN surrogate, the AD enables us to evaluate the gradients efficiently in a parallel manner and without the need of explicitly coding of the adjoint model. The second framework is the deep-learning stochastic inversion which constructs a deep-learning surrogate based on an image-oriented distance parameterization for ensemble-based seismic history matching. Instead of directly assimilating spatially dense seismic data, image-oriented distance parameterization is employed to extract valuable information from the water fronts. Inspired by the methodologies developed for image segmentation in the field of computer vision and image processing, we propose an advanced image segmentation network for accurately predicting water fronts with highly-complex spatial discontinuities. In comparison with the conventional workflows entirely based on high-fidelity simulation models, experimental results show that the proposed surrogate-supported workflow achieves an accuracy equal to or better than the conventional workflow at significantly lower cost.

The significant increase in using reservoir simulation models poses significant challenges in the design and calibration of models. Moreover, conventional model calibration, history matching, is usually performed using a trial and error process of adjusting model parameters until a satisfactory match is obtained. In addition, history matching is an inverse problem, and hence it may have non-unique solutions. In typical reservoir simulation problems with a large number of unknown parameters, the problem could be ill-posed. In recent years, assisted history matching approach has been offered to solve this problem hopefully. In this paper, proposes an efficient assisted history matching workflow is recommended, and its application in a full field reservoir simulation model of a mature gas-cap reservoir is presented. In addition, Sobol sequence experimental design technique, Kriging proxy modeling, and three novel metaheuristic optimization algorithms are used to assist the history match of 28 years of historical production and pressure data. Finally, the results are compared with those obtained using a manual history matching procedure and with those obtained using the most widely used assisted history matching workflow. Also, in the proposed workflow, a significant improvement in terms of match quality and solution time is shown.