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.