Transport in Porous Media

, Volume 102, Issue 3, pp 325–348 | Cite as

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


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


  1. Afsharpoor, A., Lee, G.S., Kam, S.I.: Mechanistic simulation of continuous gas injection period during surfactant-alternating-gas (SAG) processes using foam catastrophe theory. Chem. Eng. Sci. 65(11), 3615–3631 (2010)CrossRefGoogle Scholar
  2. Alvarez, J.M., Rivas, H.J., Rossen, W.R.: Unified model for steady-state foam behavior at high and low foam qualities. SPE J. 6(3), 325–333 (2001)CrossRefGoogle Scholar
  3. Andrianov, A., Farajzadeh, R., Mahmoodi Nick, M., Talanana, M., Zitha, P.: Immiscible foam for enhancing oil recovery: bulk and porous media experiments. Ind. Eng. Chem. Res. 51(5), 2214–2226 (2012)CrossRefGoogle Scholar
  4. Ashoori, E., Rossen, W.R.: Can formation relative permeabilities rule out a foam EOR process? SPE J. 17(2), 340–351 (2012)CrossRefGoogle Scholar
  5. Ashoori, E., van der Heijden, T.L.M., Rossen, W.R.: Fractional-flow theory of foam displacements with oil. SPE J. 15(2), 260–273 (2010)CrossRefGoogle Scholar
  6. Aster, R.C., Thurber, C.H., Borchers, B.: Parameter Estimation and Inverse Problems. International geophysics series, vol. 90. Elsevier Academic Press, Amsterdam (2005)CrossRefGoogle Scholar
  7. Avriel, M.: Nonlinear Programming: Analysis and Methods. Prentice-Hall series in automatic computation. Prentice-Hall, Englewood Cliffs (1976)Google Scholar
  8. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear programming: theory and algorithms, 3rd edn. Wiley, New York (2006)CrossRefGoogle Scholar
  9. Blaker, T., Aarra, M.G., Skauge, A., Rasmussen, L., Celius, H.K., Martinsen, H.A., Vassenden, F.: Foam for gas mobility control in the Snorre field: the FAWAG project. SPE Reserv. Eval. Eng. 5(4), 317–323 (2002)Google Scholar
  10. Boeije, C.S., Rossen, W.R.: Fitting foam simulation model parameters to data. Paper presented at the IOR 2013–17th European symposium on improved oil recovery, St. Petersburg, 2013Google Scholar
  11. Bruges, E.A., Latto, B., Ray, A.K.: New correlations and tables of coefficient of viscosity of water and steam up to 1000 bar and 1000 \(\circ \)C. Int. J. Heat Mass Transf. 9(5), 465–480 (1966)CrossRefGoogle Scholar
  12. Cheng, L., Reme, A.B., Shan, D., Coombe, D.A., Rossen, W.R.: Simulating foam processes at high and low foam qualities. Paper presented at the SPE/DOE improved oil recovery symposium, Tulsa, Oklahoma (2000)Google Scholar
  13. Computer Modeling Group: STARS\(^{\rm TM}\) user’s guide, Calgary (2007)Google Scholar
  14. Dholkawala, Z.F., Sarma, H.K., Kam, S.I.: Application of fractional flow theory to foams in porous media. J. Pet. Sci. Eng. 57(1–2), 152–165 (2007)Google Scholar
  15. Dong, Y., Rossen, W.: Insights from Fractional-Flow Theory for Models for Foam IOR. In: 14th European symposium on improved oil recovery (2007)Google Scholar
  16. Falls, A.H., Hirasaki, G.J., Patzek, T.W., Gauglitz, D.A., Miller, D.D., Ratulowski, T.: Development of a mechanistic foam simulator: the population balance and generation by snap-off. SPE Reserv. Eng. 3(3), 884–892 (1988)CrossRefGoogle Scholar
  17. Farajzadeh, R., Andrianov, A., Krastev, R., Hirasaki, G.J., Rossen, W.R.: Foam–oil interaction in porous media: implications for foam assisted enhanced oil recovery. Adv. Colloid Interf. Sci. 183, 1–13 (2012a)CrossRefGoogle Scholar
  18. Farajzadeh, R., Wassing, B.M., Boerrigter, P.M.: Foam assisted gas-oil gravity drainage in naturally-fractured reservoirs. J. Pet. Sci. Eng. 94–95, 112–122 (2012b)CrossRefGoogle Scholar
  19. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)Google Scholar
  20. Friedmann, F., Chen, W.H., Gauglitz, P.A.: Experimental and simulation study of high-temperature foam displacement in porous media. SPE Reserv. Eng. 6(1), 37–45 (1991). doi:10.2118/17357-pa CrossRefGoogle Scholar
  21. Heller, J.P.: CO2 foams in enhanced oil-recovery. In: Foams: Fundamentals and Applications in the Petroleum Industry, vol. 242, pp. 201–234. American Chemical Society, Washington, DC (1994)Google Scholar
  22. Kam, S.I., Nguyen, Q.P., Li, Q., Rossen, W.R.: Dynamic simulations with an improved model for foam generation. SPE J. 12(1), 35–48 (2007)CrossRefGoogle Scholar
  23. Kam, S.I., Rossen, W.R.: A model for foam generation in homogeneous media. SPE J. 8(4), 417–425 (2003)CrossRefGoogle Scholar
  24. Khatib, Z.I., Hirasaki, G.J., Falls, A.H.: Effects of capillary pressure on coalescence and phase mobilities in foams flowing through porous media. SPE Reserv. Eng. 3(3), 919–926 (1988)CrossRefGoogle Scholar
  25. Kovscek, A.R., Patzek, T.W., Radke, C.J.: A mechanistic population balance model for transient and steady-state foam flow in Boise sandstone. Chem. Eng. Sci. 50(23), 3783–3799 (1995)CrossRefGoogle Scholar
  26. Kovscek, A.R., Radke, C.J.: Fundamentals of foam transport in porous-media. In: Foams: Fundamentals and Applications in the Petroleum Industry, vol. 242, pp. 115–163. American Chemical Society, Washington, DC (1994)Google Scholar
  27. Lee, H.O., Heller, J.P., Hoefer, A.M.W.: Change in apparent viscosity of \({\text{ CO }}_2\) foam with rock permeability. SPE Reserv. Eng. 6(4), 421–428 (1991). doi: 10.2118/20194-pa CrossRefGoogle Scholar
  28. Lemmon, E.W., Jacobsen, R.T.: Viscosity and thermal conductivity equations for nitrogen, oxygen, argon, and air. Int. J. Thermophys. 25(1), 21–69 (2004)CrossRefGoogle Scholar
  29. Li, R.F., Yan, W., Liu, S.H., Hirasaki, G.J., Miller, C.A.: Foam mobility control for surfactant enhanced oil recovery. SPE J. 15(4), 934–948 (2010)CrossRefGoogle Scholar
  30. Liu, M., Andrianov, A., Rossen, W.R.: Sweep efficiency in \({\text{ CO } }_2\) foam simulations with oil (SPE 142999). Paper Presented at the SPE EUROPEC/EAGE Annual Conference and Exhibition, Vienna (2011)Google Scholar
  31. Lopez-Salinas, J.L., Ma, K., Puerto, M.C., Miller, C.A., Biswal, S.L., Hirasaki, G.J.: Estimation of parameters for the simulation of foam flow through porous media: effects of surfactant concentration and fluid velocity (in preparation)Google Scholar
  32. Ma, K.: Transport of surfactant and foam in porous media for enhanced oil recovery processes. Ph.D. Thesis, Rice University (2013)Google Scholar
  33. Ma, K., Lopez-Salinas, J.L., Puerto, M.C., Miller, C.A., Biswal, S.L., Hirasaki, G.J.: Estimation of parameters for the simulation of foam flow through porous media. Part 1: the dry-out effect. Energy Fuels 27(5), 2363–2375 (2013)CrossRefGoogle Scholar
  34. Masalmeh, S.K., Wei, L., Blom, C.P.A.: Mobility control for gas injection in heterogeneous carbonate reservoirs: comparison of foams versus polymers (SPE 142542). Paper presented at the SPE middle east oil and gas show and conference, Manama (2011)Google Scholar
  35. Namdar Zanganeh, M., Kam, S.I., LaForce, T., Rossen, W.R.: The method of characteristics applied to oil displacement by foam. SPE J. 16(1), 8–23 (2011). doi:10.2118/121580-pa CrossRefGoogle Scholar
  36. Namdar Zanganeh, M., Kraaijevanger, J.F.B.M., Buurman, H.W., Jansen, J.D., Rossen, W.R.: Adjoint-Based Optimization of a Foam EOR Process. Paper presented at the 13th European conference on the mathematics of oil recovery, Biarritz (2012)Google Scholar
  37. Patzek, T.W.: Description of foam flow in porous media by the population balance method. ACS Symp. Ser. 373, 326–341 (1988)CrossRefGoogle Scholar
  38. Renkema, W.J., Rossen, W.R.: Success of foam SAG processes in heterogeneous reservoirs (SPE 110408). Paper presented at the SPE annual technical conference and exhibition, Anaheim (2007)Google Scholar
  39. Roostapour, A., Kam, S.I.: Anomalous foam-fractional-flow solutions at high-injection foam quality. SPE Reserv. Eval. Eng. 16(1), 40–50 (2013). doi:10.2118/152907-pa Google Scholar
  40. Rossen, W.: Numerical challenges in foam simulation: a review. Paper presented at the SPE annual technical conference and exhibition (SPE 166232), New Orleans (2013)Google Scholar
  41. Rossen, W.R.: Foams in Enhanced Oil Recovery. Foams: theory, measurements, and applications, vol. 57, pp. 413–464. Marcel Dekker, New York (1996)Google Scholar
  42. Rossen, W.R., Zeilinger, S.C., Shi, J.X., Lim, M.T.: Simplified mechanistic simulation of foam processes in porous media. SPE J. 4(3), 279–287 (1999). doi:10.2118/57678-pa CrossRefGoogle Scholar
  43. Shan, D., Rossen, W.R.: Optimal Injection strategies for foam IOR. SPE J. 9(2), 132–150 (2004). doi:10.2118/88811-pa CrossRefGoogle Scholar
  44. Spirov, P., Rudyk, S., Khan, A.: Foam assisted WAG, Snorre revisit with new foam screening model (SPE 150829). In: North Africa technical conference and exhibition, Cairo (2012)Google Scholar
  45. The MathWorks Inc.: MATLAB User’s Guide, Natick (2012)Google Scholar
  46. Vassenden, F., Holt, T., Ghaderi, A., Solheim, A.: Foam propagation on semi-reservoir scale. SPE Reserv. Eval. Eng. 2(5), 436–441 (1999)Google Scholar

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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