Transport in Porous Media

, Volume 113, Issue 3, pp 491–510 | Cite as

Simulation of Viscous Fingering in Rectangular Porous Media with Lateral Injection and Two- and Three-Phase Flows

  • Bertrand LagréeEmail author
  • Stéphane Zaleski
  • Igor Bondino


Understanding multiphase flow in porous media is of tremendous importance for many industrial and environmental applications at various spatial and temporal scales. The present study consequently focuses on modeling multiphase flows by the volume-of-fluid method (sharp interface) in porous media with a simplified Darcy-scale approach and shows simulations of Saffman–Taylor fingering. The simplification of the Darcy-scale approach is performed by assuming sharp interfaces between pure phases. The volume-of-fluid method with octree mesh refinement is used. It is implemented in the Gerris (Popinet in J Comput Phys 190(2):572–600, 2003. doi: 10.1016/S0021-9991(03)00298-5; J Comput Phys 228(16):5838–5866, 2009. doi: 10.1016/ code which allows efficient parallel computations. We measure the scaling properties of the fractal viscous fingering patterns that appear in the numerical simulations. One of these properties is the fractal or Hausdorff dimension \(D_\mathrm{F}\). The other is the variation of the area A of the viscous fingering cluster with the length L of its perimeter, which varies as a simple power law \(A \sim L^{\alpha }\). The injection of an intermediate-viscosity Newtonian fluid as a second step is also simulated. We are thus able to observe an increase of recovery of the high-viscosity fluid behind the fingering front, due to the reduction of the viscosity contrast. Some of these results are compared to waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sandstone.


Volume-of-fluid Viscous fingering Multiphase flows Fractal patterns Invasion at breakthrough 



We thank TOTAL for the financial support and permission to publish this study.


  1. Afkhami, S., Renardy, Y.: A volume-of-fluid formulation for the study of co-flowing fluids governed by the Hele–Shaw equations. Phys. Fluids 25(8), 082001 (2013). doi: 10.1063/1.4817374.
  2. Chuoke, R.L., van Meurs, P., van der Poel, C.: The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media. Trans. AIME 216, 188 (1959)Google Scholar
  3. Davidovitch, B., Levermann, A., Procaccia, I.: Convergent calculation of the asymptotic dimension of diffusion limited aggregates: scaling and renormalization of small clusters. Phys. Rev. E 62, R5919–R5922 (2000). doi: 10.1103/PhysRevE.62.R5919 CrossRefGoogle Scholar
  4. Fabbri, C., De Loubens, R., Skauge, A., Ormehaug, P.A., Vik, B., Bourgeois, M., Morel, D., Hamon, G.: Comparison of History-Matchedwater Flood, Tertiary Polymer Flood Relative Permeabilities and Evidence of Hysteresis During Tertiary Polymer Flood in Very Viscous oils. Society of Petroleum Engineers (SPE), Conference Proceedings, 174682 (2015).
  5. Fast, P., Shelley, M.J.: Moore’s law and the Saffman-Taylor instability. J. Comput. Phys. 212(1), 1–5 (2006). doi: 10.1016/ ISSN: 0021-9991
  6. Feder, J.: Fractals. Plenum, New York (1988)CrossRefGoogle Scholar
  7. Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346–349 (1983). doi: 10.1103/PhysRevLett.50.346 CrossRefGoogle Scholar
  8. Lagrée, B., Zaleski, S., Bondino, I., Josserand, C., Popinet, S.: Scaling Properties of Viscous Fingering. ArXiv, arXiv preprint (2014).
  9. Lagrée, P.-Y., Staron, L., Popinet, S.: The granular column collapse as a continuum: validity of a two-dimensional Navier-Stokes model with a \(\mu \)(I)-rheology. J. Fluid Mech. (686):378–408 (2011). doi: 10.1017/jfm.2011.335. ISSN: 1469-7645
  10. Li, S., Lowengrub, J.S., Leo, P.H.: A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell. J. Comput. Phys. 225(1), 554–567 (2007). doi: 10.1016/ ISSN: 0021-9991
  11. Måløy, K.J., Feder, J., Jøssang, T.: Viscous Fingering fractals in porous media. Phys. Rev. Lett. 55, 2688–2691 (1985). doi: 10.1103/PhysRevLett.55.2688 CrossRefGoogle Scholar
  12. Popinet, S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190(2), 572–600 (2003). doi: 10.1016/S0021-9991(03)00298-5. ISSN: 0021-9991
  13. Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009). doi: 10.1016/ ISSN: 0021-9991
  14. Popinet, S.: Gerris Flow Solver (2014).
  15. Praud, O., Swinney, H.L.: Fractal dimension and unscreened angles measured for radial viscous fingering. Phys. Rev. E 72, 011406 (2005). doi: 10.1103/PhysRevE.72.011406 CrossRefGoogle Scholar
  16. Saffman, P.G., Taylor, G.I.: The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 245(1242), 312–329 (1958)CrossRefGoogle Scholar
  17. Skauge, A., Ormehaug, P.A., Gurholt, T., Vik, B., Bondino, I., Hamon, G.: 2-D visualisation of unstable waterflood and polymer flood for displacement of heavy oil. In: SPE Improved Oil Recovery Symposium Society of Petroleum Engineers (2012)Google Scholar
  18. Stauffer, D.: Scaling theory of percolation clusters. Phys. Rep. 54(1), 1–74 (1979). ISSN: 0370-1573Google Scholar
  19. Tryggvason, G., Scardovelli, R., Zaleski, S.: Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Bertrand Lagrée
    • 1
    • 2
    • 3
    • 4
    Email author
  • Stéphane Zaleski
    • 2
    • 3
    • 4
  • Igor Bondino
    • 1
    • 5
  1. 1.TOTAL SACourbevoieFrance
  2. 2.UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d’AlembertSorbonne UniversitésParisFrance
  3. 3.CNRS, UMR 7190Institut Jean le Rond d’AlembertParisFrance
  4. 4.Institut Jean le Rond d’AlembertUPMC Univ Paris 06Paris Cedex 05France
  5. 5.TOTAL - CSTJFPau CedexFrance

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