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Pajonk O.,SPT Group GmbH | Rosic B.V.,TU Braunschweig | Matthies H.G.,TU Braunschweig
ECMOR 2012 - 13th European Conference on the Mathematics of Oil Recovery

Bayesian estimation has become an important topic for inverse problems in the context of hydrocarbon recovery. The conceptual and computational advantages due to direct integration with uncertainty quantification workflows are appealing. Especially, linear Bayesian techniques like the ensemble Kalman filter (EnKF) have been successfully used in numerous cases. However, such techniques have difficulties in some applications which are often caused by sampling errors, a limited ensemble size, or the sometimes large number of required samples. In this work we present and discuss a closely related linear Bayesian technique which is based on orthogonal expansions of the stochastic spectrum of the involved random variables and random fields. Basically being a family of fully deterministic implementations of the well-known projection theorem of Hilbert spaces, the technique is conceptually simple, yet powerful. Since they are fully deterministic, these methods avoid all sampling errors. First combined parameter and state estimation results with a low-dimensional chaotic model are presented, using a specific choice of orthogonal expansion. These are compared to results obtained with EnSRF, since it is a close relative to these spectral estimation methods. Challenges and opportunities for applications to the inverse problem of identification for hydrocarbon reservoirs are discussed. Source

Rosic B.V.,TU Braunschweig | Rosic B.V.,University of Kragujevac | Kucerova A.,Czech Technical University | Sykora J.,Czech Technical University | And 4 more authors.
Engineering Structures

The parameters to be identified are described as random variables, the randomness reflecting the uncertainty about the true values, allowing the incorporation of new information through Bayes's theorem. Such a description has two constituents, the measurable function or random variable, and the probability measure. One group of methods updates the measure, the other group changes the function. We connect both with methods of spectral representation of stochastic problems, and introduce a computational procedure without any sampling which works completely deterministically, and is fast and reliable. Some examples we show have highly nonlinear and non-smooth behaviour and use non-Gaussian measures. © 2013 Elsevier Ltd. Source

Pajonk O.,SPT Group GmbH | Pajonk O.,TU Braunschweig | Rosic B.V.,TU Braunschweig | Matthies H.G.,TU Braunschweig
Computers and Geosciences

We present a sampling-free implementation of a linear Bayesian filter based on a square root formulation. It employs spectral series expansions of the involved random variables, one such example being Wiener's polynomial chaos. The method is compared to several related methods, as well as a full Bayesian update, on a simple scalar example. Additionally it is applied to a combined state and parameter estimation problem for a chaotic system, the well-known Lorenz-63 model. There, we compare it to the ensemble square root filter (EnSRF), which is essentially a probabilistic implementation of the same underlying estimator. The spectral method is found to be more robust than the probabilistic one, especially for variance estimation. This is to be expected due to the sampling-free implementation. © 2012 Elsevier Ltd. Source

Pajonk O.,TU Braunschweig | Schulze-Riegert R.,SPT Group GmbH | Krosche M.,SPT Group GmbH | Matthies H.G.,TU Braunschweig
ECMOR 2010 - 12th European Conference on the Mathematics of Oil Recovery

The ensemble Kalman filter (EnKF) has become very popular in the field of assisted history matching for its appealing features. Nonetheless, problems can result from so-called spurious correlations due to the finite ensemble size [e.g. Evensen, 2009], which are considered as unphysical. The result of these correlations is a reduction of ensemble spread at model locations where no related data is available. This may cause an underestimation of the uncertainty and can result in a collapsed ensemble [Hamill, 2001]. Two methods are commonly used to address the unwanted reduction of variance: covariance infla-tion and localization. This contribution presents a new covariance localization approach based on multiscale (or multiresolution) wavelet analysis [Daubechiers, 1992]: the model state vector is transformed to a multiscale wavelet space. Correlations are computed in this space, not in the model space. This procedure allows the application of a new localization scheme, i.e., a different covariance localization function can be applied for each of the scale levels using a standard Schur product approach. Especially it allows us to filter unphysical long range correlations from fine scales while retaining longer correlations on coarser scales. Afterwards EnKF updates are computed and the transformation back to model space is applied. This contribution explains our wavelet-based localization approach and presents numerical results for the application of a synthetic model. The results are compared to standard localization approaches. The application to a real field simulation model is discussed. Source

Gubik A.,SPT Group GmbH | Baffoe J.,SPT Group GmbH | Schulze-Riegert R.,SPT Group GmbH
Oil Gas European Magazine

Gas storages define a key contribution for building a reliable gas supply chain from production to consumers. In a competitive gas market with short reaction times to seasonal and other gas input/output requirements, gas storages also receive a strong focus on availability and precise prediction estimates for future operation scenarios. Reservoir management workflows are built increasingly on reservoir simulation support for optimizing production schemes and estimating the impact of subsurface uncertainties on field development scenarios. Simulation models for gas storages are calibrated to geological data and accurate reproduction of historical production data is defined as a prerequisite for reliable production and performance forecasts. The underlying model validation process is called history matching, which potentially generates alternative simulation models due to prevailing geological uncertainties. In the past, a single base case reference model was used to predict production capacities of a gas storage. The working gas volume was precisely defined over a contracted plateau delivery and the required cushion gas volume maintains the reservoir pressure during the operation. Cushion and working gas volume are strongly dependent on reservoir parameters. In this article an existing depleted gas reservoir and the operation target as a gas storage are described. Key input data to the reservoir model description and simulation are reviewed, including production history and geological uncertainties based on large m-ell spacing, limited core and well data and a limited seismic resolution. Target delivery scenarios of the prospected gas storage are evaluated under uncertainty. As one key objective, optimal working gas and cushion gas volumes are described in a probabilistic context reflecting geological uncertainties. Several work steps are defined and included in an integrated workflow design. Equiprvbable geological models are generated and evaluated based on key performance indicators. Models are simulated and ranked according to their histoiy matching quality. One representative simulation model is used in a constraint optimization process predicting the optimal balance between working gas and cushion gas volume as well as target plateau rate duration. The robustness of the optimized operation scenario is verified based on alternative geological realizations. Results are presented and practical work steps are described in detail. The work has shown the importance of extending gas storage modeling and simulation workflows to a full uncertainly description. A traditional deterministic interpretation of alternative operation scenarios of a gas storage with variable working gas and cushion gas volumes is re-formulated as an optimization process under uncertainty. Results and conclusions for gas storage management processes under uncertainty are presented. © 2013 URBAN-CERLAG Hamburg/Wien GmbH. Source

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