Application of Physics-Informed Neural Networks for Estimation of Saturation Functions from Countercurrent Spontaneous Imbibition Tests
Authors: Abbasi Jassem , Andersen Pål Østebø 
Published: January 2024

Summary
In this work, physics-informed neural networks (PINNs) are used for history matching data from core-scale countercurrent spontaneous imbibition (COUCSI) tests. To our knowledge, this is the first work exploring the variation in saturation function solutions from COUCSI tests. 1D flow was considered, in which two phases flow in opposite directions driven by capillary forces with one boundary open to flow.
The partial differential equation (PDE) depends only on a saturation-dependent capillary diffusion coefficient (CDC). Static properties such as porosity, permeability, interfacial tension, and fluid viscosities are considered known. In contrast, the CDC or its components [relative permeability (RP) and capillary pressure (PC)], are considered unknown. We investigate the range of functions (CDCs or RP/PC combinations) that explain different (synthetic or real) experimental COUCSI data: recovery from varying extents of early-time and late-time periods, pressure transducers, and in-situ saturation profiles.
History matching was performed by training a PINN to minimize a loss function based on observational data and terms related to the PDE, boundary, and initial conditions. The PINN model was generated with feedforward neural networks, Fourier/inverse-Fourier transformation, and an adaptive tanh activation function, and trained using full batching. The trainable parameters of both the neural networks and saturation functions (parameters in RP and PC correlations) were initialized randomly.
The PINN method successfully matched the observed data and returned a range of possible saturation function solutions. When a full observed recovery curve was provided (recovery data reaching close to its final value), unique and correct CDC functions and correct spatial saturation profiles were obtained. However, different RP/PC combinations composing the CDC were calculated. For limited amounts of recovery data, different CDCs matched the observations equally well but predicted different recovery behavior beyond the collected data period. With limited recovery data, when all points were still following a square root of time trend, a CDC with a low magnitude and peak shifted to high saturations gave the same match as a CDC with a high magnitude and peak shifted to low saturations. Recovery data with sufficient points not being proportional to the square root of time strongly constrained how future recovery would behave and thus which CDCs could explain the results. Limited recovery data combined with an observed in-situ profile of saturations allowed for accurate determination of CDC and prediction of future recovery, suggesting in-situ data allowed for shortened experiments. With full recovery data, in-situ PC data calibrated the PC toward unique solutions matching the input. The RPs were determined, where their phase had much lower mobility than the others. The CDC is virtually independent of the highest fluid mobility, and RPs could not be matched at their high values. Adding artificial noise in the recovery data increased the variation of the estimated CDCs.