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).