Empirical or theoretical extensions of Darcy's law for immiscible two‐phase flow have shown significant limitations in properly modeling the flow at the continuum scale. We tackle this problem by proposing a set of upscaled equations based on pore‐scale flow regimes, that is, the topology of flowing phases. The incompressible Navier‐Stokes equation is upscaled by means of multiple‐scale expansions and its closures derived from the mechanical energy balance for different flow regimes at the pore scale. We also derive the applicability conditions of the upscaled equations based on the order of magnitude of relevant dimensionless numbers, that is, Eotvos, Reynolds, capillary, Froude numbers, and the viscosity and density ratio of the system, as well as a set of closures valid for the basic flow regimes of low Eotvos number systems, that is, core‐annular and plug and drop traffic flows. We provide analytical expressions for the relative permeability of the wetting and nonwetting phases in different flow regimes and demonstrate that the effect of the flowing‐phases topology on the relative permeabilities is significant. Finally, we show that the classical two‐phase Darcy law is recovered for a limited range of operative conditions, while specific terms accounting for interfacial and wall interactions should be incorporated to accurately model ganglia or drop traffic flow.

Relative permeability measurements are commonly fitted with the Corey and Brooks‐Corey correlations. Despite such correlations fit experimental data generally well, they are mostly of an empirical nature. Here, we propose a semiempirical model to determine relative permeabilities of the wetting and the nonwetting phases in real 3‐D porous media that accounts for pore‐scale flow regimes. The starting point is the homogenization framework proposed by Picchi and Battiato (2018, https://doi.org/10.1029/2018WR023172), where the upscaling is conducted for different spatial distributions of the flowing phases in the capillary tube setting. First, we extend the approach to realistic media by allowing pore‐scale flow regimes to coexist in a complex geometry while accounting for capillary and viscous limits in the dynamics. Then, we discuss the scaling behavior of normalized relative permeabilities in terms of the phases viscosity ratio and identify three classes which govern their scaling. We also derive an analytical expression for the fractional flow. Finally, we provide a detailed validation of the proposed model for both relative permeabilities and fractional flow against data from numerical simulations and experiments available in the literature. The data set used for validation covers a wide range of systems, ranging from brine‐CO₂ to oil‐water flows. The equations derived capture well the trend of both numerical and experimental data.

The motion of a gaseous Taylor bubble in a capillary tube is typical of many biological and engineering systems, such as small-scale reactors and microfluidic devices. Although the dynamics of a bubble in a Newtonian liquid has been the subject of several studies since the seminal works of Taylor (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 161–165) and Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188), the case where the fluid exhibits a shear-thinning behaviour is much less understood. To fill this gap, we study the dynamics of a bubble that moves in a shear-thinning fluid whose viscosity is described by the Ellis viscosity model. With this aim, we derive a lubrication model in the film region to identify the scaling laws for the bubble speed, the film thickness and the pressure drop as a function of the Ellis number and the degree of shear thinning. Our model generalizes Bretherton's results to shear-thinning fluids by identification of a universal scaling law for the effective viscosity that accounts for the interplay of the zero-shear-rate and shear-thinning effects. The film thickness follows a scaling law with respect to the capillary number based on the proposed effective viscosity. The ratio between the bubble speed and the average velocity of the fluid ahead of the bubble is a function of the effective capillary number only. We show that some portions of the bubble are dominated by the zero-shear-rate effect discussing the extent to which the use of the power-law viscosity model can be legitimized. Finally, we study the location of the recirculating vortices ahead of the bubble.