Deep Learning of Two-Phase Flow in Porous Media via Theory-Guided Neural Networks

A theory-guided neural network (TgNN) is proposed as a prediction model for oil/water phase flow in this paper. The model is driven by not only labeled data, but also scientific theories, including governing equations, boundary and initial conditions, and expert knowledge. Two independent neural networks (NNs) are built in the TgNN for oil/water phase flow problems, with one approximating pressure and the other approximating saturation. The two networks are connected by loss functions, which include a data mismatch term, as well as theory-guided terms. The desired parameters in NNs are trained by a certain optimization algorithm to decrease the value of the loss function. The training process uses a two-stage strategy as follows: (1) after one of the two NNs obtains a satisfactory result, parameters in the network with better performance are fixed in calculating the nonlinear terms and (2) the other NN continues to be trained until satisfactory performance is also obtained. The proposed TgNN offers an effective way to solve the coupled nonlinear two-phase flow problem. Numerical results demonstrate that the proposed TgNN achieves better accuracy than the traditional deep neural network (DNN). This is because the governing equation can constrain spatial and temporal derivatives, and other physical constraints (i.e., boundary and initial conditions, expert knowledge) can make the outputs more scientifically consistent. The effect of sparse data (including labeled data and collocation points) is tested, and the results show that more labeled data and collocation points lead to improved long-term prediction performance. However, the TgNN can also be successfully trained in the absence of labeled data by merely adhering to the above-mentioned scientific theories. In addition, several more complicated scenarios are tested, including the existence of data noise, changes in well condition, transfer learning, and the existence of different levels of dynamic capillary pressure. Compared with the traditional DNN, TgNN possesses superior stability with the guidance of theories for the considered complex situations.