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Probability density function approach for modelling multi-phase flow with ganglia in porous media

Published online by Cambridge University Press:  17 October 2011

Manav Tyagi  and
Patrick Jenny
Manav Tyagi*
Affiliation:
Institute of Fluid Dynamics, Sonneggstrasse 3, ETH Zurich, Zuerich, CH8092, Switzerland
Patrick Jenny
Affiliation:
Institute of Fluid Dynamics, Sonneggstrasse 3, ETH Zurich, Zuerich, CH8092, Switzerland
*
†Email address for correspondence: tyagi@ifd.mavt.ethz.ch
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Abstract

A probabilistic approach to model macroscopic behaviour of non-wetting-phase ganglia or blobs in multi-phase flow through porous media is proposed. The key idea is to consider a set of stochastic Markov processes that can mimic the microscopic multi-phase dynamics. These processes are characterized by equilibrium probability density functions (PDFs) and correlation times, which can be obtained from micro-scale simulation studies or experiments. A Lagrangian viewpoint is adopted, where stochastic particles represent infinitesimal fluid elements and evolve in the physical and probability space. Ganglion mobilization and trapping are modelled by a two-state jump process with transition probabilities given as functions of ganglion size. Coalescence and breakup of ganglia influence the ganglion size distribution, which is modelled by a Langevin type equation. The joint probability density function (JPDF) of the chosen stochastic variables is governed by a high-dimensional Chapman–Kolmogorov equation. This equation can be used to derive moment (e.g. saturation, mean mobility etc.) transport equations, which in general do not form a closed system. However, in some special cases, which arise in the limit of one time scale being smaller or larger than the others, a closed set of moment transport equations can be obtained. For slowly varying and quasi-uniform flows, the saturation transport equation appears in closed form with the mean mobility fully determined, if the equilibrium PDFs are known. Furthermore, it is shown how statistical parameters such as mobilization and trapping rates and equilibrium PDFs can be obtained from the birth–death type approach, in which ganglia breakup and coalescence are explicitly considered. A two-equation transport model (one equation for the total saturation and one for the trapped saturation) is obtained in the limit of very fast coalescence and breakup processes. This model is employed to mimic hysteresis in relative permeability–saturation curves; a well known phenomenon observed in the successive processes of imbibition and drainage. For the general case, the JPDF-equation is solved using the stochastic particle method, which was proposed in our previous paper (Tyagi et al. J. Comput. Phys. 227, 2008, 6696–6714). Several one- and two-dimensional numerical simulation results are presented to show the influence of correlation times on the averaged macroscopic flow behaviour.

JFM classification

Drops and Bubbles: Breakup/coalescence Multiphase and Particle-laden Flows: Multiphase flow Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 688 , 10 December 2011 , pp. 219 - 257
Copyright
Copyright © Cambridge University Press 2011

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References

1. Ahlstrom, S. W, Foote, H. P., Arnett, R. C., Cole, C. R. & Serne, R. J. 1977 Multi-component mass transport model: theory and numerical implementation. Rep. Battle Pacific Northwest Laboratories, Richaland, Washington 99352 (USA)-2127..CrossRefGoogle Scholar
2. Amili, P. & Yortsos, Y. C. 2006 Darcian dynamics: a new approach to the mobilization of a trapped phase in porous media. Trans. Porous Med. 64, 25–49.CrossRefGoogle Scholar
3. Avraam, D. G., Kolonis, G. B., Roumeliotis, T. C., Constantinides, G. N. & Payatakes, A. C. 1994 Steady-state two-phase flow through planar and nonplanar model porous media. Trans. Porous Med. 16, 75–101.CrossRefGoogle Scholar
4. Avraam, D. G. & Payatakes, A. C. 1995a Flow rigimes and relative permeabilties during steady-state two-phase flow in porous media. J. Fluid Mech. 293, 207–236.CrossRefGoogle Scholar
5. Avraam, D. G. & Payatakes, A. C. 1995b Generalized relative permeability coefficients during steady-state two-phase flow in porous media, and correlation with the flow mechanisms. Trans. Porous Med. 20, 135–168.CrossRefGoogle Scholar
6. Barenblatt, G. I., Patzek, T. W. & Silin, D. B. 2003 The mathematical model of non-equilibrium effects in water–oil displacement. Soc. Petrol. Engng J. 8, 409–416.Google Scholar
7. Bear, J. 1972 Dynamics of Fluids in Porous Media. Dover.Google Scholar
8. 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, 1069–1089.CrossRefGoogle Scholar
9. Cercignani, C. 1988 The Boltzmann Equation and its Application. Springer.CrossRefGoogle Scholar
10. Dahle, H. K., Espendal, M. S., Ewing, R. E & varied, O. Sæ 1990 Characteristic adaptive subdomain methods for reservoir flow problems. Numer. Meth. Partial Differ. Equ. 6, 279–309.CrossRefGoogle Scholar
11. Dahle, H. K., Ewing, R. E. & Russell, T. F. 1995 Eulerian–Lagrangian localized adjoint method for a nonlinear advection-diffusion equation. Comput. Meth. Appl. Mech. Engng 34, 1051–1069.Google Scholar
12. Dias, M. M. & Payatakes, A. C. 1986a Network models for two-phase flow in porous media. Part 1. Immiscible microdisplacement of non-wetting fluids. J. Fluid Mech. 164, 305–336.CrossRefGoogle Scholar
13. Dias, M. M. & Payatakes, A. C. 1986b Network models for two-phase flow in porous media. Part 2. Motion of oil ganglia. J. Fluid Mech. 164, 337–358.CrossRefGoogle Scholar
14. Dullien, F. A. L. 1992 Porous Media: Fluid Transport and Pore Structure. Academic.Google Scholar
15. Gardiner, C. W. 2004 Handbook of Stochastic Methods. Springer.CrossRefGoogle Scholar
16. de Gennes, P. G. 1983 Hydrodynamic dispersion in unsaturated porous media. J. Fluid Mech. 136, 189–200.CrossRefGoogle Scholar
17. de Gennes, P.-G., Brochard-Wyart, & Quere, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.CrossRefGoogle Scholar
18. Gillespie, D. T. 1991 Markov Processes: An Introduction for Physical Scientists. Academic.Google Scholar
19. Gioia, F., Alfani, G., Andreutti, S. & Murena, F. 2003 Oil mobility in saturated water-wetted bed of glass-beads. J. Hazard. Mater. B97, 315–327.CrossRefGoogle Scholar
20. Hassanizadeh, S. M., Celia, M. A. & Dahle, H. K. 2002 Dynamic effect in the capillary pressure-saturation relationship and its impact on unsaturated flow. Vadose Zone J. 1, 38–57.CrossRefGoogle Scholar
21. Heinz, S. 2004 Molecular to fluid dynamics: the consequences of stochastic molecular motion. Phys. Rev. E 70 (3), 036308.CrossRefGoogle ScholarPubMed
22. Hesse, M. A. Jr., Orr, F. M. & Tchelepi, H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 35–60.CrossRefGoogle Scholar
23. Hewett, T. A. & Yamada, T. 1997 Theory for the semianalytic calculation of oil recovery and effective relative permeability using streamtubes. Adv. Water Resour. 20, 279–292.CrossRefGoogle Scholar
24. Hilfer, R. 2006a Macroscopic capillarity without a constitutive capillary pressure function. Phyisca A 371, 209–225.CrossRefGoogle Scholar
25. Hilfer, R. 2006b Macroscopic capillary and hysteresis for flow in porous media. Phys. Rev. E 73, 016307.CrossRefGoogle ScholarPubMed
26. Hinkley, R. E., Dias, M. M. & Payatakes, A. C. 1987 On the motion of oil ganglia in porous media. PhysicoChem. Hydrodyn. 8 (2), 185–211.Google Scholar
27. Ide, S. T., Jessen, K. Jr. & Orr, F. M. 2007 Storage of CO2 in saline aquifers: effects of gravity, viscous and capillary forces on amount and timing of trapping. Intl J. Greenh. Gas Control 1, 481–491.Google Scholar
28. Jenny, P., Pope, S. B., Muradoglu, M. & Caughey, D. A. 2001 A hybrid algorithm for the joint p.d.f. equation of turbulent reactive flows. J. Comput. Phys. 166, 218–252.CrossRefGoogle Scholar
29. Jenny, P., Torrilhon, M. & Heinz, S. 2010 A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion. J. Comput. Phys. 229, 1077–1098.CrossRefGoogle Scholar
30. Jerauld, G. R. & Salter, S. J. 1990 The effect of pore-structure on hysteresis in relative permeability and capillary pressure: pore-level modelling. Trans. Porous Med. 5, 103–151.CrossRefGoogle Scholar
31. Juanes, R., Spiteri, E. J. Jr., Orr, F. M. & Blunt, M. J. 2006 Impact of relatively permeability hysteresis on geological CO2 storage. Water Resour. Res. 42, W12418.CrossRefGoogle Scholar
32. Kats, F. M. Van & Dujin, C. J. Vam 2001 A mathematical model for hysteric two-phase flow in porous media. Trans. Porous Med. 43, 239–263.CrossRefGoogle Scholar
33. Kinzelbach, W. 1992 Numerische Methoden zur Modellierung des Transports von Schadstoffen im Grundwasser. Oldenbourg.Google Scholar
34. Koch, D. L. & Brady, J. F. 1987 A non-local description of advection-diffusion with application to dispersion in porous media. J. Fluid Mech. 180, 387–403.CrossRefGoogle Scholar
35. Kumar, A., Ozah, R., Noh, M., Pope, G. A., Bryant, S., Sepehrnoori, K. & Lake, L. W. 2005 Reservoir simulation of CO2 storage in deep saline aquifers. Soc. Petrol. Engng J. 10, 336–348.Google Scholar
36. Larson, R. G., Scriven, L. E. & Davis, H. T. 1977 Percolation theory of residual phases in porous media. Nature 268, 409–413.CrossRefGoogle Scholar
37. Lenhard, R. J., Parker, J. C. & Kaluarcchchi, J. J. 1989 A model for hysteretic constitutive relations governing multi-phase flow 3. Refinements and numerical simulation. Water Resour. Res. 25, 1727–1736.CrossRefGoogle Scholar
38. Lenormand, R., Zarcone, C. & Sarr, A. 1983 Mechanisms of the displacement of one fluid by another in a network of capillary ducts. J. Fluid Mech. 135, 337–353.CrossRefGoogle Scholar
39. LeVeque, R. J. 1992 Numerical Methods for Conservation Laws. Lectures in Mathematics: ETH Zurich , Birkhäuser.CrossRefGoogle Scholar
40. Muskat, M. 1949 Physical Princples of Oil Production. McGraw-Hill.Google Scholar
41. Naud, B. 2003 p.d.f. modelling of turbulent sprays and flames using particle stochastic method. PhD thesis, Delft University of Technology, The Netherlands.Google Scholar
42. Ng, K. M., Davis, H. T. & Scriven, L. E. 1978 Visualization of blob mechanics in flow through porous media. Chem. Engng Sci. 33, 1009–1017.CrossRefGoogle Scholar
43. Ng, K. M. & Payatakes, A. C. 1980 Stochastic simulation of the motion, breakup and stranding of oil ganglia on water-wet granular porous media during immiscible displacement. AIChE J. 26, 419–429.CrossRefGoogle Scholar
44. Panfilov, M. B. & Panfilova, I. V. 1995 Macrokinetic model of the trapping process in two-phase fluid displacement in porous media. Fluid Dyn. 30, 409–416.CrossRefGoogle Scholar
45. Payatakes, A. C. 1982 Dyanamics of oil ganglia during immiscible displacement in water-wet porous media. Annu. Rev. Fluid Mech. 14, 365–393.CrossRefGoogle Scholar
46. Payatakes, A. C., Nag, K. M. & Flumerfelt, R. W. 1980 Oil ganglion dynamics during immiscible displacement: model formulation. AIChE J. 26, 430–443.CrossRefGoogle Scholar
47. Piri, M. & Blunt, M. J. 2005 Three-dimensional mixed-wet random pore-scale network modelling of two- and three-phase flow in porous media. ii. results. Phys. Rev. E 71, 026302–026313.CrossRefGoogle ScholarPubMed
48. Pope, S. B. 1985 P.d.f. methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.CrossRefGoogle Scholar
49. Pope, S. B. 2003 Turbulent Flows. Cambridge University Press.Google Scholar
50. Prickett, T. A., Naymik, T. G & Longquist, C. G 1981 A random walk solute transport model for selected groundwater quality evaluations. Illinois State Water Survey Bulletin 65, Champaign, IL.Google Scholar
51. Rembold, B. & Jenny, P. 2006 A multiblock joint p.d.f. finite-volume hybrid algorithm for the computation of turbulent flows in complex geometries. J. Comput. Phys. 220, 59–87.CrossRefGoogle Scholar
52. Richards, L. A. 1931 Capillary conduction of liquid through porous mediums. Physics 1, 318–333.CrossRefGoogle Scholar
53. Roof, J. G. 1970 Snap-off of oil droplets in water-wet pores. Soc. Petrol. Engng J. 10, 85–90.CrossRefGoogle Scholar
54. Schembre, J. M. & Kovscek, A. R. 2006 Estimation of dynamic relative permeability and capillary pressure from countercurrent imbibition experiments. Trans. Porous Med. 65, 31–51.CrossRefGoogle Scholar
55. Silin, D. B. & Patzek, T. W. 2004 On barenblatt’s model of spontaneous countercurrent imbibition. Trans. Porous Med. 54, 297–322.CrossRefGoogle Scholar
56. Spiteri, E. J., Juanes, R., Blunt, M. J. & Orr, F. M. 2008 A new model of trapping and relative permeability hysteresis for all wettability characteristics. Soc. Petrol. Engng J. 13, 277–288.Google Scholar
57. Tyagi, M. & Jenny, P. 2011 Probability density function modelling of multi-phase flow in porous media with density-driven gravity currents. Trans. Porous Med. 87, 602–623.CrossRefGoogle Scholar
58. Tyagi, M., Jenny, P., Lunati, I. & Tchelepi, H. A. 2008 A stochastic, Lagrangian modelling framework for multi-phase flow in porous media. J. Comput. Phys. 227, 6696–6714.CrossRefGoogle Scholar
59. Valavanides, M. S., Constantinides, G. N. & Payatakes, A. C. 1998 Mechanistic model of steady-state two-phase flow in porous media based on ganglion dynamics. Trans. Porous Med. 30, 267–299.CrossRefGoogle Scholar
60. Wardlaw, N. C. & McKellar, M. 1985 Oil blob populations and mobilization of trapped oil in unconsolidated packs. Can. J. Chem. Engng 63, 525–532.CrossRefGoogle Scholar