This paper presents convolutional neural network architectures for integration of dynamic flow response data to reduce the uncertainty in geologic scenarios and calibrate subsurface flow models. The workflow consists of two steps, where in the first step the solution search space is reduced by eliminating unlikely geologic scenarios using distinguishing salient flow data trends. The first step serves as a pre-screening to remove unsupported scenarios from the full model calibration process in the second step. For this purpose, a convolutional neural network (CNN) with a cross-entropy loss function is designed to act as a classifier in predicting the likelihood of each scenario based on the observed flow responses. In the second step, the selected geologic scenarios are used in another CNN with an 2 -loss function (as a regression model) to perform model calibration. The regression CNN model (step 2) learns the inverse mapping from the production data space to the lowrank representation of the model realizations within the feasible set. Once the model is trained off-line, a fast feed-forward operation on the observed historical production data (input) is used to reconstruct a calibrated model. The presented approach offers an opportunity to utilize flow data in identifying plausible geologic scenarios, results in an off-line implementation that is conveniently parallellizable, and can generate calibrated models in real time, i.e., upon availability of data and without in-depth technical expertise about model calibration. Several synthetic Gaussian and non-Gaussian examples are used to evaluate the performance of the method.

Summary 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.

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