When a wetting fluid displaces a non-wetting fluid from a porous medium, a residual fraction of the non-wetting phase remains trapped as discrete ganglia spanning one to many pores. Confocal-microscopy experiments by Datta, Ramakrishnan and Weitz [Phys. Fluids 26, 022002 (2014)] in a model 3D water-wet bead pack showed that, above a threshold capillary number Ca≈2×10−4Ca\approx 2\times10^{-4}, the largest trapped ganglia are mobilized and removed, quantitatively consistent with a mean-field balance between the Darcy viscous pressure drop across a ganglion and the pore-scale capillary threshold for pushing it through a pore throat; the measured largest trapped ganglion tracked Lmax⁡≈2.5 L∗L_{\max}\approx 2.5\,L^{*}, where L∗L^{*} is the predicted mobilization length. The experiments observed no significant ganglion breakup, the criterion omits buoyancy, ganglion–ganglion interactions, local flow heterogeneity and inertia, and the origin of the factor 2.5 was not explained. This problem asks for the geometric and physical conditions under which trapped ganglia break up versus mobilize intact, for an extension and validation of the viscous–capillary mobilization criterion incorporating the omitted effects, and for an explanation of the empirical prefactor. The composite question of breakup versus intact mobilization, the extended criterion, and the prefactor remains unresolved.

Background

In immiscible two-phase flow through a porous medium, the fluid that preferentially wets the solid (the wetting phase, e.g. water in a water-wet medium) can displace the non-wetting phase (e.g. oil) in a process called imbibition, the reverse of drainage. Imbibition typically leaves a finite fraction of the non-wetting phase trapped: snap-off, in which the wetting phase pinches off the non-wetting phase at pore throats via corner films, disconnects it into ganglia — discrete clusters occupying from a single pore up to hundreds of pores. The trapped fraction is quantified by the residual non-wetting saturation SorS_{or}, and percolation theory predicts a broad, scale-free distribution of ganglion sizes. The strength of the viscous driving relative to capillary trapping is measured by the capillary number Ca≡μwQw/(Aγ)Ca\equiv\mu_{w}Q_{w}/(A\gamma), where μw\mu_{w} is the wetting-fluid viscosity, Qw/AQ_{w}/A its Darcy velocity through cross-section AA, and γ\gamma the interfacial tension; typical values span 10−710^{-7}–10−310^{-3}. Whether and how trapped ganglia can be remobilized by increasing CaCa is a central question for enhanced oil recovery, remediation of non-aqueous-phase liquid contaminants in groundwater, and geological CO2_{2} storage, where the residual phase must be either recovered or securely immobilized.

Datta, Ramakrishnan and Weitz (2014) visualized this process at pore-scale resolution in a model 3D porous medium — a disordered packing of a=19 μma=19\,\mu\text{m} radius glass beads, index- and refractive-matched for confocal microscopy — with oil as the non-wetting phase. After secondary imbibition at Ca=6.4×10−7Ca=6.4\times10^{-7}, Sor≈9%S_{or}\approx 9\%, and the ganglion-size distribution followed the percolation prediction. As CaCa was raised, SorS_{or} and the largest trapped ganglion length Lmax⁡L_{\max} stayed constant until Ca≈2×10−4Ca\approx 2\times10^{-4}, then decreased together: the largest ganglia were mobilized and swept out. This behavior was quantitatively captured by a mean-field force balance. The viscous pressure drop across a ganglion of length LL is taken, via Darcy's law (the linear relation between flow rate and pressure gradient in a porous medium, with permeability k=1.67 μm2k=1.67\,\mu\text{m}^{2} and end-point relative permeability κ≤1\kappa\le 1 quantifying the reduced wetting-phase transport caused by the trapped oil), to be Pv=(μw/κk)(Qw/A)LP_{v}=(\mu_{w}/\kappa k)(Q_{w}/A)L. Mobilizing the ganglion requires pushing it into a downstream pore throat of radius at≈0.16aa_{t}\approx 0.16a while withdrawing its tail from an upstream pore body of radius ab≈0.24aa_{b}\approx 0.24a, a capillary pressure threshold Pt−Pb=(2γcos⁡θ/ab)(ab/at−1)P_{t}-P_{b}=(2\gamma\cos\theta/a_{b})(a_{b}/a_{t}-1), with θ\theta the contact angle. Balancing the two gives the predicted mobilization length

L∗=2cos⁡θCa(abat−1)κkab,L^{*}=\frac{2\cos\theta}{Ca}\left(\frac{a_{b}}{a_{t}}-1\right)\frac{\kappa k}{a_{b}},

so that, at a given CaCa, ganglia longer than L∗L^{*} should be mobilized. Indeed, above Ca≈2×10−4Ca\approx 2\times10^{-4} the measured Lmax⁡L_{\max} and L∗L^{*} decreased in parallel — but with a constant offset, Lmax⁡≈2.5 L∗L_{\max}\approx 2.5\,L^{*}, whose origin the mean-field picture does not explain.

Three features of this result were left open by the authors. First, no significant ganglion breakup was observed, contrary to earlier suggestions in the literature, and the authors noted it is unclear why; they conjectured that breakup may depend sensitively on pore-space geometry, e.g. becoming appreciable in more disordered media with larger pore-body/pore-throat aspect ratios. Second, the analysis deliberately keeps only viscous and capillary forces — omitting buoyancy, ganglion–ganglion interactions, detailed local flow heterogeneity, and possible inertia — and does not incorporate interactions between different ganglia; the authors stated that including other factors would be a useful direction for future work. Third, the empirical prefactor relating Lmax⁡L_{\max} to L∗L^{*} was not derived. Adjacent results frame these gaps: in a companion study of simultaneous flow of both phases through similar media (Datta, Dupin and Weitz 2014), the non-wetting phase does break up into discrete ganglia above thresholds organized in a state diagram in the capillary numbers of both fluids; synchrotron imaging has identified droplet fragmentation as a pore-scale process induced by the complex, heterogeneous pore structure of carbonates (Pak et al. 2015) and a ganglion-dynamics regime of correlated breakup and coalescence in sandstone (Rücker et al. 2015); and pore-scale simulation and velocimetry work has begun to characterize flow within and around trapped ganglia (Zarikos et al. 2018; Yiotis et al. 2019; Gao et al. 2020; Imani et al. 2022; Nhunduru et al. 2024; Ferraroni et al. 2026). What is still missing is a predictive, validated account of when trapped ganglia in 3D media break up versus mobilize intact, and of how far the mean-field viscous–capillary criterion extends once the omitted effects are included.

Problem Statement

For a trapped non-wetting phase in a three-dimensional porous medium through which the wetting phase flows, establish a predictive and experimentally validated account of ganglion mobilization that goes beyond the mean-field viscous–capillary balance of Datta, Ramakrishnan and Weitz (2014). Concretely: (1) determine the geometric and physical conditions — pore-space disorder, pore-body/pore-throat aspect ratio, wettability, viscosity ratio, interfacial tension, capillary-number regime, buoyancy — under which trapped ganglia break up into smaller clusters during mobilization, versus mobilize intact as observed in the 2014 bead-pack experiments, thereby explaining why breakup was absent there while fragmentation is observed in more heterogeneous media and in other regimes; (2) extend the mobilization criterion, whose present form predicts mobilization of ganglia longer than L∗=(2cos⁡θ/Ca)(ab/at−1)κk/abL^{*}=(2\cos\theta/Ca)(a_{b}/a_{t}-1)\kappa k/a_{b}, to incorporate the effects it omits — ganglion–ganglion interactions, local flow heterogeneity, buoyancy, and inertia — and validate the extended criterion against pore-scale experiments or resolved simulations; and (3) explain the origin of the empirical prefactor in the observed relation Lmax⁡≈2.5 L∗L_{\max}\approx 2.5\,L^{*}, i.e. why the largest trapped ganglion persists at roughly 2.5 times the predicted mobilization length in the mobilization regime. These three deliverables serve one overarching objective: a validated, quantitative theory of how a trapped non-wetting phase is remobilized — with or without breakup — from 3D porous media. Answers may be theoretical, computational, experimental, or combinations thereof.

References

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