# Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms

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## Abstract

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.

## Keywords

Foam generation mechanisms Trapped gas Snap-off Lamella division Flowing foam fraction## List of Symbols

- \(\bar{r}_{\mathrm{th}}\)
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\(^{-1}\))

- \(\phi \)
Porosity

- \(\sigma \)
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)

- \(f_\mathrm{c}\)
Percolation threshold

- \(f_\mathrm{f}\)
Percolation fraction of flowing foam

- \(f_{\mathrm{LD}}\)
Lamella division probability

- \(f_{\mathrm{so,th}}\)
A randomly drawn number in a throat

- \(f_{\mathrm{so}}\)
Snap-off probability

- \(f_{\mathrm{so}}^*\)
Critical snap-off probability

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

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

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

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

- \(p^{+}_i\)
Pressure needed to open pore

*i*from the right-side boundary (Pa)- \(p^{-}_i\)
Pressure needed to open pore

*i*from the left-side boundary (Pa)- \(p_\mathrm{c}\)
Bond percolation threshold

- \(p_\mathrm{m}\), \(\bar{\tau }\)
Mean pressure difference needed to mobilize a single lamella (Pa)

- \(p_{ij}\)
Pressure at which throat

*ij*opens to flow (Pa)- \(Q_\mathrm{b}\)
Source term for the number of bubbles (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))

*R*A characteristic pore radius (m)

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

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

- \(r_{\mathrm{g}_{\mathrm{LD}}}\)
Rate of generation due to lamella division (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))

- \(r_{\mathrm{g}_{\mathrm{SO}}}\)
Rate of generation due to snap-off (\(\hbox {m}^{-3}\) \(\hbox {s}^{-1}\))

- \(r_{\mathrm{th}}\)
Throat radius (m)

- \(S_\mathrm{g}\)
Total gas saturation

- \(S_{\mathrm{gf}}\)
Flowing gas saturation

- \(S_{\mathrm{gt}}\)
Trapped gas saturation

*T*Curvilinear pathway tortuosity in the network representation of flowing foam

- \(u_\mathrm{g}\)
Darcy gas velocity (m s\(^{-1}\))

- \(X_\mathrm{f}\)
Flowing foam fraction

- \(X_\mathrm{t}\)
Trapped foam fraction

- \(X_{\mathrm{L}_{\mathrm{ss}}}\)
Maximum (steady-state) flowing lamellae fraction

## Notes

### Acknowledgements

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.

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