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.
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The figure is modified after Ransohoff and Radke (1988)
The figure is modified after Kharabaf and Yortsos (1997)
The figure is modified after Kharabaf and Yortsos (1997)
The figure is modified after Kovscek and Bertin (2003)
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Abbreviations
- \(\bar{r}_{\mathrm{th}}\) :
-
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
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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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Appendices
Appendix A: Sweep Efficiency Sensitivity to Network Size
In this appendix, we perform multiple simulations using increasing network size and see how the displacement efficiency changes with the network size. Figure 17 shows the results. Networks larger than \(25\times 25\times 25\) have similar results with some acceptable discrepancy. We chose to work with a 3D network of size \(30\times 30\times 30\) because it is big enough to capture the physics of foam flow.
Sensitivity of displacement efficiency to network size. Results converge around network size of \(25\times 25\times 25\) or larger
Appendix B: Sensitivity of Pressure Exponent to Eqs. 3 and 4
In this appendix, we test the sensitivity of the exponent of the pressure gradient to different variables in the distributions used for the throats and thresholds in Eqs. 3 and 4, respectively. Figure 18a shows the sensitivity of the exponent to the average throat size used to populate the throats. The pressure gradient exponent ranges from 0.31 to 0.32. Figure 18b is viewed as a sensitivity of the flowing foam fraction to the permeability of the network, assuming that the permeability scales as \(r_{\mathrm{th}}^2\) according to the Carman–Kozeny relationship. Hence, as permeability increases, the figure shows that the minimum pressure required to move foam is reduced, but the absolute value of the flowing foam fraction in the network is not affected. The snap-off probability, \(f_{\mathrm{so}}\), used was 0.75.
Sensitivity of the flowing foam fraction to the average throat size used to populate the throat distribution, \(\mathcal {P}(r_{\mathrm{th}})\), in Eq. 3
Sensitivity of the flowing foam fraction to the average threshold value used to populate the threshold distribution, \(\mathcal {P}(\tau )\), in Eq. 4
Sensitivity of the flowing foam fraction to size of the network used
The effect of increasing the average threshold populated in the network is shown in Fig. 19. We observe from Fig. 19a that the exponent ranges from 0.29 to 0.32. Additionally, Fig. 19b shows how the minimum pressure increases as the average value of the threshold in the network increases. The snap-off probability, \(f_{\mathrm{so}}\), used was 0.75. This is a reconfirmation of the importance of lamellae to blocking and diverting foam flow in the porous medium.
The effect of the network size on the pressure gradient exponent is shown in Fig. 20a. The pressure gradient exponent ranges from 0.32 to 0.38 depending on the network size. We anticipate that the most representative network size is the largest one shown, which is \(30\times 30\times 30\). The effect of the network size on the minimum pressure is insignificant as shown in Fig. 20b.
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Almajid, M.M., Kovscek, A.R. Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms. Transp Porous Med 131, 289–313 (2020). https://doi.org/10.1007/s11242-018-01224-4
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DOI: https://doi.org/10.1007/s11242-018-01224-4