Enhancing model parameterization with linearly constrained deep generative network for ensemble-based history matching

Multi-source information fused generative adversarial network model and data assimilation based history matching for reservoir with complex geologies

Ensemble-based data assimilation methods often suffer sampling errors due to limited ensemble sizes and model errors, which can result in filter divergence. One method to avoid filter divergence is inflation, which inflates ensemble perturbations to increase ensemble spread and account for model errors. The commonly applied inflation methods, including the multiplicative inflation, relaxation to prior spread (RTPS), and additive inflation, often use a constant inflation parameter. To capture different error growths at different scales, a scale-dependent inflation is proposed here, which applies different inflation magnitudes for variables associated with different scales. Results from the two-scale Lorenz05 model III show that for the ensemble square root filter (EnSRF) and integrated hybrid ensemble Kalman filter with ensemble mean updated by hybrid background error covariances (IHEnKF-Mean), scale-dependent inflation is superior to constant inflation. Constant inflation overinflates small-scale variables and results in increased small-scale errors, which then propagate to large-scale variables through the coupling between large- and small-scale variables and lead to increased large-scale errors. Scale-dependent inflation applies larger inflation for large-scale variables and imposes no inflation for small-scale variables, since large-scale errors have larger magnitudes than small-scale ones and small-scale errors grow faster than large-scale ones. But IHEnKF-Ensemble that updates both the ensemble mean and perturbations with hybrid background error covariances is much less sensitive to scale-dependent inflation, compared to EnSRF and IHEnKF-Mean, because updating ensemble perturbations with hybrid background error covariances can play a role similar to the scale-dependent inflation. Significance Statement Ensemble-based data assimilation and ensemble forecasts often have smaller ensemble spread than errors. Strategies used by ensemble-based data assimilation to combat insufficient ensemble spread usually focus on the short-term ensemble forecasts, rather than considering the whole ensemble forecasts over different lead times. Thus, there are obvious gaps between the ensemble-based data assimilation and ensemble forecasts. A scale-dependent inflation that can capture different error growths at different scales is proposed, which obtains improved consistency between ensemble spread and errors at different lead times and effectively links the ensemble-based data assimilation and ensemble forecasts.

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.

In current practice history matching is often performed on upscaled reservoir models without updating the original geological model. Taking into account new production data in an existing model is often problematic because of the lack of appropriate techniques. This problem becomes crucial when using geostatistical models. In such a case, the integration of new static and dynamic constraints requires the modification of an initial realization without destroying the initial history match and the overall geostatistical properties. This paper presents a new methodology based on advanced history matching techniques for updating dynamically 3D stochastic reservoir models to account for new production data. The proposed approach is based on the gradual deformation method. It allows local or global smooth transformations of the model while conserving the overall statistical characteristics. History matching is performed by coupling this method with an optimization process which integrates geostatistical modeling, upscaling, and fluid flow simulation in the same loop. This approach provides great flexibility in history matching various types of production data. In addition, inversion parameters can be selected throughout the entire modeling loop, from the facies spatial distribution, to petrochemical parameters and fluid flow parameters. Finally, local transformations may be performed to account for new wells or to update the reservoir model with new dynamic information. A successful application to a 3D stochastic reservoir model inspired from a real field is presented. The first step was to perform a history match using the production history available from an initial group of production wells. Then, new dynamic information is acquired from infill wells and over one year of additional production data. The heterogeneity distribution is locally modified by gradual deformation to update the model with this new information. The benefit of the proposed approach for improving the model characterization and reducing uncertainty on production forecasts is demonstrated.

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

This paper presents a procedure to estimate relative permeability curves together with gridblock porosities and permeabilities by automatic history matching three-phase flow production data using the LBFGS algorithm for optimization and the adjoint method for generating sensitivity coefficients. Both power law and B-splines are considered for the representation of relative permeability curves. Power law is a simple representation; however, B-splines have the advantage of being able to accurately represent any set of relative permeability curves. In the B-spline representation, we provide a simple procedure based on a transformation of parameters (control points) defining the B-splines to ensure that monotonic curves are obtained. Without this transformation, the automatic history-matching process fails. If endpoint saturations are included as parameters, we show how to modify the adjoint procedure to account for the fact that initial conditions are sensitive to endpoint saturations. While the history-matching process is inherently non-unique, we show that reasonable estimates of relative permeability curves and porosity/ permeability fields can be obtained for a synthetic reservoir example. We also show the uncertainty quantification of the model parameters and reservoir performance prediction using the Randomized Maximum Likelihood (RML) method. Introduction Relative permeabilities are important parameters in reservoir performance calculations. Usually the relative permeability curves are obtained from labs through core flood tests. They can also be incorporated into the history matching procedure as model parameters. ARCHER and WONG (1973), and YANG and WATSON (1991) were among the early researchers to consider the estimation of relative permeability curves by history matching laboratory core flood data. A detailed literature review is given in REYNOLDS et al. (2004). REYNOLDS et al. (2004) also discussed the estimation of three-phase relative permeabilities by history matching roduction data. The current work is a natural extension of REYNOLDS et al. (2004). In this paper, we provide an alternative method for relative permeability estimation based on B-spline approximation. Compared to power law representation, Bsplines ave the advantage of being able to accurately represent any set of relative permeability curves. In the B-spline representation, we provide a simple procedure based on logtransformation of parameters (control points) defining the Bsplines to ensure the B-spline curves are monotonic or convex as used in CHEN et al. (2005). Since a reservoir simulator requires monotonic relative permeability curves, without this transformation, the automatic history-matching process may generate non-monotonic relative permeability curves and result in failed reservoir simulation run. A prior model for porosity and absolute permeability and relative permeability parameters is assumed to provide regularization, i.e., Bayesian estimation is applied to generate estimates. Model parameters, which are estimated by automatic history matching of production data, consist of gridblock porosties, gridblock absolute logpermeabilities and the parameters defining the relative permeability curves. We do not assume the endpoint saturations (i.e. irreducible water saturation, critical gas saturation and residual oil saturation) to be known and consider them as independent model parameters in history matching procedure.

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.

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.

Generative deep learning is becoming a widely used approach in geological modelling, especially in problems that involve optimization processes under uncertainty like history matching. The basic idea of reservoir modelling and history matching by generative deep learning is to train some kind of reservoir generator and provide optimization under different constraints. There are different approaches that have been developed to utilize various modifications of Generative Adversarial Networks or Variational Autoencoders to implement reservoir generators, but most of them have "conventional" convolution neural networks, hence all data have to be regular (rectangular) in structure. This should be recognized as a limitation. In this work, we introduce a novel approach of reservoir modelling with Variational Autoencoders based on graph convolutions as opposed to "conventional" convolutions. In our approach reservoir model is considered as a graph, i.e., not a structured data type. Graph convolutions can deal with these data types, and connection with Variational Autoencoders provides us with the capability to generate initially unstructured reservoirs. Variational Autoencoders (VAE) demonstrate the ability to implicitly parameterize geological representations into a latent space of reduced dimensionality and provide ways to uncertainty quantification and production profiling among various geological concepts. In the first part, we introduce the motivation to Graph Variational Autoencoders (GVAE) as opposed to conventional Generative Deep learning models. The rest of the paper is dedicated to experiments on a synthetic dataset with two different geological scenarios. We show that trained GVAE performs the generation of reservoir models with reliable geology. A latent space structure between Encoder and Decoder of GVAE is represented, and interpretation considering geology under investigation is performed. Finally, we show how latent space can help to estimate the uncertainty of production and an optimization workflow under well data constraints.

The application of elastic stress simulation for fracture modeling provides a more realistic description of fracture distribution than conventional statistical and geostatistical techniques, allowing the integration of geomechanical data and models into reservoir characterization. The geomechanical prediction of the fracture distribution accounts for the propagation of fracture caused by stress perturbation associated with faults. However, the challenge lies in estimating the past remote stress conditions which induced structural deformation and fracturing, the limited applicability of the elasticity assumption, and the latent uncertainty in the structural geometry of faults. The integration of historical production data and welltest permeability into geomechanical fracture modeling is a practical way to reduce such uncertainty. We propose to combine geostatistical algorithms for history matching with geomechanical elastic simulation model for developing an integrated yet efficient fracture modeling tool. This paper presents an integrated approach to history matching of naturally fractured reservoirs which includes 1) fracture trend prediction through elastic stress simulation, 2) geostatistical population of fracture density based on fracture trend map, 3) fracture permeability modeling integrating fracture density, matrix permeability and welltest permeability, and 4) numerical flow simulation and history matching. All of these implementations are incorporated into a single forward modeling process and iterated in the automatic history matching scheme. To obtain a history match on a reservoir model, we jointly perturb the large-scale fracture trend and local-scale geostatistical fluctuations of fracture densities rather than perturbing permeability calibrated from fractures. This strategy enables us to preserve the geological/geomechanical consistency throughout the history matching process. The geomechanically simulated fracture trend model is calibrated to both production data and geological structure map (faults and horizons) by finding the optimum remote stress condition for elastic stress-field simulation. The latter is achieved by matching the actually observed structural deformation trend with the simulated one. The smaller-scale fluctuation of fracture density is simultaneously history matched through the probability perturbation method of Caers (2003)1,2. The result of synthetic reservoir application is presented.

The aim of this work is to present the effectiveness of a fully integrated approach for ensemble-based history matching on a complex real field application. We show that the predictive ability of the ensemble of models is greatly enhanced through an integrated workflow promoting multidisciplinary collaboration between all subsurface disciplines. Consistent integration of geological and engineering understanding within the dynamic data conditioning phase of a reservoir study is a challenging task. The ensemble-based approach offers an efficient solution to this challenge, especially when it is tied in with an appropriate reservoir modeling approach, suitable parameterization and repeatable workflows. All these are key factors to ensure geological consistency while improving the match quality and reliability of the generated reservoir models. One key feature of the ensemble-based method is that it overcomes the typical limitation of the traditional approaches where the number of uncertainty parameters often has to be reduced resulting from practical or algorithm constraints. This is especially important on complex reservoirs, where the subsurface uncertainties cannot be represented in a handful of scalar multipliers while honoring the static and dynamic data measurements. We organize the proposed methodology in four main steps: firstly, an initial ensemble of models able to capture the model uncertainties in all parts of the modeling processes are generated. Next, the match level for different observed data specifying the likelihood error is assigned. Then the uncertainty parameters that should be modified in the history matching process are identified. Finally, we perform a computational intensive step on large-scale computing facilities, where a state-of-art iterative ensemble algorithm, tightly connected with standard geo-modeling tools, is used to consistently update the full set of models. The result is an ensemble of reservoir models where we can quantify both the uncertainty in the production forecast and in the reservoir model parameters. The methodology has been applied on a challenging History Matching problem for a turbiditic channel complex reservoir. We demonstrate the efficiency of the method, by generating an ensemble of reservoir models that offer geologically consistent explanations of the currently measured static and dynamic data. The model forecasts reliability is confirmed through a validation process using dynamic data that have been left out of the history matching process. More importantly, however, is that the proposed workflow, and resulting models provides a good starting point for a more efficient update of the reservoir models as new static and dynamic data become available. With the proposed approach is possible to move from sporadic updates of reservoir models towards frequent model updates when the new data arrive. Utilizing all available data in a consistent manner - when the data are collected - is the key to making more robust reservoir management decisions, especially for complex reservoirs.

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.

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.

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.

SummaryWe present the development and field application of a workflow for multiscale reservoir-model calibration that seamlessly integrates production data into the reservoir description from the facies to the grid-cell scale. To start with, the permeability field is parameterized using a novel grid-connectivity-based transformation basis that can be applied with any model geometry, including unstructured and corner-point grids. The parameterization basis functions emerge from spectral decomposition of the grid-connectivity Laplacian and are related to the structural harmonics of the grid. To reconcile data with model resolution during history matching, we first use the coarsest-scale basis functions to identify the large-scale variability. Additional smaller-scale basis elements are then adaptively incorporated to successively refine the model to a level supported by data resolution. During refinement, the inclusion of more detailed basis functions into the parameterization is determined by generic modal frequency when the prior model is unavailable or by using prior information when available. In the final step of the workflow, a streamline-based inversion is performed to locally adjust the reservoir model at grid-cell resolution along preferential-flow paths defined during the coarser-scale parameterization.We demonstrate the suitability and effectiveness of the developed workflow through application to an offshore turbidite reservoir with frequent well intervention, including shut-ins and recompletions. The static model has over 300,000 cells, a complex channelized interpretation with faults, four injector/producer pairs with deviated wells, and over eight years of production history, including water cut and pressure data. The grid-connectivity-based parameterization effectively updates the prior regional permeability at scales and in locations warranted by the data, while preserving the geologic continuity and avoiding ad hoc redefinition of regions given the sparse well pattern. The multiscale calibrated-permeability field indicates flow communication previously unrecognized in static geologic interpretation or manual history matching.