  The application of Physics-Informed Neural Networks (PINNs) is investigated
for the first time in solving the one-dimensional Countercurrent spontaneous
imbibition (COUCSI) problem at both early and late time (i.e., before and after
the imbibition front meets the no-flow boundary). We introduce utilization of
Change-of-Variables as a technique for improving performance of PINNs. We
formulated the COUCSI problem in three equivalent forms by changing the
independent variables. 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 early time). 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. All three formulations
could closely approximate the correct solutions, with water saturation mean
absolute errors around 0.019 and 0.009 for XT and XY formulations and 0.012 for
the Z formulation at early time. The Z formulation perfectly captured the
self-similarity of the system at early time. This was less captured by XT and
XY formulations. The total variation of saturation was preserved in the Z
formulation, and it was better preserved with XY- than XT formulation.
Redefining the problem based on the 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.
