Modeling and simulation of multiphase flow and transport in underground porous media is an essential component in many scientific and engineering applications. Now, more and more focus has been paid on the Diffuse Interface models, which describes the interface as a continuum three-dimensional entity separating the two bulk single phase fluids.
Realistic equation of state (e.g. Peng-Robinson equation of state), has been proved to be a more accurate choice to model the interface tension, compared to a simple double-well potential. To handle the unconditionally energy stable requirement, a method called invariant energy quadratization (IEQ) is proposed. This approach allows us to construct a linear and unconditional energy stable scheme. However, the coefficients in this method are variables, which hinders the application of the fast fourier transformation method. To overcome it, a stabilized predictor-corrector approach is proposed and introduced, which is also called scalar auxiliary variable (SAV) scheme, to construct schemes which are second –order accurate, easy to implement, fast to divergence and maintaining the stability of the first-order stabilized schemes.
In this paper, the coupled model combined with diffuse interface model and Peng-Robinson equation of state is applied to describe the phenomenon around the interface. In this scheme, the Helmholtz free energy is used as the kernel, and a gradient term is added as well. The Cahn-Hilliard equation is the H-1 gradient flow of the total free energy. Unlike the Allen-Cahn equation, the Cahn-Hilliard itself can preserve the total mass because the function is based on the mass conservation law. As a result, if we treat the equilibrium in the fourth order form, we will not need to deal the Lagrange multiplier and we can get a Cahn-Hilliard equation-like model as
The main idea of the SAV scheme is to introduce the following term,the problem transforms to
We first simulate the separation of two phases of isobutene (nC4) at the temperature around 250K to 350K. Figure 3 shows the evolution of the solution of the fourth-order model.
We can see that a gas-liquid interface is formed around each liquid drops and the shape of the single drops become circle at the early stage. Then, the gas-liquid interface of each drops touch each other and these 4 drops start to mix together. Finally, a unit one is formed at the center of the whole domain. Figure 4 illustrates the energy dissipation trend during the evolution history and the mass conservation property.
Figure 5 shows the comparison of the surface tension obtained by the numerical experiment and the data from laboratory and our work meets the experiment data very well.
To show the numerical efficiency of such scheme, a 3D simulation on a 200200200 mesh is conducted and we can still get reasonable result.
|Acceptance of Terms and Conditions||Click here to agree|