Bubble snap-off

Arriola (8) and Ni (5) have observed a second mechanism for snap off in strongly constricted square capillaries. At low liquid flow rates, a bubble is trapped in the converging section of the constriction and liquid flows past the bubble. As liquid flow rate increases, waves developed in the film profile and at some critical liquid flow rate these oscillations become unstable and bubbles snap off. In these experiments, the bubble front is located upstream of the constriction neck. Therefore, no driving force for the drainage mechanism exists. Bubbles formed by this mechanism are produced at a high rate and have a radius on the order of the constriction neck. No attempt has previously been made to model snap-off rate by this mechanism in noncircular constrictions. 

In addition to this thread breakup mechanism, gas fingers and large bubbles can also experience bubble snap-off when passing through narrow pore constrictions (20,21). Although snap-off phenomena can be quite complex (21-24), the static analysis of Roof (20) indicates that the resulting bubble diameters are at least twice the pore constriction diameter. 

A porous medium shapes foam to its own liking as confined, porefilling bubbles and lamellae. Foam in porous media is not a continuous fluid. The three mechanisms of foam generation (snap-off, division, and leave-behind) are all pore geometry specific. Snap-off is a mechanical process that occurs in multiphase flow without surfactant. For successful gas-bubble snap-off, the pore-body to pore-throat constriction ratio must be sufficiently large (roughly 2) and gently sloped. Otherwise stable wet-... 

Barring direct measurement of foam texture, we adopt the following reasoning. Because of the generation of foam bubbles by the snap-off and division mechanisms (4), bubble sizes are expected to be approximately that of pore bodies. Thus, the linear bubble density should scale roughly as n 6/Dwhere... 

In a third paper by the Bernard and Holm group, visual studies (in a sand-packed capillary tube, 0.25 mm in diameter) and gas tracer measurements were also used to elucidate flow mechanisms ( ). Bubbles were observed to break into smaller bubbles at the exits of constrictions between sand grains (see Capillary Snap-Off, below), and bubbles tended to coalesce in pore spaces as they entered constrictions (see Coalescence, below). It was concluded that liquid moved through the film network between bubbles, that gas moved by a dynamic process of the breakage and formation of films (lamellae) between bubbles, that there were no continuous gas path, and that flow rates were a function of the number and strength of the aqueous films between the bubbles. As in the previous studies (it is important to note), flow measurements were made at low pressures with a steady-state method. Thus, the dispersions studied were true foams (dispersions of a gaseous phase in a liquid phase), and the experimental technique avoided long-lived transient effects, which are produced by nonsteady-state flow and are extremely difficult to interpret. 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

A study of the effect of pore geometry on foam formation mechanisms shows that snap-off" bubble formation is dominant in highly heterogeneous pore systems. The morphology of the foams formed by the two mechanisms are quite different. A comparison of two foam injection schemes, simultaneous gas/surfactant solution injection (SI) and alternate gas/surfactant solution injection (GDS), shows that the SI scheme is more efficient at controlling gas mobility on a micro-scale during a foam flood. 

The majority of the bubbles in the GDS experiment were formed in this manner. Bubbles formed by this second mechanism are several times larger than the pore radius, whereas bubbles formed by snap-off tend to be the same size as the pore throat radius. 

Snap off by the instability mechanism may occur in the following way. A bubble in an angular constriction such as a square channel will flatten against the walls as shown in Figure 2. The radius of the circular arcs in the corners for a static bubble is about one half of the tube half width. At low liquid flow rates, a bubble trapped behind a constriction has this nonaxisymmetric shape. As liquid flow rate increases, the bubble moves farther into the constriction and the fraction of cross sectional area open to liquid flow at the front of the bubble increases until the thread becomes axisymmetric at some point near the bubble front. 

If Cq is known as a function of the capillary number and the surfactant properties, the functional form of the frequency and bubble volume can be approximated from the linear results. However, a model for Cq in constricted angular tubes does not exist. If one assumes that snap off occurs as soon as the thread becomes axisymmetric, then the base state thread radius is approximately the half width of the channel at the point snap off occurs. The experimental observations of Arriola and Ni along with the theoretical predictions of Ransohoff and Radke indicate that snap off takes place very near the constriction neck. Therefore, the radius of the bubbles formed should be slightly larger than the half width of the constriction neck. In fact, approximating Cq by the constriction half width, one observes from equations 14 and 15, that the snap off frequency and bubble volume are independent of the liquid flow rate once the critical liquid flow rate has been exceeded. Ni measured the dependence of snap off on the bubble velocity, the velocity of... 

The scope of possible foam applications in the field warrants extensive theoretical and experimental research on foam flow in porous media. A lot of good work has been done to explain various aspects of the microscopic foam behavior, such as apparent foam viscosity, bubble generation by capillary snap-off, etc.. However, none of this work has provided a general framework for modeling of foam flow in porous media. This paper attempts to describe such a flow with a balance on the foam bubbles. 

A simplified one-dimensional transient solution of the bubble population balance equations, verified by experiments, has been presented elsewhere (5) for a special case of bubble generation by capillary snap-off. 

Foam (5) is a collection of gas bubbles with sizes ranging from microscopic to infinite for a continuous gas path. These bubbles are dispersed in a connected liquid phase and separated either by lamellae, thin liquid films, or by liquid slugs. The average bubble density, related to foam texture, most strongly influences gas mobility. Bubbles can be created or divided in pore necks by capillary snap-off, and they can also divide upon entering pore branchings (5). Moreover, the bubbles can coalesce due to instability of lamellae or change size because of diffusion, evaporation, or condensation (5,8). Often, only a fraction of foam flows as some gas flow is blocked by stationary lamellae (4). 

Snap-off. Snap-off is a very significant mechanism for bubble generation in porous media. This phenomenon was first identified and explained by Roof (54) to understand the origin of residual oil. Snap-off is not restricted to the creation of trapped oil globules. It repeatedly occurs during multiphase flow in porous media regardless of the presence or absence of surfactant. Hence, snap-off is recognized as a mechanical process. 

On the right of equation 4, the generation and coalescence rates, rg and rc, are expressed on a per volume of gas basis. These two terms are fundamental, for they control bubble texture. At steady state, far from any sources or sinks, and where rock properties are constant (e.g., absolute permeability, relative permeability, and capillary-pressure functions), bubble size is set by rg = rc. That is, the rate of bubble generation by snap-off balances the rate of bubble coalescence by capillary-pressure suction (20). To proceed, kinetic expressions are needed for rg and rc. 

Lamella Division A mechanism for foam lamella generation in porous media. Typically, when a foam lamella reaches a branch point in a flow channel, then the lamella may divide into two lamellae (bubbles) rather than simply follow one of the two available pathways. See also Lamella Leave-Behind, Snap-Off. 

Snap-Off A mechanism for foam lamella generation in porous media. When gas enters and passes through a constriction in a pore, a capillary pressure gradient is created and causes liquid to flow toward the region of the constriction, where it accumulates and may cause the gas to pinch-off or snap-off to create a new gas bubble separated from the original gas by a liquid lamella. See also Lamella Division, Lamella Leave-Behind. 

Roof investigated the conditions that must be met in order that the oil emerging from a water-wet constriction seperate (Choke-off, Snap-off or pinch-off) into a droplet in a larger channel. Such flow of water and oil in a water-wet system is similar to the di lacement of an aqueous surfactant solution by air in a water-wet porous medium. If snap-off occurs in the latter case, seperate air bubbles and a network of air-liquid interfaces will be produced. 

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