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Analytical Solution for the Population-Balance Model Describing Foam Displacement

Abstract

We investigate and classify possible analytical solutions for a simplified version of the foam bubble population model, by varying injection conditions and kinetic foam generation parameter. We prove that the behavior of the analytical solutions changes at the transition between two regions, similar to rarefaction-shock solutions for the Buckley-Leverett equation. In one region (region I), the solutions are in the form of traveling waves, in rather good agreement with CT scanned foam experiments and numerical simulations reported in Simjoo and Zitha (2015). In region II, however, corresponds to solutions as a sequence of waves: one spreading wave and one traveling wave. These corresponding flow profiles are different from those found so far in the experiments.

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Acknowledgements

The current work was conducted in association with the R&D project ANP \(n^\circ\) 20715-9, “Modelagem matemática e computacional de injeção de espuma usada em recuperação avançada de petróleo” (UFJF/Shell Brazil/ANP). Shell Brazil funds it in accordance with ANP’s R&D regulations under the Research, Development, and Innovation Investment Commitment. This project is carried out in partnership with Petrobras. G.C. was supported in part by CNPq Grant 303245/2019-0.

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Correspondence to Grigori Chapiro.

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Zavala, R.Q., Lozano, L.F., Zitha, P.L.J. et al. Analytical Solution for the Population-Balance Model Describing Foam Displacement. Transp Porous Med (2021). https://doi.org/10.1007/s11242-021-01589-z

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Keywords

  • Foam flow
  • Traveling waves
  • Porous media
  • Partial Differential Equations