Ostwald-ripening equilibrium in two-phase flow through porous media: predicting and validating equilibrium trapped saturation, capillary pressure, and relative permeability beyond the permanent-trapping picture

When a gas and brine share the pore space of a rock, dissolved gas diffuses through the brine and equalizes capillary pressure between connected and trapped gas clusters — Ostwald ripening — so that, given sufficient time (days to months at subsurface conditions), two-phase displacement behaves as percolation without trapping. Adebimpe, Foroughi, Bijeljic and Blunt (Phys. Rev. E 110, 035105, 2024) used pore-network models to show that conventional immiscible measurements then over-estimate residual capillary trapping by 20–25%, and proposed scaling transforms to convert such measurements; they and subsequent work left the predictive core of this correction open. The open problem is to construct and validate a predictive framework that maps pore-scale geometry and wettability — including spatially heterogeneous wettability — to the equilibrium trapped saturation, capillary pressure, and relative permeability; to test the equilibrium model's assumptions, especially whether ripening can nucleate new non-wetting clusters and when the path-independent equilibrium limit applies; to quantify equilibration timescales and transport parameters for representative media; and to validate the predictions and the measurement-conversion procedure in experiments that reach full capillary-and-concentration equilibrium. Experiments so far (12–58 h imaging of hydrogen and CO2 ripening) and a 2026 time-dependent pore-network model for one sandstone confirm the mechanism and reveal path dependence, but none of these elements is settled.

Background

Two-phase flow through porous media — the simultaneous movement of a wetting fluid (brine) and a non-wetting fluid (gas) through a rock's pore space — is the setting for subsurface gas storage (CO2, hydrogen, natural gas). Its macroscopic description uses relative permeabilities (mobility-reduction factors for each phase in Darcy's law) and the capillary pressure $P_c$, the pressure difference between the phases set at the pore scale by entry pressures of order $2\sigma\cos\theta/r$ for interfacial tension $\sigma$, contact angle $\theta$, and throat radius $r$; the contact angle measures wettability, i.e., which phase preferentially adheres to the rock surface. When brine re-invades a gas-filled rock (imbibition, following the initial drainage in which gas invaded), blobs (ganglia) of gas become surrounded by brine. The classical picture — implemented in invasion-percolation-with-trapping models (Wilkinson and Willemsen 1983; Lenormand, Zarcone and Sarr 1983), in which the invading phase advances through the lowest-entry-pressure throat accessible from its connected cluster — treats these ganglia as permanently immobile, so the residual (trapped) gas saturation $S_{2t}$ is a fixed rock property. It is routinely measured on centimetre-scale cores with completely immiscible fluids over days to weeks, and such measurements feed assessments of capillary trapping as a CO2-storage security mechanism and of residual losses in hydrogen storage.

Ostwald ripening breaks this permanence. By Henry's law, the concentration of dissolved gas in the brine adjacent to a meniscus is proportional to the gas pressure, which is higher inside a small, high-curvature ganglion than a large one; molecular diffusion of dissolved gas through the brine therefore shrinks small ganglia and feeds large ones, transferring mass between disconnected clusters. Blunt (Phys. Rev. E 106, 045103, 2022) estimated the diffusive equilibration time $t_d \sim l^2 r \phi S_{gr}\rho_g/(2DH\sigma m_g)$ (with diffusion distance $l$, throat radius $r$, porosity $\phi$, residual gas saturation $S_{gr}$, gas density $\rho_g$ and molecular mass $m_g$, diffusivity $D$ and Henry's constant $H$), of order 17–100 days for CO2, hydrogen and methane over 1 mm at subsurface conditions — slow for laboratory core floods, fast on field timescales. Adebimpe, Foroughi, Bijeljic and Blunt (Phys. Rev. E 110, 035105, 2024) then argued that once local capillary equilibrium is reached — a single uniform capillary pressure shared by connected and disconnected clusters — two-phase displacement must be modelled as percolation without trapping: trapped clusters can shrink, grow, and even serve as displacement sources. Using pore-network models (pore-space representations extracted from X-ray micro-CT images) — five networks derived from four rock types (Bentheimer sandstone, two images of Estaillades limestone, Ketton oolitic limestone, and a reservoir carbonate sample) — they predicted that conventional immiscible measurements over-estimate residual capillary trapping by 20–25%, with reduced capillary pressure hysteresis (the history-dependence of $P_c$ between drainage and imbibition) and altered relative permeability, and proposed scaling transforms converting trapping-based measurements to equilibrium values. Time-resolved imaging supports the mechanism: hydrogen ganglia in Bentheimer sandstone rearrange within 12–16 h of flow stopping (Zhang et al., Geophys. Res. Lett. 2023; Goodarzi et al., Int. J. Hydrogen Energy 56, 1139, 2024), and trapped CO2 reconnects, its residual saturation falling from 22.8% to 15.6% after 58 h (Chai et al., Environ. Sci. Technol. 2025).

What the 2024 paper did not provide is the predictive content of this correction, and it said so explicitly. Its equilibrium model assumes ripening never nucleates new non-wetting clusters inside pore space occupied by the wetting phase — an assumption left untested. The reported 20–25% reductions are empirical results for five specific networks, with the change said to depend on contact angle and pore structure in a way that is 'difficult to predict a priori'. The simulations used specific wettability choices (zero receding contact angle for drainage, $45.6^\circ\pm20.1^\circ$ distributed for imbibition), not arbitrary or spatially heterogeneous wettability. No direct experiment had reached full ripening equilibrium (days to months). And the timescales and transport parameters governing equilibration in representative media were not quantified. A 2026 follow-up by the same authors (Adebimpe et al., Phys. Rev. E 2026; arXiv 2605.10955) added a time-dependent pore-network model of ripening in Bentheimer and showed that the approach to equilibrium is path-dependent — rapid imbibition followed by rest produces both shrinkage and growth events and a final fluid arrangement different from the equilibrium model — sharpening the question of when the equilibrium idealization applies. The open problem is to turn the percolation-without-trapping insight into a validated predictive framework.

Problem Statement

The objective is a validated predictive framework for the two-phase flow properties of porous media at Ostwald-ripening (local capillary) equilibrium — the regime in which dissolved-gas transport through the wetting phase has equalized capillary pressure between connected and disconnected non-wetting clusters and displacement is percolation without trapping. The framework should establish, with predictive and experimental force rather than per-rock empiricism:

1. Property prediction: relations or models that take pore-scale geometry (pore/throat size distributions and connectivity, e.g., from micro-CT images) and wettability — including spatially heterogeneous contact-angle distributions — as inputs, and predict a priori the equilibrium trapped saturation $S_{2t}^{n}$ after imbibition, the equilibrium capillary pressure and relative permeability functions, and the conversion of conventional immiscible (trapping-based) measurements to these equilibrium values.

2. Assumption audit of the equilibrium model: whether Ostwald ripening can nucleate new disconnected clusters of the non-wetting phase within regions occupied by the wetting phase (assumed absent in the equilibrium model), and under what conditions of displacement rate, rest protocol, and initial state the path-independent equilibrium limit applies — given that time-dependent simulations show finite-rate histories yield a different final fluid arrangement than the equilibrium model.

3. Equilibration timescales: a quantitative characterization, for representative porous media and gas/brine systems, of the timescales and transport parameters (diffusivities, Henry's-law solubilities, pore lengths) that control the approach to full capillary-and-concentration equilibrium, refining the days-to-months estimates currently available.

4. Direct experimental validation: experiments that reach, or demonstrably bracket, full ripening equilibrium and test the predicted changes in trapped saturation, capillary pressure, and relative permeability, including the proposed conversion of conventional measurements — beyond the 12–58 h rest periods imaged to date.

These components jointly constitute one framework: prediction (1) is credible only within a validated domain of applicability (2, 3) and is accepted only once confronted with equilibrium experiments (4). The components are judged independently, so a partial answer resolves the components it addresses, but within a component every output it lists is required: delivering the equilibrium trapped saturation alone, or the nucleation answer without the conditions for the path-independent limit, leaves that component open. Component 4 is met by designing or analysing experiments that reach, or demonstrably bracket, full ripening equilibrium and test the predicted changes — not necessarily by performing them.

References

1. A. I. Adebimpe, S. Foroughi, B. Bijeljic, M. J. Blunt, 'Percolation without trapping: how Ostwald ripening during two-phase displacement in porous media alters capillary pressure and relative permeability', Physical Review E 110, 035105 (2024). DOI: 10.1103/PhysRevE.110.035105; arXiv: 2404.01313. https://arxiv.org/abs/2404.01313
2. A. I. Adebimpe, S. Foroughi, B. Bijeljic, M. J. Blunt, 'Time-dependent pore-network modelling of Ostwald ripening in porous media', Physical Review E (2026; published online 19 August 2026, volume assignment pending). DOI: 10.1103/kbb7-wbln; arXiv: 2605.10955. https://arxiv.org/abs/2605.10955
3. S. Goodarzi, G. Zhang, B. Bijeljic, M. J. Blunt, 'Ostwald ripening leads to less hysteresis during hydrogen injection and withdrawal: a pore-scale imaging study', International Journal of Hydrogen Energy 114, 475–485 (2025). DOI: 10.1016/j.ijhydene.2025.02.066. https://doi.org/10.1016/j.ijhydene.2025.02.066
4. R. Chai, S. Foroughi, S. Goodarzi, A. Patmonoaji, F. Y. Yow, B. Bijeljic, M. J. Blunt, 'Pore-scale imaging to quantify the evolution and reduction in trapped CO2 due to Ostwald ripening', Environmental Science & Technology 59, 26419-26427 (2025). DOI: 10.1021/acs.est.5c06424. https://doi.org/10.1021/acs.est.5c06424
5. S. Goodarzi, Y. Zhang, S. Foroughi, B. Bijeljic, M. J. Blunt, 'Trapping, hysteresis and Ostwald ripening in hydrogen storage: a pore-scale imaging study', International Journal of Hydrogen Energy 56, 1139–1151 (2024). DOI: 10.1016/j.ijhydene.2023.12.029. https://doi.org/10.1016/j.ijhydene.2023.12.029
6. Y. Zhang, B. Bijeljic, Y. Gao, S. Goodarzi, S. Foroughi, M. J. Blunt, 'Pore-scale observations of hydrogen trapping and migration in porous rock: demonstrating the effect of Ostwald ripening', Geophysical Research Letters 50, e2022GL102383 (2023). DOI: 10.1029/2022GL102383. https://doi.org/10.1029/2022GL102383
7. M. J. Blunt, 'Ostwald ripening and gravitational equilibrium: implications for long-term subsurface gas storage', Physical Review E 106, 045103 (2022). DOI: 10.1103/PhysRevE.106.045103. https://doi.org/10.1103/PhysRevE.106.045103
8. J. A. de Chalendar, C. Garing, S. M. Benson, 'Pore-scale modelling of Ostwald ripening', Journal of Fluid Mechanics 835, 363–392 (2018). DOI: 10.1017/jfm.2017.720. https://doi.org/10.1017/jfm.2017.720
9. A. AlZaabi, W. Dokhon, B. Bijeljic, M. J. Blunt, 'A comparative review of pore-scale nitrogen, carbon dioxide, hydrogen, and methane transport, hysteresis, and Ostwald ripening in storage reservoirs', SPE ADIPEC, SPE-229055-MS (2025). DOI: 10.2118/229055-MS. https://doi.org/10.2118/229055-ms
10. D. Wilkinson, J. F. Willemsen, 'Invasion percolation: a new form of percolation theory', Journal of Physics A: Mathematical and General 16, 3365–3376 (1983). DOI: 10.1088/0305-4470/16/14/028. https://doi.org/10.1088/0305-4470/16/14/028
11. R. Lenormand, C. Zarcone, A. Sarr, 'Mechanisms of the displacement of one fluid by another in a network of capillary ducts', Journal of Fluid Mechanics 135, 337–353 (1983). DOI: 10.1017/S0022112083003110. https://doi.org/10.1017/S0022112083003110