Hysteretic upscaled constitutive relationships for vertically integrated porous media flow

Florian Doster, Jan M. Nordbotten, Michael Anthony Celia

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Subsurface two-phase flow in porous media often takes place in reservoirs with a high ratio between the associated lateral and vertical extent and the lateral and vertical flow time scales. This allows for a two-scale approach with effective quantities for two-dimensional horizontal flow equations obtained from reconstructed hydrostatic vertical pressure and saturation distributions. Here, we derive explicit expressions for the two dimensional constitutive relationships for a play-type hysteretic Brooks-Corey capillary pressure function with a pore-size distribution index of 2 and quadratic relative permeabilities. We obtain an explicit hysteretic parametrization for the upscaled capillary pressure function and the upscaled relative permeabilities. The size of the hysteresis loop depends on the ratio between buoyancy and the entry pressure, i.e. it scales with the reservoir height and the ratio between drainage and imbibition capillary pressure. We find that the scaling for the relative permeability is non-monotonic and hysteresis vanishes for both small and large reservoirs.

Original languageEnglish (US)
Pages (from-to)147-161
Number of pages15
JournalComputing and Visualization in Science
Issue number4
StatePublished - Aug 2012

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Software
  • Modeling and Simulation
  • General Engineering
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics


  • CO storage
  • Hysteresis
  • Multiphase flow
  • Porous media
  • Upscaling


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