A deep learning approach for solving the stationary compositional two-phase equilibrium problems

Abstract

In this paper, we propose and investigate a deep neural network approach for solving the stationary compositional two-phase equilibrium problems in porous media. A recent approach is the unified formulation advocated by Lauser et al. (2011) which contains the complementarity conditions. The advantage of this formulation lies in its potential to handle the appearance and disappearance of phases automatically. To solve numerically the system of equations, a new strategy called NPIPM (NonParametric Interior-Point Method) is proposed by Vu et al. (2021). However, the method still has some disadvantages preventing the convergence to a solution. Taking inspiration from the work of Raissi et al. (2019) for the study of nonlinear partial differential equations using Physics-Informed Neural Networks (PINN), we realize that we can apply a deep neural network to obtain the solution of the stationary compositional model containing gas and liquid. We design a deep neural network structure and define a new loss function for the problem. The effectiveness of the proposed method is validated through numerical results obtained from a series of systems with increasing components. This machine learning approach provides a promising idea to solve the curse of dimensionality issue in the sense of the increasing number of components in stationary problems.