Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T00:31:25.907Z Has data issue: false hasContentIssue false
  • English
  • Français

A continuum-scale representation of Ostwald ripening in heterogeneous porous media

Published online by Cambridge University Press:  21 February 2020

Yaxin Li Open the ORCID record for Yaxin Li [Opens in a new window] ,
Charlotte Garing  and
Sally M. Benson
Yaxin Li*
Affiliation:
Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA
Charlotte Garing
Affiliation:
Department of Geology, University of Georgia, Athens, GA 30602, USA
Sally M. Benson
Affiliation:
Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: yaxinl@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Ostwald ripening is a pore-scale phenomenon that coarsens a dispersed phase until thermodynamic equilibrium. Based on our previous finding that multi-bubble equilibrium is possible and likely in complex porous media, we develop a new continuum-scale model for Ostwald ripening in heterogeneous porous media. In this model, porous media with two different capillary pressure curves are put into contact, allowing only diffusive flow through the aqueous phase to redistribute a trapped gas phase. Results show that Ostwald ripening can increase the gas saturation in one medium while decreasing the gas saturation in the other, even when the gas phase is trapped in pore spaces by capillary forces. We develop an analogous retardation factor to show that the characteristic time for Ostwald ripening is about $10^{5}$ times slower than a single-phase diffusion problem due to the fact that separate-phase gas requires a much larger amount of mass transfer before equilibrium is established. An approximate solution has been developed to predict the saturation redistribution between the two media. The model has been validated by numerical simulation over a wide range of physical parameters. Millimetre to centimetre-scale systems come to equilibrium in years, ranging up to 10 000 years and longer for metre-scale systems. These findings are particularly relevant for geological $\text{CO}_{2}$ storage, where residual trapping is an important mechanism for immobilizing $\text{CO}_{2}$. Our work demonstrates that Ostwald ripening due to heterogeneity in porous media is slow and on a similar time scale compared to other processes that redistribute trapped $\text{CO}_{2}$ such as convective mixing.

JFM classification

Drops and Bubbles: Bubble dynamics Multiphase and Particle-laden Flows: Multiphase flow Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A14
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrew, M., Bijeljic, B. & Blunt, M. J. 2014 Pore-by-pore capillary pressure measurements using X-ray microtomography at reservoir conditions: curvature, snap-off, and remobilization of residual CO2. Water Resour. Res. 50 (11), 8760–8774.CrossRefGoogle Scholar
Bear, J. 2013 Dynamics of Fluids in Porous Media. Dover Publications.Google Scholar
Burnside, N. M. & Naylor, M. 2014 Review and implications of relative permeability of CO2 /brine systems and residual trapping of CO2. Intl J. Greenh. Gas Control 23, 1–11.CrossRefGoogle Scholar
Cadogan, S. P., Maitland, G. C. & Trusler, J. P. M. 2014 Diffusion coefficients of CO2 and N2 in water at temperatures between 298.15 K and 423.15 K at pressures up to 45 MPa. J. Chem. Engng Data 59 (2), 519–525.CrossRefGoogle Scholar
de Chalendar, J. A., Garing, C. & Benson, S. M. 2018 Pore-scale modelling of Ostwald ripening. J. Fluid Mech. 835, 363–392.CrossRefGoogle Scholar
Crank, J. 1975 The Mathematics of Diffusion, 2nd edn. Clarendon Press.Google Scholar
Dominguez, A., Bories, S. A. & Prat, M. 2000 Gas cluster growth by solute diffusion in porous media. Experiments and automaton simulation on pore network. Intl J. Multiphase Flow 26 (12), 1951–1979.CrossRefGoogle Scholar
Fokas, A. S. & Yortsos, Y. C. 1982 On the exactly solvable equation s t = (𝛽s+𝛾)-2 s x]x +𝛼 (𝛽s+𝛾)-2s x occurring in two-phase flow in porous media. SIAM J. Appl. Maths 42 (2), 318–415.CrossRefGoogle Scholar
Garing, C., de Chalendar, J. A., Voltolini, M., Ajo-Franklin, J. B. & Benson, S. M. 2017 Pore-scale capillary pressure analysis using multi-scale x-ray micromotography. Adv. Water Resour. 104, 223–241.CrossRefGoogle Scholar
Goldobin, D. S. & Brilliantov, N. V. 2011 Diffusive counter dispersion of mass in bubbly media. Phys. Rev. E 84, 056328.Google ScholarPubMed
Greenwood, G. W. 1956 The growth of dispersed precipitates in solutions. Acta Metall. 4, 243–248.CrossRefGoogle Scholar
Hesse, M. A., Orr, F. M. & Tchelepi, H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 35–60.CrossRefGoogle Scholar
Kashchiev, D. & Firoozabadi, A. 2003 Analytical solutions for 1D countercurrent imbibition in water-wet media. SPE J. 8, 401–408.CrossRefGoogle Scholar
Krevor, S. C. M., Pini, R., Zuo, L. & Benson, S. M. 2012 Relative permeability and trapping of CO2 and water in sandstone rocks at reservoir conditions. Water Resour. Res. 48 (2), W02532.CrossRefGoogle Scholar
Li, X. & Yortsos, Y. C. 1995 Theory of multiple bubble growth in porous media by solute diffusion. Chem. Engng Sci. 50 (8), 1247–1271.CrossRefGoogle Scholar
Lifshitz, I. M. & Slyozov, V. V. 1961 The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19 (1-2), 35–50.CrossRefGoogle Scholar
McClure, J. E., Berrill, M. A., Gray, W. G. & Miller, C. T. 2016 Influence of phase connectivity on the relationship among capillary pressure, fluid saturation, and interfacial area in two-fluid-phase porous medium systems. Phys. Rev. E 94, 033102.Google ScholarPubMed
Narasimhan, T. N., Witherspoon, P. A. & Edwards, A. L. 1978 Numerical model for saturated–unsaturated flow in deformable porous media: 2. The algorithm. Water Resour. Res. 14 (2), 255–261.CrossRefGoogle Scholar
Pini, R., Krevor, S. C. M. & Benson, S. M. 2012 Capillary pressure and heterogeneity for the CO2 /water system in sandstone rocks at reservoir conditions. Adv. Water Resour. 38, 48–59.CrossRefGoogle Scholar
Plesset, M. S. & Sadhal, S. S. 1982 On the stability of gas bubbles in liquid–gas solutions. Appl. Sci. Res. 38 (1), 133–141.CrossRefGoogle Scholar
Pruess, K., Oldenburg, C. & Moridis, G.1999 TOUGH2 user’s guide, version 2.0.Google Scholar
Pruess, K. & Spycher, N. 2007 ECO2N – A fluid property module for the TOUGH2 code for studies of CO2 storage in saline aquifers. Energy Convers. Manage. 48 (6), 1761–1767.CrossRefGoogle Scholar
Spycher, N. & Pruess, K. 2005 CO2–H2O mixtures in the geological sequestration of CO2 . II. Partitioning in chloride brines at 12 100 °C and up to 600 bar. Geochim. Cosmochim. Acta 69 (13), 3309–3320.CrossRefGoogle Scholar
Tsimpanogiannis, I. N. & Yortsos, Y. C. 2002 Model for the gas evolution in a porous medium driven by solute diffusion. AIChE J. 48 (11), 2690–2710.CrossRefGoogle Scholar
Van Genuchten, M. 1980 A closed-form equation for predicting the hydraulic conductivity of unsaturated soils1. Soil Sci. Soc. Amer. J. 44, 892–898.CrossRefGoogle Scholar
Verma, A. K., Pruess, K., Tsang, C. F. & Witherspoon, P. A. 1985 A study of two-phase concurrent flow of steam and water in an unconsolidated porous medium. In Proceedings of the 23rd National Heat Transfer Conference, Denver, CO, pp. 135–143. ASME.Google Scholar
Voorhees, P. W. 1985 The theory of Ostwald ripening. J. Stat. Phys. 38 (1/2), 231–252.CrossRefGoogle Scholar
Voorhees, P. W. 1992 Ostwald ripening of two-phase mixtures. Annu. Rev. Mater. Sci. 22 (1), 197–215.CrossRefGoogle Scholar
Xu, K., Bonnecaze, R. & Balhoff, M. 2017 Egalitarianism among bubbles in porous media: an Ostwald ripening derived anticoarsening phenomenon. Phys. Rev. Lett. 119, 264502.CrossRefGoogle ScholarPubMed