Application of physics-informed neural networks for self-similar and transient solutions of spontaneous imbibition

Abstract

Spontaneous imbibition is a natural process where the wetting phase spontaneously replaces the non-wetting phase due to the capillary force in porous media. This multiphase flow mechanism happens frequently in many natural processes such as during nonaqueous phase liquids transport, CO2 storage, soil infiltration, improved oil recovery, etc. Due to the nature of the capillary dominated flow, the imbibition rate is a natural consequence of the rock-fluid interaction and remains unknown from the problem configuration. This has made the solution to spontaneous imbibition unique and differentiates it from the traditional Buckley-Leverett type of displacement problems. While many attempts have been made in the literature to solve the spontaneous imbibition governing equations, most of them are quite mathematically involved even for the self-similar solution governed by an ordinary differential equation (ODE). Only a few more complex solutions exist for the transient spontaneous imbibition problems. In this research, we have adopted the concept and workflow of the deep physics-informed neural networks (PINN) to solve the spontaneous imbibition problems, both self-similar and transient. It focuses on learning the prior knowledge embedded within the governing differential equations rather than depending on input-output data pairs. The one-dimensional unsteady state immiscible, incompressible horizontal flow equations with a saturation-dependent dispersion coefficient under Lagrangian formulation is included in the workflow as the main loss function, along with the physics governed boundary conditions such as capillary end effect. The methodology represents a robust and straightforward function approximator for the solutions of this class, and the effectiveness of the proposed framework is demonstrated through two example calculations.