Capillary snap-off

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. 

A key factor in the commercialization of surfactant-based mobility control will be the ability to create and control dispersions at distances far from the injection well (TJ ). Capillary snap-off is often considered to be the most important mechanism for dispersion formation, because it is the only mechanism that can form dispersions directly when none are present (39,40). The only alternative to snap-off is either leave-behind, or else injection of a dispersion, followed by adequate rates of thread breakup and division to maintain the injected lamellae. 

There are three important ways by which capillary pressure affects the dynamics of dispersion formation and disappearance the lower and upper limits on the range of capillary numbers over which capillary snap-off can occur in homogenous media, and an upper limit on the capillary number above which lamellae are unstable and droplets quickly coalesce. 

A field-scale simulator based on this approach would probably require simpler equations that captured the relevant phenomena without explicitly addressing many of them. Chapter 15, by Prieditis and Flumerfelt, models two-phase flow in a network of interconnected channels that consist of constricted tube segments. Work on the creation of a model that contains capillary snap-off in a network similar to that of Figure 6 has very recently been started at the University of Texas (R. Schechter, personal communication, October 26, 1987). 

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. 

As discussed above, for capillary snap-off to operate as desired, the intended noncontinuous phase must be nonwetting, and the capillary number and capillary pressure must fall within certain limits. The fluid-fluid interfacial tension is the only parameter in the capillary number and capillary pressure that is subject to effective control by the process designer. Hence, the capillary number and pressure establish limits for acceptable values of this tension. 

In summary, reservoir characteristics establish most of the parameters that control capillary snap-off, and interfacial tensions are the only controllable snap-off parameters. The dependencies of interfacial tensions on phase behavior and surfactant structure define many of the objectives of the surfactant designer. 

Since capillary snap-off disperses the nonwetting phase in the wetting fluid, wettability changes may drastically alter the effectiveness of the process. 

Because capillary snap-off produces a dispersion of the nonwetting fluid in the fluid that wets the porous medium, if the "wrong" fluid wets the solid, the "wrong" type of dispersion will be produced. As described by the Young-Dupre equation. 

The three dispersion types described by Wellington et al. are important mechanistically, in view of the apparent importance of capillary snap-off. Extant descriptions of the snap-off mechanism explicitly treat the first type of dispersion, and they should be able to accommodate the second dispersion type by addition of a second fluid that does not wet the porous medium. However, if the aqueous phase of the first two dispersion types wets the porous... 

Phase Behavior and Surfactant Design. As described above, dispersion-based mobility control requires capillary snap-off to form the "correct" type of dispersion dispersion type depends on which fluid wets the porous medium and surfactant adsorption can change wettability. This section outlines some of the reasons why this chain of dependencies leads, in turn, to the need for detailed phase studies. The importance of phase diagrams for the development of surfactant-based mobility control is suggested by the complex phase behavior of systems that have been studied for high-capillary number EOR (78-82), and this importance is confirmed by high-pressure studies reported elsewhere in this book (Chapters 4 and 5). 

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). 

Foam generation does not continue unchecked. Surfactant-stabilized lamellae are only metastable. Coalescence ensues when a translating lamella moves out of a sharply constricted pore-throat into a pore-body, and the lamella is stretched too rapidly for healing flow of foamer solution. Whereas foam generation by capillary snap-off is independent of surfactant formulation, coalescence of foam lamellae strongly depends on surfactant formulation, concentration, and salinity. 

On a microscopic scale (the inset represents about 1 - 2mm ), even in parts of the reservoir which have been swept by water, some oil remains as residual oil. The surface tension at the oil-water interface is so high that as the water attempts to displace the oil out of the pore space through the small capillaries, the continuous phase of oil breaks up, leaving small droplets of oil (snapped off, or capillary trapped oil) in the pore space. Typical residual oil saturation (S ) is in the range 10-40 % of the pore space, and is higher in tighter sands, where the capillaries are smaller. 

Effects of Capillary Number, Capillary Pressure, and the Porous Medium. Since the mechanisms of leave-behind, snap-off, lamella division and coalescence have been observed in several types of porous media, it may be supposed that they all play roles in the various combinations of oil-bearing rocks and types of dispersion-based mobility control (35,37,39-41). However, the relative importance of these mechanisms depends on the porous medium and other physico-chemical conditions. Hence, it is important to understand quantitatively how the various mechanisms depend on capillary number, capillary pressure, interfacial properties, and other parameters. 

For capillary numbers greater than the critical value, snap-off occurred even in homogenous bead packs (40). The resulting dispersions caused much greater resistance to flow than the resistance produced by leave-behind lamellae (which do not disperse the nonwetting phase). 

A lower limit on the capillary number required for snap-off arises from a static analysis of nonwetting flow into the constriction. Roof s analysis of snap-off in symmetric constrictions shows that there is a strictly geometrical requirement for the nonwetting phase to enter the constriction. Below this limit snap-off cannot occur. For example, in a circular constriction the relation... 

An upper limit on the capillary number required for snap-off arises from the dynamics of wetting fluid flow into the constriction. The capillary number must be below the upper limit for a long enough time that sufficient wetting fluid can flow back into the constriction to form a lamella (40). If the volume of wetting fluid is too small, the lamella cannot form. 

As defined by Radke and Ransohoff (Equation 7), the "snap-off" capillary number, C, contains the effective grain radius, R the permeability, K anS the relative permeability of the nonwetting phase, k ( ). In field applications, the values of all of these parameters are set by the reservoir. Also contained in C are the total superficial velocity, U, and the distance between injection and production wells, L. Within narrow limits, L can be changed by... 

The first mechanism is snap-off, studied by Roof (22), and discussed in more detail recently by Mohanty et (23), and Falls et (5). Basically a thermodynamic instability arises as curvature and hence capillary pressure variations cause the wetting fluid to... 

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. 

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... 

Fig. 6 Schematic of nonwetting phase invasion into a pore (A) schematic representation of a pore formed by four grains (B) low capillary pressure (C) critical capillary pressure for invasion (D) phase distributions after Haines jump (E) high capillary pressure (F) disconnection of the nonwetting phase after snap-off.

Fig. 1-6. A sketch illustrating liquid-vapor interfacial configurations during transition Imm adsorption to capillary-dominated imbibition in the proposed unit cell () spontaneous slit fill up (capillary condensation), (c) pore snap-off, and < /) full unit cell.

The first term describes the interfacial area per pore volume following pore emptying (while slits are still liquid filled). Liquid-vapor interfaces are composed of the curved interfaces in the corners (capillary contribution) and film interfaces in the flat areas between comers (adsorptive contribution). The second term is for the expected value of liquid-vapor interfacial area following the formation of liquid-vapor interface in the slits following slit snap-off. These include interfaces in the central pore and flat film interfaces in the slits. Detailed solutions for the integrations are given in Tuller and Or (2001). 

Knowledge of detailed liquid-vapor configurations enables separation of capillary and adsorptive contributions to the interfacial area as shown in Fig. 1-1 la (note the log-log scale). We denote liquid-vapor interfacial areas associated with menisci (curved interfaces at pore comers) as capillary contributions, and those associated with films as adsorptive contributions. The results in Fig. 1-1 la illustrate the dominant contribution of liquid films to the total liquid-vapor interfacial area of a partially saturated porous medium (Millville silt loam). Note that the flat region in Fig. 1-1 la (changes in SA with no change in p) reflects pore snap-off processes. 

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