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

  • W. G. Gray
Part of the International Centre for Mechanical Sciences book series (CISM, volume 400)


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.


Porous Medium Capillary Pressure Conservation Equation Multiphase Flow Geometric Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, T. B., and R. Jackson, A fluid mechanical description of fluidized beds, Industrial and Engineering Chemistry Fundamentals, 6, 527–539 (1967).CrossRefGoogle Scholar
  2. Anderson, W. G., Wettability literature survey Part 4. Effects of wettability on capillary pressure, Journal of Petroleum Technology, 39, 1283–1300 (1987a).Google Scholar
  3. Anderson, W. G., Wettability literature survey Part 5. Effects of wettability on relative permeability, Journal of Petroleum Technology, 39, 1453–1468 (1987b).Google Scholar
  4. 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).CrossRefGoogle Scholar
  5. Bachmat, Y., Spatial macroscopization of processes in heterogeneous systems, Israel Journal of Technology, 10 (5), 391–403 (1972).MathSciNetGoogle Scholar
  6. Bailyn, M., A Survey of Thermodynamics, AIP Press, New York (1994).Google Scholar
  7. Bennethum, L. S., Multiscale, hybrid mixture theory for swelling systems with interfaces, Center for Applied Mathematics Technical Report #259, Purdue University (1994).Google Scholar
  8. 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
  9. Boruvka, L., Y. Rotenberg, and A. W. Neumann, Free energy formulation of the theory of capillarity, Langmuir, 1 (1), 40–44 (1985).CrossRefGoogle Scholar
  10. Brooks, R. H. and A. T. Corey, Hydraulic properties of porous media, Hydrology Paper 3, Colorado State University, Fort Collins (1964).Google Scholar
  11. Callen, H. B., Thermodynamics and an Introduction to Thermostatistics, 2nd Edition, John Wiley and Sons, New York (1985).MATHGoogle Scholar
  12. 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
  13. 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).MathSciNetGoogle Scholar
  14. Derjaguin, B. V., N. V. Churaev, and V. M. Muller, Surface Forces, Consultants Bureau, New York (1987).CrossRefGoogle Scholar
  15. Edlefsen, N. E., and A. B. C. Anderson, Thermodynamics of soil moisture, Hilgardia, 15, 312–398 (1943).Google Scholar
  16. Eringen, A. C., Mechanics of Continua, 2nd Edition, Krieger Publishing, Huntington, NY (1980).Google Scholar
  17. 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
  18. 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
  19. Gray, W. G., Macroscale equilibrium conditions for two-phase flow in porous media, International Journal of Multiphase Flow (in review, 1998b ).Google Scholar
  20. 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).CrossRefMATHGoogle Scholar
  21. Gray, W. G., and S. M. Hassanizadeh, Unsaturated flow theory including interfacial phenomena, Water Resources Research, 27 (8) 1855–1863 (1991).CrossRefGoogle Scholar
  22. 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).CrossRefGoogle Scholar
  23. 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).CrossRefMATHGoogle Scholar
  24. 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).MATHGoogle Scholar
  25. 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
  26. 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).CrossRefGoogle Scholar
  27. 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).CrossRefGoogle Scholar
  28. 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).CrossRefGoogle Scholar
  29. 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).CrossRefGoogle Scholar
  30. 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
  31. 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).CrossRefGoogle Scholar
  32. 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
  33. 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).CrossRefGoogle Scholar
  34. 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).CrossRefGoogle Scholar
  35. 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
  36. Liu, I.-S., Method of Lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics and Analysis, 46, 131–148 (1972).MATHMathSciNetGoogle Scholar
  37. Miller, C. A., and P. Neogi, Interfacial Phenomena, Marcel Dekker, New York (1985).Google Scholar
  38. Moeckel, G. P., Thermodynamics of an interface, Archive for Rational Mechanics and Analysis, 57, 255–280 (1975).CrossRefMATHMathSciNetGoogle Scholar
  39. 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).CrossRefGoogle Scholar
  40. 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
  41. Müller, I., Extended Thermodynamics, Springer-Verlag, New York (1993).CrossRefMATHGoogle Scholar
  42. 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).CrossRefGoogle Scholar
  43. 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).CrossRefGoogle Scholar
  44. 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).CrossRefGoogle Scholar
  45. 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).CrossRefGoogle Scholar
  46. 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).CrossRefGoogle Scholar
  47. Slattery, J. C., Flow of viscoelastic fluids through porous media, American Institute of Chemical Engineers Journal, 3, 1066–1071 (1967).CrossRefGoogle Scholar
  48. Stankovich, J. M. and D. A. Lockington, Brooks-Corey and van Genuchten soil retention models, Journal of Irrigation and Drainage, 121, 1–7 (1995).CrossRefGoogle Scholar
  49. 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
  50. 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).CrossRefGoogle Scholar
  51. Whitaker, S., Diffusion and dispersion in porous media, AIChE Journal, 13, 420–427 (1967).Google Scholar
  52. Whitaker, S., Advances in theory of fluid motion in porous media, Industrial and Engineering Chemistry, 61 (12), 14–28 (1969).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • W. G. Gray
    • 1
  1. 1.University of Notre DameNotre DameUSA

Personalised recommendations