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Results
Abstract History matching is commonly performed in reservoir simulations to calibrate model parameters and to improve prediction accuracy. History matching problems often have non-unique solutions, i.e., there exist different combinations of parameter values that all yield the simulation results matching the measurements. In such a situation, finding a single solution matching the observations does not guarantee a correct prediction for future production. Alternatively, a more reliable prediction should be made with an uncertainty quantification based on all different possible scenarios of the model parameters. Bayesian theorem provides a theoretical foundation to represent different solutions and to quantify the uncertainty with the posterior probability density function (PDF). Lacking an analytical expression, the posterior PDF are often shown with a sample of realizations, each representing a possible scenario. This paper presents a novel sampling algorithm aiming to deal with two commonly encountered difficulties in the sampling process. First, a typical sampling method requires intensive model evaluations and hence may cause unaffordable computational burden. To alleviate this burden, our algorithm uses a Gaussian process (GP)-based surrogate as an approximation of the computationally expensive reservoir model to speed up the sampling process. The GP surrogate is adaptively refined locally such that the necessary approximation accuracy is achieved with a minimum level of computational cost. Secondly, when the dependent relationship between observation variables and input parameters is nonlinear, the posterior PDF could be in a complex form, such as multimodal, which is difficult to sample from. To tackle this difficulty, a Gaussian mixture model (GMM) is used as the proposal PDF to explore the parameter space. The GMM is flexible to approximate different distributions and is particularly efficient when the posterior is multimodal. The developed approach is tested with an illustrative history matching problem and shows its capability in handling the above-mentioned issues. Multimodal posterior of the testing problem is captured and are used to give a reliable production prediction with uncertainty quantification. The new algorithm reveals a great improvement in terms of computational efficiency comparing previously studied approaches for the sample problem.
Probabilistic Collocation Based Kalman Filter for Assisted History Matching
Li, Weixuan (University of Southern California) | Oyerinde, Dayo (University of Southern California) | Stern, Dave (University of Southern California) | Wu, Xiao-Hui (University of Southern California) | Zhang, Dongxiao (* ExxonMobil Upstream Research Company)
Abstract In this paper, we study the application of the probabilistic collocation based Kalman filter (PCKF) approach to history matching. Kalman filter is a classical approach for data assimilation and model calibration that has been widely used with success. It updates state variables (i.e., model input and output parameters) based on their correlations, which is challenging to compute efficiently for non-linear models. A popular alternative to estimating the correlations is to use an ensemble of models, which is the basis of the Ensemble Kalman filter (EnKF). The PCKF is an alternative to the EnKF in the sense that it estimates the correlations between the state variables from the Polynomial Chaos Expansion (PCE). The coefficients in the PCE are determined by the Probabilistic Collocation Method (PCM). In this paper, the PCKF method is applied to a synthetic reservoir model that mimics a deep water deposit of amalgamated channel complexes. This model exhibits complex flow patterns under different production scenarios making it useful for testing the PCKF algorithm. Acceptable history matches were obtained with a relatively small number of simulations. Our study suggests that PCKF may be a useful way to reduce the size of the ensemble required in Kalman filter methods, thus improving the efficiency of the history matching process.
Abstract In this work, an approach for estimating non-Gaussian permeability field is developed, through a probabilistic collocation based Kalman filter (PCKF). In this approach, the polynomial chaos expansion is used to parameterize the non-Gaussian permeability field. The state variables are expressed by the polynomial chaos expansion whose coefficients are sequentially updated when observations are available. The probabilistic collocation method is employed to solve for the coefficients of the polynomial chaos expansion. The probabilistic collocation method is non-intrusive to the reservoir model, thus it allows the forward simulations to be performed independently with existing reservoir simulators, as in the Monte Carlo simulation. The applicability of the approach is demonstrated with black oil problems in spatially heterogeneous reservoirs. Continuous production data are used as observations for sequentially updating the permeability field. The PCKF approach is also compared with the ensemble Kalman filter (EnKF) for investigating the accuracy, efficiency and applicability. It reveals that the PCKF other than the EnKF can be used to efficiently estimate the non-Gaussian permeability field. While keeping the advantages of EnKF, such as the sequential updating and parallelism, the PCKF is more suitable for permeability fields characterized by non-Gaussian random fields.
Abstract Reservoir simulation is subject to uncertainties, which may stem from inaccurate and imprecise measurements or inadequate characterization of spatially or temporally varying medium properties. Such uncertainties render the model parameters random and the equations describing flow and transport in the media stochastic. There exist several stochastic approaches for quantifying uncertainties, which, however, are either incompatible with existing deterministic simulators or are too demanding computationally. In this study, an alternative approach is proposed that is both accurate and efficient. In this approach, the uncertainty quantities such as permeability and porosity fields are represented by the Karhunen-Loeve expansions while the fluid saturations and reservoir production quantities are expressed by the polynomial chaos or Lagrange polynomial expansions. Two collocation-based methods, i.e. probabilistic collocation method (PCM) and Smolyak method are used to determine the coefficients of the polynomials expansions by solving for the fluid saturations and pressures at different collocation points via the original partial differential equations. This approach is non-intrusive in that it results in independent deterministic differential equations, which similar to the Monte Carlo method, can be implemented with existing codes or simulators. However, the required number of simulations in PCM or Smolyak method is much smaller than that in the Monte Carlo method. The approach is demonstrated with three-phase flow problems in heterogeneous reservoirs using Eclipse. The accuracy, efficiency, and compatibility of this approach are compared against Monte Carlo simulations. This study reveals that while their computational efforts are greatly reduced compared to Monte Carlo method, the stochastic collocation methods are able to accurately estimate the statistical moments and probability density functions of the fluid saturations, pressures, and production quantities of interest. Introduction Management of petroleum exploration and production usually relies on reservoir simulations, which involve solving mathematical equations numerically to model the dynamic behavior of multiphase flow and transport in petroleum reservoirs. It is also recognized that the spatial heterogeneity and our limited information about the formation lead to uncertainties in the process of reservoir characterization. In turn, the uncertainties render the model parameters random and the equations governing flow and transport in the media stochastic. Therefore, it is necessary to describe the reservoir properties in a stochastic point of view and employ stochastic approaches for reservoir simulations.
- North America > United States > Texas (0.46)
- North America > United States > California (0.28)
Abstract In reservoir history matching or data assimilation, dynamic data such as production rates and pressures are used to constrain reservoir models and to update model parameters. As such, even if under certain conceptualization the model parameters do not vary with time, the estimate of such parameters may change with the available observations and thus with time. In reality, the production process may lead to changes in both the flow and geomechanics fields, which are dynamically coupled. For example, the variations in the stress/strain field lead to changes in porosity and permeability of the reservoir and hence in the flow field. In weak formations such as the Lost Hills oilfield, fluid extraction may cause a large compaction to the reservoir rock and a significant subsidence at the land surface, resulting in huge economic losses and detrimental environmental consequences. The strong nonlinear coupling between reservoir flow and geomechanics possesses a challenge to construct a reliable model for predicting oil recovery in such reservoirs. On the other hand, the subsidence and other geomechanics observations can provide additional insight into the nature of the reservoir rock and help constrain the reservoir model if used wisely. In this study, the Ensemble Kalman filter (EnKF) approach is used to estimate reservoir flow and material properties by jointly assimilating dynamic flow and geomechanics observations. The resulting model can be used for managing and optimizing production operations and for mitigating the land subsidence. The use of surface displacement observations improves the match to both production and displacement data. Localization is used to facilitate the assimilation of a large amount of data and to mitigate the effect of spurious correlations resulting from small ensembles. Since the stress, strain, and displacement fields are updated together with the material properties in The EnKF, the issue of consistency at the analysis step of the EnKF is investigated. A 3D problem with reservoir fluid-flow and mechanical parameters close to those of the Lost Hills oilfield is used to test the applicability. Introduction The geomechanical behavior of a reservoir is usually only considered through rock compressibility in reservoir simulators. Rock compressibility determines the change of reservoir pore volume with respect to the change of pressure. The stress fields change, however, dramatically during depletion, water injection or different applications of enhanced oil recovery techniques. The changes in the stress field induce various geomechanical phenomena, such as land subsidence, abrupt compaction of the reservoir, induced fracturing, enhancement of natural fractures and fault activation. Among these phenomena, reservoir compaction and surface subsidence are most commonly seen. Well know examples include the sea floor subsidence in the Ekofisk field and Valhall field in the North Sea (Pattillo et al. 1998), subsidence in the Lost Hill field, California (Wallace and Pugh 1993) and in the region of the Boliva Coast and Lagunillas in Venezuela (van der Knapp and van der Vlis 1967). These complicated geomechanical situations require more sophisticated methods to take them into account in order to better predict the production and avoid facility failures. In the past ten years, extensive efforts have been made to couple the fluid-flow and geomechanics simulations to model the complex process during hydrocarbon production (Chin et al. 2000; Dean et al. 2006; Samier et al. 2006). Geomechanics module is available in several commercial reservoir simulators to facilitate the coupled modeling of fluid-flow and geomechanical processes.
- North America > United States > Texas > Stephens County (0.74)
- North America > United States > California > Kern County (0.74)
- Europe > Norway > North Sea > Central North Sea (0.54)
- North America > United States > California > San Joaquin Basin > Lost Hills Field (0.99)
- Europe > Norway > North Sea > Central North Sea > Central Graben > PL 018 > Block 2/4 > Greater Ekofisk Field > Ekofisk Field > Tor Formation (0.89)
- Europe > Norway > North Sea > Central North Sea > Central Graben > PL 018 > Block 2/4 > Greater Ekofisk Field > Ekofisk Field > Ekofisk Formation (0.89)
- (4 more...)