Transport in Porous Media

, Volume 114, Issue 1, pp 65–86 | Cite as

Study of the Well-Posedness of Models for the Inaccessible Pore Volume in Polymer Flooding

  • Sindre T. Hilden
  • Halvor Møll Nilsen
  • Xavier RaynaudEmail author


Inaccessible pore volume, also known as dead pore space, is used when simulating enhanced oil recovery by polymer injection. We show that a widely used model for inaccessible pore volume can lead to an ill-posed problem, resulting in unphysical results. By considering shock solutions of the one-dimensional problem, we derive a necessary condition that an inaccessible pore volume model must fulfill in order to obtain well-posed equations. In this derivation, we use the Rankine–Hugoniot jump condition as a selection criterion for acceptable solutions. There are other possible criteria for the one-dimensional problem, in particular \(\delta \)-shock solutions, which we also briefly describe, but these are challenging and impractical to use. Based on a heuristic understanding of relative permeability, we subsequently derive two modified models for inaccessible pore volume. The first model follows directly from the modeling assumptions, but it has limited applicability. If the inaccessible pore volume is larger than the irreducible water saturation, then the equations are ill-posed for convex relative permeabilities. A second model is derived by relaxing the assumption of inaccessibility, allowing a limited fraction of the polymer to enter the smallest pores. This second model fulfills our necessary condition for well-posedness for all values of the inaccessible pore volume and any choice of relative permeabilities. Through one- and two-dimensional numerical examples, the different models for inaccessible pore volume are compared. For our second suggested model, the polymer concentration is observed to stay below the maximum injected value, which is not the case for the conventional model. This enables a more stable implementation of the highly nonlinear system, and a reduction in the number of nonlinear iterations is also observed in some cases. As this suggested model is straightforward to implement into existing reservoir simulators and can be used for a wide range of polymer models, it serves as a possible alternative to the conventional model.


Numerical modeling Enhanced oil recovery Polymer Inaccessible pore volume 



The research is partly funded by VISTA, which is a basic research program funded by Statoil and conducted in close collaboration with The Norwegian Academy of Science and Letters, and partly by the Norwegian Research Council through the PETROMAKS2 program, Project Number 244361.


  1. Bartelds, G.A., Bruining, J., Molenaar, J.: The modeling of velocity enhancement in polymer flooding. Transp. Porous Media 26(1), 75–88 (1997)CrossRefGoogle Scholar
  2. Brooks, R., Corey, A.: Hydraulic properties of porous media. Hydrology papers, no. 3, colorado state university, ft. Collins, Colo (1964)Google Scholar
  3. Burdine, N.T., et al.: Relative permeability calculations from pore size distribution data. J. Petrol. Technol. 5(03), 71–78 (1953)CrossRefGoogle Scholar
  4. Center for Petroleum and Geosystems Engineering of The University of Texas at Austin: Technical Documentation for UTCHEM-9.0 A Three-Dimensional Chemical Flood Simulator (2000)Google Scholar
  5. Computer Modelling Group Ltd.: User’s guide STARS (2009)Google Scholar
  6. Danilov, V., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245(12), 3704–3734 (2008)CrossRefGoogle Scholar
  7. Dawson, R., Lantz, R.B., Aime, M.: Inaccessible pore volume in polymer flooding. SPE J. 253(3522), 448–452 (1972)CrossRefGoogle Scholar
  8. Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. In: Applied Mathematical Sciences, vol. 152, 2nd edn. Springer, Heidelberg (2015)Google Scholar
  9. Lake, L.W.: Enhanced Oil Recovery, reprint 2010 edn. Society of Petroleum Engineers (1989)Google Scholar
  10. Lie, K.A.: An Introduction to Reservoir Simulation Using MATLAB: User Guide for the Matlab Reservoir Simulation Toolbox (MRST), 2nd edn. SINTEF ICT, (2015)
  11. MATLAB Reservoir Simulation Toolbox, MRST 2015b (2015).
  12. Norris, U.L.: Core-scale simulation of polymer flow through porous media. Master thesis, University of Stavanger, Norway (2011)Google Scholar
  13. Pancharoen, M., Thiele, M.R., Kovscek, A.R., et al.: Inaccessible pore volume of associative polymer floods. In: SPE Improved Oil Recovery Symposium. Society of Petroleum Engineers (2010)Google Scholar
  14. Schlumberger: Eclipse Technical Description Manual, 2009.2 edn. (2009)Google Scholar
  15. Todd, M.R., Longstaff, W.J.: The development, testing, and application of a numerical simulator for predicting miscible flood performance. J. Petrol. Tech. 24(7), 874–882 (1972)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Sindre T. Hilden
    • 1
    • 2
  • Halvor Møll Nilsen
    • 2
  • Xavier Raynaud
    • 1
    • 2
    Email author
  1. 1.Department of MathematicsNTNUOsloNorway
  2. 2.Department of Applied MathematicsSINTEFOsloNorway

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