Beyond the Probability Map - Representation of Posterior Facies Probability
Y. Zhang, D.S. Oliver and Y. Chen
Event name: ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery
Session: History Matching II
Publication date: 08 September 2014
Info: Extended abstract, PDF ( 905.72Kb )
Price: € 20
Geologic facies distributions are commonly represented in geomodels by categorical variables that are intrinsically non-Gaussian and thus difficult to calibrate in ensemble Kalman filter-like algorithms. For certain types of stochastic models such as the truncated plurigaussian, it is possible to directly update model variables in such a way that the resulting realizations appear to be samples from the posterior. For other types of models, this is not possible. One common approach has been to invert flow data using the ensemble Kalman filter (EnKF) to obtain “probability maps” which are then used to condition facies realizations. Data matches obtained in this method are generally poor, however, because the probability map neglects important joint probabilities of model parameters imposed by flow data. In this paper, we propose a data assimilation method with a post-processing step that resembles the post-smoothed maximum-likelihood (ML) reconstruction method described in Nuyts et al. (2005). Disregarding the categorical feature of the facies model, reservoir properties are first updated using an EnKF-like assimilation method to honor flow data. In the post-processing step a penalty term forcing model variables to take discrete values is jointly minimized with the distance to the posterior realizations to solve for facies models that match data. The distance to posterior realizations is quantified using the ensemble representation of the posterior covariance, which represents the joint probability of model parameters. The matrix inversion lemma is used in solving the minimization problem to avoid inversion of the covariance. The ability of the ensemble to accurately represent information in data is demonstrated on two linear examples and a nonlinear reservoir flow example. Comparison is made with approaches that use only the probability map to represent the assimilated data. The results show better data matches obtained with the proposed method and reflect the importance of the information captured by the updated ensemble from the data with respect to the joint probabilities of model variables.