Fractional flow of immiscible phases occurs at the pore scale where grain surfaces and phases interfaces obstruct phase mobility. However, the larger scale behavior is described by a saturation-dependent phenomenological relationship called relative permeability. As a consequence, pore-scale parameters, such as phase topology and/ or geometry, and details of the flow regime cannot be directly related to Darcy-scale flow parameters. It is well understood that relative permeability is not a unique relationship of wetting-phase saturation and rather depends on the experimental conditions at which it is measured. Herein we use fast X-ray microcomputed tomography to image pore-scale phase arrangements during fractional flow and then forward simulate the flow regimes using the lattice-Boltzmann method to better understand the underlying pore-scale flow regimes and their influence on Darcy-scale parameters. We find that relative permeability is highly dependent on capillary number and that the Corey model fits the observed trends. At the pore scale, while phase topologies are continuously changing on the scale of individual pores, the Euler characteristic of the nonwetting phase (NWP) averaged over a sufficiently large field of view can describe the bulk topological characteristics; the Euler characteristic decreases with increasing capillary number resulting in an increase in relative permeability. Lastly, we quantify the fraction of NWP that flows through disconnected ganglion dynamics and demonstrate that this can be a significant fraction of the NWP flux for intermediate wetting-phase saturation. Rate dependencies occur in our homogenous sample (without capillary end effect) and the underlying cause is attributed to ganglion flow that can significantly influence phase topology during the fractional flow of immiscible phases.
Relative permeability quantifies saturation-dependent multiphase effects on fluid permeability relative to the absolute permeability of a material, which are key parameters required by petroleum engineers to model immiscible displacement in porous rock (Dullien 1991; Bear and Bachmat, 1990). Since relative permeability cannot be theoretically predicted within the two-phase extension of Darcy's law it must be experimentally measured by special core analysis (SCAL) or computed using direct numerical simulations. However, relative permeability experimental results often depend on the experimental protocol and many other parameters (Dullien 1991; Li et al., 2005; Hussain et al., 2012). One of the key questions in the context of direct simulations is whether quasistatic approaches capture the relevant physics well enough to accurately predict relative permeability for practical purposes (Blunt et al., 2002, 2013; Berg et al., 2016). Quasistatic approaches and current experimental protocols predict relative permeability as a function of saturation only, which implicitly assumes that for a given saturation a unique phase topology and geometry exists. However, experimental studies suggest otherwise, and relative permeability is often found to depend on capillary number, bond number, and/or the experimental protocol (Jerauld, 1997; Masalmeh et al., 2007; Masalmeh and Wei, 2010). For example, reported differences between relative permeability obtained under steady-state versus unsteady-state conditions are documented (Tsakiroglou et al., 2004) and there are known hysteretic effects between drainage and imbibition (Khayrat and Jenny, 2016).
