Transport in Porous Media

, Volume 15, Issue 3, pp 271–293 | Cite as

Nonisothermal multiphase flow of brine and gas through saline media

  • S. Olivella
  • J. Carrera
  • A. Gens
  • E. E. Alonso


We propose a general formulation for nonisothermal multiphase flow of brine and gas through saline media. The balance equations include mass balance (three species), equilibrium of stresses and energy balance (total internal energy). Salt, water and air mass balance equations are established. The balance of salt allows the establishment of the equation for porosity evolution due to solid skeleton deformation, dissolution/precipitation of salt and migration of brine inclusions. Water and air mass balance equations are also obtained. Two equations are required for water: total water in the medium and water present in solid phase brine inclusions. The mechanical problem is formulated through the equation of stress equilibrium. Finally, the balance of internal energy is established assuming thermal equilibrium between phases. Some general aspects of the constitutive theory are also presented.

Key words

brine creep deformation gas heat inclusion multiphase porosity salt 



body forces vector in equilibrium equation


elastic compliance matrix


ratio between volume and surface of the grains


grain size


effective diffusion coefficient for inclusion migration


dispersion tensor (i=h, w forα=l andi=w, a forα=g)


internal energy ofα-phase per unit mass ofα-phase


internal energy ofi-species inα-phase per unit mass ofi-species


external mass supply per unit volume of medium (i=h, w, a)


internal/external energy supply per unit volume of medium


internal sink of water in fluid inclusion equation


gravity vector


species index,h salt (halite),w water anda air (superscript)


nonadvective mass flux ofi-species inα-phase


nonadvective heat flux


advective energy flux inα-phase with respect to a fixed reference system


advective energy flux inα-phase with respect to the solid phase


total mass flux ofi-species inα-phase with respect to a fixed reference system


total mass flux ofi-species inα-phase with respect to the solid phase


permeability tensor (α=l, g)


intrinsic permeability tensor


α-phase relative permeability (α=l, g)


molecular mass of water


fluid pressure ofα-phase (α=l, g)


volumetric flux ofα-phase with respect to the solid matrix (α=l, g)


constant of gases


volumetric fraction of pore volume occupied byα-phase (α=l, g)




solid velocity vector


velocity of brine inclusions in the solid phase


phase index,s solid,l liquid andg gas (subscript)

\(\dot \varepsilon \)

strain rate tensor


(=ω α i ρα) mass ofi-species per unit volume ofα-phase


dynamic viscosity ofα-phase (α=l, g)

gradient vector


mass ofα-phase per unit volume ofα-phase

σ, σ′

stress tensor (total and net)

\(\dot \sigma \)

stress rate tensor




mass fraction ofi-species inα-phase


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  1. Abriola, L. M. and Pinder, G. F. (1985), A multiphase approach to the modeling of porous media contamination by organic compounds. 1. Equation development,Water Resour. Res. 21(1), 11–18.Google Scholar
  2. Bachmat, Y. and Bear, J. (1986), Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach,Transport in Porous Media 1, 213–240.Google Scholar
  3. Bear, J. (1979),Dynamics of Fluids in Porous Media, American Elsevier.Google Scholar
  4. Bear, J. and Bachmat, Y. (1986), Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass, momentum and energy transport,Transport in Porous Media 1, 241–269.Google Scholar
  5. Bear, J. and Bensabat, J. (1989), Advective fluxes in multiphase porous media under nonisothermal conditions,Transport in Porous Media 4, 423–448.Google Scholar
  6. Bear, J., Bensabat, J. and Nir, A. (1991), Heat and mass transfer in unsaturated porous media at a hot boundary: I. One-dimensional analytical model,Transport in Porous Media 6, 281–298.Google Scholar
  7. Bird, R. B., Stewart, W. E. and Lightfoot, E. N. (1960),Transport Phenomena, John Wiley, New York, 1960.Google Scholar
  8. Corapcioglu, M. Y. (1991), Formulation of electro-chemico-osmotic processes in soils,Transport in Porous Media 6, 435–444.Google Scholar
  9. Edlefson, N. E. and Anderson, A. B. C. (1943), Thermodynamics of soil moisture,Hilgardia 15(2), 31–298.Google Scholar
  10. Engelmann, H. J., Broochs, P. W., HÄnsel, W., and Peters, L. (1989), Dams as sealing systems in rock salt formations — Test dam construction and determination of permeability,Sealing of Radioactive Waste Repositories, Proceedings of an NEAJCEC Workshop, ISBN 92-64-03290-8, 151–162.Google Scholar
  11. Faust, C. R. and Mercer, J. W. (1979), Geothermal reservoir simulation: 1. Mathematical models for liquid- and vapour-dominated hydrothermal systems,Water Resour. Res. 15(1), 23–30.Google Scholar
  12. van Genuchten, R. (1980), A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,Soil Sci. Soc. Am. J., 892–898.Google Scholar
  13. Hassanizadeh, S. M. and Gray, W. G. (1979a), General conservation equations for multiphase systems: 1. Averaging procedure,Adv. Water Resour. 2, 131–144.Google Scholar
  14. Hassanizadeh, S. M. and Gray, W. G. (1979b), General conservation equations for multiphase systems: 2. Mass, momenta, energy and entropy equations,Adv. Water Resour. 2, 191–203.Google Scholar
  15. Hassanizadeh, S. M. (1986a), Derivation of basic equations of mass transport in porous media, Part 1. Macroscopic balance laws,Adv. Water Resour. 9, 196–206.Google Scholar
  16. Hassanizadeh, S. M. (1986b), Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws,Adv. Water Resour. 9, 207–222.Google Scholar
  17. Horvath, A. L. (1985),Aqueous Electrolyte Solutions, Physical Properties, Estimation and Correlation Methods, Ellis Horwood, Chichester.Google Scholar
  18. Langer, H. and Offermann, H. (1982), On the solubility of sodium chloride in water,J. Crystal Growth 60, 389–392.Google Scholar
  19. Lewis, R. W., Roberts, P. J. and Schrefler, B. A. (1989), Finite element modelling of two-phase heat and fluid flow in deforming porous media,Transport in Porous Media 4, 319–334.Google Scholar
  20. Milly, P. C. D. (1982), Moisture and heat transport in hysteretic, inhomogeneous porous media: A matrix head-based formulation and a numerical model,Water Resour. Res. 18(3), 489–498.Google Scholar
  21. Olivella, S., Gens, A., Alonso, E. E. and Carrera, J. (1992), Constitutive modelling of porous salt aggregates,Proc. Fourth International Symposium on Numerical Models in Geomechanics — NUMOG-IV/Swansea/UK/24–27 August 1992, A. A. Balkema, Rotterdam, pp. 179–189.Google Scholar
  22. Olivella, S., Gens, A., Carrera, J. and Alonso, E. E. (1993), Behaviour of porous salt aggregates. Constitutive and field equations for a coupled deformation, brine, gas and heat transport model,Proc. 3rd Conference on the Mechanical Behaviour of Salt, September 14–16, 1993, Ecole Polytechnique, Palaiseau, France, 255–269Google Scholar
  23. Panday, S. and Corapcioglu, M. Y. (1989), Reservoir transport equations by compositional approach,Transport in Porous Media 4, 369–393.Google Scholar
  24. Philip, J. R. and de Vries, D. A. (1957), Moisture movement in porous materials under temperature gradients,EOS Trans. AGU 38(2), 222–232.Google Scholar
  25. Pinder, G. F. and Abriola, L. M. (1986), On the simulation of nonaqueous phase organic compounds in the subsurface,Water Resour. Res. 22(9), 109S-119S.Google Scholar
  26. Pollock, D. W. (1986), Simulation of fluid flow and energy transport processes associated with high-level radioactive waste disposal in unsaturated alluvium,Water Resour. Res. 22(5), 765–775.Google Scholar
  27. Pruess, K. (1987),TOUGH User's Guide. Report LBL-20700, NUREG/CR-4640, 80 pp.Google Scholar
  28. Ratigan, J. L. (1984), A finite element formulation for brine transport in rock salt,Int. J. Numer. Anal. Meth. Geomech. 8, 225–241.Google Scholar
  29. Roedder, E. (1984), The fluids in salt,American Mineralogist 69, 413–439.Google Scholar
  30. Spiers, C. J., Schutjens, P. M. T. M., Brzesowsky, R. H., Peach, C. J., Liezenberg, J. L. and Zwart, H. J. (1990), Experimental determination of constitutive parameters governing creep of rock salt by pressure solution, Geological Society Special Publication No. 54,Deformation Mechanisms, Rheology and Tectonics.Google Scholar
  31. Sprackling, M. T. (1985),Liquids and Solids, Student Physics Series, King's College, University of London. Eds. Routledge and Kegan-Paul, London.Google Scholar
  32. Yagnik, S. K. (1983), Interfacial stability of migrating brine inclusions in alkali halide single crystals supporting a temperature gradient,J. Crystal Growth 62, 612–626.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • S. Olivella
    • 1
  • J. Carrera
    • 1
  • A. Gens
    • 1
  • E. E. Alonso
    • 1
  1. 1.Geotechnical Department, Civil Engineering SchoolTechnical University of CatalunyaBarcelonaSpain

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