Invariant extended Kalman filter

About: Invariant extended Kalman filter is a research topic. Over the lifetime, 7079 publications have been published within this topic receiving 187702 citations.

History Matching and Production Forecast Uncertainty by Means of the Ensemble Kalman Filter: A Real Field Application

Abstract: During history match reservoir models are calibrated against production data to improve forecasts reliability. Often, the calibration ends up with a handful of matched models, sometime achieved without preserving the prior geological interpretation. This makes the outcome of many history matching projects unsuitable for a probabilistic approach to production forecast, then motivating the quest of methodologies casting history match in a stochastic framework. The Ensemble Kalman Filter (EnKF) has gained popularity as Monte-Carlo based methodology for history matching and real time updates of reservoir models. With EnKF an ensemble of models is updated whenever production data are available. The initial ensemble is generated according to the prior model, while the sequential updates lead to a sampling of the posterior probability function. This work is one of the first to successfully use EnKF to history match a real field reservoir model. It is, to our knowledge, the first paper showing how the EnKF can be used to evaluate the uncertainty in the production forecast for a given development plan for a real field model. The field at hand was an on-shore saturated oil reservoir. Porosity distribution was one of the main uncertainties in the model, while permeability was considered a porosity function. According to the geological knowledge, the prior uncertainty was modeled using Sequential Gaussian Simulation and ensembles of porosity realizations were generated. Initial sensitivities indicated that conditioning porosity to available well data gives superior results in the history matching phase. Next, to achieve a compromise between accuracy and computational efficiency, the impact of the size of the ensemble on history matching, porosity distribution and uncertainty assessment was investigated. In the different ensembles the reduction of porosity uncertainty due to production data was noticed. Moreover, EnKF narrowed the production forecast confidence intervals with respect to estimate based on prior distribution. Introduction Reservoir management of modern oil and gas fields requires periodic updates of the simulation models to integrate in the geological parameterization production data collected over time. In these processes the challenges nowadays are many. First, a coherent view of the geomodel requires updating the simulation decks in ways consistent with geological assumptions. Second, the management is requiring more and more often a probabilistic assessment of the different development scenarios. This means that cumulative distribution functions, reflecting the underlying uncertainty in the knowledge of the reservoir, for key production indicators, e.g. cumulative oil production at Stock Tank condition (STC), along the entire time-life of the field, are expected outcomes of a reservoir modeling project. Moreover, production data are nowadays collected with increasing frequencies, especially for wells equipped with permanent down-hole sensors. Decision making, based on most current information, requires frequent and rapid updates of the reservoir models. The Ensemble Kalman Filter (EnKF) is a Monte-Carlo based method developed by Evensen to calibrate oceanographic models by sequential data assimilation. Since the pioneering application on near-well modeling problems by Naevdal et al., EnKF has become in the reservoir simulation community a popular approach for history matching and uncertainty assessment. This popularity is motivated by key inherent features of the method. EnKF is a sequential data assimilation methodology, and then production data can be integrated in the simulation model as they are available. This makes EnKF well suited for realtime application, where data continuously collected have to be used to improve the reliability of predictive models. EnKF maintains a Gaussian ensemble of models aligned with the most current production data by linear updates of the model parameters. In that way the statistical properties of the Gaussian ensemble, that is to say mean, variance and twopoint correlations are preserved. Because EnKF does not need either history matching gradients or sensitivity coefficients, any reservoir simulator with restarting capabilities can be used in an EnKF workflow, without modifying simulator source code. This represents an obvious advantage with respect to methods like the Randomized Maximum Likelihood (RML) method, which requires a simulator with adjoint gradient capabilities. These reasons motivate the interest on EnKF in the Upstream Industry. Nonetheless, only a few real applications were published before this work. Skjervheim et. al. compared results on using EnKF to assimilate 4D seismic data and production data, and obtained results that slightly improved the base case used for comparison. Haugen et al., see Ref. 13, report that the EnKF was used to successfully history match the simulation model of a Northern sea field, with substantial improvement compared to the reference case. In this paper we applied EnKF to history match the Zagor simulation model, quantifying also the reduction of uncertainty due to the assimilation of the production data. Different ensembles were used to investigate the connection between the effectiveness of EnKF and the size of the statistical samples. Next, we used one of the ensembles updated with EnKF to assess the uncertainty in the production forecasts. To our knowledge, this is the first paper where EnKF was used on a real reservoir from history match to uncertainty analysis of production forecasts. The paper proceeds as follows. The next section is dedicated to the discussion of the EnKF methodology, including its mathematical background and some remarks on the current limitations. Then the Zagor reservoir model is described. That includes the geological parameterization used in this work and the presentation of the different ensembles utilized in the application. The results of the application are presented in two subsequent sections. The first is dedicated to history matching and the second dedicated to the assessment of the uncertainty in the production forecasts. Finally, conclusions based on our results are drawn and some perspectives for future works are given. The Ensemble Kalman Filter The EnKF is a statistical methodology suitable to solve inverse problem, especially in cases where observed data are available sequentially in time. Assuming that the evolution of a physical system can be approximated by a numerical model, typically by the discretisation of a partial differential equation, a state vector can be used to represent the model parameters and observations. Using multiple realizations of the state vector one is able to explicitly express the model uncertainty. The EnKF can describe the evolution of the system by updating the ensemble of state vectors whenever an observation is available. In reservoir simulation, EnKF can be applied to integrate production data by updating sequentially an ensemble of reservoir models during the simulation. Each reservoir model in the ensemble is kept up-to-date as production data are assimilated sequentially. In this context every reservoir state vector comprises three types of parameters: static parameters, dynamic parameters and production data. The static parameters are the parameters that in traditional history matching do not vary with time during a simulation, such as permeability (K) and porosity (φ). The dynamic parameters include the fundamental variables of the flow simulation. These are, for black oil models, the cell pressure (p), water saturation (Sw), gas saturation (Sg) and solution gas-oil ratio (RS). In addition to the variables for each cells one add observations of the production data in each well. Production data usually include simulated data corresponding to observations such as well production rates, bottom-hole pressure values, water cut (WCT) and gas oil ratio (GOR) values. Thus, using the notation by X. H. Wen and W. H. Chen, the ensemble of state variables is modelled by multiple realizations:

Four-dimensional ensemble variational data assimilation and the unstable subspace

Abstract: The performance of (ensemble) Kalman filters used for data assimilation in the geosciences critically depends on the dynamical properties of the evolution model. A key aspect is that the error cova...

Sensor-Based Globally Asymptotically Stable Filters for Attitude Estimation: Analysis, Design, and Performance Evaluation

Abstract: This technical note presents the design, analysis, and performance evaluation of a novel globally asymptotically stable (GAS) filter for attitude estimation. The design is sensor-driven and departs from traditional solutions as no explicit representations of the attitude are considered. The proposed solution yields unique estimates and it does not suffer from drawbacks such as singularities, topological limitations for achieving global stabilization, or unwinding phenomena. The performance of the overall attitude estimation solution is evaluated with the design and implementation of an Attitude and Heading Reference System (AHRS) based on a single low-cost Inertial Measurement Unit. The performance of the proposed AHRS is assessed experimentally using a high precision motion rate table, which provides ground truth signals for comparison with the resulting estimates.

Distributed consensus extended Kalman filter: a variance-constrained approach

Abstract: This study is concerned with the distributed state estimation problem for non-linear systems over sensor networks. By using the strategy of consensus on prior estimates, a distributed extended Kalman filter (EKF) is developed for each node to guarantee an optimised upper bound on the state estimation error covariance despite consensus terms and linearisation errors. The Kalman gain matrix is derived for each node by solving two Riccati-like difference equations. It is shown that the estimation error is bounded in mean square under certain conditions. The effectiveness of the proposed filter is evaluated on an indoor localisation of a mobile robot with visual tracking systems.

Testing a decentralized filter for GPS/INS integration

Abstract: A decentralized Kalman filter strategy is presented and applied to GPS/INS (Global Positioning System/inertial navigation system) integration. Two Kalman filters are used. One is a local filter, processing GPS data and providing locally best estimates of position and velocity. The second is an INS filter which uses the results from the GPS filter as updates to the estimates obtained from the inertial data. Because of the high short-term accuracy of the inertial system, the position results from INS can be used for cycle slip detection and correction. The major advantages of this method are the flexible combination of GPS and INS and the simplicity of the implementation. Compared to centralized filtering, the decentralized filter gives globally the same optimal estimation accuracy as the centralized Kalman filter. The accuracy does not deteriorate when a suboptimal cascaded filter is used, which has some advantages in terms of computational efficiency. Extension of this method to more sensors is straightforward. Numerical results are used to illustrate the salient features of the method. >