Appendices
A. Single-phase simulation
Before conducting the two-phase flow simulations, we first ran single-phase simulations to calculate the absolute permeability of the 3D rock model. Single-phase simulations can also be used to verify the accuracy and precision of the simulation. When only one type of fluid exists in a rock, the absolute permeability can be calculated from Darcy’s law:
\begin{equation} k=\frac{\text{v\ μ\ }l}{P}\text{\ \ \ \ \ \ \ \ \ \ \ }(A.1)\nonumber \\ \end{equation}
where k is the absolute permeability (m2),v is the average fluid velocity (m/s), μ is the dynamic viscosity of the fluid (Pa s), Δl is the distance between the inlet and the outlet, and ΔP is the applied pressure difference between the inlet and the outlet. In this study, because the body force was applied only in the z direction, we calculated k in the z direction (kz ).
Because all simulation units are in lattice Boltzmann units, they must subsequently be converted into physical units. The unit conversion fork can be performed using the formula:
\begin{equation} k_{\left(\text{physical}\right)}=k_{(LB)}*\ A^{2}\text{\ \ \ \ \ \ \ \ \ }(A.2)\nonumber \\ \end{equation}
where A is the resolution of the digital rock, i.e. the physical size of a lattice grid (5.345 µm in this study).
Absolute permeability is an intrinsic property of a rock that is independent of the type of fluid. The single-phase simulations were conducted with several different values of fluid viscosity and body force to ensure the consistency of the results. When the fluid summation of velocity change in 1,000 steps difference was less than 2% for all cases, the simulations were assumed to have converged at that time. The absolute permeability results in various viscosity and body force conditions are shown in the table A.1 below.