Uncertainty quantification (UQ) of reservoir heterogeneity is essential to accurately infer fluid flow behavior in subsurface formations, and the task is often accomplished by integrating forward physics simulators with iterative data assimilation methods, and such workflows are usually computationally expensive due to the iterative nature and the prohibitive cost of physics simulations.

In this work, we develop a new Ensemble Multi-Fidelity Neural Network(EMF-Net) to mitigate the efficiency bottleneck of UQ. EMF-Net directly infers the uncertain variables (e.g., permeability) via the sparse observation of the state variables (e.g., pressure). By leveraging the regression capability of convolutional neural networks, EMF-Net directly learns the nonlinear mapping from the innovation vector to the inferred update vector without the hypothesis of a linear mapping. At the training phase, high-fidelity data computed by a physics simulator (fh) and low-fidelity data computed by a forward proxy model (fl) are used to train the EMF-Net at two different stages, respectively. At the inference phase, we first adopt the 1-stage EMF-Net to infer the update vector for the initial ensembles with a global tuning and then efficiently update the innovation vector by fl. As an option, we further refine the inference via 2-stage EMF-Net to capture the locality with enhanced alignment with the observation data. We demonstrate that EMF-Net can reach equivalent inference accuracy compared to classic data assimilation algorithms like ES-MDA, in addition to decreasing the CPU time. Therefore, the accuracy and efficiency makes it a competitive alternative for scalable real-time history-matching tasks in field-scale subsurface engineering application.