Extension of Fourier Neural Operator from Three-Dimensional (x, y, t) to Four-Dimensional (x, y, z, t) Subsurface Flow Simulation

Numerical simulation of subsurface flow in porous media is crucial for various geoscience applications. However, conducting numerical simulations for such problems, particularly in four dimensions (x, y, z, t), presents significant computational challenges and cost. These challenges are primarily due to the highly nonlinear governing partial differential equations (PDEs) and the need for refined mesh discretization. Deep learning methods offer promising alternatives to traditional simulators by leveraging neural network models. Previous studies have demonstrated that the surrogate model constructed using a Fourier neural operator (FNO) exhibits superior speed, accuracy, and data efficiency in addressing three-dimensional (x, y, t) problems compared to mainstream machine learning methods. However, applying the existing FNO algorithm to four-dimensional problems is hindered by the large number of network parameters and high GPU memory consumption. In this study, we propose a novel framework for dynamically predicting four-dimensional subsurface flow properties utilizing the FNO network and the domain decomposition method. Our approach leverages the three-dimensional FNO and time components in the x and y directions of each z-layer. The predicted results for subsequent time steps in four dimensions are generated by consolidating the results across all z-layer dimensions. We use examples of \(\hbox {CO}_2\) ${\text{CO}}_2$ injection for enhanced oil recovery (EOR) in compositional simulations for fracture systems and apply the deep learning framework that is trained from physics-based simulation data to emulate the complex flow state. Furthermore, we integrate our framework with a multilayer perceptron (MLP) network capable of processing sequential data to predict gas production under variable well control in a realistic reservoir. The key contribution of this work is the enhanced training and prediction efficiency of the FNO method for large-scale four-dimensional heterogeneous reservoir models. Consequently, our approach can be further extended to expedite workflows for reservoir optimization in closed-loop reservoir management.