Yujing Yan | Searcharxiv

arXiv2026

The PICNN-Assisted Physics-Preserving Scheme for Thermodynamically Consistent Two-Phase Flow in Porous Media

In this paper, we develop a physics-informed convolutional neural network (PICNN) assisted physics-preserving method for a thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media. Following the physics-preserving prediction-correction scheme of Li et al. \cite{li2025class}, the prediction step is performed by a PICNN trained with finite-volume residuals, where the interfacial fluxes are evaluated by the two-point flux approximation (TPFA) using two-point difference quotients of neighboring cell-centered unknowns to approximate interfacial normal gradients. The PICNN output is further corrected by a post-processing procedure to obtain energy-stable, mass-conservative, and bounds-preserving solutions. Numerical results show that the finite-volume residuals trained PICNN can replace the traditional prediction solver within the physics-preserving framework. Compared with conventional physics-informed neural networks (PINNs), the PICNN better captures local spatial interactions between each control volume and its neighboring cells, while the finite-volume residuals accommodate discontinuous permeability fields and interfacial flux continuity.

physics.flu-dyn↗

arXiv2026

A unified framework of fully decoupled, bound-preserving and energy-dissipative schemes for two-phase flow in porous media

Developing high-order numerical schemes for two-phase flow in porous media that preserve key physical properties remains a significant challenge in numerical analysis. In this article, we propose a general framework to construct fully discrete first- and second-order numerical schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media. The proposed schemes are rigorously proved to ensure five fundamental properties: (i) unique solvability; (ii) full decoupling; (iii) bound preservation for both phases; (iv) original energy dissipation; (v) local mass conservation for both phases. The key to ensure the unique solvability lies in guaranteeing the strict convexity of the discrete energy functionals associated with the constructed schemes. Departing from the coupled solution approach for the pressure and saturation variables, the proposed approach breaks traditional paradigm by subtracting the two-phase mass conservation equations to derive a fully decoupled system. In addition, the bound-preserving property for both phases is established by leveraging the singular nature of the logarithmic term around the limit values of $0$ and $1$. A rigorous error estimate for the first-order scheme, in the $\ell^{\infty}(0,T; H_h^{-1} (\Omega)) \cap \ell^{2}(0,T; \ell^2(\Omega))$ norm for the saturations of two phases, is established. Finally, various numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed schemes.

math.NA↗