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Elements of a Systematic Procedure for the Derivation of Macroscale Conservation Equations for Multiphase Flow in Porous Media

  • Conference paper
Kinetic and Continuum Theories of Granular and Porous Media

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 400))

Abstract

The framework for development of conservation equations and constitutive relations describing the flow of two fluids in a porous medium from a macroscale perspective is provided. Theorems are employed that allow global integral equations to be localized at the porous medium scale. This is a more general approach than the traditional averaging of microscale point equations. Conservation equations for mass, momentum, energy and entropy for phases, interfaces, and common lines are obtained to provide a full description of two-phase flow in porous media. The entropy inequality is developed for the three-phase system. Because of interactions among the system components (i.e. phases, interfaces, and common lines), a single entropy inequality for the entire system, rather than inequalities for each phase, interface, and common line is employed in developing the constitutive theory. The macroscale internal energy is postulated to depend thermodynamically on the extensive properties of the system, and is then decomposed to provide energy dependences for each of the system components. These decomposed forms, along with the entropy inequality, lead to the conservation equations with constitutive approximations included. Final closure of the system of equations requires a variational analysis of the mechanical behavior of the subscale geometric structure. The resulting equations show that capillary pressure is a function of interphase area per unit volume as well as saturation. The equations currently used to model multiphase flow are shown to be very restricted forms of the more general equations.

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References

  • Anderson, T. B., and R. Jackson, A fluid mechanical description of fluidized beds, Industrial and Engineering Chemistry Fundamentals, 6, 527–539 (1967).

    Article  Google Scholar 

  • Anderson, W. G., Wettability literature survey Part 4. Effects of wettability on capillary pressure, Journal of Petroleum Technology, 39, 1283–1300 (1987a).

    Google Scholar 

  • Anderson, W. G., Wettability literature survey Part 5. Effects of wettability on relative permeability, Journal of Petroleum Technology, 39, 1453–1468 (1987b).

    Google Scholar 

  • Avraam, D. G., and A. C. Payatakes, Generalized relative permeability coefficients during steady-state two-phase flow porous media, and correlation with the flow mechanisms, Transport in Porous Media, 20, 135–168 (1995).

    Article  Google Scholar 

  • Bachmat, Y., Spatial macroscopization of processes in heterogeneous systems, Israel Journal of Technology, 10 (5), 391–403 (1972).

    MathSciNet  Google Scholar 

  • Bailyn, M., A Survey of Thermodynamics, AIP Press, New York (1994).

    Google Scholar 

  • Bennethum, L. S., Multiscale, hybrid mixture theory for swelling systems with interfaces, Center for Applied Mathematics Technical Report #259, Purdue University (1994).

    Google Scholar 

  • Boruvka, L., and A. W. Neumann, Generalization of the classical theory of capillarity, Journal of Chemical Physics,66(12), 5464–5476 (June, 1977).

    Google Scholar 

  • Boruvka, L., Y. Rotenberg, and A. W. Neumann, Free energy formulation of the theory of capillarity, Langmuir, 1 (1), 40–44 (1985).

    Article  Google Scholar 

  • Brooks, R. H. and A. T. Corey, Hydraulic properties of porous media, Hydrology Paper 3, Colorado State University, Fort Collins (1964).

    Google Scholar 

  • Callen, H. B., Thermodynamics and an Introduction to Thermostatistics, 2nd Edition, John Wiley and Sons, New York (1985).

    MATH  Google Scholar 

  • Celia, M. A., P. C. Reeves, and L. A. Ferrand, Recent advances in pore scale models for multiphase flow in porous media, Reviews of Geophysics, Supplement, 1049–1057 (1995).

    Google Scholar 

  • Coleman, B. D., and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Archive for Rational Mechanics and Analysis, 13, 168–178 (1963).

    MathSciNet  Google Scholar 

  • Derjaguin, B. V., N. V. Churaev, and V. M. Muller, Surface Forces, Consultants Bureau, New York (1987).

    Book  Google Scholar 

  • Edlefsen, N. E., and A. B. C. Anderson, Thermodynamics of soil moisture, Hilgardia, 15, 312–398 (1943).

    Google Scholar 

  • Eringen, A. C., Mechanics of Continua, 2nd Edition, Krieger Publishing, Huntington, NY (1980).

    Google Scholar 

  • Gaydos, J., Y. Rotenberg, L. Boruvka, P. Chen, and A. W. Neumann, The generalized theory of capillarity, in Applied Surface Thermodynamics. (edited by A. W. Neumann and J. K. Spelt), Surfactant Science Series, 63, Marcel Dekk:.r, New York, 1–51 (1996).

    Google Scholar 

  • Gray, W G., Thermodynamics and constitutive theory for multiphase porous-media flow considering internal geometric constraints, Advances in Water Resources (in press, 1998a)

    Google Scholar 

  • Gray, W. G., Macroscale equilibrium conditions for two-phase flow in porous media, International Journal of Multiphase Flow (in review, 1998b ).

    Google Scholar 

  • Gray, W. G., and S. M. Hassanizadeh, Averaging theorems and averaged equations for transport of interface properties in multiphase systems, International Journal of Multiphase Flow, 15 (1), 81–95 (1989).

    Article  MATH  Google Scholar 

  • Gray, W. G., and S. M. Hassanizadeh, Unsaturated flow theory including interfacial phenomena, Water Resources Research, 27 (8) 1855–1863 (1991).

    Article  Google Scholar 

  • Gray, W. G., and S. M. Hassanizadeh, Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines, and common points, Advances in Water Resources, 21, 261–281 (1998).

    Article  Google Scholar 

  • Gray, W. G., and P. C. Y. Lee, On the theorems for local volume averaging of multiphase systems, International Journal of Multiphase Flow, 3, 333–340 (1977).

    Article  MATH  Google Scholar 

  • Gray, W. G., A. Leijnse, R. L. Kolar, C. A. Blain, Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press, Boca Raton, FL (1993).

    MATH  Google Scholar 

  • Hassanizadeh, S. M., Macroscopic Description of Multi-Phase Systems: Thermodynamic Theory of Flow in Porous Media, Ph.D. dissertation, Princeton University, Department of Civil Engineering (1979).

    Google Scholar 

  • Hassanizadeh, M., and W. G. Gray, General conservation equations for multi-phase systems, I. Averaging procedure, Advances in Water Resources, 2 (3), 131–144 (1979a).

    Article  Google Scholar 

  • Hassanizadeh, M., and W. G. Gray, General conservation equations for multi-phase systems, II. Mass, momenta, energy, and entropy equations, Advances in Water Resources, 2 (4), 191–203 (1979b).

    Article  Google Scholar 

  • Hassanizadeh, M., and W. G. Gray, General conservation equations for multi-phase systems, III. Constitutive theory for porous media flow, Advances in Water Resources, 3 (1), 25–40 (1980).

    Article  Google Scholar 

  • Hassanizadeh, S. M., and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interface boundaries, Advances in Water Resources, 13 (4), 169–186 (1990).

    Article  Google Scholar 

  • Hassanizadeh, S. M., and W. G. Gray, Recent advances in theories of two-phase flow in porous media, in Fluid Transport in Porous Media (edited by R du Plessis) Advances in Fluid Mechanics Series, Computational Mechanics Publications, Southampton, 105–160 (1997).

    Google Scholar 

  • Havercamp, R. and J. Y. Parlange, Prediction of water retention curve from particle size distribution 1. Sandy soils without organic matter, Soil Science, 142, 325–339 (1983).

    Article  Google Scholar 

  • Hirasaki, G. J., Thermodynamics of thin films and three-phase contact regions, in Interfacial Phenomena in Petroleum Recovery (edited by N. Morrow), Surfactant Science Series, 36, Marcel Dekker, New York, 23–75 (1991).

    Google Scholar 

  • Kool, J. B. and J. C. Parker, Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties, Water Resources Research, 23, 105–114 (1987).

    Article  Google Scholar 

  • Lenhard, R. J., J. C. Parker and S. Mishra, On the correspondence between Brooks-Corey and van Genuchten models, Journal of Irrigation and Drainage, 115, 744–751 (1989).

    Article  Google Scholar 

  • Li, D., and A. W. Neumann, Thermodynamic status of contact angles, in Applied Surface Thermodynamics, (edited by A. W. Neumann and J. K. Spelt), Surfactant Science Series, 63, Marcel Dekker, New York, 109–168 (1996).

    Google Scholar 

  • Liu, I.-S., Method of Lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics and Analysis, 46, 131–148 (1972).

    MATH  MathSciNet  Google Scholar 

  • Miller, C. A., and P. Neogi, Interfacial Phenomena, Marcel Dekker, New York (1985).

    Google Scholar 

  • Moeckel, G. P., Thermodynamics of an interface, Archive for Rational Mechanics and Analysis, 57, 255–280 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  • Montemagno, C. D., and W. G. Gray, Photoluminescent volumetric imaging: A technique for the exploration of multiphase flow and transport in porous media, Geophysical Research Letters, 22 (4), 425–428 (1995).

    Article  Google Scholar 

  • Morrow, N. R., Physics and thermodynamics of capillary action in porous media, in Flow Through Porous Media, American Chemical Society, Washington, D. C., 103–128 (1970).

    Google Scholar 

  • Müller, I., Extended Thermodynamics, Springer-Verlag, New York (1993).

    Book  MATH  Google Scholar 

  • Murad, M. A., L. S. Bennethum, and J. H. Cushman, A multi-scale theory of swelling porous media: I. Application to one-dimensional consolidation, Transport in Porous Media, 19, 93–122 (1995).

    Article  Google Scholar 

  • Reeves, P. C., and M. A. Celia, A functional relationship between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model, Water Resources Research, 32 (8), 2345–2358 (1996).

    Article  Google Scholar 

  • Russo, D., Determining soil hydraulic properties by parameter estimation: On the selection of a model for hydraulic properties, Water Resources Research, 24, 453–459 (1988).

    Article  Google Scholar 

  • Rose, W., Measuring transport coefficients necessary for the description of coupled two-phase flow of immiscible fluids in porous media, Transport in Porous Media, 3, 163–171 (1988).

    Article  Google Scholar 

  • Saripalli, K. P., H. Kim, P. S. C. Rao, and M. D. Annable, Use of interfacial tracers to measure immiscible fluid interfacial areas in porous media, Environmental Science and Technology, 31 (3), 932–936 (1996).

    Article  Google Scholar 

  • Slattery, J. C., Flow of viscoelastic fluids through porous media, American Institute of Chemical Engineers Journal, 3, 1066–1071 (1967).

    Article  Google Scholar 

  • Stankovich, J. M. and D. A. Lockington, Brooks-Corey and van Genuchten soil retention models, Journal of Irrigation and Drainage, 121, 1–7 (1995).

    Article  Google Scholar 

  • Svendsen, B., and K. Hutter, On the thermodynamics of a mixture of isotropic viscous materials with kinematic constraints, International Journal of Engineering Science, 33, 20212054 (1995).

    Google Scholar 

  • van Genuchten, M. T., A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal, 44, 892–899 (1980).

    Article  Google Scholar 

  • Whitaker, S., Diffusion and dispersion in porous media, AIChE Journal, 13, 420–427 (1967).

    Google Scholar 

  • Whitaker, S., Advances in theory of fluid motion in porous media, Industrial and Engineering Chemistry, 61 (12), 14–28 (1969).

    Article  Google Scholar 

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Authors and Affiliations

  1. University of Notre Dame, Notre Dame, IN, USA

    W. G. Gray

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  1. W. G. Gray
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Editors and Affiliations

  1. Technical University Darmstadt, Germany

    Kolumban Hutter

  2. Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany

    Krzysztof Wilmanski

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© 1999 Springer-Verlag Wien

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Gray, W.G. (1999). Elements of a Systematic Procedure for the Derivation of Macroscale Conservation Equations for Multiphase Flow in Porous Media. In: Hutter, K., Wilmanski, K. (eds) Kinetic and Continuum Theories of Granular and Porous Media. International Centre for Mechanical Sciences, vol 400. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2494-9_2

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  • DOI: https://doi.org/10.1007/978-3-7091-2494-9_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83146-5

  • Online ISBN: 978-3-7091-2494-9

  • eBook Packages: Springer Book Archive

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