Reduced-Order Modeling for Multiphase Flow Using a Physics-Based Deep Learning

Reservoir simulation is the most widely used tool for oil and gas production forecasting and reservoir management. Solving a large-scale system of nonlinear differential equations every timestep can be computationally expensive. In this work, we present a two-phase physics-constrained deep-learning reduced-order model as a surrogate model for subsurface flow production forecast. The implemented deep learning model is a physics-guided encoder-decoder, constructed based on the Embed-to-Control (E2C) framework. In our implementation, the E2C works in a way analogous to Proper Orthogonal Decomposition combined with Discrete Empirical Interpolation Method (POD-DEIM) or Trajectory Piece-Wise Linearization approach (POD-TPWL). The E2C-Reduced-order model (ROM) involves projecting the system from a high-dimensional space into a low-dimensional subspace using the encoder-decoder, approximating the progression of the system from one timestep to the next using a linear transition model, and finally projecting the system back to high-dimensional space using the encoder-decoder. To guarantee mass conservation, we adopt the Finite Elements Mixed Formulation in the neural network's loss function combined with the original data-based loss function. Training simulations are generated using a full-physics reservoir simulator (IPARS). High-fidelity pressure, velocity, and saturation solution snapshot at constant time intervals are taken as training input to the neural network. After training, the model is tested over large variations of well control settings. Accurate pressure and saturation solutions are predicted along with the injection and production well quantities using the proposed approach. Errors in the predicted quantities of interest are reduced with the increase in the number of training simulations used. Although it required a large number of training simulations for the offline (training) step, the model achieved a significant speedup in the online stage compared to the full physics model. Considering the overall computational cost, this ROM model is proper for cases when a large number of simulations are required like in the case of production optimization and uncertainty assessments. The proposed approach shows the capability of the deep-learning reduced-order model to accurately predict multiphase flow behavior such as well quantities, and global pressure and saturation fields. The model honors mass conservation and the underlying physics laws, which many existing approaches don't take into direct consideration.