1887

Abstract

Summary

Pore-scale modeling of multiphase flow of petroleum reservoir fluids through real natural porous media is a true challenge for both engineering and academic societies. Due to the fundamental aspect of pore-scale description in hydrodynamics it is widely considered in order to improve our knowledge of flow and transport phenomena. Another interest of such an approach is to derive macro-scale constitutive equations, to provide multiphase flow properties for large scale models,etc.

One of the important aspects of this problem is the construction of adequate 3D numerical models of multiphase flow through pore space. With the recent development of computed micro-tomography technique, it seems obvious that direct numerical simulations at pore-scale will be widely used in the porous medium modeling and particularly in petroleum applications.

We report 3D pore-scale simulations of invasion of porous media motivated by the analysis of waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sanstone. For a simulation of fluid distribution inside pores a model based on incompressible Navier-Stokes equations is used. We thus use a Volume-of-Fluid (VOF) method to model two-phase Stokes flows with sharp interface between the two fluids. The height function method is used to model surface tension. Accuracy and precision are tested using several levels of refinement, and comparing to reference simulations in the literature.

We focus on brine imbibition (or drainage) in an initially oil-filled porous medium (obtained by micro-tomography) at very unfavourable mobility ratio. Transient flow patterns at various viscosity ratios and capillary numbers are presented and discussed in some detail. The problems of computational speed at low capillary numbers are also addressed.

The reported methodology of a porous medium properties computation is a valuable tool for both fundamental porous media studies and applied petroleum applications.

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2014-09-08
2024-03-29
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References

  1. Agbaglah, G.G.
    [2011] Dynamique et instabilité des nappes liquides. Ph.D. thesis, Paris6.
    [Google Scholar]
  2. Ahrenholz, B., Niessner, J., Helmig, R. and Krafczyk, M.
    [2011] Pore-scale determination of parameters for macroscale modeling of evaporation processes in porous media. Water Resources Research, 47(7).
    [Google Scholar]
  3. Asinari, P., Ohwada, T., Chiavazzo, E. and Rienzo, A.F.D.
    [2012] Link-wise artificial compressibility method. Journal of Computational Physics, 231(15), 5109 – 5143, ISSN 0021-9991, doi: 10.1016/j.jcp.2012.04.027.
    https://doi.org/http://dx.doi.org/10.1016/j.jcp.2012.04.027 [Google Scholar]
  4. Bandara, U.C., Tartakovsky, A.M. and Palmer, B.J.
    [2011] Pore-scale study of capillary trapping mechanism during CO2 injection in geological formations. International Journal of Greenhouse Gas Control, 5(6), 1566 – 1577, ISSN 1750-5836, doi:10.1016/j.ijggc.2011.08.014.
    https://doi.org/http://dx.doi.org/10.1016/j.ijggc.2011.08.014 [Google Scholar]
  5. Bogdanov, I.I., Guerton, F., Kpahou, J. and Kamp, A.M.
    [2011] Direct pore-scale modeling of two-phase flow through natural media. Proceedings of the 2011 COMSOL Conference in Stuttgart; http://www.comsol.com/paper/download/83497/bogdanov_paper.pdf.
    [Google Scholar]
  6. Bondi, A.B.
    [2000] Characteristics of Scalability and Their Impact on Performance. Proceedings of the 2Nd International Workshop on Software and Performance, WOSP ‘00, ACM, New York, NY, USA, ISBN 1-58113-195-X, 195-203, doi:10.1145/350391.350432.
    https://doi.org/10.1145/350391.350432 [Google Scholar]
  7. Brinkman, H.C.
    [1947] A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A, 1, 34–.
    [Google Scholar]
  8. Cancelliere, A., Chang, C., Foti, E., Rothman, D.H. and Succi, S.
    [1990] The permeability of a random medium: Comparison of simulation with theory. Physics of Fluids A: Fluid Dynamics (1989–1993), 2(12), 2085–2088, doi:10.1063/1.857793.
    https://doi.org/http://dx.doi.org/10.1063/1.857793 [Google Scholar]
  9. Carman, P.C.
    [1937] Fluid flow through granular beds. Transactions-Institution of Chemical Engineeres, 15, 166–.
    [Google Scholar]
  10. [1956] Flow of gases through porous media. Buttherworths, London.
    [Google Scholar]
  11. Courant, R., Friedrichs, K. and Lewy, H.
    [1928] ÃIJber die partiellen differenzengleichungen der mathematischen physik. Mathematische Annalen, 100(1), 32–74, ISSN 0025-5831, doi:10.1007/BF01448839.
    https://doi.org/10.1007/BF01448839 [Google Scholar]
  12. Cummins, S.J., Francois, M.M. and Kothe, D.B.
    [2005] Estimating curvature from volume fractions. Computers & Structures, 83(6 - 7), 425 – 434, ISSN 0045-7949, doi:10.1016/j.compstruc.2004.08.017, frontier of Multi-Phase Flow Analysis and Fluid-Structure Frontier of Multi-Phase Flow Analysis and Fluid-Structure.
    https://doi.org/http://dx.doi.org/10.1016/j.compstruc.2004.08.017 [Google Scholar]
  13. Darcy, H.
    [1856] Les fontaines publiques de la ville de Dijon. Exposition et application des principes à suivre et des formules à employer dans les questions de distribution d’eau : ouvrage terminé par un appendice relatif aux fournitures d’eau de plusieurs villes au filtrage des eaux et à la fabrication des tuyaux de fonte, de plomb, de tole et de bitume. Victor Dalmont.
    [Google Scholar]
  14. Day, M.S., Colella, P., Lijewski, M.J., Rendleman, C.A. and Marcus, D.L.
    [1998] Embedded boundary algorithms for solving the poisson equation on complex domains. Tech. Rep. LBNL-41811, Lawrence Berkeley National Laboratory.
    [Google Scholar]
  15. Harvie, D.J.E., Davidson, M.R. and Rudman, M.
    [2006] An analysis of parasitic current generation in Volume of Fluid simulations. Applied Mathematical Modelling, 30(10), 1056 – 1066, ISSN 0307-904X, doi: 10.1016/j.apm.2005.08.015, special issue of the 12th Biennial Computational Techniques and Applications Conference and Workshops (CTAC-2004) held at The University of Melbourne, Australia, from 27th September to 1st October 2004.
    https://doi.org/http://dx.doi.org/10.1016/j.apm.2005.08.015 [Google Scholar]
  16. Huang, H., Meakin, P. and Liu, M.
    [2005] Computer simulation of two-phase immiscible fluid motion in unsaturated complex fractures using a volume of fluid method. Water Resources Research, 41(12), ISSN 1944-7973, doi:10.1029/2005WR004204.
    https://doi.org/10.1029/2005WR004204 [Google Scholar]
  17. Kozeny, J.
    [1927] Ueber kapillare leitung des wassers im boden. Sitzungsber. Akad. Wiss. Wien, 136, 301–.
    [Google Scholar]
  18. Kundu, P.K. and Cohen, I.M.
    [2008] Fluid Mechanics. Elsevier Academic Press, Fourth edn.
    [Google Scholar]
  19. Lagrée, P.Y., Staron, L. and Popinet, S.
    [2011] The granular column collapse as a continuum: validity of a two-dimensional Navier-Stokes model with a μ (I)-rheology. Journal of Fluid Mechanics, 686, 408–.
    [Google Scholar]
  20. Li, J.
    [1995] Calcul d’interface affine par morceaux. Comptes rendus de l’Académie des sciences. Série II, Mécanique, physique, chimie, astronomie, 320(8), 391–396.
    [Google Scholar]
  21. Martys, N.S., Torquato, S. and Bentz, D.P.
    [1994] Universal scaling of fluid permeability for sphere packings. Phys. Rev. E, 50, 403–408, doi:10.1103/PhysRevE.50.403.
    https://doi.org/10.1103/PhysRevE.50.403 [Google Scholar]
  22. Meakin, P., Feder, J., Frette, V. and Jøssang, T.
    [1992] Invasion percolation in a destabilizing gradient. Phys. Rev. A, 46, 3357–3368, doi:10.1103/PhysRevA.46.3357.
    https://doi.org/10.1103/PhysRevA.46.3357 [Google Scholar]
  23. Ohnesorge, W.v.
    [1936] Die Bildung von Tropfen an Düsen und die Auflösung flüssiger Strahlen. Z. Angew. Math. Mech, 16(4), 355–358.
    [Google Scholar]
  24. [1937] Anwendung eines kinematographischen Hochfrequenzapparates mit mechanischer Regelung der Belichtung zur Aufnahme der Tropfenbildung und des Zerfalls flüssiger Strahlen. Konrad Triltsch, Würzburg.
    [Google Scholar]
  25. Popinet, S.
    [2003] Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. Journal of Computational Physics, 190(2), 572 – 600, ISSN 0021-9991, doi: 10.1016/S0021‑9991(03)00298‑5.
    https://doi.org/http://dx.doi.org/10.1016/S0021-9991(03)00298-5 [Google Scholar]
  26. [2009] An accurate adaptive solver for surface-tension-driven interfacial flows. Journal of Computational Physics, 228(16), 5838 – 5866, ISSN 0021-9991, doi:10.1016/j.jcp.2009.04.042.
    https://doi.org/http://dx.doi.org/10.1016/j.jcp.2009.04.042 [Google Scholar]
  27. Prodanović, M. and Bryant, S.L.
    [2006] A level set method for determining critical curvatures for drainage and imbibition. Journal of Colloid and Interface Science, 304(2), 442 – 458, ISSN 0021-9797, doi: 10.1016/j.jcis.2006.08.048.
    https://doi.org/http://dx.doi.org/10.1016/j.jcis.2006.08.048 [Google Scholar]
  28. Raeini, A.Q., Blunt, M.J. and Bijeljic, B.
    [2012] Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. Journal of Computational Physics, 231(17), 5653 – 5668, ISSN 0021-9991, doi:10.1016/j.jcp.2012.04.011.
    https://doi.org/http://dx.doi.org/10.1016/j.jcp.2012.04.011 [Google Scholar]
  29. Samet, H.
    [1989] Applications of Spatial Data Structures.
    [Google Scholar]
  30. Tryggvason, G., Scardovelli, R. and Zaleski, S.
    [2011] Direct numerical simulations of gas-liquid multiphase flows. Cambridge University Press.
    [Google Scholar]
  31. Tsimpanogiannis, I.N. and Yortsos, Y.C.
    [2004] The critical gas saturation in a porous medium in the presence of gravity. Journal of Colloid and Interface Science, 270(2), 388 – 395, ISSN 0021-9797, doi: 10.1016/j.jcis.2003.09.036.
    https://doi.org/http://dx.doi.org/10.1016/j.jcis.2003.09.036 [Google Scholar]
  32. van Meurs, P. and van der Poel, C.
    [1958] A Theoretical Description of Water-Drive Processes Involving Viscous Fingering. Trans. AIME, 213, 112–.
    [Google Scholar]
  33. Weissberg, H.L. and Prager, S.
    [1970] Viscous Flow through Porous Media. III. Upper Bounds on the Permeability for a Simple Random Geometry. Physics of Fluids (1958-1988), 13(12), 2958–2965, doi: 10.1063/L1692887.
    https://doi.org/http://dx.doi.org/10.1063/L1692887 [Google Scholar]
  34. Whitaker, S.
    [1986] Flow in porous media II: The governing equations for immiscible, two-phase flow. Transport in Porous Media, 1, 105–125, ISSN 0169-3913, doi:10.1007/BF00714688.
    https://doi.org/10.1007/BF00714688 [Google Scholar]
  35. Wilkinson, D.
    [1984] Percolation model of immiscible displacement in the presence of buoyancy forces. Phys. Rev. A, 30, 520–531, doi:10.1103/PhysRevA.30.520.
    https://doi.org/10.1103/PhysRevA.30.520 [Google Scholar]
  36. [1986] Percolation effects in immiscible displacement. Phys. Rev. A, 34, 1380–1391, doi: 10.1103/PhysRevA.34.1380.
    https://doi.org/10.1103/PhysRevA.34.1380 [Google Scholar]
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