Emulsions Fundamentals and Applications in the Petroleum Industry

6 Flow of Emulsions in Porous Media Downloaded by NORTH CAROLINA STATE UNIV on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch006

Sunil L . Kokal, Brij B. Maini, and Roy Woo Petroleum Recovery Institute, 3512 33rd Street N.W., Calgary, Alberta, Canada T 2 L 2A6

A comprehensive review of the important factors that affect the flow of emulsions in porous media is presented with particular emphasis on petroleum emulsions. The nature, characteristics, and properties of porous media are discussed. Darcy's law for the flow of a single fluid through a homogeneous porous medium is introduced and then extended for multiphase flow. The concepts of relative permeability and wettability and their influence on fluid flow are discussed. The flow of oil-in-water (O/W) and water-in-oil (W/O) emulsions in porous media and the mechanisms involved are presented. The effects of emulsion characteristics, porous medium characteristics, and the flow velocity are examined. Finally, the mathematical models of emulsion flow in porous media are also reviewed. THE

F L O W O F E M U L S I O N S I N P O R O U S M E D I A is e n c o u n t e r e d i n the p r o d u c

t i o n o f o i l f r o m u n d e r g r o u n d reservoirs c o n t a i n i n g o i l , water, a n d gas. E m u l sions may f o r m n a t u r a l l y d u r i n g s i m u l t a n e o u s flow o f o i l a n d w a t e r i n p o r o u s rock f o r m a t i o n s , o r they may b e p r o m o t e d b y i n j e c t i o n o f external c h e m i c a l s . In e m u l s i o n flooding f o r h e a v y - o i l recovery {see C h a p t e r 7), externally g e n erated e m u l s i o n s are i n j e c t e d into t h e reservoir. E m u l s i o n flow t h r o u g h a p o r o u s m e d i u m m a y also be e n c o u n t e r e d i n t h e c h e m i c a l process i n d u s t r y i n fixed-bed catalytic reactors i n v o l v i n g t w o i m m i s c i b l e l i q u i d s . T h e physics o f such flows is v e r y c o m p l e x because i t involves flow o f a c o m p l e x a n d unstable fluid i n an e x t r e m e l y c o m p l e x geometry. T o d e v e l o p a w o r k i n g k n o w l e d g e for s o l v i n g p r o b l e m s i n v o l v i n g t h e flow o f e m u l s i o n s i n p o r o u s m e d i a , o n e must possess a k n o w l e d g e o f t h e nature a n d p r o p e r t i e s o f e m u l s i o n s , a n u n d e r s t a n d i n g o f the characteristics o f p o r o u s m e d i a , a n d a w o r k i n g k n o w l edge o f the basic m e c h a n i s m s i n v o l v e d i n t h e flow o f s i m p l e r fluids i n p o r o u s m e d i a . B e c a u s e t h e o t h e r chapters o f this b o o k p r o v i d e a n i n - d e p t h coverage 0065-2393/92/0231-0219 $012.00/0 © 1992 American Chemical Society

In Emulsions; Schramm, L.; Advances in Chemistry; American Chemical Society: Washington, DC, 1992.

220

EMULSIONS IN THE PETROLEUM INDUSTRY

o f the nature a n d p r o p e r t i e s o f e m u l s i o n s , w e w i l l n o t discuss t h e b u l k behavior of emulsions.

Properties of Porous Media I n this section w e w i l l define some o f the terms u s e d to characterize a p o r o u s m e d i u m a n d b r i e f l y discuss those p r o p e r t i e s o f p o r o u s materials that m a y Downloaded by NORTH CAROLINA STATE UNIV on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch006

have relevance to the flow o f e m u l s i o n s .

Porous Medium. A p o r o u s m e d i u m is s i m p l y a s o l i d c o n t a i n i n g holes o r v o i d spaces. H o w e v e r , a m e t a l b l o c k w i t h a f e w holes d r i l l e d t h r o u g h it is n o t a p o r o u s m e d i u m , at least not t h e k i n d o f p o r o u s m e d i u m that c o n c e r n s us. T h e p o r o u s m e d i u m is a s o l i d c o n t a i n i n g a large n u m b e r o f voids d i s p e r s e d t h r o u g h o u t i n e i t h e r a r e g u l a r o r r a n d o m m a n n e r , p r o v i d e d that these voids o c c u r f r e q u e n t l y e n o u g h so that e v e n a s m a l l s u b v o l u m e (small c o m p a r e d to the b u l k d i m e n s i o n s o f the solid) w i l l c o n t a i n some v o i d s . Porosity, T h e f r a c t i o n o f total v o l u m e o c c u p i e d b y the voids is c a l l e d the p o r o s i t y o f the p o r o u s m e d i u m . A d i s t i n c t i o n c a n b e m a d e b e t w e e n the pores that a r e i n t e r c o n n e c t e d a n d the pores that are totally i s o l a t e d . T h e absolute o r total p o r o s i t y is the f r a c t i o n o f b u l k v o l u m e o c c u p i e d b y a l l v o i d s , c o n n e c t e d o r not. T h e effective p o r o s i t y is t h e f r a c t i o n o f b u l k v o l u m e occupied by interconnected pores. T h e p o r o u s m e d i a c a n also b e classified b y the type o f p o r o s i t y i n v o l v e d , d e p e n d i n g o n the size a n d shape o f voids. I n sandstones a n d u n c o n s o l i d a t e d sands, t h e voids are b e t w e e n t h e a d j o i n i n g sand grains, a n d this type o f p o r o s i t y is c a l l e d i n t e r g r a n u l a r . C a r b o n a t e rocks are g e n e r a l l y m o r e c o m p l e x and m a y c o n t a i n m o r e t h a n one type o f p o r o s i t y . T h e s m a l l voids b e t w e e n the crystals o f calcite o r d o l o m i t e constitute i n t e r c r y s t a l l i n e porosity. O f t e n carbonate rocks a r e n a t u r a l l y f r a c t u r e d . T h e v o i d v o l u m e f o r m e d b y frac tures constitutes the fracture porosity. C a r b o n a t e rocks sometimes c o n t a i n relatively large holes, c a l l e d vugs, a n d these constitute the v u g u l a r p o r o s i t y . A t the extreme e n d o f the p o r e size scale, some carbonate formations m a y c o n t a i n v e r y large channels a n d cavities (several meters i n size), w h i c h constitute the cavernous p o r o s i t y . Permeability. T h e p e r m e a b i l i t y o f a p o r o u s m e d i u m is a measure o f the ease w i t h w h i c h a fluid c a n flow t h r o u g h i t . I n o t h e r w o r d s , i t is a measure o f the fluid c o n d u c t i v i t y o f the m e d i u m that d e t e r m i n e s the flow rate o f a g i v e n fluid f o r a g i v e n pressure gradient. T h e e q u a t i o n that defines p e r m e a b i l i t y was d i s c o v e r e d b y D a r c y (I ) a n d is c a l l e d D a r c y ' s l a w . F o r l i n e a r , h o r i z o n t a l , i s o t h e r m a l f l o w o f a fluid, t h e e q u a t i o n is


6.

KOKAL ET AL.

221

Flow of Emulsions in Porous Media

q = — (dP/dL) μ

(1)

w h e r e q is the flow rate (mL/s), k is p e r m e a b i l i t y (darcies), A is crosss e c t i o n a l area o f the m e d i u m ( c m ) , μ is viscosity o f the fluid (mPa-s), a n d 2

dP/dL

is the p r e s s u r e g r a d i e n t (atm/cm).

Pore Size Distribution.

T h e p o r e size d i s t r i b u t i o n is a m e a s u r e o f

Downloaded by NORTH CAROLINA STATE UNIV on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch006

the average size o f the pores a n d the v a r i a b i l i t y o f p o r e sizes. It is u s u a l l y d e t e r m i n e d b y m e r c u r y p o r o s i m e t r y . T h i s t e c h n i q u e is b a s e d o n a s i m p l e c o n c e p t u a l m o d e l o f the pores that treats the pores as c a p i l l a r y tubes. T h e p r e s s u r e r e q u i r e d to force m e r c u r y i n t o a p o r e ( a s s u m i n g that the p o r e behaves l i k e a c i r c u l a r capillary) c a n be r e l a t e d to the radius o f the p o r e b y ρ _ 2 σ cos (θ)

( 2 )

w h e r e P is the c a p i l l a r y pressure r e q u i r e d to f o r c e m e r c u r y i n t o the p o r o u s c

m e d i u m , r is the radius o f the p o r e b e i n g i n v a d e d , σ is the surface t e n s i o n o f m e r c u r y , a n d Θ is the contact angle. B y m e a s u r i n g the v o l u m e o f m e r c u r y e n t e r i n g i n t o a s a m p l e o f the p o r o u s m e d i u m as a f u n c t i o n o f the a p p l i e d c a p i l l a r y p r e s s u r e , the p o r e size d i s t r i b u t i o n c a n b e d e t e r m i n e d . T h e size m e a s u r e d b y c a p i l l a r y p o r o s i m e t r y is that o f the e n t r a n c e to the v o i d space b e i n g i n v a d e d . T h e a c t u a l size o f the v o i d space c a n be m u c h l a r g e r t h a n the e n t r a n c e size. H o w e v e r , an i m p r o v e d t e c h n i q u e u s i n g v e r y sensitive pressure transducers a n d c o m p u t e r i z e d data a c q u i s i t i o n a n d analy sis has b e e n d e v e l o p e d to m e a s u r e the size d i s t r i b u t i o n o f b o t h the p o r e entrances (throats) a n d p o r e b o d i e s (2).

Specific Surface Area.

T h e specific surface area is d e f i n e d as the

area o f i n t e r n a l surfaces p e r u n i t v o l u m e (or w e i g h t ) o f the p o r o u s m a t e r i a l . T h e specific surface area o f p o r o u s materials is v e r y h i g h . P e t r o l e u m reser v o i r rocks t y p i c a l l y possess specific surface area i n the range o f 150 to 3 0 0 0 c m / c m . T h e h i g h i n t e r n a l surface area is r e s p o n s i b l e f o r m a n y i n t e r e s t i n g 2

3

characteristics o f p o r o u s m e d i a . It plays a n i m p o r t a n t r o l e i n processes involving adsorption of material from

fluids

flowing

t h r o u g h the p o r o u s

m e d i u m . It is also a n i m p o r t a n t p a r a m e t e r i n d e e p - b e d

filtration,

i o n ex

change, a n d processes i n v o l v i n g a c h e m i c a l r e a c t i o n b e t w e e n the s o l i d m a trix a n d a flowing fluid. T h e specific surface area also has a d i r e c t i n f l u e n c e o n the p e r m e a b i l i t y o f the m e d i u m (3).

Chemical Composition.

T h e c h e m i c a l c o m p o s i t i o n o f the p o r o u s

m e d i u m c a n be v e r y i m p o r t a n t i n processes i n v o l v i n g exchange o f m a t e r i a l b e t w e e n the s o l i d grains a n d the flowing fluid. S u c h w o u l d b e the case w h e n


222


the m a t e r i a l f o r m i n g the p o r o u s s o l i d dissolves i n the fluid or a c h e m i c a l r e a c t i o n occurs b e t w e e n the s o l i d a n d the fluid. H o w e v e r , i n this study w e w i l l assume that the s o l i d is totally i n e r t ; that is, the flowing e m u l s i o n does not change the p h y s i c a l o r c h e m i c a l characteristics o f the s o l i d , a n d the s o l i d does not change the c h e m i c a l c o m p o s i t i o n o f the fluids flowing t h r o u g h it.


Flow of a Single Fluid: Darcy's Law A fluid's m o t i o n is a f u n c t i o n o f the p r o p e r t i e s o f the fluid, the m e d i u m t h r o u g h w h i c h it is flowing, a n d the external forces i m p o s e d o n it. F o r o n e d i m e n s i o n a l steady l a m i n a r flow o f a single fluid t h r o u g h a h o m o g e n e o u s p o r o u s m e d i u m , the r e l a t i o n s h i p b e t w e e n the flow rate a n d the a p p l i e d external forces is p r o v i d e d b y D a r c y ' s l a w :

cj —

kA ( A P / A L + p g s i n (Θ) μ

, W/

/ Q

w h e r e ρ is density a n d g is a c c e l e r a t i o n d u e to gravity. D a r c y ' s l a w s i m p l y says that the flow rate is p r o p o r t i o n a l to the p e r m e a b i l i t y o f the m e d i u m , the cross-sectional area, a n d the s u m o f pressure g r a d i e n t a n d the g r a d i e n t o f hydrostatic h e a d a l o n g the d i r e c t i o n o f flow; a n d that the flow rate is i n v e r s e l y p r o p o r t i o n a l to the viscosity o f the l i q u i d . F o r flow i n m o r e t h a n one d i r e c t i o n , a m o r e g e n e r a l f o r m o f e q u a t i o n 3 is r e q u i r e d : v = -(k /p)(BP/dx)

(4)

v = -(k /p)(dP/dy)

(5)

x

x

9

y

O, = -(k /\L)(dP/dZ x

+ pg)

(6)

w h e r e v , v , a n d v are the s u p e r f i c i a l velocities i n the x, y, a n d ζ d i r e c t i o n s , respectively. T h e ζ d i r e c t i o n is p a r a l l e l to v e r t i c a l . A l s o , d i f f e r e n t p e r m e a b i l i t i e s (k , k , a n d k ) are u s e d i n d i f f e r e n t d i r e c t i o n s to r e c o g n i z e the fact that p o r o u s m e d i a o f t e n e x h i b i t d i f f e r e n t p e r m e a b i l i t i e s i n d i f f e r e n t d i r e c tions. x

y

x

z

y

z

I n a p p l y i n g equations 4 - 6 , the d i r e c t i o n a l p e r m e a b i l i t i e s are t r e a t e d as p o i n t f u n c t i o n s , that is, as a p r o p e r t y o f a p o i n t i n the m e d i u m . T h e p o i n t value o f the d i r e c t i o n a l p e r m e a b i l i t y c a n b e v i s u a l i z e d as a statistical average o f the fluid c o n d u c t a n c e i n the g i v e n d i r e c t i o n o f a l l pores c o n t a i n e d i n a s m a l l v o l u m e s u r r o u n d i n g the p o i n t i n q u e s t i o n . T h i s s m a l l v o l u m e m u s t b e


6.

KOKAL ET AL.

223


v i s u a l i z e d as b e i n g s m a l l c o m p a r e d to t h e size o f t h e m e d i u m b u t large c o m p a r e d to t h e size o f t h e i n d i v i d u a l flow c h a n n e l s . E q u a t i o n s 4-6 c o m b i n e d w i t h t h e l a w o f c o n s e r v a t i o n o f mass are s u f f i c i e n t to d e r i v e equations f o r t h e flow o f a single fluid i n systems o f c o m p l e x g e o m e t r y . Several excellent books a n d r e v i e w articles are available o n this


t o p i c (3-7).

Multiphase Flow in Porous Media W h e n t w o o r m o r e i m m i s c i b l e fluids are flowing s i m u l t a n e o u s l y t h r o u g h a p o r o u s m e d i u m , t h e flow process b e c o m e s m o r e c o m p l e x . W e w i l l use t h e s i m p l e r case o f two-phase flow to r e v i e w t h e basic m e c h a n i s m s i n v o l v e d i n s u c h processes. T h e p r e s e n c e o f t w o m o b i l e phases means that each fluid c a n interact w i t h b o t h t h e p o r o u s m e d i u m a n d t h e o t h e r i m m i s c i b l e fluid. I f t h e p o r o u s m e d i u m is v i s u a l i z e d as a c o l l e c t i o n o f i n t e r c o n n e c t e d flow paths, o n l y a f r a c t i o n o f the total flow paths b e c o m e available to a g i v e n fluid, the rest b e i n g o c c u p i e d b y t h e o t h e r fluid. T h i s c o n d i t i o n necessitates the i n t r o d u c t i o n o f fluid saturation as a n i m p o r t a n t p a r a m e t e r . T h e saturation o f a fluid phase is d e f i n e d as t h e f r a c t i o n o f t o t a l v o i d space o c c u p i e d b y that fluid. F o r two-phase systems, t h e s u m o f t h e t w o fluid saturations is e q u a l to u n i t y , because any v o i d space n o t o c c u p i e d b y o n e fluid m u s t b e o c c u p i e d b y t h e other

fluid.

B y analogy to single-phase flow, u n d e r steady-state c o n d i t i o n s , the flow rate o f each fluid s h o u l d b e d i r e c t l y p r o p o r t i o n a l to t h e a p p l i e d p r e s s u r e g r a d i e n t a n d t h e cross-sectional area o f the m e d i u m a n d i n v e r s e l y p r o p o r t i o n a l to t h e fluid viscosity. T h e r e f o r e , an e q u a t i o n analogous to e q u a t i o n 1 can b e w r i t t e n f o r each

9 i

=

M

fluid:

(

A

j

y

A

L

)

j

=

u

(

7

)

μ* w h e r e t h e s u b s c r i p t i refers to a specific

fluid.

U s i n g P, i n p l a c e o f Ρ is

necessary to account f o r t h e l o c a l pressure d i s c o n t i n u i t y existing at t h e interface b e t w e e n t h e t w o fluids.

Capillary Pressure.

B e c a u s e o f t h e s m a l l size o f p o r e s , t h e

fluid-

fluid interfaces w i t h i n t h e p o r o u s m e d i u m are h i g h l y c u r v e d , a n d t h e p r e s sure d i f f e r e n c e across the interface c a n b e substantial. T h i s l o c a l pressure d i f f e r e n c e across t h e

fluid-fluid

interface is c a l l e d c a p i l l a r y p r e s s u r e . I n

g e n e r a l , o n e o f the t w o fluids p r e f e r e n t i a l l y wets t h e s o l i d a n d is c a l l e d t h e


224


w e t t i n g fluid. T h e c a p i l l a r y pressure is u s u a l l y d e f i n e d as t h e pressure i n t h e n o n w e t t i n g fluid m i n u s t h e pressure i n t h e w e t t i n g fluid. Pc(S )=F w

n w

-F

(8)

w

w h e r e t h e subscripts n w a n d w r e f e r to t h e n o n w e t t i n g phase a n d t h e w e t t i n g phase, respectively; a n d S is the saturation o f the w e t t i n g phase.


w

Wettability. T h e w e t t a b i l i t y o f the p o r o u s m e d i u m refers to its p r e f e r e n c e f o r o n e o r t h e o t h e r fluid i n b e c o m i n g w e t . It is d e f i n e d as t h e " t e n d e n c y o f o n e fluid to s p r e a d o n o r adhere to a s o l i d surface i n t h e p r e s e n c e o f o t h e r i m m i s c i b l e fluids" (7). I n a r o c k - o i l - b r i n e system, i t is a measure o f t h e p r e f e r e n c e that t h e r o c k has f o r either t h e o i l o r t h e water. A w a t e r - w e t r o c k is p r e f e r e n t i a l l y w e t t e d b y t h e w a t e r phase, a n d s i m i l a r l y f o r an o i l - w e t system, the r o c k p r i m a r i l y makes contact w i t h the o i l phase. Relative Permeability. A c o m p a r i s o n o f e q u a t i o n 7 w i t h e q u a t i o n 1 also shows that f o r the two-phase system w e have u s e d k t h e effective p e r m e a b i l i t y f o r t h e fluid, i n p l a c e o f t h e absolute p e r m e a b i l i t y k, w h i c h is a p r o p e r t y o f t h e p o r o u s m e d i u m alone. T h i s effective p e r m e a b i l i t y t e r m , k d e p e n d s o n t h e absolute p e r m e a b i l i t y , t h e type o f fluid i n v o l v e d , a n d t h e saturation o f this fluid. T h e c o n t r i b u t i o n o f t h e absolute p e r m e a b i l i t y is i s o l a t e d b y m o d i f y i n g e q u a t i o n 1 as f o l l o w s : i9

if

9 l

i = l,2

= **^(AP,/AL) μ*

(9)

T h e t e r m k i n e q u a t i o n 9 is c a l l e d relative p e r m e a b i l i t y o f the fluid i. It represents t h e f r a c t i o n b y w h i c h the fluid c o n d u c t i v i t y o f the p o r o u s m e d i u m m u s t b e m o d i f i e d to a c c o u n t f o r t h e p r e s e n c e o f t h e o t h e r fluid. T h e p r e s e n c e o f t h e o t h e r fluid i m p l i e s that some o f the flow paths w o u l d n o t b e available to this fluid, thus t h e t e r m k i n e q u a t i o n 9 must always b e less t h a n , o r at most e q u a l t o , 1. F u r t h e r m o r e , w h e n m o r e o f t h e fluid i is present i n t h e m e d i u m , i t w i l l o c c u p y m o r e o f t h e available channels, a n d h e n c e its effective p e r m e a b i l i t y w i l l b e h i g h e r . T h e r e f o r e , t h e relative p e r m e a b i l i t y t e r m is e x p e c t e d to increase w i t h i n c r e a s i n g saturation o f this fluid. H

ri

A p o r o u s m e d i u m i n g e n e r a l w i l l have flow channels o f m a n y d i f f e r e n t sizes; c o n s e q u e n t l y , t h e relative p e r m e a b i l i t y o f a g i v e n fluid w i l l d e p e n d n o t o n l y o n w h a t f r a c t i o n o f the available p o r e space i t occupies b u t also o n w h a t types o f flow channels i t o c c u p i e s . I f the fluid o c c u p i e s s m a l l e r channels, its relative p e r m e a b i l i t y w i l l b e s m a l l e r . T h e r e f o r e , t h e d i s t r i b u t i o n o f t h e fluids is a n i m p o r t a n t factor i n d e t e r m i n i n g relative p e r m e a b i l i t y . I f o n l y o n e fluid o c c u p i e s any g i v e n c h a n n e l , the fluid c o n d u c t i v i t y o f this c h a n n e l w o u l d r e m a i n u n c h a n g e d . T h e r e f o r e , i f t h e t w o fluids w e r e to


6.

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o c c u p y totally separate c h a n n e l s , the s u m o f relative p e r m e a b i l i t i e s o f the t w o fluids w o u l d b e u n i t y . H o w e v e r , b o t h fluids c a n o c c u p y d i f f e r e n t parts o f the same flow c h a n n e l . I n this s i t u a t i o n , the flow o f one i m p e d e d b y the p r e s e n c e o f the o t h e r

fluid.

fluid

might be

T h e r e f o r e , the s u m o f the

relative p e r m e a b i l i t i e s o f the t w o fluids is o f t e n less t h a n u n i t y . C l e a r l y , the relative p e r m e a b i l i t y o f a fluid i n a g i v e n p o r o u s m e d i u m is d e t e r m i n e d b y its d i s t r i b u t i o n w i t h i n the p o r e space. T h i s d i s t r i b u t i o n d e p e n d s o n several factors,

i n c l u d i n g the relative a m o u n t s o f each

fluid


p r e s e n t , the past h i s t o r y o f the system (i.e., h o w the fluids w e r e i n t r o d u c e d i n t o the system), a n d the balance o f various forces a c t i n g o n the

fluids.

U n d e r static c o n d i t i o n , these forces i n c l u d e the gravitational o r b u o y a n c y forces r e s u l t i n g f r o m the d e n s i t y d i f f e r e n c e s a n d the c a p i l l a r y forces a r i s i n g f r o m the i n t e r f a c i a l tensions o r surface energies o f the fluids. W h e n

fluids

are i n m o t i o n , the viscous d r a g forces a n d i n e r t i a l effects m a y also p l a y a r o l e i n d e t e r m i n i n g the fluid d i s t r i b u t i o n w i t h i n a p o r o u s m e d i u m . T h e r e f o r e , the factors that c a n affect relative p e r m e a b i l i t y o f a g i v e n fluid i n c l u d e the following: 1. w e t t i n g p r e f e r e n c e

o f the s o l i d i n r e l a t i o n to o t h e r

fluid

present i n the p o r o u s m e d i u m 2. p o r e g e o m e t r y a n d p o r e size d i s t r i b u t i o n o f the m e d i u m 3. saturation o f the

fluid

4. saturation h i s t o r y o f the p o r o u s m e d i u m 5. densities o f d i f f e r e n t fluids present 6. viscosities o f d i f f e r e n t fluids present 7. irrterfacial t e n s i o n b e t w e e n the t w o fluids 8. relative v e l o c i t y o f the

fluids

I n most situations o f p r a c t i c a l interest i n p e t r o l e u m r e s e r v o i r e n g i n e e r i n g , the l o c a l d i s t r i b u t i o n o f fluids w i t h i n the p o r o u s r o c k is d o m i n a t e d b y c a p i l l a r y forces. T h e r e f o r e , a reasonable a s s u m p t i o n is that c a p i l l a r y e q u i l i b rium

exists b e t w e e n the t w o fluids even u n d e r d y n a m i c c o n d i t i o n s . U n d e r

these c o n d i t i o n s , the l o c a l d i s t r i b u t i o n o f fluids a n d relative p e r m e a b i l i t y d o not d e p e n d o n the last f o u r factors l i s t e d . A large v o l u m e o f p u b l i s h e d e x p e r i m e n t a l research has s h o w n that the relative p e r m e a b i l i t y is not a s t r o n g f u n c t i o n o f the fluid viscosities a n d i n t e r f a c i a l t e n s i o n , p r o v i d e d that these variables r e m a i n w i t h i n t h e i r u s u a l range. E x p e r i m e n t a l e v i d e n c e also suggests that the relative p e r m e a b i l i t y does not change significantly w i t h the m a g n i t u d e o f relative v e l o c i t y a n d the d e n s i t y d i f f e r e n c e b e t w e e n the t w o fluids. A t a g i v e n fluid saturation i n a g i v e n p o r o u s m e d i u m , the w e t t i n g p r e f e r e n c e o f the s o l i d f o r one o f the t w o fluids present d e t e r m i n e s the

fluid

d i s t r i b u t i o n w i t h i n the p o r o u s m e d i u m , a n d c o n s e q u e n t l y , it also d e t e r m i n e s


226



the relative p e r m e a b i l i t y b e h a v i o r . I n t h e context o f r e s e r v o i r e n g i n e e r i n g , either water or o i l may preferentially wet the reservoir rock. H o w e v e r , because o f t h e c o m p l e x g e o m e t r y o f the i n t e r n a l surfaces o f n a t u r a l p o r o u s m e d i a , t h e relative p e r m e a b i l i t i e s f o r a g i v e n fluids-rock system cannot b e p r e d i c t e d f r o m a k n o w l e d g e o f t h e system w e t t a b i l i t y a n d p o r e g e o m e t r y . T h e r e f o r e , e x p e r i m e n t a l l y m e a s u r e d values o b t a i n e d i n representative s a m ples o f t h e p o r o u s m e d i u m m u s t b e u s e d . T y p i c a l e x p e r i m e n t a l l y deter m i n e d o i l - w a t e r relative p e r m e a b i l i t y characteristics f o r w a t e r - w e t a n d o i l w e t rocks are s h o w n i n F i g u r e s 1 a n d 2 , respectively. T h e relative p e r m e a b i l i t y , f o r e i t h e r o f t h e t w o fluids, b e c o m e s z e r o at n o n z e r o saturations o f t h e respective fluids. I n o t h e r w o r d s , a significant saturation o f e i t h e r fluid c a n b e c o m e i m m o b i l e a n d m a y n o t b e d i s p l a c e d b y the o t h e r fluid. A l t h o u g h b o t h w e t t i n g a n d n o n w e t t i n g fluids c a n b e c o m e t r a p p e d , t h e i r d i s t r i b u t i o n w i t h i n t h e p o r o u s m e d i u m is v e r y d i f f e r e n t . I n strongly w e t t e d systems, a t h i n l a y e r o f t h e w e t t i n g fluid always covers t h e s o l i d surfaces. T h e r e f o r e , t h e w e t t i n g fluid always remains c o n t i n u o u s e v e n at t h e i r r e d u c i b l e saturation. It b e c o m e s i m m o b i l e because o f t h e s t r o n g i n t e r a c t i o n b e t w e e n t h e s o l i d a n d t h e w e t t i n g fluid a n d t h e i n a b i l i t y o f t h e n o n w e t t i n g fluid to e n t e r v e r y fine pores a n d crevices w h e r e b u l k o f t h e i r r e d u c i b l e saturation exists. B e c a u s e o f the p r e s e n c e o f this t h i n layer o f w e t t i n g fluid b e t w e e n t h e n o n w e t t i n g fluid a n d t h e s o l i d , the n o n w e t t i n g fluid is always s u r r o u n d e d b y the w e t t i n g fluid. A n a t u r a l c o n s e q u e n c e o f this c o n d i t i o n is that parts o f t h e

1 0.91

k rw

0

0.2

0.4

0.8

0.6

1

WATER SATURATION Figure 1. Relative permeabilities for a water-wet system. (k is the relative oil permeability; k is the relative water permeability.) ro

w


6.

KOKAL ET AL.

227


10.9-


5

0.7-I

WATER SATURATION Figure 2. Relative permeabilities for an oil-wet system. ( k is the relative oil permeability; k™ is the relative water permeability.) ro

n o n w e t t i n g fluid c a n b e c o m e i s o l a t e d a n d m a y stop flowing. A t t h e t e r m i n a l saturation w h e r e t h e relative p e r m e a b i l i t y o f t h e n o n w e t t i n g phase b e c o m e s z e r o , its d i s t r i b u t i o n w i t h i n t h e p o r o u s m e d i u m is d i s c o n t i n u o u s . T h e t r a p p e d saturation o f n o n w e t t i n g f l u i d exists as i s o l a t e d b l o b s c o m p l e t e l y s u r r o u n d e d b y t h e w e t t i n g phase. T h e s e b l o b s c a n b e large e n o u g h to fill several tens o f pores o r they m a y b e s m a l l a n d f u l l y c o n t a i n e d w i t h i n i n d i v i d u a l p o r e s . T h e y r e m a i n t r a p p e d because t h e p r e s s u r e g r a d i e n t r e s u l t i n g f r o m t h e flow o f the w e t t i n g phase is n o t sufficient to o v e r c o m e t h e c a p i l l a r y resistance e n c o u n t e r e d i n f o r c i n g t h e b l o b t h r o u g h n a r r o w p o r e throats. E q u a t i o n s 8 a n d 9 c o m b i n e d w i t h t h e l a w o f c o n s e r v a t i o n o f mass are sufficient f o r m a t h e m a t i c a l d e s c r i p t i o n o f o n e - d i m e n s i o n a l two-phase

flow.

T h e o n l y a d d i t i o n a l i n f o r m a t i o n n e e d e d is t h e f u n c t i o n a l r e l a t i o n s h i p b e t w e e n relative p e r m e a b i l i t y a n d fluid saturation f o r b o t h fluids. F o r flow i n m o r e t h a n o n e d i m e n s i o n , a g e n e r a l i z e d f o r m o f e q u a t i o n 9 is u s e d . C o l l i n s (5), R i c h a r d s o n (6), a n d C r a i g (7) p r e s e n t m o r e i n f o r m a t i o n o n this subject.

Flow of Oil-in-Water Emulsions in Porous Media T h e flow o f o i l - i n - w a t e r (O/W) e m u l s i o n s i n p o r o u s m e d i a is a m o r e c o m p l e x process b e c a u s e o f t h e c o m p l e x n a t u r e o f t h e e m u l s i o n i t s e l f i n a d d i t i o n t o the c o m p l e x i t i e s o f t h e p o r o u s m e d i u m . A m a j o r issue t o b e c o n s i d e r e d is



228


w h e t h e r to treat the e m u l s i o n as a h o m o g e n e o u s fluid. I f the e m u l s i o n d r o p l e t s are v e r y s m a l l c o m p a r e d to the g e o m e t r y o f the flow c h a n n e l s , a reasonable a p p r o a c h is to use a c o n t i n u u m m o d e l o f the m a t e r i a l a n d totally i g n o r e the m i c r o s c o p i c s t r u c t u r a l details. E q u i v a l e n t h o m o g e n e o u s p r o p e r ties c a n t h e n b e assigned to the fluid. U n f o r t u n a t e l y , i n most cases o f p r a c t i c a l interest, the e m u l s i o n d r o p l e t sizes are not m u c h s m a l l e r t h a n p o r e sizes. T h e r e f o r e , t r e a t i n g the e m u l s i o n s as a p s e u d o - s i n g l e - p h a s e fluid w o u l d b e objectionable i n most cases. T o d e v e l o p an u n d e r s t a n d i n g o f the e m u l s i o n flow i n p o r o u s m e d i a , it is u s e f u l to c o n s i d e r differences a n d s i m i l a r i t i e s b e t w e e n the flow o f an O / W e m u l s i o n a n d simultaneous flow o f o i l a n d w a t e r i n a p o r o u s m e d i u m . A s discussed i n the p r e c e d i n g section, i n simultaneous flow o f o i l a n d water, b o t h fluid phases are l i k e l y to o c c u p y c o n t i n u o u s , a n d to a large extent, separate n e t w o r k s o f flow channels. A s s u m i n g the p o r o u s m e d i u m to b e w a t e r - w e t , the o i l phase b e c o m e s d i s c o n t i n u o u s o n l y at the r e s i d u a l satura t i o n o f o i l , w h e r e the o i l ceases to flow. E v e n at its r e s i d u a l saturation, the o i l m a y r e m a i n c o n t i n u o u s o n a scale m u c h l a r g e r t h a n pores. I n the flow o f an O / W e m u l s i o n , the o i l exists as t i n y d i s p e r s e d d r o p l e t s that are c o m p a r a b l e i n size to p o r e sizes. T h e r e f o r e , the o i l a n d w a t e r are m u c h m o r e l i k e l y to o c c u p y the same flow channels. C o n s e q u e n t l y , at the same w a t e r saturation the relative p e r m e a b i l i t i e s to w a t e r a n d o i l are l i k e l y to b e q u i t e d i f f e r e n t i n e m u l s i o n flow. I n n o r m a l flow o f o i l a n d water, o i l droplets o r ganglia b e c o m e t r a p p e d i n the p o r o u s m e d i u m b y the process o f s n a p - o f f o f o i l filament at p o r e throats (8). I n the flow o f an O / W e m u l s i o n , an o i l d r o p l e t is l i k e l y to b e c o m e t r a p p e d b y the m e c h a n i s m o f s t r a i n i n g c a p t u r e at a p o r e throat s m a l l e r t h a n the d r o p .

Permeability Reduction by Flow of Oil-in-Water Emulsion. M c A u l i f f e (9) p r o p o s e d the c o n c e p t o f p e r m e a b i l i t y r e d u c t i o n b y e m u l s i o n flow i n a p o r o u s m e d i u m . C o n s i d e r a single d r o p l e t o f an o i l e m u l s i o n e n t e r i n g a p o r e throat s m a l l e r t h a n itself, as s h o w n i n F i g u r e 3. T h e radius o f c u r v a t u r e o f the l e a d i n g edge is s m a l l e r t h a n the radius o f c u r v a t u r e o f the t r a i l i n g edge o f the d r o p l e t i n the p o r e throat, a n d c o n s e q u e n t l y the c a p i l l a r y pressure is greater at the front o f the d r o p l e t than at its back. A c e r t a i n pressure is t h e n r e q u i r e d to force the d r o p l e t t h r o u g h the c o n s t r i c t i o n . T h i s L a p l a c i a n d i f f e r e n t i a l pressure r e q u i r e d to m o v e the d r o p l e t t h r o u g h the p o r e throat is g i v e n b y

5Ρ = 2 σ

(10)

w h e r e σ is the i n t e r f a c i a l t e n s i o n at the o i l - w a t e r interface a n d r a n d r are the r a d i i o f c u r v a t u r e at the t r a i l i n g a n d l e a d i n g edges o f the d r o p , respecx


2


6.

KOKAL ET AL.


229

Figure 3. Emulsion blockage mechanism, (η and r are the radii of curvature at the trailing and leading edges of the drop, respectively.) (Reproduced with permission from reference 9. Copyright 1973 Society of Petroleum Engineers.) 2

tively. I f the actual pressure d i f f e r e n t i a l across the p o r e is less t h a n that p r e d i c t e d b y e q u a t i o n 10, the e m u l s i o n d r o p l e t plugs the p o r e throat. T h i s c o n s i d e r a t i o n is the f u n d a m e n t a l basis f o r M c A u l i f f e ' s (9) t h e o r y f o r p e r m e a b i l i t y r e d u c t i o n i n the flow o f e m u l s i o n s i n p o r o u s m e d i a . F o r an e m u l s i o n to be an effective b l o c k i n g agent, the o i l d r o p l e t size s h o u l d be slightly l a r g e r t h a n the p o r e throat size. E m u l s i o n d r o p l e t s have a range o f sizes, as d o p o r e throats, h e n c e a s m a l l a m o u n t o f e m u l s i f i e d o i l c a n be v e r y effective i n r e s t r i c t i n g flow. H o w e v e r , p o r e throat c o n s t r i c t i o n s s h o u l d not be excessively large. F o r adverse m o b i l i t y ratios, viscous i n s t a b i l ities d e v e l o p i n s i d e the r e s e r v o i r , a n d w a t e r starts to finger t h r o u g h . W h e n a n O / W e m u l s i o n is i n j e c t e d (or f o r m e d ) , a greater a m o u n t o f e m u l s i o n enters the m o r e p e r m e a b l e zones. A s the e m u l s i o n restricts the flow o f fluids i n the m o r e p e r m e a b l e zones, the d i s p l a c i n g w a t e r (and e m u l s i o n ) begins to flow i n t o the less p e r m e a b l e zones a n d thus e f f e c t i v e l y i m p r o v e s the sweep efficiency. W h e n the pressure d r o p across the p o r e throat is l a r g e r t h a n that p r e d i c t e d b y e q u a t i o n 10, the o i l d r o p l e t u n d e r g o e s distortions to pass t h r o u g h the p o r e c o n s t r i c t i o n . T h e d r o p l e t s pass u n d i s t o r t e d t h r o u g h those p o r e c o n s t r i c t i o n s that have d i a m e t e r s larger t h a n the d r o p l e t d i a m e t e r . T h u s , w h e n the d r o p l e t d i a m e t e r is m u c h s m a l l e r t h a n p o r e throat

size

(a

m i c r o e m u l s i o n , f o r e x a m p l e ) , the r o c k " s e e s " the fluid as a h o m o g e n e o u s fluid. T h i s c o n c e p t o f e m u l s i o n flow i n p o r o u s m e d i a , h o w e v e r , is not w i t h o u t c o n t r a d i c t i o n s (10,11).

B l o c k a g e o f the p o r e throats b y o i l d r o p l e t s necessi

tates a n increase o f d i f f e r e n t i a l pressure as g i v e n b y e q u a t i o n 10. T h i s feature i m p l i e s that the i n t e r f a c i a l t e n s i o n σ and/or the d r o p l e t radius be i n c r e a s e d . H o w e v e r , f o r e m u l s i f i c a t i o n to o c c u r , i n t e r f a c i a l t e n s i o n m u s t b e d e c r e a s e d . T h e r e f o r e , to m a i n t a i n e m u l s i o n stability (low σ ) a n d p r o v i d e effective b l o c k a g e (high σ ) , the i n t e r f a c i a l t e n s i o n has to b e m i n i m i z e d to


230


some o p t i m u m value a n d t h e d r o p l e t size m a x i m i z e d to some larger t h a n t h e p o r e throat d i a m e t e r .

diameter


Soo a n d R a d k e ( I I ) c o n f i r m e d that t h e transient p e r m e a b i l i t y r e d u c t i o n o b s e r v e d b y M c A u l i f f e (9) m a i n l y arises f r o m t h e r e t e n t i o n o f drops i n pores, w h i c h they t e r m e d as s t r a i n i n g c a p t u r e o f t h e o i l d r o p l e t s . T h e y also o b served that d r o p l e t s s m a l l e r t h a n p o r e throats w e r e c a p t u r e d i n crevices o r pockets a n d s o m e t i m e s o n t h e surface o f t h e p o r o u s m e d i u m . T h e y c o n c l u d e d , o n t h e basis o f t h e i r e x p e r i m e n t s i n sand packs a n d v i s u a l glass m i c r o m o d e l observations, that stable O / W e m u l s i o n s d o n o t flow i n t h e p o r o u s m e d i u m as a c o n t i n u u m viscous l i q u i d , n o r d o they flow b y s q u e e z i n g t h r o u g h p o r e c o n s t r i c t i o n s , b u t r a t h e r b y the capture o f the o i l droplets w i t h subsequent p e r m e a b i l i t y r e d u c t i o n . T h e y u s e d deep-bed* filtration p r i n c i ples (12,13) to m o d e l this p h e n o m e n o n , w h i c h is d i s c u s s e d i n d e t a i l later i n this chapter.

Effect of Emulsion Characteristics. T h e flow o f e m u l s i o n s i n p o r o u s m e d i a is affected b y a large n u m b e r o f variables. T h i s section d e scribes t h e p r o p e r t i e s o f e m u l s i o n s , s u c h as stability, q u a l i t y , d r o p l e t size d i s t r i b u t i o n , o i l viscosity, w a t e r - o i l i n t e r f a c i a l p r o p e r t i e s , a n d t h e i r effect o n its flow i n p o r o u s m e d i a . Emulsion Stability. A n e m u l s i o n is a t h e r m o d y n a m i c a l l y unstable system a n d has a n a t u r a l t e n d e n c y to separate i n t o two phases. T h i s t e n d e n c y is d u e to the fact that w h e n o n e phase is d i s p e r s e d i n another, the i n t e r f a c i a l area increases a n d leads t o a n increase o f t h e free energy o f the system. A n y o i l - w a t e r system tends to m i n i m i z e this free energy b y r e d u c i n g t h e i n t e r f a c i a l area a n d b y i n d u c i n g coalescence o f the d i s p e r s e d o i l d r o p l e t s . H o w e v e r , apparent e m u l s i o n stability m a y b e attained f o r a c e r t a i n t i m e p e r i o d b y u s i n g s t a b i l i z i n g agents o r e m u l s i f i e r s . T h e s e agents are either a d d e d o r c o u l d b e n a t u r a l l y o c c u r r i n g i n t h e o i l reservoir, a n d t h e y suppress t h e m e c h a n i s m s (flocculation, coalescence, c r e a m i n g , phase i n v e r s i o n , etc.) that cause e m u l s i o n b r e a k d o w n . T h e stability o f e m u l s i o n s was discussed i n C h a p t e r s 1 a n d 2 a n d w i l l b e d i s c u s s e d h e r e v e r y b r i e f l y i n r e l a t i o n to t h e i r flow i n p o r o u s m e d i a . T h e D L V O (Derjaguin, L a n d a u , Verwey, and Overbeek) theory o f colloid stabil ity is o f t e n u s e d to d e s c r i b e t h e short-range i n t e r a c t i o n b e t w e e n d r o p l e t s c a u s i n g flocculation a n d coalescence. A c c o r d i n g to this theory, t h e total p o t e n t i a l energy o f i n t e r a c t i o n is t h e s u m o f t h e L o n d o n - v a n d e r W a a l s attractive energy a n d e l e c t r i c a l d o u b l e - l a y e r r e p u l s i v e e n e r g y b e t w e e n p a r t i cles. T h e total i n t e r a c t i o n e n e r g y as a f u n c t i o n o f distance o f separation is s h o w n i n F i g u r e 4. D r o p l e t interactions c a n o c c u r as a result o f h y d r o d y n a m i c effects, s u c h as m i x i n g a n d flow i n p o r o u s m e d i a , a n d n o n h y d r o d y n a m i c effects, s u c h as d i f f u s i o n a n d surface p h e n o m e n a . I n these situations, t h e d e s t a b i l i z a t i o n o f the e m u l s i o n d u e to flocculation,


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+

Figure 4. DLVO theory of colloid stability. (Reproduced with permission from reference 10. Copyright 1988 Canadian Institute of Mining, Metallurgy, and Petroleum.)

c o a l e s c e n c e , a n d c r e a m i n g is d e t e r m i n e d b y t h e shape o f t h e i n t e r a c t i o n p o t e n t i a l e n e r g y c u r v e i n F i g u r e 4 . T h i s c u r v e is d e t e r m i n e d b y the surface p o t e n t i a l , thickness o f t h e e l e c t r i c a l d o u b l e layer, i o n i c strength, d r o p l e t size, H a m a k a r constant, a n d a n u m b e r o f o t h e r factors (10, 14). I n F i g u r e 4 the p r i m a r y m i n i m u m , P , represents the p o t e n t i a l e n e r g y o f t w o d r o p l e t s i n close p r o x i m i t y d u e t o r e p u l s i o n as a result o f the o v e r l a p o f e l e c t r o n shells. T h e m a x i m u m , M , represents t h e r e p u l s i v e energy b a r r i e r t o c o a l e s c e n c e , a n d t h e secondary m i n i m u m , S, represents t h e separation distance w h e r e droplets w i l l

flocculate

i n t o aggregates b u t m a y b e d i s p e r s e d .

T h e p r e s e n c e o f surfactants, e i t h e r n a t u r a l o r a d d e d , p r o m o t e s e m u l s i o n stability b y the r e d u c t i o n o f i n t e r f a c i a l t e n s i o n a n d t h e f o r m a t i o n o f h i g h l y r i g i d films o n t h e surface o f t h e d r o p l e t s . T h i s r e d u c t i o n o f i n t e r f a c i a l t e n s i o n c a n increase t h e m a x i m u m , M , i n F i g u r e 4 significantly t h r o u g h charge s t a b i l i z a t i o n o r steric s t a b i l i z a t i o n (15). B e c a u s e the n a t u r e a n d shape o f the i n t e r a c t i o n energy c u r v e d e t e r m i n e t h e stability o f O / W (and o t h e r types) o f e m u l s i o n s , any process, p a r a m e t e r , o r p h e n o m e n o n that affects the shape o f this c u r v e w i l l u l t i m a t e l y c o n t r o l e m u l s i o n stability. S o m e o f the p a r a m e t e r s that affect t h e stability (16) are t h e f o l l o w i n g : • Temperature. T e m p e r a t u r e c a n affect e m u l s i o n stability i n a number

o f ways. T e m p e r a t u r e

affects i n t e r f a c i a l

tension,

w h i c h g e n e r a l l y decreases w i t h i n c r e a s i n g t e m p e r a t u r e (17,


232



18). A l o w e r i n t e r f a e i a l t e n s i o n w i l l l e a d t o a m o r e stable e m u l s i o n . T e m p e r a t u r e affects p h y s i c a l p r o p e r t i e s o f o i l , w a ter, i n t e r f a e i a l f i l m s , a n d surfactant s o l u b i l i t i e s i n t h e o i l a n d w a t e r phases, w h i c h c a n a l l affect e m u l s i o n stability. F u r t h e r , the r h e o l o g y o f the e m u l s i o n i t s e l f is affected significantly b y temperature. • Pressure. R e s e r v o i r pressure has a less significant effect o n e m u l s i o n stability t h a n t e m p e r a t u r e . Interfaeial t e n s i o n d e creases as t h e pressure o f the system increases. P r e s s u r e ef fects p r o b a b l y have a n i n d i r e c t effect o n e m u l s i o n stability because o f the d e p e n d e n c e o f p h y s i c a l p r o p e r t i e s o n pressure. • Surface-active agents. Surface-active agents s u c h as e m u l s i fiers a n d surfactants p l a y a v e r y significant r o l e i n the stability o f e m u l s i o n s . T h e y greatly e x t e n d the t i m e o f coalescence, a n d thus t h e y stabilize t h e e m u l s i o n s . M e c h a n i s m s b y w h i c h t h e surface-active agents stabilize t h e e m u l s i o n are d i s c u s s e d i n d e t a i l b y B e c h e r (19) a n d C o s k u n e r (14). T h e y f o r m m e c h a n i cally strong films at the o i l - w a t e r interface that act as barriers to coalescence. T h e e m u l s i o n droplets are sterically s t a b i l i z e d b y t h e asphaltene a n d r e s i n fractions o f t h e c r u d e o i l , a n d these c a n r e d u c e i n t e r f a e i a l t e n s i o n i n some systems even at v e r y l o w concentrations (17, 20). I n situ emulsifiers are f o r m e d f r o m t h e asphaltic a n d resinous materials f o u n d i n c r u d e oils c o m b i n e d w i t h ions i n the b r i n e a n d i n s o l u b l e dis p e r s e d fines that exist i n t h e o i l - b r i n e system. C e r t a i n o i l soluble organic acids s u c h as n a p h t h e n i c , fatty, a n d a r o m a t i c acids c o n t r i b u t e to e m u l s i f i c a t i o n (21). T h e interfaeial films f o r m e d b y d i f f e r e n t c r u d e oils have d i f f e r e n t characteristics. T h e p h y s i c a l characteristics o f the films are a f u n c t i o n o f the c r u d e - o i l type a n d gas content, the c o m p o s i t i o n a n d p H o f water, t h e t e m p e r a t u r e , the presence o f n o n i o n i c p o l a r m o l e c u l e s i n the water, the extent to w h i c h the a d s o r b e d film is c o m p r e s s e d , a n d the contact t i m e a l l o w e d f o r a d s o r p t i o n a n d c o n c e n t r a t i o n o f p o l a r m o l e c u l e s i n the o i l phase (14, 22,23). T h e r h e o l o g i c a l p r o p e r t i e s o f the a d s o r b e d e m u l s i f i e r film have an i m p o r t a n t effect o n the stability o f e m u l s i o n s . V e r y f e w studies have f o c u s e d o n t h e stability o f O / W e m u l s i o n s i n p o r o u s m e d i a . Sarbar et a l . (24) c o n d u c t e d a s t u d y to d e t e r m i n e the effect o f c h e m i c a l additives o n t h e stability o f O / W e m u l s i o n flow t h r o u g h p o r o u s m e d i a . T h e y i n j e c t e d 1, 5 , a n d 1 0 % O / W e m u l s i o n s i n s a n d packs w i t h v a r y i n g p H a n d surfactant concentrations a n d f o u n d that t h e r e was a n o p t i m a l v a l u e o f the surfactant c o n c e n t r a t i o n at w h i c h e m u l s i o n s w e r e the most stable. A d d i t i o n o f s o d i u m c h l o r i d e t o the aqueous phase h a d a d e t r i m e n t a l effect o n the stability o f the e m u l s i o n . F o r t h e i r system t h e y f o u n d that t h e r e


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was a n o p t i m u m v a l u e o f p H (10) at w h i c h the e m u l s i o n s w e r e the most stable. U n s t a b l e e m u l s i o n s flow i n a m a r k e d l y d i f f e r e n t m a n n e r f r o m stable e m u l s i o n s . U n s t a b l e e m u l s i o n s have r e l a t i v e l y large i n t e r f a e i a l tensions r e s u l t i n g i n large o i l d r o p l e t s that w e r e o b s e r v e d to be d e p o s i t e d at the i n l e t o f the c o r e . T h i s o i l b a n k grows u n t i l it stalls a n d r e - e m u l s i f i c a t i o n o f the o i l takes p l a c e . T h i s r e - e m u l s i f i e d o i l is c a r r i e d to a n o t h e r p o s i t i o n , a n d a n o t h e r o i l b a n k begins to f o r m . T h i s m e c h a n i s m was not o b s e r v e d f o r the stable e m u l s i o n because t h e r e was less o i l b r e a k i n g f r o m the e m u l s i o n to b e c o m e


available f o r the b a n k i n g p h e n o m e n o n . Emulsion

Quality.

T h e q u a l i t y o f an e m u l s i o n is d e f i n e d as the v o l

u m e f r a c t i o n (or p e r c e n t ) o f the d i s p e r s e d phase i n the e m u l s i o n . T h e q u a l i t y o f e m u l s i o n s strongly affects t h e i r r h e o l o g y . S e v e r a l studies have b e e n r e p o r t e d f o r the r e l a t i o n s h i p o f i s o t h e r m a l shear stress to shear rate f o r e m u l s i o n s o f d i f f e r e n t qualities. O / W e m u l s i o n s h a v i n g qualities less t h a n 0.5 (or 50%) e x h i b i t N e w t o n i a n b e h a v i o r , a n d those h a v i n g h i g h e r qualities e x h i b i t n o n - N e w t o n i a n b e h a v i o r (9, 16, 25). F i g u r e 5 (25) shows the effect o f e m u l s i o n q u a l i t y o n the p r e s s u r e d r o p i n p o r o u s m e d i a as a f u n c t i o n o f the flow rate. F o r e m u l s i o n o f q u a l i t y u p to 4 0 % , the r e l a t i o n s h i p b e t w e e n p r e s s u r e d r o p a n d flow rate is l i n e a r , an e f f e c t s h o w i n g that D a r c y ' s l a w describes the steady-state, l a m i n a r flow o f a N e w t o n i a n m a c r o e m u l s i o n t h r o u g h p o r o u s m e d i a . F o r e m u l s i o n qualities h i g h e r t h a n 4 0 % , the r e l a t i o n s h i p b e t w e e n p r e s s u r e d r o p a n d flow rate is n o t l i n e a r , a result i n d i c a t i n g n o n - N e w t o n i a n b e h a v i o r . T h i s k i n d o f b e h a v i o r is typical of non-Newtonian pseudo-plastic

fluids.

F i g u r e 6 shows a l o g - l o g

p l o t o f p r e s s u r e d r o p versus flow rate f o r a 4 0 a n d 6 0 % flowing

macroemulsion

t h r o u g h p o r o u s m e d i a . T h e 4 0 % e m u l s i o n has a l i n e a r r e l a t i o n s h i p

o n the p l o t w i t h a slope o f 1, a n d thus shows N e w t o n i a n b e h a v i o r . T h e 6 0 % m a c r o e m u l s i o n , o n the o t h e r h a n d , gives a slope o f less t h a n 1 a n d thus shows n o n - N e w t o n i a n , p s e u d o - p l a s t i c b e h a v i o r . S i m i l a r results w e r e also o b t a i n e d b y U z o i g w e a n d M a r s d e n (26). T h e y c o n d u c t e d flow tests i n c a p i l l a r y tubes a n d i n p o r o u s m e d i a u s i n g d i f f e r e n t qualities o f O / W e m u l s i o n s a n d o b s e r v e d N e w t o n i a n b e h a v i o r u p to 5 0 % qualities a n d n o n - N e w t o n i a n b e h a v i o r at h i g h e r q u a l i t i e s , as s h o w n i n F i g ures 7 a n d 8. F o r the h i g h - q u a l i t y e m u l s i o n s , N e w t o n i a n b e h a v i o r was e x h i b i t e d at l o w shear rates a n d shear stresses. T r a n s i t i o n to n o n - N e w t o n i a n b e h a v i o r o c c u r r e d after some c r i t i c a l shear rates that w e r e f o u n d to d e p e n d o n e m u l s i o n q u a l i t y . T h e h i g h e r the q u a l i t y , the l o w e r the v a l u e o f this c r i t i c a l shear rate. T h e t r a n s i t i o n shear rate values r e p o r t e d suggest a n e x p o n e n t i a l d e p e n d e n c e o n e m u l s i o n q u a l i t y . A f t e r the t r a n s i t i o n to n o n N e w t o n i a n b e h a v i o r , these h i g h - q u a l i t y e m u l s i o n s e x h i b i t e d p s e u d o - p l a s t i c shear-thinning behavior. U z o i g w e a n d M a r s d e n (26) e x p l a i n e d this b e h a v i o r u s i n g p a r t i c l e - p a r t i cle i n t e r a c t i o n s that are c a u s e d b y forces o f attraction o r r e p u l s i o n b e t w e e n t h e m . F o r d i l u t e , l o w - q u a l i t y e m u l s i o n s , the r e p u l s i o n forces are q u i t e h i g h


234


35

30

Δ Ο Ο


+

fe,% 20 40 50 60 70

0.05

CORE Βθ4Δ(2) Βθ4Δ (!) Βθ4Δ (2) Βθ4Α (I) Βθ4Δ (I)

0.10

ΟΙ5

0.20

0.25

Flow rate,q, cm /sec 3

Figure 5. Cartesian plot of Newtonian and non-Newtonian behavior of flow of O/W macroemulsions through porous media. (f is emulsion quality.) (Repro duced with permission from reference 25. Copyright 1979 Society of Petroleum Engineers.) e

relative to the attractive forces, a n d c o n s e q u e n t l y m i n i m u m aggregation takes p l a c e . H o w e v e r , i n c o n c e n t r a t e d o r h i g h - q u a l i t y e m u l s i o n s , the r e p u l sive forces are r e d u c e d , a n d the attractive forces w i l l l e a d to the f o r m a t i o n o f aggregates c a u s i n g coalescence a n d flocculation. A t l o w shear rates these aggregates rotate l i k e single p a r t i c l e s , a n d the viscosity is h i g h . A s shear is i n c r e a s e d , the aggregates b r e a k d o w n , a n d the viscosity decreases. T h e b r e a k u p o f aggregates results i n the o b s e r v e d p s e u d o - p l a s t i c b e h a v i o r . O n the o t h e r h a n d , f o r stable l o w - q u a l i t y e m u l s i o n s , the p a r t i c l e s are far apart, a n d the net response o f the e m u l s i o n s to shear is s i m i l a r to that o f the N e w t o n i a n nature o f the c o n t i n u o u s phase. F i g u r e 9 f r o m U z o i g w e a n d M a r s d e n (26) shows a p l o t o f a p p a r e n t viscosity versus e m u l s i o n quality. T h i s g r a p h shows that apparent viscosities increase s h a r p l y w i t h q u a l i t y p a r t i c u l a r l y at l o w shear rates. It also shows the N e w t o n i a n b e h a v i o r at l o w qualities a n d n o n - N e w t o n i a n b e h a v i o r at h i g h qualities.


6.

KOKAL ET AL. irk

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I

I

f

e

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1

1

1

1 t 1M

0.10

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1.00

Flow R a t e , q , cm /sec 3

Figure 6. A log-log plot of Newtonian and non-Newtonian behavior of flow of macroemulsions through porous media. (f is emulsion quality.) (Reproduced with permission from reference 25. Copyright 1979 Society of Petroleum Engi neers.) e

W h e n u s i n g stable, d i l u t e N e w t o n i a n e m u l s i o n s t h r o u g h p o r o u s m e d i a , the

flowing

p e r m e a b i l i t y , fc , m u s t b e u s e d i n D a r c y ' s l a w t o d e s c r i b e its f

b e h a v i o r i n s t e a d o f the i n i t i a l o r c o n v e n t i o n a l p e r m e a b i l i t y . W h e n p l u g g i n g d u e to t h e flow o f N e w t o n i a n m a c r o e m u l s i o n s o c c u r s , o n l y t h e p e r m e a b i l i t y o f t h e p o r o u s m e d i u m s h o u l d b e adjusted. E m u l s i o n r h e o l o g y w i t h r e s p e c t t o N e w t o n i a n a n d n o n - N e w t o n i a n b e h a v i o r w i l l b e r e v i e w e d u n d e r the section " M a t h e m a t i c a l M o d e l s o f E m u l s i o n F l o w i n Porous M e d i a " . Average Droplet

Size and Droplet

Size Distribution.

A l l practical

e m u l s i o n s s h o w s o m e f o r m o f d r o p l e t size d i s t r i b u t i o n w i t h an average v a l u e r e p r e s e n t i n g this size d i s t r i b u t i o n . T h e average d r o p l e t size a n d the d r o p l e t size d i s t r i b u t i o n affect the r h e o l o g y o f e m u l s i o n (discussed i n C h a p t e r 4). D r o p l e t size i n r e l a t i o n to p o r e throat size affects t h e flow o f fluids i n p o r o u s m e d i a , as d i s c u s s e d p r e v i o u s l y ( F i g u r e 3).


236


CAPILLARY RADIUS (cm)

0 0.0300 • 0.0380 0.0400 0.0497 0.0540


Γ=70%

Γ=60% * 0—0—0-

2000

4000

6000

8000

10000

SHEAR RATE, sec" Figure 7. Apparent viscosity vs. shear rate at various emulsion qualities (T). (Reproduced with permission from reference 26. Copyright 1970 Society of Petroleum Engineers.)

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Emulsions Fundamentals and Applications in the Petroleum Industry

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