Ensemble Smoother Research Articles

A scalable model-independent iterative data assimilation tool for sequential and batch estimation of high dimensional model parameters and states

Ensemble-based data assimilation (DA) methods have displayed strong potential to improve model state and parameter estimation across several disciplines due to their computational efficiency, scalability, and ability to estimate uncertainty in the dynamic states and the parameters. However, a barrier to adoption of ensemble DA methods remains. Namely, there is currently a lack of available tools that enable efficient and scalable DA in a non-intrusive fashion and that support implementation flexibility. This paper presents an open-source software tool (PESTPP-DA) that implements a range of data assimilation methods—Ensemble Kalman filter, Ensemble Kalman Smoother and Ensemble Smoother—using the widely known PEST model-interface protocols, to interact with any model. Two iterative solutions can be used for nonlinear and/or non-Gaussian assimilation problems. To demonstrate the broad range of PESTPP-DA applications, two synthetic case studies are presented: (1) the Lorenz model and (2) a groundwater pumping test in the presence of a non-Gaussian hydraulic conductivity field.

An Efficient Spatial-Temporal Convolution Recurrent Neural Network Surrogate Model for History Matching

Summary Surrogate modeling has shown to be effective in improving the solving efficiency for history matching in the development of oil and gas, but the traditional surrogate models are difficult to directly deal with the high-dimensional spatial uncertain parameters, such as the permeability field. In this paper, we introduce a new deep-learning-based surrogate modeling framework, image-to-sequence regression, which can directly predict the production data from the high-dimensional spatial parameters. Specifically, a spatial-temporal convolution recurrent neural network surrogate model is proposed based on a densely connected convolutional neural network (CNN) model and a stacked multilayer long short-term memory (LSTM) model. And a surrogate-based history-matching workflow is then developed by combining the proposed surrogate model with an improved ensemble smoother data assimilation algorithm. Two case studies on a 2D and a 3D reservoir model demonstrate that the proposed surrogate model can effectively predict production data and improve the efficiency of history matching.

Estimation of levee breach discharge hydrographs: comparison of inverse approaches

ABSTRACT In this paper, we present a methodology to estimate the flow hydrograph through a levee breach in an inverse framework. The procedure requires a forward model of the river reach, water levels recorded in a gauging station downstream or upstream of the failure site and an optimization method; there is no need to model either the flood-prone area outside the river or the breach evolution. In particular, we compare the results of a novel optimization method, the ensemble smoother with multiple data assimilation (ES-MDA), with those of the Bayesian geostatistical approach (BGA). The methods are applied to the real case of the breach that occurred on the Secchia River in Northern Italy in January 2014. The two optimization methods lead to similar results, but ES-MDA requires a number of runs of the forward model about 10 times lower than that needed by BGA, saving significant computational time.

Ensemble Smoother with Multiple Data Assimilation as a Tool for Curve Fitting and Parameter Uncertainty Characterization: Example Applications to Fit Nonlinear Sorption Isotherms

The ensemble smoother with multiple data assimilation (ES-MDA) coupled to a normal-score transformation is used to fit a Langmuir isotherm curve to estimate its parameters ( $$S_{m}$$ and b) and their uncertainty. The highlights of this work are threefold: (1) the ES-MDA can be used as a curve fitting procedure, (2) the ES-MDA provides also a full uncertainty quantification about the fitted parameters, and (3) for the specific case of the Langmuir isotherm, parameter $$S_m$$ is well identified with little uncertainty, while parameter b is well identified with a larger uncertainty, indicating that solute concentrations are more sensitive to $$S_m$$ than to b. As a by-product, the number of samples required to characterize the joint uncertainty of Langmuir isotherm parameters is also investigated; it can be concluded that the minimum number of samples to use is six, with best results obtained with eight samples, a value larger than the number recommended in the literature.

Predicting Thermal Performance of an Enhanced Geothermal System From Tracer Tests in a Data Assimilation Framework

AbstractPredicting the thermal performance of an enhanced geothermal system (EGS) requires a comprehensive characterization of the underlying fracture flow patterns from practically available data such as tracer data. However, due to the inherent complexities of subsurface fractures and the generally insufficient geological/geophysical data, interpreting tracer data for fracture flow characterization and thermal prediction remains a challenging task. The present study aims to tackle the challenge by leveraging a data assimilation method to maximize the utilization of information inherently contained in tracer data, and meanwhile maintain the flexibility to handle various uncertainties. A tracer data interpretation framework was proposed with the following three components integrated: (a) We use principal component analysis (PCA) to reduce the dimensionality of model parameter space. (b) We use ES‐MDA (ensemble smoother with multiple data assimilation) to invert for fracture aperture/flow fields and obtain posterior model ensembles for uncertainty quantification. Various data types are assimilated jointly to improve the predictive ability of the posterior ensemble. (c) The inverted fracture aperture fields are then incorporated into reservoir models to predict thermal performance. We developed a field‐scale EGS model to verify the ability of the framework to characterize highly heterogeneous fracture aperture/flow fields and predicting thermal performance. We also applied the framework to a mesoscale field experiment to demonstrate its potential application in real‐world geothermal reservoirs. The results indicate that the proposed framework can effectively retrieve fracture flow information from tracer data for thermal prediction and uncertainty quantification, and thus provide informative guidance for EGS optimization and risk management.

Assimilating time-lapse seismic data in the presence of significant spatially correlated model errors

Time-lapse seismic data is becoming a common information source in reservoir model calibration workflows to improve production forecasts. The standard process compares a forward model's results with the observed data to update the model's parameters from the existing deviations. Ensemble-based methods are popular choices for this process. However, the so-called forward model is always a simplification of the real phenomena. These simplifications may significatively influence the relation between simulated and observed data and possibly yield inconsistent updates of the parameters and uncertainty underestimation. In the conventional approach for this problem, the so-called strong-constraint formulation neglects the model's limitations, causing unphysical updates of the parameters to reduce the distance between simulated and observed data. In this work, we propose a methodology to apply a weak-constraint formulation to the time-lapse seismic data assimilation to mitigate the above problem. We consider the forward model error with an additive term and update it during the data assimilation workflow. By adopting this approach, we reduce the impact of the model errors in the calibrated parameters. Also, the proposed methodology handles model bias as a type of general model error. The inclusion of the additive term weakens the updates of the model due to the time-lapse seismic data. We show that this procedure significantly benefits the data assimilation results when there are substantial spatially correlated model errors, and it has a minor impact when applied to a low-error case. We apply the proposed method to assimilate time-lapse seismic data using the Ensemble Smoother with Multiple Data Assimilations in a simple 2D case and in a realistic benchmark synthetic case based on a real offshore reservoir. In the former case, we consider model error related to the pressure sensitivity in the petroelastic model. In the latter, we consider a realistic synthetic time-lapse seismic and the correlated errors result from seismic modeling and inversion. Moreover, in the latter, we also assimilate well data. The results indicate that our methodology improved the reservoir characterization and the production forecast using relatively low-resolution time-lapse seismic data.

Accelerating Groundwater Data Assimilation With a Gradient‐Free Active Subspace Method

AbstractGroundwater models always involve high‐dimensional parameters, which makes computationally tractable data assimilation using surrogate models very challenging. To address this issue, one common practice is to employ dimension reduction (DR) techniques. Nevertheless, traditional DR methods are usually implemented based on prior parameter statistics, that is, without considering the inherent system dynamics. Here, we show that when significant difference in parameter sensitivity exists, further efficiency can be achieved by adopting a supervised DR method, that is, the active subspace (AS) method. To avoid non‐trivial efforts in calculating the gradient information needed in the standard AS method, a cluster‐based gradient‐free AS (GFAS) method is developed in this study. By combining GFAS with Gaussian process regression, a surrogate model for the CPU‐demanding groundwater model can be adaptively constructed to accelerate data assimilation. Furthermore, a compensation scheme is proposed to cope with uncertainty underestimation caused by DR. The developed approach is tested with numerical experiments and field cases, which illustrated that the new approach is more efficient than the previously developed unsupervised ones by incorporating sensitivity information. Although an iterative ensemble smoother is employed in this study, the proposed method can also be used in other data assimilation approaches, such as Markov chain Monte Carlo and ensemble Kalman filter.

Treatment of model error in subsurface flow history matching using a data-space method

Subsurface flow models are inherently imperfect, and the associated model error can lead to unreliable flow predictions. In this work, we introduce treatments for model error in a data-space inversion (DSI) framework. DSI is a history matching procedure that constructs posterior flow predictions (for quantities of interest) directly, within a Bayesian setting, using a large set of prior-model flow simulation results and observed data. The model error considered in this work derives from the use of upscaled/coarsened surrogate models for the prior-model flow simulations required by DSI. Our error treatment entails the simulation of a set of corresponding pairs of fine and upscaled models. These results are used to construct a principal component analysis (PCA) representation of error. A linear regression approach is introduced to capture the coupled nature of the coarse-scale simulation output and model error. To construct posterior DSI predictions, a joint inversion on coarse-scale simulation data and model error is performed using an ensemble smoother with multiple data assimilation. Both the prior simulation data and error terms are parameterized using PCA-based procedures. Results are presented for two-phase flow in 3D channelized geomodels. The corrected prior and DSI posterior results are compared to reference results generated from fine-scale simulations. Comparisons are presented for flow statistics, Mahalanobis distance, and relative error for multiple (synthetic) true models. The coupled model error treatment is shown to provide highly accurate prior results and posterior predictions that agree closely with reference results. Significant improvement relative to uncorrected coarse models is observed. The treatments developed here can be used to represent error from many different sources in a variety of subsurface flow settings.

A deep learning-accelerated data assimilation and forecasting workflow for commercial-scale geologic carbon storage

Fast assimilation of monitoring data to update forecasts of pressure buildup and carbon dioxide (CO2) plume migration under geologic uncertainties is a challenging problem in geologic carbon storage. The high computational cost of data assimilation with a high-dimensional parameter space impedes fast decision-making for commercial-scale reservoir management. We propose to leverage physical understandings of porous medium flow behavior with deep learning techniques to develop a fast data assimilation-reservoir response forecasting workflow. Applying an Ensemble Smoother Multiple Data Assimilation (ES-MDA) framework, the workflow updates geologic properties and predicts reservoir performance with quantified uncertainty from pressure history and CO2 plumes interpreted through seismic inversion. As the most computationally expensive component in such a workflow is reservoir simulation, we developed surrogate models to predict dynamic pressure and CO2 plume extents under multi-well injection. The surrogate models employ deep convolutional neural networks, specifically, a wide residual network and a residual U-Net. The workflow is validated against a flat three-dimensional reservoir model representative of a clastic shelf depositional environment. Intelligent treatments are applied to bridge between quantities in a true-3D reservoir model and those in a single-layer reservoir model. The workflow can complete history matching and reservoir forecasting with uncertainty quantification in less than one hour on a mainstream personal workstation.

Theory-guided full convolutional neural network: An efficient surrogate model for inverse problems in subsurface contaminant transport

Identification of the location and strength of a contaminant source in an aquifer is a challenging but crucial task. Efficient surrogate models can be constructed to replace traditional time-consuming simulators while solving this inverse problem. In recent years, with the rapid development of machine learning (ML) algorithms, the artificial neural network (ANN) has been proven to be an efficient way for surrogate modeling. However, it may be difficult for ANN-based algorithms to learn the convection-dispersion equation and predict the contaminant concentration field due to their point-to-point learning scheme. Because of their strong localized features, the concentration fields can be seen as images. In contrast, the convolutional neural network (CNN) can extract spatial information better due to its convolutional structure. Herein, a theory-guided full convolutional neural network (TgFCNN) model is proposed to solve inverse problems in subsurface contaminant transport. TgFCNN can construct robust and reliable surrogate models with limited training realizations, and be further utilized for inverse modeling tasks. The loss function of TgFCNN comprises the residual of governing equations of contaminant transport, as well as data mismatch. Moreover, the iterative ensemble smoother (IES) method is employed to update the parameters while solving the inverse problems. The proposed TgFCNN model is evaluated in four scenarios. The results demonstrate that the TgFCNN model possesses strong generalization and extrapolation abilities, and satisfactory accuracy when estimating unknown contaminant source parameters, as well as the permeability field. The time consumption of the TgFCNN surrogate model for inverse tasks is also greatly reduced compared to using traditional simulators directly.

Evaluation of the benefits of improved permeability estimation on high-resolution characterization of DNAPL distribution in aquifers with low-permeability lenses

High-resolution characterization of dense non-aqueous phase liquid (DNAPL) source zone architecture (SZA) is crucial for developing remediation strategies. Inverse modeling approaches are widely used to characterize DNAPL SZA. However, unknown parameters (permeability and DNAPL saturation) are usually assumed as Gaussian distribution in traditional geostatistical inverse methods (e.g., ensemble smoother with multiple data assimilation, ESMDA). This assumption may not be true in channels of fluvial deposits. The presence of clay lenses will bring significant challenges in reconstructing the DNAPL SZA using geostatistical inverse methods based on the Gaussian assumption. Moreover, the DNAPL characterization also suffers from a limited number of borehole data in the field. In this study, we use ESMDA's variant, ensemble smoother with multiple data assimilation-direct sampling (ESMDA-DS), to improve the estimation of non-Gaussian permeability field and evaluate the benefits of improved permeability estimation on the reconstruction of DNAPL source zone. The inversion technique is based on the iterative inversion framework to incorporate the clay content information into the DNAPL characterization. The performance of our proposed framework is examined and discussed through a numerical synthetic case with challenging pool-dominated SZA. To overcome data scarcity, we integrated hydrogeological and geophysical (electrical resistance tomography) datasets. Results show that the permeability is characterized well using ESMDA-DS method compared with ESMDA method. With the improved permeability estimation, the DNAPL source zone can be characterized in higher resolution. Compared with ESMDA method, the root mean square error (RMSE) of permeability is reduced by 38.5% using the ESMDA-DS method. Then the estimation RMSE of DNAPL saturation is reduced by 24.9% by improved permeability estimation. We also demonstrated the significant benefits of improved permeability estimation on DNAPL depletion behavior and plume evolution. Numerical simulations demonstrate with the improved permeability estimation, the DNAPL depletion prediction is highly improved, the prediction RMSE of the DNAPL dissolved plume is reduced by 34.5% after 50 years.

A differential evolutionary Markov chain algorithm with ensemble smoother initial point selection for the identification of groundwater contaminant sources

Groundwater contaminant source identification (GCSI) can provide support for the confirmation of responsibility and the remediation of pollution. This study has developed an innovative combination algorithm to recognize source properties and model parameters in groundwater contamination simultaneously. The research idea is based on Bayes theory and combines an ensemble smoother (ES) algorithm, differential evolutionary Markov chain (DEMC) algorithm, and adaptive kriging surrogate model (AKSM). Poor selection of initial estimates for unknown variables will slow down the convergence rate. Therefore, the initial points are not generated completely randomly, but are partially obtained by the ES algorithm. After the initial points have been determined, the DEMC algorithm is used to recognize source properties and model parameters in groundwater contamination. To improve the efficiency of the DEMC algorithm, the updating formula was adjusted by introducing information about the optimal chain into the iteration. However, the inversion process is time-consuming because both the ES and DEMC algorithms need to run the original simulation model frequently. To solve this problem, an AKSM was established for the original simulation model, which greatly accelerated the inversion process. Different hypothetical cases with different complexities were used to illustrate the validity of the combination algorithm. The identification results implied that the combination algorithm had not only faster convergence, but also higher accuracy. These improvements were more evident in the second case and third case. This implies that the proposed method will play a greater role with increasing problem complexity.

Efficient Uncertainty Quantification and Data Assimilation via Theory-Guided Convolutional Neural Network

SummaryA deep learning framework, called the theory-guided convolutional neural network (TgCNN), is developed for efficient uncertainty quantification and data assimilation of reservoir flow with uncertain model parameters. The performance of the proposed framework in terms of accuracy and computational efficiency is assessed by comparing it to classical approaches in reservoir simulation. The essence of the TgCNN is to take into consideration both the available data and underlying physical/engineering principles. The stochastic parameter fields and time matrix comprise the input of the convolutional neural network (CNN), whereas the output is the quantity of interest (e.g., pressure, saturation, etc.). The TgCNN is trained with available data while being simultaneously guided by theory (e.g., governing equations, other physical constraints, and engineering controls) of the underlying problem. The trained TgCNN serves as a surrogate that can predict the solutions of the reservoir flow problem with new stochastic parameter fields. Such approaches, including the Monte Carlo (MC) method and the iterative ensemble smoother (IES) method, can then be used to perform uncertainty quantification and data assimilation efficiently based on the TgCNN surrogate, respectively. The proposed paradigm is evaluated with dynamic reservoir flow problems. The results demonstrate that the TgCNN surrogate can be built with a relatively small number of training data and even in a label-free manner, which can approximate the relationship between model inputs and outputs with high accuracy. The TgCNN surrogate is then used for uncertainty quantification and data assimilation of reservoir flow problems, which achieves satisfactory accuracy and higher efficiency compared with state-of-the-art approaches. The novelty of the work lies in the ability to incorporate physical laws and domain knowledge into the deep learning process and achieve high accuracy with limited training data. The trained surrogate can significantly improve the efficiency of uncertainty quantification and data assimilation processes.NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

A novel sampling-free algorithm for subsurface data assimilation using Gaussian process-derived sensitivities

Accurate characterization of hydraulic parameters is vital for modeling subsurface flow and transport. In the past decade, ensemble-based methods have been widely applied in estimating unknown parameters from state measurements. However, these methods require sufficiently large ensemble sizes to guarantee the accuracy of the ensemble averaged parameter sensitivities, leading to heavy computational burdens especially in large-scale problems. Although different surrogates have been introduced to alleviate the computational burden, the sensitivity information therein is still calculated by sampling the surrogate. Therefore, the sampling error is still inevitable. In this study, we propose an adaptive Gaussian process (GP) based iterative smoother (GPIS) algorithm in which the parameter sensitivity indices are analytically derived from the GP surrogate. During the iterations, the GP surrogate is adaptively refined by taking the updated parameter realizations as new base points. Both numerical and experimental cases are conducted to test the effectiveness of GPIS. We also compare its performance in estimating the heterogeneous hydraulic conductivity field with that of its prototype iterative ensemble smoother (IES) and our previously developed GP based iterative ensemble smoother (GPIES). Results show that, using the GP-derived sensitivity indices, GPIS shows advantages over GPIES in terms of both estimation accuracy and computational efficiency. Although subsurface flow and transport problems are considered in this work, the proposed method can be equally applied in other hydrological problems.

Data-Space Inversion With a Recurrent Autoencoder for Naturally Fractured Systems

Data-space inversion (DSI) is a data assimilation procedure that directly generates posterior flow predictions, for time series of interest, without calibrating model parameters. No forward flow simulation is performed in the data assimilation process. DSI instead uses the prior data generated by performingO(1000)simulations on prior geomodel realizations. Data parameterization is useful in the DSI framework as it enables representation of the correlated time-series data quantities in terms of low-dimensional latent-space variables. In this work, a recently developed parameterization based on a recurrent autoencoder (RAE) is applied with DSI for a real naturally fractured reservoir. The parameterization, involving the use of a recurrent neural network and an autoencoder, is able to capture important correlations in the time-series data. RAE training is accomplished using flow simulation results for 1,350 prior model realizations. An ensemble smoother with multiple data assimilation (ESMDA) is applied to provide posterior DSI data samples. The modeling in this work is much more complex than that considered in previous DSI studies as it includes multiple 3D discrete fracture realizations, three-phase flow, tracer injection and production, and complicated field-management logic leading to frequent well shut-in and reopening. Results for the reconstruction of new simulation data (not seen in training), using both the RAE-based parameterization and a simpler approach based on principal component analysis (PCA) with histogram transformation, are presented. The RAE-based procedure is shown to provide better accuracy for these data reconstructions. Detailed posterior DSI results are then presented for a particular “true” model (which is outside the prior ensemble), and summary results are provided for five additional “true” models that are consistent with the prior ensemble. These results again demonstrate the advantages of DSI with RAE-based parameterization for this challenging fractured reservoir case.

A residual-driven adaptive Gaussian mixture approximation for Bayesian inverse problems

In this article, we develop a residual-driven adaptive Gaussian mixture approximation (RD-AGMA) for Bayesian inverse problems. The posterior distribution is often non-Gaussian in practical Bayesian inference. To obtain a good approximation of the posterior, we provide the adaptive Gaussian mixture approximation (GMA) based on a residual. For GMA, the clustering of ensemble samples provides the predictor of means, covariances and weights by smoothed expectation–maximization (SmEM). SmEM can overcome the singularity of covariance matrix for small ensemble size. Then the parameters of GMA are updated by an iterative ensemble smoother (IES). To enhance clustering efficiency, the ensemble samples for the clustering are updated by IES as well. Since the goal of inverse problems is to minimize the residual between the observation data and the model response, the adaptive GMA of the posterior is constructed through a residual threshold. The mixture components with large residuals will be discarded in the adaptive procedure. When the prior is incorporated into the likelihood model, small residuals can drive the AGMA close to the true posterior. In the proposed method, a large number of samples can be efficiently drawn from the posterior distribution of GMA. A few numerical examples are presented to demonstrate the efficacy of RD-AGMA with applications in multimodal inversion and channel identification for subsurface flow problems in porous media.

Probabilistic forecasting for geosteering in fluvial successions using a generative adversarial network

Quantitative workflows utilizing real-time data to constrain ahead-of-bit uncertainty have the potential to improve geosteering significantly. Fast updates based on real-time data are essential when drilling in complex reservoirs with high uncertainties in pre-drill models. However, practical assimilation of real-time data requires effective geological modeling and mathematically robust parameterization. We propose a generative adversarial deep neural network (GAN), trained to reproduce geologically consistent 2D sections of fluvial successions. Offline training produces a fast GAN-based approximation of complex geology parameterized as a 60-dimensional model vector with standard Gaussian distribution of each component. Probabilistic forecasts are generated using an ensemble of equiprobable model vector realizations. A forward-modeling sequence, including a GAN, converts the initial (prior) ensemble of realizations into EM log predictions. An ensemble smoother minimizes statistical misfits between predictions and real-time data, yielding an update of model vectors and reduced uncertainty around the well. Updates can be then translated to probabilistic predictions of facies and resistivities. The present paper demonstrates a workflow for geosteering in an outcrop-based, synthetic fluvial succession. In our example, the method reduces uncertainty and correctly predicts most major geological features up to 500 meters ahead of drill-bit.

Fast History Matching and Optimization Using a Novel Physics-Based Data-Driven Model: An Application to a Diatomite Reservoir

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.

Evaluating Lower Computational Burden Approaches for Calibration of Large Environmental Models.

Realistic environmental models used for decision making typically require a highly parameterized approach. Calibration of such models is computationally intensive because widely used parameter estimation approaches require individual forward runs for each parameter adjusted. These runs construct a parameter‐to‐observation sensitivity, or Jacobian, matrix used to develop candidate parameter upgrades. Parameter estimation algorithms are also commonly adversely affected by numerical noise in the calculated sensitivities within the Jacobian matrix, which can result in unnecessary parameter estimation iterations and less model‐to‐measurement fit. Ideally, approaches to reduce the computational burden of parameter estimation will also increase the signal‐to‐noise ratio related to observations influential to the parameter estimation even as the number of forward runs decrease. In this work a simultaneous increments, an iterative ensemble smoother (IES), and a randomized Jacobian approach were compared to a traditional approach that uses a full Jacobian matrix. All approaches were applied to the same model developed for decision making in the Mississippi Alluvial Plain, USA. Both the IES and randomized Jacobian approach achieved a desirable fit and similar parameter fields in many fewer forward runs than the traditional approach; in both cases the fit was obtained in fewer runs than the number of adjustable parameters. The simultaneous increments approach did not perform as well as the other methods due to inability to overcome suboptimal dropping of parameter sensitivities. This work indicates that use of highly efficient algorithms can greatly speed parameter estimation, which in turn increases calibration vetting and utility of realistic models used for decision making.

Efficient Dimensionality Reduction Methods in Reservoir History Matching

Production forecasting is the basis for decision making in the oil and gas industry, and can be quite challenging, especially in terms of complex geological modeling of the subsurface. To help solve this problem, assisted history matching built on ensemble-based analysis such as the ensemble smoother and ensemble Kalman filter is useful in estimating models that preserve geological realism and have predictive capabilities. These methods tend, however, to be computationally demanding, as they require a large ensemble size for stable convergence. In this paper, we propose a novel method of uncertainty quantification and reservoir model calibration with much-reduced computation time. This approach is based on a sequential combination of nonlinear dimensionality reduction techniques: t-distributed stochastic neighbor embedding or the Gaussian process latent variable model and clustering K-means, along with the data assimilation method ensemble smoother with multiple data assimilation. The cluster analysis with t-distributed stochastic neighbor embedding and Gaussian process latent variable model is used to reduce the number of initial geostatistical realizations and select a set of optimal reservoir models that have similar production performance to the reference model. We then apply ensemble smoother with multiple data assimilation for providing reliable assimilation results. Experimental results based on the Brugge field case data verify the efficiency of the proposed approach.