Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-26T17:52:38.241Z Has data issue: false hasContentIssue false

Non-equilibrium effects in capillarity and interfacial area in two-phase flow: dynamic pore-network modelling

Published online by Cambridge University Press:  05 July 2010

V. JOEKAR-NIASAR*
Affiliation:
Department of Earth Sciences, Utrecht University, PO Box 80021, 3508 TA Utrecht, The Netherlands
S. M. HASSANIZADEH
Affiliation:
Department of Earth Sciences, Utrecht University, PO Box 80021, 3508 TA Utrecht, The Netherlands
H. K. DAHLE
Affiliation:
Department of Mathematics, University of Bergen, Johannes Brunsgate 12, N-5007 Bergen, Norway
*
Email address for correspondence: joekar@geo.uu.nl

Abstract

Current macroscopic theories of two-phase flow in porous media are based on the extended Darcy's law and an algebraic relationship between capillary pressure and saturation. Both of these equations have been challenged in recent years, primarily based on theoretical works using a thermodynamic approach, which have led to new governing equations for two-phase flow in porous media. In these equations, new terms appear related to the fluid–fluid interfacial area and non-equilibrium capillarity effects. Although there has been a growing number of experimental works aimed at investigating the new equations, a full study of their significance has been difficult as some quantities are hard to measure and experiments are costly and time-consuming. In this regard, pore-scale computational tools can play a valuable role. In this paper, we develop a new dynamic pore-network simulator for two-phase flow in porous media, called DYPOSIT. Using this tool, we investigate macroscopic relationships among average capillary pressure, average phase pressures, saturation and specific interfacial area. We provide evidence that at macroscale, average capillary pressure–saturation–interfacial area points fall on a single surface regardless of flow conditions and fluid properties. We demonstrate that the traditional capillary pressure–saturation relationship is not valid under dynamic conditions, as predicted by the theory. Instead, one has to employ the non-equilibrium capillary theory, according to which the fluids pressure difference is a function of the time rate of saturation change. We study the behaviour of non-equilibrium capillarity coefficient, specific interfacial area, and its production rate versus saturation and viscosity ratio.

A major feature of our pore-network model is a new computational algorithm, which considers capillary diffusion. Pressure field is calculated for each fluid separately, and saturation is computed in a semi-implicit way. This provides more numerical stability, compared with previous models, especially for unfavourable viscosity ratios and small capillary number values.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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. K., Maloy, J., Hansen, A. & Batrouni, G. G. 1998 a A two-dimensional network simulator for two-phase flow in porous media. Transport Porous Media 32, 163186.CrossRefGoogle Scholar
Aker, E. K., Maloy, K. J. & Hansen, A. 1998 b Simulating temporal evolution of pressure in two-phase flow in porous media. Phys. Rev. E 58, 22172226.CrossRefGoogle Scholar
Al-Gharbi, M. S. & Blunt, M. J. 2005 Dynamic network modelling of two-phase drainage in porous media. Phys. Rev. E 71, 016308.CrossRefGoogle ScholarPubMed
Avraam, D. G. & Payatakes, A. C. 1995 a Generalized relative permeability coefficients during steady-state, two-phase flow in porous media and correlation with the flow mechanisms. Transport Porous Media 20, 135168.CrossRefGoogle Scholar
Azzam, M. I. S. & Dullien, F. A. L. 1977 Flow in tubes with periodic step changes in diameter: a numerical solution. Chem. Engng Sci. 32, 14451455.CrossRefGoogle Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Blunt, M., Jackson, M. D., Piri, M. & Valvatne, P. H. 2002 Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25, 10691089.CrossRefGoogle Scholar
Blunt, M. & King, P. 1990 Macroscopic parameters from simulations of the pore scale flow. Phys. Rev. A 42, 47804788.CrossRefGoogle ScholarPubMed
Blunt, M. & King, P. 1991 Relative permeabilities from two- and three-dimensional pore-scale network modelling. Transport Porous Media 6, 407433.CrossRefGoogle Scholar
Bravo, M. C., Araujo, M. & Lago, M. E. 2007 Pore network modelling of two-phase flow in a liquid–(disconnected) gas system. Physica A 375, 117.CrossRefGoogle Scholar
Brooks, R. H. & Corey, A. T. 1964 Hydraulic properties of porous media. Tech. Rep. Hydrol. Paper 3. Colorado State University.Google Scholar
Brusseau, M. L., Peng, S., Schnaar, G. & Constanza-Robinson, M. S. 2006 Relationships among air–water interfacial area, capillary pressure, and water saturation for a sandy porous medium. Water Resour. Res. 42, W03501.CrossRefGoogle Scholar
Brusseau, M. L., Popovicova, J. & Silva, J. A. K. 1997 Characterizing gas–water interfacial and bulk-water partitioning for gas phase transport of organic contaminants in unsaturated porous media. Environ. Sci. Technol. 31, 16451649.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 (S1), 10491058.CrossRefGoogle Scholar
Chen, D. Q., Pyrak-Nolte, L. J., Griffin, J. & Giordano, N. J. 2007 Measurement of interfacial area per volume for drainage and imbibition. Water Resour. Res. 43, W12504.CrossRefGoogle Scholar
Chen, L. & Kibbey, T. C. G. 2006 Measurement of air–water interfacial area for multiple hysteretic drainage curves in an unsaturated fine sand. Langmuir 22, 68746880.CrossRefGoogle Scholar
Cheng, J. T., Pyrak-Nolte, L. J. & Nolte, D. D. 2004 Linking pressure and saturation through interfacial area in porous media. Geophys. Res. Lett. 31, L08502.CrossRefGoogle Scholar
Constantinides, G. N. & Payatakes, A. C. 1991 A theoretical model of collision and coalescence of ganglia in porous media. J. Colloid Interface Sci. 141, 486504.CrossRefGoogle Scholar
Constantinides, G. N. & Payatakes, A. C. 1996 Network simulation of steady-state two-phase flow in consolidated porous media. AIChE J. 42, 369382.CrossRefGoogle Scholar
Costanza-Robinson, M. S. & Brusseau, M. L. 2002 Air–water interfacial areas in unsaturated soils: evaluation of interfacial domains. Water Resour. Res. 38, 13–1.CrossRefGoogle Scholar
Culligan, K. A., Wildenschild, D., Christensen, B. S. B., Gray, W., Rivers, M. L. & Tompson, A. F. B. 2004 Interfacial area measurements for unsaturated flow through a porous medium. Water Resour. Res. 40, W12413.CrossRefGoogle Scholar
Culligan, K. A., Wildenschild, D., Christensen, B. S. B., Gray, W., Rivers, M. L. & Tompson, A. F. B. 2006 Pore-scale characteristics of multiphase flow in porous media: a comparison of air–water and oil–water experiments. Adv. Water Resour. 29, 227238.CrossRefGoogle Scholar
Dahle, H. K. & Celia, M. A. 1999 A dynamic network model for two-phase immiscible flow. Comput. Geosci. 3, 122.CrossRefGoogle Scholar
Dahle, H. K., Celia, M. A. & Hassanizadeh, S. M. 2005 Bundle-of-tubes model for calculating dynamic effects in the capillary–pressure–saturation relationship. Transport Porous Media 58, 522.CrossRefGoogle Scholar
Das, D. B., Mirzaei, M. & Widdows, N. 2006 Non-uniqueness in capillary pressure–saturation–relative permeability relationships for two-phase flow in porous media: interplay between intensity and distribution of random micro-heterogeneities. Chem. Engng Sci. 61, 67866803.CrossRefGoogle Scholar
Dias, M. M. & Payatakes, A. C. 1986 a Network models for two-phase flow in porous media. Part 1. Immiscible microdisplacement of non-wetting fluids. J. Fluid Mech. 164, 305336.CrossRefGoogle Scholar
Dias, M. M. & Payatakes, A. C. 1986 b Network models for two-phase flow in porous media. Part 2. Motion of oil ganglia. J. Fluid Mech. 164, 337358.CrossRefGoogle Scholar
Fatt, I. 1956 The network model of porous media. Part I. Capillary pressure characteristics. Petroleum Trans. AIME 207, 144159.CrossRefGoogle Scholar
Fayers, F. J., Blunt, M. J. & Christie, M. A. 1990 Accurate Calibration of Empirical Viscous Fingering Models. pp. 4555. Technip.Google Scholar
Fong, K. W., Jefferson, T. H., Suyehiro, T. & Walton, L. 1993 Guide to the SLATEC Common Mathematical Library. Lawrence Livermore and Sandia National Laboratories, http://www.netlib.org/slatec/guide.Google Scholar
Gielen, T., Hassanizadeh, S. M., Leijnse, A. & Nordhaug, H. F. 2005 Dynamic effects in multiphase flow: a pore-scale network approach. In Upscaling Multiphase Flow in Porous Media (ed. Das, D. B. & Hassanizadeh, S. M.), pp. 217236. Springer.CrossRefGoogle Scholar
Hassanizadeh, S. M., Celia, M. A. & Dahle, H. K. 2002 Dynamic effects in the capillary pressure–saturation relationship and their impacts on unsaturated flow. Vadose Zone J. 1, 3857.CrossRefGoogle Scholar
Hassanizadeh, S. M. & Gray, W. G. 1990 Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13, 169186.CrossRefGoogle Scholar
Hassanizadeh, S. M. & Gray, W. G. 1993 a Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 33893405.CrossRefGoogle Scholar
Hassanizadeh, S. M. & Gray, W. G. 1993 b Toward an improved description of the physics of two-phase flow. Adv. Water Resour. 16, 5367.CrossRefGoogle Scholar
Hassanizadeh, S. M., Oung, O. & Manthey, S. 2004 Laboratory experiments and simulations on the significance of non-equilibrium effect in the capillary pressure–saturation relationship. In Unsaturated Soils: Experimental Studies. Proceedings of the International Conference: From Experimental Evidence towards Numerical Modelling of Unsaturated Soils (ed. Schanz, T.), vol. 1, pp. 314. Springer.Google Scholar
Held, R. J. & Celia, M. A. 2001 Modelling support of functional relationships between capillary pressure, saturation, interfacial area and common lines. Adv. Water Resour. 24, 325343.CrossRefGoogle Scholar
Hughes, R. G. & Blunt, M. J. 2000 Pore scale modelling of rate effect in imbibition. Transport Porous Media 40, 295322.CrossRefGoogle Scholar
Joekar-Niasar, V., Hassanizadeh, S. M. & Leijnse, A. 2008 Insights into the relationships among capillary pressure, saturation, interfacial area and relative permeability using pore-network modelling. Transport Porous Media 74 (2), 201219.CrossRefGoogle Scholar
Joekar-Niasar, V., Hassanizadeh, S. M., Pyrak-Nolte, L. J. & Berentsen, C. 2009 Simulating drainage and imbibition experiments in a high-porosity micromodel using an unstructured pore network model. Water Resour. Res. 45, W02430.CrossRefGoogle Scholar
Joekar-Niasar, V., Prodanović, M., Wildenschild, D. & Hassanizadeh, S. M. 2010 Network model investigation of interfacial area, capillary pressure and saturation relationships in granular porous media. Water Resour. Res. doi:10.1029/2009WR008585.CrossRefGoogle Scholar
Kalaydjian, F. & Marle, C. M. 1987 Thermodynamic aspects of multiphase flow in porous media. Collection Colloques et Séminaires – Institut Français du Pétrole 45, 513531.Google Scholar
King, P. R. 1987 The fractal nature of viscous fingering in porous media. J. Phys. A 20, L529L534.CrossRefGoogle Scholar
Knudsen, H. A., Aker, E. & Hansen, A. 2002 Bulk flow regimes and fractional flow in 2d porous media by numerical simulations. Transport Porous Media 47, 99121.CrossRefGoogle Scholar
Knudsen, H. A. & Hansen, A. 2002 Relation between pressure and fractional flow in two-phase flow in porous media. Phys. Rev. E 65, 056310-1–056310-10.CrossRefGoogle ScholarPubMed
Koplik, J. & Lasseter, T. J. 1985 Two-phase flow in random network models of porous media. Soc. Petrol. Engng J. 25, 89110.CrossRefGoogle Scholar
Koval, E. J. 1963 A method for predicting the performance of unstable miscible displacements in heterogenous media. Trans. AIME 219, 145150.Google Scholar
Ma, S., Mason, G. & Morrow, N. R. 1996 Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids Surfaces 117, 273291.CrossRefGoogle Scholar
Manthey, S., Hassanizadeh, S. M. & Helmig, R. 2005 Macro-scale dynamic effects in homogeneous and heterogeneous porous media. Transport Porous Media 58, 121145.CrossRefGoogle Scholar
Mason, G. & Morrow, N. R. 1987 Meniscus configurations and curvatures in non-axisymmetric pores of open and closed uniform cross-section. Proc. R. Soc. Lond. A 414 (1846), 111133.Google Scholar
Mayer, R. P. & Stowe, R. A. 1965 Mercury porosimetry-breakthrough pressure for penetration between packed spheres. J. Colloid Sci. 20, 891911.CrossRefGoogle Scholar
Mirzaei, M. & Das, D. B. 2007 Dynamic effects in capillary pressure–saturations relationships for two-phase flow in 3d porous media: implications of micro-heterogeneities. Chem. Engng Sci. 62 (7), 19271947.CrossRefGoogle Scholar
Mogensen, K. & Stenby, E. H. 1998 A dynamic two-phase pore-scale model for imbibition. Transport Porous Media 32, 299327.CrossRefGoogle Scholar
Nguyen, V. H., Sheppard, A. P., Knackstedt, M. A. & Pinczewski, W. 2006 The effect of displacement rate on imbibition relative permeability and residual saturation. J. Petrol. Sci. Engng 52, 5470.CrossRefGoogle Scholar
Niessner, J. & Hassanizadeh, S. M. 2008 A model for two-phase flow in porous media including fluid–fluid interfacial area. Water Resour. Res. 44, W08439.CrossRefGoogle Scholar
Nordbotten, J. M., Celia, M. A., Dahle, H. K. & Hassanizadeh, S. M. 2007 Interpretation of macroscale variables in Darcy's law. Water Resour. Res. 43 (8), W08430.CrossRefGoogle Scholar
Nordbotten, J. M., Celia, M. A., Dahle, H. K. & Hassanizadeh, S. M. 2008 On the definition of macroscale pressure for multiphase flow in porous media. Water Resour. Res. 44 (6), W06S02.CrossRefGoogle Scholar
Nordhaug, H. F., Celia, M. & Dahle, H. K. 2003 A pore network model for calculation of interfacial velocities. Adv. Water Resour. 26, 10611074.CrossRefGoogle Scholar
O'Carroll, D. M., Phelan, T. J. & Abriola, L. M. 2005 Exploring dynamic effects in capillary pressure in multistep outflow experiments. Water Resour. Res. 41, W11419.CrossRefGoogle Scholar
Oung, O., Hassanizadeh, S. M. & Bezuijen, A. 2005 Two-phase flow experiments in a geocentrifuge and the significance of dynamic capillary pressure effect. J. Porous Media 8, 247257.CrossRefGoogle Scholar
Pan, C., Hilpert, M. & Miller, C. T. 2004 Lattice-Boltzmann simulation of two-phase flow in porous media. Water Resour. Res. 40, W01501.CrossRefGoogle Scholar
Payatakes, A. C. 1982 Dynamics of oil ganglia during immiscible displacement in water-wet porous media. Annu. Rev. Fluid Mech. 14, 365393.CrossRefGoogle Scholar
Pereira, G. G., Pinczewski, W. V., Chan, D. Y. C., Paterson, L. & Øren, P. E. 1996 Pore-scale network model for drainage-dominated three-phase flow in porous media. Transport Porous Media 24, 167201.CrossRefGoogle Scholar
Porter, M. L., Schaap, M. G. & Wildenschild, D. 2009 Lattice-Boltzmann simulations of the capillary pressure–saturation–interfacial area relationship for porous media. Adv. Water Resour. 32 (11), 16321640.CrossRefGoogle Scholar
Prat, M. 2002 Recent advances in pore-scale models for drying of porous media. Chem. Engng J. 86, 153164.CrossRefGoogle Scholar
Princen, H. M. J. 1969 a Capillary phenomena in assemblies of parallel cylinders. Part I. Capillary rise between two cylinders. Colloid Interface Sci. 30, 6975.CrossRefGoogle Scholar
Princen, H. M. J. 1969 b Capillary phenomena in assemblies of parallel cylinders. Part II. Capillary rise in systems with more than two cylinders. Colloid Interface Sci. 30, 359371.CrossRefGoogle Scholar
Princen, H. M. J. 1970 Capillary phenomena in assemblies of parallel cylinders. Part III. Liquid columns between horizontal parallel cylinders. Colloid Interface Sci. 34, 171184.CrossRefGoogle Scholar
Pyrak-Nolte, L. J. 2007 Measurement of interfacial area per volume for drainage and imbibition. http://www.physics.purdue.edu/rockphys/DataImages/index.php.Google Scholar
Ransohoff, T. C. & Radke, C. J. 1988 Laminar flow of a wetting liquid along the corners of a predominantly gas-occupied noncircular pore. J. Colloid Interface Sci. 121, 392401.CrossRefGoogle Scholar
Reeves, P. C. & Celia, M. A. 1996 A functional relationship between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model. Water Resour. Res. 32, 23452358.CrossRefGoogle Scholar
Singh, M. & Mohanty, K. K. 2003 Dynamic modelling of drainage through three-dimensional porous materials. Chem. Engng Sci. 58, 118.CrossRefGoogle Scholar
Stauffer, F. 1978 Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media. In IAHR Symposium on Scale Effects in Porous Media, Thessaloniki, Greece, pp. 3.353.52.Google Scholar
Thompson, K. E. 2002 Pore-scale modelling of fluid transport in disordered fibrous materials. AIChE J. 48, 13691389.CrossRefGoogle Scholar
Valvanides, M. S., Constantinides, G. N. & Payatakes, A. C. 1998 Mechanistic model of steady-state two-phase flow in porous media based on ganglion dynamics. Transport Porous Media 30, 267299.CrossRefGoogle Scholar
Valvatne, P. H. & Blunt, M. J. 2004 Predictive pore-scale modelling of two-phase flow in mixed wet media. Water Resour. Res. 40, W07406.CrossRefGoogle Scholar
Van der Marck, S. C., Matsuura, T. & Glas, J. 1997 Viscous and capillary pressures during drainage: network simulations and experiments. Phys. Rev. E 56, 56755687.CrossRefGoogle Scholar
Vidales, A. M., Riccardo, J. L. & Zgrabli, G. 1998 Pore-level modelling of wetting on correlated porous media. J. Phys. D, Appl. Phys. 31, 28612868.CrossRefGoogle Scholar
Washburn, E. W. 1921 The dynamics of capillary flow. Phys. Rev. 17, 273283.CrossRefGoogle Scholar
Whitaker, S. 1977 Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying. Adv. Heat Transfer 13, 119200.CrossRefGoogle Scholar
Zhou, D., Blunt, M. J. & Orr, F. M. 1997 Hydrocarbon drainage along corners of noncircular capillaries. J. Colloid Interface Sci. 187, 1121.CrossRefGoogle ScholarPubMed