Deep Learning for Latent Space Data Assimilation in Subsurface Flow Systems

We present a new deep learning architecture for efficient reduced -order implementation of ensemble data assimilation in learned low -dimensional latent spaces. Specifically, deep learning is used to improve two important aspects of data assimilation workflows: (i) low -rank representation of complex reservoir property distributions for geologically consistent feature -based model updating, and (ii) efficient prediction of the statistical information that are required for model updating. The proposed method uses deep convolutional autoencoders (AEs) to nonlinearly map the original complex and high-dimensional parameters onto a low-dimensional parameter latent space that compactly represents the original parameters. In addition, a low-dimensional data latent space is constructed to predict the observable response of each model parameter realization, which can serve as a proxy model in the latent space to compute the statistical information needed for data assimilation. The two mappings are developed as a joint deep learning architecture with two variational AEs (VAEs) that are connected and trained together. The training procedure uses an ensemble of model parameters and their corresponding production response predictions. Simultaneous training of the two mappings leads to a joint data-parameter manifold that captures the most salient information in the two spaces for effective data assimilation, where only relevant data and parameter features are included. Moreover, the parameter-to -data mapping provides a fast forecast model that can be used to significantly increase the ensemble size in data assimilation, without the corresponding computational overhead. We apply the developed approach to a series of numerical experiments, including a 3D example based on the Volve field in the North Sea. For data assimilation methods that involve iterative schemes, such as the ensemble smoother with multiple data assimilation (ESMDA) or iterative forms of the ensemble Kalman filter (EnKF), the proposed approach offers a computationally competitive alternative. Our results suggest that a fully low-dimensional implementation of ensemble data assimilation in effectively constructed latent spaces using deep learning architectures could offer several advantages over the standard algorithms, including joint data-parameter reduction that respects the salient features in each space, geologically consistent feature -based updates, as well as increased ensemble size to improve the accuracy and computational efficiency of calculating the required statistics for the update step.