Numerical solutions of a diffusive interface model with Peng-Robinson equation of state

Abstract

This work is concerned with mathematical modeling and numerical simulations of the steady state and the movements of complex fluids involved in oil exploitation practice. Capillary pressure caused by surface tension at the interface between every two adjacent different phases of the mixture is viewed as the leading force in oil recovery from fractured oil reservoirs. Therefore, the interface between contiguous phases has become a critical mathematical modeling aspect. The diffuse interface theory, or the phase field model, or the gradient theory, has been widely applied to model or understand the interface between different phases of oil mixture. Based on the assumption that the density of every substance is continuous over the whole fluid region, the total Helmholtz free energy often contains the homogeneous part F0 (n) and the gradient contribution part Fv (n). The derivative of the total homogeneous free energy, f0 (n), or of the gradient part of free energy, fv(n), varies from substance to substance. Based on the original total free energy, the equilibrium state and the kinetic processes could be determined according to thermodynamic principles. As for the fluid system related to the oil recovery process, we apply the homogeneous free energy density and the parameters of the gradient part of the free energy density provided by the widely used Peng-Robinson equation of state (EOS). The fourth-order parabolic equation is derived and solved numerically by a convex-splitting scheme, the Crank-Nicolson scheme and a second order linearization scheme to describe the evolution processes of one-component, two-phase substances. The theoretical analyses of these numerical schemes have been obtained to demonstrate their mass conservation, energy stability, unique solvability and convergence.The Euler-Lagrange (E-L) equation derived from this expression of total Helmholtz free energy to determine the equilibrium state of the fluid systems has also been studied. In this study, it is solved numerically by the original Newton iteration and a convex-splitting based Newton iterative method. Its theoretical analysis remains a part of our future work. Numerical experiments have been carried out for both the fourth-order equation approach and the Euler-Lagrange equation approach. Our computational results match well with laboratory experimental data and are in good agreement with the well-known Young-Laplace equation.