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.