Physics-Informed Neural Networks for Modeling Flow in Heterogeneous Porous Media: A Decoupled Pressure-Velocity Approach

Abstract Physics-informed neural networks (PINNs) have shown success in solving physical problems in various fields. However, PINNs face major limitations when addressing fluid flow in heterogeneous porous media, related to discontinuities in rock properties. This is because automatic differentiation is inadequate for evaluating the spatial derivatives of hydraulic conductivity where it is discontinuous. This study aims to devise PINN implementations that overcome this limitation. This work proposes decoupling the mass conservation equation from Darcy's law and utilizing the residuals of these decoupled equations to train the loss function of the PINN, rather than using a single residual from the combined equation. As a result, we circumvent the need to find the spatial derivative of the discontinuous hydraulic conductivity, and instead, we impose the continuity of fluxes. This decoupling necessitates that each primary unknown (pressure and velocity components) be computed by the neural networks (NNs) rather than deriving the velocity (or fluxes) from the pressure. We examined three NN configurations and compared their performance by analyzing their accuracy and training time for various 2D scenarios. These scenarios explored various boundary conditions, different hydraulic conductivity fields, as well as different orientations of the heterogeneous media within the domain of interest. In these problems, the pressure and velocity field are the primary unknowns. The three configurations include: (a) one NN with the three unknowns as its outputs, (b) two NNs, one outputting pressure and the other outputting the velocity, and (c) three NNs, each having one primary unknown as an output. Utilizing these NN architectures, we were able to solve the heterogeneous problems with varying levels of accuracy when compared to results from numerical simulators. While maintaining a similar number of training parameters for a fair assessment, the configuration with three NNs yielded the most accurate results, with a comparable training time to the other configurations. Using this optimal configuration, we performed a sensitivity analysis to demonstrate the effect of modifying the NN(s) hyperparameters, such as the number of layers, the number of nodes per layer, and the learning rate, on the accuracy of the results. We introduce a novel PINN approach for modeling fluid flow in heterogeneous media. This proposed method not only preserves the inherent discontinuity of rock petrophysical properties but also leverages the benefits of automatic differentiation. By incorporating this PINN architecture, we have opened up new possibilities for extending the application of PINN to realistic reservoir simulations that capture the complexities of the subsurface.