Efficient energy-stable dynamic modeling of compositional grading

Compositional grading in hydrocarbon reservoirs caused by the gravity force highly affects the design of production and development strategies. In this paper, we propose a novel mathematical modeling for compositional grading based on the laws of thermodynamics. Different from the traditional modeling, the proposed model can dynamically describe the evolutionary process of compositional grading, and it satisfies the energy dissipation property, which is a key feature that real systems obey. The model is formulated for the two scales of free spaces without solids (laboratory scale) and porous media (geophysical scale). For the numerical simulation, we propose a physically convex-concave splitting of the Helmholtz energy density, which leads to an energy-stable numerical method for compositional grading. Using the proposed methods, we simulate binary and tenary mixtures in the free spaces and porous media, and demonstrate that compared with the laboratory scale, the simulation at large geophysical scales has more advantages in simulating the features of compositional grading.