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

Abstract

Abstract We investigated countercurrent spontaneous imbibition (COUCSI) of water displacing oil in a 1D linear system with one side open, and one side closed. The Physics-Informed Neural Networks (PINNs) technique was used to estimate saturation profiles along the core and recovery against time; based on the same input information as a reservoir simulator. We demonstrate the usefulness of Change-of-Variables as an approach to improve PINN solutions. The problem was first normalized, where only a saturation-dependent diffusion coefficient results in different solutions. The initial condition was zero saturation, the open boundary had a saturation equal to one, and the closed boundary had a zero saturation gradient. We formulated the problem in three equivalent ways by Change-of-Variables: XT, YZ, and Z formulations. The first is the original normalized form and describes saturation as a function of normalized position X and time T. The second defines saturation as a function of Z=X/T^0.5 and Y=T^0.5. The third considers saturation as a sole function of Z=X/T^0.5 and is valid only at early times (ET), before water meets the no-flow boundary. The COUCSI problem was solved using a feed-forward neural network trained based on a weighted loss, including the physics-informed loss term and terms corresponding to initial and boundary conditions for all the formulations. No synthetical or experimental data were involved in the training. The generalization ability is tested by applying the workflow to two imbibition cases with different displacement profile behavior. The PINN solutions were tracked to determine if they followed the flow's theoretical properties, including self-similarity, square root of time behavior, and Total Variation (TV). We investigated the ability of the applied formulations to estimate the correct solution (compared to numerical simulations) at early and late times. All the formulations could very closely converge to the correct solutions, with the water saturation mean absolute errors around 3.5 and 2.5 percent for XT and YZ formulations and 1.0 percent for the Z formulation at ET. The Z formulation almost perfectly captured the self-similarity properties of the system in the ET period (and in lower level, YZ), which only depends on X/T^0.5 at early time. The TV of saturation was successfully preserved in the Z formulation and YZ performed better than XT formulation. By performing a sensitivity analysis we demonstrate that Change-of-Variables can lead to a lower number of required collocation points and also smaller network sizes.