A physics-informed deep convolutional neural network for simulating and predicting transient Darcy flows in heterogeneous reservoirs without labeled data

Abstract

The physics-informed neural network (PINN) is a general deep learning framework for simulating physical processes and surrogate modeling without labeled data. The basic idea is to formulate the loss function according to the governing PDEs such that the neural network (NN) can be trained to minimize the PDE residual along with other misfits such as initial and boundary conditions. Following PINN, various networks have been developed for simulating steady and transient flows with or without labeled data. However, according to literature review, it is still not clear how to use NNs to simulate transient Darcy flows in highly heterogeneous reservoir models with source/sink terms in the absence of labeled data. In the current study, a physics-informed deep convolutional neural network (PIDCNN) architecture for simulating and predicting such flows is presented. Convolutional neural network is found to be more efficient than fully-connected neural network since 2D variables can be regarded as images. The finite volume discretization scheme is adopted to build the loss function to approximate the PDE residual such that flux continuity between neighboring cells of different properties can be implemented conveniently using the two-point flux approximation. Test cases are used to show that PIDCNN can accurately simulate transient Darcy flows in homogeneous and heterogeneous reservoirs. Further, it is demonstrated that PIDCNN can be trained as a surrogate to predict the transient flow fields of reservoir models not included in training. In addition, the CNN structure in the current study can be trained as a surrogate with labels for a particular output for better accuracy. A workflow is presented to demonstrate that CNN can be trained as an accurate surrogate for production rates using labels generated by the PIDCNN-based solver such that the entire workflow is external-label-free.