(Dominguez, 1997; Hirasaki et al., 1988). The models classify the pre-existing gas phase either as present as bubbles in the bulk liquid or trapped in the roughness of the solid. The models, however, do not aid in the interpretation of the experimental results available in the literature. This includes the existence of a minimum supersat- uration required to nucleate bubbles, based on the maximum size of any pre-existing bubbles (El-Yousfi,1992;El Yousfi et al.,1997;Wang and Dhir,1993;Yang and Kim, 1988)
The field of bubble nucleation has been widely researched, including both clas- sical and non-classical types of nucleation. While the literature correctly identifies the underlying thermodynamic parameters that govern classical nucleation, it has become apparent that for most practical applications bubble nucleation is governed by type III and IV nucleation events. Nevertheless, an understanding of bubble nucleation in systems and conditions relevant to oil production is lacking. Quantitative data are needed on nucleation rates of relevant gases (methane, ethane etc.) on pipe (carbon steel) walls in a variety of flow situations.
growth of single, isolated, bubbles controlled by heat transfer inside infinite liquids of constant superheat. Scriven obtained an analytical solution for spherical bubble growth controlled by the transport of mass and heat in an infinite pool of liquid using similarity analysis (Scriven, 1959). For single bubble growth in uniform temperature fields, such as during nucleate boiling, his analysis produces a simple parabolic rela- tionship for bubble growth;R ∝t1/2, whereR is the bubble radius and t the time of growth. The proportionality is related to the thermal (or molecular) diffusivity and superheat (or supersaturation) in heat transfer (or mass transfer) controlled growth.
Scrivens similarity analysis assumes an initial bubble size equal to zero, which is erro- neous for several situations. Deviations from his parabolic relation have been reported from experimental and theoretical works and depend on the liquid properties (Divinis et al., 2006; Kostoglou and Karapantsios, 2005). A systematic study of some of the effects resulting in the deviation of the parabolic profile is available (Enr´ıquez et al., 2014). Moreover, bubbles produced by coalescence of smaller neighboring bubbles do not follow the parabolic growth law (Buehl and Westwater, 1966; Westerheide and Westwater, 1961).
The interest here is primarily in mass-transfer controlled bubble growth. Anal- ysis of this process is complicated by the highly coupled and nonlinear nature of the governing mass and momentum balances and the diffusion equation. The movement of the gas-liquid interface (i.e., bubble size) is related through momentum transfer to the gas pressure inside the bubble. In addition, the requirement of mass conser- vation relates the bubble radius to the gas pressure and the rate of gas diffusion.
Finally, the rate of gas diffusion depends on the movement of the interface through the diffusion equation. These complexities mean there is no known general analytical solution (Arefmanesh et al., 1992). Nevertheless, Scrivens simple parabolic relation has proven to be useful in describing the isothermal mass-transfer controlled bubble growth from supersaturated solutions (Barker et al.,2002;Bisperink and Prins,1994;
Figure 3.3: Typical single bubble growth model schematic
Glas and Westwater, 1964). In slightly supersaturated solutions, 0 < ξ < 1, where mass transfer is expected to be diffusion-controlled, however, bubble growth slower than the parabolic relation has been reported (Enr´ıquez et al.,2013). A typical single bubble growth model is schematically shown in Figure 3.3.
Mass-transfer (diffusion) controlled bubble growth in viscous liquids has been treated in other classical papers include those of Barlow and Langlois (Barlow and Langlois, 1962), Street et al. (Street et al., 1971), Szekely and Martins (Szekely and Martins, 1971), and Rosner and Epstein (Rosner and Epstein, 1972). Barlow and Langlois and Szekely and Martins examined bubble growth in Newtonian liquids while Street et al. considered growth in an Ostwald-de-Waele power law liquid. The analysis of Street et al. is further complicated by considering the liquid surrounding the bubble to be finite and a variable liquid viscosity. Barlow and Langlois and Street et al. both solve mass and momentum transfer equations with a simplified diffusion equation, where the concentration gradients are restricted to a thin shell surrounding the bubble. Outside this boundary layer, gas concentration was assumed to be undisturbed and equal to the initial concentration. Under this assumption, the
equations can be combined into a single integro-differential equation which is solved by a finite difference method. Rosner and Epstein used the moment integral method to solve the diffusion equation by assuming a polynomial to describe the concentration profile inside the boundary layer.
C∗−C C∗−C0 =
1− r−R δ
2
(3.6) whereC∗ is the bulk concentration of the gas.
Several researchers, using different polynomial profiles, have adopted Rosner and Epsteins method for mass-transfer controlled growth (or collapse) of bubbles in both viscous Newtonian and viscoelastic liquids (Ruckenstein,1964). Patel followed a similar approach to develop a simple model with a set of ordinary differential equations describing bubble growth in viscous liquids (Patel, 1980). Payvar used a similar approach to predict the mass transfer-controlled bubble growth during the rapid decompression of a liquid (Payvar,1987).
Bubble growth takes place at the nucleating site as well as during bubble rise.
Liger-Belair et al. found, in studies of bubble production in champagne, that bubble growth rates were constant during ascension (Liger-Belair et al.,2002). Their findings corroborate those of Shafer and Zare (Shafer and Zare, 1991), who also observed the linearity of radius increase with time for bubbles rising in a glass of beer. Liger-Belair et al. developed a simple mass-transfer model for the expansion of bubble radius.
dn
dt =klA∆C (3.7)
wheren is the gas moles transfered,kl is the liquid phase mass transfer coefficient, A is the bubble area, and ∆C is the concentration driving force.
Gas-liquid mass transfer coefficients are also expected to be dependent on the liquid viscosity and superficial gas velocity (Fern´andez et al.,2015;Zhou et al.,2014).
Moreover, it has been shown that the mass transfer coefficients for gas dissolution (absorption) and evolution (desorption) processes are identical (Hamborg et al.,2010).
Understanding of bubble growth in the context of oil and gas production is limited, although general trends can be inferred from this review. The presence of surface active agents including asphaltenes will further complicate the growth process in real scenarios.