New Fast Simulation of 4D (x, y, z, t) CO2 EOR by Fourier Neural Operator Based Deep Learning Method

The training speed is slow for the convolutional neural network (CNN)-based physics-informed neural network (PINN) in surrogate models and it is difficult to be applied to large-scale engineering problems. The Fourier Neural Operator (FNO) network can speed up 100 times faster than the PINN according to current literature. But the current FNO only handles the 3D (x, y, t) spatial-temporal domain. In this work, we developed a new framework to simulate the 4D (x, y, z, t) subsurface flow problems using the FNO network and the domain decomposition method. After numerical simulation runs, the obtained results of subsurface flow field distributions in 4D spatial-temporal domain (x, y, z, t) are decomposed into multiple 3D spatial-temporal domains (x, y, t) in the z dimension. Then, multiple FNO networks are used to train 3D spatial-temporal domain (x, y, t) in parallel to predict the distributions of the flow field in subsequent time steps. Finally, the predicted results of the 4D spatial-temporal solution in subsequent time steps are obtained by re-coupling the trained 3D (x, y, t) results in the z dimension. In this way, our new framework successfully extends FNO-network from 3D (x, y, t) to 4D (x, y, z, t) to predict field distributions in subsurface flow. The new framework was successfully applied to some very complex cases of CO2 injection for enhanced oil recovery (EOR) in compositional simulations. The predicted accuracy is enough for the method to be applied to simulate the complex CO2 EOR in fractured systems. The computational speed in 4D (x, y, z, t) can be as fast as it does in 3D (x, y, t) through parallel training. The tested results show that our new framework can efficiently simulate the EOR processes by injecting CO2 into complex fracture reservoirs. For the first time, we developed a new methodology that successfully extends the current FNO network from 3D (x, y, t) to 4D (x, y, z, t). Our framework paves way for the fast FNO network to solve the large-scale spatial-temporal domain of reservoir engineering systems.