To investigate the connectivity of each fluid phase, we calculated the
Euler–Poincaré characteristic (Euler number hereafter). Euler number
has been used to investigate the effect of pore space heterogeneity on
permeability and the effect of fluid saturation on the fluid
connectivity (Schlüter et al., 2016; Zhang et al., 2022) and the
morphology of the nonwetting and wetting fluids is predicted to have a
great influence on the relative permeability. The method to calculate
Euler number in this study is described in Appendix B.
We considered the evolution of the Euler number for the nonwetting fluid
during simulations with M = 1 and M = 5, log Ca =
−0.25 ± 0.1, and S nw of 50% (Fig. 4a). In the
first 10,000 steps, the Euler number in both simulation conditions
increases, which indicates that at this time, the nonwetting fluid is
unstable and keeps forming new fluid clusters. After 10,000 steps, the
Euler number for both simulation conditions starts to decrease,
suggesting that some of the nonwetting fluid clusters become more
connected and form larger clusters as the simulation proceeds. In the
final converged condition, the Euler number is higher for M = 5
than for M = 1. This result indicates that at M = 5, the
nonwetting fluid forms more individual clusters and has less fluid
connectivity than at M = 1. This difference is the result of the
instability of the interface of the fluids due to the viscosity
stratification (Yiantsios & Higgins, 1988; Yih, 1967). When two
immiscible fluids have different viscosities, their velocities will also
be different at their interface, causing instability. The instability
causes the nonwetting fluid to create more clusters, decreasing its
fluid connectivity. In addition, as more clusters are formed, the size
of the nonwetting fluid clusters is smaller, and it is easier for them
to pass through the pore space. At M = 1, the nonwetting fluid is
more connected and its fluid clusters are larger. The larger clusters
cannot pass through the small pore throats, leading to lowerk nw.
We also calculated the evolution of the Euler number for the wetting
fluid under the same M and Ca conditions (Fig. 4b). For
both M = 1 and M = 5, the Euler number keeps increasing
during the simulation, which indicates that the wetting fluid keeps
getting more disconnected before convergence. The Euler number forM = 1 is higher than the Euler number for M = 5, because
larger number of nonwetting fluid clusters are formed at M =5
compared to at M =1. This also indicates that wetting fluid has
higher connectivity at M = 5 and therefore has a larger area of
contact with the solid and receives more drag force; thus,k w is lower at M = 5 than at M = 1.
To confirm our interpretation, we also calculated the degree of contact
between the wetting fluid and the solid wall for M = 1 andM = 5, as indicated by the number of wetting fluid voxels
attached to the solid surface. At the start of the simulation, 47.38 %
and 47.57 % of wetting fluid voxels were attached to the solid surface
at M = 1 and M = 5, respectively. When the simulations
converged, these respective numbers were 81.17 % and 82.58 %. This
result confirms that, at M = 5, the wetting phase fluid is more
strongly attached to the solid wall and receives more drag force from
the solid wall, thus the k w is lower at M= 5 than at M = 1, as shown in Fig. 2b.