Fig. (A. a) and fig. (A. b) show the velocity field of single-phase
simulation for kinematic viscosity = 0.077777 and 0.155555 with equal
body force of 0.0001. The fluid velocity for simulation with fluid
kinematic viscosity of 0.077777 nu_{LB} is significantly higher
compared to the fluid kinematic viscosity of 0.155555 nu_{LB} case.
This is because fluid average velocity is inversely proportional to
viscosity (equation A.1). Thus, fluid velocity is not proportional to
permeability, and must be normalized into dimensionless velocity field.
Fig. (A. c) and fig. (A. d) shows the normalized fluid velocity field
for both conditions. Both figures show a relatively similar results,
which indicate that a similar absolute permeability value is obtained
from both conditions.
B. Euler-Poincaré characteristic calculation
We applied Euler-Poincaré characteristic (Euler number) to investigate
the fluid connectivity of each fluid phase. Euler-Poincaré
characteristic is one of the features in Minkowski measures that
describes the topology of the fluid distribution. Euler number measures
the connectedness of a fluid phase by counting fluid clusters and
inherent loops (Schlüter et al., 2016). In order to apply Minkowski
functionals, the image of fluid distributions must be converted into
binary structures, where one fluid phase is described as foreground and
the other fluid phase and the solid rock is the background. The formula
to measure Euler number of a three-dimensional image is as follows:
\begin{equation}
\chi=N\ -\ L+O\nonumber \\
\end{equation}with \(\chi\) represents Euler-Poincare characteristic, N is number of
individual (unconnected) fluid clusters, L is the number of redundant
connections between clusters, and O is the number of isolated background
clusters enclosed by foreground clusters. High Euler number indicates
the increase of number of unconnected foreground clusters and the
decrease of number of isolated background clusters (wetting fluid
clusters for nonwetting fluid binary image and nonwetting fluid clusters
for wetting fluid binary image). As a 3D Berea digital rock is used as a
porous media in this study, the 3D-extended Euler-Poincaré
characteristic of the 3D binary nonwetting and wetting fluid images are
calculated using the approach proposed by Legland et al. (2007).