Influence of Soluble Surfactants on the Flow of Long Bubbles Through

Downloaded by OHIO STATE UNIV LIBRARIES on September 6, 2012 | http://pubs.acs.org Publication Date: July 10, 1989 | doi: 10.1021/bk-1989-0396.ch026

Chapter 26

Influence of Soluble Surfactants on the Flow of Long Bubbles Through a Cylindrical Capillary 1

G. M . Ginley and C. J. Radke Department of Chemical Engineering, University of California, Berkeley, CA 94720 Flow of trains of surfactant-laden gas bubbles through c a p i l l a r i e s i s an important ingredient of foam transport i n porous media. To understand the role of surfactants i n bubble flow, we present a regular perturbation expansion i n large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous l i q u i d phase, the pressure drop across the bubble increases with the e l a s t i c i t y number while the deposited thin f i l m thickness decreases s l i g h t l y with the e l a s t i c i t y number. Both pressure drop and thin f i l m thickness r e t a i n t h e i r 2/3 power dependence on the c a p i l l a r y number found by Bretherton f o r surfactant-free bubbles. Comparison of the proposed theory to available and new experimental data at c a p i l l a r y numbers less than 10-2 and f o r anionic aqueous surfactant above the critical micelle concentration shows good agreement with the 2/3 power prediction on c a p i l l a r y number and confirms the s i g n i f i c a n t impact of soluble surfactants on bubble-flow resistance. F i n a l l y , scaling arguments extend the single-bubble theory to predict the e f f e c t i v e v i s c o s i t y of the flowing bubble regime i n porous media. Again, comparison of the e f f e c t i v e - v i s c o s i t y p r e d i c t i o n to available pressure-drop data i n Berea sandstone demonstrates good agreement.

1Current address: Marathon Oil Company, Littleton, CO 80160

0097-6156/89/0396-0480$06.50/0 ο 1989 American Chemical Society In Oil-Field Chemistry; Borchardt, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

26. GINLEY AND RADKE

Flow of Long Bubbles Through a Capillary

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Foam i s a promising f l u i d for achieving mobility control i n underground enhanced o i l recovery (1-3). Widespread a p p l i c a t i o n of this technology to, for example, steam, CC^, enriched hydrocarbon, or surfactant flooding requires quantitative understanding of foam flow properties i n porous media. Because foam i n porous media i s a complicated dispersion of gas (or l i q u i d ) i n an aqueous surfactant phase (4), the pressure dropflow rate relationship depends c r i t i c a l l y on the pore-level microstructure or texture ( i . e . , on the bubble size and/or bubble-size d i s t r i b u t i o n ) . Foam texture i n turn depends on the dynamic interaction of the generation and breakage mechanisms (5), both of which are strong functions of pore geometry, and surfactant type and concentration. Numerous v i s u a l micromodel studies of foam generated and shaped i n o i l - f r e e , water-wet porous media with robust s t a b i l i z i n g surfactants, show that the bubble size i s variable but generally i s on the order of one to several pore-body volumes (4.6-11). As shown i n the schematic of Figure 1, the bubbles ride over t h i n - f i l m cushions of the wetting phase adjacent to the rock surfaces. They are separated from one another by surfactants t a b i l i z e d lamellae which terminate i n Plateau borders (12). The curvature i n the Plateau borders i s set p r i m a r i l y by the saturation of the continuous wetting phase which occupies the smallest pores (13,14). To a reasonable approximation the mean c a p i l l a r y suction pressure of the wetting phase i s applied to each lamella. When flowing, the lamellae transport as a contiguous bubble t r a i n which snakes through available pores not occupied by the wetting phase. The foam microstructure depicted i n Figure 1 suggests that important aspects of the hydrodynamic resistance of flowing bubble trains i n porous media can be captured i n studies of bubbles i n c a p i l l a r i e s (7). This work considers the hydrodynamic behavior of a single gas bubble translating i n a c y l i n d r i c a l c a p i l l a r y whose radius i s smaller than that of the undeformed bubble. The continuous l i q u i d phase contains a surfactant whose concentration i s near to or above the c r i t i c a l micelle concentration. S p e c i f i c a l l y , we extend Bretherton's analysis for a clean gas bubble (15) to include the effects of a soluble surfactant which i s k i n e t i c a l l y hindered from a t t a i n i n g l o c a l equilibrium at the g a s / l i q u i d interface. The shape of the bubble and the r e s u l t i n g pressure drop across the bubble are obtained numerically for small deviations i n surfactant adsorption from equilibrium. Given the dynamic pressure drop across a single bubble, we b r i e f l y show how foam-flow behavior i n porous media may be predicted using scaling arguments s i m i l a r to those adopted for non-Newtonian polymer solutions (16). Previous Work In 1961 Bretherton solved the problem of a long gas bubble, uncontaminated by surface-active impurities, flowing i n a c y l i n d r i c a l tube at low c a p i l l a r y numbers, Ca • pli/a (/z i s the

In Oil-Field Chemistry; Borchardt, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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OIL-FIELD CHEMISTRY

Newtonian l i q u i d v i s c o s i t y , U i s the bubble v e l o c i t y , and is the equilibrium surface tension), where surface tension and viscous forces dominate the bubble shape (15). Using a l u b r i c a t i o n analysis, Bretherton established that the bubble s l i d e s over a stationary, constant-thickness f i l m whose thickness divided by the radius of the tube, h^R^, varies as the


c a p i l l a r y number to the 2/3 power. The dimensionless pressure drop to drive the bubble, (-AP^R^o^, i s calculated from the altered shape of the bubble at i t s two ends and also scales as the c a p i l l a r y number to the 2/3 power. Experimental measurements by Bretherton of the f i l m thickness deposited by the bubble are i n good accord with h i s theory for c a p i l l a r y numbers ranging from -5 -2 10 to 10 . To our knowledge no corresponding measurements have been reported for the pressure drop of single, clean bubbles at low c a p i l l a r y numbers. Lawson and H i r a s a k i r e c e n t l y a n a l y z e d t h e c a s e o f s i n g l e bubbles immersed i n a s u r f a c t a n t s o l u t i o n and f l o w i n g a t low c a p i l l a r y numbers through a narrow c a p i l l a r y ( 7 ) . These a u t h o r s consider the r a t e - l i m i t i n g step i n the t r a n s f e r o f s u r f a c t a n t t o and from t h e i n t e r f a c e t o be f i n i t e a d s o r p t i o n - d e s o r p t i o n k i n e t i c s . S u r f a c e and b u l k d i f f u s i o n r e s i s t a n c e s a r e n e g l e c t e d , and s u r f a c t a n t d e p l e t i o n i n t h e t h i n f i l m r e g i o n i s shown t o be negligible. F o l l o w i n g L e v i c h ( 1 7 ) , t h e s u r f a c t a n t a d s o r p t i o n and t h e s u r f a c e t e n s i o n along t h e bubble a r e assumed t o d e v i a t e o n l y s l i g h t l y from t h e i r e q u i l i b r i u m v a l u e s . Lawson and H i r a s a k i a t t r i b u t e a l l the e f f e c t s of the surfactant to the constant f i l m t h i c k n e s s r e g i o n o f t h e bubble w h i l e t h e bubble ends a r e i g n o r e d . Because o f t h i s approach, t h e s e a u t h o r s a r e u n a b l e t o o b t a i n t h e v a l u e o f t h e d e p o s i t e d f i l m t h i c k n e s s as p a r t o f t h e i r a n a l y s i s . By a s s e r t i n g t h a t t h e f i l m t h i c k n e s s remains p r o p o r t i o n a l t o t h e 2/3 power o f t h e c a p i l l a r y number, t h e y e s t a b l i s h t h a t t h e dynamic p r e s s u r e drop f o r s u r f a c t a n t - l a d e n b u b b l e s a l s o v a r i e s w i t h t h e c a p i l l a r y number t o t h e 2/3 power but w i t h an unknown c o n s t a n t o f p r o p o r t i o n a l i t y . New p r e s s u r e - d r o p d a t a f o r a 1 wt% commercial s u r f a c t a n t , sodium dodecyl benzene s u l f o n a t e ( S i p o n a t e DS-10), i n w a t e r , a f t e r c o r r e c t i o n f o r t h e l i q u i d i n d i c e s between t h e b u b b l e s , c o n f i r m e d t h e 2/3 power dependence on Ca and r e v e a l e d s i g n i f i c a n t i n c r e a s e s o v e r t h e B r e t h e r t o n t h e o r y due t o t h e soluble surfactant. Here we also consider sorption k i n e t i c s as the masstransfer b a r r i e r to surfactant migration to and from the interface, and we follow the Levich framework. However, our analysis does not confine a l l surface-tension gradients to the constant thickness f i l m . Rather, we treat the bubble shape and the surfactant d i s t r i b u t i o n along the interface i n a consistent fashion. Problem Statement Figure 2 portrays a schematic of a long bubble flowing i n zero gravity through a tube f i l l e d with a completely wetting



26. GINLEY AND R A D K E


Figure 1. Schematic of the bubble-flow regime i n porous media. Open space corresponds to bubbles, dotted space i s the aqueous surfactant solution, and cross-hatched areas are sand grains.

-U

Figure 2. Flow of a single gas bubble through a l i q u i d - f i l l e d c y l i n d r i c a l c a p i l l a r y . The l i q u i d contains a soluble surfactant whose d i s t r i b u t i o n along the bubble interface i s sketched.


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OIL-FIELD CHEMISTRY

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surfactant solution. The reference frame i s such that the bubble i s stationary with the walls moving past at a v e l o c i t y of -U. The height of the bubble interface, h, i s measured from the tube wall. In this analysis the undistorted bubble radius i s always greater than the tube radius, and the bubble i s longer than at least twice the tube radius. Consequently, there i s a region of constant l i q u i d f i l m thickness, h , i n the middle of the bubble, even when


Q

surfactants are present (18). The o r i g i n of the a x i a l coordinate, x, i s placed near the constant thickness f i l m , but i t s exact l o c a t i o n i s i n i t i a l l y unspecified. The shape of the front and rear menisci change as a r e s u l t of the resistance to bubble flow. Calculation of t h i s deviation i n bubble shape establishes the dynamic pressure drop across the bubble. The expected surfactant d i s t r i b u t i o n i s also portrayed q u a l i t a t i v e l y i n Figure 2. At low Ca, r e c i r c u l a t i o n eddies i n the l i q u i d phase lead to two stagnation rings around the bubble, as shown by the two pairs of heavy black dots on the interface (!&»12). Near the bubble front, surfactant molecules are swept along the interface and away from the stagnation perimeter. They are not instantaneously replenished from the bulk solution. Accordingly, a surface stress, r , develops along the interface g

directed from low to high surface tension ( i . e . , from high to low surfactant adsorption). Surface stresses at the rear stagnation perimeter also occur, only now surfactant accumulates near the r i n g and there i s a net flux of surfactant away from the interface. S u f f i c i e n t l y far from the stagnation rings, the surfactant achieves sorption equilibrium, and the surface tension approaches the equilibrium value, < 7 . Thus a s t r e s s - f r e e , q

constant thickness f i l m underlies the bubble (18). Equilibrium surface tension i n the constant thickness f i l m portion of the bubble implies that a l l resistance to flow occurs near the ends. Thus, the bubble can be viewed as i n f i n i t e i n length, and the front and back menisci may be treated separately. I f the supply of surfactant to and from the interface i s very fast compared to surface convection, then adsorption equilibrium i s attained along the entire bubble. In this case the bubble achieves a constant surface tension, and the formal results of Bretherton apply, only now for a bubble with an equilibrium surface excess concentration of surfactant. The net mass-transfer rate of surfactant to the interface i s c o n t r o l l e d by the slower of the adsorption-desorption k i n e t i c s and the d i f f u s i o n of surfactant from the bulk solution. The characteris2 2 t i c time for d i f f u s i o n i s T /(Dc ) where D i s the bulk surfactant o' o' d i f f u s i o n c o e f f i c i e n t , c i s the bulk concentration, and T i s the v


26. GINLEY AND RADKE

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equilibrium adsorption at that concentration. Conversely, the c h a r a c t e r i s t i c time for adsorption i s T /(k-c T ) where k- i s an o' 1 o max 1 adsorption rate constant and ^ corresponds to monolayer coverage. Hence, low surfactant concentrations l i k e l y lead to d i f f u s i o n control whereas k i n e t i c l i m i t a t i o n s may dominate at high concentrations. Since our focus i s on concentrations above the c r i t i c a l micelle concentration, we take sorption k i n e t i c s to be rate l i m i t i n g . r


m a x

In examining either end of the bubble, the interface can be divided into three d i s t i n c t regions (15,20), as shown i n Figure 2. Region I i s characterized by a constant f i l m thickness, h . Q

In region III near the tube center, viscous stresses scale by the tube radius and for small c a p i l l a r y numbers do not s i g n i f i c a n t l y d i s t o r t the bubble shape from a spherical segment. Thus, even though surfactant c o l l e c t s near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension,

Influence of Soluble Surfactants on the Flow of Long Bubbles Through

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