Efficient numerical methods for multi-phase flow problems with Peng-Robinson equation of state

Abstract

This work concerns numerical simulations of diffuse interface models with Peng-Robinson equation of state (EOS). The motivation of our research arises from increasing attention of complex fluids flow problems in the oil industry. There are two basic concerns in the oil industry: oil exploration and oil exploitation. In oil exploration, properties of petroleum substances at the equilibrium state are mainly concerned. This requires us to construct numerical schemes that can accurately capture the interface information between hydrocarbon substances and phases. In oil exploitation, some fluids flow problems with complex boundary conditions need to be considered. Designed numerical schemes need to have a kinetic nature and a simple algorithm structure because of the huge amount of calculation required. In this thesis, based on these two specific needs of numerical algorithms, we introduce the energy stable scheme and Lattice Boltzmann method (LBM) to solve the equilibrium and fluids flow problems, respectively. Firstly, a Cahn-Hilliard type equation is derived to describe the single-component two-phase equilibrium problem. A first-order scalar auxiliary variable (SAV) scheme and a second-order SAV scheme are proposed to simulate the evolution process of single-component two-phase hydrocarbon substances. Mass conservation and energy stability in discrete sense are proved for these two schemes. Moreover, this approach has been expanded to the multi-component two-phase equilibrium case in this thesis. Based on the previous work [18], we modify this Cahn-Hilliard type model by introducing the mobility term. This improvement makes the multi-component model more physically compatible. A second-order SAV scheme is designed to solve the multi-component model. Numerical experiments have been carried out for both the single-component case and the multi-component case. Our numerical results match well with the laboratory data. It is worth mentioning that we have improved the calculation of interface tension and capillary pressure comparing with previous work. For multi-phase fluids flow problems, in order to verify the feasibility of the LBM for oil exploitation problems, we firstly use the single-relaxation-time LBM to solve a single-component equilibrium problem based on an Allen-Cahn type equation. Then, we design a multi-phase fluids dynamics model combined with Peng-Robinson EOS with a constant temperature under thermodynamics principles. Here, we use the multi-relaxation-time (MRT) LBM combining with Beam-Warming scheme to solve the proposed fluid model. Alterable CFL numbers can be used in the numerical simulation. High-order accurate numerical results have been obtained, which meet well with our expectations and have a great agreement with previous published results and laboratory data.