Data assimilation and uncertainty assessment for complex geological models using a new PCA-based parameterization

In this paper, a recently developed parameterization procedure based on principal component analysis (PCA), which is referred to as optimization-based PCA (O-PCA), is generalized for use with a wide range of geological systems. In O-PCA, the mapping between the geological model in the full-order space and the low-dimensional subspace is framed as an optimization problem. The O-PCA optimization involves the use of regularization and bound constraints, which act to extend substantially the ability of PCA to model complex (non-Gaussian) systems. The basis matrix required by O-PCA is formed using a set of prior realizations generated by a geostatistical modeling package. We show that, by varying the form of the O-PCA regularization terms, different types of geological scenarios can be represented. Specific systems considered include binary-facies, three-facies and bimodal channelized models, and bimodal deltaic fan models. The O-PCA parameterization can be applied to generate random realizations, though our focus here is on its use for data assimilation. For this application, O-PCA is combined with the randomized maximum likelihood (RML) method to provide a subspace RML procedure that can be applied to non-Gaussian models. This approach provides multiple history-matched models, which enables an estimate of prediction uncertainty. A gradient procedure based on adjoints is used for the minimization required by the subspace RML method. The gradient of the O-PCA mapping is determined analytically or semi-analytically, depending on the form of the regularization terms. Results for two-dimensional oil-water systems, for several different geological scenarios, demonstrate that the use of O-PCA and RML enables the generation of posterior reservoir models that honor hard data, retain the large-scale connectivity features of the geological system, match historical production data, and provide an estimate of prediction uncertainty. MATLAB code for the O-PCA procedure, along with examples for three-facies and bimodal models, is included as online Supplementary Material.