Model of foam flow

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. 

As mentioned before. Equations (5) and (6) are the differential transport equations of average bubbles and could be written from scratch without the convoluted derivations invoked here. Unfortunately, modeling of foam flow in porous media is a lot more complicated than Equations (3) and (6) lead us to believe. Having started from a general bubble population balance, we discovered that flow of foams in porous media is governed by Equations (2) and (3), and that Equations (5) and (6) are but the first terms in an infinite series that approximates solutions of (2) and (3). 

Population-Balance Modeling of Foam Flow in Porous Media... 

In order to understand the nature and mechanisms of foam flow in the reservoir, some investigators have examined the generation of foam in glass bead packs (12). Porous micromodels have also been used to represent actual porous rock in which the flow behavior of bubble-films or lamellae have been observed (13,14). Furthermore, since foaming agents often exhibit pseudo-plastic behavior in a flow situation, the flow of non-Newtonian fluid in porous media has been examined from a mathematical standpoint. However, representation of such flow in mathematical models has been reported to be still inadequate (15). Theoretical approaches, with the goal of computing the mobility of foam in a porous medium modelled by a bead or sand pack, have been attempted as well (16,17). 

Finally, the self-reproducibility in time of the lattice configuration (for two-dimensional flows) must be addressed. In the elliptic streamline region (A < 0), the lattice necessarily replicates itself periodically in time owing to closure of the streamlines. For hyperbolic flows (A > 0), the lattice is not generally reproduced however, in connection with research on spatially periodic models of foams (Aubert et al., 1986 Kraynik, 1988), Kraynik and Hansen (1986, 1987) found a finite set of reproducible hexagonal lattices for the extensional flow case A = 1. It is not clear how this unique discovery can be extended, if at all. 

Important are the equations derived on the basis of various capillary foam models. For instance, Eqs. (5.57) and (5.58) [65] are obtained if the model of liquid flow through the foam films is assumed... 

The comparison of border and film hydroconductivities [7,14] shows that the contribution of liquid flowing through films can be neglected. That is why more realistic proves to be the model of liquid flow through borders. On the basis of this model the following kinetic dependence with three constants a, b and k has been proposed in one of the first papers dedicated to foam drainage [66]... 

Although the current permeability model properly reflects many of the important features of foam displacement, the authors acknowledge its limitations in several respects. First, the open pore, constricted tube, network model is an oversimplification of true 3-D porous structures. Even though communication was allowed between adjacent pore channels, the dissipation associated with transverse motions was not considered. Further, the actual local displacement events are highly transient with the bubble trains moving in channels considerably more complex than those used here. Also, the foam texture has been taken as fixed the important effects of gas and liquid rates, displacement history, pore structure, and foam stability on in situ foam texture were not considered. Finally, the use of the permeability model for quantitative predictions would require the apriori specification of fc, the fraction of Da channels containing flowing foam, which at present is not possible. Obviously, such limitations and factors must be addressed in future studies if a more complete description of foam flow and displacement is to be realized. 

Foams usually possess a finite low-frequency elastic modulus, along with static and dynamic yield stresses. These and other aspects of foam flow and rheology can be captured qualitatively and even semiquantitatively by cellular foam models. 

In this chapter, we discuss much of the work accomplished since Fried, but without attempting a complete review. Useful synopses are available in the articles and reports of Hirasaki (2, 3), Marsden (4), Heller and Kuntamukkula (5), Baghidikian and Handy (6), and Rossen (7). Our goals are to present a unified perspective of foam flow in porous media to delineate important pore-level foam generation, coalescence, and transport mechanisms and to propose a readily applicable one-dimensional mechanistic model for transient foam displacement based upon gas-bubble size evolution [i.e., bubble or lamella population-balance (8, 9)]. Because foam microstructure or texture (i.e., the size of individual foam bubbles) has important effects on flow phenomena in porous media, it is mandatory that foam texture be accounted for in understanding foam transport. 

Figure 18. A bond percolation model of gas flow in a pore network containing foam lamellae blocking pore throats. Line intersections represent pore bodies. (Reproduced with permission from reference 14. Copyright 1993.)...

Several rheological aspects of thermoplastic foam extrusion have been described as a phenomenological model of the flow in an extrusion die, growth and physical properties of thermoplastic material with respect to extrusion foaming. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a soHd plug, with a constant velocity. AH the shear flow occurs near the waHs, where the yield stress has been exceeded and the foam behaves like a viscous Hquid. At the waH, foams can exhibit waH sHp such that bubbles adjacent to the waH have nonzero velocity. The amount of waH sHp present has a significant influence on the overaH flow rate obtained for a given pressure gradient. 

The thin liquid films bounded by gas on one side and by oil on the other, denoted air/water/oil are referred to as pseudoemulsion films [301], They are important because the pseudoemulsion film can be metastable in a dynamic system even when the thermodynamic entering coefficient is greater than zero. Several groups [301,331,342] have interpreted foam destabilization by oils in terms of pseudoemulsion film stabilities [114]. This is done based on disjoining pressures in the films, which may be measured experimentally [330] or calculated from electrostatic and dispersion forces [331], The pseudoemulsion model has been applied to both bulk foams and to foams flowing in porous media. 

The correctness of the above hydrodynamic model of a polyhedral foam with border hydroconductivity and constant radius of border curvature can be confirmed by comparing the velocity Q calculated from Eqs. (5.9) and (5.11) with the volumetric liquid rate obtained by Krotov s theory [7]. Thus, the liquid flow under gravity at r = const is... 

The border profile was studied in order to analyse qualitatively the influence of various foam parameters (surfactant kind and foam film type, foam column height, pressure drop, etc.) on the drainage process as well to check the validity of drainage models [12], The foam was placed in a cylindrical vessel (diameter 2.5 to 4 cm), similar to vessel 6 in Fig. 1.4. It was covered with a lid to prevent evaporation. The pressure above the foam was equal to the atmospheric pressure. The border profile was determined by simultaneous measurement of the capillary pressure at various levels of the foam column, i.e. the r(H) dependence in the direction of liquid flow was evaluated. Thus it was found that the best approximation (among the discussed in Section 5.3.3) appears to be the parabolic model of border profile. 

The expression for foam viscosity Eq. (8.1 la) contains two terms x 0/y which is the elasticity component, related to the demolition of foam structure, and r, which is a dissipation term, related to the liquid flow through films and borders during the deformation process. The models of Khan and Armstrong [14] and Kraynik and Hansen [43] imply that the continuous phase is in the films, no liquid exchange occurs and the film surfaces are mobile, thus predicting a very small contribution of the viscous dissipation in the films, rj = 13[Pg.584]

Foam films are usually used as a model in the study of various physicochemical processes, such as thinning, expansion and contraction of films, formation of black spots, film rupture, molecular interactions in films. Thus, it is possible to model not only the properties of a foam but also the processes undergoing in it. These studies allow to clarify the mechanism of these processes and to derive quantitative dependences for foams, O/W type emulsions and foamed emulsions, which in fact are closely related by properties to foams. Furthermore, a number of theoretical and practical problems of colloid chemistry, molecular physics, biophysics and biochemistry can also be solved. Several physico-technical parameters, such as pressure drop, volumetric flow rate (foam rotameter) and rate of gas diffusion through the film, are based on the measurement of some of the foam film parameters. For instance, Dewar [1] has used foam films in acoustic measurements. The study of the shape and tension of foam bubble films, in particular of bubbles floating at a liquid surface, provides information that is used in designing pneumatic constructions [2], Given bellow are the most important foam properties that determine their practical application. The processes of foam flotation of suspensions, ion flotation, foam accumulation and foam separation of soluble surfactants as well as the treatment of waste waters polluted by various substances (soluble and insoluble), are based on the difference in the compositions of the initial foaming solution and the liquid phase in the foam. Due ro this difference it is possible to accelerate some reactions (foam catalysis) and to shift the chemical equilibrium of some reactions in the foam. The low heat... 

This gives rise to a mobilization pressure which is higher than gas-liquid flow with no surfactant. Viscous effects of the Bretherton type are included in a network model to derive permeability expressions cor responding to different interfacial mobilities. The significant reduction in gas permeability of foams is attributed to (1) the significant decrease in the fraction of channels containing flowing gas (compared to gas-liquid flow with no surfactant), and (2) the increase in viscous and capillary effects associated with bubble train lamellae. 

These observations, coupled with the effects of bubble texture (1,13-15) and various history dependent phenomena, clearly demonstrate the inadequacy of conventional fractional flow approaches to describe foam flow in porous media. Also, early approaches which treated the foam simply as a fluid of modified viscosity are also inadequate in explaining the above characteristics. To achieve a fuller understanding of such phenomena, a detailed description of the pore level events is required. In what follows, a simple pore level model is utilized to explain some of the above macroscopic features and to identify some of the key pore level mechanisms. 

---