Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Times Using Physics-Informed Neural Networks

Countercurrent spontaneous imbibition (COUCSI) is a process in porous materials in which a wetting phase displaces non-wetting phase. In this work, we investigate for the first time the application of Physics-Informed Neural Networks (PINNs) in solving the 1D COUCSI problem in both early (ET) and late (LT) times. Also novel, we examine the Change-of-Variables technique for improving the performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables: XT-, XY-, and Z-formulations. The first describes saturation as function of normalized position X and time T; the second as function of X and Y=T^0.5; and the third as a sole function of Z=X/T^0.5 (valid only at ET). The PINN model was generated using a feed-forward neural network and trained based on minimizing a weighted loss function, including the physics-informed loss term and terms corresponding to the initial and boundary conditions. No synthetical or experimental data were involved in the training. All three formulations could closely approximate the correct solutions (obtained by fine-grid numerical simulations), with water saturation mean absolute errors (MAE) around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at ET. The Z formulation perfectly captured the self-similarity of the system at ET. This was less captured by XT and XY formulations. The total variation (TV) of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. It was demonstrated that redefining the problem based on physics-inspired variables reduced the non-linearity of the problem and allowed higher solution accuracies, a higher degree of loss-landscape convexity, a lower number of required collocation points, smaller network sizes, and more computationally efficient solutions.