Simulating two-phase flow using Physics-informed neural networks with capillary heterogeneity and hydraulic discontinuities

Simulations of two-phase flow in heterogeneous porous media are crucial for several applications, such as CO₂ sequestration, efficient oil and gas recovery, and groundwater pollution remediation. Modeling of two-phase flow systems becomes very challenging when capillary heterogeneity and hydraulic discontinuities are considered. Traditional models use numerical techniques such as finite difference, finite element, and finite volume for solving the partial differential equations of the system. Although numerical methods have been shown to produce reliable solutions for complex flow problems, they can become computationally expensive. This emphasizes the high computational demand for solving the inverse problem. The use of DNNs (deep neural networks) has become more common in predicting subsurface flow behavior. DNNs is a data-driven approach that enables the learning of a system by linking input and output parameters and provides fast predictions of dynamic, complex systems. Nevertheless, when data is extremely scarce, particularly in subsurface systems, standard DNNs are unable to yield robust results. Recent advancements enable the integration of physical constraints as partial differential equations (PDEs) into the DNNs scheme. Such a class of deep learning techniques is generally referred to as physics-informed neural networks (PINNs). PINNs are also capable to provide forward solutions for PDEs.  In this work, we examined PINNs' capabilities to provide forward solutions of a 1D steady-state two-phase flow with capillary heterogeneity at the sub-core scale. Here, we trained a PINNs system that incorporates high variability in the hydraulic properties and boundary conditions implemented as input parameters. We compared the PINNs results with numerical solutions to test the efficiency of the developed PINNs system. Results have shown that the trained PINNs system could reproduce both capillary pressure and phase saturation profiles for altering fractional flows, injection rates, hydraulic properties, and domain lengths with high accuracy and within a single training. Training the extended PINNs system was obtained in a few hours, and the post-trained system provided unlimited solutions for variable structures and boundary conditions within a few seconds.