Origin of pressure-flow non-linearity in two-phase intermittent flow in porous media

This study presents a first-principles model to predict the two-phase pressure drop in gas-liquid intermittent flow through round capillaries, which serve as the simplest analogous of a porous medium. Building upon the classical capillary flow theory, we derive a model for a train of elongated bubbles, and rigorously quantify its validity in terms of the dimensionless parameters of the problem (capillary number, number of bubbles, gas volume fraction, and channel length-to-diameter ratio). The total two-phase pressure drop is found to be non-linear with respect to the mean liquid slug velocity, varying in discrete steps due to its explicit dependence on the number of bubbles, and also influenced by the motion of both phases. Moreover, formulating the model in dimensionless terms reveals key scaling laws that govern the two-phase flow rheology. Perturbation theory shows that a single bubble induces an excess pressure drop in a liquid-filled slender channel, enabling the introduction of a pressure-dependent effective viscosity to describe the system. Finally, our analytical framework is extended from a single capillary to a bundle of cylindrical tubes. In nearly homogeneous bundles, inviscid bubble trains exhibit a smooth transition from a Bretherton-like regime with an apparent flow exponent of 2/3 at low pressure drops to weaker sub-linear regimes (with exponents between 2/3 and 1) as the pressure drop increases. Introducing the smallest pores disrupts the monotonic scaling of the flow exponent, reflecting a more complex rheological response. Deviations in two-phase flow behavior from the Darcy law are examined across the model parameter space and in relation to geometrical heterogeneity, offering new insights into pore-scale multiphase transport.