The Influence of Bimodal Heterogeneity on Viscous Fingering of a Miscible Interface in Porous Media

Document Type : Research Paper


School of Chemical Engineering, College of Engineering, University of Tehran, Iran


The flow regimes and the dynamics of the front in miscible displacements are controlled by the interactions between the mechanisms of instability involved in such processes. The instabilities may be driven by unfavorable gravity or mobility ratios or by the heterogeneity of the medium providing favorable paths for the more mobile fluid. In this work, the effect of porous medium heterogeneity with two scales of permeability variations on the frontal instability and fluid mixing have been investigated. The base mode of permeability variations has a smaller wavelength and higher frequency while the imposed mode has a larger wavelength. The effect of such a bimodal heterogeneity on the growth of mixing zone length (MZL) has been studied and the development of the previously recognized flow regimes in layered porous media have been examined. Compared to the unimodal medium comprising the base wave, in the bimodal cases with large contrast between the wavelengths of the two periodic profiles the dominance of each wave length at a different time scale predictably enhances the growth of fingers in the early and late stages. Interestingly and less intuitively, even in cases with close wave numbers between the combined modes faster growth of the mixing zone length is observed. In such cases, the coherence of equal layers in a unimodal layered medium is disturbed by the second wave number resulting in fading of the lateral diffusion regime. However, bimodal heterogeneity may attenuate the instability compared to the unimodal system with the imposed wave’s frequency.


  1. Abriola LM. Modeling contaminant transport in the subsurface: An interdisciplinary challenge. Reviews of Geophysics. 1987 Mar;25(2):125-34.
  2. Broyles BS, Shalliker RA, Cherrak DE, Guiochon G. Visualization of viscous fingering in chromatographic columns. Journal of Chromatography A. 1998 Oct 2;822(2):173-87.
  3. Fanchi JR. Principles of applied reservoir simulation. Elsevier; 2005 Dec 8.
  4. Amooie MA, Soltanian MR, Xiong F, Dai Z, Moortgat J. Mixing and spreading of multiphase fluids in heterogeneous bimodal porous media. Geomechanics and Geophysics for Geo-Energy and Geo-Resources. 2017 Sep;3(3):225-44.
  5. Enin A, Kalinin K, Petrova Y, Tikhomirov S. Optimal polymer slugs injection curves. arXiv preprint arXiv:2012.03114. 2020 Dec 5.
  6. De Wit A, Homsy GM. Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations. The Journal of chemical physics. 1997 Dec 8;107(22):9619-28.
  7. Sajjadi M, Azaiez J. Scaling and unified characterization of flow instabilities in layered heterogeneous porous media. Physical Review E. 2013 Sep 23;88(3):033017.
  8. Nijjer JS, Hewitt DR, Neufeld JA. Stable and unstable miscible displacements in layered porous media. Journal of Fluid Mechanics. 2019 Jun;869:468-99.
  9. Cantrell DL, Hagerty RM. Microporosity in arab formation carbonates, Saudi Arabia. GeoArabia. 1999 Apr 1;4(2):129-54.
  10. Rubin Y. Flow and transport in bimodal heterogeneous formations. Water Resources Research. 1995 Oct;31(10):2461-8.
  11. Lu Z, Zhang D. On stochastic modeling of flow in multimodal heterogeneous formations. Water Resources Research. 2002 Oct;38(10):8-1.
  12. Pritchard D. The instability of thermal and fluid fronts during radial injection in a porous medium. Journal of Fluid Mechanics. 2004 Jun;508:133-63.
  13. Sajjadi M, Azaiez J. Dynamics of fluid flow and heat transfer in homogeneous porous media. The Canadian Journal of Chemical Engineering. 2013 Apr;91(4):687-97.
  14. Islam MN, Azaiez J. Miscible thermo-viscous fingering instability in porous media. Part 2: Numerical simulations. Transport in porous media. 2010 Sep;84(3):845-61.
  15. Meiburg E, Chen CY. High-accuracy implicit finite-difference simulations of homogeneous and heterogeneous miscible-porous-medium flows. Spe Journal. 2000 Jun 1;5(02):129-37.
  16. Tan CT, Homsy GM. Simulation of nonlinear viscous fingering in miscible displacement. The Physics of fluids. 1988 Jun;31(6):1330-8.
  17. Manickam O, Homsy GM. Fingering instabilities in vertical miscible displacement flows in porous media. Journal of Fluid Mechanics. 1995 Apr;288:75-102.
  18. Bracewell R. The Fourier Transform and Its Applications. McGraw Hill, 2000 (34):712.
  19. Bracewell R. Discrete Hartley transform. Journal of Optical Society of America, vol. 73, no. 12, pp. 1831-1835, 1983.
  20. Shahnazari MR, Maleka Ashtiani I, Saberi A. Linear stability analysis and nonlinear simulation of the channeling effect on viscous fingering instability in miscible displacement. Physics of Fluids. 2018 Mar 27;30(3):034106.
  21. Sheng JJ, editor. Enhanced oil recovery field case studies. Gulf Professional Publishing; 2013 Apr 10.
Volume 55, Issue 2
December 2021
Pages 369-383
  • Receive Date: 29 August 2021
  • Revise Date: 30 September 2021
  • Accept Date: 01 October 2021
  • First Publish Date: 03 October 2021