HPCCprojects:Three-dimensional miscible displacements in porous media or Hele-Shaw cells


Three-dimensional miscible displacements in porous media or Hele-Shaw cells

Project description

Viscous fingering instability can occur when a less viscous fluid displaces a more viscous fluid. This hydrodynamic instability has several applications including groundwater flows and oil recovery, and it has been under investigation for many decades following the works of Hill (1952) and Saffman & Taylor (1958), cf. also Homsy (1987). Due to its similarities to porous media flows, many viscous fingering studies have been performed in a Hele-Shaw cell, an apparatus that consists of two parallel plates placed closely together. These are low Reynolds number flows, and a two-dimensional modeling that comes from averaging the Stokes equations across the gap of the Hele-Shaw cell is usually applied in the form of Darcy's law (Tryggvason 1983, Meiburg & Homsy 1988, Tan & Homsy 1988, Miranda & Widom 1998).


This project intends to perform three-dimensional Navier-Stokes simulations of miscible displacements in porous media or Hele-Shaw cells. A three-dimensional description of the problem have access to all three vorticity components, different from a two-dimensional formalism that includes only gapwise vorticity. This component of vorticity drives finger formation, and the results from this project have shown that the streamwise vorticity is responsible for the emergence of additional hydrodynamic instabilities.


The project started in 2010 during my PhD studies at UCSB under the supervision of Prof. Eckart Meiburg. We have published three papers with the aid of the computational facilities of CSDMS. There are still many open questions.

Models in use

The numerical code used for this project, pllcart3d, was developed by the Computational Fluid Dynamics Laboratory of the Department of Mechanical Engineering at UCSB. It is a C code parallelized with MPI.


Three-dimensional Navier-Stokes simulations of unstable, miscible displacements in Hele-Shaw cells have shown that in neutrally buoyant flows, the formation of streamwise vorticity quadrupoles results in the formation of a novel inner splitting mechanism. In the presence of density differences, the interaction of these quadrupoles with a Rayleigh-Taylor instability can lead to the formation of an 'anchor'-like structure in vertical displacements. Gravitational effects in horizontal displacements break the up/down symmetry of the flow and result in streamwise vorticity dipoles that can split the finger lengthwise.


List the CSDMS HPCC users of your project:

  • Rafael M. Oliveira
  • Michael O. John
  • Felix H. C. Heussler


E. Meiburg acknowledges financial support through NSF grant CBET-0651498, DOE grant DE-FG02-08ER15991, and from the Petroleum Research Fund, administered by the American Chemical Society, under grant 45175-AC9. R.M.Oliveira would like to thank the CAPES Foundation in Brazil and the Fulbright program for financial support through grants BEX 2615/06-1 and IIE 15073695.

Publications and presentations

The following three journal articles have been published:

- Rafael M. Oliveira and Eckart Meiburg. Miscible displacements in Hele-Shaw cells: three-dimensional Navier-Stokes simulations. J. Fluid Mech. (2011), vol. 687, pp. 431-460.

- M. O. John, R. M. Oliveira, F. H. C. Heussler and E. Meiburg. Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations. J. Fluid Mech. (2013), vol. 721, pp. 295-323.

- Rafael M. Oliveira and Eckart Meiburg. Three-dimensional vorticity configurations in miscible Hele-Shaw displacements. Procedia IUTAM 7 (2013) 203-212.



https://sites.google.com/site/ucsbcfdlab/people/rafael .

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