Efficient application of stochastic Discrete Well Affinity (DiWA) proxy model with adjoint gradients for production forecast

In this paper, we describe adjoint gradient formulation for the Operator-Based Linearization modeling approach. Adjoint gradients are implemented in Delft Advanced Research Terra Simulator (DARTS) framework and applied for history matching using a proxy methodology. Due to the application of adjoint gradients, the computational efficiency of the discrete well affinity (DiWA) proxy model for production forecast is significantly improved. That allows us to derive several important extensions. The proxy methodology is further extended and validated for 3D three-phase black-oil problems. The results show that the gradient-based regression can provide good history matching and reconstruct a true petrophysical characterization when the initial guess is generated based on highly reliable geological information. For cases with a limited or not sufficient geological characterization, an efficient stochastic application of DiWA proxy model is proposed. This approach consists of massive sampling procedures for collecting different realizations based on high-fidelity statistics with filtering. These realizations are generated stochastically because they are not conditioned to any production information but the basic geological statistics of the reservoir. The trained DiWA proxy model demonstrates a small deviation between the model response and the observation data. When applying the refined DiWA model for the training, the error between the model response and observation data can be further reduced. The forecast based on the trained model has slightly larger variability but the deviation is still reasonable. The enhanced DiWA methodology presents an efficient and robust technique for creating an ensemble of stochastic proxy models that can be used in production forecast, flow diagnostic, and optimization.