Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T11:08:27.380Z Has data issue: false hasContentIssue false
  • English
  • Français

Wettability and Lenormand's diagram

Published online by Cambridge University Press:  02 August 2021

Bauyrzhan K. Primkulov Open the ORCID record for Bauyrzhan K. Primkulov [Opens in a new window] ,
Amir A. Pahlavan Open the ORCID record for Amir A. Pahlavan [Opens in a new window] ,
Xiaojing Fu Open the ORCID record for Xiaojing Fu [Opens in a new window] ,
Benzhong Zhao Open the ORCID record for Benzhong Zhao [Opens in a new window] ,
Christopher W. MacMinn Open the ORCID record for Christopher W. MacMinn [Opens in a new window]  and
Ruben Juanes Open the ORCID record for Ruben Juanes [Opens in a new window]
Bauyrzhan K. Primkulov
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Amir A. Pahlavan
Affiliation:
Department of Mechanical Engineering and Materials Science, Yale University, New Haven, CT 06511, USA
Xiaojing Fu
Affiliation:
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Benzhong Zhao
Affiliation:
Department of Civil Engineering, McMaster University, Hamilton, ON, L8S 4L7, Canada
Christopher W. MacMinn
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
Ruben Juanes*
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: juanes@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Fluid–fluid displacement in porous media has been viewed through the lens of Lenormand's phase diagram since the late 1980s. This diagram suggests that the character of the flow is controlled by two dimensionless parameters: the capillary number and the viscosity ratio. It is by now well known, however, that the wettability of the system plays a key role in determining the pore-scale displacement mechanisms and macroscopic invasion patterns. Here, we endow Lenormand's diagram with the impact of wettability using dynamic and quasi-static pore-network models. By using the fractal dimension and the ratio of characteristic viscous and capillary pressures we delineate the five principal displacement regimes within the extended phase diagram: stable displacement, viscous fingering, invasion percolation, cooperative pore filling and corner flow. We discuss the results in the context of pattern formation, displacement-front dynamics, pore-scale disorder and displacement efficiency.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A34
Check for updates
Copyright
© The Author(s), 2021. 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

REFERENCES

Aker, E., Måløy, K.J., Hansen, A. & Batrouni, G. 1998 A two-dimensional network simulator for two-phase flow in porous media. Transp. Porous Media 32 (2), 163–186.CrossRefGoogle Scholar
Al-Gharbi, M.S. & Blunt, M.J. 2005 Dynamic network modeling of two-phase drainage in porous media. Phys. Rev. E 71 (1), 016308.CrossRefGoogle ScholarPubMed
Ben Amar, M. 1991 a Exact self-similar shapes in viscous fingering. Phys. Rev. A 43 (10), 5724–5727.CrossRefGoogle ScholarPubMed
Ben Amar, M. 1991 b Viscous fingering in a wedge. Phys. Rev. A 44 (6), 3673–3685.CrossRefGoogle Scholar
Blunt, M.J. 2001 Flow in porous media – pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6 (3), 197–207.CrossRefGoogle Scholar
Blunt, M.J. & Scher, H. 1995 Pore-level modeling of wetting. Phys. Rev. E 52 (6), 6387–6403.CrossRefGoogle ScholarPubMed
Borgman, O., Darwent, T., Segre, E., Goehring, L. & Holtzman, R. 2019 Immiscible fluid displacement in porous media with spatially correlated particle sizes. Adv. Water Resour. 128, 158–167.CrossRefGoogle Scholar
Bretherton, F.P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Celia, M.A., Reeves, P.C. & Ferrand, L.A. 1995 Recent advances in pore scale models for multiphase flow in porous media. Rev. Geophys. 33 (S2), 1049–1057.CrossRefGoogle Scholar
Chandler, R., Koplik, J., Lerman, K. & Willemsen, J.F. 1982 Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249–267.CrossRefGoogle Scholar
Chen, J.D. 1987 Radial viscous fingering patterns in Hele-Shaw cells. Exp. Fluids 5 (6), 363–371.CrossRefGoogle Scholar
Chen, J.D. 1989 Growth of radial viscous fingers in a Hele-Shaw cell. J. Fluid Mech. 201, 223–242.CrossRefGoogle Scholar
Chen, J.D. & Wilkinson, D. 1985 Pore-scale viscous fingering in porous media. Phys. Rev. Lett. 55 (18), 1892–1895.CrossRefGoogle ScholarPubMed
Chuoke, R.L., van Meurs, P. & van der Poel, C. 1959 The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media. Petrol. Trans. AIME 216, 188–194.CrossRefGoogle Scholar
Cieplak, M., Maritan, A. & Banavar, J.R. 1996 Invasion percolation and Eden growth: geometry and universality. Phys. Rev. Lett. 76 (20), 3754–3757.CrossRefGoogle ScholarPubMed
Cieplak, M. & Robbins, M.O. 1988 Dynamical transition in quasistatic fluid invasion in porous media. Phys. Rev. Lett. 60 (20), 2042–2045.CrossRefGoogle ScholarPubMed
Cieplak, M. & Robbins, M.O. 1990 Influence of contact angle on quasistatic fluid invasion of porous media. Phys. Rev. B 41 (16), 11508–11521.CrossRefGoogle ScholarPubMed
Constantinides, G.N. & Payatakes, A.C. 2000 Effects of precursor wetting films in immiscible displacement through porous media. Transp. Porous Media 38 (3), 291–317.CrossRefGoogle Scholar
Conti, M. & Marconi, U.M.B. 2010 Diffusion limited propagation of burning fronts. In WIT Transactions on Ecology and the Environment (ed. G. Perona & C.A. Brebbia), vol. 137, pp. 37–45. WIT.CrossRefGoogle Scholar
Cox, R.G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Datta, S.S., Ramakrishnan, T.S. & Weitz, D.A. 2014 Mobilization of a trapped non-wetting fluid from a three-dimensional porous medium. Phys. Fluids 26 (2), 22002.CrossRefGoogle Scholar
Fatt, I. 1956 The network model of porous media. Petrol. Trans. AIME 207, 144–177.CrossRefGoogle Scholar
Feder, J., Hinrichsen, E.L., Måløy, K.J. & Jøssang, T. 1989 Geometrical crossover and self-similarity of DLA and viscous fingering clusters. Physica D 38 (1–3), 104–111.CrossRefGoogle Scholar
Ferer, M., Ji, C., Bromhal, G.S., Cook, J., Ahmadi, G. & Smith, D.H. 2004 Crossover from capillary fingering to viscous fingering for immiscible unstable flow: experiment and modeling. Phys. Rev. E 70 (1), 016303.CrossRefGoogle ScholarPubMed
Furuberg, L., Måløy, K.J. & Feder, J. 1996 Intermittent behavior in slow drainage. Phys. Rev. E 53 (1), 966–977.CrossRefGoogle ScholarPubMed
Gjennestad, M.A., Vassvik, M., Kjelstrup, S. & Hansen, A. 2018 Stable and efficient time integration of a dynamic pore network model for two-phase flow in porous media. Front. Phys. 6, 56.CrossRefGoogle Scholar
Haines, W.B. 1930 Studies in the physical properties of soil. V. The hysteresis effect in capillary properties, and the modes of moisture distribution associated therewith. J. Agr. Sci. 20 (1), 97–116.CrossRefGoogle Scholar
Hill, S. 1952 Channeling in packed columns. Chem. Engng Sci. 1 (6), 247–253.CrossRefGoogle Scholar
Hinrichsen, E.L., Måløy, K.J., Feder, J. & Jøssang, T. 1989 Self-similarity and structure of DLA and viscous fingering clusters. J. Phys. A: Math. Gen. 22 (7), 271–277.CrossRefGoogle Scholar
Hoffman, R.L. 1975 A study of the advancing interface. I. Interface shape in liquid–gas systems. J. Colloid Interface Sci. 50 (2), 228–241.CrossRefGoogle Scholar
Holtzman, R. 2016 Effects of pore-scale disorder on fluid displacement in partially-wettable porous media. Sci. Rep. 6 (1), 36221.CrossRefGoogle ScholarPubMed
Holtzman, R. & Juanes, R. 2010 Crossover from fingering to fracturing in deformable disordered media. Phys. Rev. E 82 (4), 046305.CrossRefGoogle ScholarPubMed
Holtzman, R. & Segre, E. 2015 Wettability stabilizes fluid invasion into porous media via nonlocal, cooperative pore filling. Phys. Rev. Lett. 115 (16), 164501.CrossRefGoogle ScholarPubMed
Holtzman, R., Szulczewski, M.L. & Juanes, R. 2012 Capillary fracturing in granular media. Phys. Rev. Lett. 108 (26), 264504.CrossRefGoogle ScholarPubMed
Homsy, G.M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid. Mech. 19 (1), 271–311.CrossRefGoogle Scholar
Hu, R., Lan, T., Wei, G.J. & Chen, Y.F. 2019 Phase diagram of quasi-static immiscible displacement in disordered porous media. J. Fluid Mech. 875, 448–475.CrossRefGoogle Scholar
Hu, R., Wan, J., Yang, Z., Chen, Y.-F. & Tokunaga, T. 2018 Wettability and flow rate impacts on immiscible displacement: a theoretical model. Geophys. Res. Lett. 45 (7), 3077–3086.CrossRefGoogle Scholar
Jain, A.K. & Juanes, R. 2009 Preferential mode of gas invasion in sediments: grain-scale mechanistic model of coupled multiphase fluid flow and sediment mechanics. J. Geophys. Res. 114 (B8), B08101.Google Scholar
Joekar-Niasar, V. & Hassanizadeh, S.M. 2012 Analysis of fundamentals of two-phase flow in porous media using dynamic pore-network models: a review. Crit. Rev. Environ. Sci. Technol. 42 (18), 1895–1976.CrossRefGoogle Scholar
Joekar-Niasar, V., Hassanizadeh, S.M. & Dahle, H.K. 2010 Non-equilibrium effects in capillarity and interfacial area in two-phase flow: dynamic pore-network modelling. J. Fluid Mech. 655, 38–71.CrossRefGoogle Scholar
Jung, M., Brinkmann, M., Seemann, R., Hiller, T., Sanchez de La Lama, M. & Herminghaus, S. 2016 Wettability controls slow immiscible displacement through local interfacial instabilities. Phys. Rev. Fluids 1 (7), 074202.CrossRefGoogle Scholar
Kenkel, N.C. & Walker, D.J. 1996 Fractals in the biological sciences. Coenoses 11, 77–100.Google Scholar
Lan, T., Hu, R., Yang, Z., Wu, D.-S. & Chen, Y.-F. 2020 Transitions of fluid invasion patterns in porous media. Geophys. Res. Lett. 47 (20), e2020GL089682.CrossRefGoogle Scholar
Landau, L. & Levich, B. 1988 Dragging of a liquid by a moving plate. In Dynamics of Curved Fronts (ed. P. Pelcé), pp. 141–153. Elsevier.CrossRefGoogle Scholar
Lee, S., Lee, J., Le Mestre, R., Xu, F. & MacMinn, C.W. 2020 Migration, trapping, and venting of gas in a soft granular material. Phys. Rev. Fluids 5 (8), 084307.CrossRefGoogle Scholar
Léger, A., Weber, L. & Mortensen, A. 2015 Influence of the wetting angle on capillary forces in pressure infiltration. Acta Mater. 91, 57–69.CrossRefGoogle Scholar
Lenormand, R. 1990 Liquids in porous media. J. Phys.: Condens. Matter 2 (S), SA79–SA88.Google Scholar
Lenormand, R., Touboul, E. & Zarcone, C. 1988 Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165–187.CrossRefGoogle Scholar
Lenormand, R. & Zarcone, C. 1985 Invasion percolation in an etched network: measurement of a fractal dimension. Phys. Rev. Lett. 54 (20), 2226–2229.CrossRefGoogle Scholar
Levaché, B., Azioune, A., Bourrel, M., Studer, V. & Bartolo, D. 2012 Engineering the surface properties of microfluidic stickers. Lab on a Chip 12 (17), 3028–3031.CrossRefGoogle ScholarPubMed
Levaché, B. & Bartolo, D. 2014 Revisiting the Saffman–Taylor experiment: imbibition patterns and liquid-entrainment transitions. Phys. Rev. Lett. 113 (4), 044501.CrossRefGoogle ScholarPubMed
Li, S., Lowengrub, J.S., Fontana, J. & Palffy-Muhoray, P. 2009 Control of viscous fingering patterns in a radial Hele-Shaw cell. Phys. Rev. Lett. 102 (17), 174501.CrossRefGoogle Scholar
Løvoll, G., Méheust, Y., Toussaint, R., Schmittbuhl, J. & Måløy, K.J. 2004 Growth activity during fingering in a porous Hele-Shaw cell. Phys. Rev. E 70 (2), 026301.CrossRefGoogle Scholar
Måløy, K.J., Feder, J. & Jøssang, T. 1985 Viscous fingering fractals in porous media. Phys. Rev. Lett. 55 (24), 2688–2691.CrossRefGoogle Scholar
Måløy, K.J., Furuberg, L., Feder, J. & Jøssang, T. 1992 Dynamics of slow drainage in porous media. Phys. Rev. Lett. 68 (14), 2161–2164.CrossRefGoogle Scholar
Meakin, P., Tolman, S. & Blumen, A. 1989 Diffusion-limited aggregation. Proc. R. Soc. Lond. A 423 (1864), 133–148.Google Scholar
Meng, Y., Primkulov, B.K., Yang, Z., Kwok, C.Y. & Juanes, R. 2020 Jamming transition and emergence of fracturing in wet granular media. Phys. Rev. Res. 2 (2), 022012.CrossRefGoogle Scholar
Møller, P.C.F. & Bonn, D. 2007 The shear modulus of wet granular matter. Europhys. Lett. 80 (3), 38002.CrossRefGoogle Scholar
Niemeyer, L., Pietronero, L. & Wiesmann, H.J. 1984 Fractal dimension of dielectric breakdown. Phys. Rev. Lett. 52 (12), 1033–1036.CrossRefGoogle Scholar
Odier, C., Levaché, B., Santanach-Carreras, E. & Bartolo, D. 2017 Forced imbibition in porous media: a fourfold scenario. Phys. Rev. Lett. 119 (20), 208005.CrossRefGoogle ScholarPubMed
Øren, P.E., Bakke, S. & Arntzen, O.J. 1998 Extending predictive capabilities to network models. SPE J. 3 (4), 324–336.CrossRefGoogle Scholar
Park, C.W. & Homsy, G.M. 1984 Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139, 291–308.CrossRefGoogle Scholar
Paterson, L. 1984 Diffusion-limited aggregation and two-fluid displacements in porous media. Phys. Rev. Lett. 52 (18), 1621–1624.CrossRefGoogle Scholar
Patmonoaji, A., Muharrik, M., Hu, Y., Zhang, C. & Suekane, T. 2020 Three-dimensional fingering structures in immiscible flow at the crossover from viscous to capillary fingering. Intl J. Multiphase Flow 122, 103147.CrossRefGoogle Scholar
Patzek, T.W. 2001 Verification of a complete pore network simulator of drainage and imbibition. SPE J. 6 (2), 144–156.CrossRefGoogle Scholar
Primkulov, B.K., Pahlavan, A.A., Fu, X., Zhao, B., MacMinn, C.W. & Juanes, R. 2019 Signatures of fluid-fluid displacement in porous media: wettability, patterns and pressures. J. Fluid Mech. 875, R4.CrossRefGoogle Scholar
Primkulov, B.K., Talman, S., Khaleghi, K., Rangriz Shokri, A., Chalaturnyk, R., Zhao, B., MacMinn, C.W. & Juanes, R. 2018 Quasistatic fluid-fluid displacement in porous media: invasion-percolation through a wetting transition. Phys. Rev. Fluids 3 (10), 104001.CrossRefGoogle Scholar
Richefeu, V., El Youssoufi, M.S. & Radjaï, F. 2006 Shear strength properties of wet granular materials. Phys. Rev. E 73 (5), 051304.CrossRefGoogle ScholarPubMed
Saffman, P.G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Sandnes, B., Flekkøy, E.G., Knudsen, H.A., Måløy, K.J. & See, H. 2011 Patterns and flow in frictional fluid dynamics. Nat. Commun. 2 (1), 288.CrossRefGoogle ScholarPubMed
Sheppard, A.P., Knackstedt, M.A., Pinczewski, W.V. & Sahimi, M. 1999 Invasion percolation: new algorithms and universality classes. J. Phys. A: Math. Gen. 32 (49), 521–529.CrossRefGoogle Scholar
Singh, K., Scholl, H., Brinkmann, M., Di Michiel, M., Scheel, M., Herminghaus, S. & Seemann, R. 2017 The role of local instabilities in fluid invasion into permeable media. Sci. Rep. 7 (1), 444.CrossRefGoogle ScholarPubMed
Stokes, J.P., Weitz, D.A., Gollub, J.P., Dougherty, A., Robbins, M.O., Chaikin, P.M. & Lindsay, H.M. 1986 Interfacial stability of immiscible displacement in a porous medium. Phys. Rev. Lett. 57 (14), 1718–1721.CrossRefGoogle Scholar
Strang, G. 2007 Computational Science and Engineering. Wellesley-Cambridge Press.Google Scholar
Trojer, M., Szulczewski, M.L. & Juanes, R. 2015 Stabilizing fluid-fluid displacements in porous media through wettability alteration. Phys. Rev. Appl. 3 (5), 054008.CrossRefGoogle Scholar
Tzimas, G.C., Matsuura, T., Avraam, D.G., Van Der Brugghen, W., Constantinides, G.N. & Payatakes, A.C. 1997 The combined effect of the viscosity ratio and the wettability during forced imbibition through nonplanar porous media. J. Colloid Interface Sci. 189 (1), 27–36.CrossRefGoogle Scholar
Valvatne, P.H. & Blunt, M.J. 2004 Predictive pore-scale modeling of two-phase flow in mixed wet media. Water Resour. Res. 40 (7), W07406.CrossRefGoogle Scholar
Van Meurs, P. 1957 The use of transparent three-dimensional models for studying the mechanism of flow processes in oil reservoirs. Trans. AIME 210 (01), 295–301.CrossRefGoogle Scholar
Vizika, O., Avraam, D.G. & Payatakes, A.C. 1994 On the role of the viscosity ratio during low-capillary-number forced imbibition in porous media. J. Colloid Interface Sci. 165 (2), 386–401.CrossRefGoogle Scholar
Voinov, O.V. 1977 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.CrossRefGoogle Scholar
Wilkinson, D. & Willemsen, J.F. 1983 Invasion percolation: a new form of percolation theory. J. Phys. A: Math. Gen. 16 (14), 3365–3376.CrossRefGoogle Scholar
Zhao, B., MacMinn, C.W. & Juanes, R. 2016 Wettability control on multiphase flow in patterned microfluidics. Proc. Natl Acad. Sci. USA 113 (37), 10251–10256.CrossRefGoogle ScholarPubMed
Zhao, B., et al. , 2019 Comprehensive comparison of pore-scale models for multiphase flow in porous media. Proc. Natl Acad. Sci. USA 116 (28), 13799–13806.CrossRefGoogle ScholarPubMed