A Probabilistic Collocation-Based Kalman Filter for History Matching

SPE Journal ◽  
2011 ◽  
Vol 16 (02) ◽  
pp. 294-306 ◽  
Author(s):  
Lingzao Zeng ◽  
Haibin Chang ◽  
Dongxiao Zhang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Initial Conditions ◽  
Collocation Point ◽  
Computational Effort ◽  
Production Data ◽  
Sampling Errors ◽  
Probabilistic Collocation

Summary The ensemble Kalman filter (EnKF) has been used widely for data assimilation. Because the EnKF is a Monte Carlo-based method, a large ensemble size is required to reduce the sampling errors. In this study, a probabilistic collocation-based Kalman filter (PCKF) is developed to adjust the reservoir parameters to honor the production data. It combines the advantages of the EnKF for dynamic data assimilation and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, all the system parameters and states and the production data are approximated by the PCE. The PCE coefficients are solved with the probabilistic collocation method (PCM). Collocation realizations are constructed by choosing collocation point sets in the random space. The simulation for each collocation realization is solved forward in time independently by means of an existing deterministic solver, as in the EnKF method. In the analysis step, the needed covariance is approximated by the PCE coefficients. In this study, a square-root filter is employed to update the PCE coefficients. After the analysis, new collocation realizations are constructed. With the parameter collocation realizations as the inputs and the state collocation realizations as initial conditions, respectively, the simulations are forwarded to the next analysis step. Synthetic 2D water/oil examples are used to demonstrate the applicability of the PCKF in history matching. The results are compared with those from the EnKF on the basis of the same analysis. It is shown that the estimations provided by the PCKF are comparable to those obtained from the EnKF. The biggest improvement of the PCKF comes from the leading PCE approximation, with which the computational burden of the PCKF can be greatly reduced by means of a smaller number of simulation runs, and the PCKF outperforms the EnKF for a similar computational effort. When the correlation ratio is much smaller, the PCKF still provides estimations with a better accuracy for a small computational effort.

scholarly journals DATA ASSIMILATION BY ENSEMBLE KALMAN FILTER WITH THE LORENZ EQUATIONS

2016 ◽  
Vol 38 ◽  
pp. 190
Author(s):  
Regis Sperotto de Quadros ◽  
Fabrício Pereira Harter ◽  
Daniela Buske ◽  
Larri Silveira Pereira
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Initial Conditions ◽  
Background Field ◽  
Lorenz Equations ◽  
Integration Period ◽  
Numerical Weather Forecasting ◽  
Chaotic Nature

Data Assimilation is a procedure to get the initial condition as accurately as possible, through the statistical combination of collected observations and a background field, usually a short-range forecast. In this research a complete assimilation system for the Lorenz equations based on Ensemble Kalman Filter is presented and examined. The Lorenz model is chosen for its simplicity in structure and the dynamic similarities with primitive equations models, such as modern numerical weather forecasting. Based on results, was concluded that, in this implementation, 10 members is the best setting, because there is an overfitting for ensembles with 50 and 100 members. It was also examined if the EnKF is effective to track the control for 20% and 40% of error in the initial conditions. The results show a disagreement between the “truth” and the estimation, especially in the end of integration period, due the chaotic nature of the system.  It was also concluded that EnKF have to be performed sufficiently frequently in order to produce desirable results.

Data Assimilation for Strongly Nonlinear Problems by Transformed Ensemble Kalman Filter

SPE Journal ◽  
2014 ◽  
Vol 20 (01) ◽  
pp. 202-221 ◽  
Author(s):  
Qinzhuo Liao ◽  
Dongxiao Zhang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Homogeneous Medium ◽  
Forward Model ◽  
Dynamic State ◽  
Initial Time ◽  
Static State ◽  
Strongly Nonlinear

Summary The ensemble Kalman filter (EnKF) has been widely used for data assimilation. It is challenging, however, when the relation of state and observation is strongly nonlinear. For example, near the flooding front in an immiscible flow, directly updating the saturation by use of the EnKF may lead to nonphysical results. One possible solution, which may be referred to as the restarted EnKF (REnKF), is to update the static state (e.g., permeability and porosity) and rerun the forward model from the initial time to obtain the updated dynamic state (e.g., pressure and saturation). However, it may become time-consuming, especially when the number of assimilation steps is large. In this study, we develop a transformed EnKF (TEnKF), in which the state is represented by displacement as an alternative variable. The displacement is first transformed from the forecasted state, then updated, and finally transformed back to obtain the updated state. Because the relation between displacement and observation is relatively linear, this new method provides a physically meaningful updated state without resolving the forward model. The TEnKF is tested in the history matching of multiphase flow in a 1D homogeneous medium, a 2D heterogeneous reservoir, and a 3D PUNQ-S3 model. The case studies show that the TEnKF produces physical results without the oscillation problem that occurs in the traditional EnKF, whereas the computational effort is reduced compared with the REnKF.

scholarly journals Improvement of ozone forecast over Beijing based on ensemble Kalman filter with simultaneous adjustment of initial conditions and emissions

2011 ◽  
Vol 11 (24) ◽  
pp. 12901-12916 ◽  
Author(s):  
X. Tang ◽  
J. Zhu ◽  
Z. F. Wang ◽  
A. Gbaguidi
Keyword(s):  
Kalman Filter ◽  
Air Quality ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Urban Site ◽  
Ozone Forecast ◽  
Regional Air Quality ◽  
Simultaneous Adjustment ◽  
Mean Square Errors

Abstract. In order to improve the surface ozone forecast over Beijing and surrounding regions, data assimilation method integrated into a high-resolution regional air quality model and a regional air quality monitoring network are employed. Several advanced data assimilation strategies based on ensemble Kalman filter are designed to adjust O3 initial conditions, NOx initial conditions and emissions, VOCs initial conditions and emissions separately or jointly through assimilating ozone observations. As a result, adjusting precursor initial conditions demonstrates potential improvement of the 1-h ozone forecast almost as great as shown by adjusting precursor emissions. Nevertheless, either adjusting precursor initial conditions or emissions show deficiency in improving the short-term ozone forecast at suburban areas. Adjusting ozone initial values brings significant improvement to the 1-h ozone forecast, and its limitations lie in the difficulty in improving the 1-h forecast at some urban site. A simultaneous adjustment of the above five variables is found to be able to reduce these limitations and display an overall better performance in improving both the 1-h and 24-h ozone forecast over these areas. The root mean square errors of 1-h ozone forecast at urban sites and suburban sites decrease by 51% and 58% respectively compared with those in free run. Through these experiments, we found that assimilating local ozone observations is determinant for ozone forecast over the observational area, while assimilating remote ozone observations could reduce the uncertainty in regional transport ozone.

History Matching Using the Ensemble Kalman Filter With Multiscale Parameterization: A Field Case Study

SPE Journal ◽  
2010 ◽  
Vol 16 (02) ◽  
pp. 307-317 ◽  
Author(s):  
Yanfen Zhang ◽  
Dean S. Oliver
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Multiple Scales ◽  
Gaussian Random Fields ◽  
Rock Properties ◽  
Real Field ◽  
Production Data ◽  
Reservoir Properties ◽  
Water Cut

Summary The increased use of optimization in reservoir management has placed greater demands on the application of history matching to produce models that not only reproduce the historical production behavior but also preserve geological realism and quantify forecast uncertainty. Geological complexity and limited access to the subsurface typically result in a large uncertainty in reservoir properties and forecasts. However, there is a systematic tendency to underestimate such uncertainty, especially when rock properties are modeled using Gaussian random fields. In this paper, we address one important source of uncertainty: the uncertainty in regional trends by introducing stochastic trend coefficients. The multiscale parameters including trend coefficients and heterogeneities can be estimated using the ensemble Kalman filter (EnKF) for history matching. Multiscale heterogeneities are often important, especially in deepwater reservoirs, but are generally poorly represented in history matching. In this paper, we describe a method for representing and updating multiple scales of heterogeneity in the EnKF. We tested our method for updating these variables using production data from a deepwater field whose reservoir model has more than 200,000 unknown parameters. The match of reservoir simulator forecasts to real field data using a standard application of EnKF had not been entirely satisfactory because it was difficult to match the water cut of a main producer in the reservoir. None of the realizations of the reservoir exhibited water breakthrough using the standard parameterization method. By adding uncertainty in large-scale trends of reservoir properties, the ability to match the water cut and other production data was improved substantially. The results indicate that an improvement in the generation of the initial ensemble and in the variables describing the property fields gives an improved history match with plausible geology. The multiscale parameterization of property fields reduces the tendency to underestimate uncertainty while still providing reservoir models that match data.

scholarly journals Assessment of multilevel ensemble-based data assimilation for reservoir history matching

2019 ◽  
Vol 24 (1) ◽  
pp. 217-239
Author(s):  
Kristian Fossum ◽  
Trond Mannseth ◽  
Andreas S. Stordal
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Computational Cost ◽  
Ensemble Size ◽  
Matching Problems ◽  
Reservoir Models ◽  
Reservoir History Matching ◽  
Computational Resources

AbstractMultilevel ensemble-based data assimilation (DA) as an alternative to standard (single-level) ensemble-based DA for reservoir history matching problems is considered. Restricted computational resources currently limit the ensemble size to about 100 for field-scale cases, resulting in large sampling errors if no measures are taken to prevent it. With multilevel methods, the computational resources are spread over models with different accuracy and computational cost, enabling a substantially increased total ensemble size. Hence, reduced numerical accuracy is partially traded for increased statistical accuracy. A novel multilevel DA method, the multilevel hybrid ensemble Kalman filter (MLHEnKF) is proposed. Both the expected and the true efficiency of a previously published multilevel method, the multilevel ensemble Kalman filter (MLEnKF), and the MLHEnKF are assessed for a toy model and two reservoir models. A multilevel sequence of approximations is introduced for all models. This is achieved via spatial grid coarsening and simple upscaling for the reservoir models, and via a designed synthetic sequence for the toy model. For all models, the finest discretization level is assumed to correspond to the exact model. The results obtained show that, despite its good theoretical properties, MLEnKF does not perform well for the reservoir history matching problems considered. We also show that this is probably caused by the assumptions underlying its theoretical properties not being fulfilled for the multilevel reservoir models considered. The performance of MLHEnKF, which is designed to handle restricted computational resources well, is quite good. Furthermore, the toy model is utilized to set up a case where the assumptions underlying the theoretical properties of MLEnKF are fulfilled. On that case, MLEnKF performs very well and clearly better than MLHEnKF.

Structural Surface Uncertainty Modeling and Updating Using the Ensemble Kalman Filter

SPE Journal ◽  
2010 ◽  
Vol 15 (04) ◽  
pp. 1062-1076 ◽  
Author(s):  
A.. Seiler ◽  
S.I.. I. Aanonsen ◽  
G.. Evensen ◽  
J.C.. C. Rivenæs
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Reservoir Characterization ◽  
Production Data ◽  
Base Case ◽  
Future Production ◽  
Fixed Dimension ◽  
Structural Surface ◽  
Bottom Reservoir

Summary Although typically large uncertainties are associated with reservoir structure, the reservoir geometry is usually fixed to a single interpretation in history-matching workflows, and focus is on the estimation of geological properties such as facies location, porosity, and permeability fields. Structural uncertainties can have significant effects on the bulk reservoir volume, well planning, and predictions of future production. In this paper, we consider an integrated reservoir-characterization workflow for structural-uncertainty assessment and continuous updating of the structural reservoir model by assimilation of production data. We address some of the challenges linked to structural-surface updating with the ensemble Kalman filter (EnKF). An ensemble of reservoir models, expressing explicitly the uncertainty resulting from seismic interpretation and time-to-depth conversion, is created. The top and bottom reservoir-horizon uncertainties are considered as a parameter for assisted history matching and are updated by sequential assimilation of production data using the EnKF. To avoid modifications in the grid architecture and thus to ensure a fixed dimension of the state vector, an elastic-grid approach is proposed. The geometry of a base-case simulation grid is deformed to match the realizations of the top and bottom reservoir horizons. The method is applied to a synthetic example, and promising results are obtained. The result is an ensemble of history-matched structural models with reduced and quantified uncertainty. The updated ensemble of structures provides a more reliable characterization of the reservoir architecture and a better estimate of the field oil in place.

scholarly journals Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation

2015 ◽  
Vol 22 (6) ◽  
pp. 645-662 ◽  
Author(s):  
M. Bocquet ◽  
P. N. Raanes ◽  
A. Hannart
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Ad Hoc ◽  
Finite Size ◽  
Practical Interest ◽  
Sampling Errors ◽  
Jeffreys Prior ◽  
Error Statistics ◽  
Additional Information

Abstract. The ensemble Kalman filter (EnKF) is a powerful data assimilation method meant for high-dimensional nonlinear systems. But its implementation requires somewhat ad hoc procedures such as localization and inflation. The recently developed finite-size ensemble Kalman filter (EnKF-N) does not require multiplicative inflation meant to counteract sampling errors. Aside from the practical interest in avoiding the tuning of inflation in perfect model data assimilation experiments, it also offers theoretical insights and a unique perspective on the EnKF. Here, we revisit, clarify and correct several key points of the EnKF-N derivation. This simplifies the use of the method, and expands its validity. The EnKF is shown to not only rely on the observations and the forecast ensemble, but also on an implicit prior assumption, termed hyperprior, that fills in the gap of missing information. In the EnKF-N framework, this assumption is made explicit through a Bayesian hierarchy. This hyperprior has so far been chosen to be the uninformative Jeffreys prior. Here, this choice is revisited to improve the performance of the EnKF-N in the regime where the analysis is strongly dominated by the prior. Moreover, it is shown that the EnKF-N can be extended with a normal-inverse Wishart informative hyperprior that introduces additional information on error statistics. This can be identified as a hybrid EnKF–3D-Var counterpart to the EnKF-N.

Importance of Distributed Temperature Sensor Data for Steam Assisted Gravity Drainage Reservoir Characterization and History Matching Within Ensemble Kalman Filter Framework

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Amit Panwar ◽  
Japan J. Trivedi ◽  
Siavash Nejadi
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Reservoir Characterization ◽  
Temperature Data ◽  
Gravity Drainage ◽  
Production Data ◽  
Facies Distribution ◽  
Steam Assisted Gravity Drainage ◽  
Temperature Observations

Distributed temperature sensing (DTS), an optical fiber down-hole monitoring technique, provides a continuous and permanent well temperature profile. In steam assisted gravity drainage (SAGD) reservoirs, the DTS plays an important role to provide depth-and-time continuous temperature measurement for steam management and production optimization. These temperature observations provide useful information for reservoir characterization and shale detection in SAGD reservoirs. However, use of these massive data for automated SAGD reservoir characterization has not been investigated. The ensemble Kalman filter (EnKF), a parameter estimation approach using these real-time temperature observations, provides a highly attractive algorithm for automatic history matching and quantitative reservoir characterization. Due to its complex geological nature, the shale barrier exhibits as a different facies in sandstone reservoirs. In such reservoirs, due to non-Gaussian distributions, the traditional EnKF underestimates the uncertainty and fails to obtain a good production data match. We implemented discrete cosine transform (DCT) to parameterize the facies labels with EnKF. Furthermore, to capture geologically meaningful and realistic facies distribution in conjunction with matching observed data, we included fiber-optic sensor temperature data. Several case studies with different facies distribution and well configurations were conducted. In order to investigate the effect of temperature observations on SAGD reservoir characterization, the number of DTS observations and their locations were varied for each study. The qualities of the history-matched models were assessed by comparing the facies maps, facies distribution, and the root mean square error (RMSE) of the predicted data mismatch. Use of temperature data in conjunction with production data demonstrated significant improvement in facies detection and reduced uncertainty for SAGD reservoirs. The RMSE of the predicted data is also improved. The results indicate that the assimilation of DTS data from nearby steam chamber location has a significant potential in significant reduction of uncertainty in steam chamber propagation and production forecast.

scholarly journals Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation

2015 ◽  
Vol 2 (4) ◽  
pp. 1091-1136 ◽  
Author(s):  
M. Bocquet ◽  
P. N. Raanes ◽  
A. Hannart
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Finite Size ◽  
Practical Interest ◽  
Sampling Errors ◽  
Jeffreys Prior ◽  
Error Statistics ◽  
Perfect Model ◽  
Key Points

Abstract. The ensemble Kalman filter (EnKF) is a powerful data assimilation method meant for high-dimensional nonlinear systems. But its implementation requires fixes such as localization and inflation. The recently developed finite-size ensemble Kalman filter (EnKF-N) does not require multiplicative inflation meant to counteract sampling errors. Aside from the practical interest of avoiding the tuning of inflation in perfect model data assimilation experiments, it also offers theoretical insights and a unique perspective on the EnKF. Here, we revisit, clarify and correct several key points of the EnKF-N derivation. This simplifies the use of the method, and expands its validity. The EnKF is shown to not only rely on the observations and the forecast ensemble but also on an implicit prior assumption, termed hyperprior, that fills in the gap of missing information. In the EnKF-N framework, this assumption is made explicit through a Bayesian hierarchy. This hyperprior has been so far chosen to be the uninformative Jeffreys' prior. Here, this choice is revisited to improve the performance of the EnKF-N in the regime where the analysis strongly relaxes to the prior. Moreover, it is shown that the EnKF-N can be extended with a normal-inverse-Wishart informative hyperprior that additionally introduces climatological error statistics. This can be identified as a hybrid 3D-Var/EnKF counterpart to the EnKF-N.

Data Assimilation Using the Constrained Ensemble Kalman Filter

SPE Journal ◽  
2010 ◽  
Vol 16 (02) ◽  
pp. 331-342 ◽  
Author(s):  
Hemant A. Phale ◽  
Dean S. Oliver
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Model Parameters ◽  
State Variables ◽  
Reservoir Properties ◽  
Compositional Model ◽  
Highly Nonlinear ◽  
Nonnegativity Constraints

Summary When the ensemble Kalman filter (EnKF) is used for history matching, the resulting updates to reservoir properties sometimes exceed physical bounds, especially when the problem is highly nonlinear. Problems of this type are often encountered during history matching compositional models using the EnKF. In this paper, we illustrate the problem using an example in which the updated molar density of CO2 in some regions is observed to take negative values while molar densities of the remaining components are increased. Standard truncation schemes avoid negative values of molar densities but do not address the problem of increased molar densities of other components. The results can include a spurious increase in reservoir pressure with a subsequent inability to maintain injection. In this paper, we present a method for constrained EnKF (CEnKF), which takes into account the physical constraints on the plausible values of state variables during data assimilation. In the proposed method, inequality constraints are converted to a small number of equality constraints, which are used as virtual observations for calibrating the model parameters within plausible ranges. The CEnKF method is tested on a 2D compositional model and on a highly heterogeneous three-phase-flow reservoir model. The effect of the constraints on mass conservation is illustrated using a 1D Buckley-Leverett flow example. Results show that the CEnKF technique is able to enforce the nonnegativity constraints on molar densities and the bound constraints on saturations (all phase saturations must be between 0 and 1) and achieve a better estimation of reservoir properties than is obtained using only truncation with the EnKF.

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