Summary
Foams are routinely used worldwide to divert acid in well-stimulation treatments, and to divert gas flow in improved oil recovery (IOR) projects on a pilot basis. The complexity of foam behavior and the apparent contradictions among foam studies have bedeviled attempts to understand foams and to design effective field treatments.
In 1992, Osterloh and Jante1 identified two distinct foam-flow regimes: a high-quality (dry) regime, in which the steady-state pressure gradient is independent of the gas flow rate, and a low-quality (wet) regime, in which the pressure gradient is independent of the liquid flow rate. New data for strong foams in a variety of porous media, with various surfactant formulations, and over a range of flow rates, show this behavior to be general. In each regime, foam behavior is dominated by a single mechanism: at high qualities by capillary pressure and coalescence, and at low qualities by bubble trapping and mobilization. Based on these insights, a new, unified model unites foam behavior in acid-well stimulation and gas-diversion IOR and reconciles apparently contradictory data. This model correlates the effects of parameters, such as rock permeability and surfactant formulation, on foam behavior in the two flow regimes and on the transition between the two regimes. A variety of laboratory data supports the model's assumptions.
The implications of these results for designing foam processes for IOR and acid diversion are discussed. These insights can guide the selection of foam formulation and injection quality for foam applications.
Introduction
Foams are used on a pilot basis for gas diversion in IOR processes,2 aquifer remediation,3 and, routinely, for the diversion of acid in well-stimulation treatments worldwide.4 However, the effectiveness of foam continues to be unreliable and unpredictable. Predictive modeling of foam applications is hampered by the complexity of foam behavior and the apparent contradictions in published foam studies. For instance, the apparent rheology of foam is either Newtonian,5–7 shear-thinning,8–10 shear-thickening,11 or a mixture of Newtonian and shear-thinning.12,13 Foam strength increases as foam quality decreases14 or increases.15 Foam diversion compensates for differences in permeability between layers completely, partially, or not at all.16,17 The pressure gradient in foam flow depends on liquid velocity and is independent of gas velocity, or it depends on gas velocity and is independent of liquid velocity.1 Predictive modeling and design of foam applications is not possible until a consistent mechanistic framework is found for the wide range of observed foam behavior.
Pc* Regime.
Khatib et al.18 found that foam stability in porous media is limited by capillary pressure. In studying foam in beadpacks, they identified an abrupt transition from strong foam to no foam (or weak foam) in a very narrow range of capillary pressure around the limiting capillary pressure. Supporting observations were reported for sandpacks,19 Boise sandstone,5 and fired Berea sandstone.6 In the Pc* regime, foam bubbles change size as needed to maintain foam at the limiting capillary pressure. As a result, liquid saturation, Sw, is constant and equal to Sw*, its value at Pc*. If Pc* and Sw* are independent of flow rates, as in the fixed-Pc* model,20 then krw(Sw) is constant during foam flow; as a result, the pressure gradient is proportional to the liquid flow rate, or liquid superficial velocity Uw, and independent of the gas flow rate.5,6
Second Flow Regime.
Osterloh and Jante1 identified two foam-flow regimes in a sandpack study. In one regime, the pressure gradient was reasonably independent of the gas flow rate, as predicted by the Pc* model. In the second regime, however, the pressure gradient was reasonably independent of the liquid flow rate and dependent on the gas flow rate. The transition zone between the two regimes was characterized by a specific value of the gas fractional flow fg (gas velocity divided by total velocity), fg*. Foam quality is fg rescaled from the range of 0 to 1 to the range of 0 to 100%. Fig. 1 shows the two regimes and the transition foam quality at fg*=0.94. Therefore, fg>0.94 corresponds to the high-quality regime (also called the coalescence regime6); foam in this regime obeys the Pc* model. On the other hand, for the second, low-quality regime, the Pc* model does not apply. Osterloh and Jante1 pointed out that the transition from one regime to the other most likely occurs when the limiting capillary pressure is reached. Fig. 2 is a simplified representation of a contour plot, similar to Fig. 1, that shows the important parameters and definitions used throughout this study.
Supporting observations are provided by Parlar et al.,12 Robert and Mack,21 and Rossen and Wang,7 all of whom used Berea cores. Rossen and Wang7 also proposed a model for foam behavior in the low-quality regime. Their model is based on the assumption that the bubble size in this regime is fixed (perhaps at roughly pore size) and, therefore, independent of liquid and gas flow rates. This conjecture is based on the idea that diffusion causes bubbles smaller than the pore size to disappear and, with bubbles smaller than pores, foam-creation processes, like snap-off and lamella division, are inhibited. Foam with a fixed bubble size is then modeled as a Bingham plastic22 that is trapped or flows depending on the pressure gradient.
The assumption of a fixed bubble size implies that the pressure gradient in the low-quality regime should be a strong function of the porous medium. The pressure gradient is set by bubble trapping, which, with the bubble size fixed, depends primarily on pore geometry and surface tension9,23-26; however, surface tension in many cases does not vary greatly with surfactant formulation. For instance, at room temperature, many surfactant formulations give surface tensions with air between approximately 25 and 40 dyne/cm. Therefore, a rock's petrophysical properties should play the key role in determining the pressure gradient in the low-quality regime, while surfactant formulation is the most important parameter in the high-quality regime because of its effect on foam-coalescence processes. At the time, there was no direct evidence that bubble size is fixed in the low-quality regime.
