Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms

  • Muhammad M. AlmajidEmail author
  • Anthony R. Kovscek


The mobility of gas is greatly reduced when the injected gas is foamed. The reduction in gas mobility is attributed to the reduction in gas relative permeability and the increase in gas effective viscosity. The reduction in the gas relative permeability is a consequence of the larger amount of gas trapped when foam is present while the increase in gas effective viscosity is explicitly a function of foam texture. Therefore, understanding how foam is generated and subsequent trapped foam behavior is of paramount importance to modeling of gas mobility. In this paper, we push the envelope to enlighten our decisions of which descriptions are most physical to foam flow in porous media regarding both the flowing foam fraction and the rate of generation. We use a statistical pore network interwoven with the invasion percolation with memory algorithm to model foam flow as a drainage process and investigate the dependence of the flowing foam fraction on the pressure gradient and to shed light on foam generation mechanisms. A critical snap-off probability is required for strong foam to emerge in our network. The pressure gradient and, hence, the gas mobility reduction are very low below this critical snap-off probability. Above this snap-off probability threshold, we find that the steady-state flowing lamellae fraction scales as \((\nabla \tilde{p})^{0.19}\) in 2D lattices and as \((\nabla \tilde{p})^{0.32}\) in 3D lattices. Results obtained from our network were convolved with percolation network scaling ideas to compare the probabilities of snap-off and lamella division mechanisms in the network during the initial gas displacement at the leading edge of the gas front. At this front, during strong foam flow, lamella division is practically nonexistent in 2D lattices. In 3D lattices, lamella division occurs, but the probability of snap-off is always greater than the probability of lamella division.


Foam generation mechanisms Trapped gas Snap-off Lamella division Flowing foam fraction 

List of Symbols


Mean throat radius (m)

\(\beta \)

Trapping parameter

\(\Delta p\)

Pressure drop (Pa)

\(\Delta p_\mathrm{F}\)

Pressure drop needed to mobilize one pathway in the presence of foam (Pa)

\(\Delta p_{\mathrm{cap}}\)

A pore throat capillary resistance (Pa)

\(\epsilon \)

Discrepancy between the two least resistant paths in the network

\(\mathcal {L}\)

Curvilinear pathway length in the network representation of flowing foam (m)

\(\nabla p^{\mathrm{min}}\)

Minimum pressure gradient for mobilizing foam (Pa m\(^{-1}\))

\(\nabla p_{\mathrm{ss}}\)

Maximum (steady-state) applied pressure gradient (Pa m\(^{-1}\))

\(\phi \)


\(\sigma \)

Interfacial tension between the wetting and nonwetting phases (N m\(^{-1}\))

\(\sum _i r_{\mathrm{g}_{i}}\)

Sum of the contribution of other foam generation mechanisms (m\(^{-3}\) s\(^{-1}\))

\(\tau _{ij}\)

Threshold that a lamella would have if it is present in throat ij (Pa)

\(\varepsilon \)

A small perturbation to the percolation threshold

\(\zeta \)

Mesh size (m)


Percolation threshold


Percolation fraction of flowing foam


Lamella division probability


A randomly drawn number in a throat


Snap-off probability


Critical snap-off probability


Permeability (\(\hbox {m}^2\))


Distance between lamellae in the network representation of flowing foam (m)


Number density of foam bubbles per unit volume of flowing gas (\({\mathrm{m}^{-3}}\))


Number density of foam bubbles per unit volume of trapped gas (\({\mathrm{m}^{-3}}\))


Pressure needed to open pore i from the right-side boundary (Pa)


Pressure needed to open pore i from the left-side boundary (Pa)


Bond percolation threshold

\(p_\mathrm{m}\), \(\bar{\tau }\)

Mean pressure difference needed to mobilize a single lamella (Pa)


Pressure at which throat ij opens to flow (Pa)


Source term for the number of bubbles (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))


A characteristic pore radius (m)


Rate of foam coalescence (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))


Rate of foam generation (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))


Rate of generation due to lamella division (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))


Rate of generation due to snap-off (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))


Throat radius (m)


Total gas saturation


Flowing gas saturation


Trapped gas saturation


Curvilinear pathway tortuosity in the network representation of flowing foam


Darcy gas velocity (m s\(^{-1}\))


Flowing foam fraction


Trapped foam fraction


Maximum (steady-state) flowing lamellae fraction



We would like to thank Saudi Aramco for providing the graduate fellowship that made this work possible. Many thanks are also due to the SUPRI-A Industrial Affiliates for providing additional financial support. The authors also thank Wonjin Yun for reviewing this paper internally.


  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. Almajid, M.M., Kovscek, A.R.: Pore-level mechanics of foam generation and coalescence in the presence of oil. Adv. Colloid Interface Sci. 233, 65–82 (2016)CrossRefGoogle Scholar
  3. Balan, H.O., Balhoff, M.T., Nguyen, Q.P., Rossen, W.R.: Network modeling of gas trapping and mobility in foam enhanced oil recovery. Energy Fuels 25(9), 3974–3987 (2011)CrossRefGoogle Scholar
  4. Bertin, H.J., Quintard, M.Y., Castanier, L.M.: Development of a bubble-population correlation for foam-flow modeling in porous media. SPE J. 3(04), 356–362 (1998)CrossRefGoogle Scholar
  5. Chen, M., Rossen, W., Yortsos, Y.C.: The flow and displacement in porous media of fluids with yield stress. Chem. Eng. Sci. 60(15), 4183–4202 (2005)CrossRefGoogle Scholar
  6. Chen, M., Yortsos, Y.C., Rossen, W.R.: Pore-network study of the mechanisms of foam generation in porous media. Phys. Rev. E 73(3), 036,304 (2006)CrossRefGoogle Scholar
  7. Chou, S.I.: Percolation theory of foam in porous media. In: SPE/DOE Enhanced Oil Recovery Symposium. Society of Petroleum Engineers (1990)Google Scholar
  8. Cohen, D., Patzek, T.W., Radke, C.J.: Onset of mobilization and the fraction of trapped foam in porous media. Transp. Porous Media 28(3), 253–284 (1997)CrossRefGoogle Scholar
  9. de Gennes, P.G.: Conjectures on foam mobilization. Revue de l’Institut français du pétrole 47(2), 249–254 (1992)CrossRefGoogle Scholar
  10. Dholkawala, Z.F., Sarma, H., Kam, S.: Application of fractional flow theory to foams in porous media. J. Pet. Sci. Eng. 57(1–2), 152–165 (2007)CrossRefGoogle Scholar
  11. 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(03), 884–892 (1988)CrossRefGoogle Scholar
  12. Friedmann, F., Chen, W.H., Gauglitz, P.A.: Experimental and simulation study of high-temperature foam displacement in porous media. SPE Reserv. Eng. 6(01), 37–45 (1991)CrossRefGoogle Scholar
  13. Gillis, J.V., Radke, C.J.: A dual gas tracertechnique for determining trapped gas saturation during steady foamflow in porous media. In: SPE Annual Technical Conference andExhibition. Society of Petroleum Engineers (1990)Google Scholar
  14. Hirasaki, G.J., Lawson, J.B.: Mechanisms of foam flow in porous media: apparent viscosity in smooth capillaries. SPE J. 25(2), 176–190 (1985)Google Scholar
  15. Jones, S., Getrouw, N., Vincent-Bonnieu, S.: Foam flow in a model porous medium: II. The effect of trapped gas. Soft Matter 14(18), 3497–3503 (2018)CrossRefGoogle Scholar
  16. Kam, S.I., Rossen, W.R.: A model for foam generation in homogeneous media. SPE J. 8(04), 417–425 (2003)CrossRefGoogle Scholar
  17. Kam, S.I., Nguyen, Q.P., Li, Q., Rossen, W.R.: Dynamic simulations with an improved model for foam generation. SPE J. 12(01), 35–48 (2007)CrossRefGoogle Scholar
  18. Kharabaf, H., Yortsos, Y.C.: Invasion percolation with memory. Phys. Rev. E 55(6), 7177 (1997)CrossRefGoogle Scholar
  19. Kharabaf, H., Yortsos, Y.C.: A pore-network model for foam formation and propagation in porous media. SPE J. 3(01), 42–53 (1998)CrossRefGoogle Scholar
  20. Kovscek, A.R., Bertin, H.J.: Foam mobility in heterogeneous porous media. Transp. Porous Media 52(1), 17–35 (2003)CrossRefGoogle Scholar
  21. Kovscek, A.R., Radke, C.J.: Fundamentals of foam transport in porous media. ACS Adv. Chem. Ser. 242, 115–164 (1994)CrossRefGoogle Scholar
  22. 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
  23. Kovscek, A.R., Tang, G.Q., Radke, C.J.: Verification of roof snap off as a foam-generation mechanism in porous media at steady state. Colloids Surf. A Physicochem. Eng. Asp. 302(1), 251–260 (2007)CrossRefGoogle Scholar
  24. Kovscek, A.R., Chen, Q., Gerritsen, M.: Modeling foam displacement with the local-equilibrium approximation: theory and experimental verification. SPE J. 15(01), 171–183 (2010)CrossRefGoogle Scholar
  25. Laidlaw, W.G., Wilson, W.G., Coombe, D.A.: A lattice model of foam flow in porous media: a percolation approach. Transp. Porous Media 11(2), 139–159 (1993)CrossRefGoogle Scholar
  26. Ma, K., Ren, G., Mateen, K., Morel, D., Cordelier, P.: Modeling techniques for foam flow in porous media. SPE J. 20(03), 453–470 (2015)CrossRefGoogle Scholar
  27. Myers, T.J., Radke, C.J.: Transient foam displacement in the presence of residual oil: experiment and simulation using a population-balance model. Ind. Eng. Chem. Res. 39(8), 2725–2741 (2000)CrossRefGoogle Scholar
  28. Nelson, P.H.: Pore-throat sizes in sandstones, tight sandstones, and shales. AAPG Bull. 93(3), 329–340 (2009)CrossRefGoogle Scholar
  29. Nguyen, Q.P., Rossen, W.R., Zitha, P.L.J., Currie, P.K.: Determination of gas trapping with foam using X-ray computed tomography and effluent analysis. SPE J. 14(02), 222–236 (2009)CrossRefGoogle Scholar
  30. Patzek, T.W.: Description of Foam Flow in Porous Media by the Population Balance Method, Chap 17, pp. 326–341. Society of Petroleum Engineers, Richardson (1988). Google Scholar
  31. Ransohoff, T.C., Radke, C.J.: Mechanisms of foam generation in glass-bead packs. SPE Reserv. Eng. 3(02), 573–585 (1988)CrossRefGoogle Scholar
  32. Roof, J.G.: Snap-off of oil droplets in water-wet pores. Soc. Pet. Eng. J. 10, 85–90 (1970)CrossRefGoogle Scholar
  33. Rossen, W.R., Gauglitz, P.A.: Percolation theory of creation and mobilization of foams in porous media. AIChE J. 36(8), 1176–1188 (1990)CrossRefGoogle Scholar
  34. Rossen, W.R., Mamun, C.K.: Minimal path for transport in networks. Phys. Rev. B 47(18), 11,815 (1993)CrossRefGoogle Scholar
  35. Sochi, T., Blunt, M.J.: Pore-scale network modeling of Ellis and Herschel–Bulkley fluids. J. Pet. Sci. Eng. 60(2), 105–124 (2008)CrossRefGoogle Scholar
  36. Stauffer, D., Aharony, A.: Introduction to Percolation Theory. CRC Press, Boca Raton (1994)Google Scholar
  37. Tang, G.Q., Kovscek, A.R.: Trapped gas fraction during steady-state foam flow. Transp. Porous Media 65(2), 287–307 (2006)CrossRefGoogle Scholar
  38. Wardlaw, N., Li, Y., Forbes, D.: Pore-throat size correlation from capillary pressure curves. Transp. Porous Media 2(6), 597–614 (1987)CrossRefGoogle Scholar
  39. Wong, H., Radke, C.J., Morris, S.: The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95–110 (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Energy Resources EngineeringStanford UniversityStanfordUSA

Personalised recommendations