Since the 1980s, experimental and field studies have found an anomalously slow propagation of foam (Friedmann et al. 1991, 1994; Patzek 1996), a phenomenon that cannot be fully explained by surfactant adsorption. Friedmann et al. (1994) conducted foam‐propagation experiments in a cone‐shaped sandpack and concluded that foam, once formed in the narrow inlet, was unable to propagate at all at lower superficial velocities near the wider outlet. They concluded that long‐distance foam propagation in radial flow from an injection well is in doubt.
Ashoori et al. (2012) provide a theoretical explanation for a slower and/or nonpropagation of foam front at decreasing superficial velocity. By linking foam propagation to the minimum superficial velocity (or minimum pressure gradient ∇Pmin) required for foam generation in homogeneous porous media (Rossen and Gauglitz 1990; Gauglitz et al. 2002), the study reveals that the minimum velocity for maintaining the propagation of foam is far less than that for creating foam but greater than the minimum velocity for maintaining foam in place. Lee et al. (2016) and Izadi and Kam (2019) find a minimum velocity for foam propagation from analysis of a similar population‐balance model but associate it with the minimum velocity for foam stability.
In this study, we extend the experimental approach of Friedmann et al. (1991) in the context of the theory of Ashoori et al. (2012). We observe dynamic propagation of foam in a cylindrical core with stepwise increasing diameter such that the superficial velocity decreases from inlet to outlet (in a ratio of 16:1). Previously (Yu et al. 2019), we mapped the conditions for foam generation (at large superficial velocities) in a Bentheimer sandstone core, in relation to surfactant concentration and injected gas fraction (foam quality). In this study, we enrich the map with the conditions for downstream propagation of foam (at significantly smaller superficial velocities). We also interpret our results for both foam generation and propagation in terms of local pressure gradient (following the implications of Ashoori et al. 2012), which plays a dominant role in the mobilization and creation of foam.
Our results suggest that the minimum superficial velocities for both foam generation and propagation increase with increasing foam quality and decreasing surfactant concentration, in agreement with theory (Rossen and Gauglitz 1990). In addition, the minimum velocity for propagation of foam is much less than that for foam generation, as has been predicted by Ashoori et al. (2012). Implications of our laboratory results for field application of foam are briefly discussed.
Applications of foam in porous media range from enhanced oil recovery (Schramm 1994; Rossen 1996) and acid diversion in well stimulation (Burman and Hall 1986; Kennedy et al. 1992) to aquifer‐ and soil‐remediation processes (Hirasaki et al. 1997). For petroleum reservoir engineers, foam‐enhanced oil recovery is of interest because foam significantly improves the volumetric sweep efficiency of injected gas. Foam in porous media comprises liquid films (called lamellae) restricting the flow of gas in the pore network. The presence of lamellae greatly decreases gas mobility, resulting in improved gas sweep. The number of lamellae per unit volume of gas (inversely related to bubble size) determines the mobility decrease (also called the strength) of foam. The population of lamellae and therefore properties of the foam are the result of processes creating and destroying lamellae.
Propagation of foam over long distances far from the injection well is needed to divert gas flow deep into a reservoir. The conditions that dominate both creation and propagation of foam in porous media, therefore, have been a concern to foam researchers for decades. Various theories and experimental results cast light on the mechanisms of foam generation and propagation.
Theory suggests that a minimum pressure decrease ΔPmin is required to mobilize a static lamella blocking a pore throat (Bikerman 1973; Rossen and Gauglitz 1990). Mobilized lamellae can multiply by lamella division (Rossen 1996), triggering foam generation. A percolation theory for foam generation in steady gas‐liquid flow (Rossen and Gauglitz 1990) relates the minimum pressure gradient ∇Pmin or minimum superficial velocity for foam generation to rock and fluid properties such as permeability k, surface tension, and injected liquid volume fraction fw. This theory fits experimental data (Rossen and Gauglitz 1990; Friedmann et al. 1994; Yu et al. 2019) regarding the impact of injected liquid fraction fw on the minimum velocity for foam generation. A greater injected liquid fraction fw contributes to lamella creation (Rossen and Gauglitz 1990) and decreases the rate of lamella coalescence (Khatib et al. 1988). Therefore, foam generation becomes easier as fw increases because of effects on both lamella creation and destruction. More recent work (Yu et al. 2019) indicates an effect of surfactant concentration on the minimum velocity for foam generation, which reflects the link between lamella stability and foam generation. The surfactant concentrations used in that study (Yu et al. 2019) are far greater than the critical micelle concentration (Jones et al. 2016).
Gauglitz et al. (2002) conducted three different types of foam‐generation experiments. In fixed‐injection‐rate experiments (Fig. 1a), foam is generated by fixing total superficial velocity and foam quality [gas fractional flow, fg = (1 − fw)]. Superficial velocity is first set at a low initial value and is then increased in steps to larger values. On the triggering of foam generation at , the pressure gradient along the core increases abruptly (Gauglitz et al. 2002; Yu et al. 2019) to a much larger value reflecting strong foam. If superficial velocity is decreased after strong foam is created, strong foam can be maintained at superficial velocities at which it would not be created from a state of no foam or coarse foam.
In fixed‐pressure‐difference experiments (Fig. 2a), foam is generated by maintaining the pressure decrease across the core at a set value (Gauglitz et al. 2002). These experiments reveal a third, unstable steady state over a range of superficial velocities, with a pressure gradient intermediate between the coarse‐foam and strong‐foam states. The values of ∇Pmin and in an experiment at fixed injection rate correspond to the point where the plot of ∇P bends backward toward smaller values of ut with increasing ∇P.
Others have reported foam hysteresis in strong‐foam behavior, in the sense that, once strong foam is created, ∇P remains somewhat greater if superficial velocity ut is decreased (Lotfollahi et al. 2016) than its value at the same superficial velocity when ut was increasing. In contrast, the triggering of foam generation is an abrupt increase in ∇P by more than an order of magnitude starting from a state of coarse foam or no foam, a distinct phenomenon from the hysteresis in strong foam.
The population‐balance model of Kam and Rossen (2003) and its variants (Kam et al. 2007; Kam 2008; Lee et al. 2016) are designed to explain the experiments of Gauglitz et al. (2002). This model introduces a relation between pressure gradient, lamella creation, and foam generation. It is the only foam model demonstrated to represent and explain the trigger and multiple steady states seen in foam‐generation experiments (Gauglitz et al. 2002). Its predicted behavior is shown schematically in Fig. 2b.
Successful foam propagation in the field resembles a radial flow pattern with stable displacement of gas because of mobility control at the gas‐displacement front. In the near‐well region, the large velocities of gas and liquid, as well as large pressure gradient, favor the generation and propagation of foam (Rossen and Gauglitz 1990). At large distances away from an injection well, however, both superficial velocity and pressure gradient are low, and the issue of foam propagation further from the well comes into question. Some field applications of steam foam in the 1980s reported slow foam propagation to limited distances from the injector and therefore raise concerns about long‐distance foam propagation. Observation wells in a steam‐foam pilot in the Mecca Lease, Kern River, reported foam propagation to a distance of 43 m (140 ft) from the injector over 4.5 years (Patzek 1996). This was slower than the rate that would be predicted from surfactant adsorption and the high‐foam quality in that test. In Section 26 of the steam‐foam pilot in the Midway‐Sunset field, two observation wells 12 m (39 ft) from the injector reported a breakthrough of steam foam after 8 months of surfactant injection (Friedmann et al. 1994; Patzek 1996). Based on the estimated foam propagation rate to 12 m, foam should have arrived at the observation well 21 m (69 ft) from the injector after approximately 24 months of surfactant injection (Friedmann et al. 1994; Patzek 1996). Unfortunately, surfactant injection continued for a period of only 18 months before it was shut off, leaving the hypothesis untested (Patzek 1996).
Friedmann et al. (1994) conducted a foam‐propagation experiment in a cone‐shaped sandpack to seek explanations for what they interpreted as stalled propagation of steam foam in the Midway‐Sunset pilot. Surfactant (Chevron Chaser SD1020; 0.3 wt%) and nitrogen (N2) were coinjected at constant fg = 0.987 from the narrow inlet of the cone‐shaped sandpack with a 1.25:5.00 ratio of diameters between the injection and exit face. Foam generated near the inlet section then propagated to increasingly wider downstream sections, with six pressure‐difference measurements in total. According to the pressure response, strong foam stalled in the fifth section and did not reach the end of the sandpack, even after 300 pore volumes (PVs) of foam injection.
Friedmann et al. (1991) combined coreflooding experiments and numerical simulation. They developed a population‐balance simulation model and fit the model's coefficients to the results of six separate coreflooding experiments. Afterward, they conducted a foam‐displacement experiment in a Berea sandstone core with three different, increasing diameters (0.95, 2.5, and 5.0 cm) along the core length and compared the data with the simulation results of their population‐balance model. Surfactant solution and N2 were coinjected (at fg = 95%, and T = 100°C) into the vertically mounted core at a velocity that creates strong foam only in the narrow section (d = 0.95 cm). Foam propagation in the widest portion (d = 5.0 cm) was then observed and documented based on the pressure‐difference measurements across the three sections in the widest portion.
Their population‐balance simulation model assumes a minimum velocity for lamella creation. Coalescence depends in the model on surfactant concentration but not on capillary pressure or water saturation. Their simulation and experimental results agree well with each other and show that the breakthrough of strong foam at the core outlet was delayed by approximately 3.3 PVs compared with the breakthrough of surfactant. With a minimum velocity for lamella generation, the model should predict a minimum velocity for propagation and for maintaining foam, but this is not explored in the paper.
Ashoori et al. (2012) used the population‐balance model of Kam (2008) to explain and predict long‐distance foam displacement in the context of multiple foam steady states. They combined fractional‐flow analysis (also called the method of characteristics) and numerical simulation to study long‐distance foam propagation at various superficial velocities in a one‐dimensional linear porous medium. At superficial velocities greater than the minimum velocity for generation (Rossen and Gauglitz 1990; Gauglitz et al. 2002; Yu et al. 2019), strong foam can be created in situ and propagate downstream (Fig. 2b). As superficial velocity decreases to smaller values, an intermediate state of weak foam propagates ahead of the strong‐foam state, whose propagation rate slows. This may explain the delay in breakthrough of strong foam seen in the experiment of Friedmann et al. (1991). With a further decrease in superficial velocity to a minimum velocity for propagation, which we here call , the characteristic velocity of strong foam decreases to zero; foam stops moving forward. However, the model indicates that strong foam remains stable in place at . At a yet‐lower superficial velocity , foam becomes unstable and collapses. Their analysis implies that the failure of foam propagation at is a result of insufficient lamella creation at the leading edge of the foam front (from insufficient pressure gradient there), instead of complete destruction/collapse of foam. In other words, the flux of lamellae to the foam‐displacement front is quenched by the rate of lamella coalescence at the front.
Ashoori et al. (2012) solved for the traveling wave at the leading edge of the foam bank to conclude that the conditions for foam propagation are more stringent than for foam stability. Lee et al. (2016) and Izadi and Kam (2019) analyzed a population‐balance foam model incorporating trapped gas and a minimum ∇P for foam flow. They found that strong‐foam mobility can vary with distance from the injection well and reported a minimum velocity for foam propagation. They associated minimum velocity for propagation with the minimum velocity for foam stability.
In this study, we focus on gathering experimental evidence on foam propagation in a core of variable diameter. The configuration of this core (Fig. 3), based on that originally designed by Friedmann et al. (1991), provides an opportunity for foam to flow at three different superficial velocities ut in the three core sections of different diameter as it is injected at a constant volumetric flow rate Qt. As described previously, the analysis of Ashoori et al. (2012) suggests the existence of three transition points for foam behavior in terms of superficial velocity, illustrated schematically in Fig. 2b: , minimum velocity for foam generation; , minimum velocity for foam propagation; , velocity at which steady state of strong foam becomes unstable and collapses. We therefore design our experimental procedures (described later) in a way that the model's implications (Ashoori et al. 2012) can be examined and verified. Furthermore, we also explore the impacts of surfactant concentration Cs and foam quality fg (plotted in terms of injected liquid volume fraction fw) on foam propagation as a function of velocity. We plot the three key velocities (, , and ) against different foam qualities and surfactant concentrations. We then analyze the trend of data and discuss the implications.
Experimental Apparatus and Materials
Fig. 4 is a schematic of the apparatus. The core is mounted vertically with the narrow section at the bottom. Aqueous solutions and N2 are coinjected from the bottom. In total, seven pressure transducers (0 to 150 bar) and six pressure‐difference meters (0 to 10 bar) are placed along the core to monitor foam propagation. The safety range of the pressure‐difference meters is between 0 and 20 bar, with the accuracy range calibrated between 0 and 10 bar. In most cases, the pressure gradient shown in the Results is calculated based on the difference in absolute pressure between two adjacent absolute‐pressure gauges. The measurements from pressure‐difference meters are used only if the flowing pressure difference is less than 10 bar, and one or more of the absolute‐pressure gauges is damaged.
We use a cylindrical core of Bentheimer sandstone (k = 2.5 darcy, ϕ = 0.25) with stepwise changing diameters (Fig. 3). All experiments are conducted at a laboratory temperature of approximately 22°C. Surfactant solutions are made by weighing and mixing BIO‐TERGE® AS‐40 (Sodium C14‐16 Olefin Sulfonate) in brine (3.0-wt% sodium chloride). The vertically mounted core is divided into three sections (Fig. 3): a narrow inlet Section 1 at the bottom, with 1‐cm diameter and 6.1‐cm length (PV ≅1.2 mL); a wider middle Section 2 with 2.67‐cm diameter and 6.9‐cm length (PV ≅9.7 mL); and the widest and longest Section 3, with 4‐cm diameter and 27.0‐cm length (PV ≅84.8 mL). The ratio of superficial velocities is 16.0:7.1:1.0 from Section 1 to Section 3. The core is drilled from one large piece of cylindrical core (40 cm long and 4 cm wide) to avoid capillary discontinuities. Fig. 3 illustrates the locations of the pressure gauges along the core.
The pressure transducers used in our experiment are placed some distance from the section boundaries (Fig. 3): ΔP1 measures the pressure decrease of the entire Section 1 and first half of (approximately 3.5 cm) of Section 2; ΔP2 measures the second half of Section 2 and the beginning of (approximately 3.0 cm) Section 3. Drilling holes directly at the section boundary would be difficult, and distortion in flow at the boundary is difficult to interpret. We can infer the presence of strong foam in the narrow section from a large pressure difference between the first two taps ΔP1; propagation through the second section from the pressure difference between the second and third taps ΔP2; and propagation through the widest section from the next three pressure differences, ΔP3, ΔP4, and ΔP5 (Fig. 3). The pressure difference near the outlet ΔP6 could be distorted by the capillary end effect. The multiple pressure taps at the same diameter in Section 3 gives the most reliable indication of conditions for foam propagation and collapse. We describe how we infer ∇P in the next two sections and compare results inferred in Section 2 and measured in Section 3 in our results.
Backpressure was held fixed in each experiment. In most experiments, it was set to 10 to 15 bar. In some experiments, we set backpressure at 40 (Cs = 0.05 wt%, fg = 88%) and 60 bar (Cs = 0.3 wt%, fg = 88%) to allow attainment of superficial velocities in the desired range. Gas compressibility is relatively unimportant in our results. In each case, whether foam is advancing or retreating, our focus is on behavior at the leading edge of the foam bank. The pressure difference between this edge and the outlet of the core is insignificant.
Experimental Method: Defining Criteria and Procedure
Analogous to the criteria defined by Yu et al. (2019) for foam‐generation experiments, we define here the criteria and procedures for foam‐propagation experiments. The details of these criteria could change for studies in other porous media or with different foam compositions. The experiments proceed in three steps designed to determine the conditions for foam generation in Section 1, and then for propagation in Sections 2 and 3, and finally for foam collapse in Sections 3 and 2.
The first stage of our experiment is designed to measure in Section 1 and establish stable foam in that section before propagation into Sections 2 and 3. The criteria for determining are taken from Yu et al. (2019). A steady state of low ΔP1 must first be established. Then superficial velocity is raised until, on such an increase, ΔP1 increases quickly to a much‐larger value. The second stage of experiment focuses on the superficial velocity at which foam propagation begins, by increasing ut in steps. After strong foam breaks through at the end of the core, we finalize our experiment by stepwise decreasing ut to obtain the superficial velocity at which foam collapses. The experimental procedures, along with associated experimental artifacts, are defined and explained in 13 steps. For clarity, we also summarize the procedures in a flow chart (Appendix A). The procedure to clean and initialize the core and apparatus is summarized in Step 13.
Step 1. To determine in the given experiment, start with an injection rate well below the expected value of . If for a particular surfactant concentration and liquid volume fraction is already known or can be estimated from available data (Yu et al. 2019), we can use that information to select an initial injection rate of surfactant and gas. If an estimate of is not available, start at a relatively low a value of superficial velocity in the first section.
Step 2. Initialize the core with steady flow of brine and N2 at this superficial velocity.
Step 3. Start coinjection of surfactant solution and N2 at this same superficial velocity and liquid volume fraction fw. Because the PV of the first section is approximately 1.2 mL, keep the injection rate constant until at least 2.0 to 3.0 mL surfactant is injected, to satisfy adsorption in that section. If no foam generation is indicated (no substantial increase in ΔP1), increase the injection rate in steps until foam generation is indicated by a sharp increase in ΔP1 by at least a factor of 10, well beyond the magnitude of the increase in superficial velocity. As noted later, it is usually not possible to allow ΔP1 to reach steady state before decreasing the injection rate to prepare for the next step. If foam is already indicated by a large value of ΔP1 at the first injection rate, then cannot be determined from this experiment. The test for propagation can continue, however. The uncertainty in is the gap between the last superficial velocity before foam generation and the superficial velocity at which foam generation occurred.
The next series of steps are designed to determine from ΔP data from Sections 2 and 3 in turn, as follows:
Step 4. After foam generation has occurred in Section 1, allow ΔP1 to increase to between 4 and 5 bar. Before any significant increase is seen in ΔP2, decrease the injection rate to a much smaller value, one that is not expected to allow propagation through Section 2. If pressure continues to increase in Section 2, it is not possible to determine in that section in this experiment. (In that case, go to Step 6 if desired to check propagation into Section 3 but first verify that propagation does not proceed immediately into Section 3.) If propagation is not indicated in Section 2 at the first superficial velocity, continue with the low injection rate for a long period (approximately 24 hours) before any further changes to verify that propagation of strong foam has not occurred into Section 2. One must begin the test at a superficial velocity less than to determine .
Step 5. If no strong foam is indicated in Section 2 after a long period of injection (approximately 24 hours), increase the superficial velocity in a series of steps to greater values and, after each step, keep the flow rate constant for a short period of time. Repeat this procedure until strong foam is indicated in Section 2 by an increase in ΔP2. The first indication of propagation into Section 2 is a steady, large increase in ΔP1, which should begin shortly after the increase in superficial velocity. (ΔP1 comprises a significant part of Section 2). As ΔP1 stabilizes, ΔP2 should start to increase to a value of up to 100 times its earlier value. [One should avoid waiting too long (more than 24 hours) for a pressure response to avoid the so‐called incubation effect, where slow accumulation of perturbations over a long period of coinjection of gas and surfactant solution can lead to foam generation under conditions in which it would not otherwise be seen (Baghdikian and Handy 1991).]
Step 6. This superficial velocity is as measured in Section 2. From this superficial velocity estimate, the injection rate at the inlet required for foam to propagate is determined in Section 3.
Step 7. After a steady‐state ΔP2 is obtained, increase the injection rate to a value somewhat less than the injection rate for propagation in Section 3 estimated in the previous step. Verify that foam does not propagate at this velocity into Section 3 (i.e., ΔP2 does not increase more than proportionately to the increase in injection rate, and ΔP3 remains low).
Step 8. If no foam is indicated in Section 3 in 1 to 2 hours, raise the superficial velocity in a sequence of steps (each lasting approximately 1 to 2 hours) until foam propagation is indicated by an increase in ΔP3. As noted, the first indication of propagation into Section 3 is a steady, large increase in ΔP2 from its previous steady value. Hold that injection rate constant until steady‐state strong foam is established throughout Section 3 (i.e., in ΔP4, ΔP5, and ΔP6). If foam does not propagate throughout Section 3, increase the velocity again in steps until strong foam is indicated throughout Section 3. This final value represents for Section 3. We exclude any cases where a large pressure gradient at the end of the core, which reflects at least in part the capillary end effect, propagates upstream into the widest section (Ransohoff and Radke 1988; Apaydin and Kovscek 2001; Nguyen et al. 2003; Simjoo and Zitha 2013).
The uncertainty in for both Sections 2 and 3 is the gap between the largest velocity for which foam propagation is not indicated and the first velocity for which it is.
Once foam is established throughout the core, collapse of foam in a given section is indicated when, on a decrease in injection rate, there is a decrease in pressure gradient in that section by a factor of 5 to 10. This decrease in pressure gradient should be complete in a relatively short period (roughly 2 to 5 hours). It is likely that this represents a transition to continuous‐gas foam (Falls et al. 1988; Rossen 1996) rather than disappearance of all foam lamellae. We proceed as follows.
Step 9. After the core is filled with strong foam, decrease the superficial velocity in Section 3 in steps. The magnitude of velocity decreases should not be less than the velocity steps used in Steps 5 to 8.
Step 10. Hold the injection rate constant for at least 2 to 3 hours to see whether the state of strong foam is maintained. Here we make our judgment based on the measurement of ΔP4 and ΔP5. ΔP3 is affected (with unknown magnitude) by the strong foam that flows out of Section 2, and ΔP6 is likely affected by end effects.
Step 11. Collapse of strong foam is indicated by a decrease of pressure difference by a factor of 5 to 10 (or greater) in Section 3. If foam collapse is indicated in Section 3 (both ΔP4 and ΔP5 decrease by a factor of 5 to 10), record this velocity as the minimum velocity to maintain strong foam, . If strong foam remains stable, keep decreasing velocity in steps until foam collapse is indicated (or until further decreases are not feasible with the apparatus). The uncertainty is the difference between this velocity and the previous velocity tested.
Step 12. Repeat Steps 9 to 11 for Section 2 (if there is sufficient pressure in N2 cylinder). Document the value of for Section 2 in the same way as for Section 3.
Step 13. At the end of each experiment, we clean the apparatus in preparation for a new experiment. After injection of surfactant solution and N2 is stopped, we inject a solution of 50-vol% 2‐proponal for 1.5 to 2 PVs. Backpressure is then decreased to 1 bar. Afterward, we inject fresh water (20 to 40 PVs) to wash surfactant out of the core and apparatus. We then flush the apparatus with carbon dioxide first and then vacuum the system for 1.5 to 2.0 hours. After vacuuming, we saturate the system with fresh water, followed by injecting 1 to 1.5 PVs of brine solution of 3.0 wt% concentration of sodium chloride.
In case any of these procedures or criteria are unsatisfied, it may not be possible to record the desired velocity for the given section. We illustrate application of this experiment approach in the following section.
Application of the Experimental Method
In this section, we illustrate the application of our experimental method and the way we interpret some of the experimental artifacts. The vertical solid lines in Figs. 5, 6, and 7 represent times when superficial velocity is increased or decreased. The labeled curves represent the measured pressure differences between individual pairs of pressure taps.
Figs. 5a and 5b show an experiment to measure in Section 1, specifically an experiment with fg = 95%, and Cs = 0.3 wt%. We start the test with coinjection of brine and N2 at ut = 22.21 ft/D in Section 1 (data not shown). After steady state is reached, we switch to coinjection of surfactant solution and N2 (at the same ut) at time 0 (Fig. 5a). Through successive increases in superficial velocity (221, 228, and 251 ft/D), strong‐foam generation is triggered at = 251 ft/D at t = 1.4 hours (Fig. 5a). Before foam reaches steady state in Section 1, we decrease superficial velocity to 13.32 ft/D and hold it constant for approximately 20.5 hours (Fig. 5b) to verify that propagation of strong foam into Section 2 has not occurred. Pressure difference across Section 1 continues to increase, despite the decreased superficial velocity, and gradually stabilizes. Clearly there had been some decrease in mobility in Section 1 (increase in ΔP1) on the increase in ut to 221 ft/D at approximately 0.9 hours, but it stabilizes at a ΔP value too small to be considered strong foam. We conclude that is 251 ft/D in this experiment. At t = 16.1 hours, ΔP2 starts to increase but then stabilizes at a value we consider to be too small to represent strong foam. At approximately 22 hours, an increase in velocity triggers foam propagation in Section 2. Our next example focuses on a different experiment to illustrate the determination of .
Figs. 6a and 6b show an experiment with fg = 95% and Cs = 0.05 wt% to illustrate the procedures for determining foam propagation in Sections 2 and 3. After foam generation is triggered in Section 1 (data not shown), we keep the superficial velocity steady at ut = 11.4 ft/D for approximately 18 hours. There is no significant increase in ΔP2, ΔP3, and ΔP4 during this period. We begin by increasing superficial velocity in Section 2. For the first two superficial velocities tested (3.1 ft/D from 19.1 to 21.7 hours and 4.6 ft/D from 21.7 to 22.9 hours), no significant increase in either ΔP1 or ΔP2 is observed (Fig. 6a). At t = 22.9 hours, we increase the superficial velocity to 6.2 ft/D. ΔP1 immediately starts to increase (Fig. 6a), indicating that strong foam begins to propagate into the first half of Section 2. After approximately 1.5 hours, ΔP1 stabilizes, and at about the same time ΔP2 starts a sharp and continuous increase to a value of approximately 400 times greater than it had been. The sharp increase of ΔP1, followed by the increase of ΔP2 to a much greater value, indicates the propagation of strong foam in Section 2 at ut = 6.2 ft/D.
We next repeat the same procedure for Section 3. As strong foam in Section 2 achieves steady state (at 34.5 hours), there is some increase in ΔP3 (Fig. 6b). ΔP3 fluctuates between 0.0 and 0.5 bar for 13 hours (Fig. 6b); eventually, the trend stops increasing. This is similar to behavior in Section 2 (ΔP2) in the experiment in Fig. 5b. In both cases, we judge that this does not indicate successful propagation of strong foam. During an increase in superficial velocity to 2.86 ft/D at approximately 46 hours, there is an unmistakable increase in ΔP3. Moreover, foam propagates at this superficial velocity to ΔP4 and further downstream (data not shown). We conclude that is 2.86 ft/D for Section 3 based on the result.
As noted, based on our criteria, a modest and stable increase in ΔP in a section is insufficient evidence for successful foam propagation. In addition, there is some scatter in results for Sections 2 and 3. For the experiment with fg = 95% and Cs = 0.3 wt%, propagation of strong foam begins at 2.94 ft/D in Section 2 (Fig. 5b) and 1.94 ft/D in Section 3 (data not shown). For the experiment with fg = 95% and Cs = 0.05 wt%, propagation of strong foam begins at 6.2 ft/D in Section 2 (Fig. 6a) and 3.57 ft/D in Section 3 (Fig. 6b). These differences may reflect the flow distortions in Section 2 arising from the change in diameter between sections (illustrated schematically in Fig. 3). We report both sets of results. Compared with the overall trend, the differences are not great.
In Figs. 7a and 7b, we illustrate the procedures for determining for an experiment with fg = 82%, Cs = 0.05 wt%. Strong foam propagates to Section 3 at 1.15 ft/D. At t = 25.4 hours, superficial velocity in Section 3 is decreased to 0.72 ft/D (Fig. 7a). ΔP5 and ΔP6 start to decrease and fall to less than 1.0 bar in 5 hours (Fig. 7a). ΔP3 and ΔP4 also start to decrease but eventually stabilize, and ΔP3 even rebounds at 42 hours (Fig. 7a). We further decrease superficial velocity to 0.58 ft/D at t = 47.8 hours (Fig. 7b). ΔP4, ΔP5, and ΔP6 decrease to less than 1 bar in approximately 4 hours, and ΔP3 lingers at approximately 1.5 bar for another 23 hours. At t = 73.6 hours, we decrease superficial velocity in Section 3 to 0.48 ft/D. ΔP3 decreases to less than 1.0 bar in 3 hours, indicating collapse of strong foam in Section 2. Although there is some ambiguity in the exact value of , we conclude that, for Section 3, is 0.48 ft/D.
Figs. 8a and 8b plot the critical superficial velocities for foam generation, propagation, and collapse against liquid fractional flow for the surfactant concentrations used. Appendix B provides the numerical values of the points plotted in Figs. 8a and 8b. The corresponding foam qualities are labeled in the plots. Values corresponding to , , and are labeled in the plot. The values of and are considerably less than for each surfactant concentration and foam quality. As in previous studies (Rossen and Gauglitz 1990; Friedmann et al. 1994; Gauglitz et al. 2002; Yu et al. 2019) (cf. Fig. 1b), there is some scatter in our results. There is an additional uncertainty caused by the stepwise increase/decrease of superficial velocity. The magnitude of this uncertainty is indicated by the error bar for each datum in Figs. 8a and 8b, which represents the difference between the recorded superficial velocity (i.e., ) and the superficial velocity tested at previous step, at which foam does not propagate). The leftward error bars represent the size of the last velocity increase for and , and the rightward error bars represent the size of velocity decrease in determining .
The dashed lines interpolated from the critical superficial velocities divide the data into four subregions (Figs. 8a and 8b). The region to the right of the dashed line connecting values of indicates the flow conditions for foam generation. The region to the right side of the dashed line connecting values of defines conditions for propagation of strong foam into a region without foam. The region between the dashed lines connecting values of and values of represents the conditions at which the stability of strong foam can be maintained. At flow conditions to the left side of the dashed line, connecting values of strong foam can neither be generated nor maintained.
We observe successful propagation of strong foam at ut = 0.385 ft/D for the experiment of fg = 82%, and Cs = 0.3 wt% (Fig. 8b). We could not decrease the superficial velocity to smaller values during this experiment because of the limitation in the range of the gas mass‐flow meter. Therefore, we could not identify a critical value of superficial velocity , below which foam does not propagate at fw = 0.18.
Although superficial velocity is fixed in our corefloods, the theory of foam generation and propagation illustrated in Figs. 1 and 2 holds that pressure gradient ∇P plays the key role in both processes. Figs. 9a through 9d plot our experimental data in terms of superficial velocity and pressure gradient in the manner of Fig. 2. The vertical dashed lines represent the range of superficial velocities for foam generation, propagation, and collapse. The boundaries of the velocity interval are determined by the critical superficial velocities, and their uncertainties are plotted in Fig. 8. The dashed curves give a qualitative illustration of the multiple steady state of foam in porous media. We plot our data in such a manner (Fig. 9) to visualize the contrast of the pressure gradient between different steady states of foam. In addition, it helps relate our experimental results to implications from previous theories on foam generation (Gauglitz et al. 2002; Kam and Rossen 2003; Kam 2008; Lee et al. 2016) and propagation (Ashoori et al. 2012).
Interpreting ∇P at the critical transitions in our experiments is complicated by four issues.
First, as for determining transition superficial velocities, there is a gap between the last datum before the transition and that at which the transition is observed, represented by the error bars in Fig. 8 as discussed previously.
Second, the transition itself is marked by a sudden, marked increase or decrease in ∇P away from the transition value, which cannot be observed directly. We estimate the pressure gradient at the onset of foam generation by linear extrapolation of data measured at superficial velocities just less than . Similarly, we estimate the pressure gradient at foam collapse from strong‐foam data at velocities just greater than .
Third, we have no pressure tap entirely within the first and second sections of our core for measuring ∇P there (Fig. 4). We derive an approximate estimate using Darcy's law for incompressible rectilinear flow in two cores of different diameter. In that case, pressure difference scales roughly with the length of the section and the inverse square of the diameter. Based on this approximation, 12 of 13 pressure differences across the first tap (ΔP1) arise from the first section and 2.2 of 3.2 of ΔP2 is from the second section. These are only approximations, of course, but allow us to obtain a rough estimate of ∇P in each section. For Section 3, ΔP3, ΔP4, and ΔP5, entirely within that section, directly reflect ∇P in that section.
Discussion and Conclusions
Our experiments demonstrate the existence of three critical superficial velocities for the generation (), propagation (), and destruction () of foam in steady gas‐liquid flow in homogeneous porous media. Consistent with previous theory (Ashoori et al. 2012) and experiment (Friedmann et al. 1991, 1994), mobilizing the displacement front of strong foam requires a minimum superficial velocity . At superficial velocities less than , the steady state of strong foam cannot move forward but can be stable until a yet‐lower superficial velocity is reached.
The critical superficial velocities needed to maintain the stability and the propagation of strong foam are considerably less than the superficial velocity required for triggering foam generation in steady flow (Figs. 8a and 8b).
As previous experiments show (Rossen and Gauglitz 1990; Gauglitz et al. 2002; Yu et al. 2019), foam generation becomes easier with increasing surfactant concentration (even far above the critical micelle concentration) and liquid volume fraction injected. The same trend applies to foam propagation and maintenance (Figs. 8a and 8b). The impact of liquid volume fraction and surfactant concentration on foam propagation is less significant compared with their impact on foam generation. However, the increase of propagation velocity as foam becomes dryer is still substantial. The difficulty of generating and maintaining foam in porous media reflects the decrease in lamella stability.
The population‐balance model of Kam and Rossen (2003), Kam et al. (2007), and Kam (2008), as applied by Ashoori et al. (2012), predicts the existence of minimum velocities and pressure gradients for foam generation, propagation, and stability in place. In this model, foam generation, propagation, and stability are the result of competing effects of lamella creation and destruction. Increasing velocity and decreasing foam quality help lamella creation, and increasing surfactant concentration [through its effect on the limiting capillary pressure (Khatib et al. 1988; Apaydin and Kovscek 2001)] and decreasing foam quality help lamella stability. The trends with velocity, foam quality, and surfactant concentration seen in Fig. 8 agree with trends predicted by that model and with theories of foam stability.
In field application, if superficial velocity is insufficient at the well, there are ways to improve foam generation. Slugs of surfactant solution and gas are often injected alternatively (surfactant alternating gas). In the near‐well region, alternative drainage and imbibition of liquid then creates favorable conditions for foam generation. At distances far away from the injection well, however, the effects of alternating slug injection are greatly damped, where the flow of surfactant solution and gas comingles as if they are being coinjected. In our experiments, surfactant and N2 are coinjected at a fixed liquid volume fraction, a condition that closely resembles the flow condition far from injection well.
Determining the implications of the trends shown here for a given field would require experiments conducted under conditions of, and with fluids from, that field. In our experiments, N2 foam is generated and flows at low temperature (average, 22°C) and low pressure (10 to 60 bar backpressure). The salinity of solutions we use is relatively low (3.0 wt% sodium chloride). The porous media used in the corefloods is homogeneous, free of oil, and highly permeable (which also implies low capillary pressure). Under reservoir conditions (much higher temperature and salinity, lower permeability, presence of oil, etc.), the difficulty of foam propagation observed in our experiments is likely to be magnified. Placing foam far from a well in a heterogeneous reservoir may not depend on direct propagation of foam from the well, however (Falls et al. 1988; Tanzil et al. 2002; Shah et al. 2019). Moreover, a process that depends on altering the injection profile in a layered reservoir may not depend on deep foam propagation.
surfactant concentration, wt%
injected liquid volume fraction
critical superficial velocity for foam generation, ft/D
critical superficial velocity for foam propagation, ft/D
critical superficial velocity for foam collapse, ft/D
minimum pressure gradient for foam generation, Pa/m
This project is sponsored by the Joint Industry Project on foam enhanced oil recovery at Delft University of Technology. We thank and acknowledge both the sponsorship and advice of sponsoring companies and their representatives. Sebastien Vincent‐Bonnieu gratefully acknowledges Shell Global Solutions International B.V. for granting permission to publish this work. In addition, we thank Michiel Slob and Jolanda van Haagen Donker for technical assistance with our experiments.