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

Abstract

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 $\ell^{\infty}(0,T; H_h^{-1} (Ω)) \cap \ell^{2}(0,T; \ell^2(Ω))$ 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.