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

Abstract

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.

Figures (12)

• Figure 1: Two-dimensional sketch of the physical problem, and qualitative plot of the wall pressure distribution along the axial direction.
• Figure 2: Comparison of the main available correlations for the dimensionless ($a$) uniform thin-film thickness and ($b$) tip-to-tip bubble pressure drop in an axisymmetric capillary of radius $r$, neglecting inertia and buoyancy effects, in the limit of inviscid bubble. For the correlation proposed by Balestra2018, see Eq. \ref{['dpb_dim_improved']}, both the curvature-induced component alone ('$c$') and the combined contribution including the jump in normal viscous stresses ('$c+n$') are displayed for comparison. The abscissa represents the capillary number, given in Eq. \ref{['capillary']}, being $\sigma$ the surface tension at the interface between the bubble and the surrounding fluid. The region of vanishing capillary numbers ($\text{Ca}\lesssim0.005$), where Bretherton1961's model is valid, is highlighted by the grey area.
• Figure 3: Schematic of Taylor flow with $N=3$ gas bubbles separated by liquid slug domains in a capillary tube. The front and rear menisci of each bubble are idealized as hemispherical caps, as highlighted by the black dashed rectangle enclosing the last bubble. The blue dashed rectangle denotes a representative unit cell -- shown separately in Fig. \ref{['fig:pressure_decomp']} -- which forms the repeating element (of varying length) of the two-phase flow Kreutzer2005Ni2017.
• Figure 4: ($a$) Validity constraint imposed by \ref{['eq:bastmin']} on the dimensionless mean bubble length $b^{\ast}$ (left ordinate axis) and relationship between the capillary numbers based on the bubble and liquid slug velocities (right ordinate axis), as given by Eq. \ref{['eq:CalCa']}. ($b$) Variation of the maximum bubble number $N_{\rm{max}}^{\rm{geom}}$ with the capillary number based on the bubble velocity, see Eq. \ref{['eq:Nmax_bubble']}: the dashed line represents the continuous trend of the ratio $L^{\ast}/b^{\ast}$, for a channel of slenderness $L^{\ast}=100$, while the superimposed staircase-like profile visualizes the effect of the floor operator, highlighting discrete changes in the bubble number. ($c$) Geometric range of variation of the volumetric gas fraction $\Phi$ with the capillary number based on the bubble velocity, showing both the lower ($\Phi_{\rm{min}}$) and upper ($\Phi_{\rm{max}}^{\rm{geom}}$) bounds -- see Eqs. (\ref{['eq:Phi_min']}, \ref{['eq:Phi_max']}) -- for $L^{\ast}=100$ and a varying number $N<N_{\rm{max}}^{\rm{geom}}$ of bubbles composing the train.
• Figure 5: Dimensionless pressure drops plotted against the capillary number based on the liquid velocity for a channel of slenderness $L^{\ast}=100$. Panels ($a$) and ($b$) show the slug- and bubble-related contributions, respectively, for varying gas saturation $\Phi$ and bubble number $N$. Panels ($c$) to ($f$) display the total pressure drop for fixed bubble numbers and varying volumetric gas fractions; for reference, the single-phase limit is shown as a continuous grey line.