Hydrostatic equilibrium is a static state that the fluid pressure is balanced with gravity. Classic finite difference and finite volume methods cannot maintain this equilibrium state on discrete scales, and produce non-physical phenomena often such as spurious currents. In this work, volume integral of the source term is implemented at cell interface, so that the balance of pressure and gravity under discrete conditions is achieved on discrete scales. A well-balanced gas kinetic scheme of the Navier-Stokes(NS) equation is therefore constructed. The scheme maintains accurately the static equilibrium state under isothermal conditions and captures small perturbation propagation near equilibrium state at machine precision. At the same time, the scheme solves non-isothermal stratified flow near equilibrium state. Apart from the non-isothermal equilibrium described by Euler's equation, the scheme can also represent balance of thermal conduction in addition to force balance. It is shown that the scheme preserves second-order accuracy with respect to both density and temperature distribution when simulating an non-isothermal equilibrium state. Numerical examples reveal that the scheme is a potential tool in simulating stratified flow of temperature and density under gravitational field.

With lattice Boltzmann method, thermal miscible displacement process in porous media was investigated numerically in pore scale. Influence of thermal viscous diffusion coefficient (β_T) and Lewis number (Le) on interface shape and sweep efficiency are quantitatively analyzed. It shows that with the increase of β_T, the instability of interface increases and the sweep efficiency decreases. As β_T>0, with increase of Le, interface instability decreases; Interface between displacement fluid and displaced fluid tends to be flat; Fingertip residual rate decreases and sweep efficiency increases. As β_T<0, effects of Le on sweep efficiency are opposite.

We extend gas kinetic scheme (GKS) to low-speed porous media flow, and test feasibility and effectiveness of the method. It shows that GKS is second-order accurate in space and is capable of predicting permeability of porous media. Compared with single-relaxation-time lattice Boltzmann method, GKS implements precisely no slip boundary condition, thus reflects correctly characteristics of viscosity-independent permeability. For complex flow in Berea sandstone slice structure, simulation results are in good agreement with experimental data, and permeability is calculated accurately. A criteria of Ma number in GKS is proposed for Darcy flow. It shows that GKS could be a promising scheme for porous media flows.