Introduction

Foam has been widely field tested in oil and gas recovery operations as a mobility control agent and profile control agent because of its property: the mobility of gas, flowing through a porous medium, can be significantly reduced by gas-foaming an aqueous surfactant solution1. The foam is applicable in multiple processes in the oil and gas industry, including enhanced oil recovery2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21, hydraulic fracturing22, gas blocking23, acid diversion during matrix stimulation24,25, clean-up of contaminated aquifer26, and CO2 storage27,

A growing body of research focuses on understanding the mechanisms of foam flow in porous media. Many aspects of the research have been conducted, such as the mechanism of bubble generation and coalescence28, minimum pressure gradient29,30, gas and liquid velocities31, the permeability and homogeneity of porous media32,33,34, salinity35,36, surfactant type36,37,38, and gas type39,40,41,42. However, among all these works is still missing, a systematic study on the effect of surfactant concentration that covers both, a concentration in the magnitude of the CMC (critical micellar concentration), and a higher concentration often utilized in the industry. Moreover, it remains challenging in the field, the accurate prediction of the pressure gradient and water saturation of a particular foam-flow system under given conditions. For a given water saturation in a porous medium or the fraction of pore space occupied by one of the fluids, there is an associated average capillary pressure. Capillary pressure is the pressure difference between two distinct phases (flowing or not) caused by surface or interfacial tension and the mean curvature of the fluid–fluid interface. Capillary pressure in porous media can be measured as a function of saturation by different methods43,44,45,46. Bubbles are thought to coalesce by lamella rupture as the limiting capillary pressure (Pc*) is approached45,46. Bubbles can be generated by capillary snap-off and lamella division. Lamella division is thought to occur above a certain minimum pressure gradient (MPG)29,30.

The aim of this study is to further advance the understanding of foam flow in porous media by examining the effects of surfactant concentration on capillary pressure, pressure gradient, and apparent viscosity, which can be compared with previous test results in the literature. To achieve this goal, seven foam-quality-scan experiments, with different gas velocities and surfactant concentrations, were conducted with pre-generated foam injected into a 145 Darcy-sandpack in a transparent polycarbonate tube (2″ D × 11.2″ L). Capillary pressure was measured by the same pressure probe constructed and validated by Vavra et al47. Direct measurements of capillary pressure were made at ambient conditions during foam-quality-scan experiments. The apparatus was utilized to characterize N2 and CO2 foam flow with in-situ capillary pressure measurements in a high-permeability homogeneous sandpack for 1% AOS concentration48. In this manuscript, the test results of additional flowrates and surfactant concentrations are disclosed with only N2 as the gas.

Contemporary mathematical reservoir simulators have used the concept of the “limiting capillary pressure” to model the mobility of the gas phase in foam flooding. It is a “mobility reduction factor, FM”, as a function of water saturation and other variables49,50. The conceptual basis of this model is that in situ generation of foam will sub-divide that gas phase into bubbles, which increases the resistance to gas flow. The assumption in this model is that the division of bubbles to smaller bubbles continue until a “limiting capillary pressure” is approached. (This assumption is challenged in this manuscript under the studied conditions.) This is modeled with FM(Sw) changing rapidly as the water saturation corresponding to the limiting capillary pressure is approached. Parameter estimation algorithms have been developed to estimate the parameters for this model from foam core floods or sandpack experiments. Usage of this model has resulted in many investigations to validate and/or improve the model5,51,52. This study is focused on understanding foam flow in porous media by examining the effects of different surfactant concentrations in addition to different flowrates. This manuscript shows that in extremely homogeneous porous media and very little contrast between pore body and pore throat, this assumption of the conception of “limiting capillary pressure” may not be valid. This study is focused on understanding foam flow in porous media by examining the effects of different surfactant concentrations in addition to different flowrates.

Materials and methods

Chemicals and porous media

Alpha Olefin Sulfonate AOS14-16 (activity = 39.03 wt.%, Lot#7653919, Stepan®) was selected as the surfactant for this study. Fresh solutions of 0.05wt%, 0.5 wt.%, and 1 wt.% AOS14-16 were prepared in 3 wt.% NaCl with Milli-Q® ultrapure water for each foam-flow test. The CMC of AOS14-16 is approximately 0.005 wt. % for AOS with 3.0 wt.% NaCl measured by Jones et al53. Dry N2 gas (Airgas, 99.999% purity) was supplied by Matheson. The experimental condition in all experiments is 20 °C and 1 atm.

The gas and surfactant solution were co-injected through a foam pre-generator before entering the sand pack, and the surfactant solution was recycled for each test. Properties of the sand pack, and the foam pre-generator for foam-flooding testing are summarized in Table 1. The sands for both porous media, foam pre-generator and sand pack, were identical. 20/40 mesh here means that 90 percent of the sand is fine enough to pass through a 20 mesh and is coarse enough to be retained on a 40 mesh sieve. Details of building the sand pack are given in Vavra et.al47.

Table 1 Sandpacks and foam pre-generator properties.
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The details of the system for conducting foam-flow tests with ambient pressure and temperature is shown in Fig. 1. The injected gas and surfactant solution first flow through a foam pre-generator, then into the sandpack. A differential pressure transducer (DP) was installed for measuring the pressure drop across the sandpack. The surfactant solution, delivered by an HPLC pump, was recycled.

Fig. 1
figure 1

Schematic of sandpack foam-flooding system.

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The schematic of the capillary pressure-probe module is detailed in Vavra et al47. The system is designed to avoid mixing the continuous and discontinuous phases in the lines directly connected to each side of the probe. That is, N2 can always be kept separate from the surfactant solution. Capillary pressures are monitored with a differential pressure transducer equipped with a range of ± 5 psi. Details of the probe construction and static validation are provided in Vavra et al47.

Foam quality scan experimental procedure at fixed gas flow rate

All foam quality scan experiments were conducted with constant gas-volumetric flow rates defined at standard conditions. Liquid flow rates were adjusted to change the foam quality (\({f}_{g}\)). All tests were conducted from high foam quality to low foam quality. The foam qualities were given by:

$${f}_{g}= {q}_{g}/({q}_{g}+{q}_{l})$$
(1)

where \({f}_{g}\) is foam quality, \({q}_{g}\)[cm3/s] is evaluated at average-pressure gas volumetric flow rate at the midpoint of the pack, and \({q}_{l}\) [cm3/s] is liquid volumetric flow rate.

N2 gas was injected with flow rates defined at standard conditions of 1 cm*3/min (1 × rate, the base rate), 3 cm*3/min (3 × rate, three times the base rate), and 9 cm*3/min (9 × rate, nine times the base rate), corresponding to pressure-compensated superficial velocities of 2.4 ft/day (8.2 × 10–6 m/s), 7 ft/day (2.5 × 10–5 m/s), and 16.1 ft/day (5.7 × 10–5 m/s) in the sandpack. Liquid rates were changed to vary the foam quality. Specifics of experiment conditions are given in Table 2. Surface tension does not vary significantly in the range of 0.05 – 1 wt.%.

Table 2 Summary of experiment conditions.
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Capillary pressure and overall pressure drop were recorded by the Validyne pressure transducer software every 5 s and averaged over a minimum of 0.25 PV of total injection at each flow condition. The apparent viscosity, \({\mu }_{app}\), was then calculated from Darcy’s law by assuming pseudo-single-phase flow:

$${\mu }_{app}=kA/({q}_{g}+{q}_{l})(\frac{\left|\Delta P\right|}{L})$$
(2)

where \({\mu }_{app}\) [cP] is apparent viscosity, \(k\) [D] is absolute permeability, A [cm2] is the cross-sectional area of the sandpack, \(\Delta P\) [atm] is the overall pressure drop, and \(L\) [cm] is the length of the sandpack.

Results and discussion

Effect of surfactant concentration and flowrate on foam flow: apparent viscosity, capillary pressure, and transition foam quality

Seven flow experiments were conducted with different surfactant concentrations and flow rates at ambient conditions (~ 20 °C, 1 atm). The summarized test results are represented in Fig. 2. Each experiment was conducted with fixed gas flowrate while only changing surfactant solution injection rate. All foam quality scan were conducted from dry to wet, and all measurements were recorded after a steady state appears to be achieved. The transition foam quality (fg*) is the quality at the peak of the apparent viscosity-foam quality curve, and all the transition foam qualities are highlighted in yellow.

Fig. 2
figure 2

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for all experiments as a function of concentration: 1wt% (black), 0.5 wt% (red), and 0.05 wt% (blue) AOS, and standard gas flowrates: 9X (squares), 3X (triangles), and 1X (circles).

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All the test experimental results in Fig. 2A) correspond to different flowrates and surfactant concentrations; black is 1%, red is 0.5%, and blue is 0.05%; circles are 1X flowrate, triangles are 3X flowrate, and squares are 9X flowrate. All of them follow the same trend. As the foam quality increases, the apparent viscosity increases, and then decreases. The point with maximum apparent viscosity is called the transition foam quality. The position of the transition foam quality depends on the flowrate and the surfactant concentration. There is little hysteresis observed during the experiment between decreasing foam quality and increasing it.

In Fig. 2B), for foam qualities less than the transition foam quality, as the gas fractional flow increases, the capillary pressure increases, and reaches the plateau capillary pressure. After the transition foam quality, the capillary pressure decreases. The value of the plateau capillary pressure is depending on the flowrate and the surfactant concentration. To better discuss each case, the test results will be compared directly for different fixed flowrate with changing AOS concentration, and fixed AOS concentrations with changing flowrate.

Effect of surfactant concentration on foam flow: fixed flowrate with changing AOS concentration

9 × flowrate with changing AOS concentration

In Fig. 3 all test results are represented for 9 × flowrate with changing AOS concentration; black is 1% and blue is 0.05%. This is the highest flowrates among all experiments. In Fig. 3A), it is observed that when the foam conditions are wetter than the transition foam quality, apparent viscosity overlay each other. When the foam qualities are greater than the transition foam quality, it is hypothesized that foam system is in the generation-limiting regime or the dry-foam regime. In this regime, whenever we have a homogeneous sandpack, at higher foam qualities, foam generation by snap-off may be absent, this will be discussed in detail in Sect. 3.4.5. At AOS lower concentration, the maximum apparent viscosity is apparently reached at a lower foam quality than that at higher concentration. However, at higher concentration the maximum apparent viscosity can go to higher foam quality before the foam gets weaker with the increasing foam quality.

Fig. 3
figure 3

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for the 9 × flowrate experiments at 1% AOS (black) and 0.05% AOS (blue) concentrations. The green dashed line is a guide to illustrate the plateau of the capillary pressure.

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In Fig. 3B), the values of transition capillary pressure are about the same at 9 × flowrate for low-and-high surfactant concentration, and both have a similar plateau indicated by the green dashed line. But at the lower concentration, the plateau ends at a lower foam quality. The value of this plateau is interpreted as the “plateau capillary pressure”. In the previous work48 and this study, the term “plateau” is used because as gas velocity increases, the rate of foam generation by lamella division might also increase and, hence, increase the value of this plateau capillary pressure.

3 × Flowrate with changing AOS concentration

Figure 4 represents all test results for 3 × flowrate with changing AOS concentration; black is 1% and blue is 0.05%. In Fig. 4A), a similar trend to the 9 × flowrate on the apparent viscosity was observed, at a lower concentration, the transition foam quality occurred at a much lower foam quality. In Fig. 4B), comparing to the 9 × flowrate test results, the capillary pressure at the plateau occurs not only at a lower foam quality, but also at a lower capillary pressure, indicated by the green dashed lines. The plateau capillary pressure is dependent on both surfactant concentration and flowrate. In both cases, the trend is the same, once the foam gets dryer than that of the transition foam quality, capillary pressure decreases because foam is hypothesized to be on the generation-limiting regime.

Fig. 4
figure 4

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for 3 × flowrate experiments at 1% AOS (black) and 0.05% AOS (blue) concentrations. The green dashed lines are guides to illustrate the plateau of the capillary pressure.

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1 × Flowrate with changing AOS concentration

In Fig. 5 are represented all test results for 1 × flowrate with changing AOS concentration. This is the lowest flowrate evaluated with changing concentration; black is 1%, red is 0.5%, and blue is 0.05%. In Fig. 5A), apparent viscosity is remarkably similar for the 1% and 0.5% concentration. However, at the lowest concentration (0.05%), the transition foam quality occurs at a much lower foam quality (fg* = 0.3). The weak foam regime starts for foam quality greater than 0.3 foam quality. In Fig. 5B), the capillary pressure test results indicates that for fg > fg*, the transition foam quality, the capillary pressure drops. Then, a plateau in capillary pressure occurs below the transitional foam quality for the two higher surfactant concentrations but not for the lowest surfactant concentration, indicated by the green dashed lines.

Fig. 5
figure 5

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for 1 × flowrate experiments at 1% AOS (black), 0.5% AOS (red), and 0.05% AOS (blue) concentrations. The green dashed lines are guides to illustrate the plateau of the capillary pressure for 1% and 0.5% AOS concentrations.

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By comparing foam-flowing test results, especially the change in maximum apparent viscosity, it is obvious that change in flowrate has a much stronger influence on the foam strength for 0.05 wt.% AOS14-16 than it does for the higher concentration. In the case of the lowest velocity, the maximum apparent viscosity at 0.05 wt.% is much lower than 0.5 and 1.0 wt.%. The reason for this could be a combination of.

  1. 1.

    The surfactant concentration of 0.05 wt.% AOS was well above its CMC in 3% NaCl solution. However, when foam is generated inside the porous media, much of surfactant molecules will be concentrated at the gas–water interface, and concentration in aqueous phase could be approaching the CMC. In this case, micelles could be absent, and the foam is less stable54,55,56.

  2. 2.

    The minimum pressure gradient (MPG)29,30 to generate strong foam increases with decreasing surfactant concentration, such that at lower flowrate, a much weaker foam is generated with 0.05 wt.% AOS.

  3. 3.

    The surfactant concentration likely has an influence on the disjoining pressure and the interfacial viscosity, and thus foam stability54,55,56.

  4. 4.

    The overlap of the apparent viscosity curves for different concentrations for foam quality lower than the transition foam quality is remarkable. This overlap implies that in this range, foam apparent viscosity is independent of surfactant concentration.

Effect of flowrate on foam flow: fixed surfactant concentration with changing flowrate

1wt% AOS concentration with changing flowrate

In Fig. 6, all foam-flowing test results are represented for 1 wt% AOS14-16 concentration with changing flowrate; circles are 1X flowrate, triangles are 3X flowrate, and squares are 9X flowrate. This is the highest concentration evaluated. In Fig. 6A), it is observed that increasing the flowrate decreases the apparent viscosity in the lower foam quality region because of the shear thinning effect57,58. However, for the higher foam qualities that are dryer than the transition foam quality, the curves fall on top of each other. In Fig. 6B), looking at the capillary pressure, it has a trend that is shown by the dashed green line for different flowrates in the low-quality regime. A similar trend was shown by Ouali et.al. who estimated capillary pressure from gas saturations measured by CT59.

Fig. 6
figure 6

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for 1wt% AOS (black) concentration at 1x, 3x, and 9 × flowrates. The green dashed line is a guide to illustrate the trend of the capillary pressure.

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0.05wt% AOS concentration with changing flowrate

Shown by the lowest tested surfactant concentration, Fig. 7 represents the results for 0.05wt% AOS14-16 concentration with changing flowrate; circles are 1X flowrate, triangles are 3X flowrate, and squares are 9X flowrate. In Fig. 7A), the transition foam quality occurs at a lower foam quality compared to the higher concentrations.

Fig. 7
figure 7

(A) Apparent viscosity and (B) Capillary pressure vs. foam quality for 0.05wt% AOS (blue) concentration at 1x, 3x, and 9 × flowrates. The green dashed line is a guide to illustrate the trend of the capillary pressure.

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Test results represented in Fig. 7B) indicate that at a higher flowrate and foam quality, a much higher capillary pressure is measured. Again, the green dashed line denotes the common trend of the capillary pressure with changing flowrate in the low-quality regime.

Discussion of results

Low-quality regime

In Fig. 6B) and 7B), the capillary pressure measurements for the foam quality lower than the plateau follow the common trend line. This trend line is independent of surfactant concentration and flowrate. In this regime, the hypothesis is that the bubbles are not limited by coalescence due to high capillary pressure, nor limited by foam generation. The evidence for this hypothesis is that the capillary pressure depends only on the foam quality (fg). In addition, the hypothesis is that foam texture (bubble size) is independent of surfactant concentration and flowrate60. However, the flowing fraction of bubbles is expected to increase with foam quality.

From Fig. 6A) and 7A), at foam quality equal to 0.4, the apparent viscosity as a function of total velocity is shown on a log–log scale in Fig. 8. The slope of the log–log plot for 1% and 0.05% AOS are -0.90 and -0.96, respectively. This slope of the log–log plot is typical for a Bingham-plastic fluid in porous media61,62,63,64,65. This rheology suggests a binary population of either flowing or stationary bubbles. Although the concentrations are 20 times different, the apparent viscosities approximately overlay each other. The implication of this similar apparent viscosity is that the bubble sizes in this low-quality regime are independent on concentrations. It may be the flowing fraction of bubbles that are changing with foam quality. Further details are provided in Vavra et al48.

Fig. 8
figure 8

Log–log plot of apparent viscosity versus total superficial velocity at fg = 0.4 foam quality; for 1 wt% AOS, the slope n = -0.90 and for 0.05 wt% AOS, the slope n = -0.96.

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High-quality regime

The high-quality regime is where foam quality is greater than the transition foam quality (fg > fg*). The capillary pressure decreases with increasing foam quality in this regime. Therefore, because the capillary pressure is less than the plateau capillary pressure, foam coalescence is unlikely the controlling mechanism, but rather due to the lack of foam generation. At the higher qualities in the studied experimental conditions of high-permeability homogeneous sandpack with little pore body to pore throat contrast. the hypothesis is that foam generation becomes the limiting factor of the foam apparent viscosity in this regime. Vavra et al.48 found that the foam texture is limited by the foam generation rather than coalescence due to limiting capillary pressure by observing the foam texture and type corresponded to foam coarsening and continuous gas channel in this high-quality regime.

Transition foam quality

The transition foam quality is the quality at the peak of the apparent viscosity-foam quality curve. For the lowest surfactant concentration (0.05%) with slowest flowrate (1x), the “high-quality regime” occurs for foam quality greater than 0.3 (fg > 0.3). At the highest flowrate (9 × and 0.05% concentration), the high-quality regime occurs for foam quality greater than 0.7 (fg > 0.7). For the 1% AOS concentration with slowest flowrate (1x), the “high-quality regime” occurs for foam quality greater than 0.5 (fg > 0.5). At the highest flowrate (9x), the high-quality regime occurs for foam quality greater than 0.87 (fg > 0.87). The transition foam quality increases with either increasing surfactant concentration and/or flowrate.

For surfactant concentration at 0.5%, only 1 × flowrate was measured. As shown in Fig. 5A), the transition foam quality of 0.5% higher than that of 1% could be an artifact and the transition foam quality of 0.5% should be no greater than that of 1% (see Fig. 2A)). Therefore, it is assumed that the transition foam quality of 0.5% and 1% AOS under 1 × flowrate are the same. This assumption is also extended to 3 × and 9 × flowrates where no measurement was made. A contour plot is made based on this assumption, shown in Fig. 9.

Fig. 9
figure 9

Contour plot of transition foam quality as a function of surfactant concentration and total fluid superficial velocity. (The measured transition foam quality at 0.5 wt% AOS and 3 ft/day is 0.54).

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Based on the trend shown in Fig. 9. If the surfactant concentration is higher than 0.5%, the transition foam quality is independent of surfactant concentration and increases with increasing total superficial velocity. If the surfactant concentration is lower than 0.5%, the transition foam quality is dependent on both surfactant concentration and total fluid superficial velocity. The transition foam quality increases with both increasing in surfactant concentration and total superficial velocity, below 0.5% surfactant concentration.

Plateau capillary pressure

The plateau capillary pressure is not changing with surfactant concentration for the highest flowrate (9x, Fig. 3B). For lower flowrates (1 × and/or 3x, Fig. 5B) and/or Fig. 4B), the plateau capillary pressure is either dependent on the surfactant concentration or degenerated to a single point. A contour plot is made for plateau capillary pressure with surfactant concentration and gas velocity multiple, shown in Fig. 10. Gas velocity multiples include the base rate, three times the base rate and nine times the base rate. The contour plot is Interpolated between 0.05% and 1% concentration. The blue-circle is posting a single value (0.37 psi) that is measured at 0.5% concentration but was not used for interpolating the contour plot. This contour plot shows the increased spacing at larger gas velocity multiples. This implies that the plateau capillary pressure has less dependence on surfactant concentration at higher gas velocities. When under a higher velocity, the generation rate of bubbles by lamella division is faster28,66. With this faster rate of foam generation, the stability of lamella due to coalescence is of less consequence, and thus, less dependent on the surfactant concentration.

Fig. 10
figure 10

Contour plot of plateau capillary pressure (psi) as a function of surfactant concentration and gas velocity multiple. The contour plot is made based on the red-cross values measured at 0.05% and 1%, and the blue-circle value of 0.37 is measured at 0.5%.

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Foam generation by capillary snap-off

Foam generation by capillary snap-off is generally accepted as the primary mechanism for foam generation in porous media67. However, it is well known that capillary snap-off does not occur in smooth water-wet capillary tubes57. The requirement for snap-off is for the pore body/ throat ratio to be sufficiently greater than unity67,68. Falls et al. found that the capillary pressure for snap-off in glass beads must be less than one half of the capillary entry pressure58. Thus, there is no capillary snap-off when surfactant solution slowly drains by gravity from a uniform bead pack28. Glass beads and well sorted sand (as used in this investigation) have pore body/throat ratios that are close to unity and also usually have residual or trapped non-wetting phase saturation less than 10%. In contrast, Berea sandstone is a consolidated rock with a large pore body/throat ratio (5 – 11)69, which results in a residual non-wetting phase residual saturation of ~ 40%. Based on previous experimental studies, we have not encountered issues with foam generation in Berea sandstone20.

Foam can also be generated by capillary snap-off in constricted glass capillaries, glass bead packs28, sandpacks70,71 and sintered glass72 if there is sufficient contrast in permeability in series with the direction of flow (e.g. > 4). In such cases, the foam generation is independent of the total fluid flow rate, an indication that capillary snap-off is the primary foam generating mechanism.

This discussion on capillary snap-off should make it clear that conditions are not favorable for foam generation by capillary snap-off at higher gas fractional flows in the homogeneous sandpack used in this investigation. For this reason, foam was pre-generated by a short section of the same sand that was retained by a frit that has finer pore size that the sand. The step change in permeability at the exit of the frit is going from finite pore size to infinite pore size. It determines the size of the pre-generated bubbles, independent of foam quality, flow rate, and surfactant concentration (provided the concentration is well above the CMC).

Foam generation by lamella or bubble division

A foam generation mechanism that is an alternative to capillary snap-off is due to lamella or bubble division when a “train” of moving bubbles splits at a branch point. This was first mentioned by Mast66 and is described and distinguished from capillary snap-off by Falls et al.28. At low pressure gradients, bubbles can propagate as a translating “train of bubbles” that is surrounded by stationary bubbles. In this case, the size of the bubbles in the train remains the same. If the pressure gradient is increased enough, some of the stationary bubbles mobilize and a bubble train may divide at a branch point to form two bubble trains. Bubbles at the branch point were observed to “divide” into two bubbles if the original bubble is sufficiently longer than the bubble diameter. This is foam generation by “lamella division”. Lamella division is always favorable for generating strong foam. The favorable conditions for lamella division require an adequate pressure gradient.

If conditions are not favorable for either capillary snap-off or lamella division, sometimes continuous gas channels are observed for form as described by Vavra et al.48. Conditions favorable for continuous gas channels are: (1) large foam quality, fg, (2) small flow rates, and (3) small surfactant concentrations.

Plateau capillary pressure vs. limiting capillary pressure

The term, “limiting capillary pressure” was not used to describe the results of this manuscript because of the reasons discussed in Vavra et al.48. The term, “limiting” implies that the foam strength is limited by the maximum stable capillary pressure. The term “plateau” is chosen to describe the flattened portion of the capillary pressure curves. Whereas “limiting” implies coalescence by lamella rupture as the dominating mechanism preventing further increase in capillary pressure with increase in foam quality (gas fractional flow). The term, “limiting capillary pressure” was originally used by Khatib et al.45. because it was thought at that time that the reason for the apparent viscosity decreasing with foam quality greater than the transition foam quality was because of foam coarsening due to coalescence at the limiting capillary pressure. Here we show that the plateau in the capillary pressure occurs at foam qualities less than or equal to the transition foam quality. This is in the range where apparent viscosity is increasing with foam quality. The decrease in the apparent viscosity for foam quality greater than the transition foam quality as shown here and by Vavra et al48. is not believed to be due to foam coalescence for reasons that we now explain.

It should be noted that the decrease in the apparent viscosity for foam quality greater than the transition foam quality corresponded to the region where the capillary pressure was less that the plateau capillary pressure. If the capillary pressure is less that the plateau capillary, then foam coalescence should not be the limiting factor in the foam strength. Thus, we infer that foam generation (or lack thereof) by bubble (or lamella) division at branch points in a “train” of mobile bubbles is the governing mechanism in this regime (fg > fg*). The evidence for this is the increase of the transition foam quality with increase in total fluid velocity.

If one wishes to design an experiment in which the foam mobility is dominated by the limiting capillary pressure, then the porous media should be selected such that capillary snap-off will continue to generate new bubbles up to large values of foam quality, fg. This can be accomplished in either of two ways: (1) use a porous medium with a large pore body/throat ratio, such as Berea sandstone, or (2) pack a sandpack with alternating bands of contrasting grain sizes of at least 2:1 (or permeability contrast of 4:1).

Interpretation of the difference of results from Khatib, et al45

A reader may interpret Fig. 2 as the capillary pressure reaching a plateau at a foam quality well before the peak in apparent viscosity, and then capillary pressure even appearing to fall with increasing foam quality in the high-quality regime. This initially appears to go against published test results45, which suggests that foam collapse happens at a capillary pressure comparable with a limiting capillary pressure. It should first be recognized that the foam quality scan experiment in the current manuscript began at high foam quality and the liquid injection was increased in steps while maintaining a constant gas injection rate. Thus, the sequence of the change in foam quality is opposite of that described by Khatib, et al45. Here, only a weak foam existed at the higher foam qualities because the liquid saturations have failed to reach its threshold for capillary snap-off conditions, and the pressure gradient was too low for bubble division at branch points in the bubble trains. As the foam quality further decreased, the apparent viscosity increased, and the capillary pressure increased until it either reached a plateau or it peaked and then decreased. The reason that the apparent viscosity and capillary pressure increased upon increasing the liquid flow at a constant gas flow rate is because adequate liquid saturation in the matrix must exist before foam can generate by snap-off. Thus, both the apparent viscosity and capillary pressure are increasing with increasing liquid injection rate (decreased foam quality), until a maximum apparent viscosity is reached. If the fixed gas injection rate is sequentially increased in a series of experiments, the peak foam viscosity occurs at a higher foam quality (gas fractional flow) because the increased total fluid flux increases the pressure gradient and promotes foam generation by bubble division at higher foam qualities73.

The experiments by Khatib, et al. differed in that the experiments began at lower foam quality (gas fractional flow) and the threshold of water saturation in the matrix was surpassed for foam to be generated by capillary snap-off. Once the foam is generated, the resulting large pressure gradients is sufficient for continuing generating strong foam by bubble division at branch points in a train of moving bubbles, even as the foam quality (gas fractional flow) increases.

Details about how the capillary pressure is measured is reported in Vavra, et al47. This paper shows that there is little hysteresis between decreasing foam quality or increasing foam quality, if the experiments are extended long enough. The experiments by Khatib, et al45. were conducted with severe time limits and there was not sufficient time to determine if the results were steady-state or if there would be hysteresis.

The capillary pressure is a function of the liquid saturation, with only minor dependence on whether or not foam is present. The contribution of the liquid in the thin lamella between foam bubbles generally contribute an insignificant amount to the liquid saturation in porous media. Thus, the capillary pressure is a good indicator of changes in liquid saturation. Therefore, the authors interpreted capillary pressure above the transition foam quality being less than that at the transition as an indicator that the liquid saturations being higher than the liquid saturation at the transition. It may appear puzzling to someone why the liquid saturation would become larger as the liquid fractional flow is less than that at the transition. The interpretation of the authors is that in homogeneous, well-sorted sandpacks, foam generation by capillary snap-off ceases above some moderate foam quality (or below some liquid fractional flow). When foam is no longer being generated, a continuous gas foam now exists in the sandpack. The continuous gas foam is a weak foam, which leads to a higher water saturation. Evidence of this conjecture is that when the gas injection rate increased (from 1x) by 3x, 9x, or 16x, the transition foam quality also increases. This increase in the transition foam quality with the increase in gas rate is interpreted as due to foam generation by bubble division at branch points on trains of moving bubbles. For a fixed fractional flow, increasing the total flux (1) increases the pressure gradient, (2) the increased pressure gradient mobilizes some stationary bubbles and thus increases the fraction of the bubbles that are mobile, (3) the increased fraction of mobilized bubbles also increases the number of branch points in the trains of moving bubbles, (4) the increased number of branch points increases the rate of bubble division at the branch points, (5) the increased rate of bubble division decreases the average size of the bubbles, (6) the reduction in the average size of the bubbles results in an increase of the apparent viscosity.

Interpretation of the difference of results from natural sedimentary rocks

The results of this manuscript are limited to well sorted sandpacks. Generally, in natural sedimentary rocks, hysteresis is less of an issue. Ettinger and Radke74 found that for 0.83 wt% AOS in 0.8 D fired Berea sandstone at a foam quality of 0.9, weak foam was observed at 3 ft/day but strong foam at 8 ft/day. Consolidated sandstones have, in some cases, shown a foam flood with initially a weak foam, then a strong-foam starts generating from the outflow end, and the strong-foam slowly propagates against the direction of flow to the inflow end51. The outflow end corresponds to a zero capillary pressure boundary condition that results in a high liquid saturation at the outflow end, and thus favorable conditions for snap-off. A more detailed examination by Almajid and Kovscek52 has shown that, in some cases, the initial strong-foam generation corresponded to flow in series across lower-permeability lamella in the middle of the core, which apparently generated foam by capillary snap-off. This heterogeneity effect has been demonstrated to be the case of sandpacks and consolidated media with heterogeneities in series71,72,75.

Conclusion

In conclusion, the experimental setup involving a 145-Darcy sandpack with an in-situ capillary pressure probe has provided valuable insights into the dynamics of foam flow. An increase in transition foam quality with rising surfactant concentration was observed for concentration below 0.5 wt% AOS, and no change between 0.5 wt% and 1 wt% AOS. When the foam qualities are greater than the transition foam quality, a "foam generation-limiting" regime is identified, characterized by a decrease in apparent viscosity and capillary pressure. The uniformity of foam texture in the low-quality regime, independent of surfactant concentration and flowrate, adds a nuanced layer to our understanding. Interestingly, the study reveals a minimum surfactant concentration requirement for robust foam generation, highlighting the critical role of concentration in foam stability. Importantly, in the homogeneous sandpack utilized, limiting capillary pressure emerges as a significant factor only at lower flowrates and higher qualities, contrasting with the dominating influence of foam generation upon surpassing a minimum pressure gradient at high foam qualities. This suggests that the foam dynamics observed are specific to the characteristics of the sandpack used. The discussion extends to propose that in porous media with a larger pore body/throat ratio and/or greater permeability contrast, limiting capillary pressure may assume a more dominant role, a phenomenon potentially more pronounced in natural, consolidated rocks. The findings presented contribute not only to our comprehension of foam behavior in porous media but also offer crucial considerations for the design and interpretation of future experiments, especially in diverse geological contexts.