Finite PINN Net: Physics-informed deep convolutional neural networks for learning 3D transient Darcy flows in heterogeneous porous media

In this study, we have developed a novel and robust physics-informed deep convolutional neural network called "Finite PINN Net" for simulating multiphase, 3D, transient, and compressible Darcy flow in heterogeneous petroleum reservoirs. In the case of porous media, simulating such flows with neural networks (NNs) in highly heterogeneous reservoirs with source/sink terms is challenging due to the non-linearity of the fluid and rock properties. Moreover, enforcing flux continuity across cell boundaries is not viable via automatic differentiation due to the discontinuity of the pressure between two neighboring cells. We have addressed this issue by enforcing the flux continuity using the finite-volume discretization scheme and two-point flux approximation (TPFA) and embedding it into the NN framework. The structure of the Finite PINN Net and training procedure is analogous to the iterative time propagation of the Euler method. The key finding here is that we show, for the first time, the possibility of simulating 3D transient multiphase flows in highly heterogeneous porous media with NNs without using any labeled data. Finally, we validate the predictions of the Finite PINN Net in spatiotemporally resolved pressure and water saturation by comparing against the traditional simulator.