Transport in Porous Media

, Volume 102, Issue 3, pp 325–348

Non-uniqueness, Numerical Artifacts, and Parameter Sensitivity in Simulating Steady-State and Transient Foam Flow Through Porous Media

  • Kun Ma
  • Rouhi Farajzadeh
  • Jose L. Lopez-Salinas
  • Clarence A. Miller
  • Sibani Lisa Biswal
  • George J. Hirasaki

DOI: 10.1007/s11242-014-0276-9

Cite this article as:
Ma, K., Farajzadeh, R., Lopez-Salinas, J.L. et al. Transp Porous Med (2014) 102: 325. doi:10.1007/s11242-014-0276-9


The uniqueness and sensitivity of foam modeling parameters are crucial for simulating foam flow through porous media. In the absence of oil in the porous medium, the local-equilibrium foam model investigated in this work uses three parameters to describe the foam quality dependence: \(fmmob,\, fmdry\), and \(epdry\). Even for a specified value of \(epdry\), in some cases, two pairs of \(fmmob\) and \(fmdry\) values can experimentally match measured transition foam quality (\(f_\mathrm{g}^{t}\)) and transition foam apparent viscosity (\(\mu _\mathrm{foam,app}^t\)). This non-uniqueness can be broken by limiting the solution such that \(fmdry\) is smaller than the transition water saturation (\(S_\mathrm{w}^t\)). In addition, a three-parameter fit using all experimental data of apparent viscosity versus foam quality was developed to simultaneously estimate \(fmmob,\, fmdry\), and \(epdry\). However, a better strategy is to conduct and match a transient experiment, in addition to steady-state experiments, in which a gas displaces the surfactant solution at 100 % water saturation. This transient foam quality scans the entire range of fractional flow, and the values of the foam parameters that best match the experiment can be uniquely determined. The numerical artifact of pressure oscillations in simulating this transient foam process was investigated by comparing the finite difference algorithm with the method of characteristics. Sensitivity analyses indicated that the estimated foam parameters were highly dependent on the parameters used for the water and gas relative permeabilities. In particular, the water relative permeability exponent and connate water saturation are important.


Foam model Porous media Surfactants Reservoir simulation Fractional flow theory Mobility control 

List of Symbols


A parameter regulating the slope of the dry-out function near \(fmdry\)


Fractional flow

\(f_\mathrm{g}^t \)

Transition foam quality at which the maximum foam apparent viscosity is achieved


A dimensionless foam function in the foam model


Critical water saturation in the foam model


Reference mobility reduction factor in the foam model


Permeability (darcy)


Relative permeability

\(k_\mathrm{rw}^0 \)

End-point relative permeability of the aqueous phase

\(k_\mathrm{rg}^0 \)

End-point relative permeability of the gaseous phase

\(L \)

Length of the porous medium (ft)

\(p \)

Pressure (psi)


Capillary pressure (psi)


Limiting capillary pressure (psi)


Superficial (Darcy) velocity (ft/day)



\(S_\mathrm{w}^t \)

Transition water saturation at which the maximum foam apparent viscosity is achieved

\(t \)

Time (s)

\(\mu \)

Viscosity (cp)

\(\mu _\mathrm{foam,app}\)

Local foam apparent viscosity (cp)

\({\overline{\mu }}_\mathrm{foam,app}\)

Average foam apparent viscosity (cp)

\(\mu _\mathrm{foam,app}^t \)

Maximum foam apparent viscosity obtained at the transition foam quality (cp)

\(\phi \)


\(\Phi _\mathrm{D}\)

Flow potential (dimensionless gas pressure)

\(\omega \)

Weighting parameter in the multi-variable, multi-dimensional search

\(\Theta \)

Penalty function in the multi-variable, multi-dimensional search

\(\sigma \)

Penalty coefficient in the multi-variable, multi-dimensional search



Boundary condition


Without foam


With foam


Exponent in the \(k_\mathrm{rg} \) curve


Exponent in the \(k_\mathrm{rw} \) curve


Transition between high- and low-quality foam





Gaseous phase


Residual gas


Aqueous phase


Connate water

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Kun Ma
    • 1
    • 2
  • Rouhi Farajzadeh
    • 3
    • 4
  • Jose L. Lopez-Salinas
    • 1
  • Clarence A. Miller
    • 1
  • Sibani Lisa Biswal
    • 1
  • George J. Hirasaki
    • 1
  1. 1.Department of Chemical and Biomolecular EngineeringRice UniversityHoustonUSA
  2. 2.Total E&P Research and TechnologyHoustonUSA
  3. 3.Shell Global Solutions InternationalRijswijkThe Netherlands
  4. 4.Delft University of TechnologyDelftThe Netherlands

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