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Non-Newtonian oil flow through porous media Fariss, Tariq Fariss al- 1984

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NON-NEWTONIAN OIL FLOW THROUGH POROUS MEDIA by TARIQ FARISS AL-jFARISS B . S c , The U n i v e r s i t y o f Baghdad, 1976 M.E., T u l a n e U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY •in THE FACULTY OF GRADUATE STUDIES (Department o f C h e m i c a l E n g i n e e r i n g ) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 @ Ta r i q F a r i s s A l - F a r i s s , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r equ i r ements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Co lumb i a , I agree t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copy i ng o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s unde r s tood t h a t copy ing o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l owed w i t hou t my w r i t t e n p e r m i s s i o n . Department o f C/te^T to. I £<iQ*l ttsrl*-*^, The U n i v e r s i t y o f B r i t i s h Co lumbia 1956 Main Mall Vancouve r , Canada V6T 1Y3 Date £</ J ¥ I IVKJf A B S T R A C T A m a t h e m a t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n was made o f t h e f l o w t h r o u g h p o r o u s m e d i a o f non-N e w t o n i a n f l u i d s w i t h y i e l d s t r e s s e s . The a n a l y s i s o f t h e p r o b l e m was made i n t h r e e p a r t s : 1) Two m o d e l s were d e v e l o p e d t o d e s c r i b e t h e f l o w t h r o u g h p o r o u s m e d i a o f a power-law f l u i d w i t h y i e l d s t r e s s , l a ) D a r c y ' s law was m o d i f i e d by c o u p l i n g t h e H e r s c h e l - B u l k l e y r h e o l o g i c a l m odel w i t h t h e c a p i l l a r y f l o w model and h y d r a u l i c g r a d i e n t d e s c r i p t i o n o f a p o r o u s medium t o g e n e r a t e a g e n e r a l i z e d f o r m o f t h e f l o w e q u a t i o n a s : y f f L kAn « " o where 1-n F e f f = I ( 3 + 1 / n ) n ( 8 e K ) 2 o r e q u a l l y as 1-n y e f f = | (3 + l / n ) n (8 C eK) 2 and a Q = f ^ — where C , 3 a r e c o n s t a n t s l b ) The f r i c t i o n f a c t o r - R e y n o l d s number f o r m o f t h e f l o w e q u a t i o n was a l t e r e d t o d e s c r i b e t h e f l o w e q u a t i o n a s : f * 248 c a l . R* e where and f * 3 R* e exp. pL v 2 1-e ° 2 12p x, 2HD e 3 V 1 1 + e 2 T C p o o n T D e 1 n+1 C = 6 ( ~ - n - ) I ^ — A (1-e) 2a) An o i l w h i c h e x h i b i t e d a r h e o l o g i c a l b e h a v i o u r w h i c h was w e l l d e s c r i b e d by t h e H e r s c h e l - B u l k l e y model was d e v e l o p e d by a d d i n g p a r a f f i n wax t o N e w t o n i a n o i l s . Two o i l s were u s e d , C l a r u s - B and C l a r u s - C o i l s s u p p l i e d by S h e l l Canada Company, w i t h 2.5%, 4% and 5% p a r a f f i n wax f o r t h e p r e l i m i n a r y t e s t s . C r u d e o i l f r o m t h e P eace R i v e r f i e l d i n A l b e r t a was a l s o f o u n d t o e x h i b i t t h e same b e h a v i o u r . 2b) The f l o w c h a r a c t e r i s t i c s o f t h e t e s t o i l s and t h e c r u d e o i l were measured u s i n g a r o t a t i o n a l v i s c o m e t e r (Haake R o t o v i s c o RV-12). - i v -3) The three types of waxy o i l s were t e s t e d i n two sand beds of 91 and 100 cm l e n g t h w i t h two s i z e s of p a r t i c l e s 0.077 and 0.128 cm r e s p e c -t i v e l y , thus g i v i n g Dc/D^ r a t i o s of 62.7 and 79.4 and p o r o s i t i e s of 0.36 and 0.44. The f l o w behaviours of the waxy o i l s through the porous medium were determined over a wide range of flow r a t e s (0.0057-17.4 m l / s e c ) , Reynolds numbers (3.38 x 1 0 ~ 8 - 0.33), p r e s s u r e drops (3.11 - 299.6 kPa) , 2 9 f r i c t i o n f a c t o r s (8.9 x 10 - 2.86 x 10 ), y i e l d s t r e s s e s (0.71 - 69.37 dyne/cm 2), and temperatures (2-20°C). The experimental r e s u l t s show agreement with the mathematical models i n the range o f flow r a t e s i n v e s t i g a t e d . An average e r r o r of 3.64% and 5.5% was o b t a i n e d w i t h C l a r u s C and B r e s p e c t i v e l y . The crude o i l gave an average e r r o r 3.9%. The l o g a r i t h m i c l e a s t square f i t of the f * - R* c o r r e l a t i o n gave a c o r r e l a t i o n c o e f f i c i e n t of 0.98 f o r the two a r t i f i c i a l waxy o i l s , w h i l e the crude o i l showed e x c e l l e n t f i t f o r the c o r r e l a t i o n i n a wide range of Reynolds numbers i n the laminar r e g i o n . There was some s c a t t e r i n the data which was ex-p l a i n e d by s e n s i t i v i t y o f the f i t t i n g parameter- (K) to smal l e r r o r s i n the measurements of the r h e o l o g i c a l parameters n, x and H. - v -TABLE OF CONTENTS Page CHAPTER 1 .; INTRODUCTION ' 1 1.1 O b j e c t i v e s o f t h i s t h e s i s 3 2 .. BACKGROUND 5 2.1 C l a s s i f i c a t i o n o f F l u i d B e h a v i o u r 5 2.1.1 N e w t o n i a n F l u i d s 5 2.1.2 Non-Newtonian F l u i d s 8 2.1.2.1 Time I n d e p e n d e n t F l u i d s . . . 8 2.1.2.2 Time Dependent F l u i d s 12 2.1.2.3 V i s c o e l a s t i c F l u i d s 14 2.2 R h e o l o g i c a l P r o p e r t i e s Measurements 14 2.2.1 Y i e l d V a l u e Measurement 14 2.2.2 S h e a r S t r e s s - S h e a r R a t e Measurements 17 2.3 P h y s i c a l P r o p e r t i e s o f C r u d e O i l s 21 2.4 P r o p e r t i e s o f P e t r o l e u m Waxes 22 2.5 Mechanism o f Wax C r y s t a l l i z a t i o n 23 2.6 O i l R e s e r v o i r C o n d i t i o n s 25 2.7 E x p e r i m e n t a l and F i e l d O b s e r v a t i o n 27 2.8 The P o r o u s Medium 35 2.8.1 U n d e r g r o u n d F o r m a t i o n 35 2.8.2 P e r m e a b i l i t y 36 2.8.3 P o r o s i t y 37 2.8.4 T o r t u o s i t y 38 2.9 F l o w T h r o u g h P o r o u s M e d i a 38 2.9.1 D a r c y ' s Law 39 2.9.2 L i m i t a t i o n o f D a r c y ' s Law 41 2.10 W a l l and End E f f e c t s o f Sand Column 42 2.11 T h e o r e t i c a l I n t e r p r e t a t i o n s 43 2.11.1 N e w t o n i a n F l u i d A n a l y s i s 43 2.11.2 Non-Newtonian F l u i d A n a l y s i s 46 3 • MATHEMATICAL MODEL ANALYSIS 53 3.1 The C o n c e p t o f H y d r a u l i c R a d i u s (R^) 54 3.2 The S u p e r f i c i a l V e l o c i t y ( V 0 ) 56 3.3 G e n e r a l i z e d D a r c y ' s Law 57 3.4 M o d i f i e d F r i c t i o n F a c t o r - Reynolds'" Number C o r r e l a t i o n 6 3 - v i -Page CHAPTER 4 EXPERIMENTAL OBJECTIVES AND FLUID SELECTION .' 66 4.1 F l u i d S e l e c t i o n 70 5 EXPERIMENTAL APPARATUS 72 5.1 R h e o l o g i c a l E x p e r i m e n t s 72 5.1.1 The V i s c o m e t e r (HAAKE R o t o v i s c o RV-12) '. 7 2 5.1.2 V i s c o s i t y , S h e a r S t r e s s and S h e a r Rate Measurement '74 5.2 F l o w E x p e r i m e n t s 8 0 5.2.1 M a t e r i a l s 81 5.2.2 E q u i p m e n t ' 82 5.2.3 E x p e r i m e n t a l T e c h n i q u e and P r o -c e d u r e 8 8 5.3 E x p e r i m e n t a l P r o b l e m s and D e v e l o p m e n t s 90 6 . RESULTS AND DISCUSSION 92 6.1 G e n e r a l C o n s i d e r a t i o n s 92 6.2 R h e o l o g i c a l E x p e r i m e n t s '95 6.2.1 R e s u l t s 95 6.2.2 A n a l y s i s 100 6.2.3 D i s c u s s i o n 100 6.3 F l o w System E x p e r i m e n t s 103 6.3.1 M o d i f i e d D a r c y ' s Law 105 6.3.1.1 R e s u l t s 105 6.3.1.2 A n a l y s i s i 109 6.3.1.3 D i s c u s s i o n 122 6.3.1.4 The S i g n i f i c a n c e o f M e a s u r e m e n t - E r r o r s 1 2 5 6.3.2 F r i c t i o n F a c t o r - R e y n o l d s Number C o r r e l a t i o n 129 6.3.2.1 R e s u l t s 129 6.3.2.2 A n a l y s i s • 133 6.3.2.3 D i s c u s s i o n 135 6.3.2.4 The S i g n i f i c a n c e o f Measurement E r r o r s 139 6.4 R e p r o d u c i b i l i t y 142 C H A P T E R 7 C O N C L U S I O N S 3 R E C O M M E N D A T I O N N O M E N C L A T U R E B I B L I O G R A P H Y A P P E N D I C E S - v i i i -APPENDICES Page APPENDIX A D e r i v a t i o n o f E q u a t i o n 3-15. The F l o w o f Non-Newtonian F l u i d w i t h a Y i e l d V a l u e i n C a p i l l a r y Tube - 165 B Summary o f V i s c o m e t r i c D a t a and C a l c u l a t i o n s . ' 172 C Summary o f t h e F l o w System D a t a and C a l c u l a t i o n s i n Terms o f P r e s s u r e Drop - F l o w R a t e R e l a t i o n s h i p ' 182 D Summary o f t h e F l o w System D a t a and C a l c u l a t i o n s i n Terms o f F r i c t i o n F a c t o r - R e y n o l d s " Number 190 E Sample C a l c u l a t i o n s 211 F M a t e r i a l P r o p e r t i e s and S p e c i f i c a t i o n s .216 - i x -LIST OF TABLES Page TABLE 2-1 M o d e l s R e l a t i n g x v x t o du/dy f o r F l u i d s w i t h a Y i e l d S t r e s s 10 2-2 M o d e l s R e l a t i n g Tyx t o du/dy f o r F l u i d s w i t h o u t a Y i e l d S t r e s s 10 2-3 P r o p e r t i e s o f N i n e Waxy C r u d e O i l s 21 2-4 M e l t i n g P o i n t s o f Some P a r a f f i n i c P e t r o l e u m Waxes '22 2-5 F l u i d P r e s s u r e s and T e m p e r a t u r e s i n some U.S. Deep W e l l s 26 2-6 C h a r a c t e r i s t i c P r o p e r t i e s o f Bombay H i g h C r u d e O i l o f I n d i a 29 2-7 Q u a l i t y o f N a h o r k a t i y a (Assam) Crud e O i l i n I n d i a 30 2-8 R h e o l o g i c a l P r o p e r t i e s o f Waxy C r u d e O i l o f C a b i n d a 32 2-9 API G r a v i t i e s and Po u r P o i n t s o f S e l e c t i v e Waxy C r u d e O i l s 3 3 2-10 T y p i c a l V a l u e s f o r P e r m e a b i l i t y and P o r o s i t y f o r V a r i o u s P o r o u s S o l i d s 38 2-11 The V a l u e s o f t h e C o n s t a n t ' C" Used i n V a r i o u s P a p e r s P u b l i s h e d S i n c e 1965 47 6-1 V a r i a b l e s and Ranges I n v e s t i g a t e d 93 6-2 A g i n g Time f o r 2^ .5% Wax i n C l a r u s - B f o r t h e S m a l l Column a t 14°C 94 6-3 The V i s c o m e t e r ( R o t o v i s c o RV-12) C o n s t a n t s . f 1^ f cinci •••••• ••••••• ••••••• 98 -x-Page TABLE 6-4 R h e o l o g i c a l P a r a m e t e r s o f t h e Waxy O i l s M e a s u r e d by R o t o v i s c o RV-12 99 6-5 R e s u l t s f o r t h e F l o w o f P u r e C l a r u s O i l s T h r o u g h t h e Sand Beds / 105 6-6 R e s u l t s o f t h e F l o w S y s t e m f o r C l a r u s - B S o l u t i o n s i n t h e l a r g e Column , 106 6-7 R e s u l t s o f t h e F l o w S y s t e m f o r C l a r u s - B O i l o f t h e S m a l l Column by A p p l y i n g E q u a t i o n 3-18 107 6-8 R e s u l t s o f t h e F l o w S y s t e m f o r C l a r u s - C S o l u t i o n s and t h e C r u d e O i l - 108 6-9 D i f f e r e n t V a l u e s f o r t h e R h e o l o g i c a l P a r a m e t e r n f o r C l a r u s - B O i l S o l u t i o n s 120 6-10 S t a t i s t i c a l A n a l y s i s o f t h e D a t a shown i n A p p e n d i x C f o r t h e F l o w System U s i n g M o d i f i e d D a r c y ' s E q u a t i o n 3-18 ' / 121 6-11 S t a t i s t i c a l C o m p a r i s o n o f F r i c t i o n F a c t o r P r e d i c t i o n s f o r a l l 235 D a t a P o i n t s o f C l a r u s - B and C l a r u s - C S o l u t i o n s . . . . 136 6-12 T y p i c a l V a l u e s o f t h e Y i e l d V a l u e Measurements f o r 4% Wax i n C l a r u s - B O i l 142 6-13 E x p e r i m e n t a l R e s u l t s f o r t h e F l o w o f 5% -Wax'in. Cl'arus-B O i l . T h r o u g h t h e L a r g e Column a t 1 6 ° C , R e p e a t e d T h r e e Times. ; . ....... ......... ......... 148 B - l R e s u l t s f r o m t h e R o t a t i o n a l V i s c o m e t e r RV-12 U s i n g S e n s o r System NV f o r 2.5% Wax i n C l a r u s - B O i l a t D i f f e r e n t T e m p e r a t u r e s . . . . 172 B-2 R e s u l t s f r o m t h e R o t a t i o n a l V i s c o m e t e r RV-12 U s i n g S e n s o r System NV f o r 4% Wax i n C l a r u s - B O i l a t D i f f e r e n t T e m p e r a t u r e s 174 B-3 R e s u l t s f r o m t h e R o t a t i o n a l V i s c o m e t e r RV-12 U s i n g S e n s o r S y s t e m NV f o r 5% Wax i n C l a r u s - B O i l a t D i f f e r e n t T e m p e r a t u r e s 176 - x i -TABLES Page B-4 R e s u l t s f r o m t h e R o t a t i o n a l V i s c o m e t e r RV-12 U s i n g S e n s o r Systems NV f o r 2.5% Wax i n C l a r u s - C a t D i f f e r e n t T e m p e r a t u r e s 178 B-5 R e s u l t s f r o m t h e R o t a t i o n a l V i s c o m e t e r RV-12 U s i n g S e n s o r System NV f o r Pe a c e R i v e r C r u d e O i l o f A l b e r t a a t D i f f e r e n t T e m p e r a t u r e s 18 0 B-6 S h e a r R a t e C o r r e c t i o n s due t o n o n - N e w t o n i a n B e h a v i o u r '1'81 C-1 R e s u l t s f o r t h e S m a l l Column U s i n g C l a r u s - B Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f P r e s s u r e D r o p s -F l o w R a t e s 18-2 C-2 R e s u l t s f o r t h e S m a l l Column U s i n g C l a r u s - C Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f P r e s s u r e D r o p s -F l o w R a t e s 185 C-3 R e s u l t s f o r t h e L a r g e Column U s i n g C l a r u s - B Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f P r e s s u r e D r o p s -F l o w R a t e s -18 6 C-4 R e s u l t s f o r t h e L a r g e Column U s i n g C l a r u s - C Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f P r e s s u r e D r o p s - . F l o w R a t e s : 18 8 C-5 R e s u l t s f o r t h e L a r g e Column U s i n g Waxy C r u d e O i l f r o m t h e Pe a c e R i v e r F i e l d o f A l b e r t a a t D i f f e r e n t T e m p e r a t u r e s i n t e r m s o f P r e s s u r e D r o p s - F l o w R a t e s 189 D - l R e s u l t s f o r t h e S m a l l Column U s i n g C l a r u s - B Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f F r i c t i o n F a c t o r s -Reynolds*' Numbers ,190 D-2 R e s u l t s f o r t h e S m a l l Column U s i n g C l a r u s - C Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f F r i c t i o n F a c t o r s -R e y n o l d s Numbers 194 D-3 R e s u l t s f o r t h e L a r g e Column U s i n g C l a r u s - B Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n Terms o f F r i c t i o n F a c t o r s -R e y n o l d s Numbers ,196 - x i i -Page TABLE D-4 R e s u l t s f o r t h e L a r g e Column U s i n g C l a r u s - C Waxy O i l a t D i f f e r e n t T e m p e r a t u r e s and Wax C o n c e n t r a t i o n s i n t e r m s o f F r i c t i o n F a c t o r s -R e y n o l d s Numbers 200 D-5 R e s u l t s f o r t h e L a r g e Column U s i n g Waxy C r u d e O i l f r o m t h e P e a c e R i v e r F i e l d o f A l b e r t a a t D i f f e r e n t T e m p e r a t u r e s i n t e r m s o f F r i c t i o n F a c t o r s - R e y n o l d s Number 202 D-6 C a l c u l a t e d and P r e d i c t e d F r i c t i o n F a c t o r s f o r b o t h Columns f o r a l l D a t a i n t e r m s o f l o g ( f * ) - l o g ( R * ) 204 D-7 R e s u l t s o f L e a s t S q u a r e F i t f o r A l l C l a r u s - B and C l a r u s - C D a t a i n t e r m s o f l o g ( f * ) as F u n c t i o n o f l o g (R*) 210 F - l L u b r i c a t i n g C l a r u s - B O i l S p e c i f i c a t i o n s 216 F-2 L u b r i c a t i n g C l a r u s - C O i l S p e c i f i c a t i o n s 217 F-3 P a r a f f i n Wax PARVAN 55 C h a r a c t e r i s t i c s 218 F-4 P h y s i c a l P r o p e r t i e s o f P e a c e R i v e r C r u d e O i l o f A l b e r t a 219.. F-5 S p e c i f i c a t i o n s o f t h e P r e s s u r e T r a n s d u c e r DPT 3 6 2 J - 5 0 f r o m D y n i s c o 220. F-6 D e n s i t i e s o f t h e Waxy O i l s a t D i f f e r e n t T e m p e r a t u r e s .221 - x i i i -LIST OF FIGURES Page FIGURE 2-1 S t e a d y s t a t e ' s h e a r i n g m o t i o n between two p a r a l l e l p l a t e s 6 2-2 R h e o l o g i c a l f l o w c u r v e s on a r i t h m e t i c c o o r d i n a t e s f o r v a r i o u s t y p e s o f t i m e -• i n d e p e n d e n t " f l u i d 6 2-3 R h e o l o g i c a l f l o w c u r v e s f o r t h i x o t r o p i c and r h e o p e c t i c f l u i d s 13 2-4 V a r i a t i o n o f a p p a r e n t y i e l d v a l u e w i t h l e n g t h / d i a m e t e r r a t i o . 17 2-5 The m e a s u r i n g p r i n c i p l e o f t h e r o t a t i o n a l v i s c o m e t e r s 19 2-6 R h e o l o g i c a l b e h a v i o u r o f two t y p e s o f N o r t h A f r i c a n waxy c r u d e o i l s 2 8 2-7 R h e o l o g i c a l b e h a v i o u r o f e x t r a - h e a v y c r u d e o i l s f r o m V e n e z u e l a 28 2-8 S h e a r s t r e s s / s h e a r r a t e f o r C a b i n d a waxy c r u d e o i l s 31 2- 9 V a r i a t i o n o f p o u r p o i n t w i t h wax c o n t e n t o f waxy f u e l o i l s 34 2-10 T w e l v e s c h e m a t i c f l o w c u r v e s f o r n o n -D a r c i a n f l o w 41 3- 1 S c h e m a t i c d i a g r a m o f t h e v e l o c i t y p r o f i l e i n a p i p e o f n o n - N e w t o n i a n f l u i d w i t h a y i e l d s t r e s s 58 4- 1 S c h e m a t i c d r a w i n g f o r y i e l d v a l u e m e a s u r e -ments • 68 5- 1 S c h e m a t i c d i a g r a m o f t h e R o t o v i s c o RV-12 s e t - u p s 73 5-2 D i m e n s i o n s and r a n g e o f measurement o f s e n s o r s y s t e m NV 75 - x i v -Page FIGURE. 5-3 S c h e m a t i c d i a g r a m o f t h e f l o w t e s t e q u i p m e n t 83 5-4 D e t a i l s o f t h e s m a l l s a n d c o l u m n d e s i g n 85 5- 5 D e t a i l s o f t h e l a r g e s a n d c o l u m n d e s i g n 86 6- 1 S h e a r s t r e s s - s h e a r r a t e c u r v e s f o r 2.5% wax i n C l a r u s - B o i l , m e a s u r e d on a r o t a t i o n a l v i s c o m e t e r , where t h e s o l i d l i n e r e p r e s e n t t h e H e r s c h e l - B u l k l e y m odel f i t 96 6-2 S h e a r s t r e s s - s h e a r r a t e c u r v e s f o r C l a r u s - B s o l u t i o n s a t T = 14°C me a s u r e d on a r o t a t i o n a l v i s c o m e t e r , where t h e s o l i d l i n e r e p r e s e n t t h e H e r s c h e l - B u l k l e y model f i t 97 6-3 Y i e l d v a l u e s o f t h r e e t y p e s o f waxy o i l s a s a f u n c t i o n o f t e m p e r a t u r e , m e a s u r e d on a r o t a t i o n a l v i s c o m e t e r a t l o w e s t s p e e d ( n 1 = 0.01 m i n ~ l ) , where t h e s o l i d l i n e i s drawn by eye 102 6-4 P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 2.5% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18 110 6-5 P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 4% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18 I l l 6-6 P r e s s u r e d r o p - f l o w c u r v e s f o r 5% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18 112 - X V -FIGURE Page 6-7 Pressure drop - flow rate curves for 2.5% wax i n Clarus-C o i l flowing through a 91 cm long sand column with Dp = 0.077 cm, where the s o l i d l i n e s represent the model f i t of equation 3-18 '113 6-8 Pressure drop - flow rate curves for 2.5% wax i n Clarus-B o i l flowing through a 100 cm long sand column with Dp = 0.128 cm, where the s o l i d l i n e s represent the model f i t of equation 3-18 114 6-9 Pressure drop - flow rate curves for 5% wax in Clarus-B flowing through a 100 cm long sand column with Dp = 0.128 cm, where the s o l i d l i n e s represent the model f i t of equation 3-18 115 6-10 Pressure drop - flow rate curves for 2.5% wax i n Clarus-C o i l flowing through a 100 cm long sand column with Dp = 0.128 cm, where the s o l i d l i n e s represent the model f i t of equation 3-18 .... 116 6-11 Pressure drop - flow rate curves for Peace River Crude o i l flowing i n a 100 cm long sand bed with Dp = 0.128 cm, where the s o l i d l i n e s represent the model f i t of equation 3-18 117 6-12 F r i c t i o n factor - Reynolds number cor-, r e l a t i o n data for solutions of Clarus-B and Clarus-C flowing through a 91 cm long sand column with Dp = 0.077 cm at d i f f e r e n t temperatures and wax content 130 6-13 F r i c t i o n factor - Reynolds number co r r e l a t i o n data for solutions of Clarus-B and Clarus-C flowing through a 100 cm long sand column with Dp = 0.12 8 cm at d i f f e r e n t temperatures and wax content 131 6-14 F r i c t i o n factor - Reynolds number cor r e l a t i o n data for Peace River crude o i l flowing through a 100 cm long sand column with Dp = 0.128 cm at d i f f e r e n t temperatures 132 -xv i -Page FIGURE 6-15 Shear stress - shear rate curves for Clarus-B solutions at d i f f e r e n t temperatures, measured on two rota-t i o n a l viscometers, Rotovisco RV-12 and RV-3 to show the r e p r o d u c i b i l i t y of the rheological data, where the s o l i d l i n e s represent the Herschel-Bulkley model f i t 14 3 6-16 Pressure drop - flow rate curves showing the r e p r o d u c i b i l i t y of the flow data, where the s o l i d l i n e s represent the model f i t of equation 3-18 145 6-17 Pressure drop - flow rate curves for 5% wax i n Clarus-B o i l at T = 16°C flowing through a 100 cm long sand column with Dp = 0.128 cm where the s o l i d l i n e represents the model f i t of equation 3-18, showing the r e p r o d u c i b i l i t y of the data 146 6-18 F r i c t i o n factor - Reynolds number co r r e l a t i o n data for 5% wax i n Clarus-B o i l at 16°C flowing through a 100 cm long sand column with Dp = 0.12 8 cm, showing the r e p r o d u c i b i l i t y of the data .147 - x v i i -ACKNOWLEDGEMENT The author wishes to express h i s a p p r e c i a t i o n to P r o f e s s o r K.L. Pinder f o r h i s guidance and suggestions which played a very c o n s i d e r a b l e r o l e i n the completion of the presen t work. S p e c i a l g r a t i t u d e and a p p r e c i a t i o n are owed P r o f e s s o r s N. E p s t e i n , A.P. Watkinson, A. Meisen, and J . de V r i e s , f o r t h e i r many c o n s t r u c t i v e comments and h e l p f u l d i s c u s s i o n s as committee members. Thanks to Carmen f o r her e x c e l l e n t t y p i n g and to my c o l l e a g u e N. Gune f o r u s e f u l d i s c u s s i o n s . Thanks a l s o to Ara b i a n American O i l Company (ARAMCO) and Chevron O i l Company f o r s u p p l y i n g the crude o i l samples. Thanks a l s o t o NSERC f o r the Research Grant which pr o v i d e d the funds f o r t h e equipment, and to the U n i v e r -s i t y o f King Saud (formerly Riyadh U n i v e r s i t y ) i n Saudi A r a b i a which pr o v i d e d my s c h o l a r s h i p . CHAPTER ONE INTRODUCTION The gradual decline i n resources of l i g h t and medium crude o i l s i n the world has caused the petroleum industry, in order to guarantee an adequate l e v e l of energy supply from hydrocarbons, to devote more and more resources and e f f o r t s to t r y to improve the present production techniques and to tes t new ones which may permit an increase i n the recovery of medium and l i g h t crude o i l s , or the economical production of most of the heavy and waxy o i l s reserves now available. Waxy crude o i l s are becoming more important since they often have a low sulfur content, making them desirable from an environmental viewpoint. Also, since known resources are available the need for new supplies has en-couraged production of these d i f f i c u l t to handle crudes. A number of investigators have reported from laboratory data the presence of a threshold gradient for flow for certa i n classes of f l u i d s including heavy and waxy o i l s . Also, f i e l d data have indicated that o i l reservoirs of p a r a f f i n - r i c h o i l s and of heavy o i l s , exhibit a y i e l d stress value such that a f i n i t e pressure gradient, c a l l e d the threshold gradient must be provided before flow i s i n i t i a t e d , (5,19,22,23,63,65, 73,81). -2-P r e v i o u s s t u d i e s o f t h e f l o w o f f l u i d s t h r o u g h p o r o u s m e d i a have been c o n c e r n e d w i t h two t y p e s o f f l u i d b e h a v i o u r . F i r s t , t h o s e f l u i d s w h i c h may be c o n s i d e r e d N e w t o n i a n ; i . e . , f l u i d s f o r w h i c h t h e r e l a t i o n between s h e a r s t r e s s and s h e a r r a t e i s a s i m p l e p r o p o r t i o n a l i t y . T h i s i n c l u d e s a l l g a s e s and most homogeneous, n o n - p o l y m e r i c l i q u i d s i n c l u d i n g l i g h t c r u d e o i l s and o i l s w i t h s m a l l wax c o n t e n t . S e c o n d , t h o s e f l u i d s w h i c h may be c o n s i d e r e d n o n - N e w t o n i a n w i t h o u t a y i e l d s t r e s s , i . e . , f l u i d s f o r w h i c h t h e r e l a t i o n between s h e a r s t r e s s and r a t e o f s h e a r i s n o t a s i m p l e p r o p o r t i o n -a l i t y . S u s p e n s i o n s and p o l y m e r s o l u t i o n s a r e t y p i c a l e x a m p l e s o f t h e s e c o n d c l a s s . T hose f l u i d s show a d e c r e a s e o f a p p a r e n t v i s c o s i t y w i t h i n c r e a s i n g s h e a r r a t e s , i . e . , s h e a r t h i n n i n g (8, 9, 15, 16, 43, 75, 7 6 ) . None o f t h e s e r e s e a r c h e r s have i n v e s t i g a t e d e x p e r i -m e n t a l l y t h e f l o w o f s h e a r - t h i n n i n g f l u i d s w i t h a y i e l d s t r e s s , ( s u c h a s waxy and h e a v y o i l s ) i n p o r o u s m e d i a . P r e v i o u s s t u d i e s on t h e s e c r u d e s were c o n c e r n e d o n l y w i t h t h e p r o b l e m s r e s u l t i n g f r o m h a n d l i n g waxy o i l s i n p i p e l i n e s (4, 5, 6, 7, 11, 13, 19, 25, 35, 36, 45, 56, 71, 74, 80, 81, 88, 89, 91) . The t h e o r i e s p r e s e n t l y u s e d t o p r e d i c t o i l w e l l c a p a -c i t i e s and t o d e s i g n f l o w p r o g r a m s a r e b a s e d on t h e assump-t i o n t h a t o i l b e h a v e s a s a N e w t o n i a n f l u i d and o b e y s D a r c y ' s law, E q u a t i o n 2-9 (page 4 0 ) . I f h e a v y o r waxy c r u d e s e x h i b i t a y i e l d v a l u e , t h e n f l o w w i l l n o t o c c u r a c r o s s t h e t o t a l a r e a a v a i l a b l e f o r f l o w , s i n c e t h e y i e l d v a l u e w i l l n o t be -3-e x c e e d e d a t some d i s t a n c e f r o m a s o l i d s u r f a c e , and s m a l l p o r e s may have no f l o w . T h u s , t h e m e a s u r e d f l o w s f o r s u c h f l u i d s a r e s m a l l e r t h a n t h e y w o u l d be i f no y i e l d v a l u e e x i s t e d . To t h e a u t h o r ' s k n o w l e d g e , t h e r e a r e no p u b l i s h e d r e s u l t s o f an e x p e r i m e n t o r p r o d u c t i o n t e s t o f t h e f l o w t h r o u g h p o r o u s m e d i a o f f l u i d s w h i c h e x h i b i t y i e l d s t r e s s p l u s n o n -N e w t o n i a n , s h e a r t h i n n i n g , o r s h e a r t h i c k e n i n g f l o w b e h a v i o u r . 1.1 O b j e c t i v e s o f T h i s T h e s i s The g e n e r a l o b j e c t i v e o f t h i s s t u d y i s t o e x t e n d t h e k nowledge o f f l u i d - p a r t i c l e s y s t e m s by e x a m i n i n g t h e f l o w t h r o u g h p o r o u s m e d i a o f n o n - N e w t o n i a n f l u i d s w i t h y i e l d s t r e s s e s . A t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e p r o b l e m w i l l be f o l l o w e d by an e x p e r i m e n t a l t e s t o f t h e t h e o r y . i ) T h e o r e t i c a l : D a r c y ' s law w i l l be m o d i f i e d t o d e s -c r i b e t h e f l o w t h r o u g h p o r o u s m e d i a o f n o n - N e w t o n i a n f l u i d s w i t h y i e l d s t r e s s e s . T h i s c a n be done by c h a r a c t e r i z a t i o n o f t h e n o n - N e w t o n i a n f l u i d s w i t h an a p p r o p r i a t e r h e o l o g i c a l e q u a t i o n o f s t a t e and c o m b i n i n g i t w i t h e q u a t i o n s o f m o t i o n t o o b t a i n an a p p r o p r i a t e m odel f o r t h e f l o w t h r o u g h p o r o u s m e d i a . The m o d i f i e d D a r c y ' s e q u a t i o n w i l l a l s o be e x p r e s s e d i n t e r m s o f a f r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n . i i ) E x p e r i m e n t a l : T e s t t h e p r o p o s e d m odel by f l o w i n g an a r t i f i c i a l f l u i d w i t h t h e d e s i r e d b e h a v i o u r i n a p o r o u s medium and m e a s u r i n g t h e p r e s s u r e d r o p and f l o w r a t e i n a s p e c i a l l y d e s i g n e d f l o w s y s t e m . T h i s w i l l be done u n d e r d i f f e r e n t e x p e r i m e n t a l c o n d i t i o n s . . V a r i a b l e s s u c h as t e m p e r a -t u r e , f l o w r a t e , p r e s s u r e d r o p , p a r t i c l e s i z e , c o l u m n s i z e , -4-o i l t y p e s , wax c o n t e n t and y i e l d v a l u e a r e t o be s t u d i e d . Once;,, t h i s has b e e n a c c o m p l i s h e d , a waxy c r u d e o i l w i l l be t e s t e d . The p r e s e n t i n v e s t i g a t i o n w i l l be r e s t r i c t e d t o : 1) V i s c o u s f l o w 2) S i n g l e p h a s e f l o w 3) I s o t h e r m a l f l o w 4) U n c o n s o l i d a t e d p o r o u s m e d i a . - 5 -CHAPTER TWO BACKGROUND I n t h i s s e c t i o n e m p h a s i s w i l l be p l a c e d on t h e r h e o l o g i c a l b e h a v i o u r o f h e a v y and waxy c r u d e o i l s a nd t h e f l o w o f n o n - N e w t o n i a n l i q u i d s t h r o u g h p o r o u s m e d i a . 2 . 1 C l a s s i f i c a t i o n o f F l u i d B e h a v i o u r 2 . 1 . 1 N e w t o n i a n f l u i d s C o n s i d e r a f l u i d l o c a t e d between two p a r a l l e l p l a t e s w h i c h a r e s e p a r a t e d by a n o r m a l d i s t a n c e Ay a s shown i n F i g u r e 2 - 1 . A f o r c e F i n t h e p l a n e of t h e u p p e r p l a t e c a u s e s i t t o m a i n t a i n a c o n s t a n t v e l o c i t y &u r e l a t i v e t o t h e s t a t i o n a r y l o w e r p l a t e . A t s t e a d y s t a t e , F i s b a l a n c e d by a s h e a r i n g f o r c e b e n e a t h t h e t o p p l a t e r e s u l t i n g f r o m t h e v i s -c o s i t y o f t h e f l u i d , w h i c h i s i n 'laminar; f l o w . The s h e a r s t r e s s x . i s d e f i n e d a s yx x = T y x , = F/A 2 - 1 where x y x i s a s h e a r s t r e s s a c t i n g i n t h e x - d i r e c t i o n on a s u r f a c e n o r m a l t o t h e y - d i r e c t i o n . - 6 -F A U £ _k_ A = surface area 1 S  — -— !-»• du F l u i d ..-Ay -h- * V stationary F i g u r e 2-1 ..'Steady s t a t e s h e a r i n g motion between two p a r a l l e l p l a t e s 6 CJ CJ w w u -P. w u Q) .fi Newtonian Pseudoplastic D i l a t a n t Bingham p l a s t i c Shear thinning with y i e l d • stress '' Shear thickening with y i e l d stress Shear rate F i g u r e 2-2 R h e o l o g i c a l flow, curves on a r i t h m e t i c c o o r d i n a t e s f o r v a r i o u s types of time-independent f l u i d -7-A m easure o f t h e d e f o r m a t i o n o f t h e f l u i d c a u s e d by T „ ; i s p r o v i d e d by t h e a n g l e y, and f o r s m a l l y Y 2-3 by a n a l o g y w i t h d e f o r m a t i o n s i n e l a s t i c s o l i d , y may be d e s c r i b e d as a s t r a i n . The r a t e o f s t r a i n o r r a t e o f d e f o r m a -t i o n i s t h e r e f o r e T h i s i s a l s o c a l l e d t h e s h e a r r a t e o r v e l o c i t y g r a d i e n t . N e w t o n i a n f l u i d s a r e d e f i n e d a s t h o s e e x h i b i t i n g a d i r e c t p r o -p o r t i o n a l i t y between s h e a r s t r e s s and s h e a r r a t e i n l a m i n a r f l o w . Thus where y i s t h e v i s c o s i t y - a p r o p o r t i o n a l i t y c o n s t a n t , w h i c h i s i n d e p e n d e n t o f s h e a r r a t e , b e i n g a f f e c t e d o n l y by^ t e m p e r a -t u r e and p r e s s u r e f o r a g i v e n f l u i d s y s t e m . A p l o t o f s h e a r s t r e s s (x) a g a i n s t s h e a r r a t e (y) i n l a m i n a r f l o w i s known i.as a " R h e o l o g i c a l f l o w c u r v e " . The s l o p e of t h i s c u r v e i s conte-s t a n t f o r a N e w t o n i a n f l u i d and i s g i v e n by u, a s shown i n F i g u r e 2 - 2 ( A ) . N e w t o n i a n b e h a v i o u r i s f o u n d i n a l l g a s e s and i n l i q u i d s and s o l u t i o n s o f low m o l e c u l a r w e i g h t . dy _ du d t " dy 2-3 T — T ~ yx 2-4 -8-2.1.2 Non-Newtonian f l u i d s A l l those f l u i d s f o r which the flow curve (x versus f ) i s not l i n e a r , nor through the o r i g i n a t a given temperature and p r e s s u r e are s a i d to be non-Newtonian. These m a t e r i a l s are commonly d i v i d e d i n t o t h r e e broad groups, although i n r e a l i t y these c l a s s i f i c a t i o n s are o f t e n by no means d i s t i n c t or s h a r p l y d e f i n e d (83). 2.1.2.1 Time-independent f l u i d s The shear r a t e of these f l u i d s i s s o l e l y dependent upon the instantaneous shear s t r e s s a t a p o i n t . •They are some-times r e f e r r e d t o as "non-Newtonian v i s c o u s f l u i d s " or a l t e r -n a t i v e l y as "pu r e l y v i s c o u s f l u i d s " . These may be c l a s s i f i e d as: a) F l u i d s w i t h y i e l d s t r e s s The p h y s i c a l behaviour of f l u i d s w i t h a y i e l d s t r e s s i s u s u a l l y e x p l a i n e d i n terms of an i n t e r n a l s t r u c t u r e i n three dimensions which i s capable of p r e v e n t i n g movement f o r v a l u e s o f shear s t r e s s l e s s than the y i e l d v a l u e , x . For x g r e a t e r than X q the i n t e r n a l s t r u c t u r e c o l l a p s e s cpmpletely, a l l o w i n g s h e a r i n g movement to occur. Some o f the e m p i r i c a l models which have been proposed f o r r e l a t i n g shear s t r e s s t o shear r a t e i n these substances are;' (1) the Bingham p l a s t i c model ,du, x = x + u (-=:—) o M p vdy' 2-5 -9-T h i s m o d e l , w h i c h c o n t a i n s two c o n s t a n t s " y i e l d s t r e s s " T q and " p l a s t i c v i s c o s i t y " u , d e s c r i b e s a l i n e a r f l o w c u r v e , F i g u r e 2-2(D) and has r e c e i v e d t h e most a n a l y t i c a l a t t e n t i o n . (2) H e r s c h e l B u l k l e y M o d e l , (37) . T T ,du, n „ , T = T + H (-5—) 2-6 o dy T h i s m odel has t h r e e c o n s t a n t s : y i e l d s t r e s s , T , f l o w b e h a v i o u r i n d e x / n, and c o n s i s t e n c y i n d e x / H. A l o g a r i t h m i c p l o t o f T - T o v e r s u s du/dy f o r t h e s e m a t e r i a l s i s o f t e n f o u n d t o be l i n e a r o v e r a w i d e r a n g e o f s h e a r r a t e s . The v a l u e s o f H and n c a n be d e t e r m i n e d f r o m t h e i n t e r c e p t and s l o p e o f t h a t p l o t . T h i s model i s more g e n e r a l t h a n t h e Bingham m o d e l , w h i c h i t e q u a l s f o r n = l , and as w e l l s i n c e i t d e s c r i b e s b o t h t h e y i e l d s t r e s s a nd power law b e h a v i o u r was s e l e c t e d f o r t h i s s t u d y . O t h e r m o d e l s a r e shown i n T a b l e 2-1. Exa m p l e s o f f l u i d s w i t h a y i e l d s t r e s s may " Include: heavy and waxy o i l s ' , c e r t a i n p l a s t i c m eits, o i l w e l l d r i l l i n g muds, most c o n c e n t r a t e d t i n o r g a n i c s l u r r i e s " peat s l u r r i e s , margarine s h o r t e n i n g s , greases, g r a i n water s u s p e n s i o n s , c h o c o l a t e m i x t u r e s , t o o t h p a s t e , soap--and detergent s l u r r i e s a n d p a p e r p u l p - : ' b) F l u i d w i t h o u t a y i e l d s t r e s s 1) Pseudo p l a s t i c f l u i d s : The m a j o r i t y o f n o n -N e w t o n i a n m a t e r i a l s a r e t o be f o u n d i n t h i s c a t e g o r y . Some o f t h e e m p i r i c a l m o d e l s w h i c h have been p r o p o s e d f o r r e l a t i n g - 1 0 -Table 2-1 Models relating TVX to dujdy for fluids with a yield stress (83) Model Form (for TVX > T „ ) (dujdy = 0 for T V I < T „ ) Empirical Constants Bingham _ _ ») Idu\ plastic T " T " T v lb force/ft 2 >) lb mass/ft sec I. Herschcl-Bulkley II. III. Crowley-Kitzes Tvx Tv .g\dy). ' VX ' V 1 + C ( T V I - T , ) " 1.2 + 5(CT; X ° - 8 + 1)» 1.2 - 2 5 ( C T ^ - ' + 1 ) ' T v lb force/ft2 JJ' (lb force)"1 - 1 lb mass f t ' - J m sec"1 in dimensionless T, lb force/ft2 r)0 lb mass/ft sec C (lb force/ft2)-" n dimensionless c (lb force/ft2)0'2 Table 2 - 2 Models re la t ing TVX to <-/;//<•/)• for f lu ids w i t h o u t a y i e ld stress ( 8 3 ) Model Form Empirical Constants 1. Power law or K 1 du Ostwald-dcWaelc T " ' = 7c [ ^ K lb mass sec" - 2 ft- 1 // dimensionless 3. De Haven 5. I'owcll-F.yring d. Rciner-PhilippoiT 7. Sisko du A + Hr"-X \dy I + C: •t. I'randll-Eyring . -r„ = /f s i n l r ' I (du C — + - s i n n - ' \dy) B I (du A [di-ll x + /!« — I'm 1 + (T„IAY Ah- sec"1 lb force"1 n f t " sec ' lb force a dimensionless / / 0 lb mass/ft sec . C (lb force/ft2)-" n dimensionless A lb force/ft2 D sec-1 A sec - 1 B ft=/lb force C lb force sec/ft2 / i n , / ' a , lb mass/ft sec /4 lb force/ft2 A lb force sec/ft2 /? lb force scc"/ft2 // dimensionless - 1 1 -s h e a r s t r e s s t o s h e a r r a t e i n p s e u d o - p l a s t i c f l u i d s a r e shown i n T a b l e 2-2 w h i l e t h e i r f l o w c u r v e s a r e shown i n c u r v e B o f F i g u r e 2-2. A l o g a r i t h m i c p l o t o f x v e r s u s du/dy f o r t h e s e m a t e r i a l s i s o f t e n f o u n d t o be l i n e a r o v e r a w i d e r a n g e o f s h e a r r a t e s a n d t h i s a c c o u n t s f o r t h e w i d e s p r e a d u s e o f t h e power law ( E q u a t i o n 2-7) t o c h a r a c t e r i z e f l u i d s o f t h i s t y p e . du n T = H ' 0 2-7 The " f l o w b e h a v i o u r i n d e x " n i s t h e s l o p e o f t h e l o g a r i t h m i c p l o t , w h i c h i s l e s s t h a n one f o r s h e a r t h i n n i n g o r p s e u d o -p l a s t i c b e h a v i o u r . The " c o n s i s t e n c y i n d e x " H" i s c a l c u l a t e d f r o m t h e i n t e r c e p t on t h e x a x i s a t u n i t s h e a r r a t e . I n g e n e r a l , H' and n a r e v e r y s e n s i t i v e t o t e m p e r a t u r e c h a n g e s . Examples o f p s e u d o - p l a s t i c f l u i d s may u s u a l l y be f o u n d i n t h e f o l l o w i n g m a t e r i a l s : r u b b e r s o l u t i o n s , a d h e s i v e s , p o l y m e r s o l u t i o n s o r m e l t s , g r e a s e s , s t a r c h s u s p e n s i o n s , c e l l u l o s e a c e t a t e , s o l u t i o n s u s e d i n r a y o n m a n u f a c t u r i n g , m a y o n n a i s e , s o a p , d e t e r g e n t s l u r r i e s , p a p e r p u l p , napalm, p a i n t s , d i s -p e r s i n g m e d i a i n c e r t a i n p h a r m a c e u t i c a l s , and b i o l o g i c a l f l u i d s . 2) D i l a t a n t f l u i d s The f l o w c u r v e (C) shown i n F i g u r e 2-2 i s c h a r a c t e r i z e d by z e r o y i e l d s t r e s s and may u s u a l l y be f i t t e d by one o f t h e m o d e l s i n T a b l e 2-2. The power law i s o f t e n a p p l i c a b l e , b u t w i t h n g r e a t e r t h a n u n i t y . The a p p a r e n t v i s c o s i t y i n c r e a s e s -12-w i t h i n c r e a s i n g s h e a r r a t e . E xamples o f m a t e r i a l s w h i c h have b e e n f o u n d t o e x h i b i t r h e o l o g i c a l d i l a t a n c y are.:; some ^aqueous s u s p e n s i o n s o f t i t a n i u m d i o x i d e , some c o r n f l o u r -s u g a r s o l u t i o n s , some gum a r a b i c - b o r a x s o l u t i o n s , s t a r c h , p o t a s s i u m s i l i c a t e , o r gum a r a b i c i n w a t e r , q u i c k s a n d , wet b e a c h s a n d , many defloG.culated, pigments, d i s p e r s i o n s c o n t a i n i n g h i g h c o n c e n t r a t i o n s o f s u s p e n d e d s o l i d s s u c h as m i c a and powdered q u a r t z , i r o n powder i n low v i s c o s i t y l i q u i d s . 2.1.2.2 T i m e - d e p e n d e n t f l u i d s T h e s e m a t e r i a l s a r e u s u a l l y c l a s s i f i e d i n t o two g r o u p s , t h i x o t r o p i c f l u i d s and r h e o p e c t i c f l u i d s , d e p e n d i n g upon w h e t h e r t h e s h e a r s t r e s s d e c r e a s e s o r i n c r e a s e s w i t h t i m e a t a g i v e n s h e a r r a t e and c o n s t a n t p r e s s u r e . a) T h i x o t r o p i c F l u i d s : T h e s e s u b s t a n c e s e x h i b i t a r e v e r s i b l e d e c r e a s e i n s h e a r s t r e s s w i t h t i m e a t a c o n s t a n t r a t e o f s h e a r and f i x e d t e m p e r a t u r e . The s h e a r s t r e s s w i l l o f c o u r s e a p p r o a c h some l i m i t i n g v a l u e c o n s i s t e n t w i t h t h e s t r u c t u r a l e q u i l i b r i u m w i t h i n t h e f l u i d c o r r e s p o n d i n g t o t h e p a r t i c u l a r s h e a r r a t e b e i n g u s e d . I f t h e f l o w c u r v e i s m e a s u r e d i n a s i n g l e e x p e r i m e n t i n w h i c h t h e s h e a r r a t e i s s t e a d i l y i n c r e a s e d from z e r o t o a maximum v a l u e and t h e n i m m e d i a t e l y d e c r e a s e d s t e a d i l y t o w a r d s z e r o , a f o r m o f h y s t e r e s i s l o o p w i l l be o b t a i n e d , as s k e t c h e d i n F i g u r e 2-3. A r o t a r y v i s c o m e t e r i s most c o n v e n i e n t f o r t h i s t y p e o f s t u d y . -13-S h e a r r a t e , du/dy F i g u r e 2-3 R h e o l o g i c a l f l o w c u r v e s f o r t h i x o t r o p i c and r h e o p e c t i c f l u i d s T h i x o t r o p i c p r o p e r t i e s a r e f o u n d i n foams, some s o l u -t i o n s o r m e l t s o f h i g h p o l y m e r s , o i l w e l l d r i l l i n g muds, g r e a s e s , m a r g a r i n e and s h o r t e n i n g , p r i n t i n g i n k s , many f o o d m a t e r i a l s , p a i n t s e t c . b) R h e o p e c t i c F l u i d s : T h e s e m a t e r i a l s , o c c a s i o n a l l y r e f e r r e d t o as a n t i t h i x o t r o p i c f l u i d s , a r e r e l a t i v e l y r a r e i n o c c u r r e n c e . T hey e x h i b i t a r e v e r s i b l e i n c r e a s e i n s h e a r s t r e s s w i t h t i m e a t a c o n s t a n t r a t e o f s h e a r u n d e r i s o t h e r m a l c o n d i t i o n s . A h y s t e r e s i s l o o p may be o b t a i n e d f o r t h e s e s u b s t a n c e s as shown i n F i g u r e 2-3. R h e o p e c t i c b e h a v i o u r i s o f t e n e x p l a i n e d i n t e r m s a n a l o g o u s t o t h o s e u s e d t o a c c o u n t f o r d i l a t a n c y b u t w i t h more p r o l o n g e d t i m e p e r i o d s f o r t h e s t r u c t u r a l c h a n g e s i n -v o l v e d . R h e o p e c t i c c h a r a c t e r i s t i c s have been o b s e r v e d i n t h e -14-f o l l o w i n g m a t e r i a l s , - b e n t o n i t e c l a y s u s p e n s i o n s , v a n a d i u m p e n t o x i d e s u s p e n s i o n s , gypsum s u s p e n s i o n s , c e r t a i n s o l s , d i l u t e s u s p e n s i o n s o f ammonium o l e a t e . 2.1.2.3. V i s c o e l a s t i c f l u i d s T h e s e m a t e r i a l s e x h i b i t b o t h v i s c o u s and e l a s t i c p r o -p e r t i e s . I n a p u r e l y e l a s t i c s o l i d t h e s t r e s s c o r r e s p o n d i n g t o a g i v e n s t r a i n i s i n d e p e n d e n t o f t i m e , whereas f o r v i s c o -e l a s t i c s u b s t a n c e s t h e s t r e s s w i l l g r a d u a l l y d i s s i p a t e . I n c o n t r a s t t o p u r e l y v i s c o u s l i q u i d s , on t h e o t h e r hand, v i s c o -e l a s t i c f l u i d s f l o w when s u b j e c t e d t o s t r e s s b u t p a r t o f t h e d e f o r m a t i o n i s g r a d u a l l y r e c o v e r e d upon r e m o v a l o f t h e s t r e s s . I t i s e v i d e n t t h a t 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 s u c h s u b -s t a n c e s a t any i n s t a n t w i l l be a f u n c t i o n o f t h e r e c e n t h i s t o r y o f t h e m a t e r i a l and c a n n o t be d e s c r i b e d by r e l a t i o n -s h i p s between s h e a r s t r e s s and s h e a r r a t e a l o n e , b u t w i l l r e q u i r e i n c l u s i o n o f t h e t i m e d e r i v a t i v e o f b o t h o f t h e s e q u a n t i t i e s . E x a m ples o f v i s c o - e l a s t i c f l u i d s i n c l u d e b i t u m e n s , f l o u r dough, n a p a l m and s i m i l a r j e l l i e s , p o l y m e r , and p o l y m e r m a l t s s u c h as n y l o n and many p o l y m e r s o l u t i o n s . I t s h o u l d be e m p h a s i z e d h e r e t h a t a l l f l u i d s u s e d i n t h i s s t u d y must n o t e x h i b i t s e c o n d o r d e r e f f e c t s s u c h as t i m e d e p e n d e n c y o r v i s c o - e l a s t i c i t y . 2.2 R h e o l o g i c a l P r o p e r t i e s M easurements 2.2.1 Y i e l d v a l u e measurement ( T Q ) T h e r e i s no u n i q u e u n i v e r s a l method f o r t h e measurement o f t h e y i e l d s t r e s s o f a f l u i d . A r e v i e w o f t h e p r i n c i p a l -15-methods w h i c h have b e e n u s e d t o e v a l u a t e t h e y i e l d s t r e s s o f f l u i d s was g i v e n by V a c a d l o and C h a r l e s ( 9 3 ) . They c l a s s i f i e d t h e methods i n two c a t e g o r i e s : 2.2.1.1 Dynamic methods: Among t h e dynamic methods are.; i ) E x t r a p o l a t i o n o f a f l o w c u r v e t o z e r o s h e a r r a t e u s i n g d a t a o b t a i n e d a t h i g h e r s h e a r r a t e s . The f l o w c u r v e c a n be o b t a i n e d e i t h e r f r o m a r o t a t i o n a l v i s c o m e t e r o r c a p i l l a r y t u b e v i s c o m e t e r . i i ) The a n a l y s i s o f t h e f l o w d e c a y c u r v e o b t a i n e d w i t h a r o t a t i o n a l v i s c o m e t e r . i i i ) The a n a l y s i s o f t h e f l o w d e c a y c u r v e o b t a i n e d w i t h a c a p i l l a r y t u b e v i s c o m e t e r . 2.2.1.2 S t a t i c methods Measurements u n d e r s t a t i c c o n d i t i o n s a r e p r e f e r a b l e f o r o b t a i n i n g more r e l i a b l e v a l u e s o±- y i e l d s t r e s s . Among t h e s t a t i c methods a r e : i ) A body, e.g. a s p h e r e o r r e c t a n g u l a r l a m i n a ( s h e e t ) i s p u l l e d upwards i n t h e s u b s t a n c e f r o m t h e end o f a h o r i z o n t a l s p r i n g by means o f a t h r e a d u n t i l i t b e g i n s t o move. i i ) U - tube a p p a r a t u s o r t e s t c o i l w h i c h i s f i l l e d w i t h t h e l i q u i d and s u b j e c t e d t o a s l o w l y i n c r e a s i n g p r e s s u r e u n t i l f l o w b e g i n s . The y i e l d v a l u e i s c a l c u l a t e d by t h e e x p r e s s i o n R A P  T o 2L 2-8 -16-i i i ) A p p l y i n g t e n s i o n t o smooth o r v e r t i c a l l y g r o o v e d c y l i n d e r s . The w i d t h s and t h e d e p t h s o f t h e g r o o v e s must be s u c h t h a t t h e s t r u c t u r e o f t h e m a t e r i a l i s p r e s e r v e d i n t h e g r o o v e s . G i l l and R u s s e l l (29) have t e s t e d two s t a t i c methods f o r t h e measurement o f t h e y i e l d v a l u e o f r e s i d u a l f u e l o i l s . The f i r s t method was t h e U.-tube method i n w h i c h t h e y t e s t e d d i f f e r e n t L/D r a t i o s , and t h e s e c o n d method was a c o r i i c y l i n -d r i c a l v i s c o m e t e r w h i c h c o n t a i n s two c y l i n d e r s : an . i n n e r c y l i n d e r c a r r i e s a b a l a n c e p o i n t e r and i s s u s p e n d e d so t h a t i t i s f r e e t o r o t a t e i n s i d e an o u t e r d r i v e n c y l i n d e r w i t h t h e o i l i n t h e a n n u l a r s p a c e . The y i e l d v a l u e was t a k e n as t h e s h e a r s t r e s s c o r r e s p o n d i n g t o t h e d e f l e c t i o n o f t h e p o i n t e r as t h e o u t e r c y l i n d e r s t a r t s t o r o t a t e . D i f f e r e n t r e s u l t s were o b t a i n e d u s i n g v a r i o u s i n n e r and o u t e r c y l i n d e r s t o a l t e r t h e a n n u l a r gap, b u t w i t h t h e w e t t e d l e n g t h k e p t c o n -s t a n t . T h e i r c o n c l u s i o n s were t h a t t h e y i e l d v a l u e depends on t h e r a t i o o f L/D f o r t h e U - t u b e , and l e n g t h t o gap r a t i o f o r t h e v i s c o m e t e r . F i g u r e 2-4 shows t h a t a r a t i o , L/D = 14 0, i s a b o u t t h e minimum v a l u e n e c e s s a r y f o r a t u b e y i e l d v a l u e t o be i n d e p e n d e n t o f L/D, and a r a t i o , l e n g t h / g a p = 70 i s t h e minimum v a l u e f o r t h e v i s c o m e t e r y i e l d v a l u e t o be i n d e p e n d e n t o f l e n g t h / g a p r a t i o . I n o u r y i e l d v a l u e m e a s u r e m e n t s , we have u s e d s e n s o r s y s t e m NV w i t h a r o t a t i o n a l v i s c o m e t e r ( R o t o -v i s c o RV-12), w h i c h h a s , l e n g t h / g a p r a t i o o f 150. - 1 7 -I L A'CONIC • LtNDRIC AL VISCOMtTCR »CJUL7J U-1 UBt RCSULT3 » 3 Y t t L D V A L U C i C L O v»t.ut 'OR VO " 300 F i g u r e 2-4 V a r i a t i o n o f Apparent Y i e l d Value w i t h Length/Diameter R a t i o (29) 1. Length/gap r a t i o = 70 2. Length/gap r a t i o = 14 3. Length/gap r a t i o = 8.7 5 The same equipment (Rotov i s c o RV-12) was used f o r the measurements of shear s t r e s s e s and r a t e s o f shear f o r the f l u i d s used i n t h i s study. 2.2.2 Shear S t r e s s - Shear Rate Measurements There are s e v e r a l types o f v i s c o m e t e r s mentioned i n the l i t e r a t u r e . Van Wazer e t a l . . ( 9 0 ) have g i v e n e x t e n s i v e des-c r i p t i o n s o f commercially a v a i l a b l e i n s t r u m e n t s . The most common types of v i s c o m e t e r s a r e : the c a p i l l a r y -18-tube viscometer, the c o n c e n t r i c c y l i n d e r r o t a r y viscometer, the r o t a t i n g c y l i n d e r i n an " i n f i n i t e " medium, and the cone-and-plate type of r o t a r y viscometer. The e s s e n t i a l f e a t u r e o f the c a p i l l a r y tube viscometers i s the measurement o f the f r i c t i o n a l p r e s s u r e drop a s s o c i a t e d w i t h the laminar flow o f f l u i d at a g i v e n r a t e through a l o n g , smooth, c y l i n d r i c a l tube of known dimensions. The flow curve f o r the f l u i d can be deduced from a s e r i e s of such measurements a t v a r i o u s known flow r a t e s . There are t h r e e c o r r e c t i o n s which must always be made to the measurements ob t a i n e d on a c a p i l l a r y tube viscometer, and these c o r r e c t i o n s a r e ; head of f l u i d above the tube e x i t , k i n e t i c energy e f f e c t s , and tube entrance e f f e c t . The c o n c e n t r i c c y l i n d e r r o t a r y viscometers are designed to shear a f l u i d l o c a t e d i n annulus between two c o n c e n t r i c c y l i n d e r s , one of which i s r o t a t i n g w h i l e the other i s h e l d s t a t i o n a r y . As shown i n F i g u r e 2-5 a motor d r i v e s the i n n e r c y l i n d e r . A v i s c o s i t y r e l a t e d torque, caused by the r e -s i s t a n c e of the sample to s h e a r i n g , a c t s on the i n n e r c y l i n d e r . T h i s torque d e f l e c t s a measuring s p r i n g p l a c e d between the motor and the i n n e r c y l i n d e r . The s i z e of the s p r i n g d e f l e c -t i o n c o r r e l a t e s l i n e a r l y w i t h the torque. The s p r i n g de-f l e c t i o n i s transformed i n t o an e l e c t r i c a l s i g n a l and shows up as e i t h e r a d i g i t a l i n d i c a t i o n or d i r e c t l y on the x - y - t r e c o r d e r . The r o t a t i n g c y l i n d e r i n an " i n f i n i t e " medium viscometer i s one of the s i m p l e s t viscometers to use, being a m o d i f i c a -t i o n o f the c o n c e n t r i c c y l i n d e r viscometer i n which the - 1 9 -Torque Measuring Springs produces "S "-Signal Tachometer Generator produces "n"-Signal Motor coaxial cylinders alternatively: cone/plate Rotor Sample F i g u r e 2-5 The -Measuring P r i n c i p l e of the R o t a t i o n a l Viscometers -20-c u p r a d i u s i s e x t e n d e d e f f e c t i v e l y t o i n f i n i t y . The f l o w c u r v e c a n be d e d u c e d f r o m measurements o f t h e t o r q u e r e q u i r e d t o r o t a t e a c y l i n d r i c a l r o d a t v a r i o u s known s p e e d s , when immersed i n an " i n f i n i t e " f l u i d . The c o n e - a n d - p l a t e v i s c o m e t e r c o n s i s t s e s s e n t i a l l y o f a f l a t , h o r i z o n t a l p l a t e and an i n v e r t e d c o n e , t h e apex o f w h i c h i s i n c o n t a c t w i t h t h e p l a t e . The a n g l e between t h e p l a t e and t h e cone s u r f a c e i s v e r y s m a l l - u s u a l l y l e s s t h a n one d e g r e e - and t h e f l u i d sample i s l o c a t e d i n t h i s s m a l l gap between t h e cone and p l a t e . The f l o w c u r v e c a n be c o n -s t r u c t e d f r o m measurements o f t h e t o r q u e r e q u i r e d t o r o t a t e t h e c o n e a t v a r i o u s speeds a s i l l u s t r a t e d i n F i g u r e 2-5. F o r a c c u r a t e measurement o f v i s c o s i t y , s h e a r s t r e s s , and s h e a r r a t e , t h e c o a x i a l c y l i n d r i c a l t y p e v i s c o m e t e r s a r e c o n -s i d e r e d t o be b e s t . HAAKE R o t o v i s c o v i s c o m e t e r s make i t p o s s i b l e t o s e l e c t a s u i t a b l e c o m b i n a t i o n o f s e n s o r s y s t e m s and m e a s u r i n g d r i v e -u n i t s . M a k i n g u s e o f t h e r o t o r s p e e d r a n g e a v a i l a b l e , a wide v i s c o s i t y r a n g e c a n be m e a s u r e d (2 < y < 1 0 8 mPa.s). The o t h e r a d v a n t a g e s o f u s i n g t h i s c o n c e n t r i c t y p e o f v i s c o -m e t e r ( R o t o v i s c o RV-12) a r e i t s s e n s i t i v i t y t o t h e m e a s u r e -ments o f s m a l l s h e a r r a t e s ' w h i c h e n a b l e d us t o measure t h e y i e l d v a l u e o f t h e f l u i d s , a s w e l l a s i t s f l e x i b i l i t y , s i n c e i t c a n a l s o be c o n v e r t e d t o a c o n e - a n d - p l a t e t y p e o f v i s c o m e t e r f o r t h e measurement o f h i g h l y v i s c o u s m a t e r i a l s . A d e t a i l d e s c r i p t i o n o f t h i s i n s t r u m e n t w i l l be g i v e n i n S e c t i o n 5.1.1. -21-2.3 P h y s i c a l P r o p e r t i e s o f C r u d e O i l s T y p i c a l examples o f c r u d e o i l p r o p e r t i e s as r e l a t e d t o wax c o n t e n t a r e g i v e n i n S e c t i o n 2.7 o f t h i s t h e s i s . T a b l e 2-3 shows n i n e waxy c r u d e o i l s r a n g i n g i n g r a v i t y f r o m 9.4 t o 3 7 ° API (1.004 t o 0.84 s p e c i f i c g r a v i t y ) and t h e i r wax c o n t e n t , p o u r p o i n t , , a n d v i s c o s i t y . T a b l e 2-3 P r o p e r t i e s o f N i n e Waxy C r u d e O i l s (81) C r u d e O i l D e n s i t y (API) Wax C o n t e n t (%) P o u r P o i n t ( ° C ) V i s c o s i t y ': (mP'a. s) 38°C. 204°C A 9.4 4.3 18- 20,000 8.0 B 12.2 0.04 -4; 1,200 3.9 C 14.0 0.05 -7 1,400 2.9 D 14.3 0.04 ;4" 1,400 3.9 E 22.2 0.02 .-40 21 1.1 F 22.9 24.4 -4 170 2.9 G 24.8 0.04 -51-' 21 1.1 H 30.0 21.50 29 85 1.8 I 37.0 38.10 29- 11 1.0 -22-2.4 P r o p e r t i e s o f P e t r o l e u m Waxes P e t r o l e u m waxes c a n be c l a s s i f i e d i n t o two c a t e g o r i e s ( 5 4 ) : v 2.4.1 C o m m e r c i a l p a r a f f i n waxes: The u s u a l c o m m e r c i a l g r a d e s o f p a r a f f i n wax a r e m a i n l y composed o f n - a l k a n e s , w h i c h have s t r a i g h t c h a i n s , b u t c o n -t a i n s m a l l amounts,between 2 and a b o u t 10 p e r c e n t , o f i s o -a n d / o r c y c l o a l k a n e s and some o i l . T a b l e 2-4 shows t h e most common h y d r o c a r b o n waxes aTTd^'their m e l t i n g p o i n t s . T a b l e 2-4 M e l t i n g P o i n t s o f Some P a r a f f i n i c P e t r o l e u m Waxes (54) N ormal A l k a n e M e l t i n g P o i n t , °C C 2 1 H 4 4 40.2 C 2 2 H 4 6 44.0 C 2 3 H 4 8 47.5 C 2 4 H 5 0 50.6 C 2 5 H 5 2 53.5 C 2 6 H 5 4 56.3 C 2 7 H 5 6 58.8 C 2 8 H 5 8 61.2 C 2 9 H 6 0 63.4 C 3 0 H 6 2 65.4 C H 74.6 35 72 C 3 6 H 7 4 75.9 -23-2.4.2 M i c r o c r y s t a l l i n e waxes: These waxes c o n s i s t mainly of h i g h m o l e c u l a r weight is.o-and/or c y c l o - p a r a f f i n s . T h e i r average m o l e c u l a r weight i s about 600-700. They are c h a r a c t e r i z e d by a h i g h p l a s t i -c i t y , which seems to be r e l a t e d not o n l y to the type of hydrocarbons p r e s e n t but a l s o t o t h e i r extremely wide range of m e l t i n g p o i n t s . They a l s o have a very f i n e c r y s t a l l i n e s t r u c t u r e . 2.5 Mechanism of Wax C r y s t a l l i z a t i o n Many o i l s d e p o s i t wax when c o o l e d . An understanding of the mechanism of wax c r y s t a l l i z a t i o n from o i l i s t h e r e f o r e very d e s i r a b l e f o r t h i s study. The waxes most common i n many waxy crude o i l s are n - p a r a f f i n s ranging from C^^ to C 2g, (39). When a waxy o i l i s c o o l e d , the wax u s u a l l y c r y s t a l l i z e s i n the form of t h i n p l a t e s or n e e d l e s . When enough wax has c r y s t a l l i z e d , i t can form a three dimensional network throughout the o i l and cause s o l i d i f i c a t i o n . That i s why " f l u i d i t y improvers" are used sometimes, to prevent t h i s n e t -work formation (4). I t i s known t h a t a s p h a l t e n e s , r e s i n s , and micro c r y s t a l l i n e wax of high m o l e c u l a r weight p l a y an important r o l e i n determining the c r y s t a l - f o r m i n g tendency of p a r a f f i n i c wax. Recent s t u d i e s (4, 11, 13, 25, 35, 36, 42, 56, 68, 81, 89, 91, 92, 39) have i n d i c a t e d t h a t d e p o s i t i o n of waxes from - 2 4 -o i l s decreases w i t h i n c r e a s e i n flow r a t e and i n c r e a s e s w i t h wax c o n c e n t r a t i o n and the temperature d i f f e r e n c e between the o i l and i t s c l o u d p o i n t . To e x p l a i n t h i s phenomenon i n the f i e l d where the underground r e s e r v o i r has s p e c i f i c con-d i t i o n s o f temperature, p r e s s u r e , and crude o i l composition, the p a r a f f i n i s e i t h e r i n suspension or i n s o l u t i o n i n the crude. As the o i l flows to the s u r f a c e , t h e r e i s g e n e r a l l y a r e d u c t i o n of temperature, p r e s s u r e and the amount o f . d i s s o l v e d gases c o n t a i n e d i n the o i l . Reduction of tempera-t u r e ,. and gas break-out w i l l reduce the s o l u b i l i t y of p a r a f f i n i n the crude. Thus, as the crude which c o n t a i n s p a r a f f i n moves through the r e s e r v o i r , r i s e s to the s u r f a c e and flows t o storage tanks a t atmospheric temperature, the s o l u b i l i t y of the p a r a f f i n may be exceeded. D e p o s i t i o n w i l l begin a t the p o i n t i n the system when the temperature of the system f a l l s below i t s c l o u d p o i n t , and w i l l c o ntinue as long as t h e r e i s a f u r t h e r decrease of the • p a r a f f i n : sblu'lDilTity i n the crude. _ The s o l u b i l i t y of p a r a f f i n waxes i n petroleum f r a c t i o n s has been s t u d i e d e x t e n s i v e l y by a number of workers ( 3 , 32, 54, 59, 8 0 ) . They found t h a t wax s o l u b i l i t y i n petroleum f r a c t i o n s i n c r e a s e s as the wax m e l t i n g p o i n t decreases and as the average b o i l i n g p o i n t o f the petroleum f r a c t i o n de-c r e a s e s . T h e o r e t i c a l l y , t h r e e main v a r i a b l e s a f f e c t s o l u b i l i t y : e q u i l i b r i u m temperature of s o l u t i o n , m o l e c u l a r weights of s o l v e n t and of s o l u t i o n s . T h e s e o b s e r v a t i o n s have been t a k e n i n t o a c c o u n t i n t h e s e l e c t i o n o f t h e s y n t h e t i c waxy o i l s w h i c h were u s e d i n t h i s s t u d y . The wax u s e d was a p a r a f f i n i c t y p e (Parvan-55) f r o m ESSO O i l Company, i t s p r o p e r t i e s a r e shown i n T a b l e F-3 i n t h e A p p e n d i x , w h i l e t h e o i l s u s e d were p a r a f f i n i c o i l s ( C l a r u s - B and Claru's-C) f r o m S h e l l O i l Company. T h e i r p r o -p e r t i e s a r e shown i n T a b l e s F - l and F-2 i n t h e A p p e n d i x . S o l u t i o n s o f d i f f e r e n t wax c o n c e n t r a t i o n s were made by d i s s o l v i n g t h e wax i n t h o s e o i l s . 2.6 O i l R e s e r v o i r C o n d i t i o n s : T e m p e r a t u r e s and p r e s s u r e s o f a r e s e r v o i r ' s f l u i d s a r e d i f f e r e n t f r o m one w e l l t o a n o t h e r , and depend on t h e d e p t h o f t h e r e s e r v o i r , t h e e l e v a t i o n f r o m s e a l e v e l , and t h e s u r -f a c e t e m p e r a t u r e . T a b l e 2-5 shows t y p i c a l w e l l c o n d i t i o n s t a k e n f r o m d i f f e r e n t p l a c e s i n t h e U n i t e d S t a t e s o f A m e r i c a . Tempera-t u r e s and p r e s s u r e s n o r m a l l y i n c r e a s e w i t h d e p t h b e l o w t h e s u r f a c e , and t h e r a t e o f i n c r e a s e w i t h d e p t h i s c a l l e d p r e s s u r e g r a d i e n t and g e o t h e r m a l g r a d i e n t . The g e o t h e r m a l g r a d i e n t i s a p p r o x i m a t e l y c o n s t a n t b e l o w an u p p e r s u r f a c e (15-122 meter)',-• where t h e t e m p e r a t u r e i s a f f e c t e d by atmos-p h e r i c t e m p e r a t u r e c h a n g e s and by t h e c i r c u l a t i o n o f g r o u n d w a t e r . W h i l e t h e t e m p e r a t u r e g r a d i e n t i s g e n e r a l l y c o n s t a n t w i t h i n any one h o l e , i t may v a r y g r e a t l y f r o m a r e a t o a r e a , e v e n i n t h e same s t r a t i g r a p h i c s e q u e n c e o f r o c k s . U n l i k e r e s e r v o i r p r e s s u r e s , w h i c h g e n e r a l l y d e c l i n e w i t h o i l and Table 2-5 F l u i d Pressures and Temperatures i n Some U.S. Deep Wells (50) Name and L o c a t i o n of Well T o t a l Formation and Pressure Formation Temperature Deepest Formation E l e v a t i o n above Sea Level Depth (Feet) Depth P s i Gradient Depth (Feet) °F Gradient(assume 60 a t Surface) Superior O i l Co., Well 34, #51-11, Sec. 11, T8N-R12W, Caddo Co., Okla, Springer Formation (Pennsyl-vanian), E l e v . 1,440 f t 17,823 17,823 13,153 74 psi/100 f t 17,738 253 1.1°F/100 f t Superior O i l Co., P a c i f i c Creek, Sec. 27, T27N-R103W, Sublette Co., Wyo., F r o n t i e r sand (Upper Cretaceous), E l e v . 7,090 f t 20,521 20,521 14,365 70 psi/100 f t 20,521 310 1.2°F/100 f t Superior O i l Co., Limoneira #1, Sec. 9, T2N-R22W, Ventura Co., C a l i f . , Upper Miocene, E l e v . 261 f t 18,734 18,734 10,678 57 psi/100 f t 18,734 250 1.0°F/100 f t Bahamas O i l L t d . , Andros #1, S t a f f o r d Creek, Andros I s , BW1, Lower Cretaceous, E l e v . 19 f t 14,585 14,585 6,315 43 psi/100 f t 10,670 98 0.3°F/100 f t -27-gas p r o d u c t i o n , t h e r e s e r v o i r t e m p e r a t u r e s r e m a i n f a i r l y -c o n s t a n t , (50) . 2.7 E x p e r i m e n t a l and F i e l d O b s e r v a t i o n Some r e c e n t s t u d i e s have been made on 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 many k i n d s o f c r u d e o i l s . T h e s e s t u d i e s have r e p o r t e d t h e e x i s t e n c e o f n o n - N e w t o n i a n b e h a v i o u r s f o r c e r t a i n c r u d e s . None o f t h e s e s t u d i e s have c o n s i d e r e d t h e f l o w o f s u c h o i l s t h r o u g h p o r o u s m e d i a e x p e r i m e n t a l l y . C o o p e r , e t a l . (19) have s t u d i e d t h e p r o b l e m o f s t a r t -up o f p i p e l i n e s c a r r y i n g g e l l i n g - t y p e c r u d e o i l s i n t h e C a n a d i a n A r c t i c . They showed t h a t s u c h o i l s e x h i b i t a y i e l d s t r e s s b e f o r e f l o w commences and n o n - N e w t o n i a n r h e o l o g i c a l b e h a v i o u r on c o n t i n u e d s h e a r i n g . S i m i l a r b e h a v i o u r was r e -p o r t e d by o t h e r w o r k e r s (68, 8 1 ) . B a r r y (5) o f M o b i l R e s e a r c h has r e p o r t e d t h a t many waxy c r u d e o i l s o f N o r t h A f r i c a b ehave as n o n - N e w t o n i a n Bingham f l u i d s w i t h f l o w c u r v e s as shown i n F i g u r e 2-6. He f o u n d t h a t a t some t e m p e r a t u r e above t h e p o u r p o i n t ( u s u a l l y a b o u t 10°C h i g h e r ) waxy c r u d e o i l s b e h a v e as N e w t o n i a n f l u i d s . When t h e waxy c r u d e s a r e c o o l e d b e l o w t h i s t e m p e r a t u r e , t h e y become non-N e w t o n i a n . B o t h s a m p l e s e x h i b i t e d a y i e l d s t r e s s a t l o w e r t e s t t e m p e r a t u r e s . B r o d , e t a l . (12) have t e s t e d c r u d e o i l s f r o m L i b y a and A l g e r i a and f o u n d a y i e l d v a l u e o f 350-400 d y n e / c m 2 a t a t e s t t e m p e r a t u r e o f 4°C by u s i n g t h e U - t u b e method f o r y i e l d v a l u e measurement. The p o u r p o i n t o f t h e c r u d e was 1 8 ° C . -28-400 350 300 250 200 I 50 100 50 < y 50 °F / / t V ./ 60 °F / / / •< V > / o / , / o 70 °F V V 0 ^ 50 100 150 200 250 \ SHEAR RATE, .SEC"' 50 100 150 200 250 SHEAR RATE, SEC"1 Figure. 2-6r:~ R h e o l o g i c a l behaviour o f two types of North A f r i c a n waxy crude o i l s (5) . 40 80 120 Shear Rate (sec - 1 ) 160 F i g u r e 2-7: R h e o l o g i c a l behaviour of extra-heavy crude o i l s of Venezuela (73) ; - ' ~ " • -- 2 9 -A r a n h a , e t a l . (4) have s t u d i e d t h e e f f e c t o f f l u i d i t y i m p r o v e r s on Bombay H i g h C r u d e O i l f l o w i n g t h r o u g h s u b m a r i n e p i p e l i n e s i n I n d i a . T h i s t y p e o f c r u d e has a wax c o n t e n t o f 16% by w e i g h t . They r e p o r t e d t h a t n o t o n l y t h e wax c o n t e n t o f t h e c r u d e s , b u t a l s o a s p h a l t e n e s and r e s i n s p l a y an i m p o r t a n t r o l e i n d e t e r m i n i n g t h e c r y s t a l - f o r m i n g t e n d e n c y o f p a r a f f i n i c wax. T h i s c r u d e a l s o shows a y i e l d v a l u e a t c e r t a i n t e m p e r a t u r e s as shown i n T a b l e 2-6. T a b l e 2-6 C h a r a c t e r i s t i c P r o p e r t i e s o f Bombay H i g h C r u d e O i l o f I n d i a (4) N a t u r e P a r a f f i n i c S p e c i f i c g r a v i t y , ° A P I 38.5 D e n s i t y a t 15°C, gm/cc 0.832 S u l p h u r c o n t e n t , wt% 0.16 Wax c o n t e n t , wt% 16 ASTM max. p o u r p o i n t , ° C 30 t o 36 ASTM min. p o u r p o i n t , ° C 24 t o 33 Y i e l d v a l u e * a t , d y n e / c m 2 20 ° C 43 25 ° C 8 30 ° C 3 * The y i e l d v a l u e s were o b t a i n e d f r o m t h e g r a p h r e l a t i n g s h e a r s t r e s s and r a t e o f s h e a r by e x t r a -p o l a t i n g t o z e r o r a t e o f s h e a r . A n o t h e r h i g h p o u r p o i n t o i l was d i s c o v e r e d i n I n d i a i n N a h q r k a t i y a , Assam ( 7 4 ) . The c r u d e i s m i x e d b a s e i n t y p e b u t i t s q u a l i t y v a r i e s c o n s i d e r a b l y between i n d i v i d u a l w e l l s . The l i g h t e r c r u d e s , w h i c h f o r m t h e b u l k o f t h e p r o d u c t i o n , have a h i g h p o u r p o i n t (29'-'30°G) , and c o n t a i n up t o 15% wt -30-o f wax. The h e a v i e r c r u d e s a r e much l o w e r i n p o u r p o i n t and wax c o n t e n t and c o n t a i n much l e s s d i s t i l l a t e . The p r o -p e r t i e s o f a t y p i c a l h i g h wax c o n t e n t c r u d e a r e shown i n T a b l e 2-7. T a b l e 2-7 Q u a l i t y o f N a h o r k a t i y a (Assam) C r u d e O i l i n I n d i a (74) S p e c i f i c g r a v i t y a t 1 2 2 ° F / 6 0 ° F 0.856 K i n e m a t i c v i s c o s i t y a t 1 2 2 ° F , CS 3.0 IP (upper) p o u r p o i n t , Op 85 Wax c o n t e n t , % wt . 14 S u l p h u r c o n t e n t , % wt 0.27 A s p h a l t e n e s c o n t e n t , % wt 0.46 I r a n i and Z a j a c (45) o f t h e G u l f R e s e a r c h and D e v e l o p -ment Company have s t u d i e d t h e h a n d l i n g o f h i g h p o u r p o i n t West A f r i c a n waxy c r u d e o i l s i n Z a i r e and C a b i n d a . They s t r e s s e d t h e u s e o f c o a x i a l r o t a t i o n a l v i s c o m e t e r s ( l i k e F e r r a n t i o r Haake v i s c o m e t e r s ) t o measure r e l e v a n t s t a t i c and dynamic r h e o l o g i c a l p a r a m e t e r s o f t h e c r u d e , s i n c e t h e p o u r p o i n t r e p r e s e n t s o n l y a q u a l i t a t i v e measure o f 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 any waxy c r u d e . Z a i r e - c r u d e has 20% wt wax, w h i l e C a b i n d a h a s 13.4% wt wax a t -18°C ( 0 ° F ) . F i g u r e 2-8 i s a p l o t o f t h e s h e a r s t r e s s d e v e l o p e d i n C a b i n d a waxy c r u d e o i l as a f u n c t i o n o f i n c r e a s i n g s h e a r r a t e . I t i s s i g n i f i c a n t t h a t a t 29.4 and 26.7°C (85 and 80°F) t h e c r u d e s behave as Bingham p l a s t i c s . E x t r a p o l a t i o n -31-S H E A R S T R E S S ( D Y N E S / C M 1 ) F i g u r e 2-8 Shear s t r e s s / s h e a r r a t e f o r Cabinda waxy Crude O i l (45) of the l i n e a r s e c t i o n to i t s i n t e r s e c t i o n w i t h the shear s t r e s s a x i s g i v e s the y i e l d v a l u e of an i d e a l Bingham p l a s t i c . T h i s non-Newtonian behaviour i s brought about by the p r e -c i p i t a t e d waxes forming an i n t e r a c t i v e s t r u c t u r e t h a t must be broken down b e f o r e the crude w i l l flow. However, at temperatures below 26.7°C (80°F), the r h e o l o g y of the waxy crude change s u b s t a n t i a l l y . At h i g h shear r a t e s , the v i s c o -s i t y d e c r e a s e s with i n c r e a s i n g shear ( i . e . shear t h i n n i n g ) , a f e a t u r e t h a t c o u l d i n f l u e n c e the behaviour o f the crude i n the f i e l d . -32-T a b l e 2-8 R h e o l o g i c a l P r o p e r t i e s o f Waxy C r u d e O i l o f C a b i n d a (45) T e m p e r a t u r e P l a s t i c V i s c o s i t y (mPa.s)-Y i e l d V a l u e (dyne/cm 2) OC °F 29.4 85 128 11.25 26.7 80 271 36.56 23.9 75 379 42.18 21.1 70 742 62.81 D a v e n p o r t and Somper (22) h a v e s t u d i e d t h e h a n d l i n g o f waxy c r u d e o i l s o f L i b y a and N i g e r i a . They i n v e s t i g a t e d t h e p r o b l e m o f r e s t a r t i n g o i l f l o w a f t e r a shutdown, when t h e o i l w a s - c o o l e d q u i e s c e n t l y and had d e v e l o p e d a g e l s t r u c t u r e . They r e p o r t e d t h a t waxy c r u d e o i l s e x h i b i t Bingham b e h a v i o u r , i . e . t h e e x i s t e n c e o f a y i e l d v a l u e . I t was r e p o r t e d t h a t one o f t h e b i g p r o b l e m s i n c r u d e o i l r h e o l o g y i s t h e d i f f i -c u l t y i n o b t a i n i n g r e p e a t a b l e r e s u l t s . A s i m i l a r c o m p l a i n t was made by Uhde and Kopp (89) , who t e s t e d c r u d e o i l s w i t h a l a r g e amount o f wax and a h i g h p o u r p o i n t . S i f f e r m a n (81) o f C o n o c o , I n c . h a s i n v e s t i g a t e d t h e e f f e c t s o f f l o w i m p r o v e r s ( w a x c r y s t a l m o d i f i e r s ) on t h r e e p r o p e r t i e s o f waxy o i l s ; v i s c o s i t y , g e l s t r e n g t h ( y i e l d v a l u e ) , and p o u r p o i n t . H i s s t u d y i n c l u d e d t h r e e t y p e s o f -33-waxy c r u d e o i l , D i c k i n s o n ( N o r t h D a k o t a ) , Udang ( I n d o n e s i a ) , and Amna ( L i b y a ) . T h e i r p o u r p o i n t s and s p e c i f i c g r a v i t i e s a r e shown i n T a b l e 2-9. He e m p h a s i z e d t h e i m p o r t a n c e o f t h e g e l T a b l e 2-9 API> G r a v i t i e s and Pour P o i n t s o f S e l e c t i v e Waxy C r u d e O i l s (81) Sample C r u d e O i l G r a v i t y P o u r P o i n t 0 A P I , sp. q r . °F °C Amna ( L i b y a ) D i c k i n s o n ( N o r t h Dakota) Udang ( I n d o n e s i a ) 36 0.845 34 0.855 40 0.825 75 24 95 35 100 38 s t r e n g t h ( y i e l d v a l u e ) and t h e a p p a r e n t v i s c o s i t y i n t h e c h a r a c t e r i z a t i o n o f t h e f l o w b e h a v i o u r o f waxy o i l s . C u s t o m a r i l y , t h e p o u r p o i n t i s t h e s o l e c r i t e r i o n f o r d e t e r -m i n i n g a c c e p t a b i l i t y o f a c r u d e o i l f o r p i p e l i n e o p e r a t i o n s . The same b e h a v i o u r was r e p o r t e d by G o n z a l o Rogas, e t a l . (73) i n t h e i r s t u d i e s o f e x t r a - h e a v y c r u d e o i l s f r o m t h e O r i n o c o o i l b e l t i n V e n e z u e l a . They c o n c l u d e d t h a t e x t r a -h e a v y c r u d e o i l s e x h i b i t n o n - N ewtonian Bingham p l a s t i c t y p e o f r h e o l o g i c a l b e h a v i o u r s , a s shown i n F i g u r e 2-7. The d e g r e e o f d i v e r g e n c e f r o m N e w t o n i a n b e h a v i o u r d e pends on t h e API g r a v i t y o f t h e c r u d e o i l and i t s t e m p e r a t u r e . They have -3 4-a l s o r e p o r t e d h i g h y i e l d p o i n t s v a l u e s o f up to 95.7 Pa at the ambient temperature. Crude o i l s w i t h s p e c i f i c g r a v i t y l e s s than 0.S86 have a y i e l d v a l u e of l e s s than 7.2 Pa. R e s i d u a l f u e l o i l s a l s o show non-Newtonian behaviour below c e r t a i n temperatures because o f the wax c o n t e n t of such f u e l s . G i l l and R u s s e l l (29) s t u d i e d the p u m p a b i l i t y of f u e l o i l s and found a rheology-temperature r e l a t i o n s h i p s i m i l a r t o t h a t r e p o r t e d f o r waxy c r u d e s . They a l s o r e -j e c t e d the pour p o i n t as a p u m p a b i l i t y i n d i c a t o r . F i g u r e 2-9 shows the v a r i a t i o n o f pour p o i n t w i t h wax content f o r a waxy f u e l o i l . T h i s f i g u r e a l s o i n d i c a t e s F i g u r e 2-9 V a r i a t i o n o f pour p o i n t w i t h wax content of waxy f u e l o i l s (29) - 3 5 -t h a t t h e r e m o v a l o f 1% wax c o r r e s p o n d s t o a change " i n p o u r p o i n t f r o m 11 ° t o 12°F. 2.8 The P o r o u s Medium A p o r o u s medium may be v i s u a l i z e d a s made up o f i n n u m e r a b l e f l o w c o n s t r i c t i o n s a nd e x p a n s i o n s w i t h i n t e r -c o n n e c t i n g c u r v e d p o r e c h a n n e l s . The d e t a i l s o f t h e s h a p e s , s i z e s , a nd i n t e r c o n n e c t i o n s o f t h e h o l e s a r e s e l d o m known, b u t t h e medium c a n be c h a r a c t e r i z e d by i t s a v e r a g e p r o p e r -t i e s , s u c h a s i t s a v e r a g e r e s i s t a n c e t o f l o w o f f l u i d s , i t s p o r o s i t y , e t c . The b a s i c l a w d e s c r i b i n g t h e f l o w o f N e w t o n i a n f l u i d s t h r o u g h p o r o u s m e d i a i s D a r c y ' s Law, t h e e q u i v a l e n t o f Ohm's Law i n f l o w o f e l e c t r i c c u r r e n t . An e x t r e m e l y l a r g e a r r a y o f m a t e r i a l s c a n be c l a s s i f i e d a s p o r o u s m e d i a . P o r o u s m e d i a a r e c l a s s i f i e d a s u n c o n -s o l i d a t e d o r c o n s o l i d a t e d and as o r d e r e d o r random. Examples o f u n c o n s o l i d a t e d m e d i a a r e b e a c h s a n d , g l a s s b e a d s , c a t a l y s t p e l l e t s , s o i l , g r a v e l , and t h e p a c k i n g s s u c h a s c h a r c o a l . E x a m ples o f c o n s o l i d a t e d m e d i a a r e most o f t h e n a t u r a l l y o c c u r r i n g r o c k s , s u c h a s s a n d s t o n e s , l i m e s t o n e s , e t c . I n a d d i t i o n , c o n c r e t e , cement, b r i c k s , p a p e r , c l o t h , and so f o r t h a r e man-made c o n s o l i d a t e d m e d i a . 2.8.1 U n d e r g r o u n d f o r m a t i o n (72) The p o r o u s m e d i a o f p r i n c i p l e i n t e r e s t h e r e a r e t h e -36-r o c k s f r o m u n d e r g r o u n d f o r m a t i o n s t h a t c o n t a i n n a t u r a l g a s , o i l , o r w a t e r . T h e s e r o c k s a r e f i r s t c l a s s i f i e d as t o t h e i r c h e m i c a l c o m p o s i t i o n , s a n d s t o n e , l i m e s t o n e , c h e r t , o r s e r p e n t i n e . R o u g h l y 40 p e r . c e n t o f o i l r e s e r v e s o f t h e w o r l d a r e c o n t a i n e d i n s a n d s t o n e f o r m a t i o n s . S a n d s t o n e c o n s i s t s o f g r a i n s o f q u a r t z , u s u a l l y cemented t o g e t h e r w i t h a r g i l l a c e o u s m a t e r i a l s . R o u g h l y 6 0 p e r c e n t o f t h e o i l r e s e r v e s o f t h e w o r l d a r e c o n t a i n e d i n l i m e s t o n e r e s e r v o i r s , w h i c h i n c l u d e t h o s e f o r m a t i o n s made o f l i m e s t o n e ( c a l c i u m c a r b o n a t e ) and t h o s e made o f d o l o m i t e , t h e d o u b l e c a r b o n a t e o f c a l c i u m and magnesium. O n l y o c c a s i o n a l l y i s o i l f o u n d i n c h e r t , a f i n e l y c r y s t a l l i n e q u a r t z , s e r p e n t i n e , a h y d r o u s magnesium s i l i c a t e , o r f r a c t u r e s i n d e n s e r o c k s s u c h a s g r a n i t e . 2.8.2 P e r m e a b i l i t y The p e r m e a b i l i t y o f a p o r o u s medium i s i t s most u s e -f u l f l u i d - f l o w p r o p e r t y . The p e r m e a b i l i t y i s a measure o f t h e e a s e w i t h w h i c h a f l u i d w i l l f l o w t h r o u g h t h e medium; t h e h i g h e r t h e p e r m e a b i l i t y , t h e h i g h e r t h e f l o w r a t e f o r a g i v e n p r e s s u r e g r a d i e n t . "The p e r m e a b i l i t y i s a s t a t i s t i -c a l a v e r a g e o f t h e f l u i d c o n d u c t i v i t i e s o f a l l t h e f l o w c h a n n e l s i n t h e medium. T h i s a v e r a g e c o n d u c t i v i t y t a k e s i n t o a c c o u n t t h e v a r i a t i o n s i n s i z e , s hape, d i r e c t i o n , and i n t e r c o n n e c t i o n s o f a l l t h e f l o w c h a n n e l s " (72) . S i n c e , o b v i o u s l y , a number o f p o r e s o r f l o w c h a n n e l s must be c o n s i d e r e d i n o b t a i n i n g a s t a t i s t i c a l " a v e r a g e " p e r m e a b i l i t y , i t i s o f t e n c o n v e n i e n t f o r m a t h e m a t i c a l p u r p o s e s t o c o n s i d e r t h e p e r m e a b i l i t y a s t h e p r o p e r t y a t a p o i n t i n t h e medium. I n a homogenous medium t h e p e r m e a b i l i t y a t e a c h p o i n t c o -i n c i d e s w i t h t h e a v e r a g e p e r m e a b i l i t y . I n a h e t e r o g e n e o u s medium t h e p e r m e a b i l i t y v a r i e s f r o m p o i n t t o p o i n t . The most commonly u s e d u n i t o f p e r m e a b i l i t y i s t h e d a r c y . The A m e r i c a n P e t r o l e u m I n s t i t u t e d e f i n e s a d a r c y as f o l l o w s : "A p o r o u s medium has a p e r m e a b i l i t y o f one d a r c y when a s i n g l e p h a s e f l u i d o f one c e n t i p o i s e v i s c o -s i t y t h a t c o m p l e t e l y f i l l s t h e v o i d s o f t h e medium w i l l f l o w t h r o u g h i t u n d e r c o n d i t i o n s o f v i s c o u s f l o w a t t h e r a t e o f one c u b i c c e n t i m e t e r . ' p e r s e c o n d p e r s q u a r e c e n t i -m e t e r o f c r o s s - s e c t i o n a l a r e a u n d e r a p r e s s u r e g r a d i e n t o f one a t m o s p h e r e p e r c e n t i m e t e r , " (20) (one d a r c y = 9 . 8 7 x 1 0 cm ) . 2.8.3 P o r o s i t y The p o r o s i t y o f a p o r o u s medium i s d e f i n e d a s t h e v o i d volume, o r volume o f p o r e s p a c e d i v i d e d by t h e t o t a l volume o f t h e medium. V o i d o r p o r e v o l u m e s a r e u s u a l l y d e t e r m i n e d by m e a s u r i n g e i t h e r g r a v i m e t r i c a l l y o r v o l u -m e t r i c a l l y t h e amount o f l i q u i d n e e d e d t o s a t u r a t e t h e d r y medium. T a b l e 2-10 shows t y p i c a l v a l u e s f o r permea-b i l i t y and p o r o s i t y . - 3 8 -T a b l e 2 - 1 0 T y p i c a l V a l u e s f o r P e r m e a b i l i t y a n d P o r o s i t y f o r V a r i o u s P o r o u s S o l i d s ( 7 2 ) P o r o u s S o l i d P e r m e a b i l i t y P o r o s i t y ( d a r c y ) ( f r a c t i o n ) S a n d 2 + 1 8 0 0 . 3 1 - 0 . 5 0 S a n d s t o n e 1 0 - 6 + i i 0 . 0 8 - 0 . 4 0 L i m e s t o n e 1 0 ~ 6 + 2 0 . 0 1 5 - 0 . 2 0 B r i c k 0 . 0 0 4 8 -y 0 . 2 2 0 . 1 2 - 0 . 3 4 S o i l 0 . 2 9 + 1 4 0 . 4 3 - 0 . 5 4 B e r l s a d d l e s 1 3 0 , 0 0 0 + 3 9 0 , 0 0 0 0 . 6 8 - 0 . 8 3 W i r e c r i m p s 3 , 8 0 0 -y 1 0 , 0 0 0 0 . 6 8 - 0 . 7 6 S i l i c a p o w d e r 0 . 0 1 3 + 0 . 0 5 1 0 . 3 7 2 . 8 . 4 T o r t u o s i t y I t i s t h e r e l a t i v e a v e r a g e l e n g t h o f a f l o w p a t h , i . e . t h e r a t i o o f t h e a v e r a g e l e n g t h o f t h e f l o w p a t h s t o t h e l e n g t h o f t h e m e d i u m . T h e t o r t u o s i t y i s a m a c r o s c o p i c m e a s u r e o f b o t h t h e s i n u o u s n e s s o f t h e f l o w p a t h a n d t h e v a r i a t i o n i n p o r e s i z e a l o n g t h e f l o w p a t h . L i k e p o r o s i t y , t o r t u o s i t y c o r r e l a t e s w i t h p e r m e a b i l i t y b u t c a n n o t b e u s e d a l o n e t o p r e d i c t p e r m e a b i l i t y e x c e p t i n s o m e l i m i t i n g c a s e s . 2 . 9 Flow Through Porous Media F l u i d flow through porous media depends on the pro-perties of both the medium and the f l u i d , the physical law governing such flow and the i n i t i a l and boundary conditions imposed. Consequently, i f the laws governing flow are known and may be expressed mathematically, the flow behaviour can be p r e d i c t e d f o r any o i l r e s e r v o i r f o r which the porous medium and o i l p r o p e r t i e s are known and f o r which the i n i t i a l and boundary c o n d i t i o n s may be s p e c i f i e d . One of the most important p r o p e r t i e s of a porous medium i s i t s p e r m e a b i l i t y , which determines the e f f e c t i v e flow c a p a c i t y of the o i l r e s e r v o i r . O i l s are g e n e r a l l y assumed to be Newtonian i n t h e i r flow behaviour, but as has been.shown t h i s i s not a l -ways the case. Secondary and t e r t i a r y o i l r e c o v e r y a l s o may r e s u l t i n s i m i l a r flow c a l c u l a t i o n s . Non-Newtonian f l u i d s , e s p e c i a l l y polymer s o l u t i o n s , and emulsions, o f t e n are i n j e c t e d i n t o the r e s e r v o i r i n v a r i o u s enhanced o i l r e c o v e r y p r o c e s s e s . In a d d i t i o n , foams sometimes are c i r c u l a t e d d u r i n g d r i l l i n g . Thermal r e c o v e r y of o i l by steam and a i r i n j e c t i o n may l e a d to the flow o f n a t u r a l emulsions and foams through porous media. Some enhanced o i l r e c o v e r y p r o j e c t s i n v o l v i n g the i n j e c t i o n o f non-Newtonian f l u i d s have been s u c c e s s f u l , but most of these p r o j e c t s e i t h e r f a i l e d or performed below e x p e c t a t i o n . These r e s u l t s suggest the need f o r a thorough study of the s t a b i l i t y and behaviour of non-Newtonian f l u i d s a t r e s e r v o i r c o n d i t i o n s , and a l s o a new look a t the flow of non-Newtonian f l u i d s i n porous media. 2.9.1 Darcy's Law In 18 56, as a r e s u l t of experimental s t u d i e s on the flow o f water'through u n c o n s o l i d a t e d sand f i l t e r beds, Henry - 4 0 -D a r c y e m p i r i c a l l y f o r m u l a t e d a l a w w h i c h b e a r s h i s n a m e ( 5 8 ) . T h i s l a w h a s b e e n e x t e n d e d t o d e s c r i b e , w i t h s o m e l i m i t a t i o n s , t h e m o v e m e n t o f o t h e r f l u i d s i n c l u d i n g t w o o r m o r e i m m i s c i b l e f l u i d s , i n c o n s o l i d a t e d r o c k s a n d o t h e r p o r o u s m e d i a ( 7 7 ) . D a r c y ' s L a w s t a t e s t h a t t h e a v e r a g e a p p a r e n t v e l o c i t y o f a h o m o g e n o u s " f l u i d i n a . p o r o u s m e d i u m i s p r o p o r t i o n a l t o .-the p r e s s u r e g r a d i e n t , ~(y a n d K t e r m s w e r e i n t r o d u c e d t o D a r c y 1 s.- l a w i n 1 9 3 0 ( 7 7 ) ) t o g i v e V = - - 2-9 o y d x f o r o n e - d i m e n s i o n a l f l o w . T h e n e g a t i v e s i g n i n d i c a t e s t h a t , i f t h e f l o w i s t a k e n a s p o s i t i v e i n t h e p o s i t i v e x - d i r e c t i o n , t h e n t h e p r e s s u r e d e c r e a s e s i n t h a t d i r e c t i o n , s o t h a t t h e s l o p e d p / d x i s n e g a t i v e . ( D a r c y u s e d h e a d i n s t e a d o f p r e s s u r e ) D a r c y ' s L a w d o e s n o t a p p l y t o t h e f l o w w i t h i n i n d i v i d u a l p o r e c h a n n e l s , b u t i s a s t a t i s t i c a l l a w w h i c h a v e r a g e s t h e b e h a v i o u r o f m a n y p o r e c h a n n e l s . O w i n g t o t h e p o r o s i t y o f t h e r o c k , t h e t o r t u o s i t y o f t h e f l o w p a t h s , a n d t h e a b s e n c e o f f l o w i n s o m e o f t h e " d e a d " p o r e s p a c e s , t h e a c t u a l f l u i d v e l o c i t y w i l l v a r y f r o m o n e p o i n t t o a n o t h e r w i t h i n t h e r o c k , a n d m a i n t a i n a n a v e r a g e a p p a r e n t v e l o c i t y . B e c a u s e a c t u a l v e l o c i t i e s a r e i n g e n e r a l n o t m e a s u r a b l e , a n d t o k e e p p o r o s i t y a n d p e r m e a b i l i t y s e p a r a t e d , a p p a r e n t v e l o c i t i e s f o r m t h e b a s i s o f D a r c y ' s L a w . T h i s m e a n s t h a t t h e a c t u a l a v e r a g e f o r w a r d v e l o c i t y - 4 1 -o f a f l u i d w i l l be t h e a p p a r e n t v e l o c i t y d i v i d e d by t h e p o r o s i t y , when t h e f l u i d c o m p l e t e l y s a t u r a t e s t h e r o c k ( 9 ) . 2 . 9 . 2 L i m i t a t i o n o f D a r c y ' s Law D a r c y ' s Law (Eqn. 2 - 9 ) r e p r e s e n t s a l i n e a r r e l a t i o n -s h i p b etween t h e f l o w r a t e (q) and t h e p r e s s u r e g r a d i e n t (g~) /" m o r e o v e r , t h e s t r a i g h t l i n e r e p r e s e n t i n g t h e r e l a t i o n -s h i p p a s s e s t h r o u g h t h e o r i g i n o f c o o r d i n a t e s . Any d e v i a -t i o n f r o m t h i s t y p e o f r e l a t i o n s h i p r e p r e s e n t " n o n - D a r c i a n f l o w s " a s i n d i c a t e d i n F i g u r e 2 - 1 0 . g r a d p - — » - g r a d p F i g u r e 2 - 1 0 T w e l v e s c h e m a t i c f l o w c u r v e s f o r n o n - D a r c i a n f l o w (77) -42-Sorae o f t h e more o b v i o u s d e v i a t i o n s f r o m D a r c y ' s Law c a u s e d by i g n o r i n g i t s b a s i c a s s u m p t i o n s have been d i s c u s s e d by S c h e i d e g g e r (77) . T h e s e a s s u m p t i o n s may be summarized as f o l l o w s : 1) N e w t o n i a n f l u i d . 2) V i s c o u s - f l o w . 3) The p o r o u s medium i s homogeneous and i s o t r o p i c , w i t h r e s p e c t t o p e r m e a b i l i t y . 4) Homogeneous f l o w , i . e . one p h a s e f l o w . 5) No s h r i n k a g e o r e x p a n s i o n o f t h e san d b e d . 6) The f l o w i s i s o t h e r m a l . 7) The p o r o u s medium i s c o m p l e t e l y s a t u r a t e d by t h e f l u i d . 8) The f l o w i s u n d e r s t e a d y s t a t e c o n d i t i o n s . 9) No i o n i c e f f e c t s o r c h e m i c a l r e a c t i o n s between t h e f l o w i n g f l u i d and t h e s u r f a c e o f t h e p o r o u s medium. Such a n o m a l i e s have been o b s e r v e d i n s o i l s . The p r o p o s e d r e s e a r c h encompasses t h e m o d i f i c a t i o n o f D a r c y 1 s Law t o a c c o u n t f o r t h e no n - N e w t o n i a n b e h a v i o u r o f waxy and hea v y o i l s , w h i c h have a y i e l d s t r e s s and power l a w b e h a v i o u r . 2.10 W a l l and End E f f e c t o f Sand Column Cohen and M e t z n e r (17) have made an e x t e n s i v e s t u d y on t h e c h a n n e l l i n g e f f e c t o f p a c k e d b e d s . They recommended -43-t h a t w a l l e f f e c t s c o u l d be a v o i d e d i f t h e p a c k e d c o l u m n s have bed t o p a r t i c l e d i a m e t e r r a t i o s g r e a t e r t h a n 30 f o r N e w t o n i a n f l u i d s , and g r e a t e r t h a n a b o u t 50 f o r n o n -N e w t o n i a n f l u i d s . I n t h i s t h e s i s , r a t i o s (D /D ) o f 62.7 ' c' p and 7 9.4 (column t o p a r t i c l e d i a m e t e r ) were u s e d . They have a l s o s u g g e s t e d t h a t any c o r r e c t i o n f o r t h e w a l l e f f e c t , i n p r e d i c t i n g p r e s s u r e d r o p - f l o w r a t e r e l a t i o n s h i p , w i l l be s e n s i t i v e t o t h e method o f p a c k i n g , p a r t i c l e shape and s i z e d i s t r i b u t i o n s , and t o 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 t h e f l u i d u n d e r c o n s i d e r a t i o n . W h i l e i n t h e c a s e o f end e f f e c t o f s a n d c o l u m n , i t has b e en f o u n d t h a t f o r b e d t o p a r t i c l e d i a m e t e r o f more t h a n 60, o n l y v e r y s m a l l end e f f e c t s were p r e s e n t , w h i c h c a n be n e g l e c t e d ( 5 2 ) . F o r o u r s y s t e m , we have u s e d two c o l u m n s w i t h 91 and 100 cm between t h e p r e s s u r e t a p s , and t h e p r e s s u r e t a p s were i n s t a l l e d b e f o r e t h e end o f t h e column t o a v o i d s u c h c o r r e c t i o n s . 2.11 T h e o r e t i c a l I n t e r p r e t a t i o n 2.11.1 N e w t o n i a n f l u i d a n a l y s i s The most d e f i n i t i v e s t u d y o f f l o w o f N e w t o n i a n f l u i d s t h r o u g h p o r o u s m e d i a a p p e a r s t o be t h a t o f E r g u n ( 2 6 ) , w h i c h p r o v i d e s t h e b a s i s f o r many m o d e l s f o r n o n - N e w t o n i a n f l o w . Good d i s c u s s i o n s o f t h e s e a n a l y s e s and i t s e x t e n s i o n t o p u r e l y v i s c o u s n o n - N e w t o n i a n f l u i d s a r e p r o v i d e d by B i r d , e t . a l . ( 9 ) , C h r i s t o p h e r and M i d d l e m a n ( 1 6 ) , C h r i s -t o p h e r ( 1 5 ) , G a i t o n d e and M i d d l e m a n ( 2 8 ) , S a d o w s k i ( 7 5 ) , and S a d o w s k i and B i r d ( 8 ) . As d i s c u s s e d by C h r i s t o p h e r -44-and Middleman, a number of equations are d e r i v a b l e f o r p u r e l y v i s c o u s non-Newtonian f l u i d s , a l l o f which reduce c o r r e c t l y to the Blake-Kozeny or Ergun form i n the case o f Newtonian f l u i d s , but d i f f e r i n the non-Newtonian case. These equations are based on the c o u p l i n g o f a c a p i l l a r i c or h y d r a u l i c r a d i u s model wi t h an assumed r e l a t i o n s h i p between shear r a t e and shear s t r e s s f o r t h a t f l u i d . For a Newtonian f l u i d , the shear v i s c o s i t y i s gi v e n by: y = T / Y 2-10 The flow o f t h i s type o f f l u i d i n a c i r c u l a r tube i s known as H a g e n - P o i s e u i l l e Law (9) < v > = o—T~ 2-11 8 y L Now i f the s u p e r f i c i a l v e l o c i t y , V Q (equation 3-7) and mean h y d r a u l i c r a d i u s (equation 3-6) are i n t r o d u c e d i n t o equation 2-11, we get; D 2 e 3 V = 2 ^ 2-12 O L 72 C ( 1 - e ) 2 y Where C' i s a con s t a n t which accounts f o r t o r t u o s i t y e f f e c t s , t h i s constant must be determined e x p e r i m e n t a l l y . B i r d (9) by u s i n g C = 25/12, converted equation 2-12 i n t o what i s known as the Blake .r-Kozeny equation: -45-D2 e 3  v = P £E ° 150 d - e ) - 2 y L 2 - 1 3 E q u a t i o n 2-13 i s t h e most s u c c e s s f u l s e m i - e m p i r i c a l e x p r e s s i o n u s e d t o d e s c r i b e V i s c o u s - f l o w o f N e w t o n i a n f l u i d s t h r o u g h p a c k e d b e d s . As m e n t i o n e d p r e v i o u s l y , D a r c y ' s Law may be s t a t e d a s : V = - * ^ 2-9 o u L By c o m p a r i n g e q u a t i o n s 2-12, 2-13, and 2-9 t h e permea-b i l i t y , K, o f t h e p o r o u s medium i s g i v e n a s -o '2 - 3 D' K = 2 2-14a C" ( l - e ) r 2 • o r D2 £ 3 K = 2 : 2-l4b 150 ( 1 - e ) 2 where K i s a c h a r a c t e r i s t i c p r o p e r t y o f t h e medium. B l a k e (10) was t h e f i r s t t o p r e s e n t t h e above e q u a t i o n s i n t h e f o r m o f a f r i c t i o n f a c t o r - R e y n o l d s number r e l a t i o n -s h i p f o r p a c k e d beds a s f o l l o w s : f * = C"/R| 2-15 -46-where C" = c o n s t a n t ' AP D f * = £ 2-16 2 L 1-e R* = e 2-17 The above e q u a t i o n s were l a t e r r e v i v e d by E r g u n (26) i n 1952. T a b l e 2-11 summarizes r e c e n t p a p e r s (1965-1979) on t h e v i s c o u s f l o w o f N e w t o n i a n and n o n - N e w t o n i a n f l u i d s t h r o u g h s t a n t (C") i n e q u a t i o n 2-15 by v a r i o u s a u t h o r s . I t a l s o shows t h e f l u i d s u s e d , t h e f l u i d r h e o l o g i c a l m o d e l , the' t y p e o f p o r o u s medium, and t h e r a n g e o f a v e r a g e p a r t i c l e d i a m e t e r (D ) u s e d i n t h e i r s t u d i e s . P 2.11.2 Non-Newtonian f l u i d a n a l y s i s Many s t u d i e s (8, 9, 14, 15, 16, 17, 26, 28, 30, 31, 38, 43, 47, 52, 61, 64, 75, 76, 85, 8 6 ) , have been made on t h e f l o w o f n o n - N e w t o n i a n f l u i d s t h r o u g h p o r o u s m e d i a . However t h e y have a l l i g n o r e d t h e y i e l d v a l u e p r o b l e m and have c o n -c e n t r a t e d on o t h e r f l o w b e h a v i o u r s . S a v i n s (76) has r e v i e w e d a l a r g e number o f t h e t h e o r i e s w h i c h a r e p a r t i a l l y s u c c e s s f u l i n d e s c r i b i n g some o f t h e o b s e r v e d f e a t u r e s o f t h e v i s c o -e l a s t i c phenomena. None o f them seem t o be a b l e t o s i m u l a t e , w i t h o u t a m b i g u i t y a l l f e a t u r e s o f t h e v i s c o e l a s t i c phenomena and c e r t a i n o t h e r e f f e c t s o b s e r v e d i n complex f l u i d s . Most o f t h e s e t h e o r i e s and s p e c i a l c a s e s o f t h e more g e n e r a l p o r o u s m e d i a . I t shows t h e e x p r e s s i o n s u s e d f o r t h e c o n --47-T a b l e 2-11 The V a l u e s o f t h e C o n s t a n t C" Used i n P a p e r s P u b l i s h e d S i n c e 1965 (47) Reference C " Experimental media Granular Tiling Fluid model (or porous dp [mm] medium) Sadowski, Bird (0) 180 Christopher, Middleman (16) 150 Gregory, Griskey ; •( 31 ) 180 Gaiionde, Middleman (28) 150 Marshal, Metzner (52) 150 Kozicki, Hsu, Tiu Yu, Wen, Bailie 173 150 Siskovic, Gregory, Griskey 180 water, glycerine, poly-ethylene glycol in water polyvinyl alcohol in water hydroxyethylcellulose in water polyethylene glycol in water carboxymethylcellulose in water polyisobutylene in toluene molten polyethylene polyisobutylene in toluene polyisobutylene in decalin E T 597 in water, Carbopol sugar in water polyox in water molten polyethylene Newtonian beads, lead shot Ellis beads, lead shot power law beads 0.524 2.807 0.524 -r 2.807 0.723 power law spheres 1.37 -r* 6.0 power law spheres, sand — power law sintered bronze 0.1266 porous disk Newtonian spheres 3.18 and 3.97 power law spheres, cubes 2.03 -f- 8.89 power law spheres 1.37 -f- 6.0 Wampler, Gregory , (31) Kemblowski, Mertl Mishra, Singh, Mishra (82) Brea. Edwards. Wilkinson Hanna, Kozicki. Tin Michele Kcmblowski, Dziubinski Kemblowski, MichnieWcz (47) 180 molten po!y(ethylene terephthalate) 150 glycerine in water, polyvinyl alcohol in water polyacrilamide in water and others 150 water, mobile oil polyvinyl alcohol in water 160 water, glycerol slurry 180 water, polyox in water 180 water, glucose-sirup carboxymethylcellulose in water separan in water 150 molten polyethylene, molten polypropylene 180 molten poly(cthylcne terephthalate) Newtonian beads Newtonian spheres, beads, sand power law spheres, beads, sand Newtonian beads power law beads Newtonian spheres power law spheres Newtonian spheres Newtonian spheres power law •spheres power law spheres Newtonian spheres sand 0.12 0.83 0:265 - 3.246 0.265 -f- 3.246 4.27 + 7.41 4.27 H- 7.41 1.1 -;- 3.1 1.1 + 3.1 2.373 H- 3.975 1 -=- 20 1 -4- 20 2.45 and 3.0 1.0 and 1.2 0.111 and 0.176 -48-t h e o r i e s a r e s u b j e c t t o a s s u m p t i o n s , s u c h a s , t h e a b s e n c e o f a y i e l d s t r e s s o r no t i m e - d e p e n d e n t b e h a v i o u r . B i r d (9) was t h e f i r s t t o s u g g e s t t h a t t h e c a p i l l a r y o r h y d r a u l i c r a d i u s c o n c e p t c o u l d be e x t e n d e d t o d e s c r i b e t h e f l o w o f r h e o l o g i c a l l y c omplex f l u i d s i n p o r o u s m e d i a . T h e s e e a r l y i d e a s were t h e p r e c u r s o r t o t h e g e n e r a l i z a t i o n o f D a r c y ' s Law f o r no n - N e w t o n i a n f l u i d s i n t e r m s o f t h e E l l i s m o del (8, 75) and t h e power law model (15, 1 6 ) . The a p p r o a c h e s o f B i r d and S a d o w s k i ( 8 ) , C h r i s t o p h e r and M i d d l e -man ( 1 6 ) , and G r e g o r y and G r i s k e y (31) a r e t y p i c a l . B o t h , S a d o w s k i and Middleman have c o n s i d e r e d t h e f l o w o f non-Ne w t o n i a n p o l y m e r s o l u t i o n s t h r o u g h p o r o u s m e d i a , w h i l e G r e g o r y s t u d i e d m o l t e n p o l y m e r s . Muskat (57, 58) has a n a l y z e d D a r c y ' s work and d i d an e x t e n s i v e s t u d y on t h e f l o w o f homogenous f l u i d s t h r o u g h p o r o u s m e d i a b a s e d on t h e a s s u m p t i o n t h a t D a r c y 1 s Law i s v a l i d . R e c e n t l y a r e v i e w o f t h e e x p e r i m e n t a l and t h e o r e t i c a l work u s i n g t h e c a p i l l a r y model a p p r o a c h was made by Kemblowski and M i c h n i e w i c z ( 4 7 ) . The r e s u l t s o f v a r i o u s a u t h o r s were p r e s e n t e d i n T a b l e 2-11 a s a m o d i f i e d f r i c t i o n f a c t o r ( f * ) v e r s u s a m o d i f i e d R e y n o l d s number ( R * ) , by u s i n g t h e r e l a t i o n s h i p o f E q u a t i o n 2-15. One example o f a no n - N e w t o n i a n f l u i d i s p o l y m e r s o l u t i o n s and e m u l s i o n s . T h e s e f l u i d s c a n be m o d e l l e d by t h e Ostwald-desWaele "Power Law", i n w h i c h s h e a r s t r e s s i s r e l a t e d t o t h e r a t e o f s h e a r by E q u a t i o n 2-7 o r a s : -49-x = H Y n 2-7 where the "apparent v i s c o s i t y " (sometimes c a l l e d " e f f e c t i v e v i s c o s i t y " ) can be gi v e n as: y = H' Y n 1 2-18 app C h r i s t o p h e r and Middleman (15, 16) s t a r t e d with the flow of a power law f l u i d i n a c i r c u l a r tube; i . e . e Then they i n t r o d u c e d the s u p e r f i c i a l v e l o c i t y , V , h y d r a u l i c r a d i u s , R h, and " t o r t u o s i t y " e f f e c t , <2<' = 2 5/12 to g et: D e 1+1/n g. A T 5 1/n V = n £ (_£ ) ( 6 A p ) 2-20 o 3n+l l3 (l-eT ;25H'L J 1 Z V Now i f the e x p r e s s i o n f o r the p e r m e a b i l i t y , K, from Equation 2-14b i s i n t r o d u c e d i n t o Equation 2-20 and s i m p l i -f i e d : v n = _ K _ AP 2 _ 2 1 o u e f f L where 1-n V f f = I T ( 9 + 3 / n ) n ( 1 5 0 K e > 2 2 - 2 2 -50-which i s c a l l e d the "non-Newtonian bed f a c t o r " , which accounts f o r the dependence o f V D on K and e due to non-Newtonian behaviour. For non-Newtonian f l u i d s , most i n v e s t i g a t o r s i n c l u d i n g the author of t h i s study have used the same d e f i n i t i o n of the m o d i f i e d f r i c t i o n f a c t o r , Equation 2-16, w h i l e the m o d i f i e d Reynolds number may be d i f f e r e n t from Equation 2-17. I t s d e f i n i t i o n depends upon the r h e o l o g i c a l behaviour of the f l u i d used. For example C h r i s t o p h e r and Middleman (16) used the power law model Equation 2-7 and they d e f i n e d the m o d i f i e d Reynolds number as: TT 2-n „ V D p R* = 2 P 2-2^ e 150 y e f f ; (1-e) A 1 6 Another s i m i l a r approach was made by B i r d and Sadowski (8, 75), f o r the flow of non-Newtonian f l u i d s through porous media. They have used the E l l i s model f o r the rheo-l o g i c a l f i t t i n g i n s t e a d of the power law model. The E l l i s model may be w r i t t e n as: Y = Q Q T + Q 1 x a 2-24 where Q^i Q-^r and a are determined e x p e r i m e n t a l l y . T h i s model i s used to account f o r v i s c o e l a s t i c behaviour o f p o l y -mer s o l u t i o n s . The E l l i s analog of the H a g e n - P o i s e u i l l e equation f o r the flow o f non-Newtonian f l u i d s through c i r c u l a r tubes i s given as: - S I -'S? <V> = Q o 2-25 S a d o w s k i ' s e x p r e s s i o n f o r t h e p e r m e a b i l i t y i s g i v e n b y E q u a t i o n 2-14b. H i s c o r r e l a t i o n between t h e m o d i f i e d f r i c t i o n f a c t o r and t h e m o d i f i e d R e y n o l d s number i s b a s e d on where f * i s g i v e n by E q u a t i o n 2-16 and t h e R e y n o l d s number i s g i v e n a s : S a d o w s k i (75) c l a i m e d t h a t t h e e x p e r i m e n t a l r e s u l t s f o r p o l y m e r s o l u t i o n s o f low and medium m o l e c u l a r w e i g h t s were s u c c e s s f u l l y c o r r e l a t e d f o r v a l u e s o f t h e e f f e c t i v e R e y n o l d s number l e s s t h a n 10. R e c e n t l y , P a s c a l (62, 63, 64, 65, 66) has a n a l y z e d D a r c y ' s law t h e o r e t i c a l l y and i t s a p p l i c a t i o n t o non-New-t o n i a n f l u i d s by i n t r o d u c i n g t h e e f f e c t o f t h e y i e l d v a l u e . He has i n v e s t i g a t e d t h e e f f e c t o f t h e t h r e s h o l d g r a d i e n t i n s t e a d y and u n s t e a d y s t a t e f l o w t h r o u g h p o r o u s m e d i a , and f * = I 8 O / R 4 2-26 2-27 where 2-28 -52-has o b t a i n e d s o l u t i o n s f o r w e l l f l o w t e s t a n a l y s e s w h i c h i n c l u d e t h e t h r e s h o l d g r a d i e n t e f f e c t , i . e . p r e s s u r e draw down and p r e s s u r e b u i l d up. A c o m p a r i s o n between t h e p r e s s u r e r e s p o n s e s was made w i t h and w i t h o u t t h e t h r e s h o l d g r a d i e n t t o c l a r i f y t h e i m p o r t a n c e o f t h i s p a r a m e t e r i n t h e a c c u r a c y o f t h o s e t e s t s . H i s i n v e s t i g a t i o n s have i n -d i c a t e d t h a t t h e t h r e s h o l d g r a d i e n t has a s i g n i f i c a n t e f f e c t i n some s i t u a t i o n s , and as a r e s u l t , i t s h o u l d be c o n s i d e r e d i n t h e a n a l y s i s o f o i l w e l l f l o w t e s t s f o r an a c c u r a t e measurement o f t h e p o r o u s medium p r o p e r t i e s i n p r o d u c i n g r e s e r v o i r s w i t h o i l s w i t h a t h r e s h o l d g r a d i e n t . A l l t h e a s s u m p t i o n s u s e d i n d e v e l o p i n g D a r c y ' s law (p. 4 2 ) , t h e h y d r a u l i c r a d i u s and t h e s u p e r f i c i a l v e l o c i t y a p p l y t o t h e d e f i n i t i o n o f p e r m e a b i l i t y g i v e n i n E q u a t i o n 2-14a. -53-CHAPTER THREE M A T H E M A T I C A L MODEL A N A L Y S I S T h e r e a r e t h r e e b a s i c t h e o r e t i c a l a p p r o a c h e s u s e d t o d e s c r i b e t h e f l o w t h r o u g h p o r o u s m e d i a . Common t o e a c h a p p r o a c h i s t h e a s s u m p t i o n t h a t t h e f l o w may be a n a l y z e d on t h e b a s i s o f m i c r o s c o p i c f l o w w i t h i n p o r e s . T h e o r e t i c a l m o d e l s w h i c h d e s c r i b e t h e m i c r o s c o p i c s t r u c t u r e o f p o r e s a r e p r o p o s e d and t h e m i c r o s c o p i c f l o w p r o p e r t i e s a r e t h e n d e r i v e d . T h e s e t h r e e a p p r o a c h e s a r e : i ) F i r s t method: The h y d r a u l i c r a d i u s t h e o r y : The p o r o u s medium i s assumed t o be e q u i v a l e n t t o an a s s e m b l a g e o f c h a n n e l s . The N a v i e r - s t o k e s e q u a t i o n s a r e s o l v e d f o r a s i n g l e c h a n n e l and t h e n t h e r e s u l t s a r e a p p l i e d t o t h e c o l l e c t i o n . i i ) S e c o n d method: The d r a g t h e o r y , c o n s i d e r s t h e p a r t i c l e s m a k i n g up t h e p o r o u s medium t o be o b s t a c l e s t o an o t h e r w i s e s t r a i g h t f l o w o f t h e f l u i d . The d r a g on e a c h p a r t i c l e i s c a l c u l a t e d f r o m t h e N a v i e r - S t o k e s e q u a t i o n s and t h e sum o f t h e r e s i s t a n c e s o f a l l t h e p a r t i c l e s i s t h o u g h t t o e q u a l t h e t o t a l r e s i s t a n c e o f t h e bed t o t h e f l o w o f t h e f l u i d . -54-i i i ) T h i r d method: The s t a t i s t i c a l t h e o r y s u g g e s t s t h a t t h e p o r o u s medium s h o u l d be c o n s i d e r e d i n t r i n s i c a l l y d i s o r d e r e d . T h i s i s t h e o p p o s i t e v i e w f r o m t h e o t h e r methods w h i c h c o n s i d e r t h e p o r o u s medium t o be i n t r i n s i c a l l y o r d e r e d . The N a v i e r - S t o k e s e q u a t i o n s c a n n o t be u s e d f o r s u c h d i s -o r d e r e d m e d i a b u t t h e methods o f s t a t i s t i c a l m e c h a n i c s have t o be a p p l i e d i n s t e a d . I n p r a c t i c e , t h e h y d r a u l i c r a d i u s t h e o r y has g i v e n s u c c e s s f u l c o r r e l a t i o n s f o r d a t a f r o m f l o w e x p e r i m e n t s f o r d e n s e b e d s . C o r r e l a t i o n s d e r i v e d f r o m t h e d r a g t h e o r i e s have been p a r t i c u l a r l y u s e f u l f o r s e d i m e n t a t i o n e x p e r i m e n t s . The s u c c e s s o f t h e s t a t i s t i c a l t h e o r i e s l i e s w i t h t h e s t u d y o f m u l t i p h a s e f l o w . A r a t h e r c o m p l e t e r e v i e w o f the. h y d r a u l i c r a d i u s t h e o r y i s g i v e n by S c h e i d e g g e r ( 7 7 ) . F o r t h i s s t u d y , t h e f i r s t method was c h o s e n f o r t h e a n a l y s i s o f t h e f l o w t h r o u g h p o r o u s m e d i a o f s h e a r t h i n n i n g f l u i d s w i t h y i e l d s t r e s s e s , b e c a u s e o f i t s p r e v i o u s s u c c e s s i n c o r r e l a t i n g e x p e r i m e n t a l d a t a . B e f o r e p r o c e e d i n g t o t h e m a t h e m a t i c a l a n a l y s i s , l e t us d e f i n e t h e h y d r a u l i c r a d i u s and t h e s u p e r f i c i a l v e l o c i t y . T h e s e two t e r m s were u s e d i n t h i s s t u d y . 3.1 The C o n c e p t o f H y d r a u l i c R a d i u s (R h) The b a s i c c o n c e p t o f t h e h y d r a u l i c r a d i u s i s due t o d i m e n s i o n a l c o n s i d e r a t i o n s . I t i s o b s e r v e d t h a t t h e p e r m e a b i l i t y , K, i n a b s o l u t e u n i t s , has t h e d i m e n s i o n o f -55-l e n g t h s q u a r e d . I t i s r e a s o n a b l e t h a t t h e r e e x i s t s a c h a r a c t e r i s t i c l e n g t h w h i c h may be u s e d t o c h a r a c t e r i z e t h e p e r m e a b i l i t y o f a p o r o u s medium. T h i s l e n g t h i s t e r m e d t h e " H y d r a u l i c R a d i u s " o f t h e p o r o u s medium. A p o s s i b l e m easure o f t h i s l e n g t h , R^, w o u l d be t h e r a t i o o f volume t o s u r f a c e o f t h e p o r e s p a c e . B i r d (9) has d e f i n e d t h e mean h y d r a u l i c r a d i u s (R^) as t h e r a t i o o f c r o s s s e c t i o n a l a r e a o f a s t r e a m t o i t s w e t t e d p e r i m e t e r . I t i s g e n e r a l l y u s e d t o d e s c r i b e f l o w i n n o n - c i r c u l a r t u b e s . F o r example, t h e h y d r a u l i c r a d i u s o f a c i r c u l a r t u b e may be g i v e n a s : R 2 R ID = I ?_ _ _ c _ D h 2TT R 2 4 c A l s o , t h e h y d r a u l i c r a d i u s may be e x p r e s s e d i n t e r m s o f t h e p o r o s i t y , e , and w e t t e d s u r f a c e p e r u n i t volume (a) o f a: bed i n t h e f o l l o w i n g way: R, = c r o s s s e c t i o n a v a i l a b l e f o r f l o w h w e t t e d p e r i m e t e r volume a v a i l a b l e f o r f l o w t o t a l w e t t e d s u r f a c e _ volume o f v o i d s / v o l u m e o f b e d _ e_ 2-2 ~ w e t t e d s u r f a c e / v o l u m e o f bed - a The q u a n t i t y , a, i s r e l a t e d t o t h e " s p e c i f i c s u r f a c e " , a v , by: a = a v (1-e) 3-3 - 5 6 -where a v i s the t o t a l p a r t i c l e surface/volume of p a r t i c l e . T h i s q u a n t i t y may be used t o d e f i n e the mean p a r t i c l e d i a -meter D : P D = — 3 - 4 P a ^ v Th i s d e f i n i t i o n i s chosen because, f o r spheres, Equation 3 - 4 g i v e s equal to the diameter of a sphere. By combin-i n g Equations 3 - 2 , 3 - 3 , and 3 - 4 , one can get e D R = P_ 3 _ 5 h 6 ( l - e ) J 5 s u b s t i t u t i n g Equation 3 - 5 i n t o 3 - 1 we get: e D R c = Tafe 3 - 6 The sand bed was modelled as a tube of very complicated c r o s s s e c t i o n with h y d r a u l i c r a d i u s (R^). 3 . 2 The S u p e r f i c i a l V e l o c i t y (V ): T h i s i s the average l i n e a r v e l o c i t y a f l u i d would have i n the column i f no packing were pres e n t . I t was de-f i n e d by B i r d (9) as: V o = < V > e 3 - 7 Another d e f i n i t i o n f o r the s u p e r f i c i a l v e l o c i t y may be given as, ( 6 1 , 1 7 ) : -57-V = <V> e L/L 3-8 o ' e where L /L = C' = e ^ f e c t i v e l e n g t h o f t h e f l o w path. e' s t r a i g h t l e n g t h o f t h e p o r o u s medium The d i f f e r e n c e between t h e s e two d e f i n i t i o n s (Eq. 3-7 and 3-8) i s t h a t t h e t o r t u o s i t y e f f e c t ( C ) was i n t r o -d u c e d i n E q u a t i o n 3-8 b u t n o t i n E q u a t i o n 3-7, w h i c h means <V> i n E q u a t i o n 3-8 i s t h e a v e r a g e i n t e r s t i t i a l v e l o c i t y , w h i l e <V> i n E q u a t i o n 3-7 i s t h e a v e r a g e i n t e r s t i t i a l v e l o c i t y component p a r a l l e l t o t h e bed a x i s . B o t h d e f i n i -t i o n s were c o n s i d e r e d i n t h e model a n a l y s i s o f t h i s s t u d y . 3.3 G e n e r a l i z e d D a r c y 1 s Law A s i m i l a r a n a l y s i s t o t h a t u s e d t o d e v e l o p t h e N e w t o n i a n f l u i d , E q u a t i o n 2-12, and power-law f l u i d , E q u a t i o n 2-21, has been made f o r t h e f l o w o f n o n - N e w t o n i a n f l u i d s w i t h y i e l d s t r e s s t h r o u g h p o r o u s m e d i a . I n e a c h c a s e a s u i t a b l e r h e o l o g i c a l model h a s t o be s e l e c t e d . T h i s s e l e c t e d model must r e p r e s e n t t h e b e h a v i o u r o f t h e f l u i d w i t h r e a s o n a b l e a c c u r a c y o v e r t h e r a n g e o f i n t e r e s t . F o r t h i s s t u d y , t h e H e r s c h e l - B u l k l e y model ( E q u a t i o n 2-6) was s e l e c t e d , b e c a u s e i t i s more g e n e r a l t h a n t h e power-law model ( E q u a t i o n 2-7) o r Bingham model ( E q u a t i o n 2 - 5 ) , s i n c e i t i n c l u d e s b o t h t h e power law b e h a v i o u r s and t h e y i e l d s t r e s s e s o f f l u i d s . -58-3.3.1 M o d e l d e v e l o p m e n t C o n s i d e r t h e v i s c o u s f l o w i n a p i p e o f a v i s c o u s n o n -N e w t o n i a n f l u i d whose r h e o l o g i c a l b e h a v i o u r i s d e s c r i b e d by t h e H e r s c h e l - B u l k l e y M o d e l ( 3 7 ) . T = T q+ H y n where T>T Q k- 2 - 6 The v e l o c i t y p r o f i l e f o r t h e f l o w i n a t u b e o f a n o n -N e w t o n i a n f l u i d w i t h y i e l d s t r e s s may be r e p r e s e n t e d s c h e m a t i c a l l y a s shown i n F i g u r e 3-1. Ap = p - p k o F i g u r e 3-1 S c h e m a t i c d i a g r a m o f t h e v e l o c i t y p r o f i l e i n a p i p e o f n o n - N e w t o n i a n f l u i d w i t h a y i e l d s t r e s s From a f o r c e b a l a n c e we f i n d 2TT R L T = IT R AP e 3-9 s u b s t i t u t e E q u a t i o n 2-6 i n t o E q u a t i o n 3-9 t o g e t - 5 9 -2TTRL ( H ^ ) 1 1 + T ) = TTR 2 A P e dR o d v dR 1,R A P . H 2 L O e 1/n 3-10 3-11 By i n t e g r a t i n g Equation 3-11, the v e l o c i t y p r o f i l e i s : V(R) = n (n+1)g x 1+1/n x 1+1/n ( a R - _°, _ ( a R c - -I) 3-12 such t h a t V(R ) = 0 a t R = R c c where a A P 2HL 3-13 The average v e l o c i t y i s determined as Rr <V> = n . • ^ 2TT R V(R) dR v o l u m e t r i c flow r a t e _ o ;  c r o s s s e c t i o n a l area « rRn _ 2TT c R dR o-* A f t e r i n t e g r a t i o n and s i m p l i f i c a t i o n we get: <V> = l+3n R c 1+1/n A P . o 2HL ~HR e c 1/n 3-14 3-15 Equation 3-15 was obtained i n c o l l a b o r a t i o n w i t h P r o f e s s o r H. P a s c a l of the U n i v e r s i t y o f A l b e r t a . D e t a i l s of the de-r i v a t i o n are shown i n Appendix A. Equation 3-15 r e p r e s e n t s i n a s i m p l i f i e d . f o r m the flow o f a shear t h i n n i n g or shear t h i c k e n i n g Bingham f l u i d i n l o n g s t r a i g h t c a p i l l a r y tubes. T h i s equation i s analogous -60-t o t h e H a g e n - P o i s e u i l l e e q u a t i o n f o r N e w t o n i a n f l u i d s , E q u a t i o n 2-11. F o r n = l and = o t h e n H = u and E q u a t i o n 3-15 w i l l be r e d u c e d t o H a g e n - P o i s e u i l l e e q u a t i o n R 2 <V> = ^ |* 2-11 8u L e F o r n o n - N e w t o n i a n f l u i d f l o w t h r o u g h p o r o u s m e d i a we s h a l l l o o k a t two d i f f e r e n t methods f o r i n t r o d u c i n g t h e t o r t u o s i t y . 3.3.2 F i r s t method ( P a s c a l ' s a p p r o a c h ) D e f i n e t h e ' s u p e r f i c i a l v e l o c i t y ' V q i n t e r m s o f t h e a v e r a g e v e l o c i t y a s (17, 61): <V> = 3-8 The e f f e c t i v e p r e s s u r e g r a d i e n t due t o s h e a r i s AP _ AP < L L L * L e e I f we i n t r o d u c e E q u a t i o n s 3-8 and d e t e r m i n e K by c o m p a r i s o n e q u a t i o n ) we f i n d 3-16 and 3-16 i n t o E q u a t i o n 2-11 w i t h E q u a t i o n 2-9 ( D a r c y ' s 3-17 Now s u b s t i t u t e E q u a t i o n s 3-8, 3-16 and 3-17 i n t o -61-E q u a t i o n 3-15 and t h e r e s u l t i s t h e f l o w e q u a t i o n f o r t h e f l o w o f a H e r s c h e l - B u l k l e y f l u i d AP = Q n + a L K A n 3-11 where e f f 1-n H i n 2 j (±- + 3) (8eK) 3-19 and a - (—) 1/2 _ T o T O = P/r 3-20 n 1-n -1-n £ £ £ - H , i + 3 ) " ( 8 E ) 2 R 2 3-21 In t h i s c a s e t h e r a t i o L / L e / w h i c h a l l o w s f o r t h e n o n - a x i a l component o f t h e p o r e , was i n t r o d u c e d on b o t h s i d e s o f t h e e q u a t i o n and c a n c e l l e d o u t . 3.3.3 S e c o n d method ( t h e a u t h o r ' s a p p r o a c h ) Let,us;afiow i n t r o d u c e t h e d e f i n i t i o n s o f s u p e r f i c i a l v e l o c i t y ( E q u a t i o n 3-7) and h y d r a u l i c r a d i u s ( E q u a t i o n 3-6) i n t o E q u a t i o n 3-15, t h e n V_ = en o 3n+l D £ 3 ( l - e ) 1+1/n r AP 3(1 2HC'L e D n l / n _ H J 3-22 -62-Now, i f the p e r m e a b i l i t y given by Equation 2-14a i s i n t r o d u c e d i n t o the m o d i f i e d Blake-Kozeny Equation 3-22 and i t i s s i m p l i f i e d the r e s u l t w i l l be: ^ = r# Q n + V 3 " 2 3 where n 1-n y e f f = I ( 3 + 1 / n ) ( 8 C ' e K ) 2 3 - 2 4 E C - 1 / 2 T ° Vpf f H N 2 2 = £ (3 + i/n) (8 C ' E ) K 3-26 1-n ' -1-n K 4In t h i s case the e f f e c t o f pore c o n f i g u r a t i o n as d e f i n e d by the t o r t u o s i t y , C , has been r e t a i n e d i n the model. The above equations can a l s o be presented i n the e n g i n e e r i n g form of a f r i c t i o n f a c t o r , — Reynolds number r e l a t i o n s h i p f o r packed beds. These equations w i l l be de-veloped i n the next s e c t i o n . I t i s worthwhile to i n d i c a t e t h a t Equation 3-22 can be c o n s i d e r e d as M o d i f i e d Blake-Kozeny equation f o r non-Newtonian f l u i d s w i t h y i e l d s t r e s s e s w h i l e Equations 3-18 and 3-23 can be c o n s i d e r e d as M o d i f i c a t i o n s o f Darcy's law. For example, a Newtonian f l u i d has n = 1, T = 0, so u e f f = H = y and both equations reduce to Darcy's law, -63-Equation 2-9. 3.4 M o d i f i e d F r i c t i o n F a c t o r - Reynolds Number C o r r e l a t i o n (the author's approach) To be c o n s i s t e n t w i t h other s t u d i e s i n t h i s area o f f l u i d mechanics, the flow behaviour w i l l be presented i n the u s u a l e n g i n e e r i n g form. T h i s form o f c o r r e l a t i o n can be used t o estimate the flow behaviour of d i f f e r e n t f l u i d s i n a wide range o f porous media. These c o r r e l a t i o n s normally have two dime n s i o n l e s s q u a n t i t i e s , a bed f r i c t i o n f a c t o r (f*) and bed Reynolds number (R*). B i r d (9) has d e f i n e d the f r i c t i o n f a c t o r as: F = A K f 3-27 where f i s a dime n s i o n l e s s q u a n t i t y known as the f r i c t i o n f a c t o r , F i s the f o r c e a s s o c i a t e d w i t h the k i n e t i c behaviour of the f l u i d , A i s a c h a r a c t e r i s t i c area, and K i s a c h a r a c t e r i s t i c k i n e t i c energy per u n i t volume. Blake (10) and Ergun (26) were the f i r s t to propose the f o l l o w i n g d e f i n i t i o n o f f r i c t i o n f a c t o r f o r porous media, which gave s a t i s f a c t o r y agreement wi t h experimental r e s u l t s and has gained wide acceptance f o r Newtonian and non-Newtonian f l u i d s f l o w i n g through porous media. D _P_ 2-16 1-e -64-The generalized Darcy's law Equation 3-22 may be rearranged to AP L v 1 1 o D e K3n+l' v3(l-e) n+1 .3 (1-e) T p 2C'H 3-28 For viscous flow f* = 2-15 If Equation 3-28 i s substituted into Equation 2-16 for (AP/L) and applying Equation 2-15 after s i m p l i f i c a t i o n and manipulation, we get the d e f i n i t i o n of a Modified Reynolds number as: Re* 12 p V 5 K o * 2HD e 3 V n + e 2 T % p o o 3-29 where n * D e C = 6 ( ^ ) 3n+l ; v3(l-e) n+1 ) (1-e) 3-30 and C" = 72 C 3-31 The permeability expression equation (2-l4a) was not substituted-for the-permeability term' (K) as was found by other investigators (15, 16) for non-Newtonian flow through porous m e d i a and i t i s a l s o a u n i q u e number w h i c h i n c l u d e s t h e e f f e c t o f t h e y i e l d s t r e s s (T ) i n t h e d e n o m i n a t o r . A l s o i t i s a more g e n e r a l f o r m o f t h e R e y n o l d s . number s i n c e i f T q = 0 and n = 1 i t w i l l r e d u c e t o E q u a t i o n 2-17, f o r t h e f l o w o f N e w t o n i a n f l u i d s t h r o u g h p o r o u s m e d i a . The a v e r a g e v a l u e o f C" w i l l be d e t e r m i n e d e x p e r i m e n t a l l y . A l l t h e a n a l y s e s d e v e l o p e d i n t h i s C h a p t e r a r e b a s e d on t h e same a s s u m p t i o n s u s e d i n f o r m u l a t i n g D a r c y ' s law S e c t i o n 2.9.2, page ( 4 2 ) , e x c e p t a s s u m p t i o n number one where t h e f l u i d s h e r e a r e n o n - N e w t o n i a n . The a s s u m p t i o n s u n d e r l y i n g t h e d e r i v a t i o n o f t h e Kozeny e q u a t i o n ( E q u a t i o n 2-12) and i t s c o n v e r s i o n f r o m a c a p i l l a r y t u b e model t o a p o r o u s m e d i a model were d i s c u s s e d by S c h e i d e g g e r ( 7 7 ) . -66-CHAPTER FOUR E X P E R I M E N T A L O B J E C T I V E S A N D F L U I D S E L E C T I O N B e f o r e a model c a n be a c c e p t e d i t s h o u l d be p r o v e n e x p e r i m e n t a l l y . To d e t e r m i n e b o t h t h e f l o w b e h a v i o u r i n a p o r o u s m e d i a o f a H e r s c h e l - ' B u l k l e y f l u i d and a l s o t o c h a r a c t e r i z e i t r h e o l o g i c a l l y s e v e r a l s t e p s were n e c e s s a r y . a) D e v e l o p p r o p e r t e c h n i q u e s o r e q u i p m e n t t o measure t h e r h e o l o g i c a l f l u i d b e h a v i o u r s , s u c h a s , x, x , f , H, and n. b) Make o r s e l e c t a s u i t a b l e f l u i d o r m i x t u r e w h i c h c a n behave a c c o r d i n g t o H e r s c h e l - B u l k l e y m odel ( E q u a t i o n 2 - 6 ) ; i . e . , t h e f l u i d s h o u l d have a y i e l d s t r e s s and s h e a r t h i n n i n g o r t h i c k e n i n g b e h a v i o u r . c) Once t h e f l u i d i s c h o s e n , a f l o w s y s t e m must be d e s i g n e d and c o n s t r u c t e d t o t e s t t h e f l o w o f t h e s e l e c t e d f l u i d t h r o u g h p o r o u s m e d i a . I n c a r r y i n g o u t t h e above t h r e e p o i n t s , i t was d e c i d e d f i r s t t o d e v e l o p a method t o t e s t d i f f e r e n t k i n d s o f s o l u t i o n s -67-f o r y i e l d s t r e s s b e h a v i o u r , s i n c e t h e new v i s c o m e t e r ( R o t o v i s c o RV-12) was n o t a v a i l a b l e u n t i l n e a r t h e end o f t h e s t u d y . One o f t h e methods u s e d , was t o a l t e r a t o r s i o n b a l a n c e f o r y i e l d v a l u e (T ) measurement. The p r i n c i p l e o f t h i s method i s t h a t s h e e t s o f d i f f e r e n t s i z e s and s h a p e s a r e s u s p e n d e d and b a l a n c e d i n t h e f l u i d ( o i l ) a t t h e d e s i r e d t e m p e r a t u r e by a t h r e a d f o r a c h o s e n p e r i o d o f t i m e . Then an upward f o r c e i s a p p l i e d on t h e s h e e t g r a d u a l l y u n t i l i t s t a r t s m o v i n g . The y i e l d v a l u e i s c a l c u l a t e d f r o m v T q = F o r c e A p p l i e d / T o t a l S u r f a c e A r e a o f t h e S h e e t . The d i s a d v a n t a g e o f t h i s method, i s t h a t t h e v a l u e o f y i e l d s t r e s s (T ) i s v e r y s e n s i t i v e t o t h e s p e e d a t w h i c h t h e f o r c e i s a p p l i e d and a l s o d e p ends on t h e shape and s i z e o f t h e s h e e t u s e d . • The s e c o n d method d e v e l o p e d f o r y i e l d v a l u e measurements u s e d d i f f e r e n t s i z e s and l e n g t h s o f g l a s s U - t u b e s immersed i n a c o n s t a n t t e m p e r a t u r e b a t h . The p r i n c i p l e o f t h i s method i s t h a t t h e sample ( o i l - w a x m i x t u r e ) i s k e p t i n a c o n s t a n t t e m p e r a t u r e b a t h f o r a c e r t a i n p e r i o d o f t i m e , t h e n f o r c e i s a p p l i e d g r a d u a l l y a t one end w i t h t h e o t h e r end open t o t h e a t m o s p h e r e , u n t i l f l o w o f o i l b e g i n s . F l o w was o b s e r v e d v i s i b l y b y m a r k i n g t h e o i l l e v e l . The p r e s s u r e d r o p r e q u i r e d t o s t a r t t h e f l o w was m e a s u r e d by means o f a manometer. The y i e l d s t r e s s (T ) i s c a l c u l a t e d f r o m x = AP D/4L o 2-8 -68-T h i s method i s w i d e l y a c c e p t e d by many i n v e s t i g a t o r s i n o i l r e s e a r c h l a b s ( 9 3 ) , b u t we a l s o d i s c o v e r e d t h a t t h e me a s u r e d y i e l d v a l u e d e pends on t h e r a t i o (L/D) o f t h e U-tube u s e d . A s c h e m a t i c d r a w i n g i s shown b e l o w o f t h e s e two methods 1*' P i s t o n /l^y^£orce s c a l e c o o l i n g J>.-w a t e r t h r e a d s h e e t j a c k e t mano-m e t e r c o o l i n g w a t e r -> — -> U-t u b e (a) (b) F i g u r e 4-1 S c h e m a t i c d r a w i n g f o r y i e l d v a l u e measurements a) s u s p e n d e d s h e e t method b) U - t u b e method --69-The d a t a f r o m t h e s e two methods were n o t u s e d i n t h e c o r r e l a t i o n s , b u t t h e y were u s e d i n t e s t i n g and s e l e c t i n g t h e o i l s u s e d i n t h e f l o w t e s t s , and t h e c o n d i t i o n s u n d e r w h i c h t h e o i l s f o r m a s t r u c t u r e and r e s i s t f l o w ( i . e . h a v e y i e l d s t r e s s e s ) . A l s o , t h e y were u s e d t o e s t i m a t e t h e a g i n g t i m e r e q u i r e d f o r s t r u c t u r e t o f o r m . D u r i n g t h e s e l e c t i o n o f t h e t e s t f l u i d s and, l a t e r , d u r i n g t h e a c t u a l f l o w measurements i t was n e c e s s a r y t o a c c u r a t e l y d e t e r m i n e t h e s h e a r s t r e s s - s h e a r r a t e c u r v e s u n d e r t h e t e s t c o n d i t i o n s . E q u i p m e n t a v a i l a b i l i t y c a u s e d many p r o b l e m s d u r i n g t h e e a r l y p a r t o f t h i s s t u d y . The R o t o v i s c o RV1 v i s c o m e t e r i n t h e d e p a r t m e n t a f t e r 17 y e a r s was no l o n g e r a c c u r a t e . The RV3 m odel w h i c h was k i n d l y l e n t by Dr. M o d i i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g was newer and more a c c u r a t e b u t was o n l y a v a i l -a b l e f o r s h o r t p e r i o d s and was n o t e q u i p p e d w i t h a c o o l i n g s y s t e m . Thus t h e low t e m p e r a t u r e a g i n g p e r i o d s n e e d e d were n o t a t t a i n a b l e . F i n a l l y , i n February,: 1983 a new R o t o v i s c o RV-12 was p u r c h a s e d . W i t h t h i s i n s t r u m e n t t h e r e q u i r e d a c c u r a c y and a g i n g p e r i o d s c o u l d be a t t a i n e d and a l l t h e r h e o l o g i c a l m easurements were t h e n r e p e a t e d . The p r o c e d u r e u s e d w i t h t h i s i n s t r u m e n t a r e d e s c r i b e d i n S e c t i o n 5.1.1. S i n c e r h e o l o g i c a l e x p e r i m e n t s were c a r r i e d o u t , a f t e r t h e f l o w s y s t e m t e s t s , b e c a u s e o f t h e d e l a y i n g e t t i n g t h e new v i s c o m e t e r , a b o u t 300 ml o f o i l was t a k e n f r o m t h e -70-s a m p l i n g l i n e d u r i n g e a c h f l o w t e s t and s t o r e d u n t i l t h e new v i s c o m e t e r was a v a i l a b l e . 4.1 F l u i d S e l e c t i o n S i n c e t h e p h y s i c a l p r o p e r t i e s o f c r u d e o i l s t e n d t o change when s t o r e d i n c o n t a c t w i t h a i r , and a l s o s i n c e t h e y a r e complex m i x t u r e s w h i c h a r e h a r d t o d e f i n e , i t was d e c i d e d t o t e s t t h e m o d e l s f i r s t w i t h a r t i f i c i a l f l u i d s w i t h t h e c o r r e c t r h e o l o g i c a l b e h a v i o u r . S i n c e e v e r y c r u d e o i l w h e t h e r h e a v y o r l i g h t h a s some wax c o n t e n t ( s e v e r a l e xamples o f t h e s e o i l s a r e g i v e n i n S e c t i o n 2.7 o f t h i s t h e s i s ) , i t was d e c i d e d t o f i r s t t e s t a r t i f i c i a l o i l - p a r a f f i n s o l u t i o n s . When t h e model had b e e n r e f i n e d , i t was t e s t e d w i t h a waxy c r u d e o i l - f r o m t h e - P e a c e R i v e r f i e l d i n A l b e r t a . S y n t h e t i c n o n - N e w t o n i a n o i l s have p r e v i o u s l y b een made u s i n g l u b r i c a t i n g o i l s a s b a s e and, t o e n a b l e t h e d e v e l o p -ment o f a m a c r o m o l e c u l a r s t r u c t u r e , m a t e r i a l s s u c h as f i n e l y d i v i d e d c a r b o n p a r t i c l e s , p a r a f f i n waxes, p o l y m e r s o r complex o r g a n i c s a l t s h a v e b e e n a d d e d ( 6 ) . I n t h i s s t u d y p a r a f f i n i c o i l s w i t h a w i d e r a n g e o f y i e l d v a l u e s and f l o w b e h a v i o u r s were p r o d u c e d by m i x i n g p a r a f f i n i c wax ( P a r v a n 55 f r o m I m p e r i a l O i l Co., T a b l e F-3) a t c o n c e n t r a t i o n s o f 2.5, 4 and 5% wt i n two t y p e s o f o i l ( C l a r u s - B and C l a r u s - C f r o m S h e l l O i l L t d . , T a b l e s F - l , F - 2 ) w h i c h h a v e d i f f e r e n t N e w t o n i a n v i s c o s i t e s and d e n s i t i e s . -71-The p a r a f f i n waxes e n a b l e t h e m i x t u r e t o d e v e l o p a m a c r o m o l e c u l a r c r y s t a l l i n e s t r u c t u r e w h i c h c a u s e s t h e non-i d e a l Bingham p l a s t i c b e h a v i o u r . C r y s t a l s i z e and shape o f t h e wax c r y s t a l s aire a f u n c t i o n o f t h e h e a t i n g and c o o l i n g h i s t o r y o f t h e f l u i d . However, i f t h e f l u i d i s m a i n t a i n e d a t a s p e c i f i c t e m p e r a t u r e f o r a p e r i o d o f t i m e t h e c r y s t a l s w i l l t a k e t h e s i z e and shape c h a r a c t e r i s t i c s a p p r o p r i a t e f o r t h a t t e m p e r a t u r e . An a g i n g t i m e o f a t l e a s t 2 h o u r s was n e c e s s a r y f o r o u r m i x t u r e s t o f o r m a m a c r o m o l e c u l a r s t r u c t u r e - b u t i n o u r t e s t s t h e samp l e s were m a i n t a i n e d a t t h e t e s t t e m p e r a t u r e f o r more t h a n 12 h o u r s i n t h e v i s c o -m e t e r and i n t h e s a n d column, b e f o r e any measurement was made. To s t u d y as w i d e a r a n g e o f r h e o l o g i c a l p a r a m e t e r s as t h e p r e s s u r e r a t i n g o f t h e e q u i p m e n t w o u l d a l l o w , t e s t s were made a t 2 ° , 8 ° , 1 0 ° , . 1 2 ° , 1 4 ° , 1 6 ° , 18® and 20°C. -72-CHAPTER FIVE E X P E R I M E N T A L A P P A R A T U S 5.1 R h e o l o g i c a l E x p e r i m e n t s The g o o d n e s s o f f i t o f t h e e x p e r i m e n t a l d a t a and t h e f l o w model d e v e l o p e d i n C h a p t e r F o u r d e pends on t h e c l o s e a g r e e m e n t between t h e H e r s c h e l - B u l k l e y model and t h e v i s c o m e t r i c d a t a f o r t h e t e s t f l u i d . The s h e a r s t r e s s -s h e a r r a t e d a t a were d e t e r m i n e d u s i n g a r o t a t i o n a l v i s c o -m e t e r . 5.1.1 The V i s c o m e t e r (HAAKE R o t o v i s c o RV-12) The v i s c o m e t e r u s e d i n t h i s t h e s i s t o measure s h e a r s t r e s s , s h e a r r a t e and y i e l d v a l u e o f t h e waxy o i l s was a HAAKE R o t o v i s c o M o d e l RV-12, a r o t a t i n g bob t y p e c o a x i a l c y l i n d e r v i s c o m e t e r . A s c h e m a t i c d i a g r a m o f t h e s e t - u p , i s shown i n F i g u r e 5-1. The o u t e r s t a t i o n a r y c y l i n d e r (cup) i s s u r r o u n d e d by a t e m p e r a t u r e c o n t r o l l e d j a c k e t o f w a t e r . I n s i d e t h e c u p i s an i n n e r c y l i n d e r (bob) o f s m a l l e r d i a m e t e r w h i c h r o t a t e s t o s h e a r t h e m a t e r i a l i n t h e gap between cup and bob. -73-... .. F i g u r e 5-1 Schemat ic d iagram o f the Ro tov i s co RV-12 set-ups (78) © Basic instrument ROTOVISCO RV 12 © Recorder: xy/t ® Speed programmer PG 142 j © Measuring-drive-units: M 150, M 500, M 1500 - choose one or more to cover the full range of your samples. © Stand © Temperature vessel , • : < 3 r " ' •"' ® Thermal liquid constant temperature circulator. A refrigerated circulator model is best suited for viscosity measurements at or below room tempera-ture. © Sensor system: 40 alternatives to choose from for optimal test conditions and results. ' " 's -74-T h e r e a r e many c o m b i n a t i o n s p o s s i b l e i n s i z e and t y p e o f bobs and c u p s f o r t h i s v i s c o m e t e r . The s e t o f NV cup and bob was c h o s e n b e c a u s e i t h a s t h e b i g g e s t s u r f a c e a r e a o f any HAAKE c o a x i a l c y l i n d e r s e n s o r s y s t e m , h e n c e t h e g r e a t e s t s e n s i t i v i t y . As shown i n F i g u r e 5-2, i t i s b e l l s h a p e d and s h e a r s t h e sample on b o t h t h e i n s i d e and o u t s i d e . T h i s b i g s u r f a c e a r e a has e n a b l e d us t o measure t h e y i e l d v a l u e a t t h e l o w e s t r o t o r s p e e d . The RV-12 v i s c o m e t e r has 30 f i x e d r o t o r s p e e d s ( f r o m 0.01 t o 512 min "*") t o c h o o s e f r o m o r by u s i n g t h e s p e e d p r o -grammer (PG-142), t h e s p e e d may be i n c r e a s e d o r d e c r e a s e d c o n t i n u o u s l y and s m o o t h l y i n a p r e d e t e r m i n e d p r o g r a m . The i n s t r u m e n t u s e d i n t h e s e s t u d i e s h as a maximum t o r q u e o f 150 cm. gm. The a c c u r a c y o f t h e s h e a r s t r e s s measurements o f t h i s d e v i c e i s a b o u t ± 2 % , w h i c h depends on many f a c t o r s s u c h asy t h e s i z e o f t h e s t r e s s s i g n a l S, gap s i z e , t h e t e m p e r a t u r e c o n t r o l and t h e r o t o r d e s i g n . 5.1.2 V i s c o s i t y , S h e a r S t r e s s and S h e a r R a t e Measurement Two i n s t r u m e n t f a c t o r s (A' and M) a r e needed t o c a l c u l a t e t h e t h r e e p a r a m e t e r s ( v i s c o s i t y , , ( y ) / s h e a r s t r e s s (T) and s h e a r r a t e (j)) f r o m t h e f o l l o w i n g t h r e e e q u a t i o n s : -75-SENSOR SYSTEM NV Rotor (BOB) 1 r a d i u s R2; R 3 (mm) he i gh t L (mm) 17 ,85 ; 20,1 60 t STATOR (CUP) ; \ r a d i u s R] ; R^  (mm) 17,5 ; 20 ,5 i RADII RATIO R a / R i 1,02 I "- SAMPLE VOLUME V ( cm 3 ) 9 . -Ri « I i TEMPERATURE: max. (°C) m i n . (°C) 150 - 30 •—in CALCULATION FACTORS A ( P a / s c » l e g r a d . ) M (m in / s ) G (mPa - s / s c a l e g r a d . -m i n ) 0,5336 5,41 98,63 Viscosity 7] [mPa«] 10° 10' 10" 10T 10* 10' K) 4 ~1 —1 ' — I ' — I ' — I • — I r—1 1 l» 10"1 10"' 10° 10' K) 1- 10' 10* tO* Shear rale [«~'J F i g u r e 5-2 Dimensions and range of measurement o f sensor system NV (44) -76-S h e a r S t r e s s S h e a r R a t e V i s c o s i t y x = A'.S ( d y n e / c m ) 5-1 y = M.n' ( s e c - 1 ) 5-2 y = T/Y (m Pa.s o r cp) 5-3 where S and n' a r e t h e s h e a r s t r e s s i n d i c a t o r r e a d i n g and t h e r o t o r s p e e d r e s p e c t i v e l y . The u n i t has 10 b a s i c s p e e d s f o r n 1 (1, 2, 4, 8,16, 32, 64, 128, 256, 512 min "*") , and e a c h s p e e d c a n be r e d u c e d by a f a c t o r o f 10 o r 100, w h i c h g i v e s t h e 30 v a l u e s f o r n 1 . The s h e a r s t r e s s i n d i c a t o r S has a s c a l e r a n g e o f 1 t o 100 and e a c h r e a d i n g c a n be i n c r e a s e d by a f a c t o r o f 3 f o r s m a l l v a l u e s o f S. F o r r o t a t i o n a l v i s c o m e t e r s w i t h a c o a x i a l c y l i n d e r s e n s o r s y s t e m , s h e a r r a t e c a n be m a t h e m a t i c a l l y d e f i n e d a s : R Y, = 2 0) 2 2 R -RT a l 5-4 where 0) = Y i . = 2TT • n' 60 15 R 2 2 R Z - RT a I n 5-5 5-6 o r s i m p l y •y^ = M « n ' (s "*") 5-2 -77-where R. = 1 R = a 60 = n' = M = shear r a t e a t r a d i u s R^ (s r a d i u s o f the i n n e r c y l i n d e r - r o t o r (cm) r a d i u s of the outer c y l i n d e r - c u p (cm) angular v e l o c i t y ( r a d i a n per second) -1 -1 r o t o r speed (min ) shear r a t e f a c t o r (sec-rpm) J"; i t combines a l l f a c t o r s given i n the b r a c k e t [ ] i n Equation 5-6; i . e . i t depends o n l y on the r a d i i of cup and r o t o r . The shear s t r e s s a t the r o t o r r a d i u s R^ can be c a l c u l a t e d from the torque on the bob as measured by the m e a s u r i n g - d r i v e - u n i t . The s i g n a l " S " , p r o p o r t i o n a l to the torque, i s e i t h e r recorded on a X-time p l o t t e r or i n d i c a t e d on the b a s i c instrument RV-12. x . = Md 2TT • h • R. l 2 T T . h - R l Md 5-7 x . = f• Md i 5-8 f = 2Tr-h-R: = f. [Md L S J < 5-9 5-10 a' = Md 5-11 = f - a, s 5-12 -78-x . f - a * A 1 -S 5-13 5-1 where 2 = shear s t r e s s a t the r a d i u s (Pa or dyne/cm ) = torque (dyne, cm) h = h e i g h t of the r o t o r (cm) R. = r a d i u s of the r o t o r (cm) l -3 f = shape f a c t o r (cm ) - i t combines a l l f a c t o r s i n equation 5-7 i n the above b r a c k e t {}. I t i s a constant f o r a p a r t i c u l a r sensor system and d e f i n e d by the geometry of the r o t o r . a' = the "compliance" f a c t o r i n the b r a c k e t Uc/ d e f i n e s the l i n e a r r e l a t i o n s h i p between the torque Md and the S - s i g n a l and c h a r a c t e r i z e s , the measuring s p r i n g i n the m e a s u r i n g - d r i v e - u n i t . I t i s a c o n s t a n t value obtained by c a l i b r a t i o n as i n -d i c a t e d by the s u b s c r i p t c. A 1 = shear s t r e s s f a c t o r . I t combines both f a c t o r s above, c h a r a c t e r i z i n g the instrument w i t h r e s p e c t to sensor system and measuring d r i v e u n i t . The shear s t r e s s of the f l u i d a c t i n g on a r a d i u s R^ when s u b j e c t e d to a shear r a t e y^, i s simply c a l c u l a t e d by m u l t i p l y i n g the s i g n a l "S" by the shear s t r e s s f a c t o r "A 1" (Equation 5-1). The Newtonian v i s c o s i t y of a t e s t sample i s c a l c u l a t e d by d i v i d i n g the shear s t r e s s by the p r e s e t shear r a t e . -79 -U = T./Y• 5-3 The values of A' and M for sensor system NV were given i n the l i t e r a t u r e that comes with the viscometer (Figure 5-2), and were ca l i b r a t e d i n the laboratory by using standard v i s c o s i t y sample Canon S-200 . 5.1.3 Experimental Procedure The viscometer (Rotovisco RV-12) with the NV attachments was c a l i b r a t e d against a c e r t i f i e d standard v i s c o s i t y o i l (Canon S-200) each time before i t was used for measurements. The shear stress-shear rate measure-ments were made on the solutions of p a r a f f i n wax i n o i l , and the crude o i l samples which were taken from the sampling l i n e of the flow system. Zero adjustment and maximum in d i c a t i o n adjustments of the measuring drive unit were made before c a l i b r a t i o n with the standard o i l by setting the s e n s i t i v i t y E=0.3, speed n=64 min \ reduction R=l and damping D=0 (44). After repeating t h i s adjustment, the instrument i s ready for making measurements. Listed below i s the step-by-step program followed i n a l l experiment. 1) The viscometer was turned on. A f i f t e e n minute warm-up period was allowed for the e l e c t r o n i c s . The bob was mounted on the measuring drive unit, and allowed to turn f r e e l y i n a i r at a speed of 64 rpm -80-(n=64 min ) w h i l e t h e z e r o a d j u s t m e n t o f t h e b a s i c u n i t was made by t u r n i n g t h e knob, w h i c h s e r v e s t o c o r r e c t t h e z e r o s e t t i n g f o r t h e d i g i t a l c o u n t e r . 2) D e p r e s s p u s h b u t t o n "STOP". 3) A q u a n t i t y o f t h e sample (9 ml o f waxy o i l ) were p l a c e d i n t h e NV-cup and i n s e r t e d i n t h e v i s c o m e t e r a t room t e m p e r a t u r e and t h e l i d was p l a c e d on t h e c u p . 4) The w a t e r c i r c u l a t o r w h i c h c o n t r o l s t h e t e m p e r a t u r e was s e t t o t h e d e s i r e d t e m p e r a t u r e t h e n t u r n e d o n . 5) The sample was aged f o r a t l e a s t 12 h o u r s t o a l l o w t h e w a x - o i l s o l u t i o n t o e q u i l i b r a t e i n t h e v i s c o -m e t e r a t t h e d e s i r e d t e m p e r a t u r e . 6) A f t e r a g i n g , t h e v i s c o m e t e r was s e t a t t h e l o w e s t s p e e d (0.01 min ^) and a r e a d i n g was t a k e n o f t h e s i g n a l S w h i c h r e p r e s e n t s t h e y i e l d v a l u e (T ). 7) The s p e e d was t h e n i n c r e a s e d g r a d u a l l y and t h e e q u i l i b r i u m v a l u e o f t o r q u e (S) a t e a c h s p e e d was r e -c o r d e d . T h i s p r o c e d u r e was r e p e a t e d u n t i l e i t h e r t h e maximum s p e e d o f 512 min 1 was r e a c h e d o r t h e d i g i t a l •, c o u n t e r s i g n a l ("S") e x c e e d e d maximum s c a l e r e a d i n g . 5.2 F l o w E x p e r i m e n t s I n t h i s s e c t i o n , t h e f l o w s y s t e m w i l l be d e s c r i b e d i n d e t a i l . The p r o p e r t i e s o f t h e f l u i d s and a l l m a t e r i a l s u s e d i n t h e s e e x p e r i m e n t s w i l l be m e n t i o n e d . A d e s c r i p -t i o n o f t h e e x p e r i m e n t a l a p p a r a t u s d e s i g n as w e l l a s an -81-a c c o u n t o f t h e e x p e r i m e n t a l p r o c e d u r e and p r o b l e m s w i l l be e x p l a i n e d . 5.2.1 M a t e r i a l s T h r e e t y p e s o f f l u i d s (waxy o i l s ) were t e s t e d i n t h e f l o w s y s t e m i n two d i f f e r e n t s i z e s o f p a c k e d c o l u m n s . Two o f t h e s e f l u i d s were l u b r i c a t i n g o i l s ( C l a r u s - B and C l a r u s - C ) f r o m S h e l l O i l L t d . , w h i c h have d i f f e r e n t N e w t o n i a n v i s c o s i t i e s and d e n s i t i e s , t h e p h y s i c a l p r o -p e r t i e s o f t h e s e o i l s a r e shown i n t h e A p p e n d i x i n T a b l e s F - l and F-2. P a r a f f i n wax ( P a r v a n 55 f r o m I m p e r i a l O i l Co., T a b l e F-3 shows t h e p h y s i c a l p r o p e r t i e s o f t h i s wax) was added t o t h e two l u b r i c a t i n g o i l s a t t h r e e d i f f e r e n t c o n c e n t r a t i o n s (2.5, 4 and 5% w t ) . The t h i r d o i l t e s t e d was c r u d e o i l f r o m t h e P e a c e . R i v e r f i e l d i n A l b e r t a . I t s p h y s i c a l p r o p e r t i e s a r e shown i n T a b l e F-4. The p a r t i c l e s u s e d f o r t h e bed p a c k i n g were, Ottawa R i v e r s a n d o f s i z e r a n g e -20+25 U.S. s t a n d a r d s i e v e a n a l y s i s ( a v e r a g e p a r t i c l e d i a m e t e r 0.077 cm) i n t h e s m a l l column (91 cm l o n g , D c = 4.83 cm) and a s i z e r a n g e -14+16 U.S. s t a n d a r d s^ieve a n a l y s i s ( a v e r a g e p a r t i c l e d i a m e t e r 0.128 cm) i n t h e l a r g e c o l u m n . (100 cm l o n g , D c = 10.16 cm). T h i s gave a r a t i o o f Dc/D^= 62.7 f o r t h e s m a l l c o l u m n , and a r a t i o o f D /D = 7 9 . 4 f o r c p t h e l a r g e c o l u m n . T h e s e two column t o p a r t i c l e d i a m e t e r r a t i o s gave p o r o s i t i e s o f 0.36 and 0.44 f o r t h e s m a l l -82-and l a r g e c o l u m n r e s p e c t i v e l y . P o r o s i t i e s was me a s u r e d by w a t e r d i s p l a c e m e n t . The two columns were c o n s t r u c t e d o f P e r s p e x t u b i n g , w h i l e t h e p i p i n g s y s t e m was made o f c o p p e r t u b i n g o f 3/4" and 1/2" d i a m e t e r . A l l t h e t u b i n g was i n s u l a t e d t o r e d u c e t h e h e a t l o s s e s . A n t i f r e e z e s o l u t i o n was u s e d i n t h e two c o n s t a n t t e m p e r a t u r e b a t h s t o p r e v e n t f r e e z i n g a t l o w e r t e m p e r a t u r e s . 5.2.2 E q u i p m e n t A s c h e m a t i c d i a g r a m o f t h e f l o w s y s t e m i s p r e s e n t e d i n F i g u r e 5-3. B a s i c a l l y , t h e f l u i d t o be t e s t e d was pumped f r o m a 60 l i t r e c a p a c i t y r e s e r v o i r by a g e a r pump, a p a r t g o i n g t o t h e t e s t s e c t i o n , i . e . t h e p a c k e d c o l u m n , and t h e r e s t o f t h e f l u i d was c i r c u l a t e d b a c k t o t h e r e s e r v o i r t o keep t h e mass s t i r r e d and t o m a i n t a i n a c o n s t a n t o i l t e m p e r a t u r e . The t e m p e r a t u r e o f t h e i n l e t s t r e a m t o t h e column was m e a s u r e d by a D i g i t a l Omega the r m o m e t e r Model 199-P2 w i t h a RTD t y p e s e n s o r and r e s i s t a n c e t h e r m o m e t e r p r o b e (RTD), w h i c h was a c c u r a t e t o ± 0.1°C. The o u t l e t s t r e a m t e m p e r a t u r e was me a s u r e d by a c a l i b r a t e d m e r c u r y t h e r m o -m e t e r . A l s o two c a l i b r a t e d m e r c u r y t h e r m o m e t e r s were u s e d t o measure t h e i n l e t and o u t l e t t e m p e r a t u r e s o f t h e c o o l i n g w a t e r i n t h e j a c k e t o f t h e co l u m n . Two' j a c k e t e d p e r s p e x c o l u m n s were u s e d i n t h i s s t u d y . E a c h column was c o n n e c t e d i n t u r n t o t h e f l o w s y s t e m . T h e s e c o l u m n s were f i l l e d w i t h s a n d and s e t 1£ (reservoir) pressurized vessel drainage feed tank temperature controller high pressure a i r ; ' . cylinder waxy oi l \>-thermometer pump s o constant temp, bath gear pump sampling line ,^ , thermo.couple vent jacket--sand column (porous media) regulator thermometer transducer cooling water O O recorder oooo o drainage constant temp, bath Figure 5-3 Schematic diagram of the flow test equipment -84-i n a v e r t i c a l p o s i t i o n t o a v o i d s e t t l i n g o f t h e s a n d i n t h e d i r e c t i o n p e r p e n d i c u l a r t o f l o w . U p l i f t i n g o f t h e s a n d was p r e v e n t e d by a t o p r e t a i n e r s c r e e n . The i n t e r n a l d i a m e t e r s were 4.83 cm and 10.16 cm, w i t h p r e s s u r e t a p s e p a r a t i o n o f 91 and 100 cm r e s p e c t i v e l y . E a c h c o l u m n was s u r r o u n d e d by a p e r s p e x t u b i n g w a t e r j a c k e t i n w h i c h c o n s t a n t t e m p e r a t u r e w a t e r was c i r c u l a t e d . F i g u r e s 5-4 and 5-5 show t h e d e t a i l s o f t h e d e s i g n o f t h o s e c o l u m n s . The t o p and b o t t o m f l a n g e s were g a s k e t e d w i t h 0 - r i n g s and b o l t e d t o t h e c o n n e c t i o n p i e c e s w h i c h h o l d t h e s c r e e n s i n p l a c e . The m a i n d i f f e r e n c e i n t h e d e s i g n o f t h e two c o l u m n s was t h e l o c a t i o n o f t h e two p r e s s u r e t a p s . They were c o n n e c t e d t h r o u g h t h e j a c k e t i n t h e s m a l l c o l u m n , and t h r o u g h t h e end f l a n g e s i n t h e l a r g e r c o l u m n t o a v o i d d i r e c t c o n t a c t between t h e s e t u b e s and t h e c o o l i n g w a t e r w h i c h c o u l d c a u s e wax s o l i d i f i c a t i o n a t t h e e n t r a n c e o f t h o s e t u b e s . A l s o two s c r e e n s were a t t a c h e d t o t h e ends o f p r e s s u r e t a p s ( i n s i d e t h e column) t o p r e v e n t t h e sand f r o m e n t e r i n g t h e " t r a n s d u c e r t u b e s . A cone o f 6.0 cm b a s e and 1.2 7 cm h e i g h t i n t h e s m a l l c o l u m n (10.16 cm b a s e and 2.54 cm h e i g h t i n t h e l a r g e column) were d r i l l e d i n e a c h f l a n g e f a c i n g t h e s a n d bed t o p r o v i d e a c o l l e c t i o n a r e a f o r t h e f l u i d b e f o r e e n t e r i n g and a f t e r l e a v i n g t h e s a n d b e d , so t h a t t h e f l o w i n t o and o u t o f t h e s a n d bed i s a c r o s s t h e e n t i r e c r o s s - s e c t i o n . The d i s t a n c e between t h e b o t t o m p r e s s u r e t a p s and t h e l o w e r s c r e e n s were 5.5 and 2.0 cm f o r t h e J s m a l l a n d l a r g e column r e s p e c t i v e l y . -85-i n l e t s t r e a m Dimensions in centimeters F i g u r e 5-4 D e t a i l s o f the sma l l sand column d e s i g n -86-24 .5 i outlet I inlet all dimensions ; cm. F i g u r e 5-5 D e t a i l s o f the l a r g e sand column d e s i g n -87-E a c h column was f i l l e d w i t h s a n d i n e s s e n t i a l l y t h e same manner. The col u m n was f i r s t a s s e m b l e d and t h e b o t t o m f l a n g e was b o l t e d t o g e t h e r w i t h t h e column i n a v e r t i c a l p o s i t i o n . Sand was p o u r e d i n t o t h e col u m n t o a h e i g h t o f a b o u t 5 cm. The p a r t i a l l y f i l l e d b e d was t h e n tamped w i t h a p l u n g e r . T h i s p r o c e s s was r e p e a t e d f o r e a c h 5 cm o f bed h e i g h t u n t i l t h e b e d was c o m p l e t e . The bed was j u s t b a r e l y o v e r f i l l e d so t h a t i t w o u l d be f i x e d r i g i d l y when t h e r e t a i n i n g s c r e e n was t i g h t e n e d down. By t h i s method a s t a b l e b e d was p r o d u c e d . The c o l u m n t h e n was i n s e r t e d i n t o t h e f l o w s y s t e m and s l o w l y f i l l e d w i t h t h e t e s t o i l a t room t e m p e r a t u r e . A l l a i r b u b b l e s were f l u s h e d f r o m t h e bed t h r o u g h t h e v e n t a t t h e t o p o f t h e c o l u m n , b e f o r e t h e t e s t s were made. Water a t ±0.1°C o f t h e t e s t t e m p e r a t u r e was t h e n c i r c u l a t e d t h r o u g h t h e j a c k e t f r o m a s e c o n d c o n s t a n t t e m p e r a t u r e b a t h . Two r o t o m e t e r s were a t t a c h e d t o t h e o u t l e t s t r e a m t o i n d i c a t e s t e a d y s t a t e f l o w and one r o t a m e t e r was c o n -n e c t e d i n t h e c i r c u l a t i n g l i n e t o c h e c k t h e c i r c u l a t i n g r a t e . The f l o w r a t e t h r o u g h t h e column was m e a s u r e d i n e a c h c a s e by t h e t i m e t o c o l l e c t a known volume o f o i l . The p r e s s u r e d r o p i n t h e bed was me a s u r e d and r e c o r d e d f r o m a s e n s i t i v e d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r ( t y p e : DPT 362J-50 f r o m D y n i s c o ) c o n n e c t e d a c r o s s t h e . t w o p r e s s u r e t a p s . I t was c a p a b l e o f m e a s u r i n g p r e s s u r e s up t o H- 345 kPa, T a b l e F-5 shows t h e s p e c i f i c a t i o n s o f t h i s t r a n s d u c e r . - 8 8 -5.2.3 Experimental Technique and Procedure S o l u t i o n s o f 2.-5, 4 and 5% wt. wax i n C l a r u s o i l s were made by s t i r r i n g the r e q u i r e d amount o f wax i n t o the o i l a t about 60°C. Some of t h i s s o l u t i o n was used to f l u s h the equipment from the pre v i o u s runs and was then d i s c a r d e d . The r e s e r v o i r was f i l l e d with the new s o l u t i o n from the feed tank by means of the gear pump. When the r e s e r v o i r was f u l l o f o i l , the system was p r e s s u r i z e d to about 34 5 kPa- with a i r and the o i l was c i r c u l a t e d through the c o o l i n g c o i l which i s immersed i n a constant temperature bath t i l l the temperature was to w i t h i n +0.2°C of the d e s i r e d temperature. The temperature was con-t r o l l e d i n the flow system i n two ways; e i t h e r by keeping the c i r c u l a t i n g r a t e o f the o i l i n the system c o n s t a n t and a d j u s t i n g the temperature o f the bath u n t i l the d e s i r e d temperature was reached, or by keeping the temperature o f the bath c o n s t a n t , and a d j u s t i n g the flow r a t e o f the c i r c u l a t i n g o i l manually by a v a l v e l o c a t e d j u s t b e f o r e the rotometer. P r e l i m i n a r y t e s t s had shown t h a t a t l e a s t 2 hours were needed f o r the waxy o i l t o form a s t a b l e s t r u c t u r e a t any gi v e n temperature. T h e r e f o r e , the flow t e s t s were s t a n d a r d i z e d by a l l o w i n g , i n every run, a 12 hour aging p e r i o d . When the r e s e r v o i r o i l had come to w i t h i n +0.2°C of the d e s i r e d temperature and a f t e r the aging p e r i o d , -89-t h e f l o w r e g u l a t o r v a l v e ( l o c a t e d a t t h e e n t r a n c e o f t h e s a n d column) was opened v e r y s l o w l y t o a l l o w t h e a i r p r e s s u r e ( a r o u n d 345 kPa) i n t h e t a n k t o f o r c e t h e o i l t h r o u g h t h e san d c o l u m n . The p r e s s u r e d i f f e r e n c e a c r o s s t h e column was r e c o r d e d on a c h a r t r e c o r d e r w h i c h was c o n n e c t e d t o t h e p r e s s u r e t r a n s d u c e r . The t h r e s h o l d g r a d i e n t o f f l o w had t o be e x c e e d e d b e f o r e f l o w w o u l d s t a r t , t h e n t h e f l o w r a t e was a d j u s t e d and t h e p r e s s u r e d r o p was r e c o r d e d a t e a c h s t e a d y f l o w r a t e . S t e a d y f l o w was i n d i c a t e d b y a r o t a m e t e r and m e a s u r e d i n a g r a d u a t e d c y l i n d e r . The t r a n s d u c e r was c a l i b r a t e d a g a i n s t a m e r c u r y manometer. The g r a v i t a t i o n a l head o f o i l was z e r o e d - o u t a t t h e t r a n s d u c e r . A f t e r e a c h r u n t h e w a t e r j a c k e t t e m p e r a t u r e was i n c r e a s e d t o 40°C f o r s e v e r a l h o u r s , t h e n t h e o i l was d r a i n e d and t h e column was f l u s h e d , r e a d y f o r t h e n e x t r u n . T h i s was done t o a v o i d any wax b u i l d up i n t h e s a n d column. B e f o r e f i l l i n g t h e sand column w i t h waxy o i l , t h e p r e s s u r e t a p s were f l u s h e d w i t h p u r e C l a r u s o i l f r o m a n o t h e r p r e s s u r i z e d 4 - l i t r e c o n t a i n e r , t o remove any wax i n t h e s e t u b e s . The c l e a r p e r s p e x c o l u m n s a l l o w e d t h e e x p e r i m e n t e r t o o b s e r v e t h e f l o w o f f l u i d s w i t h i n t h e sand b e d s . T h i s was p a r t i c u l a r l y u s e f u l f o r c h e c k i n g t h a t a i r b u b b l e s were n o t t r a p p e d w i t h i n t h e b e d . The f i r s t f l o w e x p e r i m e n t was made w i t h p u r e C l a r u s - B and C l a r u s - C o i l t o measure t h e p e r m e a b i l i t y and t o r t u o s i t y e f f e c t s . Then waxy o i l s o f C l a r u s - B , C l a r u s - C -90-and f i n a l l y t h e c r u d e o i l were t e s t e d . E a c h f l o w t e s t t o o k a b o u t two d a y s . 5.3 E x p e r i m e n t a l P r o b l e m s and D e v e l o p m e n t s The main p r o b l e m s may be summarized a s f o l l o w s : 1. C o r r e c t i o n f a c t o r o f S was a p p l i e d f o r a l l o s h e a r s t r e s s d a t a , s i n c e i t was f o u n d t h a t e v e n f o r a N e w t o n i a n f l u i d , t h e l i n e o f s h e a r s t r e s s - s h e a r r a t e d i d n o t p a s s t h r o u g h t h e o r i g i n . The c o r r e c t i o n p r o c e d u r e was g i v e n i n t h e v i s c o m e t e r manual (44) and i t had a d i f f e r e n t v a l u e f o r e a c h f l u i d . 2. A m e r c u r y manometer was u s e d i n i t i a l l y t o measure p r e s s u r e d r o p i n t h e f l o w t e s t s , b u t i t had a s l o w r e s p o n s e and t h e waxy o i l f l o w e d i n t o t h e p r e s s u r e t a p s and b l o c k e d them, so i t was r e p l a c e d w i t h a p r e s s u r e d i f f e r e n t i a l t r a n s d u c e r , w h i c h r e s p o n d e d g u i c k l y w i t h a n e g l i g i b l e o i l f l o w . 3. The p r e s s u r e t a p s f o r t h e l a r g e column were d r i l l e d i n t h e end f l a n g e s i n o r d e r t o a v o i d any wax s o l i -d i f i c a t i o n i n t h e s e t u b e s , s i n c e t h i s p r o b l e m was n o t i c e d w i t h t h e s m a l l c o l u m n , i n w h i c h t h e p r e s s u r e t a p s were d r i l l e d t h r o u g h t h e j a c k e t . 4. I t was v e r y d i f f i c u l t t o r e a c h t e m p e r a t u r e s b e l o w 10°C f o r C l a r u s o i l s and 2°C f o r t h e c r u d e o i l i n t h e f l o w s y s t e m , b e c a u s e o f wax p r e c i p i t a t i o n on t h e w a l l o f t h e t u b e s , e x p e c i a l l y t h e c o o l i n g b a t h c o i l . T h i s p r o b l e m c a u s e d a d r o p i n f l o w r a t e i n t h e c i r c u l a t i n g l o o p , -91-and the temperature of the o i l increased. The d i f f i c u l t y was removed by replacing the 3/8" c o i l by a 1/2" c o i l i n the cooling bath. 5. It was very d i f f i c u l t to measure the threshold gradient ( o t 0 L ) experimentally, because of i t s s e n s i t i v i t y to the rate at which the pressure increased i n the column. It also depends on the past history of the f l u i d i n the porous medium. Hence, i t was decided to determine t h i s value by extrapolating the values for the lowest three flow rates using the least square method. 6. In some cases, e s p e c i a l l y at higher flow rates, the temperature of the o i l i n the piping system changed by more than +0.3°C. In these circumstances, the data were discarded and the test was repeated. CHAPTER SIX R E S U L T S AND D I S C U S S I O N The v a r i a b l e s and t h e i r r a n g e s i n v e s t i g a t e d i n t h i s s t u d y a r e summarized i n T a b l e 6-1. 6.1 G e n e r a l C o n s i d e r a t i o n Over 26 3 f l o w r a t e - p r e s s u r e d r o p r e a d i n g s were made w i t h t h e f l o w s y s t e m u s i n g two d i f f e r e n t c o l u m n s , w h i l e 218 s h e a r r a t e - s h e a r s t r e s s r e a d i n g s were t a k e n w i t h t h e r o t a t i o n a l v i s c o m e t e r . The e x p e r i m e n t s were p e r -f o r m e d on t h r e e t y p e s o f o i l ( C l a r u s - B , C l a r u s - C , and c r u d e o i l ) w i t h d i f f e r e n t wax c o n t e n t s , v i s c o s i t i e s , and d e n s i t i e s . A l l r e s u l t s a r e c o r r e l a t e d a s p r e s s u r e d r o p -f l o w r a t e r e l a t i o n s h i p s and as a f r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n . 6.1.1 A g i n g T e s t S i n c e t h e s l o w e v o l u t i o n o f c r u d e o i l i n u n d e r g r o u n d f o r m a t i o n s i s i m p o s s i b l e t o s i m u l a t e i n t h e l a b , and s i n c e t h e H e r s c h e l - B u l k l e y m odel i s a t i m e - i n d e p e n d e n t f o r m u l a , t i m e i s n o t a c r i t i c a l f a c t o r i n o u r s t u d y . However, i n -93-T a b l e 6 - 1 V a r i a b l e s a n d R a n g e s I n v e s t i g a t e d V a r i a b l e U n i t . R ange I . F L U I D PARAMETERS Wax c o n c e n t r a t i o n s (% w t . ) 2 . 5 - 5 T e m p e r a t u r e ( ° C ) 2 - 20 O i l d e n s i t y ( g m / c c . ) 0 . 8 6 4 2 - 0 . 8 9 1 8 I I . RHEOLOGICAL PARAMETERS S h e a r s t r e s s (x) 2 ( d y n e / c m ) 0 . 7 1 - 5 0 2 . 1 2 S h e a r r a t e Cy) ( s - 1 ) ' 0 . 0 5 4 - 2 7 7 0 Y i e l d v a l u e (x ) o 2 ( d y n e / c m ) 0 . 7 1 - 6 9 . 3 7 H ( d y n e . s n / c m 2 ) 0 . 4 0 5 4 - 1 7 . 6 2 n (-) 0 . 4 1 - 1 . 0 8 I I I . POROUS MEDIUM PARAMETERS P r e s s u r e d r o p ( A P ) ( k P a ) 3 . 1 1 - 2 9 9 . 6 F l o w r a t e (Q) ( m l / s e c ) 0 . 0 0 5 7 - 1 7 . 4 0 F r i c t i o n f a c t o r ( f * ) (-) 8 . 9 0 x . ' . l O 2 - 2 . 8 6 . x 1 0 9 R e y n o l d s n u m b e r ( R* ) (-) 3 . 3 8 x 1 0 ~ 8 - 0 . 3 3 P o r o s i t y (e) (-) 0 . 3 6 - 0 . 4 4 B e d l e n g t h ( L ) ( cm) 9 1 - 1 0 0 B e d d i a m e t e r (D ) c ( cm) 4 . 8 3 - 1 0 . 1 6 P a r t i c l e d i a m e t e r (D ) n (cm) 0 . 0 7 7 - 0 . 1 2 8 r D /D r a t i o (-) 6 2 . 7 3 - 7 9 . 4 c p P e r m e a b i l i t y ( K ) ( cm ) 3 . 1 5 x 1 0 ~ 6 - 1 5 . 8 x 1 0 " 6 -94-o r d e r f o r t h e waxy o i l s t o have a y i e l d s t r e s s , a c e r t a i n p e r i o d o f t i m e must be a l l o w e d f o r i t t o r e a c h a u n i f o r m t e m p e r a t u r e , and t o d e v e l o p a w a x - o i l g e l s t r u c t u r e i n t h e medium. T h i s t i m e was e s t i m a t e d e x p e r i m e n t a l l y t o be more t h a n 2 h o u r s , as shown i n T a b l e 6-2 f o r t h e s m a l l c olumn. A l s o t h e y i e l d v a l u e s were m e a s u r e d as a f u n c t i o n o f t i m e by t h e s u s p e n d e d p l a t e method, C h a p t e r F o u r , and s i m i l a r r e s u l t s were f o u n d . I t was d e c i d e d t o s t a n d a r d i z e t h e e x p e r i m e n t s by a l l o w i n g an a g i n g p e r i o d o f 12 h o u r s minimum t o g i v e t h e o i l s u f f i c i e n t t i m e t o f o r m a g e l -s t r u c t u r e and t o r e a c h an u n i f o r m t e m p e r a t u r e t h r o u g h o u t t h e p o r o u s medium. The i n t e n t i o n i s n o t t o i n v e s t i g a t e T a b l e 6-2 A g i n g Time E s t i m a t i o n f o r 2.5% Wax i n C l a r u s - B f o r t h e S m a l l Column a t 14°C Time S t a r t i n g P r e s s u r e (min) (kPa) 5 10.58 10 13.51 20 14.05 30 15.48 60 (1 hr) 17.39 120 (2 hr) 20.93 180 (3 hr) 21.34 600 (10 hr) 21.68 1440 (24 hr) 20.19 a l l t h e p a r a m e t e r s w h i c h e f f e c t f l o w i n p o r o u s m e d i a , b u t t o d e v e l o p a H e r s c h e l - B u l k l e y f l u i d and t h e n t e s t -95-t h e t h e o r e t i c a l model b e h a v i o u r w i t h i t . 6.2 R h e o l o g i c a l E x p e r i m e n t s N e c e s s a r y f o r i n t e r p r e t a t i o n o f t h e r e s u l t s were a c c u r a t e measurements o f t h e r h e o l o g i c a l p a r a m e t e r s , s h e a r s t r e s s ( x ) , s h e a r r a t e ( f ) , and y i e l d v a l u e (x ) . 6.2.1 R e s u l t s The r e s u l t s o f t h e v i s c o m e t r i c e x p e r i m e n t s a r e g i v e n i n T a b l e s B - l , B-2, B-3, B-4, and B-5 i n t h e A p p e n d i x f o r a l l t y p e s o f o i l s t e s t e d f o r t h i s s t u d y . T y p i c a l d a t a a r e p l o t t e d i n F i g u r e 6-1 as s h e a r s t r e s s (x) v e r s u s s h e a r r a t e ( y ) . F i g u r e 6-2 shows t h e e f f e c t o f wax c o n t e n t on t h e r h e o l o g i c a l b e h a v i o u r o f t h e waxy o i l s . The r h e o grams f o r e a c h f l u i d were d e t e r m i n e d f r o m t h e v a l u e s o f t o r q u e and o f bob RPM t h r o u g h t h e u s e o f t h e f o r m u l a g i v e n i n S e c t i o n 5. The c o n s t a n t s o f t h e i n s t r u m e n t (M, A', and G) f o r t h e NV bob and cup c o m b i n a t i o n were s u p p l i e d by t h e m a n u f a c t u r e r , and were c h e c k e d i n o u r l a b by u s i n g s t a n d a r d v i s c o s i t y f l u i d (Canon S - 2 0 0 ) . T a b l e 6-3 shows i n s t r u m e n t c o n s t a n t s m e a s u r e d a t d i f f e r e n t t e m p e r a t u r e s . S i n c e t h e d i f f e r e n c e between measured and g i v e n v a l u e s o f t h e s e c o n s t a n t s i s l e s s t h a n 1%, i t was d e c i d e d t o u s e t h e v a l u e s w h i c h were g i v e n by t h e m a n u f a c t u r e r . -96-50 100 150 200 SHEAR RATE (sec"1) 250 F i g u r e 6-1 Shear s t r e s s - shear r a t e curves f o r 2.5% wax i n C la rus-B o i l , measured on a r o t a t i o n a l v i s c o m e t e r , where the s o l i d l i n e r ep r e sen t the H e r s c h e l - B u l k l e y model f i t . -97-SHEAR RATE (sec - 1) F i g u r e 6-2: Shear s t r e s s - shear r a t e curves f o r C la rus-B s o l u t i o n s at T = 14°C measured on a r o t a t i o n a l v i s c o m e t e r , where the s o l i d l i n e r e p r e s e n t the He r s che l -B u l k l e y model f i t . -98-T a b l e 6-3 The V i s c o m e t e r ( R o t o v i s c o RV-12) C o n s t a n t s C h e c k e d W i t h S t a n d a r d F l u i d (Canon S-200) A' P a / s c a l e g r a d . M min/S G C P / s c a l e g r a d . m i n 20°C 0.5357 5.41 99.02 25°C 0.530 5.41 97.96 4 0°C 0.525 5.41 97.02 Me a s u r e d ( a v e r a g e ) 0.5302 5.41 98.0 M a n u f a c t u r e r -'•<. V a l u e s 0.5336 5.41 98.63 S e v e r a l s a m p l e s o f t h e waxy o i l s were t a k e n d u r i n g e a c h f l o w s y s t e m e x p e r i m e n t and t h e i r r h e o g r a m s were d e t e r m i n e d i n t h e r o t a t i o n a l v i s c o m e t e r . The y i e l d v a l u e s (T ) were me a s u r e d w i t h t h e NV bob and cup c o m b i n a t i o n i n t h e R o t o -v i s c o a t i t s l o w e s t s p e e d (0.01 min "*") . The measurements were made a f t e r 12 h o u r s t o a l l o w t h e w a x - o i l s o l u t i o n t o e q u i l i b r a t e i n t h e v i s c o m e t e r a t t h e d e s i r e d t e m p e r a t u r e . T a b l e 6-4 shows t h e H e r s c h e l - B u l k l e y r h e o l o g i c a l p a r a m e t e r s T , n, and H f o r a l l t h e o i l s u s e d i n t h e t e s t s , a s measured i n t h e r o t a t i o n a l v i s c o m e t e r . - 9 9 -T a b l e 6 - 4 R h e o l o g i c a l P a r a m e t e r s o f t h e O i l s M e a s u r e d b y R o t o v i s c o R V - 1 2 T e m p . (°c> Wax (%wt) To 2 ( d y n e / c m ) H 2 ( d y n e s / c m ) n O i l T y p e 10 2 . 5 6 . 0 5 6 . 7 5 0 . 8 9 C l a r u s - B 12 2 . 5 2 . 3 1 3 . 8 2 7 0 . 9 6 14 2 . 5 1 . 4 2 3 . 0 0 2 0 . 9 6 16 2 . 5 1 . 0 7 3 . 0 8 5 0 . 9 2 " 18 2 . 5 0 . 7 1 2 . 0 1 0 0 . 9 7 " 12 4 . 0 3 3 . 6 2 1 2 . 2 2 4 0 . 7 7 " 14 4 . 0 3 1 . 5 0 3 . 3 5 4 0 . 9 7 16 4 . 0 1 6 . 3 6 4 . 6 1 3 0 . 8 8 " 18 4 . 0 4 . 8 0 4 . 3 5 6 0 . 8 5 " 20 4 . 0 1 . 9 6 2 . 8 5 0 . 9 0 " 12 5 . 0 6 9 . 3 7 7 . 3 6 7 0 . 8 3 14 5 . 0 4 8 . 0 2 5 . 3 6 9 0 . 8 7 " 16 5 . 0 3 5 . 7 5 4 . 6 3 2 0 . 8 7 " 18 5 . 0 2 6 . 5 0 5 . 6 7 5 0 . 8 0 " 2 0 5 . 0 1 9 . 2 1 3 . 0 1 8 0 . 9 0 10 2 . 5 1 1 . 7 4 1 7 . 6 3 0 . 8 8 C l a r u s - C 12 2 . 5 8 . 0 1 4 . 5 8 0 . 8 6 II 14 2 . 5 2 . 6 7 9 . 9 7 0 . 9 3 II 16 2 . 5 2 . 3 1 7 . 2 7 6 0 . 9 4 II 18 2 . 5 1 . 4 2 4 . 8 9 0 0 . 9 6 II 2 2 1 . 0 6 6 . 7 2 6 0 . 5 4 C r u d e O i l 8 - 9 . 4 3 2 . 7 7 9 0 . 6 1 it 10 - 6 . 7 6 1 . 2 7 4 0 . 7 0 it 14 3 . 5 6 0 . 4 0 5 4 0 . 8 1 II -100-The measured shear r a t e readings taken from the viscometer were not found to be d i f f e r e n t from the t r u e shear r a t e s due to non-Newtonian behaviour, because of the small gap between NV cup and bob, t h e r e f o r e no c o r r e c t i o n s were made. (See Table B-6 i n the Appendix f o r the d e t a i l s ) . 6 . 2 . 2 A n a l y s i s The r h e o l o g i c a l parameter n and H of Table 6-4 were determined by a l e a s t square f i t of the experimental data (T versus y) by t a k i n g the l o g a r i t h m of the H e r s c h e l -B u l k l e y model equation (2-6) as: l o g d - x ^ ) = l o g H + n l o g ( y ) 6-1 where H was determined from the i n t e r c e p t , w h i l e n was taken as the slope o f the l i n e . The H e r s c h e l - B u l k l e y model p r o v i d e d a s a t i s f a c t o r y r h e o l o g i c a l f i t (Figures 6-1 and 6-2) to the v i s c o m e t r i c data and was t h e r e f o r e assumed to d e s c r i b e the steady s t a t e r h e o l o g i c a l behaviours of these f l u i d s . A sample c a l c u l a -l a t i o n i s g i v e n i n Appendix E of the d e t e r m i n a t i o n of T , H and n. 6 . 2 . 3 D i s c u s s i o n I t i s c l e a r from F i g u r e 6-1 t h a t as the temperature decreases, the y i e l d s t r e s s i n c r e a s e s and the flow behaviour becomes more non-Newtonian. -101 F i g u r e 6-2 shows t h a t the higher the wax content, the more non-Newtonian the o i l behaves. T h i s k i n d of behaviour i s due to the presence of a wax s t r u c t u r e which r e q u i r e s a y i e l d s t r e s s to i n i t i a t e flow o f the f l u i d s . The e f f e c t of these two f a c t o r s , wax content and temperatures are shown i n F i g u r e 6-3. S i m i l a r behaviour was r e p o r t e d by Rojas (73) f o r heavy o i l s , i n which case the y i e l d v alue i n c r e a s e d w i t h d e c r e a s i n g API g r a v i t y . Table 6-4 i l l u s t r a t e s i n mathematical terms the same ge n e r a l c o n c l u s i o n s about the r h e o l o g i c a l behaviour of these waxy o i l s . As the temperature decreases the o i l s become more non-Newtonian. T h i s c o u l d be seen from the i n c r e a s e i n the y i e l d v alue and the decrease i n the value of the parameter n. T h i s k i n d of behaviour a l s o i s t r u e w i t h i n c r e a s e i n wax content. There were some smal l d i s c r e p a n c i e s from these g e n e r a l i z a t i o n s p o s s i b l y due to the f i t of the H e r s c h e l -Bukley model, to the experimental e r r o r s and to the accuracy of the method used i n the d e t e r m i n a t i o n of these f a c t o r s , i . e . the c a l c u l a t i o n of the slope and i n t e r c e p t . Another p o s s i b l e reason f o r these d e v i a t i o n s i s the dynamic method used to measure the y i e l d s t r e s s ( f ) , although the s t a t i c p l a t e method gave very s i m i l a r r e s u l t s . Table 6-4, a l s o shows t h a t the value of the parameter H decreases w i t h i n c r e a s i n g temperature. T h i s i s what would be expected s i n c e H r e p r e s e n t s the v i s c o s i t y when the o i l becomes Newtonian, i . e . when x =0 and n = 1. -102-Figure 6-3 12 V> 16 18 TEMPERATURE (°C) Y i e l d values of three types of waxy o i l s as a funct ion of temperature, measured on a r o t a t i o n a l viscometer at lowest speed (n' = 0.01 min --'-), where the s o l i d l i n e i s drawn by eye. -103-T h e s e p a r a m e t e r s n and H w h i c h were c a l c u l a t e d f r o m t h e H e r s c h e l - B u l k l e y model ( E q u a t i o n 2-6) a r e v e r y s e n s i t i v e t o t h e s h e a r r a t e r a n g e a t w h i c h t h e y were d e t e r m i n e d . S i n c e i t i s v e r y d i f f i c u l t t o measure o r e s t i m a t e s h e a r r a t e s i n p o r o u s m e d i a , we have u s e d t h e r a n g e o f low s h e a r r a t e s m e a s u r e d by s e n s o r s y s t e m NV. A l s o , t h e low f l o w r a t e s t h r o u g h t h e s a n d beds u s e d i n t h e f l o w e x p e r i -ments j u s t i f y t h e r a n g e o f s h e a r r a t e s m e a s u r e d i n t h e r h e o l o g i c a l measurements. In a d d i t i o n , t h e s h e a r r a t e r a n g e s c o v e r e d h e r e depend on t h e N e w t o n i a n v i s c o s i t y o f t h e b a s e o i l . F o r example, P e a c e R i v e r C r u d e O i l b a s e has l o w e r v i s c o s i t y t h a n C l a r u s - B and t h e l a t t e r has l e s s v i s c o s i t y t h a n C l a r u s - C , h e n c e , h i g h e r s h e a r r a t e m e a s u r e -ments were t a k e n w i t h t h e c r u d e t h a n w i t h C l a r u s - C o i l . A l s o t h e s h e a r r a t e r a n g e s u s e d depend on t h e t e m p e r a t u r e s i n v o l v e d . The h i g h e r t h e t e m p e r a t u r e , t h e l o w e r t h e v i s c o s i t y , i . e . h i g h e r s h e a r r a t e s c o u l d be m e a s u r e d A p p e n d i x B e x p l a i n s t h i s p o i n t more f u l l y . 6.3 F l o w S y s t e m E x p e r i m e n t s The a n a l y s i s o f t h e f l o w s y s t e m e x p e r i m e n t s i s b a s e d upon: (1) A m o d i f i e d D a r c y ' s law, S e c t i o n 3.3.2 and 3.3.3. The f l o w d a t a a r e p r e s e n t e d as p r e s s u r e d r o p v e r s u s f l o w r a t e c u r v e s , (2) F r i c t i o n f a c t o r - R e y n o l d s number a n a l y s i s as d e v e l o p e d i n S e c t i o n 3.4. B o t h a p p r o a c h e s w i l l be d i s c u s s e d s e p a r a t e l y . -104-F o r t h e i n t e r p r e t a t i o n o f t h e r e s u l t s an a c c u r a t e measurement o f p o r o s i t y , p a r t i c l e d i a m e t e r (D ) , d e n s i t y , P p e r m e a b i l i t y , p r e s s u r e d r o p and f l o w r a t e were n e c e s s a r y . The p e r m e a b i l i t y was m e a s u r e d w i t h a p u r e C l a r u s - B o i l f o r t h e s m a l l column and w i t h p u r e C l a r u s - C f o r t h e l a r g e c olumn a t t e m p e r a t u r e s o f 25°C and 3 0 ° C . S i n c e t h e o i l s were N e w t o n i a n (no s h e a r t h i n n i n g o r y i e l d s t r e s s were o b s e r v e d ) a t t h e s e t e m p e r a t u r e s , D a r c y ' s l a w was a p p l i e d and t h e p e r m e a b i l i t y was d e t e r m i n e d f r o m t h e s l o p e o f t h e l i n e Ap v e r s u s Q; Ap = H i Q 2-9 The v a l u e o f K f o r a N e w t o n i a n f l u i d was c a l c u l a t e d — 6 2 — 6 t o be 3.15 x 10 cm f o r t h e s m a l l column and 15.8 x 10 2 cm f o r t h e l a r g e c o l u m n . The s m a l l s a n d bed has a l o w e r v a l u e o f K t h a n t h e l a r g e one, s i n c e s a n d w i t h a mean p a r t i c l e s p h e r i c a l d i a m e t e r D^ = 0.077 cm was u s e d i n t h e s m a l l bed and D = 0.128 cm i n t h e l a r g e one. The P r e s u l t s a r e shown i n T a b l e 6-5. The p o r o s i t y (e) o r v o i d f r a c t i o n was d e t e r m i n e d by w a t e r d i s p l a c e m e n t . The s m a l l column gave e = 0.36 w h i l e t h e l a r g e one gave e = 0.44. I t i s u s u a l t h a t t h e p o r o -s i t y o f a p o r o u s medium i n c r e a s e s ' w i t h p e r m e a b i l i t y , ' a s ,. shown i n T a b l e 2-10. -105-T a b l e 6-5 R e s u l t s f o r t h e F l o w o f P u r e C l a r u s O i l s T h r o u g h t h e Sand Beds P u r e C l a r u s - B O i l P u r e C l a r u s -C O i l T = 25°C y =103 mPa. s T = 30°C y =,. 18 0, mPa.s D /D = 62 .7 L = -91 cm D /D = 79.4 d p L = 1.00 cm F l o w R a t e P r e s s u r e Drop F l o w R a t e P r e s s u r e Drop ( c . c . / s e c ) (kPa) ( c . c . / s e c ) (kPa) 0.158 25.697 0.456 5.39 0.334 55.282 1.42 19.90 0.545 88.618 2.40 34.55 0.708 119.176 3.333 46.99 0.997 160.704 4.50 62.19 The f l u i d d e n s i t y was m e a s u r e d i n a p y c n o m e t e r , by w e i g h i n g a known volume o f f l u i d a t t h e d e s i r e d t e m p e r a -t u r e . I t was f o u n d t h a t t h e wax c o n t e n t had no s i g n i f i -c a n t e f f e c t on t h e d e n s i t i e s , s i n c e low wax c o n c e n t r a t i o n s were u s e d . T a b l e F-6 shows t h e d e n s i t i e s o f t h e waxy o i l s u s e d , a t d i f f e r e n t t e m p e r a t u r e s . 6.3.1 M o d i f i e d D a r c y ' s Law 6.3.1.1 R e s u l t s The r e s u l t s f o r a l l t h e d a t a a r e p r e s e n t e d h e r e a s p r e s s u r e d r o p v e r s u s f l o w r a t e i n t h r e e d i f f e r e n t ways: a) T a b l e s : As shown i n t h e A p p e n d i x , T a b l e s C - l , C-2, C-3, C-4, and C-5. b) E q u a t i o n s : T h e s e a r e summarized i n T a b l e s 6-6, 6-7, and 6-8 i n w h i c h E q u a t i o n s 3-18 and 3-2 3 were a p p l i e d . - 1 0 6 -T a b l e 6 - 6 R e s u l t s o f t h e F l o w S y s t e m f o r C l a r u s - B S o l u t i o n s i n t h e L a r g e C o l u m n F i t t o E q u a t i o n 3 - 1 8 Temp o „ Wax % w t . K x l O 2 H e f f . - n , 1+r i dyne, s- /cm ( a Q L ) ' k P a ^ c a l k P a 1 0 12 14 1 6 1 8 12 14 16 18 2 0 2 . 5 2 . 5 2 . 5 2 . 5 2 . 5 5 5 5 5 5 8 . 6 6 1 1 . 2 5 1 2 . 3 7 4 1 5 . 9 2 1 3 . 5 8 0 . 5 3 0 . 8 5 1 . 0 6 . 5 8 1 1 . 8 1 2 . 7 7 8 1 . 9 3 3 1 . 9 5 0 6 1 . 9 0 3 8 1 . 4 8 0 7 1 . 9 7 7 0 . 8 8 6 1 . 0 5 2 . 7 8 3 3 . 0 1 8 2 2 . 4 2 1 6 . 4 2 5 . 7 0 1 . 8 4 1 . 7 3 3 6 . 5 7 3 . 9 2 8 . 3 8 2 2 . 6 9 1 0 . 3 7 7 3 . 2 0 Q ° ' 8 6 + 2 2 . 4 2 3 4 . 3 6 3 Q ' 8 9 + 1 6 . 4 2 Q ° - 9 3 + 5 . 7 0 2 6 . 4 4 2 0 . 9 5 9 Q ° - 9 2 + 1 . 8 4 0 9S 1 6 . 7 4 9 5 Q + 1 . 7 3 9 7 1 . 0 5 3 6 7 . 8 2 5 3 1 1 . 8 7 8 4 . 5 8 7 3 1 . 5 1 Q ° - 8 3 + 3 6 . 5 Q 0 . 7 6 + ? 3 > 9 2 Q 0 . 8 0 + 8 . 3 8 Q ° - 8 9 + 2 2 . 6 9 Q 1 ' 0 + 1 0 . 3 7 F i t t o E q u a t i o n 3 - 2 3 1 0 2 . 5 9 . 5 1 3 . 0 4 9 12 2 . 5 1 2 . 0 9 2 . 0 7 7 14 2 . 5 1 2 . 9 4 2 . 0 4 0 16 2 . 5 1 6 . 7 6 2 . 0 0 4 18 2 . 5 1 4 . 0 2 1 . 5 2 8 1 2 5 0 . 3 3 1 . 2 3 5 14 5 0 . 5 9 0 . 6 1 8 16 5 1 . 1 4 3 1 . 2 0 4 1 8 5 7 . 0 7 1 2 . 9 9 1 20 5 1 1 . 8 1 3 . 0 1 8 2 2 . 4 2 1 6 . 4 2 5 . 7 0 1 . 8 4 1 . 7 3 3 6 . 5 7 3 . 9 2 8 . 3 8 2 2 . 6 9 1 0 . 3 7 7 3 . 1 6 3 4 . 3 6 2 6 . 4 4 2 0 . 9 5 5 1 6 . 7 4 4 9 6 8 . 4 0 3 6 8 . 9 9 3 1 2 . 8 4 8 4 . 5 8 9 3 1 . 5 1 Q ° -86 + 2 2 . 4 2 Q ° -89 + 16 . 4 2 0 ° ' 9 3 + 5 . 7 0 Q ° -92 + 1 . 8 4 Q ° -9 5 + 1 . 7 3 Q ° -8 3 + 36 . 5 Q 0 ' 76 + 73 . 9 2 Q ° -8 + 8 . 3 8 Q ° -89 + 22 . 6 9 Q 1 . 0 + 10 . 3 7 Determined b y l e a s t square f i t t i n g of-the points for- the• three lowest flow, rates. -107-Table 6-7 Results of the Flow System f or Clarus-B O i l of the Small Column by Applying Equation 3-18 Temp. °C Wax %wt KxlO' ? cnr-e f f dyne ssn/cm ( a L ) " o kPa ^ c a l . (kPa) 10 12 14 18 12 14 16 18 20 16 18 20 2.5 2.5 2.5 2.5 4 4 4 4 4 5 5 5 0.60 2.55 2.68 3.07 2.508 1.091 1.40 1.872 4.43 0.72 4.65 1.037 3.917 2.507 2.407 1.749 3.95 2.888 3.98 5.81 4.278 1.839 5.305 0.537 1.68 7.39 3.27 1.41 18.92 12.96 5.10 3.82 0.0 17.93 43.56 12.26 3975.32 581.47 Q 0' 9 3 + 1.68 Q ° ' 9 4 + 7.39 515.98 Q' 9 5 + 3.27 299.93 Q 0 , 9 8 + 1.41 1246.0 Q ° - 8 4 + 18.92 1393.83 Q 0 - 9 8 + 12.96 1496.68 Q 0 - 9 8 + 5.10 1 04** 1372.36 Q + 3.82 402.87 Q 1 , 0 ^ * + 0.0 1850 583.42 Q0.88 + ± 7 9 3 0 99 Q U , y y + 43.56 502.3 Q 0 - 7 7 + 12.26 Determined by lea s t square f i t t i n g of the points for the three lowest flow rates. Some values of n i n Tables 6-7 and 6-8 were greater than one, possibly due to wax buildup i n the entrance of the pressure taps i n the small column. - 1 0 8 -Table 6-8 (a) Results of the Flow System for Clarus-C O i l of the Small Column by Applying Equation 3-18 Temp. (°C) Wax % wt K x 1 0 6 2 cm V e f f . 1 + n dyne;sn/cm (OL)* KPa ^ P c a l kPa 12 2 .5 1 .266 1 0 . 0 7 1 6 .3 0 95 4570 Q + 6 .3 ** 1 03 2 2 1 6 . 4 5 Q + 4 . 3 3 14 2 .5 2 . 5 3 1 2 . 3 2 4 . 3 3 16 2 .5 2 .202 7 .813 0 . 0 1 01** 1 7 1 1 . 9 8 Q + 0 . 0 18 2 .5 2 .857 8 .553 0 .4 i afft* 1 1 7 8 . 3 7 Q + 0 .4 (b) Results of the flow system f o r Clarus -C o i l of the large column 10 2 .5 1 3 . 7 1 1 . 4 9 6 9 . 8 0 93 140 Q + 9 . 8 12 2 .5 1 4 . 3 2 1 2 . 9 1 8 .22 1 2 1 . 3 7 3 Q 0 , 9 8 + 8 .22 14 2 .5 1 2 . 6 0 7 .33 9 . 9 8 0 9S 8 9 . 6 4 Q + 9 . 9 8 16 2 .5 1 3 . 3 7 .276 4 . 1 5 6 7 . 5 2 7 Q 1 , 0 + 4 . 1 5 18 2 .5 1 0 . 4 6 3 .579 2 .14 0 QS 5 2 . 5 6 Q + 2 .14 (c) Results of the flow system f o r crude o i l of the large column 2 - 8 .3 4 .31 6 . 2 5 8 8 6 . 6 5 Q' 9 3 + 6 . 2 5 8 8 - 9 . 4 8 5 0 .162 1 3 . 4 7 5 1 5 . 2 5 Q 0' 5 5 + 1 3 . 4 7 5 10 - 2.64 0 .021 4 . 4 9 2 1 3 . 0 8 2 5 Q 0 , 4 1 + 4 . 4 9 2 14 - 5 .10 0 . 0 1 8 1.382 3 .4473 Q 0 - 5 3 + 1 .382 Determined by l e a s t square f i t t i n g of the points for the three lowest flow rates. See page 107 -109-c) G r a p h s : T h e s e a r e d i v i d e d i n t o two g r o u p s . The f i r s t g r o u p r e p r e s e n t t h e f l o w o f waxy o i l s ( C l a r u s - B and C l a r u s - C ) t h r o u g h t h e s m a l l column (L = 91 cm, D = 0.077 cm, D = 4.83 cm, and e = 0.36), and a r e p c shown i n F i g u r e s 6-4, 6-5, 6-6, and 6-7 where t h e s o l i d l i n e r e p r e s e n t t h e model p r e d i c t i o n , i . e . m o d i f i e d D a r c y ' s law E q u a t i o n 3-18. The s e were t h e f i r s t d a t a t o be c o l l e c t e d b e f o r e any m o d i f i c a t i o n s were made t o t h e f l o w s y s t e m . The s e c o n d g r o u p r e p r e s e n t s t h e f l o w o f waxy o i l s ( C l a r u s -B, C l a r u s - C , and t h e Pe a c e R i v e r C r u d e o i l ) t h r o u g h t h e l a r g e c o l u m n (L = 100 cm, D = 0.128 cm, D =10.16 cm, P c and e = 0.44). Th e s e a r e shown i n F i g u r e s 6-8, 6-9, 6-10, and 6-11. A l s o h e r e t h e s o l i d l i n e r e p r e s e n t s t h e model p r e d i c t i o n , i . e . E q u a t i o n 3-18. The s e e x p e r i m e n t s were made a f t e r t h e s m a l l column e x p e r i m e n t s . 6.3.1.2 A n a l y s i s The g o a l i n m o d e l i n g t h i s s y s t e m was t o p r e d i c t t h e f l o w b e h a v i o u r f r o m r h e o l o g i c a l measurements and f o r m a t i o n p r o -p e r t i e s . T h e r e f o r e , t h e f i r s t a t t e m p t a t f i t t i n g t h e c u r v e s were made u s i n g t h e N e w t o n i a n p e r m e a b i l i t y a s mea s u r e d w i t h t h e s o l v e n t o i l s . The r e s u l t s were n o t good. On r e c o n s i d e r -i n g t h e m o d e l , i t was r e a l i z e d t h a t a s t h e y i e l d v a l u e i n -c r e a s e s t h e p e r m e a b i l i t y must, d e c r e a s e s i n c e t h e r e was i n -s u f f i c i e n t d r i v i n g f o r c e . t o overcome t h e y i e l d v a l u e and c a u s e f l o w i n a l l t h e p o r e s . -110-FLOW RATE (ml/sec) F i g u r e 6-4: P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 2.5% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e m odel f i t o f e q u a t i o n 3-18. -111-I I I I FLOW RATE (ml/sec) F i g u r e 6-5: P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 4% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. CL O cr o 220-180-140 U J 100 cr ZD 10 10 LU cr CL 60-20 F i g u r e 6-6 FLOW RATE (ml/sec) I I P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 5% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a 91 cm l o n g sand column w i t h D p = 0.077 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. -113-F i g u r e 6-7: 0.04 0.08 0.12 0.16 FLOW RATE (ml/sec) P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 2.5% wax i n C l a r u s - C o i l f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.07 7 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e m odel f i t o f e q u a t i o n 3-18. 140 120 £ 1 0 0 b 80 LT Q L U 60 tr LO LU AO tr CL 206 0 F i g u r e 6-8 i h- 1 I—1 I FLOW RATE (ml /sec) P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 2.5% wax i n C l a r u s - B o i l f l o w i n g t h r o u g h a l 0 0 cm l o n g sand column w i t h Dp = 0.128 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. -115-FLOW RATE (ml/sec) F i g u r e 6-9: P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 5% wax i n C l a r u s - B f l o w i n g t h r o u g h a 100 cm l o n g s a n d column w i t h Dp = 0.12 8 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e m odel f i t o f e q u a t i o n 3-18. -116-1 1 1 1 I FLOW RATE (ml/sec) F i g u r e 6-10: P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 2.5% wax i n C l a r u s - C o i l f l o w i n g t h r o u g h a 100 cm l o n g s a n d column w i t h Dp = 0.128 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. Figure 6-11 FLOW RATE (ml/sec) P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r P e a c e R i v e r C r u d e o i l f l o w i n g i n a 100 cm l o n g s a n d bed w i t h D p = 0.128 cm, where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. - 1 1 8 -It "was then decided to hold - the t o r t u o s i t y -factor':. ( C ) i n Equation 3 - 2 3 constant and to allow the perme-a b i l i t y to vary. Similar behaviour was discovered by Sadowski ( 7 5 ) when he was investigating the flow behaviour of polymer solutions through packed beds, i n which the permeability was found to be a function of the volumetric flow of f l u i d through the bed. He explained the behaviour as being caused by adsorption of polymer on the p a r t i c l e surfaces and gel formation. He has concluded the following ( 7 5 ) : a) The permeability of a porous medium decreases with increasing f l u i d throughput. While the rate of change of permeability increases with increasing polymer concentration. b) The rate of change of the permeability with volumetric flow rate, decreases with increasing p a r t i c l e diameter. The above discussion and the fact that the flow model permeability i s for a non-Newtonian f l u i d j u s t i f y the decision to calculate the actual permeability for each flow t e s t . The values of the non-Newtonian permeabilities are given i n Tables 6 - 6 , 6 - 7 , and 6 - 8 . A least square f i t of the r e s u l t s were made to Equations 3 - 1 8 and 3 - 2 3 . -119-Ap = e r Q n + a L 3-23 K A n where V>eff ®0 i ^ g f f a n & a 0 a r e d e f i n e d b y E q u a t i o n s 3-19, 3-20, 3-24, and 3-25, pages ( 6 1 , 6 2 ) . The f i t was made o f l o g (Ap - a Q L ) a g a i n s t l o g Q o f t h e above E q u a t i o n s 3-18 and 3-23. The s l o p e i s n and t h e i n t e r c e p t i s t h e l o g o f t h e c o e f f i c i e n t o f Q n. The t h r e s -h o l d g r a d i e n t - l e n g t h t e r m ( a Q L ) was d e t e r m i n e d by l e a s t s q u a r e f i t t i n g o f t h e p o i n t s f o r t h e t h r e e l o w e s t f l o w r a t e s , ( t h e _ r e a s o n i s g i v e n i n S e c t i o n 5 . 3 ) . From t h e y e f f L ^ e f f i n t e r c e p t (—^ ) t h e t e r m — - — c a n be d e t e r m i n e d and K A n K s u b s t i t u t e d i n t o E q u a t i o n 3-21 o r E q u a t i o n 3-26 t o d e t e r -mine t h e v a l u e o f t h e p e r m e a b i l i t y , K. - l ^ n -1-n _ | f f = H ( 1 + 3 ) n ( 8 £ ) 2 R 2 3 _ 2 1 i \ 4 n The v a l u e s o f n u s e d i n t h e s e a n a l y s e s were t h o s e o f t h e s l o p e o f t h e p l o t l o g (Ap - o t QL) a g a i n s t l o g Q. The v a l u e s o f n f o u n d by t h e v i s c o m e t e r were v e r y c l o s e t o t h o s e f o u n d f r o m t h e f l o w c u r v e s . T y p i c a l v a l u e s o f n a r e g i v e n i n T a b l e 6-9. The r e s t a r e shown i n T a b l e s 6-4, 6-6, 6-7, and 6-8. The d e c i s i o n t o u s e t h e n f r o m t h e f l o w d a t a was made a t t h e same t i m e a s i t was d e c i d e d t o u s e t h e non-N e w t o n i a n p e r m e a b i l i t y t e r m , s i n c e t h i s v a l u e was a l s o c a l c u l a t e d f r o m t h e f l o w d a t a . T h i s d e c i s i o n was r e i n -f o r c e d by t h e f a c t t h a t t h e v a l u e s o f n c a l c u l a t e d f o r -120-T a b l e 6-9 D i f f e r e n t V a l u e s f o r t h e P a r a m e t e r (n) f o r C l a r u s - B O i l T e m p e r a t u r e wax %wt n, f r o m °C V i s c o m e t e r S m a l l Bed L a r g e Bed 10 2.5 0.89 0.93 0.86 12 2.5 0.96 0.94 0.89 14 2.5 0.96 0.95 0.93 16 2.5 0.92 - 0.92 18 2.5 0.97 0.98 0.95 e a c h s y s t e m by t h e two t e c h n i q u e s were v e r y s i m i l a r . I n a way t h i s a g reement i n d i c a t e s t h a t t h e f l o w model i s a good r e p r e s e n t a t i o n o f t h e f l o w b e h a v i o u r o f t h e c h o s e n f l u i d s . The a g r e e m e n t between t h e e x p e r i m e n t a l and p r e d i c t e d r e s u t s , and between e x p e r i m e n t s was t e s t e d by u s i n g a) Root Mean:. S q u a r e E r r o r (69) ,'.. Root Mean / S q u a r e E r r o r - ' = J J > P e x p - ^ c a l j N - l where N i s t h e number o f d a t a p o i n t s i n e a c h e x p e r i m e n t , w h i l e AP , was c a l c u l a t e d u s i n g E q u a t i o n 3-18. T a b l e cax • 6-10 summarizes t h e e v a l u a t i o n o f t h e r e s u l t s , b) P e r c e n t a g e E r r o r s , % D e v i a t i o n AP . - AP . exp c a l AP exp x 100 - ' • Table 6 - 1 0 : "' . . S t a t i s t i c a l Analysis of the Data Shown i n Appendix C for the Flow Sys Using Modified Darcy's Equation 3 - 1 8 RUN F r o m N O . T o T ° ( d y n e / cm^) N TEMPERATURE ( ° C ) WAX CONTENT (%:.wt.) ROOT MEAN SQUARE ' ERROR" B .1 B 8 B 1 5 B 25 B 32 B 4 1 B 49 B 5 6 B 64 B 74 B 79 B 86 B 7 B 14 B 24 B 3 1 B 4 0 B 4 8 B 55 B 63 B 73 B 78 B 85 B 94 6 . 0 5 2 . 3 1 1 . 4 2 0 . 7 1 3 3 . 6 3 1 . 5 0 1 6 . 3 0 4 . 8 1 . 9 6 3 5 . 7 2 6 . 5 1 9 . 2 0 7 7 1 0 7 9 8 7 8 1 0 5 7 9 1 0 . 0 1 2 . 0 1 4 . 0 1 8 . 0 1 2 . 0 1 4 . 0 1 6 . 0 1 8 . 0 2 0 . 0 1 6 . 0 1 8 . 0 2 0 . 0 2 . 5 2 . 5 2 . 5 2 . 5 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 5 . 0 5 . 0 5 . 0 1 4 . 4 1 1 . 3 8 1 . 9 4 1 . 6 0 3 . 8 1 4 . 0 3 4 . 4 2 4 . 0 0 4 . 3 7 . 1 3 . 5 1 9 . 5 4 8 . 9 1 C 95 c i o i C 1 0 8 C 1 1 5 C 1 0 0 C 1 0 7 C 1 1 4 C 1 2 3 8 2 . 6 7 2 . 3 1 1 . 4 2 6 7 7 9 1 2 . 0 1 4 . 0 1 6 . 0 1 8 . 0 2 . 5 2 . 5 2 . 5 2 . 5 7 . 5 4 2 . 4 2 2 . 9 0 6 . 8 1 B 1 2 4 B 1 3 2 B 1 4 1 B 1 5 1 B 1 5 8 B 1 6 4 B 1 7 1 B 1 7 7 B 1 8 4 B 1 9 1 B 1 3 1 B 1 4 0 B 1 5 0 B 1 5 7 B 1 6 3 B 1 7 0 B 1 7 6 B 1 8 3 B 1 9 0 B 1 9 8 6 . 0 5 2 . 3 1 1 . 4 2 1 . 0 7 0 . 7 1 6 9 . 3 5 8 . 7 3 5 . 7 2 6 . 5 1 9 . 2 8 9 1 0 7 6 7 6 7 7 8 1 0 . 0 1 2 . 0 1 4 . 0 1 6 . 0 1 8 . 0 1 2 . 0 1 4 . 0 1 6 . 0 1 8 . 0 2 0 . 0 2 . 5 2 . 5 2 . 5 2 . 5 2 . 5 S..0-5 . 0 5 . 0 5 . 0 5 . 0 1 8 . 4 4 3 . 6 8 2 . 4 9 1 . 1 7 1 . 0 5 2 0 . 9 4 3 0 . 4 5 2 5 . 3 5 2 2 . 7 2 3 . 0 9 C 1 9 9 C 2 0 7 C 2 1 4 C 2 2 2 C 2 2 9 C 2 0 6 C 2 1 3 C 2 2 1 C 2 2 8 C 2 3 5 1 1 . 7 4 8 2 . 6 7 2 . 3 1 1 . 4 2 8 7 8 7 7 1 0 . 0 1 2 . 0 1 4 . 0 1 6 . 0 1 8 . 0 2 . 5 2 . 5 2 . 5 2 . 5 2 . 5 5 . 0 3 5 . 3 9 2 . 6 4 1 . 7 2 1 . 1 7 CRUDE •CRUDE CRUDE CRUDE 1 - _8 9 - 1 4 15 - 22 2 3 - 28 2 1 . 0 9 . 4 3 6 . 7 6 3 . 5 6 8 6 8 6 2 . 0 8 . 0 1 0 . 0 1 4 . 0 2 . 8 7 1 . 3 6 1..16 0 . 5 9 . 'OVERALL R E S U L T S B 1 C 95 B 1 2 4 C199 : CRUDE B 94 C 1 2 3 B 1 9 8 : \C235: . . 1 - 2 8 94 29 . 75 37 28 6 . 4 3 5 . 1 4 1 5 . 0 9 3:44 i s 70 -122-The v a l u e s of these d e v i a t i o n s are shown i n Tables C-1, C-2, C-3, C-4, and C-5 i n the Appendix. 6.3.1.3 D i s c u s s i o n Table 6-6 shows t y p i c a l experimental r e s u l t s c a l c u -l a t e d from the two models (Equations 3-18 and 3-23, m o d i f i e d Darcy's law e q u a t i o n s ) . Since the b a s i s f o r each model was the same, the two s o l u t i o n s gave the same model equations. These are the s o l i d l i n e s on F i g u r e s 6-8 and 6-9. The d i f f e r e n c e i n the two equations comes from the d e f i n i t i o n s o f the t h r e s h o l d g r a d i e n t , aQ (Equations 3-20 and 3-25) and VQff (Equations 3-19 and 3-24). These two parameters c o n t a i n the t o r t u o s i t y C i n Equations 3-24 and 3-2 5, w h i l e Equations 3-19 and 3-20 do not have t h i s term. In g e n e r a l , the p e r m e a b i l i t y decreases w i t h decreas-i n g temperature and i n c r e a s i n g wax c o n c e n t r a t i o n , s i n c e the y i e l d v alue i s a l s o i n c r e a s i n g w i t h d e c r e a s i n g temperature. There are some d i s c r e p e n c i e s from t h i s g e n e r a l i z a t i o n due t o the model f i t , experimental e r r o r , and the method of e s t i m a t i o n the p e r m e a b i l i t y , as mentioned p r e v i o u s l y . As the temperature i n c r e a s e s the p e r m e a b i l i t y —6 2 approaches the v a l u e of 15.8 x 10 cm found f o r the l a r g e column with pure Clarus-C o i l and a value of — 6 2 3.15 x 10 cm found f o r the small column w i t h pure Clarus-B o i l as can be seen by the data shown i n Table 6-5. From F i g u r e 6-8, i t can be seen t h a t the 10°C curve f i t i s -123-much worse t h a n i t i s f o r t h e h i g h e r t e m p e r a t u r e s . ,:. A root, mean s q u a r e e r r o r -. of-. ' IS .44 i s . shown i n T a b l e 6-10. A t 10°C t h e o i l has a l a r g e r y i e l d v a l u e w h i c h may i n d i c a t e a l i m i t a t i o n t o t h e model a t h i g h e r y i e l d v a l u e s . A l s o i n F i g u r e 6-9 w h i c h g i v e s t h e r e s u l t s f o r t h e l a r g e c o l u m n , a s i m i l a r b e h a v i o u r i s s e e n b e c a u s e o f a h i g h e r wax c o n t e n t , hence a h i g h e r y i e l d v a l u e . The c u r v e f o r 14°C shows t h e w o r s t f i t , ( r o o t mean s q u a r e e r r o r o f 30.45, T a b l e 6-10), w h i l e t h e p o i n t s f o r t h e h i g h e s t f l o w r a t e a t 16°C o f t h e same f i g u r e shows t h e w o r s t d e v i a t i o n i n a l l t h e d a t a (Run No. 183, d e v i a t i o n = 41.19%, T a b l e C - 3 ) . T h i s h i g h d e v i a t i o n i s p r o b a b l y due t o t h e method o f a n a l y s i s and t h e s e n s i t i v i t y o f t h e f i t t i n g p a r a m e t e r s t o any s m a l l e r r o r i n t h e e x p e r i m e n t a l measurements o f n, x , H and Q, as w i l l be shown i n t h e o n e x t s e c t i o n . T a b l e 6-10 shows f o r e a c h c u r v e t h e r o o t mean s q u a r e - e r r o r s , ' t h e y i e l d v a l u e , t h e number o f e x -p e r i m e n t a l p o i n t s i n v o l v e d , t h e wax c o n c e n t r a t i o n s and t h e t e m p e r a t u r e a t w h i c h t h e e x p e r i m e n t was p e r f o r m e d . Runs No. B l t o C123 were p e r f o r m e d i n t h e s m a l l c o l u m n , and r u n s No. B124 t o C2 35 were made i n t h e l a r g e column, as were t h e c r u d e o i l t e s t s . G e n e r a l l y , t h i s t a b l e i n d i c a t e s more d e v i a t i o n f r o m t h e m odel f o r l o w e r t e m p e r a t u r e s and h i g h e r wax c o n c e n t r a t i o n b o t h o f w h i c h c a u s e h i g h e r y i e l d v a l u e s . In T a b l e 6-7 and F i g u r e s 6-4, 6-5 and 6-6, a r e t h e e x p e r i m e n t a l r e s u l t s c a l c u l a t e d by E q u a t i o n 3-18 f o r t h e -124-s m a l l sand b ed, w i t h C l a r u s - B s o l u t i o n s o f d i f f e r e n t wax c o n c e n t r a t i o n s . The p e r m e a b i l i t y i n c r e a s e s w i t h t e m p e r a t u r e t o a p p r o a c h t h e N e w t o n i a n v a l u e o f — 6 2 3.15 x 10 cm . F o r t h e s e r e s u l t s , t h e p e r m e a b i l i t y c a n be e x p e c t e d t o be l e s s a c c u r a t e t h a n f o r t h e l a r g e c olumn due t o t h e s m a l l e r p a r t i c l e d i a m e t e r used" -(D = 0.077 cm), and t o t h e e f f e c t o f a s m a l l amount o f P wax w h i c h may b u i l d up on them e s p e c i a l l y a t t h e e n t r a n c e t o p r e s s u r e t a p s . The i n c r e a s e i n s c a t t e r o f t h e d a t a c a n be s e e n i n F i g u r e 6-5 and 6-6 o r i n Table ; 6-7. B e c a u s e o f t h e above p r o b l e m s , some m o d i f i c a t i o n s were made i n t h e e q u i p m e n t . The l o c a t i o n o f t h e p r e s s u r e t a p s was c h a n g e d t o t h e f l a n g e s and c l e a r o i l b a c k -f l u s h i n g was u s e d t o r e d u c e c o n s t r i c t i o n o f t h e p r e s s u r e t a p s . The l a r g e r s a n d s i z e i n t h e s e c o n d column a l s o l e a d t o more c o n s i s t e n t r e s u l t s . The r e s u l t s o f t h e s e m o d i f i c a t i o n s c a n be s e e n by c o m p a r i n g F i g u r e s 6-6 and 6-9 f o r t h e h i g h e s t wax c o n t e n t o f 5%wt. S a d o w s k i (75) f a c e d s i m i l a r p r o b l e m s w i t h p o l y m e r s o l u t i o n s , " I t w o u l d n o t be e x p e c t e d t h a t t h e ' A d s o r p t i o n -g e l l i n g ' phenomenon w o u l d be t h e same f o r a column w i t h Dp = 0.1315 cm, a s f o r b e d s o f much s m a l l e r p a r t i c l e s i z e (Dp= 0.0715 cm), i f o n l y b e c a u s e i t i s d o u b t f u l i f any o f t h e p o r e s o f b e d s o f h i g h p e r m e a b i l i t y w o u l d e v e r be b l o c k e d o f f " . T a b l e 6-8 shows t h e r e s u l t s f o r t h e o t h e r two t y p e s o f waxy o i l s , i . e . C l a r u s - C s o l u t i o n s f o r t h e s m a l l and l a r g e column and t h e c r u d e o i l . The d a t a o f t h i s T a b l e a r e g r a p h e d i n F i g u r e s 6-7, 6-10, and 6-11. F i g u r e 6-7 -125-shows t h e f l o w b e h a v i o u r o f 2.5% wax i n C l a r u s - C o i l i n t h e s m a l l c o l u m n . C l o s e a g r e e m e n t was f o u n d between e x p e r i m e n t a l and p r e d i c t e d v a l u e s , and t h e p e r m e a b i l i t y i n c r e a s e s w i t h t e m p e r a t u r e a s w o u l d be e x p e c t e d . F i g u r e 6-10 shows t h e same t y p e o f o i l f l o w i n g i n t h e l a r g e c o l u m n , w h i c h a l s o shows c l o s e a g r e e m e n t between e x p e r i -m e n t a l and p r e d i c t e d v a l u e s , e x c e p t i n t h i s c a s e t h e p e r m e a b i l i t y d o e s n o t change i n t h e e x p e c t e d manner, due t o i t s s e n s i t i v i t y t o any s m a l l e r r o r s i n measurements. A l s o t h i s u n e x p e c t e d r e s u l t may be due t o t h e h i g h e r f l o w r a t e s , s i n c e more w a x - o i l s o l u t i o n w i l l p a s s t h r o u g h t h e p o r o u s medium and may c a u s e a t h i n wax l a y e r t o b u i l d up on t h e s u r f a c e o f t h e s a n d p a r t i c l e s , t h u s r e d u c i n g t h e p e r m e a b i l i t y . T h i s phenomenon was r e p o r t e d by S a d o w s k i ( 7 5 ) , S e c t i o n 6.3.1.2 ,page 1 1 8 ) . F i g u r e 6-11 shows t h e r e s u l t s f o r t h e low v i s c o s i t y and low d e n s i t y P eace R i v e r waxy c r u d e o i l w h i c h g i v e s an e x c e l l e n t f i t t o t h e t h e o r y . However, T a b l e 6-8 a l s o shows t h e p e r m e a b i l i t y d o e s n o t c hange i n t h e e x p e c t e d manner as e x p l a i n e d a b o v e . 6.3.1.4 The S i g n i f i c a n c e o f Measurement E r r o r s I n o r d e r t o u n d e r s t a n d t h e u n e x p e c t e d c h a n g e s i n K, y f f and t h e s c a t t e r o f some d a t a f r o m t h e m o d e l , t h e s e n s i t i v i t y o f t h e m o d i f i e d D a r c y ' s e q u a t i o n t o s m a l l e r r o r s on t h e measurements o f T , n, H, and Q f o r a t y p i c a l r u n were t e s t e d . -126-F o r Run C199 t o C206 T = 10°C O i l = C l a r u s - C L = 100 cm Wax = 2.5% wt A = 81.1 cm 2 p = 0.8884 gm/cm 3 D p= 0.128 cm x 0 = 11.74 (dyne/cm 2) D c= 10.16 cm n 0.93 H ='17.63 (dyne/cnT) s*1 AP = 140.75 Q 0 ' 9 3 + 9.8 kPa a 0 L = 9 . 8 kPa a Q = 9 8 0 dyne/cm 3 6 = 0.309 y e f f = 11.496 d y n e . s n / c m 1 + n ^ e f f 5 — £ — = 8.363 x 10 f r o m t h e e x p e r i m e n t — 6 2 K = 13.7 x 10 cm a v e r a g e p e r m e a b i l i t y f o r t h i s e x p e r i m e n t L e t us c h o o s e - t w o - d i f f e r e n t f l o w r a t e s Run No. AP exp (kPa) AP . •. c a l (kPa) Q ( c c / s ) C199 C202 14.31 59.31 15.20 56.45 0.03 0.305 The f o l l o w i n g e q u a t i o n w i l l be u s e d AP = ^ ^ Q n + a L K A n where a« = 3 Z°-o -127-1-n * e t t = f ( r 7 + 3 ^ ( 8 e K ) 2 • l ^ f - | ( I + 3 ) n ( 8 e ) ^ K ~ ^ ~ d 2 1) I f T q i s r e d u c e d b y 10%, i . e . T q = 10.566 dyne/cm Run No. A p c a l . % e r r o r i n A P c a l (kPa.) ( E q u a t i o n a) C199 15.2 - 6.5% C202 56.45 - 1.7% T h i s i n d i c a t e s a t l o w e r f l o w r a t e s , t h e y i e l d v a l u e has g r e a t e r i n f l u e n c e i n t h e p r e d i c t e d d a t a . T q h a s no d i r e c t e f f e c t on t h e e s t i m a t i o n o f V-eff o r K, however i t i s r e l a t e d t o K i n E q u a t i o n b b u t t h i s e q u a t i o n was n o t u s e d t o d e t e r m i n e K. I n a d d i t i o n , any change i n x 0 w i l l be a s s o c i a t e d w i t h c h a n g e s i n n and H by t h e H e r s c h e l - B u l k l e y m o d e l . 2) I f n i s r e d u c e d by 10%, i . e . f o r n = 0.84 i n s t e a d o f 0.9 3 we g e t : Run No. % e r r o r i n AP % e r r o r ' i n K ( E q u a t i o n a) ( E q u a t i o n d) C199 36.5 -44.8 -45 C202 53.5 -44.8 -45 % e r r o r i-n \x Cf. err ( E q u a t i o n c) -128-The sign ( + )' i n the error means increase and (-) means decrease. This shows how sensitive the model prediction, K and u e££ are to a small change i n : n and t h i s factor may explain the scatter of the data from the model. For t h i s reason a better f i t to the model should be obtained by selecting n from the flow data rather than from the viscometric ex-periments . 3) I f H i s reduced by 10% i . e . H = 15.87 an error of -10% i n K and -10.3% error i n the value of u~ M e f f r e s u l t s . 4) If the flow rate (Q) i s reduced by 10%, Run No. Q (cc/sec) AP , / c,al (kPa) % error i n AP Equation a C199 C202 0.027 0.2745 15.20 56.45 - 3.33 - 7.71 This shows higher percentage error i n the predicted pressure i s associated with higher flow rates, when ex-perimental measurements have the same percentage error i n Q. Q has no d i r e c t e f f e c t on K or y csz i n the above ^ef f equations, but any errors i n Q w i l l give d i f f e r e n t experi-y e f f mental value of — , hence w i l l e f f e c t these values. From the above four s e n s i t i v i t y t e s t s , i t i s clear that the model prediction and the f i t t i n g parameter K are very sensitive to the value of n, while T has more e f f e c t -129-on t h e model a t l o w e r f l o w r a t e s t h a n a t h i g h e r f l o w r a t e s . T h e s e t e s t s g i v e an e x p l a n a t i o n f o r t h e v a r i a t i o n s o f K and V e f f / and t h e d e v i a t i o n o f some o f t h e r e s u l t s f r o m t h e m o d e l , e s p e c i a l l y a t h i g h wax c o n t e n t and low t e m p e r a -t u r e s . 6.3.2 F r i c t i o n F a c t o r - R e y n o l d s Number C o r r e l a t i o n 6.3.2.1 R e s u l t s A l l t h e d a t a a r e p r e s e n t e d on t h e b a s i s o f a f r i c t i o n f a c t o r - R e y n o l d s Number c o r r e l a t i o n f o r t h e l a m i n a r f l o w r e g i o n . The d e v e l o p m e n t o f t h i s c o r r e l a t i o n was d i s c u s s e d i n S e c t i o n 3.4 o f t h i s t h e s i s . The r e s u l t s a r e shown a s : a) T a b l e s D - l , D-2, D-3, D-4, D-5, and D-6 i n t h e A p p e n d i x . b) G r a p h s : A l l d a t a a r e shown i n t h r e e g r a p h s o f t h e f * - R* c o r r e l a t i o n s , e • - •_ • The f i r s t . g r a p h , F i g u r e 6-12 . i s - f o r t h e s m a l l c o l u m n . The s o l i d l i n e r e p r e s e n t s t h e model p r e d i c t i o n . The s e c o n d g r a p h , F i g u r e 6-13 shows t h e d a t a f o r s o l u t i o n s o f C l a r u s - B and C l a r u s - C i n t h e l a r g e column, w h i l e t h e t h i r d g r a p h , F i g u r e 6-14 i s f o r t h e f l o w o f t h e waxy c r u d e o i l i n t h e l a r g e c o l u m n . T h r e e g r a p h s were u s e d b e c a u s e o f t h e l a r g e number o f d a t a p o i n t s . T a b l e D-7 shows t h e l e a s t s q u a r e f i t o f t h e e x p e r i m e n t a l d a t a . -130-F i g u r e 6-12: F r i c t i o n f a c t o r - R e y n o l d s number, c o r r e l a t i o n d a t a f o r s o l u t i o n s o f C l a r u s - C and C l a r u s - C f l o w i n g t h r o u g h a 91 cm l o n g s a n d column w i t h Dp = 0.077 cm a t d i f f e r e n t t e m p e r a t u r e s and wax c o n t e n t . - 1 3 1 -REYNOLDS NUMBER (Re*) F i g u r e 6-13: F r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n d a t a f o r s o l u t i o n s o f C l a r u s - B and C l a r u s - C f l o w i n g t h r o u g h a 100 cm l o n g s a n d column w i t h Dp = 0.128 cm a t d i f f e r e n t t e m p e r a t u r e s and wax c o n t e n t . -132-id 7 i d 6 i d 5 i d 4 103 i d 2 id 1 io° 10 REYNOLDS NUMBER (Re") F i g u r e 6-14: F r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n d a t a f o r Peace R i v e r c r u d e o i l f l o w i n g t h r o u g h a 100 cm l o n g sand column w i t h Dp = 0.12 8 cm a t d i f f e r e n t t e m p e r a t u r e s . -133-6.3.2.2 A n a l y s i s T h i s a n a l y s i s o f t h e e x p e r i m e n t a l r e s u l t s i s b a s e d on t h e f r i c t i o n f a c t o r - R e y n o l d s Number c o r r e l a t i o n i n t h e l a m i n a r r e g i o n ; i . e . t h e f l o w d a t a a r e p r e s e n t e d as C" e' where t h e c o n s t a n t C" comes f r o m t h e e x p r e s s i o n f o r t h e p e r m e a b i l i t y c o e f f i c i e n t e q u a t i o n 2-14a: 2 .2 D 3 «D 3 K = 2—5 = —-ELS 2-14a 72 C (1-e) 2 C" (1-e) 2 T a b l e s 2-11 page (47) summarizes r e c e n t p a p e r s and t h e v a l u e s o f C" u s e d by p r e v i o u s i n v e s t i g a t o r s . Carman (14) r e p o r t e d a v a l u e o f 180, E r g u n (26) 150, and L a r k i n s e t a l . (49) f o u n d i t t o be 118. T h e r e i s no u n i v e r s a l v a l u e f o r t h i s c o n s t a n t . V a l u e s o f 180 and 150 a p p e a r t o be more common f o r f i t t i n g s i m i l a r e x p e r i m e n t s w i t h d i f f e r e n t k i n d s o f f l u i d s and p a c k i n g s . The v a l u e o f C" depends on many f a c t o r s , s u c h a s ; s h a p e , t y p e and s i z e o f p a c k i n g s , a n d t h e t e c h n i q u e u s e d f o r l o a d -i n g t h e p a c k i n g . I n o u r c a s e , i t was d e c i d e d t o d e t e r m i n e t h e v a l u e C" e x p e r i m e n t a l l y by u s i n g N e w t o n i a n f l u i d s i . e . , C l a r u s - B and C l a r u s - C o i l s w i t h o u t wax, i n t h e column a t r e l a t i v e l y h i g h e r t e m p e r a t u r e s (25 and 3 0 ° C ) . The d a t a a r e shown i n -134-Table 6-5. Darcy's law f o r Newtonian f l u i d s (Equation 2-9) was a p p l i e d to determine the p e r m e a b i l i t y , then Equation 2-14a was used to determine the value of C". The c o n s t a n t was c a l c u l a t e d t o be 214.5 f o r the s m a l l sand bed and 281.5 f o r the l a r g e one. I t was decided to use an average value of 248 i n our model a n a l y s i s . The main equations which were used to i n t e r p r e t the data of Appendix D a r e : a) F r i c t i o n f a c t o r f o r the porous medium f * exp A P ) V 5 D _P L 1-e 2-16 f * . c a l . 248/R* e 6-1 b) M o d i f i e d Reynolds Number which was d e r i v e d i n S e c t i o n 3.4 i s 12p V 2 % R* = 3-29 2HD e 3 V n + e 2 x % p o • where n D .e n+1 ? = ^SH+V ( i r f e ) ) ( 1 _ £ ) 3 - 3 0 To t e s t the f i t of the experimental data the f o l l o w i n g methods are used: -135-I) Root Mean S q u a r e E r r o r , g i v e n i n S e c t i o n 6.3.1.2. I I ) P e r c e n t a g e E r r o r , g i v e n i n T a b l e D-6. I l l ) L e a s t S q u a r e F i t . The r e s u l t s a r e p r e s e n t e d i n T a b l e D-7 i n t h e A p p e n d i x . 6.3.2.3 D i s c u s s i o n I n most o f t h e r e p o r t e d s t u d i e s ( T a b l e 2-11) t h e p a r t i c l e s u s e d were s p h e r i c a l i n shape, w h i l e o u r s were r o u n d e d s a n d p a r t i c l e s . The v a l u e s o f C" (118, 150, 180, and 248) were c o r r e l a t e d f o r o u r d a t a i n t h e f o r m o f E q u a t i o n 2-15, i . e . f * = C/R* (2-15) The v a l u e o f C" = 248, w h i c h was d e t e r m i n e d i n t h i s s t u d y shows good a g r e e m e n t w i t h o u r d a t a as shown i n T a b l e 6-11, e s p e c i a l l y i f t h e 8 l o w e s t R e y n o l d s numbers a r e e x -c l u d e d . T h i s t a b l e i n c l u d e s a l l d a t a e x c e p t t h e c r u d e o i l d a t a ( i . e . f o r C l a r u s - B and C l a r u s - C s o l u t i o n s f o r t h e s m a l l and l a r g e c o l u m n s ) , t h u s 235 d a t a p o i n t s were t e s t e d . As c a n be s e e n i n t h e t a b l e t h e more p o i n t s a t low R e y n o l d s numbers a r e e x c l u d e d , t h e b e t t e r t h e f i t f o r t h e m o d e l . F o r a l l - 1 3 6 -T a b l e 6 - 1 1 S t a t i s t i c a l C o m p a r i s o n o f F r i c t i o n F a c t o r s ( f * = C " / R | ) f o r a l l 2 3 5 D a t a P o i n t s o f C l a r u s - B a n d C l a r u s - C S o l u t i o n s c" L o w e r L i m i t f o r R* e N o . o f D a t a P o i n t s f . c a l Ma x imum R o o f M e a n S q u a r e E r r o r I n c l u d e d E x c l u d e d 1 1 8 . 0 0 . 0 2 3 5 0 3 . 4 9 1 x l 0 9 ' 0 . 1 5 5 1 x l 0 + 9 1 5 0 . 0 0 . 0 2 3 5 0 4 . 4 3 8 x l 0 9 0 . 2 1 2 2 x l 0 + 9 1 8 0 . 0 0 . 0 235 0 5 . 3 2 5 x l 0 9 0 . 2 8 0 2 x l 0 + 9 2 4 8 . 0 0 . 0 2 3 5 0 7 . 3 3 7 x l 0 9 0 . 4 5 1 5 x l 0 + 9 1 1 8 . 0 0 . 5 x l 0 - 6 227 8 2 . 3 6 x l 0 8 -I Q 0 . 5 1 3 4 x 1 0 1 5 0 . 0 0 . 5 x l 0 " 6 227 8 3 . 0 x 1 0 8 " 0 . 4 6 0 4 x 1 0 1 8 0 . 0 0 . 5 x l 0 ~ 6 227 8 3 . 6 x l 0 8 0 . 4 2 6 0 x l 0+ 8 2 4 8 . 0 0 . 5 x l 0 ~6 227 8 4 . 9 6 x l 0 8 0 . 4 1 9 7 x l 0 + 8 1 1 8 . 0 O . l x l O - 5 2 2 1 14 l . L S x l O8 0 . 2 1 5 8 x l 0 + 8 1 5 0 . 0 0 . 1 x l 0 ~ 5 2 2 1 14 1 . 5 0 x l 0 8 0 . 1 8 7 9 x l 0 + 8 1 8 0 . 0 O . l x l O - 5 2 2 1 14 1 . 8 0 x l 0 8 0 . 1 7 2 1 x l 0+ 8 2 4 8 . 0 O . l x l O- 5 2 2 1 14 2 . 4 8 x l 0 8 0 . 1 8 4 5 x l 0 + 8 1 1 8 . 0 0 . 5 x l 0- 5 2 0 0 35 2 . 3 6 x l 0 7 0 . 9 1 6 3 x l 0 + 7 1 5 0 . 0 0 . 5 x l 0- 5 2 0 0 35 3 . 0 x l 07 0 . 8 1 2 6 x l 0 + 7 1 8 0 . 0 0 . 5 x l 0- 5 2 0 0 35 • 3 . 6 x l 07 0 . 7 3 2 1 x l 0 + 7 2 4 8 . 0 0 . 5 x l 0- 5 2 0 0 35 4 . 9 6 x I 0 7 0 . 6 3 9 8 x l 0 + 7 1 1 8 . 0 O . l x l O- 4 1 8 5 50 1 . 1 8 x l 07 0 . 4 9 8 7 x l 0 + ? 1 5 0 . 0 O . l x l O- 4 1 8 5 50 1 . 7 7 x l 07 •• 0 . 4 4 4 6 x l 0 + ? 1 8 0 . 0 O . l x l O- 4 1 8 5 50 1 . 8 0 x l 07 0 . 4 0 2 4 x l 0 + 7 . 2 4 8 . 0 O . l x l O - 4 1 8 5 50 2 . 4 8 x l 0 7 0 . 3 5 2 8 x l 0+ 7 -137-b u t t h e v e r y low R e y n o l d s number d a t a t h e v a l u e o f C" = 248 i s b e t t e r t h a n o t h e r c o n s t a n t s g i v e n i n t h e l i t e r a t u r e . I t s l a r g e - v a l u e i s most l i k e l y a shape f a c t o r e f f e c t . The maximum p e r c e n t a g e e r r o r i n t h e c a l c u l a t e d v a l u e s o f t h e l o g a r i t h m i c f r i c t i o n f a c t o r , b a s e d upon C" = 248 was 7.59% f o r 235 e x p e r i m e n t a l p o i n t s , ( T a b l e D-6 i n t h e A p p e n d i x ) . Hence f o r t h , i t was d e c i d e d t o u s e t h e v a l u e 248 f o r t h i s s t u d y , i . e . t h e f l o w d a t a i s t o be p r e s e n t e d i n t h e f o l l o w i n g c o r r e l a t i o n f * , = 248/R* 6-1 c a l c . '  se The r e s u l t s o f a l e a s t s q u a r e s l o g a r i t h m i c f i t o f t h e d a t a i s g i v e n i n T a b l e D-7 i n t h e A p p e n d i x . The l e a s t s q u a r e s l o p e i s -0.933 and t h e i n t e r c e p t i s 2.667, s h o w i n g s a t i s -f a c t o r y a g r e e m e n t w i t h t h e t h e o r e t i c a l v a l u e s o f -1.0 and 2.395, r e s p e c t i v e l y . T h e s e two v a l u e s c o r r e s p o n d t o an e r r o r o f 6.7% f o r t h e s l o p e and 11.36% f o r t h e i n t e r c e p t . T a b l e D-6 shows t h a t a maximum e r r o r o f 7.69% was o b t a i n e d f o r t h e c r u d e o i l a t t h e l o w e s t v o l u m e t r i c f l o w r a t e . I n g e n e r a l , t h e s c a t t e r o f t h e d a t a o c c u r s a t l o w e r v a l u e s o f R e y n o l d s number f o r e a c h s e t o f d a t a . The s c a t t e r o f t h e d a t a d o e s n o t seem t o e x h i b i t any p a r t i c u l a r p a t t e r n . I t i s assumed t h a t t h e s c a t t e r a t t h e l o w e r R e y n o l d s numbers a r i s e s f r o m e x p e r i m e n t a l e r r o r and t h e s e n s i t i v i t y t o e r r o r s i n m e a s u r e -ments r a t h e r t h a n some anomalous b e h a v i o u r o f t h e f l u i d s u s e d . -138-The s c a t t e r o f t h e d a t a a t low R e y n o l d s numbers were a l s o n o t i c e d by C h r i s t o p h e r (15) when u s i n g t h e power-law m o d e l . S a d o w s k i (7 5) e x p l a i n e d t h i s s c a t t e r as b e i n g due t o t h e f a i l u r e o f t h e power-law model t o d e s c r i b e t h e s h e a r s t r e s s - s h e a r r a t e b e h a v i o u r a t low s h e a r s t r e s s e s o r what has been c a l l e d b elow t h e " l o w e r l i m i t i n g v i s c o s i t y " when t h e v i s c o s i t y s h o u l d a p p r o a c h a c o n s t a n t v a l u e a t low s h e a r s t r e s s e s . I n t h e c a s e o f t h e H e r s c h e l - B u l k l e y model t h e Ejn t e r m a t low s h e a r r a t e s becomes s m a l l compared t o T Q and t h e model becomes i n s e n s i t i v e t o i t . R e s u l t s o f t h e s m a l l c o l u m n F i g u r e 6-12 show more s c a t t e r f o r C l a r u s - B o i l t h a n C l a r u s - C o i l b e c a u s e o f t h e h i g h e r wax c o n t e n t i n t h e C l a r u s - B o i l . A l s o , t h e s c a t t e r o f t h e d a t a f o r t h e l a r g e c o l u m n F i g u r e 6-13 i s l e s s t h a n t h e d a t a f o r t h e s m a l l column F i g u r e 6-12, b e c a u s e o f l a r g e r p a r t i c l e s i z e s u s e d i n t h e l a r g e s a n d bed. The c r u d e o i l d a t a F i g u r e 6-14, show t h e b e s t f i t o f a l l t h e d a t a , e v e n t h o u g h t h e c r u d e o i l d i d n o t p l a y any r o l e i n e v a l u a t i n g t h e c o n s t a n t 248. A l s o h i g h e r R e y n o l d s numbers were u s e d f o r t h e c r u d e o i l . The a d v a n t a g e s o f p r e s e n t i n g o u r r e s u l t s i n t e r m s o f a f r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n o v e r m o d i f i e d D a r c y ' s law a r e t h e f o l l o w i n g : a) I t i s more g e n e r a l and a c c e p t e d t o a g r e a t e r d e g r e e by many i n v e s t i g a t o r s , and i t i s t h e r e f o r e , e a s i e r t o compare o u r r e s u l t s w i t h t h e i r s . -139-b) i t was n o t n e c e s s a r y t o f i n d t h r e s h o l d g r a d i e n t ( a Q L ) t e r m by e x t r a p o l a t i o n of•'_the l o w e s t t h r e e p o i n t s . c) T h i s c o r r e l a t i o n g ave a b e t t e r f i t f o r : t h e e x p e r i m e n t a l d a t a a t h i g h e r f l o w r a t e s , w i t h i n t h e r a n g e i n v e s t i g a t e d . 6.3.2.4 The S i g n i f i c a n c e o f F i t t i n g P a r a m e t e r s I n o r d e r t o u n d e r s t a n d t h e s c a t t e r o f t h e d a t a -f r o m t h e m o d e l , e s p e c i a l l y a t low R e y n o l d s numbers, t h e s e n s i t i v i t y o f f * - R* c o r r e l a t i o n was c a l c u l a t e d f o r s m a l l e r r o r s i n t h e measurement o f T , n, H, and Q f o r t h e same o e x p e r i m e n t u s e d i n S e c t i o n 6.3.1.4. Run No V Q ( c m / s ) f * exp. f * . c a l . R* e C199 C202 0.3699 x 10~ 3 0.3761 x 10~ 2 2.292 x 10 8 9.190 x 10 6 4.547 x 10 8 9.541 x 10 6 5.454 x 10~ 7 2.599 x 10~ 5 f * exp. AP D p 1-e f * 1 c a l , 248  Re* where R* = e 12p ? 2H D„ e 3 V n + e 2 T P -140-a n d n+1 5 = .6 ( 3 ^ ) \ ^ r ^ : A U-e) 2 1) I f x Q i s r e d u c e d by 10%, i . e . T q = 10.566 dyne/cm % E r r o r i n % E r r o r i n Run NO. R*. R* , . f C a l . CI 99. 5.454 x I O - 7 8.95 -8.2 C202 2.599 x 1 0 _ 5 3.93 -3.8 T h i s t a b l e e x p l a i n s t h e s c a t t e r o f t h e d a t a a t low ReynoldsV.number, ' when t h e y i e l d v a l u e has a g r e a t e f f e c t o n . t h e r e s u l t s . 2) I f n i s r e d u c e d by 10%, i . e . n = 0.84 % E r r o r i n % E r r o r i n Run No. R* R*: f * ^ " •- e e c a i . C199 5.454 x I O - 7 17.67 -15 C202 2.599 x 1 0 ~ 5 -1.4 1.4 T h i s a l s o shows how s e n s i t i v e t h e m o d e l i s t o t h e v a l u e o f n. I t i n d i c a t e s t h a t any s m a l l change i n n w i l l r e s u l t i n a l a r g e e r r o r i n f * , e s p e c i a l l y a t l o w e r f l o w c a l . r a t e s o r R* and l e s s e r r o r s a t h i g h e r f l o w r a t e s w h i c h i s •e' e x a c t l y what we have f o u n d f r o m t h e e x p e r i m e n t s , as c a n be s e e n f r o m t h e t h r e e g r a p h s o f f * v e r s u s R*. -141-3) I f H i s r e d u c e d by 10%, i . e . H = 1 5 . 8 7 Run No. % E r r o r i n % E r r o r i n R* f * ,. Q c a l . C199 1.3 -1.24 C202 6.33 -5.8 T h i s i n d i c a t e s t h a t t h e e r r o r i n H has l i t t l e e f f e c t on R* o r f * e s p e c i a l l y a t low R*. 4) I f Q i s r e d u c e d by 10%, i . e . , Run No. V (cm/sec) % E r r o r i n , % E r r o r i n % E r r o r i n -f * exp. f * , c a l -4 C199 3.3291 x 10 23.5 -18 22 -3 C202 3.3849 x 10 23.5 -14.2 16.6 T h i s t a b l e shows t h a t any s m a l l e r r o r s i n f l o w r a t e measurements w i l l r e s u l t i n l a r g e r e r r o r s i n t h e model p r e d i c t i o n . I t i s c l e a r f r o m t h e s e s e n s i t i v i t y t e s t s t h a t t h e model i s v e r y s e n s i t i v e t o any e r r o r s i n T q and n a t low R e y n o l d s number and l e s s s e n s i t i v e t o x and n a t h i g h e r R*. On t h e o 3 e o t h e r hand e r r o r i n H has a v e r y s m a l l b u t o p p o s i t e e f f e c t on R* and f * , w h i l e i t h a s a g r e a t e r e f f e c t a t h i g h e r R* t h a n a t low R*. F i n a l l y any s m a l l e r r o r s i n Q w i l l r e s u l t -142-i n l a r g e r e r r o r s i n f * , f * n , and R*. ^ exp. c a l . e Th e s e t e s t s were v e r y u s e f u l i n e x p l a i n i n g t h e s c a t t e r o f t h e d a t a a t l o w e r R e y n o l d s " numbers and h i g h wax c o n t e n t ( i . e . h i g h e r y i e l d v a l u e ) . 6.4 R e p r o d u c i b i l i t y Some o f t h e y i e l d v a l u e s m e a s u r e d w i t h t h e v i s c o -m e t e r were compared w i t h t h e v a l u e s o b t a i n e d w i t h t h e s u s -pended p l a t e y i e l d v a l u e a p p a r a t u s ( S e c t i o n 4) and c l o s e a g reement was o b t a i n e d . Some o f t h e s e v a l u e s a r e shown below. T a b l e 6-12 T y p i c a l V a l u e s o f t h e Y i e l d V a l u e Measurements f o r 4% Wax i n C l a r u s - B . O i l Type T e m p e r a t u r e Y i e l d V a l u e (T ) (dyne/cm 2) (°C) S u s p e n d e d • P l a t e ' R o t a t i o n a l — V i s c o m e t e r 4% wax i n C l a r u s - B o i l 16 18 20 16.66 4.44 1.85 16.36 4.80 1.96 Measurements o f s h e a r s t r e s s - s h e a r r a t e were r e p e a t e d w i t h a n o t h e r v i s c o m e t e r , R o t o v i s c o RV-3 and v e r y good a g r e e -ment was o b t a i n e d w i t h t h e measurements made w i t h t h e R o t o -v i s c o t y p e RV-12. T y p i c a l e x a m ples a r e shown i n F i g u r e 6-15. -143-ft 20 UJ tr i — CO LT < LLl CO F i g u r e 6-15: 120 160 200 SHEAR RATE (sec"1) S h e a r s t r e s s - s h e a r r a t e c u r v e s f o r C l a r u s - B s o l u -t i o n s a t d i f f e r e n t t e m p e r a t u r e s , m e a s u r e d on two r o t a t i o n a l v i s c o m e t e r s , R o t o v i s c o RV-12 and RV-3 t o show t h e r e p r o d u c i b i l i t y o f t h e r h e o l o g i c a l d a t a , where t h e s o l i d l i n e s r e p r e s e n t t h e H e r s c h e l - B u l k l e y m o del f i t . -144-In t h e s t u d y o f t h e f l o w t h r o u g h t h e p o r o u s m e d i a , s e v e r a l r u n s were r e p r o d u c e d i n t h e s m a l l and l a r g e s a n d b e d s , as shown i n F i g u r e s 6-16 and 6-17 and 6-18, and T a b l e 6-13. The r e p r o d u c i b i l i t y o f t h e f l o w s y s t e m r e s u l t s i s d e p e n d e n t on t h e t e c h n i q u e u s e d i n p e r f o r m i n g t h e e x p e r i -ments and on t h e s y s t e m i t s e l f . I n g e n e r a l , p o o r r e p r o d u c i b i l i t y i s a s s o c i a t e d w i t h h i g h wax c o n c e n t r a t i o n s and s m a l l s i z e s o f p a r t i c l e d i a m e t e r . -145-FLOW RATE (ml/sec) F i g u r e 6-16: P r e s s u r e d r o p - f l o w r a t e c u r v e s s h owing t h e r e p r o d u c i b i l i t y o f t h e f l o w d a t a , where t h e s o l i d l i n e s r e p r e s e n t t h e model f i t o f e q u a t i o n 3-18. *The numbers 91 and 100 r e p r e s e n t t h e s a n d c o l u m n s ' l e n g t h i n c e n t i m e t e r s . -146-i 1 1 r 0.1 0.2 0.3 0.4 0.5 0.6 FLOW RATE (ml/sec) F i g u r e 6-17: P r e s s u r e d r o p - f l o w r a t e c u r v e s f o r 5% wax i n C l a r u s - B o i l a t T = 16°C f l o w i n g t h r o u g h a 100 cm l o n g s a n d column w i t h Dp = 0.128 cm where t h e s o l i d l i n e r e p r e s e n t s t h e m odel f i t o f e q u a t i o n 3-18, s h o w i n g t h e r e -p r o d u c i b i l i t y o f t h e d a t a . io 8 id 7 1 6 6 1 6 5 1 6 4 REYNOLDS NUMBER (Re*) F i g u r e 6-18: F r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n d a t a f o r 5% wax i n C l a r u s - B o i l a t 16°C f l o w i n g t h r o u g h a 10 0 cm l o n g s a n d column w i t h Dp = 0.128 cm, showing t h e r e p r o d u c i b i l i t y o f t h e d a t a . - 1 4 8 -T a b l e 6 - 1 3 E x p e r i m e n t a l R e s u l t s f o r t h e F l o w o f 5% Wax i n C l a r u s - B O i l T h r o u g h t h e L a r g e C o l u m n a t 1 6 ° C , R e p e a t e d T h r e e T i m e s ( D a t a a r e P l o t t e d i n F i g u r e s 6 - 1 7 a n d 6 - 1 8 ) ( 1 ) R e p o r t e d AP e x p ( k P a ) V x 1 0 4 o ( c m / s e c ) f * x 10 6 e x p R* x 1 0 8 e 1 4 . 5 8 1 . 4 9 2 1 4 3 4 . 3 . 3 9 6 3 6 . 1 1 4 . 4 1 4 4 0 5 . 9 2 9 . 3 0 5 5 . 1 4 8 . 7 3 0 1 5 8 . 5 1 1 2 . 6 0 7 5 . 2 8 1 3 . 9 7 0 8 4 . 4 8 2 8 3 . 1 0 9 5 . 4 2 2 0 . 4 4 4 5 0 . 0 5 9 3 . 8 0 1 1 1 . 9 5 3 1 . 5 1 6 6 2 5 . 2 7 1 3 6 . 7 0 1 4 1 . 7 0 6 7 . 1 0 2 3 6 . 8 9 3 5 6 9 1 . 0 ( 2 ) R e p e a t e d 2 3 . 2 0 1 . 3 9 3 2 6 1 9 2 . 9 6 2 3 6 . 2 5 2 . 1 2 1 1 7 6 5 6 . 8 3 9 5 0 . 8 4 3 . 7 3 6 7 9 7 . 8 2 1 . 0 6 6 8 . 4 8 9 . 2 4 8 1 7 5 . 4 1 2 6 . 2 8 4 . 0 3 1 5 . 0 0 6 8 1 . 7 3 3 2 5 . 5 1 0 2 . 0 9 2 8 . 0 5 2 2 8 . 4 1 1 0 9 3 1 2 0 . 4 3 4 3 . 1 5 7 1 4 . 1 6 2 4 8 7 ( 3 ) R e p e a t e d 3 2 . 3 6 3 . 3 5 4 6 3 0 . 0 1 7 . 0 4 8 . 6 2 7 . 1 8 9 2 0 6 . 0 7 6 . 8 3 7 0 . 0 1 1 2 . 8 4 8 9 2 . 8 9 2 4 0 . 4 0 9 1 . 9 5 2 0 . 1 3 6 4 9 . 6 7 5 7 6 . 6 0 1 1 2 . 7 9 3 2 . 3 6 7 2 3 . 5 8 1 4 3 9 . 0 1 2 9 . 4 5 5 2 . 4 0 4 1 0 . 3 2 3 5 8 7 . 0 -149-CHAPTER SEVEN C O N C L U S I O N The r e s u l t s o f t h e p r e s e n t i n v e s t i g a t i o n may be summarized as f o l l o w s : 1) The t h r e e - p a r a m e t e r H e r s c h e l - B u l k l e y m odel ( E q u a t i o n 2-6) was shown t o d e s c r i b e w e l l t h e s t e a d y -s t a t e n o n - Newtonian b e h a v i o u r o f waxy o i l s w i t h y i e l d s t r e s s e s . The p a r a m e t e r s o f t h i s model were d e t e r m i n e d by t h e g e n e r a l p r o c e d u r e p r e s e n t e d on S e c t i o n 6.2.2. The model gave an e x c e l l e n t f i t t o v i s c o m e t r i c d a t a f o r v a l u e s o f s h e a r r a t e up t o 27 70 s--'-. The a d v a n t a g e o f t h i s m o d e l i s t h a t i t c a n d e s c r i b e t h e f l o w b e h a v i o u r o f Bingham f l u i d s as w e l l as t h o s e w h i c h show power-law b e h a v i o u r . The waxy o i l s t e s t e d showed more n o n - N e w t o n i a n b e h a v i o u r as t h e wax c o n t e n t i n c r e a s e d and t h e t e m p e r a t u r e d e c r e a s e d . 2) The d a t a f r o m t h e e x p e r i m e n t s on t h e f l o w o f n o n - N e w t o n i a n f l u i d s w i t h y i e l d s t r e s s e s t h r o u g h t h e s a n d -150-b e d s , g e n e r a l l y f i t t e d t h e d e v e l o p e d m a t h e m a t i c a l m o d e l s w e l l . The r e s u l t s were p r e s e n t e d i n two f o r m s . 2a) M o d i f i e d D a r c y ' s Law E q u a t i o n s (3-18, 3-23). T h i s m odel shows h i g h s e n s i t i v i t y t o t h e r h e o l o g i c a l p a r a m e t e r s , p o r o u s medium p r o p e r t i e s and method o f a n a l y s i s . F o r s u f f i c i e n t l y s m a l l p a r t i c l e s and s u f f i c i e n t l y l a r g e wax c o n c e n t r a t i o n s , t h e o b s e r v e d r e s u l t s were p o o r , w h i l e f o r low wax c o n c e n t r a t i o n and l a r g e r p a r t i c l e s t h e r e s u l t s were good. A l s o i t was o b s e r v e d t h a t t h e a c t u a l perme-a b i l i t y f o r n o n - N e w t o n i a n f l u i d s i s s m a l l e r t h a n f o r N e w t o n i a n f l u i d s and t h e d e g r e e o f d i v e r g e n c e f r o m t h e New-t o n i a n p e r m e a b i l i t y i n c r e a s e s w i t h t h e y i e l d s t r e s s o f t h e f l u i d , t h e v o l u m e t r i c f l o w r a t e and t h e s i z e o f p a r t i c l e s u s e d . T h i s phenomenon was a l s o o b s e r v e d by S a d o w s k i and B i r d (75) when t h e y i n v e s t i g a t e d t h e f l o w o f p o l y m e r s o l u t i o n s t h r o u g h p o r o u s m e d i a . 2b) F r i c t i o n f a c t o r - R e y n o l d s number c o r r e l a t i o n ( E q u a t i o n 2 - 1 5 ) . T h i s f o r m o f t h e model a n a l y s i s showed e x c e l l e n t a g r e e m e n t between t h e e x p e r i m e n t a l and p r e d i c t e d r e s u l t s e s p e c i a l l y a t h i g h e r R e y n o l d s number. The i n c r e a s e d s c a t t e r i n t h e d a t a a t low R e y n o l d s numbers was e x p l a i n e d by t h e i n c r e a s e d s e n s i t i v i t y o f t h e m odel t o any s m a l l e r r o r i n t h e measurement o f t h e y i e l d v a l u e (T q) and t h e r h e o -l o g i c a l f l o w b e h a v i o u r i n d e x n as f l o w d e c r e a s e d . T h i s c o r r e l a t i o n d o e s n o t c o n t a i n t h e p e r m e a b i l i t y p a r a m e t e r K, but t h e f i t t i n g c o n s t a n t C"=248 w h i c h was d e t e r m i n e d e x p e r i m e n t a l l y w i t h N e w t o n i a n oil. The p e r m e a b i l i t y , however, i s r e l a t e d t o C". -151-CHAPTER EIGHT RECOMMENDATIONS S i n c e t h i s s t u d y has a l r e a d y c o v e r e d many o f t h e v a r i a b l e s w h i c h m i g h t e f f e c t t h e f l o w o f n o n -N e w t o n i a n f l u i d w i t h y i e l d s t r e s s e s t h r o u g h t h e p o r o u s medium, i t i s recommended t o : 1) T e s t t h e model w i t h h e a v y c r u d e o i l s o f d i f f e r e n t ° A P I g r a v i t i e s , s i n c e t h e s e c r u d e s showed s i m i l a r r h e o l o g i c a l b e h a v i o u r s t o t h o s e o f waxy o i l s , whose r h e o l o g i c a l b e h a v i o u r s c a n be d e s c r i b e d by t h e H e r s c h e l - B u l k l e y m o d e l . 2) Do f u r t h e r s t u d i e s on t h e p o s s i b i l i t y o f t h e p r e s e n c e o f wax a d s o r p t i o n on t h e p a r t i c l e s u r f a c e and i t s e f f e c t on t h e change i n t h e v a l u e o f t h e perme-a b i l i t y . 3) Do more s t u d i e s on t h e f l o w o f N e w t o n i a n f l u i d s t h r o u g h s a n d beds t o d e t e r m i n e t h e e x a c t v a l u e f o r t h e f a c t o r C" i n E q u a t i o n 2-15. T h i s c a n be done -152-by u s i n g d i f f e r e n t D /D r a t i o s f o r t h e s a n d and c p d i f f e r e n t t e c h n i q u e s f o r p a c k i n g . S i n c e i t was o b s e r v e d f r o m t h e l i t e r a t u r e s u r v e y t h a t most i n -v e s t i g a t o r s u s e a v a l u e w h i c h s u i t s t h e i r d a t a b e s t , i n t h i s s t u d y we p r o p o s e d t h e v a l u e o f 248 w h i c h gave us t h e b e s t f i t f o r o u r d a t a . -153-N O M E N C L A T U R E a c o n s t a n t i n e q u a t i o n (3-2) (cm ) a* c o n s t a n t i n e q u a t i o n (5-11) a V s p e c i f i c s u r f a c e i n e q u a t i o n (3-3) ( c m - 1 ) A c r o s s s e c t i o n a l a r e a o f t h e column (cm ) A' s h e a r s t r e s s f a c t o r i n e q u a t i o n (5-1) A s u r f a c e a r e a '(cm 2) C c o n s t a n t i n e q u a t i o n (2-12) C" c o n s t a n t i n e q u a t i o n (2-15) C R c o r r e c t i o n f a c t o r i n t a b l e (B-6) D d i a m e t e r (cm) D C c o l u m n d i a m e t e r (cm) D P p a r t i c l e d i a m e t e r (cm) E power o f t e n i n t h e computer p r i n t o u t f shape f a c t o r i n e q u a t i o n (5-8) (cm ) f * f r i c t i o n f a c t o r f o r p o r o u s m e d i a (-) f * exp e x p e r i m e n t a l f r i c t i o n f a c t o r (-) c a l c a l c u l a t e d f r i c t i o n f a c t o r (-) f f r i c t i o n f a c t o r i n t u b e s (-) F f o r c e (dyne) G v i s c o s i t y f a c t o r i n t a b l e 6-3 H c o n s i s t e n c y i n d e x i n e q u a t i o n (2-6) (dyne n , 2, s /cm ) H* c o n s i s t e n c y i n d e x i n e q u a t i o n (2-7) (dyne n , 2. s /cm ) -154-h h e i g h t o f t h e r o t o r K p e r m e a b i l i t y o f p o r o u s medium K c h a r a c t e r i s t i c k i n e t i c e n e r g y p e r u n i t volume i n e q u a t i o n ( 3 - 2 7 ) L l e n g t h o f t u b e o r column L g e f f e c t i v e l e n g t h o f f l o w b a t h M s h e a r r a t e f a c t o r i n e q u a t i o n (5-2) t o r q u e n f l o w b e h a v i o u r i n d e x i n e q u a t i o n (2-6) n' r o t o r s p e e d i n e q u a t i o n (5-2) N number o f d a t a p o i n t s P p r e s s u r e P^ p r e s s u r e i n f i g u r e (3-1) P Q p r e s s u r e i n f i g u r e (3-1) q v o l u m e t r i c f l o w r a t e Q v o l u m e t r i c f l o w r a t e Q Q v o l u m e t r i c f l o w r a t e i n s i d e t h e c o r e Q Q c o n s t a n t i n e q u a t i o n (2-24) Q 1 c o n s t a n t i n e q u a t i o n (2-24) Q L v o l u m e t r i c f l o w r a t e between c o r e and w a l l (cm) (cm ) -1 (cm) (cm) ( s . rpm) (dyne, cm) (-) (rpm) (-) 2 (dyne/cm ) 2 (dyne/cm ) 2 (dyne/cm ) ( c c / s ) ( c c / s ) ( c c / s ) ( c c / s ) Q T t o t a l v o l u m e t r i c f l o w r a t e ( c c / s ) R r a d i u s o f t u b e (cm) R a r a d i u s o f o u t e r c y l i n d e r (cm) R. l r a d i u s o f i n n e r c y l i n d e r (cm) R C r a d i u s o f c a p i l l a r y t u b e (cm) R h h y d r a u l i c r a d i u s i n e q u a t i o n (3-5) (cm) -155-K e. S T u V V o <v> V max w x y Y i Y Y. a a a c a c u app e f f e f f m o d i f i e d R e y n o l d s number f o r p o r o u s m e d i a s h e a r s t r e s s i n d i c a t o r t e m p e r a t u r e v e l o c i t y v e l o c i t y s u p e r f i c i a l v e l o c i t y a v e r a g e v e l o c i t y i n t u b e s maximum v e l o c i t y a n g u l a r v e l o c i t y s h e a r s t r e s s y i e l d v a l u e s h e a r s t r e s s a t r a d i u s R. 1 s h e a r s t r e s s i n e q u a t i o n (2-1) s t r a i n s h e a r r a t e (du/dy) c o n s t a n t d e f i n e d i n e q u a t i o n (2-28) s h e a r r a t e a t r a d i u s R. l c o n s t a n t d e f i n e d i n e q u a t i o n (3-12) c o n s t a n t d e f i n e d i n e q u a t i o n (2-24) t h r e s h o l d g r a d i e n t i n e q u a t i o n (3-25) t h r e s h o l d g r a d i e n t i n e q u a t i o n (3-20) N e w t o n i a n v i s c o s i t y p l a s t i c v i s c o s i t y i n e q u a t i o n (2-5) a p p a r e n t v i s c o s i t y i n e q u a t i o n (2-18) n o n - N e w t o n i a n b e d f a c t o r i n e q u a t i o n (3-24) n o n - N e w t o n i a n b e d f a c t o r i n e q u a t i o n (3-19) (-) ° C ) cm/s) cm/s) cm/s) cm/s) cm/s) r a d . / s ) 2 dyne/cm ) 2 dyne/cm ) 2 dyne/cm ) 2 dyne/cm ) -) s " 1 ) s " 1 ) -) 3 dyne/cm ) 3 dyne/cm ) mPa.s) mPa.s) mPa.s) d y n e . s n / c m 1 + n ) -, n / 1+n. d y n e . s /cm ) -156-du/dy v e l o c i t y gradient ( y ) (s" 1) AP pressure drop 2 (dyne/cm ) AP exp experimental pressure drop (dyne/cm2) AP , c a l calculated pressure drop 2 (dyne/cm ) e porosity (-) ? factor defined i n equation (3 -30) (cm ) P density (gm/cc) e constant defined i n equation (3-25) (-) F constant defined i n equation (3-20) (-) X parameter defined i n table ( B -6) (-) t time (s) -157-B I B L I O G R A P H Y 1. A l - F a r i s s , T a r i q F., "Non-Newtonian O i l F l o w T h r o u g h P o r o u s M e d i a " , R e s e a r c h p r o p o s a l s u b m i t t e d f o r Ph.D. Degree i n Ch.Eng., UBC, Van. (Feb. 1 9 8 2 ) . 2. A l - F a r i s s , T., P a s c a l , H., and P i n d e r , K.L. "Flow T h r o u g h P o r o u s M e d i a o f a S h e a r - T h i n n i n g L i q u i d w i t h a Y i e l d V a l u e " , P a p e r p r e s e n t e d a t t h e 5 5 t h A n n u a l M e e t i n g o f t h e S o c i e t y o f R h e d l o g y , K n o x v i l l e , T e n n e s s e e ( O c t o b e r 1983) . 3. A l l e n , A.B., and Work, L . T . / " S o l u b i l i t y o f R e f i n e d P a r a f f i n Waxes i n P e t r o l e u m F r a c t i o n " , I n d . Eng. Chem., V o l . 30, No. 7 J u l y ( 1 9 3 8 ) . 4. A r a n h a , H.P., e t . a l . , " S t u d y o f F l o w I m p r o v e r s f o r T r a n s p o r t a t i o n o f Bombay H i g h C r u d e O i l T h r o u g h Submarine P i p e l i n e s " , J...P.T. , 2539 (Dec. 1 9 8 1 ) . 5. B a r r y , E.G. , "Pumping o f Non-^Newtonian-Waxy Crude' O i l s " , J . I n s t . P e t . , ( M o b i l R e s e a r c h and D e v e l o p m e n t C o r p n . , N.J.) V o l . 57, No. 554 (March 1 9 7 1 ) . 6. B a t y c k y , J.P.* " P i n p e l i n e F l o w o f T h i x o t r o p i c O i l " , Ph.D. T h e s i s i n C h e m i c a l E n g i n e e r i n g , U n i v e r s i t y o f C a l g a r y , (1972).. 7- B i l d e r b a c k , C A . and M c D o u g u l l , L.A., "Complete P a r a f f i n C o n t r o l i n P e t r o l e u m P r o d u c t i o n " , J . P . T . (Sept.1969) 8. B i r d , B., and S a d o w s k i , T . J . , "Non-Newtonian F l o w T h r o u g h P o r o u s M e d i a " , T r a n s . S o c . R h e o l o g y , 9:2, 243-271 (1965) . 9. B i r d , R.B., S t e w a r t , W.E. and L i g h t f o o t , E.N., T r a n s p o r t Phenomena, J o h n W i l e y & Sons, I n c . N.Y. ( 1 9 6 0 ) . 10. B l a k e , F.C., "The R e s i s t a n c e o f P a c k i n g t o F l u i d F l o w " , P a p e r p r e s e n t e d a t A..I-.Ch_.E. M e e t i n g i n Richmond, Dec. 7, (1922). 11. B o t t , T.R., and Gudmundsson, J.S., " D e p o s i t i o n o f P a r a f f i n Wax From F l o w i n g S y s t e m s " , U n i v e r s i t y o f B i r m i n g h a m , I n s t i t u t e o f P e t r o l e u m , IP77-007 (1977) .. -158-12. B r o d , M. et,, a l . , " F i e l d E x p e r i e n c e w i t h t h e Use o f A d d i t i v e s i n t h e P i p e l i n e T r a n s p o r t a t i o n o f Waxy C r u d e s " , J . I n s t . P e t . , V o l . 57, No. 554, 111 (March 1971). 13. B u r g e r , E.D., P e r k i n s , T.K. and S t r i e g l e r , J.H. (SPE, ARCO O i l and Gas Co.) " S t u d i e s o f Wax D e p o s i t i o n i n t h e T r a n s A l a s k a P i p e l i n e s " ; J . P.T. , pp. 107-5-1.086 (June 19 81). 14. Carman, P.C. " F l u i d F l o w T h r o u g h G r a n u l a r B e d s " , T r a n s . . I n s . Chem. E n g r s . , London, 15, 50 (1937). 15. C h r i s t o p h e r , R.H., "Power-Law F l o w T h r o u g h P o r o u s M e d i a " , M.Sc. T h e s i s , Chem. Eng., U n i v . o f R o c h e s t e r , R o c h e s t e r , N.Y. (1965). 16. C h r i s t o p h e r , R. and Midd l e m a n , S., "Power-Law F l o w T h r o u g h A P a c k e d Tube", I n d and Eng. Chem. Fund., V o l . 4 , No. 4 (Nov. 1965). 17. Cohen, T., and M e t z n e r , A.B., " W a l l E f f e c t s i n L a m i n a r Flow o f F l u i d s T h r o u g h P a c k e d B e d s " , A . I . C h . E . J . , V o l . 27, No. 5, 705 ( S e p t . 1981). 18. C o l l i n s , R.E., and G r u e s b e c k , C., " E n t r a i n m e n t and D e p o s i t i o n o f F i n e P a r t i c l e s i n P o r o u s M e d i a " , S . P . E . J . , 847 (Dec. 1982). 19. C o o p e r , D.F., S m i t h , J.W., Ryan, E . J . and A l e x a n d e r , G., " T r a n s i e n t T e m p e r a t u r e E f f e c t s i n P r e d i c t i n g S t a r t -up C h a r a c t e r i s t i c s o f G e l l i n g - T y p e C r u d e O i l s " , H e a t T r a n s f e r , P u b l i s h e d by N a t i o n a l R e s e a r c h C o u n c i l o f Canada 1978. 20. C r a f t , B.C. and Hawkins, M.F., 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 , P r e n t i c e - H a l l , I n c . N . J . (1959,). 21. Dake, L , P . , F u n d a m e n t a l s o f R e s e r v o i r E n g i n e e r i n g , E l s e v i e r S c i e n t i f i c P u b l i s h i n g Company, N.Y. 1978. 22. D a v e n p o r t , T.C. and Somper, R.S.,"The Y i e l d V a l u e and Breakdown o f C r u d e O i l G e l s " , ( B r i t i s h P e t r o l e u m Co. L t d . , BP R e s e a r c h C e n t r e , M i d d l e s e x ) J . I n s t . P e t . 57, 554 (March 1971). 23. D e a l y , J . M . , " V i s c o s i t y o f O i l Sands L i q u i d s " , P u b l i s h e d i n The O i l Sands o f Canada - V e n e z u e l a , pp. 303-306 (1977). 24. D u e r k s e n , J.H., and Hsueh, L., "Steam,., D i s t i l l a t i o n o f Cr u d e O i l s " , S . P . E . J . , 265 ( A p r i l 1983). -159-25. E l l s , J.W., and Brown, V.R., "The D e s i g n o f P i p e l i n e s t o H a n d l e Waxy C r u d e O i l s " , J . I n s t . P e t . , V o l . 57, No. 555 (May 1971). (The B r i t i s h P e t . Comp. L t d . ) 26. E r g u n , S., " F l u i d F l o w T h r o u g h P a c k e d Columns", Chem. Eng. P r o g . , V o l . 48, No. 2, 89 (19 52). 27. E v g e n ' e v , A . E . and T u r n i e r , V . N . , " R h e o l o g i c a l P r o p e r t i e s o f Foam i n a P o r o u s Medium", I n t . Ch. Eng., V o l . 9, No. 2 ( A p r i l 1969). •".&> 28. G a i t o n d e , N.).Y., and Midd l e m a n , S., "Flow o f V i s c o e l a s t i c F l u i d s T h r o u g h P o r o u s M e d i a " , I.E.C . Fund., V o l . 6, No. 1, 145 (Feb. 1967). 29. G i l l , F., and R u s s e l l , R.J., " P u m p a b i l i t y o f R e s i d u a l F u e l O i l s " , I n d . Eng. Chem., V o l . 46, No. 6, 1264 (June 1954). 30. G r e e n k o r n , R.A., " S t e a d y F l o w T h r o u g h P o r o u s M e d i a " , A . I . C h . E . J . , V o l . 27, No. 4, 529 ( J u l y 1981). 31. G r e g o r y , D.R., et-* a l . , "Flow o f M o l t e n P o l y m e r s T h r o u g h P o r o u s M e d i a " , A . I . C h . E . J . , V o l . 13, No. 1 ( J a n u a r y 1967) . 32. Gudmundsson, J . S . and B o t t , T.R., " S o l u b i l i t y o f P a r a f f i n Wax i n K e r o s e n e " , F u e l , V o l . 56 ( J a n u a r y 1977). 33. Haber, S. and M a u r i , R., "Boundary C o n d i t i o n s f o r D a r c y ' s F l o w T h r o u g h P o r o u s M e d i a " , I n t . J . M u l t i p h a s e F l o w , V o l . 9, No. 5, 561 (1983). 34. Hanks, R.W., and R i c k s , B.L., " L a m i n a r - T u r b u l e n t T r a n s i -t i o n i n F l o w o f P s e u d o p l a s t i c F l u i d s w i t h Y i e l d S t r e s s e s " , J . H y d r o n a u t i c s , V o l . 8, No. 4, 163 ( O c t . 1974). 35. H a r v e y , A.H., e t a l . , " P i p e l i n e D e s i g n f o r G e l l i n g O i l s " , P a r t I , The O i l and Gas J . ( A u g u s t 23, 1971). 36. H a r v e y , A.H., and A r n o l d , M.D., " P i p e l i n i n g O i l s Below T h e i r Pour P o i n t " , P a r t I I , The O i l and Gas J . (Au g u s t 30, 1971) . 37. H e r s c h e l , W.H., and B u l k l e y , R . , P r o c . Am. S o c . T e s t . Mat., XXVI, 61 (1926) . 38. H e r z i g , J . P . , e t a l . , "Flow o f S u s p e n s i o n s T h r o u g h P o r o u s M e d i a " , I n d . Eng. Chem.. V o l . 62, No. 5, (May 1970) - 1 6 0 -H o l d e r , G.A., and W i n k l e r , J . , "Wax C r y s t a l l i z a t i o n f r o m D i s t i l l a t e F u e l s " , J . I n s t . P e t . , V o l . 51, No. 499, ( J u l y 1965) . H o r i e , M., "Time-Dependent S h e a r F l o w o f A r t i f i c i a l S l u r r i e s ' ' , Ph.D. T h e s i s , Ch. Eng., UBC (March 1 9 7 8 ) . H u b b e r t , M.K., " D a r c y ' s Law and t h e F i e l d E q u a t i o n s o f t h e F l o w o f U n d e r g r o u n d F l u i d s " , P e t . T r a n s . AIME, V o l . 207, 222 (1956) Hunt, E . B . , " L a b o r a t o r y S t u d y o f P a r a f f i n D e p o s i t i o n " , P e t r o l e u m T r a n s a c t i o n , 1259-1269 (Nov. 1 9 6 2 ) . •(Pan A m e r i c a n P e t . C o r p , T u l s a , O k l a . ) I k o k u , C U . and Ramey, H.J., " T r a n s i e n t F l o w o f Non-N e w t o n i a n Power-Law F l u i d s i n P o r o u s M e d i a " , (SPE-AIME, S t a n f o r d U . ) , S . P . E . J . (June 1 9 7 9 ) , (SPE-AIME, SPE-AIME, S t a n f o r d U.)' I n s t r u c t i o n M a n u a l , R o t o v i s c o RV-12, HAAKE M e s s - T e c h n i k GmbH U. Co., W. Germany. I r a n i , C. and Z a j a c , J . , " H a n d l i n g o f H i g h P o u r P o i n t West A f r i c a n C r u d e O i l s " , J.P.T., 289 (Feb. 1 9 8 2 ) . Kawase, Y. and U l b r e c h t , J . , "Drag and Mass T r a n s f e r i n Non-Newtonian F l o w s T h r o u g h M u l t i p a r t i c l e S ystems a t Low and H i g h R e y n o l d s Numbers", p a p e r p r e s e n t e d a t t h e 7 4 t h AIChE A n n u a l M e e t i n g i n New O r l e a n s , (1981) . Kem b l o w s k i , Z., and M i c h n i e w i c z , M., "A New Look a t t h e L a m i n a r F l o w o f Power-Law F l u i d s T h r o u g h G r a n u l a r B e d s " , R h e o l . A c t a , V o l . 18, No. 6, 730 ( 1 9 7 9 ) . L a i r d , W.M., " S l u r r y and S u s p e n s i o n T r a n s p o r t " , I n d . E n g . Chem., V o l . 49, No. 1 ( J a n u a r y 1 9 5 7 ) . L a r k i n s , R.P., W h i t e , R.R. and J e f f r e y , D.W. "Two-Phase C o n c u r r e n t F l o w i n P a c k e d B e d s " , A .T-VCh. E . J . , 7, 231 (1961) . " ~ " L e v o r s e n , A. 1>., G e o l o g y o f P e t r o l e u m , 2nd. E d i t i o n , W.H. Freeman and Company, San F r a n c i s c o (1967)„. M a r s d e n , S.S. and S u h a i l , A.K.; "The F l o w o f Foam T h r o u g h S h o r t P o r o u s M e d i a and A p p a r e n t V i s c o s i t y M e a s u r e -ments", S J^P_ 1E^J. (March 1 9 6 6 ) . -161-52. M a r s h a l l , R . J . , and M e t z n e r , A.B., "Flow o f V i s c o -e l a s t i c F l u i d s T h r o u g h P o r o u s M e d i a " , I.E,C. Fund., V o l . 6, No. 3, 393 ( A u g u s t 1967). 53. M a v i s , F.T., "A S t u d y o f t h e P e r m e a b i l i t y o f Sand", P u b l i s h e d by t h e U n i v e r s i t y o f Iowa, Iowa C i t y , Iowa (1936). 54. Mazee, W.M., " P h y s i c a l and C h e m i c a l P r o p e r t i e s o f P e t r o l e u m Waxes", J . I n s t . P e t . , V o l . 44, No. 419 (Nov. 1958). 55. McKay, W.N., e t . a l . , "The P r e d i c t i o n o f P r e s s u r e G r a d i e n t s f o r a Non-Newtonian C r u d e O i l F l o w i n g i n a P i p e l i n e " , J o u r n a l o f C a n a d i a n P e t r o l e u m , T e c h n o l o g y , C a l g a r y ( S p r i n g 1964). 56. M e n d e l l , J . L . , and J e s s e n , F.W., " P a r a f f i n I n h i b i t i o n and F l o w Improvement i n C r u d e O i l S y s t e m s " , J . Can. P e t . T e c h . , M o n t r e a l ( A p r i l - J u n e , 1972). 57. Muskat, M., P h y s i c a l P r i n c i p l e s o f O i l P r o d u c t i o n , M c G r a w - H i l l Book Company, I n c . N.Y.,(1949). 58. Muskat, M v The F l o w o f Homogeneous F l u i d s T h r o u g h P o r o u s Media,. M c G r a w - H i l l Book Company, I n c . N.Y., (1937). 59. N a t h a n , C . C , " S o l u b i l i t y S t u d i e s on H i g h M o l e c u l a r W e i g h t P a r a f f i n H y d r o c a r b o n s O b t a i n e d f r o m P e t r o l e u m Rod Waxes", P e t r o l e u m T r a n s a c t i o n s , AIME, V o l . 204,(1955). 60. N i e l d , D.A., " A l t e r n a t i v e M o d e l f o r W a l l E f f e c t i n L a m i n a r F l o w o f a F l u i d T h r o u g h a P a c k e d Column", A . I . C h . E . J . , V o l . 29, No. 4, 688 ( J u l y 1983). 61. Odeh, A.S., and Yang, H.T., "Flow o f Non-Newtonian Power-Law F l u i d s T h r o u g h P o r o u s M e d i a " , S . P . E . J . , (June 1979). 62. P a s c a l , F., P a s c a l , H. and M u r r a y , D.W., " C o n s o l i d a t i o n w i t h T h r e s h o l d G r a d i e n t s " I n t e r n a t i o n a l J . f o r  N u m e r i c a l and A n a l y t i c a l Methods i n G e o m e c h a n i c s , V o l . 5, 247-261 (1981). 63. P a s c a l , H., " E f f e c t o f T h r e s h o l d G r a d i e n t on P r e s s u r e B u i l d u p and F l o w T e s t s i n W e l l s " , Revue de  L ' i n s t i t u t F r a n c a i s du P e t r o l e , ( M a i - J u i n 1979). 64. P a s c a l , H., "Non-Steady F l o w o f Non-Newtonian F l u i d s . T h r o u g h P o r o u s M e d i a " , T n t V J . : E n g . S c i . 21, 199-210 (1983). -162-65. P a s c a l , H., " N o n - s t e a d y F l o w T h r o u g h P o r o u s M e d i a i n t h e P r e s e n c e o f a T h r e s h o l d G r a d i e n t " , ACTA  M e c h a n i c a . 39, 207-224 (1981) 66. P a s c a l , H., " R h e o l o g i c a l B e h a v i o r E f f e c t o f Non-N e w t o n i a n F l u i d s on S t e a d y and U n s t e a d y F l o w T h r o u g h A P o r o u s Medium", I n t . J . N u m e r i c a l and  A n a l . Methods i n G e o m e c h a n i c s , V o l . 7, 289 ( 1 9 8 3 ) . 67. P e r k i n s , T.K. and J o h n s t o n , O.C., "A Review o f D i f f u s i o n and D i s p e r s i o n i n P o r o u s M e d i a " , S . P . E . J . (March 1963) . 68. P e r k i n s , T.K., and T u r n e r , J.B., " S t a r t i n g B e h a v i o r o f G a t h e r i n g L i n e s and P i p e l i n e s F i l l e d w i t h G e l l e d P r u d h o e Bay O i l " , J . P.T. (March 1971) (SPE-AIME, A t l a n t i c R i c h f i e l d Co.) 69. P e r r y , R.H., and C h i l t o n , C.H., C h e m i c a l E n g i n e e r s Handbook, M c G r a w - H i l l Kogakusha, L t d . , 5 t h E d i t i o n , T o k y o ( 1 9 7 3 ) . 70. P o n d e r , R.M. and M i l e s , A . J . , " P e r m e a b i l i t y - P o r o s i t y R e l a t i o n s h i p f o r U n c o n s o l i d a t e d Sand", P u b l i c a t i o n  o f M i s s o u r i S c h o o l o f M i n e s , R o l l a , M i s s o u r i (1954) 71. P r i c e , R . C , "Flow I m p r o v e r s f o r Waxy C r u d e s " , J . I n s t . P e t . , V o l . 57, No. 554 (March 1971) ( E s s o C h e m i c a l s R e s e a r c h C e n t r e , B e r k s h i r e ) . 72. R i c h a r d s o n , J.G., "Flow T h r o u g h P o r o u s M e d i a " , F l u i d D ynamics Handbook, 1 s t E d i t i o n , M c G r a w - H i l l Book Company, I n c . N.Y. (1961) . 73. R o j a s , G. , e t . * a l . , " R h e o l o g i c a l B e h a v i o r o f E x t r a - H e a v y C r u d e O i l s f r o m t h e O r i n o c o O i l B e l t , V e n e z u e l a " , P u b l i s h e d i n The O i l Sands o f C a n a d a - V e n e z u e l a 284-302 ( 1 9 7 7 ) . 74. R u s s e l l , R . J . , and Chapman, E.D. "The Pumping o f 85°F P o u r P o i n t Assam C r u d e O i l a t 65°F", J . I n s t . P e t . , V o l . 57, No. 554 (March 1 9 7 1 ) . 75. S a d o w s k i , T.J., 7 "Non-Newtonian F l o w T h r o u g h P o r o u s M e d i a " , Ph.D. T h e s i s , Ch.Eng., U n i v . o f W i s c o n s i n (1963) . 76. S a v i n s , J.G., "Non-Newtonian F l o w T h r o u g h P o r o u s M e d i a " , I n d . Eng. Chem., V o l . 61, No. 10 ( O c t o b e r 1 9 6 9 ) . p 77. S c h e i d e g g e r , A.E., The P h y s i c s o f F l o w T h r o u g h P o r o u s M e d i a , 3 r d E d i t i o n , U n i v e r s i t y o f T o r o n t o P r e s s , T o r o n t o and B u f f a l o (1974 ). -163-78. Schramm, G., O p t i m i z a t i o n o f R o t o v i s c o T e s t s , HAAKE M e s s - T e c h n i k GmbH U. Co/, West Germany (1981)-79. Schumacher, M.M., E n h a n c e d R e c o v e r y o f R e s i d u a l and Heavy O i l s , . 2nd E d . Noyes D a t a C o r p . New J e r s e y 1980. 8 0 . Shock, D.A., e t a l . , " S t u d i e s o f t h e Mechanism o f P a r a f f i n D e p o s i t i o n and I t s C o n t r o l " , J . P . T . ( S e p t . 1955) . 81. S i f f e r m a n , T.R., "Flow P r o p e r t i e s o f D i f f i c u l t - T o - H a n d l e Waxy C r u d e O i l s " , SPE-AIME, Conoco I n c . J.P.T. ( A u g u s t 1979) . 82. S i n g h , D. e t . a l . , "Flow o f Bingham F l u i d T h r o u g h F i x e d and F l u i d i z e d B e d s " , I n d i a n J . T e c h . , V o l . 14, 591 (Dec. 1976) . 83. S k e l l a n d , A.H.P., Non-Newtonian F l o w and H e a t T r a n s f e r , J o h n W i l e y and Sons, I n c . , N.Y. ( 1 9 6 7 ) . 84. S u l l i v a n , R.R., and H e r t e l , K.L., "The F l o w o f A i r T h r o u g h P o r o u s M e d i a " , J . A p p l i e d P h y s i c s , V o l . 11, 761 (Dec. 1 9 4 0 ) . 85. S w a r t z e n d r u b e r , D., "Non-Darcy F l o w B e h a v i o r i n L i q u i d -S a t u r a t e d P o r o u s M e d i a " , J . G e o p h y s i c a l R e s e r c h , V o l . 67, No. 13, 5205 (Dec. 1 9 6 2 ) . 86. Teeuw, D., et» a l . , "Power-Law F l o w and H y d r o d y n a m i c B e h a v i o u r o f B i o p o l y m e r S o l u t i o n s i n P o r o u s M e d i a " , P a p e r p r e s e n t e d a t t h e S.P.E. F i f t h I n t . Symp. on o i l f i e l d and G e o t h e r m a l Chem. S t a n f o r d , C a l . (May 1980) . 87. T e m p l i n , P.R., " C o e f f i c i e n t o f Volume E x p a n s i o n f o r P e t r o l e u m Waxes and P u r e n - P a r a f f i n s " / M e l l o n I n s t . P i t t s b u r g h , Pa., I n d . Eng. Chem., V o l . 48, No. 1 J a n u a r y (19 5 6 ) . 88. T u t t l e , R.N., " H i g h - P o u r - P o i n t and A s p h a l t i c C r u d e O i l s and C o n d e n s a t e s " , . . L L P ^ I . , 1192 (June 1 9 8 3 ) . 89. UHDE, A. and Kopp, G., "Waxy C r u d e s i n R e l a t i o n t o P i p e l i n e O p e r a t i o n s " , J . I n s t , P e t . , V o l . 57, No. 554 (March 1971) . 9 0 • Van Wazer, J.R., e t ; a l . , V i s c o s i t y and F l o w Measurement, J o h n W i l e y , N.Y.(1963). V e r s c h u u r , E . , e t , a l . , " P i l o t - S c a l e S t u d i e s on R e - s t a r t i n g P i p e l i n e s C o n t a i n i n g G e l l e d Waxy C r u d e s " , J . I n s t . P e t . . V o l . 57, No. 555 (May 1971) . V e r s c h u u r , E . , H a r t o g , A.P. and V e r h e u l , CM., "The E f f e c t o f T h e r m a l S h r i n k a g e and C o m p r e s s i b i l i t y on t h e Y i e l d i n g o f G e l l e d Waxy C r u d e O i l i n P i p e -l i n e s " , ( S h e l l R e s e r a c h N.V.) J . I n s t . P e t . , 57, 555 (May 19 7 1 ) . V o c a d l o , J . J . , and C h a r l e s , M.E., "Measurement o f Y i e l d S t r e s s o f F l u i d - L i k e V i s c o p l a s t i c S u b s t a n c e s " , Cand. J . Chem. Eng., V o l . 49, 576 ( O c t . 1 9 7 1 ) . 16 5-APPENDIX A The flow of non-Newtonian f l u i d with a y i e l d value i n c i r c u l a r tub The der i v a t i o n of equation (3-15) Le H R r T R I c — -P \iax • L - " ^ * AP = P, ' -Fig. A - l The v e l o c i t y p r o f i l e i n a pipe of non-Newtonian f l u i d with a y i e l d stress By doing a force balance from F i g . A - l 2irRLeT = irR2AP (A-l) for a v i s c o - i n e l a s t i c f l u i d with y i e l d stress T R AP ,dV.n, T = 2~ Le = dR To (A-2) hence the v e l o c i t y p r o f i l e dV dR 1 ,R AP H 4 Le x ) o 1/n (A-3) V(R) R ; av = / o R_ 1 /R AP H 4 Le 1/n dR (A-4) - 1 6 6 -n 2LeH V(R) = n+1 AP , „ 1+1/n 1+1/n APR T APR. x °) - ( - — ) 2HLe H k2LeH H ( A - 5 ) i . e . V(R) = (n+1) a 1+1/n 1+1/n (aR - ^p> (A-6) where a = AP/2HLe Equation A-6 may be written as 1+1/n V(R) = - (n+1)a ( a R c " H } a l -1 - (" ^o 1+1/n -V) a.R - — c H (A-7) The maximum v e l o c i t y corresponds at R R , where o V(R ) = V o max (A-8) n 1+1/n V max (n+l)a ( 0 l R c ." H } OR 1 - ( - ^ T ) aR - — c . H (A-9) On the other hand, from condition s p e c i f i e d at the core, we have 2TTR LT = TTR 2AP o o o ( A-10 ) or R APH T = R AP/2Le = ° „ = R Ha ( A - l l ) o o 2LeH o v J where aR = x /H and a = ( A"12) o o zLeH As a r e s u l t , Eq. (A-9) becomes 1+1/n V = - , (aR - x /H) (A-13) max (n+1) a c o i-°> Eq. (A-13) may be expressed as 1+1/n V = - -,—TTT— (aR - aR ) f A - 1 4 " ) max (n+1) a c o  1'*J To determine the average v e l o c i t y , the following equation must be used ^Total " Q o + Q L CA-15) where Q = TTR 2 V (A-16) xo o max 0 = The flow rate at the core o •R Q = 2TT . c R V(R) dR (A-17) Ll J R' o -168-from Eq. (A-9) and Eq. (A-16) we get; TTR 2 n o o (n+1)a 1+1/n 1+1/n ( a R o - r> - <aRc - ir> (A-18) from Eq. (A-6) and Eq. (A-17) we get: 2i\n R QT = c R L (n+1)a R L_ 1+1/n 1+1/n (aR - dR (A-19) substitute Eq. (A-18) and Eq. (A-19) into Eq. (A-15) irR 2 n <T (n+1)a 1+1/n 1+1/n i rn 1+1/n T 2TTO _ , ^ . . v (R 2-R 2 ) ( a R - T-°-) + , / c R(aR - TT2-)-(n+l)a c o c H (n+1)a R H o 1+1 /n dR the l a s t term i n Eq. A-20 can be integrated by part 1A-20) /vdu = uv - /udv (A-21) Let's V = R then dV = dR (A-22) and 1+11 XL du = (aR - — ) dR ri -169-x 2+1/n t h e n U = ^2tflT ^ " r f ) (A-23) substitute into Eq. (A-21) 1+1/n „ 2+1/n T nR x ' n / c R (aR - -2) dR = , . (aR - -2.) - / (aR -H a(2n+l) H a(2n+l) R o R. T ..2+1/n 2 , _ o, 3+1/n T ' n^(aR - — ) nR _ _o _ H a(2n+l) H a 2(2n+l)(3n+l) R o by s i m p l i f i c a t i o n and s u b s t i t u t i o n of Eq. (A-12) we get; R o 1+1/n _ . _ V 2+1/n T J x nR (aR - — ) R ^ / _ -O. .„ c c H / c R (aR - — ) dR = / 0  H a(2n+l) x :3+l/n n 2 ( a R c ' a 2(2n+l)(3n+l) ( A _ 2 4 ) Now substitute Eq. (A-24) into Eq. (A-20) such that Eq. (A-12) i s v a l i d we get; 9 1+1/n „ o „ , „ o N 2+1/n irnR 2 x 2im 2R (aR - — ) • c o c c H Q T = ." (n+1)a ( a R c " rr } + a z ( n + l ) ( 2 n + l ) ~ T 3+1/n 2Trn3(aR - r-5-) ' _ c H (A-25) a 3(n+1)(2n+l)(3n+l) -170-substitute Eq. (A-12) into Eq. (A-25) TrnR 2 x 1 / n 2un 2R (aR - ^ ) 1 / n c o c c H 3™ = ~ ~,—TT^— (aR _ TT~) (aR - aR ) H T T — , ., ,„—TT^  <T (n+1)a c H c o a 2(n+1)(2n+l) 1/n x 2irn 3(aR -) (aR - aR ) 2 - -g-. — 7 - — 7 — — — (aR - aR ) 3 (A-26) c o a d(n+l)(2n+l)(3n+l) c o since R q i s very small i n c a p i l l a r y tubes i . e . R^ >> R q then (aR - aR ) = aR (1 - R /R ) = aR (A-27) C O c o c c (aR - aR ) 2 = a 2R 2 ( 1 - R /R ) 2= a 2R 2 (A-28) C O c o c c (aR - aR ) 3 = a 3R 3(1-R /R ) 3 = a 3R 3 (A-29) C O c o c c where we assume (1 - R /R ) P i 1 o c substitute these equations into Eq. (A-26) we get o 1/n „ 9 „ / „ o. 1/n nirR " T 2un 2R j aR - — ) c , _o c c H  Q T = " n+1 c " H } + (n+l)(2n+l) 1/n T 2-rrn3R 3(aR - — ) 9 c H (A-30) (n+1)(2n+l)(3n+l) -171-m r R ; _ c_ Q T " n+1 1 / n 2 n 2 n 2 1 + 2 n + l ( 2 n + l ) ( 3 n + l ) ( A - 3 1 ) n i rR 1 / n C ( a R - z?) n+1 v c H 2 n ( 3 n + l ) - 2 n 2 ( 2 n + l ) ( 3 n + l ) - 1 m r R 3 n+1 c is,/n r^n+ifi H  ) L3n+1 J n i rR 1 / n h " l*t ( a R c " r> ( A - 3 2 ) Q T n R T <v> = — = —4 ( a R - — ) A l + 3 n v c H 1 / n ( A - 3 3 ) n R R 1 / n x 1 / n C C / o . <v> = — — ( a - -^nr) l + 3 n R H y c ( A - 3 4 ) 1 + 1 / n 1 / n n R AP x <v> = l + 3 n 2 L e H R H ' c C A - 3 5 ) - 1 7 2 -A P P E N D I X B SUMMARY OF V I S C O M E T R I C DATA AND C A L C U L A T I O N S T a b l e B - l R e s u l t s o f t h e R o t a t i o n a l V i s c o m e t e r R V - 1 2 U s i n g S e n s o r S y s t e m NV f o r 2 .5% Wax i n C l a r u s - B O i l a t D i f f e r e n t T e m p e r a t u r e s T = 1 0 ° C ; T G = 6 . 0 5 d y n e / c m 2 T = 1 2 ° C 2 ; T 0 = 2 . 3 1 d y n e / c m n = 0 . 8 9 ; H = 6 . 7 5 d y n e s n / c m 2 n = 0 . 9 6 ; H = 3 . 8 2 7 <3yne s n / c m 2 t ( s _ 1 ) 2 T ( d y n e / c m ) Y ( s - 1 ) 2 T ( d y n e / c m ) 5 . 4 1 3 5 . 7 5 1 0 . 8 2 4 0 . 5 5 8 . 6 6 5 2 . 8 3 2 1 . 6 4 7 4 . 1 7 1 0 . 8 2 6 4 . 0 3 3 4 . 6 2 1 1 7 . 3 9 1 7 . 3 1 9 1 . 7 8 4 3 . 2 8 1 4 6 . 2 1 2 1 . 6 4 1 1 0 . 9 9 6 9 . 2 5 2 2 8 . 9 1 3 4 . 6 2 1 6 4 . 8 8 8 6 . 5 6 2 8 3 . 8 8 4 3 . 2 8 2 0 0 . 6 3 1 3 8 . 5 0 4 3 7 . 0 2 6 9 . 2 5 3 0 2 . 0 2 8 6 . 5 6 3 6 6 . 5 8 T = 1 4 ° C ; T Q = 1 . 4 2 d y n e / c m T = 1 6 ° C 2 ; x 0 = 1 . 0 7 d y n e / c m n = 0 . 9 6 ; H = 3 . 0 0 2 d y n e s n / c m 3 n = 0 . 9 2 H = 3 . 0 8 5 d y n e s n / c m 2 Y ( s ) 2 x ( d y n e / c m ) Y ( s ) 2 T ( d y n e / c m ) 1 0 . 8 2 3 1 . 4 8 1 0 . 8 2 2 9 . 3 5 2 1 . 6 4 5 9 . 2 3 2 1 . 6 4 5 3 . 3 6 3 4 . 6 2 9 2 . 3 1 3 4 . 6 2 7 9 . 5 1 4 3 . 2 8 1 1 5 . 2 6 4 3 . 2 8 9 7 . 6 5 6 9 . 2 5 1 8 0 . 8 9 6 9 . 2 5 1 4 7 . 8 1 8 6 . 5 6 2 2 5 . 7 1 8 6 . 5 6 1 8 4 . 6 3 1 3 8 . 5 0 3 5 1 . 6 4 1 3 8 . 5 0 2 8 7 . 6 1 1 7 3 . 1 2 4 3 2 . 2 2 • 1 7 3 . 1 2 3 5 6 . 9 8 2 7 6 . 9 9 5 5 0 . 6 8 -173-T a b l e B - l ( c o n t i n u e d ) T = 1 8 ° C ; 2 x 0 = 0 . 7 1 d y n e / c m n = 0 . 9 7 ; H = 2 . 0 1 d y n e s n / c m 3 Y ( s ) 2 T ( d y n e / c m ) 2 1 . 6 4 4 0 . 5 5 3 4 . 6 2 6 1 . 9 0 4 3 . 2 8 7 6 . 8 4 6 9 . 2 5 1 1 8 . 4 6 8 6 . 5 6 1 4 9 . 4 1 1 3 8 . 5 0 2 3 5 . 3 2 1 7 3 . 1 2 2 9 4 . 5 5 2 7 6 . 9 9 4 6 2 . 1 0 - 1 7 4 -T a b l e B - 2 R e s u l t s o f t h e R o t a t i o n a l V i s c o m e t e r R V - 1 2 U s i n g S e n s o r S y s t e m NV f o r 4% Wax i n C l a u r s - B O i l a t D i f f e r e n t T e m p e r a t u r e s T = 1 2 ° C n = 0 . 7 7 2 , T Q = 3 3 . 6 2 d y n e / c m , H = 1 2 . 2 2 4 d y n e s n / c m T = 1 4 ° C n = 0 . 9 7 2 ; x Q = 3 1 . 5 d y n e / c m ; H = 3 . 3 5 4 d y n e s n / c m Y (s ) X 2 ( d y n e / c m ) • , " I N Y ( s ) X 2 ( d y n e / c m ) 0 . 5 4 1 1 . 0 8 2 2 . 1 6 4 4 . 3 2 8 8 . 6 6 1 7 . 3 1 3 4 . 6 2 6 9 . 2 5 8 6 . 5 6 3 6 . 8 2 5 4 . 9 6 6 4 . 5 7 8 3 . 2 4 1 0 7 . 2 5 1 3 9 . 2 7 1 9 5 . 3 0 3 1 4 . 2 9 3 6 1 . 7 8 5 . 4 1 1 0 . 8 2 2 1 . 6 4 3 4 . 6 2 4 3 . 2 8 6 9 . 2 5 8 6 . 5 6 1 3 9 . 5 0 1 7 3 . 1 2 4 6 . 9 6 6 7 . 7 7 1 0 2 . 4 5 1 3 7 . 6 7 1 6 3 . 2 8 2 3 9 . 0 5 2 8 7 . 0 8 4 2 0 . 4 8 5 0 1 . 5 8 T = 1 6 ° C • n = 0 . 8 8 ; 2 x D = 1 6 . 3 6 d y n e / c m H = 4 . 6 1 3 d y n e s n / c m T = 1 8 ° C n = 0 . 8 5 2 ; x D = 4 . 8 d y n e / c m ; H = 4 . 3 5 6 d y n e s n / c n f Y ( s _ 1 ) X 2 ( d y n e / c m ) Y ( s " 1 ) X 2 ( d y n e / c m ) 5 . 4 1 1 0 . 8 2 2 1 . 6 4 3 4 . 6 2 4 3 . 2 8 6 9 . 2 5 8 6 . 5 6 1 3 8 . 5 0 1 7 3 . 1 2 3 4 . 6 8 5 7 . 1 0 8 7 . 5 1 1 1 9 . 5 3 1 4 0 . 8 7 2 0 3 . 3 0 2 4 3 . 3 2 3 5 9 . 1 1 4 2 9 . 5 5 1 0 . 8 2 2 1 . 6 4 3 4 . 6 2 4 3 . 2 8 6 9 . 2 5 8 6 . 5 6 1 3 8 . 5 0 1 7 3 . 1 2 3 7 . 8 9 6 5 . 6 3 9 3 . 9 1 1 1 2 . 5 9 1 6 5 . 9 5 1 9 7 . 9 7 3 ,00 .00 3 5 7 . 5 1 -175-T a b l e B - 2 ( c o n t i n u e d ) T = 2 0 ° C 2 ; T 0 = 1 . 9 6 d y n e / c m n =• 0 . 9 0 ; H = 2 . 8 5 d y n e s n / c m 2 Y ( s ) 2 X ( d y n e / c m ) 2 1 . 6 4 4 7 . 4 9 3 4 . 6 2 6 9 . 9 0 4 3 . 2 8 8 4 . 8 4 6 9 . 2 5 1 2 6 . 9 9 8 6 . 5 6 1 5 5 . 2 8 1 3 8 . 5 0 2 4 1 . 1 9 1 7 3 . 1 2 2 9 0 . 8 1 2 7 6 . 9 9 4 4 2 . 8 9 T a b l e B - 3 R e s u l t s o f t h e R o t a t i o n a l V i s c o m e t e r R V - 1 2 U s i n g S e n s o r S y s t e m NV f o r 5% Wax i n C l a r u s - B O i l a t D i f f e r e n t T e m p e r a t u r e s T = 1 2 ° C ; T o _ 2 6 9 . 3 7 dyne/cm T = 1 4 ° C To _ f 4 8 . 0 2 dyne/cm 2 '• n = 0 . 8 3 ; H = 7 . 3 6 7 dyne sn/cm 2 n = 0 . 8 7 H = 5 . 3 6 9 dyne s n/cm Y (s ) T (dyne/cm 2) Y ( s ) T (dyne/cm 2 5 . 4 1 1 0 4 . 0 5 1 0 . 8 2 9 2 . 8 5 8 . 6 6 1 1 3 . 6 6 1 7 . 3 1 1 1 3 . 1 2 1 0 . 8 2 1 2 1 . 1 3 2 1 . 6 4 1 2 2 . 1 9 1 7 . 3 1 1 4 4 . 0 7 3 4 . 6 2 1 5 7 . 9 5 2 1 . 6 4 1 5 0 . 4 6 4 3 . 2 8 1 8 5 . 6 9 3 4 . 6 2 2 0 5 . 9 7 6 9 . 2 5 2 6 3 . 0 6 4 3 . 2 8 2 4 0 . 1 2 8 6 . 5 6 3 1 1 . 0 9 6 9 . 2 5 3 3 9 . 3 7 1 3 8 . 5 0 4 4 8 . 2 2 8 6 . 5 6 3 9 0 . 6 0 T = 1 6 ° C ; T o — 2 3 5 . 7 5 dyne/cm T = 1 8 ° C I T o — 2 2 6 . 5 dyne/cm n = 0 . 8 7 ; H 4 . 6 3 2 dyne s n / c m 2 n = 0 . 8 0 H = 5 . 6 7 5 dyne s n / c m 2 Y ( s ) T 2 (dyne/cm ) Y ( s _ 1 ) T 2 (dyne/cm ) 5 . 4 1 5 6 . 0 3 5 . 4 1 4 7 . 4 9 8 . 6 6 6 5 . 1 0 8 . 6 6 5 9 . 2 3 1 0 . 8 2 7 0 . 4 4 1 0 . 8 2 6 5 . 6 3 1 7 . 3 1 9 1 . 7 8 1 7 . 3 1 8 3 . 7 8 2 1 . 6 4 1 0 3 . 5 2 2 1 . 6 4 9 3 . 9 1 3 4 . 6 2 1 3 6 . 6 0 3 4 . 6 2 1 2 0 . 0 6 4 3 . 2 8 1 5 9 . 0 1 4 3 . 2 8 1 3 7 . 6 7 6 9 . 2 9 2 1 2 . 3 7 6 9 . 2 5 1 8 9 . 4 3 8 6 . 5 6 2 4 9 . 7 2 8 6 . 5 6 2 2 1 . 4 4 1 3 8 . 5 0 3 7 2 . 4 5 1 3 8 . 5 0 3 2 3 . 8 9 1 7 3 . 1 2 4 3 1 . 6 8 1 7 3 . 1 2 3 8 7 . 9 3 -177-Table B-3 (continued) T = 20°C ; T Q = 19.21 dyne/cm2 n = 0.90 ; H =3.018 dyne s n/cm 2 Y ( s _ 1 ) 2 T,(dyne/cm ) 5.41 29.88 8.66 40.55 10.82 48.02 17.31 63.50 21.64 71.50 34.62 94.98 43.28 106.18 69.25 147.27 86.56 174.49 138.50 264.67 173.12 320.16 276.99 483.44 -178-T a b l e B - 4 R e s u l t s o f t h e R o t a t i o n a l V i s c o m e t e r R V - 1 2 U s i n g S e n s o r S y s t e m NV f o r 2 .5% Wax i n C l a r u s - C a t D i f f e r e n t T e m p e r a t u r e s T = 1 0 ° C 2 ; = 1 1 . 7 4 d y n e / c m T = 1 2 ° C 2 , x Q = 8 . 0 d y n e / c m n = 0 . 8 8 ; H = 1 7 . 6 2 d y n e s n / c m 2 n = 0 . 8 6 , H = 1 4 . 5 8 d y n e s n / c m 2 Y (s ) 2 X ( d y n e / c m ) Y ( s _ 1 ) 2 X ( d y n e / c m ) 1 . 0 8 3 1 . 4 8 2 . 1 6 3 3 . 6 2 2 . 1 6 4 3 . 7 6 4 . 3 3 6 2 . 9 6 5 . 4 1 9 2 . 3 1 5 . 4 1 7 5 . 7 7 8 . 6 6 1 2 5 . 9 3 8 . 6 6 1 0 2 . 9 8 1 0 . 8 2 1 5 5 . 8 1 1 0 . 8 2 1 2 3 . 8 0 1 7 . 3 1 2 2 9 . 9 8 1 7 . 3 1 1 7 6 . 0 9 2 1 . 6 4 2 7 4 . 8 0 2 1 . 6 4 2 1 2 . 3 7 3 4 . 6 2 4 0 8 . 2 0 3 4 . 6 2 3 1 8 . 5 6 4 3 . 2 8 4 8 5 . 0 4 4 3 . 2 8 3 7 7 . 7 9 T = 1 4 ° C 2 ; x Q = 2 . 6 7 d y n e / c m T = 1 6 ° C 2 , X D = 2 . 3 1 d y n e / c m n = 0 . 9 3 ; H = 9 . 9 7 d y n e s n / c m 2 n = 0 . 9 4 H = 7 . 2 7 6 d y n e s n / c m 2 f ( s _ 1 ) 2 x ( d y n e / c m ) Y ( s ) 2 X ( d y n e / c m ) 5 . 4 1 5 0 . 1 6 5 . 4 1 3 6 . 2 8 8 . 6 6 7 6 . 3 0 8 . 6 6 5 7 . 6 3 1 0 . 8 2 9 7 . 1 2 1 0 . 8 2 7 3 . 1 0 1 7 . 3 1 1 4 1 . 9 4 1 7 . 3 1 1 0 7 . 7 9 2 1 . 6 4 1 7 7 . 1 6 2 1 . 6 4 1 3 1 . 8 0 3 4 . 6 2 2 7 3 . 7 4 3 4 . 6 2 2 0 4 . 3 7 4 3 . 2 8 3 3 2 . 9 7 4 3 . 2 8 2 4 9 . 7 2 6 9 . 2 5 5 0 2 . 1 2 6 9 . 2 5 3 8 6 . 8 6 8 6 . 5 6 4 7 1 . 7 0 -179-T a b l e B - 4 ( c o n t i n u e d ) T = 1 8 ° C 2 ; T 0 = 1 . 4 2 d y n e / c m n = 0 . 9 6 ; H = 4 . 8 9 d y n e s n / c m 2 t ( s _ 1 ) 2 x ( d y n e / c m ) 8 . 6 6 3 9 . 4 9 1 0 . 8 2 5 2 . 2 9 1 7 . 3 1 7 5 . 7 7 2 1 . 6 4 9 3 . 3 8 3 4 . 6 2 1 4 6 . 2 1 4 3 . 2 8 1 8 3 . 0 2 6 9 . 2 5 2 8 9 . 2 1 8 6 . 5 6 3 6 1 . 2 5 -180-T a b l e B -5 R e s u l t s o f t h e R o t a t i o n a l V i s c o m e t e r R V - 1 2 U s i n g S e n s o r S y s t e m NV f o r P e a c e R i v e r C r u d e O i l o f A l b e r t a a t D i f f e r e n t T e m p e r a t u r e s T = 2 °C ; T o = 2 2 1 . 0 6 dyne/cm T = 8 °C ; To 2 = 9 . 4 3 dyne/cm n = 0 . 5 4 ; H 6 . 7 2 6 dyne.s n/cm 2 n = 0 . 6 1 . ; H = 2 . 7 7 9 dyne s n / c m 2 Y ( s 1 ) (dyne/cm2) Y ( s ) T-.Xdyne/cm2) 5 . 4 1 3 7 . 6 2 1 . 6 4 2 7 . 9 9 8 . 6 6 4 1 . 5 1 3 4 . 6 2 3 3 . 6 9 1 0 . 8 2 4 5 . 7 8 4 3 . 2 8 3 8 . 6 7 1 7 . 3 1 5 5 . 0 3 6 9 . 2 5 4 7 . 2 1 2 1 . 6 4 5 8 . 0 6 8 6 . 5 6 5 3 . 2 5 3 4 . 6 2 6 7 . 4 8 1 3 8 . 5 0 6 6 . 0 6 4 3 . 2 8 7 1 . 5 7 1 7 3 . 1 2 7 2 . 8 2 6 9 . 2 5 8 4 . 9 1 2 7 6 . 9 9 9 1 . 4 9 8 6 . 5 6 9 1 . 1 4 3 4 6 . 2 4 9 5 . 0 5 1 3 8 . 5 0 1 0 9 . 8 1 6 9 2 . 4 8 1 5 0 . 1 9 1 7 3 . 1 2 1 2 1 . 7 3 1 3 8 4 . 9 6 2 4 3 . 7 5 2 7 6 . 9 9 1 5 0 . 9 0 2 7 6 9 . 9 2 4 0 8 . 1 0 3 4 6 . 2 4 1 7 1 . 7 1 6 9 2 . 4 8 2 5 8 . 6 9 1 3 8 4 . 9 6 3 7 6 . 0 8 T = 1 0 ° C ; T c ) -2 6 . 7 6 dyne/cm ^ T = 1 4 ° C ; T o 2 = 3 . 5 6 dyne/cm n = 0 . 7 0 ; H = 1 . 2 7 4 dyne s n / c m n = 0 . 8 1 ; H = 0 . 4 0 5 4 dyne s n / c m Y ( s ) T 2 (dyne/cm ) • / - I x Y ( s ) T 2 (dyne/cm ) 6 9 . 2 5 3 2 . 0 8 7 2 7 6 . 9 9 4 1 . 8 5 8 6 . 5 6 3 6 . 3 5 6 3 4 6 . 2 4 5 0 . 2 1 1 3 8 . 5 0 4 7 . 5 6 2 6 9 2 . 4 8 8 4 . 8 9 1 7 3 . 1 2 5 4 . 1 4 3 1 3 8 4 . 9 6 1 4 5 . 5 5 2 7 6 . 9 9 7 0 . 8 6 2 3 4 6 . 2 4 7 9 . 2 2 2 6 9 2 . 4 8 1 2 8 . 2 4 2 1 3 8 4 . 9 6 2 0 2 . 6 6 2 7 6 9 . 9 2 3 5 7 . 4 1 -181-T a b l e B-6 S h e a r Rate C o r r e c t i o n s due t o n o n - Newtonian B e h a v i o u r S k e l l a n d (83) has g i v e n theviCorrecti'Ori f a c t o r as: 15 1 -R. l R n' C T i . e . (Y • ) v 1 i ' t r u e M n 1 'R and < V R i > - 1 i C R = 1 + - V 1 ; «s - » 2(ir> 1 + | i n ( ^ ) + + k Tt _ JL \ 3 + 2 \ 5 _ 1 3 45 945 4725 A7 + . where A = ( i - 1) i n ( ^ ) i C R = R. = R = a n' = n = c o r r e c t i o n f a c t o r r a d i u s o f i n n e r c y l i n d e r r a d i u s o f o u t e r c y l i n d e r r o t o r s p e e d , rpm t h e s l o p e o f a l o g a r i t h m i c p l o t o f t o r q u e v e r s u s r o t a t i o n a l s p e e d n' F o r t h e NV-system : R /R. = 1 . 0 2 J a l F o r n = 0.93 f o r n = 0.80 C R = 1.0015 C R = 1.0101 -182-TABLE C-1 R e s u l t s f o r the small column using c l a r u s - B o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . (L=91.0 cm, Dp=0.077 cm, Dc=4.83 cm, porosity=0.36) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN I NO. (C) %WAX (c.c./s) EXP. CAL. %DEV B 1 10.0 2.5 0.0078 41 .90 45. 1 3 7.71 B 2 10.0 2.5 0.0110 59.60 61 .84 3.76 B 3 10.0 2.5 0.0167 88.30 89.94 1 .86 B 4 10.0 2.5 0.0220 121.00 1 1 5.92 4.20 B 5 10.0 2.5 0.0317 169.60 161.95 4.51 B 6 10.0 2.5 0.0440 221.90 219.34 1.15 B 7 10.0 2.5 0.0663 286.80 320.51 1 1 .75 B 8 12.0 2.5 0.0452 38.51 39.07 1 .44 B 9 12.0 2.5 0.0750 58.67 58.33 0.57 B 10 12.0 2.5 0.1069 80.80 78.47 2.88 B 1 1 12.0 2.5 0.1638 1 12.60 1 13.52 0.82 B 12 12.0 2.5 0.2325 153.20 154.95 1.14 B 1 3 12.0 2.5 0.3104 201.40 201.01 0.19 B 14 12.0 2.5 0.4012 252.60 253.84 0.49 B 1 5 14.0 2.5 0.0354 24.41 24.87 1 .88 B 16 14.0 2.5 0.0531 34.62 34.98 1 .03 B 1 7 14.0 2.5 0.0750 47.91 47.32 1 .23 B 18 14.0 2.5 0.0992 60.65 60.71 0.09 B 1 9 14.0 2.5 0.1311 77.66 78. 1 5 0.64 B 20 14.0 2.5 0.1717 98.40 100.01 1 .63 B 21 14.0 2.5 0.2063 115.20 1 18.43 2.81 B 22 14.0 2.5 0.2592 143.60 146.34 1 .91 B 23 •14.0 2.. .5 0.3167 1 77.. 1 0 176.34 0.43 B 24 14.0 2.5 0.4008 223.20 219.76 1 .54 B 25 18.0 2.5 0.0663 22. 1 6 22.40 1.10 B 26 18.0 2.5 0.1256 40.69 40.68 0.03 B 27 18.0 2.5 0. 1867 59.83 59.32 0.86 B 28 18.0 2.5 0.2773 85.44 86.74 1 .52 B 29 18.0 2.5 0.3833 120.50 118.60 1 .58 B 30 18.0 2.5 0.5583 167.90 170.82 1 .74 B 31 18.0 2.5 0.7777 236.90 235.84 0.45 B 32 12.0 4.0 0.0117 46.33 48.56 4.81 B 33 12.0 4.0 0.0182 63.62 61 .99 2.56 B 34 12.0 4.0 0.0257 79.95 76.39 4.45 B 35 12.0 4.0 0.0363 99.95 95.72 4.24 B 36 12.0 4.0 0.0481 119.70 116.31 2.83 B 37 12.0 4.0 0.0683 152.80 149.72 2.02 B 38 12.0 4.0 0.0979 193.50 195.87 1 .22 B 39 12.0 4.0 0.1411 252.40 259.44 2.79 B 40 12.0 4.0 0.1600 284.90 286.21 0.46 -183-TABLE C-1 (continued) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c.c,/s) EXP. CAL. %DEV B 41 14.0 4.0 0.0150 35.00 35.70 2.00 B 42 14.0 4.0 0.0298 57.60 57.52 0.14 B 43 14.0 4.0 0.0421 75.20 75.54 0.45 B 44 14.0 4.0 0.0578 97. 1 0 98.22 1.15 B 45 14.0 4.0 0.0717 121.30 1 18.26 2.50 B 46 14.0 4.0 0.0967 153.80 154.15 0.23 B 47 14.0 4.0 0.1256 196.60 195.38 0.62 B 48 14.0 4.0 0.1783 260.20 270.24 3.86 B 49 16.0 4.0 0.0227 41.12 41 .76 1 .56 B 50 16.0 4.0 0.0317 56.88 55.89 1 .74 B 51 16.0 4.0 0.0558 94.92 93.62 1 .37 B 52 16.0 4.0 0.071 1 119.40 117.31 1 .75 B 53 16.0 4.0 0.0900 150.70 146.45 2.82 B 54 16.0 4.0 0.1128 188.00 181 .43 3.50 B 55 16.0 4.0 0.1587 244.50 251.48 2.86 B 56 18.0 4.0 0.0325 42. 55 42.71 0.37 B 57 18.0 4.0 0.0467 59.61 60.48 1 .46 B 58 18.0 4.0 0.0589 76.55 75.98 0.74 B 59 18.0 4.0 0.0772 97.64 99.47 1 .88 B 60 18.0 4.0 0.0906 117.10 1 16.72 0.33 B 61 18.0 4.0 0.1106 145.30 1 42.75 1 .75 B 62 18.0 4.0 0. 1322 176.40 171.17 2.97 B 63 18.0 4.0 0. 1875 235.90 .244 . 47 3.63 B 64 20.0 4.0 0.0588 20.70 20. 00 3.40 B 65 20.0 4.0 0.0917 32. 44 32. 00 1 .36 B 66 20.0 4.0 0.1200 42.75 42.57 0.42 B 67 20.0 4.0 0.1556 55.52 56.05 0.95 B 68 20.0 4.0 0.2039 70. 15 74.67 6.44 B 69 20.0 4.0 0.2667 97. 10 99.24 2.21 B 70 20.0 4.0 0.3239 119.40 121.95 2.14 B 71 20.0 4.0 0.4139 158.30 158.15 0.10 B 72 20.0 4.0 0.5222 206.20 202.34 1 .87 B 73 20.0 4.0 0.6056 247.90 236.73 4.51 B 74 16.0 5.0 0.0094 43.91 48.49 1 0.43 B 75 16.0 5.0 0.0165 74.50 67.88 8.88 B 76 16.0 5.0 0.0289 105.80 99.71 5.76 B 77 16.0 5.0 0.0400 131.60 126.82 3.63 B 78 16.0 5.0 0.0725 177.10 201 .70 13.89 -184-TABLE C-1 ( cont inued) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c.c./s) EXP. CAL. %DEV B 79 18.0 5.0 0.0185 54.09 54.79 1 .30 B 80 18.0 5.0 0.0639 87.43 81 .87 6.35 B 81 18.0 5.0 0.1133 115.30 111.13 3.61 B 82 18.0 5.0 0.1806 141.20 150.72 6.74 B 83 18.0 5.0 0.2667 189.10 201 .21 6.40 B 84 18.0 5.0 0.3125 225.70 228.01 1 .02 B 85 18.0 5.0 0.3637 273.90 257.93 5.83 B 86 20.0 5.0 0.0057 18.99 21 .62 13.84 B 87 20.0 5.0 0.0154 38.79 32.42 16.42 B 88 20.0 5.0 0.0293 55.52 45.43 18.17 B 89 20.0 5.0 0.0587 74.50 68.84 7.60 B 90 20.0 5.0 0. 1020 99.68 98.87 0.81 B 91 20.0 5.0 0.1400 120.10 122.79 2.24 B 92 20.0 5.0 0.2413 164.40 180.36 9.71 B 93 20.0 5.0 0.3313 213.00 226.79 6.47 B 94 20.0 5.0 0.4042 261.60 262.30 0.27 -185-TABLE C-2 R e s u l t s f o r the small column using c l a r u s - C o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . (L= 91.0 cm, Dp=0. 077 cm, Dc=4 .83 cm, porosity=0.36) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c. c . / s ) EXP. CAL. %DEV C 95 12.0 2.5 0.0088 54. 17 57.25 5.69 C 96 12.0 2.5 0.0136 83.34 83.35 0.01 C 97 12.0 2.5 0.0196 114.20 1 15.33 0.99 C 98 12.0 2.5 0.0308 177.00 173.81 1 .80 C 99 12.0 2.5 0.0460 243.00 251.51 3.50 C1 00 12.0 2.5 0.0583 299.60 313.42 4.61 C101 14.0 .2.5 0.0133 30.56 30.23 1 .09 C102 14.0 2.5 0.0255 55.84 54.96 1 .58 C1 03 14.0 2.5 0.0404 84.73 85.66 1 .09 C104 14.0 2.5 0.0525 112.50 110.85 1 .47 C105 14.0 2.5 0.0771 160.00 162.57 1.61 C106 14.0 2.5 0.1047 224.60 221 .20 1 .51 C107 14.0 2.5 0.1313 281.70 278.15 1 .26 C108 16.0 2.5 0.0183 30. 1 4 30. 1 0 0.13 C1 09 16.0 2.5 0.0335 55.56 55.44 0.22 C1 1 0 16.0 2.5 0.0513 84.73 85.25 0.62 C1 1 1 16.0 2.5 0.0692 113.30 1 15.35 1.81 CI 1 2 16.0 2.5 0.1044 171.10 174.74 2.13 CI 1 3 16.0 2.5 0.1354 227. 10 227.21 0.05 C1 1 4 16.0 2.5 0.1688 289.60 283.89 1 .97 C1 1 5 18.0 2.5 0.0238 22.22 21 .20 4.61 CI 1 6 18.0 2.5 0.0500 46.39 46.76 0.80 C1 1 7 18.0 2.5 0.0808 74.59 78.25 4.91 C1 1 8 18.0 2.5 0.1033 96.54 101.91 5.56 C1 1 9 18.0 2.5 0.1244 120.80 124.48 3.04 C120 18.0 2.5 0.1583 156.20 161.36 3.30 CI 21 18.0 2.5 0.1783 188.40 183.43 2.64 C122 18.0 2.5 0.2146 227.40 223.98 1 .50 C123 18.0 2.5 0.2528 283. 10 267.26 5.60 -186-TABLE C-3 R e s u l t s f o r the l a r g e column using c l a r u s - B o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . (L= 100.0 cm, Dp=0. 128 cm, Dc=10. 16 cm, porosity=0.44) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c.c./s) EXP. CAL. %DEV B1 24 10.0 2.5 0.0200 23.49 24.95 6.22 B1 25 10.0 2.5 0.0690 36.35 29.76 18.12 B1 26 10.0 2.5 0.1570 45.74 37.31 18.42 B1 27 10.0 2.5 0.2880 55.63 47.52 14.59 B1 28 10.0 2.5 0.4880 68.76 61 .92 9.95 B1 29 10.0 2.5 0.7960 79.87 82.58 3.39 B1 30 10.0 2.5 1.2080 90.29 108.54 20.21 B1 31 10.0 2.5 1.9000 107.00 149.55 39.77 B1 32 12.0 2.5 0. 1170 20. 38 21.51 5.55 B1 33 12.0 2.5 0.2750 29.85 27.31 8.50 B1 34 12.0 2.5 0.5000 36.97 34.96 5.43 B1 35 12.0 2.5 0.7250 44.92 42.23 5.99 B1 36 12.0 2.5 1 .0920 54.59 53.58 1 .84 B1 37 12.0 2.5 1 .3580 60.81 61 .54 1 .20 B1 38 12.0 2.5 1.7920 72.92 74. 17 1 .72 B1 39 12.0 2.5 2.4670 88.20 93. 1 8 5.64 B1 40 12.0 2.5 3.1830 104.90 112.72 7.45 B1 41 14.0 2.5 0.1380 9.54 9.89 3.68 B1 42 14.0 2.5 0.3750 1 6.72 16.32 2. 39 B1 43 14.0 2.5 0.6000 22.80 22. 1 4 2.89 B1 44 14.0 2.5 0.8400 29.51 28. 1 8 4.50 B1 45 14.0 2.5 1 . 1830 37.66 36.61 2.78 B1 46 14.0 2.5 1.6750 51.13 48.42 .5.31 B1 47 14.0 2.5 2.1560 58.74 59.72 1 .67 B1 48 14.0 2.5 2.6250 68.76 70.57 2.63 B1 49 14.0 2.5 3.5420 86.81 91 .42 5.31 B1 50 14.0 2.5 4.3750 105.60 1 1 0.02 4.19 B1 51 16.0 2.5 0.3540 9.67 9.90 2.40 B1 52 16.0 2.5 0.6630 1 6. 03 16.20 1 . 06 B1 53 16.0 2.5 0.9830 23.36 22.47 3.81 B1 54 16.0 2.5 1.4330 32. 1 3 31 .02 3.45 B1 55 16.0 2.5 1.8330 39.39 38.44 2.41 B1 56 16.0 2.5 2.4170 47. 1 3 49.04 4.06 B1 57 16.0 2.5 2.8750 55.97 57.22 2.23 B1 58 18.0 2.5 0.3130 7.12 7.29 2.33 B1 59 18.0 2.5 0.7170 14.17 1 3.94 1 .62 B1 60 18.0 2.5 1.1600 21 .77 21 .02 3.46 BI 61 18.0 2.5 1.8170 30.75 31 .27 1 .69 B1 62 18.0 2.5 2.3500 40.42 39.45 2.41 B1 63 18.0 2.5 3.2500 51.13 53.05 3.76 -187-TABLE C-3 (continued) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c.c./s) EXP. CAL. %DEV B1 64 12.0 5.0 0.0183 66.70 71 .58 7.32 B1 65 12.0 5.0 0.0411 108.90 105.16 3.43 B1 66 12.0 5.0 0.0642 145.20 135.93 6.39 B167 12.0 5.0 0.0900 183.20 168.10 8.24 B1 68 12.0 5.0 0. 1233 221.20 207.40 6.24 B169 • 1 2.0 5.0 0.1808 261.60 271.31 3.71 B1 70 12.0 5.0 0.2542 303.40 348.06 1 4.72 B171 14.0 5.0 0.0238 88.47 95.39 7.82 B1 72 14.0 5.0 0.0638 136.50 119.35 12.57 B173 14.0 5.0 0.1292 179.40 151.58 15.51 B1 74 14.0 5.0 0.2625 222.20 207.02 6.83 B175 14.0 5.0 0.4792 261.30 284.22 8.77 B176 14.0 5.0 0.6917 299.30 351.88 17.57 B177 16.0 5.0 0.0121 14.58 17.50 20.06 B1 78 16.0 5.0 0.0358 36.11 30.11 16.61 B1 79 16.0 5.0 0.0708 55. 1 4 45.88 1 6.80 B1 80 16.0 5.0 0.1133 75.28 63.00 16.31 B1 81 16.0 5.0 0.1658 95.42 82.45 13.59 B1 82 16.0 5.0 0.2556 112.00 113.10 0.98 B1 83 16.0 5.0 0.5442 141.70 200.06 41.19 B1 84 18.0 5.0 0.0267 24.32 26.05 7.13 B1 85 18.0 5.0 0.1000 44 . 92 33.59 25.23 B1 86 18.0 5.0 0.2200 59.08 44.67 24.39 B1 87 18.0 5.0 0.5808 77.65 74.84 3.61 B1 88 18.0 5.0 1.0080 101.10 107.88 6.71 B1 89 18.0 5.0 1.2833 119.20 128.30 7.64 B1 90 18.0 5.0 2.3555 152.80 204.01 33.52 B1 91 20.0 5.0 0.3150 21 .08 20.30 3.72 B1 92 20.0 5.0 0.6460 30.06 30.73 2.21 B1 93 20.0 5.0 1 .0170 40.08 42.42 5.83 B1 94 20.0 5.0 1.3670 49.89 53.44 7.12 B1 95 20.0 5.0 1.5940 60. 12 60.60 0.79 B1 96 20.0 5.0 2.0170 75.84 73.93 2.52 B1 97 20.0 5.0 2.4750 93.90 88.36 5.90 B1 98 20.0 5.0 3.2500 116.40 112.78 3.11 -188-TABLE C-4 R e s u l t s f o r the l a r g e column using c l a r u s - C o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . (L= 100.0 cm, Dp=0. 128 cm, DC=10. 16 cm, porosity=0.44) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) %WAX (c.c,/s) EXP. CAL. %DEV C1 99 10.0 2.5 0.0300 14.31 1 5.20 6.20 C200 10.0 2.5 0.1056 30.84 27.20 11.81 C201 10.0 2.5 0.2000 45.56 41 .31 9.34 C202 10.0 2.5 0.3050 59.31 56.45 4.82 C203 10.0 2.5 0.4208 72.23 72.73 0.69 C204 10.0 2.5 0.5275 83.34 87.45 4.93 C205 10.0 2.5 0.6383 95. 15 102.51 7.73 C206 10.0 2.5 0.7625 111.10 119.18 7.27 C207 12.0 2.5 0.0342 12.08 12.66 4.81 C208 12.0 2.5 0.1044 24.03 21 .48 1 0.62 C209 12.0 2.5 0.2096 36.53 34.47 5.65 C21 0 12.0 2.5 0.2917 46.25 44.51 3.77 C21 1 12.0 2.5 0.4292 58.34 61 .20 4.91 C21 2 12.0 2.5 0.5458 70.28 75.27 7.10 C21 3 12.0 2.5 0.7975 94. 17 105.45 1 1 .98 C21 4 14.0 2.5 0.0417 1 3.89 1 4.35 3.30 C21 5 14.0 2.5 0.1650 28.20 26. 1 2 7.38 C21 6 14.0 2.5 0.2747 38.20 36. 17 5.31 C21 7 14.0 2.5 0.3975 48.62 47. 1 9 2.95 C218 14.0 2.5 0.5600 61 .53 61.51 0.04 C21 9 14.0 2.5 0.7083 72.23 74.39 2.99 C220 14.0 2.5 0.8444 82.51 86.09 4.34 C221 14.0 2.5 1.0333 97.65 102.19 4.. 64 C222 16.0 2.5 0.0458 7.08 7.24 2.30 C223 16.0 2.5 0.3033 26.39 24.63 6.67 C224 16.0 2.5 0.4778 37.50 36.41 2.89 C225 16.0 2.5 0.6250 46.25 46.35 0.23 C226 16.0 2.5 0.7750 56.25 56.48 0.41 C227 16.0 2.5 0.9330 65.28 67. 1 5 2.87 C228 16.0 2.5 1.1083 75.84 78.99 4. 15 C229 18.0 2.5 0.0944 7.36 7.72 4.93 C230 18.0 2.5 0.2322 16.11 1 5.27 5.22 C231 18.0 2.5 0.4111 25.84 24.73 4.30 C232 18.0 2.5 0.6000 35. 1 4 34.49 1 .84 C233 18.0 2.5 0.7944 43.75 44.38 1 .43 C234 18.0 2.5 1.0000 52.78 54.70 3.64 C235 18.0 2.5 1.2667 66.67 67.94 1 .90 -189-TABLE C-5 R e s u l t s f o r the l a r g e column using waxy crude o i l from the Peace R i v e r f i e l d of A l b e r t a at d i f f e r e n t temperatures. (L= 100.0 cm, Dp=0.128 cm, DC=10. 16 cm, porosity=0.44) TEMP. FLOW RATE PRESSURE DROP(kPa) RUN NO. (C) (c.c./s) EXP. CAL. %DEV CRUDE 1 2.0 0.0375 9.72 10.35 6.42 CRUDE 2 2.0 0.1320 21.81 19.44 10.88 CRUDE 3 2.0 0.2430 32.23 29.51 8.45 CRUDE 4 2.0 0.4330 46.67 46.04 1 .35 CRUDE 5 2.0 0.6170 60. 14 61 .56 2.36 CRUDE 6 2.0 0.7830 72.37 75.28 4.02 CRUDE 7 2.0 0.9000 82.92 84.82 2.29 CRUDE 8 2.0 . 1.0830 94. 1 3 99.58 5.79 CRUDE 9 8.0 0.2110 20.04 19.95 0.45 CRUDE 10 8.0 0.7250 25.77 26.25 1 .85 CRUDE 11 8.0 1.4500 32.48 32. 18 0.93 CRUDE 12 8.0 2. 1830 37.04 36.90 0.38 CRUDE 13 8.0 3.0670 43.53 41 .72 4.17 CRUDE 14 8.0 4.6250 46.50 48.88 5.11 CRUDE 15 10.0 0.0900 ' 8.84 9.37 5.90 CRUDE 16 10.0 0.2850 12.44 12.31 1 .03 CRUDE 17 10.0 0.4440 14.51 13.87 4.41 CRUDE 18 10.0 0.7880 1 6.86 16.36 2.98 CRUDE 19 10.0 1.3000 19.69 1 9.06 3.20 CRUDE20 10.0 2.1330 23.49 22.34 4.90 CRUDE21 10.0 3.0500 24.88 25. 1 6 1.12 CRUDE22 10.0 4.8670 26.95 29. 52 9.55 CRUDE2 3 14.0 0.2880 3.11 3.16 1 .74 CRUDE24 14.0 1.1170 5. 18 5.04 2.81 CRUDE25 14.0 3.0670 7 . 8 1 7.63 2.34 CRUDE26 14.0 7.3330 10.71 1 1 .29 5.44 CRUDE27 14.0 13.1000 14.51 ' 14.86 2.41 CRUDE28 14.0 17.4000 18.17 17.05 6.17 -190-TABLE D-1 Re s u l t s f o r the small column using c l a r u s - B o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . TEMP= 10.0 C %WAX= 2.5 OIL=CLARUS-B POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8910 gm/cc. Dp= 0.077 cm Dc= 4.83 cm YIELD= 6.05 dyne/cm.sq. n= 0.93 H= 6.750 dyne*s**n/cm.sq. 2 RUN NO. DELP(dyne/cm ) Vo (cm/s) F* Re* B 1 • 0. 4190E+06 0. 4247E- 03 0. 1608E+09 0. 1755E- 05 B 2 0. 5960E+06 0. 6026E- 03 0. 1136E+09 0. 3183E- 05 B 3 0. 8830E+06 0. 9099E- 03 0. 7384E+08 0. 6220E- 05 B 4 0. 1210E+07 0. 1201E- 02 0. 5809E+08 0. 9570E- 05 B 5 0. 1-696E+07 0. 1728E- 02 0. 3930E+08 0. 1643E- 04 B 6 0. 2219E+07 0. 2402E- 02 0. 2663E+08 0. 2613E- 04 B 7 . 0.. 2868E+07 0. 3621E- 02 0. 1515E+08 0. 4532E- 04 TEMP= 12.0 C POROSITY= 0.36 Dp= 0.077 cm n= 0.94 %WAX= 2.5 LENGTH= 91 cm Dc= 4.83 cm H= 3.827 dyne*s**n/cm.sq. 2 OIL=CLARUS-B DENSITY= 0.8904 gm/cc. YIELD= 2.31 dyne/cm.sq, RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B 8 0. 3851E+06 0. 2469E- 02 0. 4376E+07 0. 5340E- 04 B 9 0. 5867E+06 0. 4094E- 02 0. 2425E+07 0. 1006E- 03 B 10 0. 8080E+06 0. 5835E- 02 0. 1644E+07 0. 1538E- 03 B 1 1 0. 1126E+07 0. 8938E- 02 0. 9764E+06 0. 2523E- 03 B 1 2 0. 1532E+07 0. 1269E- 01 0. 6589E+06 0. 3751E- 03 B 1 3 0. 2014E+07 0. 1694E- 01 0. 4859E+06 0. 5176E- 03 B 1 4 0. 2526E+07 0. 2190E- 01 0. 3648E+06 0. 6869E- 03 TEMP= 14.0 C %WAX= 2.5 OIL=CLARUS-B POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8897 g m / c c Dp= 0.077 cm Dc= 4.83 cm YIELD= 1.42 dyne/cm.sq n= 0.95 H= 3.002 dyne*s**n/cm.sq. 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B 1 5 0. 2441E+06 0. 1933E- 02 0. 4529E+07 0 .5259E- 04 B 1 6 0. 3462E+06 0. 2896E- 02 0. 2862E+07 0 .8714E- 04 B 1 7 0. 4791E+06 0. 4094E- 02 0. 1982E+07 0 .1320E- 03 B 18 0. 6065E+06 0. 5413E- 02 0. 1435E+07 0 .1827E- 03 B 19 0. 7766E+06 0. 7157E- 02 0. 1051E+07 0 .2513E- 03 B 20 0. 9840E+06 0. 9371E- 02 0. 7769E+06 0 .3400E- 03 B 21 0. 1152E+07 0. 1126E- 01 0. 6300E+06 0 .4167E- 03 B 22 0. 1436E+07 0. 1415E- 01 0. 4973E+06 0 .5356E- 03 B 23 0. 1771E+07 0. 1728E- 01 0. 4112E+06 0 .6660E- 03 B 24 0. 2232E+07 0. 2188E- 01 0. 3232E+06 0 .8599E- 03 3 u n ^ I W ll tJ ow t a w r a w m t d W t d i C "WS 1 2 o O 13 tf*tf*tf*tf*tf*tf*tf*tf*i . o to II c o - j c n u i t f * G j r o — • i 2 vo • i O C D o i-3 —• I • K < tf* -J II • o i a n o o o o o o o o o i C d 3 • n i tr-1 C J ro —. —. vo cn 0 0 1 T J cn cn vo cn ro cn -j U l 1 -o CTl co —» —* to cn o l Qi K> en CD CO o o o o i K : Cd Cd Cd pi RJ Cd Cd Cd 1 3 + + + + + + + + i ro O o o o o o o o i \ - J -j -j o> cn cn cn 1 o I D C * g ii n Cd S! to II 2 > -—- O X co tf* p-3 II • • O o o o o o o o 0 0 0 0 ll < U l to tf* vo cn cn CO CO to —- CD o tf* vo • —J co ro vo —fc co cn —» -—. o -- o O J ai -j —» U l o ro CD o CL g tf* tf* -j ro tf* o -»j CD g ^< o Cd Cd Cd Cd RJ PJ Pl Pl \ 3 g 1 1 I 1 1 i i 1 cn n> o o o o o o o O —- * ro to ro to to ro ro 0 0 * * D O o o o o o o o O \^ n Cd f-< o cd 2 f —» ro co U l cn vo —» 0 0 • g tr* CO II vo vo CD tf* CD U l cn • a >-< o o o CO vo cn U l o -> cn II ro o U l CO tf* vo VO K j > pa cd Cd Cd Cd Cd Cd Cd • 0 0 II 33 + + + + + + + + —• a o o o o o o o o • O CO -j —i -j - J - 0 0 0 0 0 U l i O CO CO c o o o o o o o o o Qi VO •< —' -o cn CO ro —» cn — 3 vo —» o o —* —' cn ro in o tf* —» tf* cn o tf* \ g o cn VO CO 0 0 cn U l ro n \ cd i a i Cd 1 a i H i Pl 1 PJ 1 Cd 1 * g o o o CD o o o o O cn n co tf* tf* tf* tf* tf* U l U l iQ • tn td to- td to- td tn td td tf* co co oo co oo oo oo co o VO 0 0 - 0 cn cn tf* oo ro o o o o o o o o o ro to _. _. vo -o CTi tf* co U l VD cn —' vo VD 0 0 cn tf* ro co ro vo vo VO cn 0 0 vo tf* U l CD ~o U l cn to 0 0 Cd Cd Cd Cd Cd Cd Pl Cd Cd + + + + + + + + + o o o o o o o o o o -j ~o CTi CTi cn CTl o o o o o O O o O CD - J U l 0 0 ro _» vo cn - J oo - J cn VO tf* vo CO 0 0 o tf* oo to - J o tf* - J tf* 0 0 U l o cn VD —' o o Cd i Cd i Cd 1 Cd i Cd i Cd i Cd ( Cd i Cd i o o o o o o o o o ro to ro to ro to to 0 0 CO o o o o o o o o o ro ro tf* -J _. ro tf* U l vo CTl cn to - J co tf* VO CD tf* VO o o cn ro CTi —• ~o ro CD oo VD —» —• o Cd Pl Cd Cd Cd Cd Cd Cd Cd + + + + + + + + + o o o o o o o o o - J -j -j oo CD CO 0 0 CD o o o o o O o o O cn U l oo CTi 0 0 CD cn tf* —» o to tf* oo ro —> —' to 0 0 O tf* oo CTl tf* —* 0 0 CD O U l tf* tf* ro 0 0 Cd 1 cd 1 Cd 1 Cd i ts 1 Pl Cd • • Cd l Cd 1 1 o o o o o o o o o tf* tf* tf* tf* tf* U l U l U l CTl 3 a *V >-3 W ll TJ O Cd C II w z td td td td td td td o O TJ • O CO II . oo co ro to ro ro ro 2 CO • >-i —* o vo co CT> on O tf* o >-g — • --j K J ro II • o a n o PI g • o o o o o o o o tr1 oo TJ CTi to —» —' co U l tf* ro * -. CO cn ro U l vo o ro a CTi - J o tf* 0 0 cn —• •*<: VO VD cn tf* 0 0 VD CTl 3 Cd Pl Cd Cd Cd Cd Pl ro + + + + + + + \ o o o o o o O o - J - J -o CTi cn cn (Tl g II o cd £ . ro II 2 > - — — O X ro tf* t-g II • • S3 co II o o o O o o O < ro co tf* o tf* vo • tf* oo ro —' —» CTl 0O > o o to o o U l o ca CTi o a g tf* tf* vo —fc —• cn —» g <^ o U l ~o ro tf* VD CTi VO \ 3 g RJ Cd Cd Cd Cd Cd Cd cn fD 1 i i i i i i — * o o o o o o o tn ro ro * * 3 K a o - o o o o o o o O Cd 2 t"1 g tr1 tn II vo —»• —* ro tf* CTi —* • a i-i o —» to vo U l o o —» in ll i-3 F ro cn — vo o —' —j * K > vo ui ro o o —fc U l • oo II » Cd Cd Cd Pl Cd Cd Cd oo a + + + + + + + • O CO o o o o o o o CTl • 1 U l CTi CTi CTl tTi CTi to oo td VD ' o D J O O O O O O o •< tf* 3 ro —* —* 0 0 U l co —fc fD iQ co CTi —• oo U l ~J \ g CD VD tf* U l cn o co fD o \ —* O tf* oo 0 0 CD —fc * g o Cd i Cd Cd 1 l Cd 1 Cd Cd Cd • i • in o o 1 1o o o o o o £1 • to ro ro oo oo oo co 3 a TJ -3 ll T J O td a II w s 2 o O TJ • O CO II 2 VD • i-H O ca o >-3 -• « K) CD -j II • o a O O cd g • o f 0 0 13 CTi — v a •< 3 fD n x a f <H> •-3 g II o P J s; > ro » 2 > to — ro O X t r 1 • tf* i-3 II Cd o • rc — CO II a < oo ro i o vo • —* ^ — . Qi o -» Ul o ^ g - — -g 3 O . o \ fD g o cn * 3 • — tn rt * hi-* 3 VO 3 C \ K O O fD I (1 H H H Qi i g cd 2 c — i • t r 1 in II i tn a n o i xQ II i-3 tr1 i * Kj > i II W i o G i • o co I - 0 • I —• co td CO Qi CO 0 0 3 fD iQ \ 3 fD n \ * 3 o • • U) o i i n r I i Q • -192-TABLE D-J ( cont inued) TEMP= 16.0 C %WAX= 4.0 OIL=CLARUS-B . POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8890 Dp= 0.077 cm Dc= 4.83 cm YIELD= 16.36 dyne/cm.sq . n= 0.98 H= 4.613 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B 49 0. 4112E+06 0. 1239E- 02 0. 1859E+08 0 •6097E- 05 B 50 0. 5688E+06 0. 1729E- 02 0. 1320E+08 0 .1097E- 04 B 51 0. 9492E+06 0. 3047E- 02 0. 7094E+07 0 .2832E- 04 B 52 0. 1194E+07 0. 3882E- 02 0. 5498E+07 0 .4156E- 04 B 53 0. 1507E+07 0. 4913E- 02 0. 4332E+07 0 .5954E- 04 B 54 0. 1880E+07 0. 6156E- 02 0. 3442E+07 0 .8298E- 04 B 55 0. 2445E+07 0. 8661E- 02 0. 2262E+07 0 .1341E- 03 TEMP= 18.0 G P0R0SITY= 0.36 Dp= 0.077 cm n= 1.04 RUN NO. DELP(dyne/cm ) Vo(cm/s) %WAX= 4.0 LENGTH= 91 cm Dc= 4.83 cm H= 4.356 dyne* s * *n/cm . sq . 2 F* OIL=CLARUS-B DENSITY^ 0.8883 g m / c c . YIELD= 4.80 dyne/cm.sq, Re* B 56 0. 4255E+06 0. 1774E- 02 0 .9389E+07 0. 2405E- 04 B 57 0. 5961E+06 0. 2548E- 02 0 .6376E+07 0. 3959E- 04 B 58 0. 7655E+06 0. 3215E- 02 0 .5143E+07 0. 5359E- 04 B 59 0. 9764E+06 0. 4215E- 02 0 . 3816E + 07 0. 7508E- 04 B 60 0. 1171E+07 0. 4943E- 02 0 .3328E+07 0. 9091E- 04 B 61 0. 1453E+07 0. 6035E- 02 0 .2769E+07 0. 1148E- 03 B 62 0. 1764E+07 0. 7217E- 02 0 .2352E+07 0. 1407E- 03 B 63 0. 2359E+07 0. 1024E- 01 0 .1564E+07 0. 2067E- 03 TEMP= 20.0 C %WAX= 4.0 OIL=CLARUS"B POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8875 g m / c c . Dp= 0.077 cm Dc= 4.83 cm YIELD= 1.96 dyne/cm.sq, n= 1.06 H= 2.850 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B 64 0. 2070E+06 0. 3211E- 02 0. 1395E+07 0 .9126E- 04 B 65 0. 32 44E+06 0. 5004E- 02 0. 9004E+06 0 .1504E- 03 B 66 0. 4275E+06 0. 6550E- 02 0. 6926E+06 0 .2008E- 03 B 67 0. 5552E+06 0. 8491E- 02 0. 5352E+06 0 .2634E- 03 B 68 0. 7015E+06 0. 1113E- 01 0. 3937E+06 0 .3472E- 03 B 69 0. 971OE+06 0. 1456E- 01 0. 3185E+06 0 .4544E- 03 B 70 0. 1194E+07 0. 1768E- 01 0. 2655E+06 0 .5508E- 03 B 71 0. 1583E+07 0. 2259E- 01 0. 2155E+06 0 •7003E- 03 B 72 0. 2062E+07 0. 2850E- 01 0. 1763E+06 0 .8776E- 03 B 73 0. 2479E+07 0. 3305E- 01 0. 1577E+06 0 .1013E- 02 -193-TABLE D-1 (cont inued) TEMP= 16.0 C POROSITY= 0.36 Dp= 0.077 cm n= 0.88 RUN NO. %WAX= 5.0 OIL=CLARUS-B. LENGTH= 91 cm DENSITY= 0.8890 g m / c c . Dc= 4.83 cm YIELD= 35.75 dyne/cm.sq, H= 4.632 d y n e * s * * n / c m . s q . 2 DELP(dyne/cm ) Vo(cm/s) F* Re* B 74 0.4391E+06 B 75 0.7450E+06 B 76 0.1058E+07 B 77 0.1316E+07 B 78 0.1771E+07 0.5153E-03 0.9007E-03 1577E-02 2183E-02 0 0 0.3957E-02 0.1147E+09 0.6372E+08 0.2952E+08 0.1915E+08 0.7845E+07 0.5780E-06 0.1705E-05 0.4950E-05 0.9077E-05 0.2663E-04 TEMP= 18.0 C POROSITY= 0.36 Dp= 0.077 cm n= 0.99 RUN NO. H= 5 2 DELP(dyne/cm ) %WAX= . 5.0 OIL=CLARUS-B LENGTH= 91 cm DENSITY= 0.8883 g m / c c Dc= 4.83 cm YIELD= 26.50 dyne/cm.sq, 675 dyne* s * *n/cm . sq . B 79 0.5409E+06 B 80 0.8743E+06 B 81 0.1153E+07 B 82 0.1412E+07 B 83 0.1891E+07 B 84 0.2257E+07 B 85 0.2739E+07 Vo(cm/s) 0.101OE-02 0.3487E-02 0.6186E-02 0.9856E-02 0.1456E-01 0.1706E-01 0.1985E-01 F* Re' 0.3682E+08 0.4993E+07 0.2093E+07 0.1009E+07 0.6196E+06 0.5385E+06 0.4825E+06 0.2712E-05 0.2403E-04 0.5920E-04 0.1162E-03 0.1964E-03 0.241OE-03 0.2919E-03 TEMP= 20.0 C %WAX= 5.0 OIL=CLARUS"B POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8875 g m / c c Dp= 0.077 cm Dc= 4.83 cm YIELD= 19.21 dyne/cm.sq, n= 0.77 H= 3.018 d y n e * s * * n / c m . s q . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B 86 0. 1 899E+06 0. 3095E- 03 0. 1378E+09 0 .3886E- 06 B 87 0. 3879E+06 0. 8384E- 03 0. 3836E+08 0 .2690E- 05 B 88 0. 5552E+06 0. 1601E- 02 0. 1505E+08 0 .9179E- 05 B 89 0. 7450E+06 0. 3202E- 02 0. 5049E+07 0 .3294E- 04 B 90 0. 9968E+06 0. 5568E- 02 0. 2235E+07 0 .8790E- 04 B 91 0. 1201E+07 0. 7642E- 02 0. 1429E+07 0 .1515E- 03 B 92 0. 1644E+07 0. 1317E- 01 0. 6585E+06 0 .3735E- 03 B 93 0. 2130E+07 0. 1808E- 01 0. 4528E+06 0 .6187E- 03 B 94 0. 2616E+07 0. 2206E- 01 0. 3736E+06 0 .8431E- 03 -194-TABLE D-2 R e s u l t s fo r the sma l l column us ing c l a r u s - C o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . TEMP= 12.0 C POROSITY= 0.36 Dp= 0.077 cm n= 0.95 C C C C C 95 96 97 98 99 %WAX= 2.5 OIL=CLARUS-C LENGTH= 91 cm DENSITY= 0.8870 g m / c c . Dc= 4.83 cm YIELD= 8.00 dyne/cm.sq, H= 14.580 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) C100 0, 0, 0, 0 0 0. 5417E+06 8334E+06 1142E+07 1770E+07 2430E+07 2996E+07 0, 0, 0, 0, 0, 0, 4803E-03 7424E-03 1070E-02 1681E-02 2511E-02 3182E-02 F* Re* 0.1633E+09 0.1052E+09 0.6937E+08 0.4354E+08 0.2680E+08 0.2058E+08 0.1379E-05 0.2715E-05 0.4637E-05 0.8644E-05 0.1454E-04 0.1954E-04 TEMP= 14.0 C POROSITY= 0.36 Dp= 0.077 cm n= 1.03 %WAX= 2.5 LENGTH= 91 cm Dc= 4.83 cm H= 9.970 d y n e * s * * n / c m . s q . QIL=CLARUS-C DENSITY= 0.8856 g m / c c YIELD= 2.67 dyne/cm.sq. RUN NO. PRESSURE DROP VELOCITY F* Re* CI 01 0. 3056E+06 0 .7260E- 03 0. 4038E+08 0. 5334E- 05 CI 02 0. 5584E+06 0 .1392E- 02 0. 2008E+08 0. 1195E- 04 CI 03 0. 8473E+06 0 .2205E- 02 0. 1214E+08 0. 2021E- 04 C1 04 0. 1125E+07 0 .2866E- 02 0. 9543E+07 0. 2692E- 04 CI 05 0. 1600E+07 0 .4208E- 02 0. 6293E+07 0. 4049E- 04 C1 06 0. 2246E+07 0 .5715E- 02 0. 4790E+07 0. 5560E- 04 CI 07 0. 2817E+07 0 .7167E- 02 0. 3819E+07 0. 7006E- 04 TEMP= 16.0 C %WAX= 2.5 OIL=CLARUS"C POROSITY= 0.36 LENGTH= 91 cm DENSI TY= 0.8841 g m / c c Dp= 0.077 cm Dc= 4.83 cm YIELD= 2.31 dyne/cm.sq. n= 1.01 H= 7.276 dyne* s * *n/cm. sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* CI 08 0. 3014E+06 0 .9989E- 03 0 .2108E+08 0. 1037E- 04 CI 09 0. 5556E+06 0 .1829E- 02 0 . 1159E+08 0. 2180E- 04 CI 10 0. 8473E+06 0 .2800E- 02 0 .7539E+07 0. 3552E- 04 CI 1 1 0. 1133E+07 0 .3777E- 02 0 .5542E+07 0. 4942E- 04 C1 12 0. 1711E+07 0 .5699E- 02 0 .3675E+07 0. 7680E- 04 CI 1 3 0. 2271E+07 0 .7391E- 02 0 .2900E+07 0. 1009E- 03 C1 14 0. 2896E+07 0 .9214E- 02 0 .2380E+07 0. 1268E- 03 -195-TABLE D-2 ( cont inued) TEMP= 18.0 C %WAX= 2.5 OIL=CLARUS~C POROSITY= 0.36 LENGTH= 91 cm DENSITY= 0.8826 gm/c . c . Dp= 0.077 cm Dc= 4.83 cm YIELD= 1.42 dyne/cm.sq . n= 1.08 H= 4.890 d yne * s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* C1 1 5 0. 2222E+06 0. 1299E- 02 0. 9202E+07 0. 2221E- 04 C1 1 6 0. 4639E+06 0. 2729E- 02 0. 4352E+07 0. 5030E- 04 C1 1 7 0. 7459E+06 0. 4411E- 02 0. 2680E+07 0. 8212E- 04 C1 18 0. 9654E+06 0. 5639E- 02 0. 2122E+07 0. 1047E- 03 C1 1 9 0. 1208E+07 0. 6790E- 02 0. 1832E+07 0. 1256E- 03 C120 0. 1562E+07 0. 8641E- 02 0. 1462E+07 0. 1585E- 03 C121 . 0. 1884E+07 0. 9732E- 02 • 0. 1390E+07 0. 1776E- 03 C1 22 0. 2274E+07 0. 1171E- 01 0. 1158E+07 0. 2119E- 03 CV23 0. 2831E+07 0. 1380E- 01 0. 1039E+07 0. 2474E- (13 -196-TABLE D-3 R e s u l t s f o r the l a rge column us ing c l a ru s-B o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . TEMP= 10.0 C %WAX= 2.5 OIL=CLARUS-B POROSITY= 0.44 LENGTH=100 cm DENSITY= 0.8910 g m / c c . Dp= 0.128 cm DC=10.16 cm YIELD= 6.05 dyne/cm.sq . n= 0.86 H= 6.750 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 24 0. 2349E+06 0. 2470E- 03 0. 8414E+09 0. 5004E- 06 B1 25 0. 3635E+06 0. 8510E- 03 0. 1097E+09 0. 4981E- 05 B1 26 0. 4574E+06 0. 1936E- 02 0. 2667E+08 0. 2057E- 04 B1 27 0. 5563E+06 0. 3551E- 02 0. 9641E+07 0. 5435E- 04 B1 28 0. 6876E+06 0. 6017E- 02 0. 4150E+07 0. 1198E- 03 B1 29 0. 7987E+06 0. 9815E- 02 0. 1812E+07 0. 2390E- 03 B1 30 0. 9029E+06 0. 1490E- 01 0.. 8893E+06 0. 4188E- 03 B1 31 0. 1070E+07 0. 2343E- 01 0. 4258E+06 0. 7520E- 03 IMP= 12.0 C %WAX= 2 .5 OIL =CLARUS-B POROSITY= 0.4 4 Dp= 0 . 128 cm n= 0.89 RUN NO. DELP(dyne/cm ) Vo(cm/s) LENGTH= 100 cm Dc=10.16 cm H= 3.827 dyne*s**n/cm.sq 2 F* DENSITY= 0.8904 g m / c c YIELD= 2.31 dyne/cm.sq, Re* B1 32 0. 2038E+06 0. 1443E- 02 0. 2140E+08 0. 2862E- 04 B1 33 0. 2985E+06 0. 3391E- 02 0. 5677E+07 0. 1065E- 03 B1 34 0. 3697E+06 0. 6165E- 02 0. 2127E+07 0. 2461E- 03 B1 35 0. 4492E+06 0. 8940E- 02 0. 1229E+07 0. 4025E- 03 B1 36 0. 5459E+06 0. 1347E- 01 0. 6579E+06 0. 6782E- 03 B1 37 0. 6081E+06 0. 1674E- 01 0. 4745E+06 0. 8876E- 03 B1 38 0. 7292E+06 0. 2210E- 01 0. 3265E+06 0. 1244E- 02 B1 39 0. 8820E+06 0. 3042E- 01 0. 2084E+06 0. 1822E- 02 B1 40 0. 1049E+07 0. 3925E- 01 0. 1489E+06 0. 2459E- 02 TEMP= 14.0 C %WAX= 2.5 OIL=CLARUS~B POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8897 g m / c c Dp= 0.128 cm Dc=l0.16 cm YIELD= 1.42 dyne/cm.sq . n= 0.93 H= 3.002 dyne* s * *n/cm. sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 41 0. 9540E+05 0. 1700E- 02 0. 7224E+07 0. 5483E- 04 B1 42 0. 1672E+06 0. 4620E- 02 0. 1714E+07 0. 2269E- 03 B1 43 0. 2280E+06 0. 7400E- 02 0. 9112E+06 0. 4161E- 03 B1 44 0. 2951E+06 0. 1036E- 01 0. 6017E+06 0. 6295E- 03 B1 45 0. 3766E+0.6 0. 1458E- 01 0. 3877E+06 0. 9462E- 03 B1 46 0. 5113E+06 0. 2065E- 01 0. 2624E+06 0. 1418E- 02 B1 47 0. 5874E+06 0. 2658E- 01 0. 1820E+06 0. 1892E- 02 B1 48 0. 6876E+06 0. 3237E- 01 0. 1436E+06 0. 2363E- 02 B1 49 0. 8681E+06 0. 4367E- 01 0. 9962E+05 0. 3302E- 02 B1 50 0. 1056E+07 0. 5395E- 01 0. 7937E+05 0. 4174E- 02 -197-TABLE D-3 ( cont inued) TEMP= 16.0 C POROSITY= 0.44 Dp= 0.128 cm n= 0.92 %WAX= 2.5 OIL=CLARUS-B RUN NO. DELP(dyne/cm ) B151 0.9670E+05 B152 0.1603E+06 B153 0.2336E+06 B154 0.3213E+06 B155 0.3939E+06 B156 0.4713E+06 B157 0.5597E+06 LENGTH= 100 cm DENSITY= 0.8890 g m / c c . DC=10.16 cm YIELD= 1.07 dyne/cm.sq, H= 3.085 dyne* s * *n/cm. sq . 2 . Vo(cm/s) F* Re* 0.4365E-02 0.8175E-02 0. 1212E-01 0.1767E-01 0.2260E-01 0.2980E-01 0.3545E-01 0.1112E+07 0.5253E+06 0.3482E+06 0.2254E+06 0.1689E+06 0.11 62E+.06 0.9754E+05 0.2213E-03 0.4836E-03 0.7730E-03 0.1197E-02 0.1586E-02 0.2167E-02 0.2632E-02 TEMP= 18.0 C P O R O S I T Y = 0.44 Dp= 0.128 cm n= 0.95 %WAX= 2.5 OIL=CLARUS-B B1 58 B1 59 B1 6 0 B1 61 B1 62' B1 6 3 L E N G T H = 100 cm D E N S I T Y = 0.8883 g m / c c DC=10.16 cm Y I E L D = 0.71 dyne/cm.sq, H= 2.010 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) 0, 0, 0 0 0 0 7 1 2 0 E + 0 5 1 4 1 7 E + 0 6 2 1 7 7 E + 0 6 3 0 7 5 E + 0 6 4 0 4 2 E + 0 6 5 1 1 3 E + 0 6 0 . 3 8 5 9 E - 0 2 0 . 8 8 4 1 E - 0 2 0 . 1 4 3 0 E - 0 1 0 . 2 2 4 0 E - 0 1 0 . 2 8 9 8 E - 0 1 0 . 4 0 0 7 E - 0 1 F* RE* 0.1048E+07 0.3974E+06' 0.2333E+06 0.1343E+06 0.1055E+06 0.6980E+05 0.2892E-03 0.7999E-03 0.1393E-02 0.2301E-02 0.3055E-02 0.4347E-02 TEMP= 12.0 C %WAX= 5.0 OIL=CLARUS"B POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8904 g m / c c ,Dp= 0.128 cm DC=10.16 cm YIELD= 69.37 dyne/cm.sq. n= 0.83 H= 7.367 dyne* s * *n/cm . sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 64 0. 6670E+06 0. 2256E- 03 0 . 2866E+10 0. 4010E- 07 BI 65 0. 1 089E + 07 0. 5068E- 03 0 .9268E+09 0. 2003E- 06 B1 66 0. 1452E+07 0. 7916E- 03 0 .5065E+09 0. 4841E- 06 B1 67 0. 1832E+07 0. 1109E- 02 0 .3254E+09 0. 9417E- 06 B1 68 0. 2212E+07 0. 1520E- 02 0 .2093E+09 0. 1748E- 05 B1 69 0. 2616E+07 0. 2229E- 02 0 .1151E+09 0. 3689E- 05 B170 0. 3034E+07 0. 3134E- 02 0 .6754E+08 0. 7133E- 05 -198-TABLE D-3 (cont inued) TEMP= 14.0 C %WAX= 5.0 OIL=CLARUS-B POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8897 g m / c c . Dp= 0.128 cm DC=10.16 cm YIELD= 48.02 dyne/cm.sq, n= 0.76 H= 5.369 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 71 0. 8847E+06 0. 2935E- 03 0. 2248E+10 0. 9728E- 07 B1 72 0. 1365E+07 0. 7867E- 03 0. 4827E+09 0. 6859E- 06 B173 0. 1794E+07 0. 1593E- 02 0. 1547E+09 0. 2744E- 05 B1 74 0. 2222E+07 0. 3237E- 02 0. 4641E+08 0. 1087E- 04 B1 75 0. 2613E+07 0. 5909E- 02 0. 1638E+08 0. 3430E- 04 B1 76 0. 2993E+07 0. 8529E- 02 0. 9004E+07. 0. 6827E- 04 TEMP= 16.0 C %WAX= 5.0 OIL=CLARUS~B POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8890 g m / c c Dp= 0.128 cm Dc=10.16 cm YIELD= 35.75 dyne/cm.sq, n= 0.80 H= 4.632 dyne* s * *n/cm. sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 77 0. 1458E+06 0. 1492E- 03 0. 1434E+10 0. 3400E- 07 B1 78 0. 3611E+06 0. 4414E- 03 0. 4059E+09 0. 2934E- 06 B1 79 0. 5514E+06 0. 8730E- 03 0. 1585E+09 0. 1128E- 05 B1 80 0. 7528E+06 0. 1397E- 02 0. 8448E+08 0. 2835E- 05 B1 81 0. 9542E+06 0. 2044E- 02 0. 5002E+08 0. 5944E- 05 B1 82 0. 1120E+07 0. 3115E- 02 0. 2528E+08 0. 1338E- 04 B1 83 0. 1417E+07 0. 6710E- 02 0. 6893E+07 0. 5697E- 04 TEMP= 18.0 C %WAX= 5.0 OIL=CLARUS~B POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8883 g m / c c Dp= 0.128 cm DC=10.16 cm YIELD= 26.50 dyne/cm.sq . n= 0.89 H= 5.675 dyne* s * *n/cm. sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 84 0. 2432E+06 0. 3292E- 03 0. 4919E+09 0. 2195E- 06 B1 85 0. 4492E+06 0. 1233E- 02 0. 6476E+08 0. 2917E- 05 B1 86 0. 5908E+06 0. 2713E- 02 0. 1760E+08 0. 1310E- 04 B187 0. 7765E+06 0. 7162E- 02 0. 3319E+07 0. 7627E- 04 B188 0. 1011E+07 0. 1243E- 01 0. 1435E+07 0. 1947E- 03 B189 0. 1192E+07 0. 1582E- 01 0. 1043E+07 0. 2883E- 03 B1 90 0. 1528E+07 0. 2904E- 01 0. 3969E+06 0. 7361E- 03 -199-TABLE D-3 (cont inued) TEMP= 20.0 C POROSITY= 0.44 Dp= 0. 128 cm n= 1.00 %WAX= 5.0 OIL=CLARUS-B LENGTH= 100 cm DENSITY= 0.8875 g m / c c . Dc=l0.16 cm YIELD= 19.21 dyne/cm.sq. H= 3.018 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* B1 91 0 . 2108E + 06 0. 3884E- 02 0. 3066E+07 0. 3708E- 04 B1 92 0 .3006E+06 0. 7965E- 02 0. 1 040E+07 0. 1357E- 03 B1 93 0 .4008E+06 0. 1254E- 01 0. 5592E+06 0. 2936E- 03 B1 94 0 .4989E+06 0. 1686E- 01 0. 3852E+06 0. 4737E- 03 B1 95 0 .6012E+06 0. 1965E- 01 0. 3414E+06 0. 6023E- 03 B1 96 0 .7584E+06 0. 2487E- 01 0. 2690E+06 0. 8603E- 03 B1 97 0 .9390E+06 0. 3052E- o r 0. 2212E+06 0. 1160E- 02 B198" 0 . 1 1 64E+0.7 0. 4007E- 01 0. 1 590E+06 0. 1699E- 02 -200-TABLE D-4 R e s u l t s fo r the l a rge column us ing c l a r u s - C o i l at d i f f e r e n t temperatures and wax c o n c e n t r a t i o n s . TEMP=. 10.0 -C %WAX= 2.5 OIL=CLARUS-C POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8884 g m / c c . Dp= 0.128 cm Dc=1 0.1-6 cm YIELD= 11.74 dyne/cm. sq . n= 0.93 H= 17.630 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* C1 99 0. 1431E+06 0. 3699E- 03 0. 2292E+09 0. 5454E- 06 C200 0. 3084E+06 0. 1302E- 02 0. 3987E+08 0. 5043E- 05 C201 0. 4556E+06 0. 2466E- 02 0. 1642E+08 0. 1393E- 04 C202 0. 5931E+06 0. 3761E- 02 0. 9190E+07 0. 2599E- 04 C203 0. 7223E+06 0. 5189E- 02 0. 5880E+07 0. 4079E- 04 C204 0. 8334E+06 0. 6504E- 02 0. 4317E+07 0. 5533E- 04 C205 0. 9515E+06 0. 7870E- 02 0. 3366E+07 0. 7105E- 04 C206 0. 1T11E+07 0. 9402E- 02 . 0. 2755E+07 0. 8926E- 04 TEMP= 12.0 C P O R O S I T Y = 0.44 Dp= 0.128 cm n= 0.98 %WAX= 2.5 O I L = C L A R U S - C L E N G T H = 100 cm D E N S I T Y = 0.8870 g m / c c DC=10.16 cm Y I E L D = 8.00 dyne/cm.sq. H= 14.580 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re" C207 0. 1208E+06 0. 4217E- 03 0. 1491E+09 0. 1003E- 05 C208 0. 2403E+06 0. 1287E- 02 0. 3183E+08 0. 6890E- 05 C209 0. 3653E+06 0. 2584E- 02 0. 1200E+08 0. 2002E- 04 C21 0 0. 4625E+06 0. 3597E- 02 0. 7848E+07 0. 3190E- 04 C21 1 0. 5834E+06 0. 5292E- 02 0. 4572E+07 0. 5332E- 04 C21 2 0. 7028E+06 0. 6730E- 02 0. 3406E+07 0. 7234E- 04 C21 3 0. 9417E+06 0. 9834E- 02 0. 2138E+07 0. 1148E- 03 TEMP= 14.0 C %WAX= 2.5 OIL=CLARUS-C POROSITY= 0.44 LENGTH= 100 cm DENSITY= 0.8856 g m / c c Dp= 0.128 cm Dc=l0.16 cm YIELD= 2.67 dyne/cm.sq, n= 0.95 H= 9.970 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) F* Re* C 2 1 4 0. 1 3 8 9 E + 0 6 0 . 5 1 4 2 E - 0 3 0. 1 1 5 5 E + 0 9 0. 3 4 3 7 E - 0 5 C 2 1 5 0. 2 8 2 0 E + 0 6 0 . 2 0 3 4 E - 0 2 0. 1 4 9 8 E + 0 8 6 . 2 7 0 3 E - 04 C 2 1 6 0. 3 8 2 0 E + 0 6 0 . 3 3 8 7 E - 0 2 0. 7 3 2 0 E + 0 7 0. 5 2 5 6 E - 04 C 2 1 7 0. 4 8 6 2 E + 0 6 0 . 4 9 0 1 E - 0 2 0. 4 4 5 0 E + 0 7 0. 8 2 9 5 E - 04 C 2 1 8 0. 6 1 5 3 E + 0 6 0 . 6 9 0 5 E - 0 2 0. 2 8 3 7 E + 0 7 0. 1 2 4 7 E - 0 3 C 2 1 9 0. 7 2 2 3 E + 0 6 0 . 8 7 3 4 E - 0 2 0. 2 0 8 2 E + 0 7 0. 1 6 3 8 E - 0 3 C 2 2 0 0. 8 2 5 1 E + 0 6 0 . 1 0 4 1 E - 01 0. 1 6 7 3 E + 0 7 0. 2 0 0 2 E - 0 3 C 2 2 1 0. 9 7 6 5 E + 0 6 0 . 1 2 7 4 E -01 0. 1 3 2 3 E + 0 7 0. 2 5 1 4 E - 0 3 -201-T A B L E D-4 (cont inued) TEMP= 16.0 C POROSITY= 0.44' Dp= 0. 128 cm n= 1.00 %WAX= 2.5 OIL=CLARUS-C L E N G T H = 100 cm D E N S I T Y = 0.8841 gm/c .c . DC=10.16 cm Y I E L D = 2.31 dyne/cm.sq . H= 7.276 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) C222 0.7080E+05 C223 0.2639E+06 C224 0.3750E+06 C225 0.4625E+06 C226 0.5625E+06 C227 0.6528E+06 C228 0.7584E+06 0.5647E-03 0.3740E-02 0.5891E-02 0.7706E-02 0.9556E-02 0.1150E-01 0.1367E-01 0, 0, 0, 0, 0, 0, 0, F* Re* 4890E+08 4155E+07 2379E+07 1715E+07 1357E+07 1086E+07. 8943E+06. 0.5103E-05 0 .791OE-04 0.1365E-03 0.1858E-03 0.2364E-03 0.2900E-03. 0.3496E-03 TEMP= 18.0 C %WAX= 2.5 POROSITY= 0.44 Dp= 0.128 cm n= 0.95 OIL=CLARUS-C RUN NO. DELP(dyne/cm ) Vo(cm/s) C229 C230 C231 C232 C233 C234 C235 LENGTH= 100 cm Dc= 1 0.16 cm H= 4.890 dyne*s * *n/cm.sq . 2 F* 0 0 0 0 0 0 0 7360E+05 1611E+06 2584E+06 3514E+06 4375E+06 5278E+06 6667E+06 0 0 0, 0 0 0 0 1164E-02 2863E-02 5069E-02 7398E-02 9795E-02 1233E-01 1562E-01 DENSITY= 0.8826 gm/c .c . YIELD= 1.42 dyne/cm.sq, Re* 0, 0, 0, 0, 0, 0, 0, 1198E+08 4335E+07 2218E+07 1416E+07 1006E+07 7658E+06 6029E+06 0, 0, 0, 0, 0, 0, 0, 241OE-04 8441E-04 1732E-03 2719E-03 3764E-03 4891E-03 6376E-03 - 2 0 2 -TABLE D-5 R e s u l t s fo r the l a rge column us ing waxy crude o i l from the Peace R i ve r f i e l d of A l b e r t a . TEMP= 2.0 C POROSITY= 0.44 Dp= 0 . 128 cm n= 0.93 OI.L=CRUDE L E N G T H = 100 cm D E N S I T Y = 0.8736 gm/c .c . DC=10.16 cm Y I E L D = 21.06 dyne/cm.sq, H= 6.726 dyne*s * *n/cm.sq . 2 RUN NO DELP(dyne/cm ) Vo(cm/s) F* Re* CRUDE 1 0. 9723E+05 0 .4624E- 03 0. 1014E+09 0. 5264E- 06 CRUDE 2 0. 2181E+06 0 .1628E- 02 0. 1835E+08 0. 5928E- 05 CRUDE .3 0. 3223E+06 0 .2996E- 02 0. 8000E+07 0. 1825E- 04 CRUDE 4 0. 4667E+06 0 .5339E- 02 0. 3649E+07 0. 5035E- 04 CRUDE 5 0. 6014E+06 0 .7608E- 02 0. 2316E+07 0. 9105E- 04 CRUDE 6 0. 7237E+06 0 .9655E- 02 0. 1730E+07 0. 1337E- 03 CRUDE 7 0. 8292E+06 0 .1110E- 01. 0. 1501E+07 0. 1664E- 03 CRUDE 8 0. 9413E+06 0 .1335E- 01 0. 1.1 77E+07 0. 2212E- 03 TEMP= 8.0 C POROSITY= 0.44 Dp= 0 . 1 28 cm n= 0.55 RUN NO. DELP(dyne/cm ) O I L = C R U D E L E N G T H = 100 cm D E N S I T Y = 0.8688 gm/c .c . DC=10.16 cm Y I E L D = 9.43 dyne/cm.sq. H= 2.779 dyne* s * *n/cm. sq . 2 Vo(cm/s) F* Re* CRUDE 9 CRUDE 10 CRUDE11 CRUDE 12 CRUDE 13 CRUDE 14 0.2004E+06 0.2577E+06 0.3248E+06 0.3704E+06 0.4353E+06 0.4650E+06 0 . 2 6 0 2 E - 0 2 0 . 8 9 4 0 E - 0 2 0 . 1 7 8 8 E - 0 1 0 . 2 6 9 2 E - 0 1 0 . 3 7 8 2 E - 0 1 0 . 5 7 0 3 E - 0 1 0 . 6 6 3 4 E + 0 7 0 . 7 2 2 6 E + 0 6 0 . 2 2 7 7 E + 0 6 0 . 1 1 4 5 E + 0 6 0 . 6 8 2 0 E + 0 5 0 . 3 2 0 4 E + 0 5 0.3046E-04 0.2981E-03 0.1027E-02 0.2096E-02 0.3756E-02 0.7513E-02 TEMP= 10.0 C POROSITY= 0.4 4 Dp= 0 . 128 cm n= 0.41 O I L = C R U D E L E N G T H = 100 cm D E N S I T Y = 0.8672 gm/c . c . DC=10.16 cm Y I E L D = 6.76 dyne/cm.sq, H= 1.274 dyne*s * *n/cm.sq . 2 RUN NO. DELP(dyne/cm ) Vo(cm/s) CRUDE 15 CRUDE 16 CRUDE 17 CRUDE 18 CRUDE 19 CRUDE20 CRUDE21 CRUDE22 0.8845E+05 0.1244E+06 0.1451E+06 0.1686E+06 0.1969E+06 0.2349E+06 0.2488E+06 0.2695E+06 0.1110E-02 0.3514E-02 0.5475E-02 0.9716E-02 0.1603E-01 0.2630E-01 0.3761E-01 0.6001E-01 F* Re* 0 . 1612E + 08 0.2262E+07 0.1087E+07 0.4010E+06 0.1720E+06 0.7625E+05 0.3949E+05 -0. 1680E+05 0.8665E-05 0.8117E-04 0.1905E-03 0.5695E-03 0.1469E-02 0.3720E-02 0.7238E-02 0.1714E-01 - 2 0 3 -T A B L E D-5 ( cont inued) TEMP= 1 4 . 0 C P O R O S I T Y = 0.44 Dp= 0 . 1 2 8 cm n= 0 . 5 3 O I L = C R U D E L E N G T H = 1 0 0 cm D E N S I T Y = 0 . 8 6 4 2 g m / c c . D C = 1 0 . 1 6 cm . Y I E L D = 3 . 5 6 dyne/cm.sq. H= 0 . 4 0 5 4 d yne* s * *n/cm. sq . 2 RUN NO. DELP ( dyne/cm ) Vo(cm/s) F* Re* C R U D E 2 3 C R U D E 2 4 C R U D E 2 5 C R U D E 2 6 C R U D E 2 7 C R U D E 2 8 0 . 3 1 1 O E + 0 5 0 . 5 1 8 3 E + 0 5 0 . 7 8 0 8 E + 0 5 0. 1 0 7 1 E + 0 6 0 . 1 4 5 1 E + 0 6 0. 1 8 1 7 E + 0 6 0. 3 5 5 1 E - 0 2 0 . 1 3 7 7 E - 0 1 0 . 3 7 8 2 E - 0 1 0 . 9 0 4 2 E - 0 1 0. 1 6 1 5 E - 0 0 0 . 2 1 4 5 E - 0 0 0 . 5 5 5 7 E + 0 6 0 . 6 1 5 9 E + 0 5 0 . 1 2 3 0 E + 0 5 0 . 2 9 5 2 E + 0 4 0 . 1 2 5 3 E + 0 4 0 . 8 8 9 7 E + 0 3 1 6 8 8 E - 0 3 2 2 7 4 E - 0 2 1 5 0 0 E - 0 1 7 2 7 8 E - 0 1 2 0 2 5 E - 0 0 0 . 3 3 1 3 E - 0 0 -204-TABLE D-6 C a l c u l a t e d a n d ~ P . r . e d i c , t e d _ F r i c t i o n , L F a c t o r _ s . f o r .both Columns f o r a l l D a t a in-. Terms o f ' l o q "(f*) - l o g ~(R*) RUN [ NO. TEMP(C) %WAX LogRe* LogF*exp . L o g F * c a l c . %Dev. B 1 10.0 2.5 -5.7557 8.2063 8.1501 -0.69 B 2 10.0 2.5 -5.4972 8.0554 7.8916 -2.08 B 3 10.0 2.5 -5.2062 7.8683 7.6006 -3.52 B 4 10.0 2.5 -5.0191 7.7641 7.4135 -4.73 B 5 10.0 2.5 -4.7844 7.5944 7.1787 -5.79 B 6 10.0 2.5 -4.5829 7.4254 6.9773 -6.42 B 7 10.0 2.5 -4.3437 7. 1804 6.7381 -6.56 B 8 12.0 2.5 -4.2725 6.6411 6.6669 0.39 B 9 12.0 2.5 -3.9974 6.3847 6.3918 0.11 B 10 12.0 2.5 -3.8130 6.2159 6.2074 -0.14 B 1 1 12.0 2.5 -3.5981 5.9896 5.9926 0.05 B 1 2 12.0 2.5 -3.4259 5.8188 5.8203 0.03 B 13 12.0 2.5 -3.2860 5.6865 5.6804 -0.11 B 1 4 12.0 2.5 - 3 . 1631 5.5621 5.5575 -0.08 B 15 14.0 2.5 -4.3226 6.6560 6.7170 0.91 B 1 6 14.0 2.5 -4.1067 6.4567 6.5012 0.68 B 17 14.0 2.5 -3.9289 6.2971 6.3233 0.41 B 18 14.0 2.5 -3.7891 6.1569 6.1836 0.43 B 19 14.0 2.5 -3.6519 6.0216 6.0465 0.41 B 20 14.0 2.5 -3.5216 5.8904 5.9160 0.43 B 21 14.0 2.5 -3.4338 5.7994 5.8283 0.50 B 22 14.0 2.5 -3.3254 5.6968 5.7198 0.40 B 23 14.0 2.5 -3.2310 5.6138 5.6254 0.21 B 24 14.0 2.5 - 3 . 1205 5.5096 5.5149 0.10 B 25 18.0 2.5 -3.7617 6.0700 6.1562 1 .40 B 26 18.0 2.5 -3.4549 5.7789 5.8494 1 .20 B 27 18.0 2.5 -3.2702 5.6021 5.6646 1.10 B 28 18.0 2.5 -3.0887 5.4133 5.4832 1 .27 B 29 18.0 2.5 -2.9416 5.2815 5.3361 1 .02 B 30 18.0 2.5 -2.7721 5.0986 5.1664 1.31 B 31 18.0 2.5 -2.6232 4.9604 5.0179 1.14 B 32 12.0 4.0 -6.0839 7.8982 8.4784 6.84 B 33 12.0 4.0 -5.7300 7.6494 8.1245 5.85 B 34 12.0 4.0 -5.4642 7.4504 7.8587 5.19 B 35 12.0 4.0 -5.2045 7.2477 7.5990 4.62 B 36 12.0 4.0 -4.9978 7.0803 7.3923 4.22 B 37 12.0 4.0 -4.7496 6.881 3 7.1440 3.68 B 38 12.0 4.0 -4.5047 6.6714 6.8992 3.30 B 39 12.0 4.0 -4.2662 6.4694 6.6606 2.87 B 40 12.0 4.0 -4.1864 6.4128 6.5808 2.55 -205-TABLE D-6 ( cont inued ) RUN NO. TEMP(C) %WAX LogRe* LogF*exp . L o g F * c a l c . %Dev B 41 14.0 4.0 -5.7846 7.5586 8.1790 7.59 B 42 14.0 4.0 -5.2143 7.1787 7.6087 5.65 B 43 14.0 4.0 -4.9337 6.9936 7.3282 4.57 B 44 14.0 4.0 -4.6834 6.8305 7.0777 3.49 B 45 14.0 4.0 -4.5167 6.7400 6.9112 2.48 B 46 14.0 4.0 -4.2908 6.5832 6.6853 1 .53 B 47 14.0 4.0 -4.0999 6.4627 6.4943 0.49 B 48 14.0 4.0 -3.8539 6.2797 6.2482 -0.50 B 49 16.0 4.0 -5.2057 7.2693 7.6002 4.35 B 50 16.0 4.0 -4.9481 7. 1206 7.3426 3.02 B 51 16.0 4.0 -4.5308 6.8509 6.9252 1 .07 B 52 16.0 4.0 -4.3617 6.7402 6.7562 0.24 B 53 16.0 4.0 -4.2029 6.6368 6.5974 -0.60 B 54 16.0 4.0 -4.0563 6.5369 6.4507 -1 .34 B 55 16.0 4.0 -3.8444 6.3543 6.2388 -1 .85 B 56 18.0 4.0 -4.6189 6.9726 7.0133 0.58 B 57 18.0 4.0 -4.4024 6.8045 6.7969 -0.11 B 58 18.0 4.0 -4.2709 6.7112 6.6654 -0.69 B 59 18.0 4.0 -4.1245 6.5816 6.5189 -0.96 B 60 18.0 4.0 -4.0414 6.5222 6.4358 -1 .34 B 61 18.0 4.0 -3.9401 6.4423 6.3345 -1 .70 B 62 18.0 4.0 -3.8517 6.3714 6.2463 -2.00 B 63 18.0 4.0 -3.6847 6.1942 6.0792 -1 .89 B 64 20.0 4.0 -4.0397 6.1446 6.4342 4.50 B 65 20.0 4.0 -3.8228 5.9544 6.2172 4.23 B 66 20.0 • 4.0 -3.6972 5.8405 6.0917 4.12 .B 67 20.0 4.0 -3.5794 5.7285 5.9738 4.11 B 68 20.0 4.0 -3.4594 5.5952 5.8539 4.42 B 69 20.0 4.0 -3.3426 5.5031 5.7370 4.08 B 70 20.0 4.0 -3.2590 5.4241 5.6535 4.06 B 71 20.0 4.0 -3.1547 5.3334 5.5491 3.89 B 72 20.0 4.0 -3.0567 5.2463 5.4512 3.76 B 73 20.0 4.0 -2.9944 5.1978 5.3888 3.54 B 74 16.0 5.0 -6.2381 8.0596 8.6326 6.64 B 75 16.0 5.0 -5.7683 7.8043 8.1629 4.39 B 76 16.0 5.0 -5.3054 7.4701 7.6998 2.98 B 77 16.0 5.0 -5.0421 7.2822 7.4365 2.08 B 78 16.0 5.0 -4.5746 6.8946 6.9691 1 .07 B 79 18.0 5.0 -5.5667 7.5661 7.9612 4.96 B 80 18.0 5.0 -4.6192 6.6984 7.0137 4.50 B 81 18.0 5.0 -4.2277 6.3208 6.6221 4.55 B 82 18.0 5.0 -3.9348 6.0039 6.3292 5.14 B 83 18.0 5.0 -3.7069 5.7921 6.1014 5.07 B 84 18.0 5.0 -3.6180 5.7312 6.0124 4.68 B 85 18.0 5.0 -3.5348 5.6835 5.9292 4.14 TABLE D-6 RUN NO. TEMP(C) %WAX LogRe* B 86 20.0 5.0 -6.4105 B 87 20.0 5.0 -5.5702 B 88 20.0 5.0 -5.0372 B 89 20.0 5.0 -4.4823 B 90 20.0 5.0 -4.0560 B 91 20.0 5.0 -3.8196 B 92 20.0 5.0 -3.4277 B 93 20.0 5.0 -3.2085 B 94 20.0 5.0 -3.0741 C 95 12.0 2.5 -5.8604 C 96 12.0 2.5 -5.5662 C 97 12.0 2.5 -5.3338 C 98 12.0 2.5 -5.0633 C 99 12.0 2.5 -4.8374 C1 00 12.0 2.5 -4.7091 CI 01 14.0 2.5 -5.2729 C1 02 14.0 2.5 -4.9226 C1 03 14.0 2.5 -4.6944 C1 04 14.0 2.5 -4.5699 C1 05 14.0 2.5 -4.3927 C1 06 14.0 • 2.5 -4.2549 C1 07 14.0 2.5 -4.1545 CI 08 16.0 2.5 -4.9842 C1 09 16.0 2.5 -4.6615 C1 1 0 16.0 2.5 -4.4495 CI 1 1 16.0 2.5 -4.3061 Gl 1 2 16.0 2.5 -4.1146 CI 1 3 16.0 2.5 -3.9961 C1 1 4 16.0 2.5 -3.8969 CI 1 5 18.0 2.5 -4.6535 CI 1 6 18.0 2.5 -4.2984 C1 1 7 18.0 2.5 -4.0856 CI 18 18.0 2.5 -3.9801 CI 1 9 18.0 2.5 -3.9010 CI 20 18.0 2.5 -3.8000 CI 21 18.0 2.5 -3.7506 C1 22 18.0 2.5 , -3.6739 C1 23 18.0 2.5 -3.6066 B1 24 10.0 2.5 -6.3007 B1 25 10.0 2.5 -5.3027 B1 26 10.0 2.5 -4.6868 B1 27 10.0 2.5 -4.2648 B1 28 10.0 2.5 -3.9215 B1 29 10.0 - 2.5 -3.6216 B1 30 10.0 2.5 -3.3780 B1 31 10.0 2.5 - 3 . 1238 (cont inued) LogF*exp. L o g F * c a l c . %Dev. 8. 1392 8.8050 7.56 7.5839 7.9647 4.78 7. 1775 7.4317 3.42 6.7032 6.8767 2.52 6.3493 6.4504 1 .57 6.1550 6.2140 0.95 5.8186 5.8222 0.06 5.6559 5.6029 -0.95 5.5724 5.4686 -1 .90 8.2130 8.2548 0.51 8.0220 7.9607 -0.77 7.8412 7.7282 -1 .46 7.6389 7.4577 -2.43 7.4281 7.2320 -2.71 7.3134 7.1035 -2.96 7.6062 7.6674 0.80 7.3028 7.3170 0.19 7.0842 7.0888 0.07 6.9797 6.9644 -0.22 6.7989 6.7871 -0.17 6.6803 6.6493 -0.47 6.5819 6.5490 -0.50 7.3239 7.3788 0.74 7.0641 7.0561 -0.11 6.8773 6.8440 -0.49 6.7437 6.7005 -0.64 6.5653 6.5091 -0.86 6.4624 6.3906 -1.12 6.3766 6.2914 -1 .35 6.9639 7.0481 1.19 6.6387 6.6928 0.81 6.4281 6.4800 0.80 6.3267 6.3746 0.75 6.2629 6.2956 0.52 6.1649 6.1945 0.48 6.1430 6.1449 0.03 6.0637 6.0682 0.07 6.0166 6.0009 -0.26 8.9250 8.6951 -2.64 8.0402 7.6971 -4.46 7.4260 7.0813 -4.87 6.9841 6.6593 -4.88 6.6180 6.3160 -4.78 6.2582 6.0162 -4.02 5.9490 5.7725 -3.06 5.6292 5.5183 -2.01 TABLE D-6 RUN NO. TEMP(C) %WAX LogRe* B1 32 12.0 2.5 -4.5284 B1 33 12.0 2.5 -3.9666 B1 34 12.0 2.5 -3.6148 B1 35 12.0 2.5 -3.4106 B1 36 12.0 2.5 -3.1959 B1 37 12.0 2.5 -3.0852 B1 38 12.0 2.5 -2.9473 B139 12.0 2.5 -2.7918 B140 12.0 2.5 -2.6696 B1 41 14.0 2.5 -4.2610 B1 42 14.0 2.5 -3.6442 B1 43 14.0 2.5 -3.3808 B1 44 14.0 2.5 -3.2010 B145 14.0 2.5 -3.0240 B146 14.0 2.5 -2.8483 B147 14.0 2.5 -2.7231 B1 48 14.0 2.5 -2.6265 B149 14.0 2.5 -2.4812 B1 50 14.0 2.5 -2.3794 B1 51 16.0 2.5 -3.6550 B152 16.0 2.5 -3.3155 B1 53 16.0 2.5 -3.1118 B154 16.0 2.5 -2.9219 B1 55 16.0 2.5 -2.7997 B1 56 16.0 2.5 -2.6641 B1 57 16.0 2.5 -2.5797 B1 58 18.0 2.5 -3.5388 B159 18.0 2.5 -3.0970 B1 60 18.0 2.5 -2.8560 B1 61 1.8.0 2.5 -2.6381 B1 62 18.0 2.5 -2.5150 B1 63 18.0 2.5 -2.3618 B164 12.0 5.0 -7.3969 B165 12.0 5.0 -6.6983 B166 12.0 5.0 -6.3151 B1 67 12.0 5.0 -6.0261 B1 68 12.0 5.0 -5.7575 B1 69 12.0 5.0 -5.4331 B1 70 12.0 5.0 -5.1467 B171 14.0 5.0 -7.0954 B172 14.0 5.0 -6.2428 B173 14.0 5.0 -5.6351 B1 74 14.0 5.0 -5.0281 B1 75 14.0 5.0 -4.5171 B176 14.0 5.0 -4.2084 207-(cont inued ) LogF*exp . L o g F * c a l c . %Dev 7.3304 6.9229 -5.89 6.7541 6.3610 -6.18 6.3278 6.0090 -5.30 6.0896 5.8051 -4.90 5.8185 5.5904 -4.08 5.6760 5.4797 -3.58 5.5140 5.3418 -3.22 5.3189 5.1864 -2.55 5. 1729 5.0641 -2.15 6.8588 6.6554 -3.06 6.2340 6.0386 -3.24 5.9596 5.7752 -3.19 5.7794 5.5955 -3.29 5.5885 5.4185 -3.14 5.4190 5.2428 -3.36 5.2601 5.1176 -2.78 5.1572 5.0212 -2.71 4.9983 4.8757 -2.52 4.8997 4.7739 -2.63 6.0461 6.0496 0.06 5.7204 5.7099 -0.18 5.5418 5.5062 -0.65 5.3530 5.3164 -0.69 5.2276 5.1942 -0.64 5.0652 5.0584 -0.13 4.9892 4.9741 -0.30 6.0204 5.9332 -1 .47 5.5992 5.4914 -1 .96 5.3679 5.2504 -2.24 5.1281 5.0326 -1 .90 5.0233 4.9094 -2.32 4.8439 4.7563 -1 .84 9.4573 9.7913 3.41 8.9670 9.0927 1 .38 8.7046 8.7095 0.06 8.5124 8.4206 -1 .09 8.3208 8.1520 -2.07 8.0611 7.8276 -2.98 7.8296 7.5412 -3.82 9.3518 9.4898 1 .45 8.6837 8.6373 -0.54 8.1895 8.0294 -1 .99 7.6667 7.4226 -3.29 7.2143 6.9116 -4.38 6.9544 6.6029 -5.32 -208-TABLE D-6 ( cont inued) RUN NO. TEMP(C) %WAX LogRe* LogF*exp . L o g F * c a l c . %Dev B177 16.0 5.0 -7.4710 9.1565 9.8654 7.19 B178 16.0 5.0 -6.5363 8.6084 8.9307 3.61 B1 79 16.0 5.0 -5.9519 8.2000 8.3464 1 .75 B180 16.0 5.0 -5.5518 7.9268 7.9462 0.24 B1 81 16.0 5.0 -5.2296 7.6990 7.6241 -0.98 B182 16.0 5.0 -4.8758 7.4026 7.2702 -1 .82 B18.3 16.0 5.0 -4.2406 6.8384 6.6351 -3.06 B184 18.0 5.0 -6.6586 8.6919 9.0531 3.99 B1 85 18.0 5.0 -5.5351 7.8113 7.9295 1 .49 B1 86 18.0 5.0 -4.8827 7.2455 7.2772 0.43 B1 87 18.0 5.0 -4.1176 6.5210 6.5122 -0.14 B188 18.0 5.0 -3.7106 6.1569 6.1052 -0.85 B189 18.0 5.0 -3.5402 6.0183 5.9346 -1.41 B1 90 18.0 5.0 -3.1331 5.5987 5.5275 -1 .29 B1 91 20.0 5.0 -4.4309 6.4866 6.8253 4.96 B1 92 20.0 5.0 -3.8674 6.0170 6.2620 3.91 B1 93 20.0 5.0 -3.5322 5.7476 5.9267 3.02 B1 94 20.0 5.0 -3.3245 5.5857 5.7189 2.33 B1 95 20.0 5.0 -3.2202 5.5333 5.6147 1 .45 B1 96 20.0 5.0 -3.0654 5.4298 5.4598 0.55 B1 97 20.0 5.0 -2.9355 5.3448 5.3300 -0.28 B1 98 20.0 5.0 -2.7698 5.2014 5.1644 -0.72 C1 99 10.0 2.5 -6.2633 8.3602 8.6577 3.44 C200 10.0 2.5 -5.2973 7.6006 7.6918 1.18 C201 10.0 2.5 -4.8560 7.2154 7.2504 0.48 C202 10.0 2.5 -4.5852 6.9633 6.9796 0.23 C203 10.0 2.5 -4.3894 6.7694 6.7839 0.21 C204 10.0 2.5 -4.2570 6.6352 6.6515 0.24 C205 10.0 2.5 -4 . 1484 6.5271 6.5428 0.24 C206 10.0 2.5 -4.0493 6.4401 6.4437 0.06 C207 12.0 2.5 -5.9987 8.1735 8.3932 2.62 C208 12.0 2.5 -5.1618 7.5028 7.5562 0.71 C209 12.0 2.5 . -4.6985 7.0792 7.0931 0.20 C210 12.0 2.5 -4.4962 6.8948 6.8906 -0.06 C21 1 12.0 2.5 -4.2731 6.6601 6.6675 0.11 C212 12.0 2.5 -4.1406 6.5322 6.5350 0.04 C21 3 12.0 2.5 -3.9401 6.3300 6.3345 0.07 C21 4 14.0 2.5 -5.4638 8.0626 7.8583 -2.60 C21 5 14.0 2.5 -4.5682 7.1755 6.9626 -3.06 C21 6 14.0 2.5 -4.2793 6.8645 6.6738 -2.86 C217 14.0 2.5 -4.0812 6.6484 6.4757 -2.67 C218 14.0 2.5 -3.9041 6.4529 6.2986 -2.45 C21 9 14.0 2.5 -3.7857 6.3185 6.1801 -2.24 C220 14.0 2.5 -3.6985 6.2235 6.0931 -2.14 C221 14.0 2.5 -3.5996 6.1216 5.9941 -2.1 3 -2 09-TABLE D-6 ( cont inued) RUN NO. TEMP(C) %WAX LogRe* LogF*exp. L o g F * c a l c . %Dev. C222 1.6.0 2.5 -5.2922 7.6893 7.6866 -0.03 C223 16.0 2.5 -4.1018 6.6186 6.4962 -1 .88 C224 16.0 2.5 -3.8649 6.3764 6.2594 -1 .87 C225 16.0 2.5 -3.7310 6.2343 6.1255 -1.78 C226 16.0 2.5 -3.6264 6.1326 6.0208 -1 .86 C227 16.0 2.5 -3.5376 6.0358 5.9321 -1 .75 C228 16.0 2.5 -3.4564 5.9515 5.8509 -1 .72 C229 18.0 2.5 -4.6180 7.0785 7.0124 -0.94 C230 18.0 2.5 -4.0736 6.6370 6.4681 -2.61 C231 18.0 2.5 -3.7615 6.3460 6.1559 -3.09 C232 18.0 2.5 -3.5656 6.1511 5.9600 -3.21 C233 18.0 2.5 -3.4244 6.0026 5.8188 -3.16 C234 18.0 2.5 -3.3106 5.8841 5.7051 -3.14 C235 18.0 2.5 -3.1955 5.7802 5.5899 -3.40 RUN NO. TEMP(C) LogRe* LogF*exp . L o g F * c a l c . %Dev. CRUDE 1 2 0 -6. 2787 8. 0060 8. 6731 7 69 CRUDE 2 2 0 -5. 2271 7. 2636 7. 6216 4 70 CRUDE 3 2 0 -4. 7387 6 9031 7. 1 332 3 23 CRUDE 4 2 0 -4. 2980 6 5622 6. 6925 1 95 CRUDE 5 2 0 -4. 0407 6 3647 6. 4352 1 10 CRUDE 6 2 0 -3. 8739 6 2380 6. 2683 0 48 CRUDE 7 2 .0 -3. 7788 6 1764 6. 1732 -0 05 CRUDE 8 2 0 -3 . 6552 6 0708 6. 0496 - o .35 CRUDE 9 8 0 -4. 4962 6 8217 6. 8906 1 .00 CRUDE 10 8 .0 -3. 4923 5 8590 5. 8867 0 .47 CRUDE 11 8 .0 -2 9458 5 3574 5. 3402 - o .32 CRUDE 12 8 .0 -2 6302 5 0592 5. 0245 - o .69 CRUDE 13 8 .0 -2 3720 4 8339 4. 7665 -1 .41 CRUDE 14 8 .0 -2 0650 4 5057 4. 4595 -1 03 CRUDE 15 10 .0 -5 0388 7 2074 7. 4333 3 04 CRUDE 16 10 .0 -4 0550 6 3545 6. 4495 1 47 CRUDE 17 10 .0 -3 6786 6 0362 6 0730 0 .61 CRUDE 18 10 .0 -3 1 940 5 6031 5. 5885 - o .26 CRUDE 19 10 .0 -2 7737 5 .2358 5 1682 -1 .31 CRUDE20 10 .0 -2 3603 4 .8823 4 7547 -2 .68 CRUDE21 10 .0 -2 0636 4 .5965 4 4580 -3 . 1 1 CRUDE22 10 .0 -1 6786 4 .2253 4 0730 -3 .74 CRUDE23 14 .0 -3 7726 5 .7448 6 1670 6 .85 CRUDE24 14 .0 -2 6432 4 7895 5 0378 4 .93 CRUDE25 14 .0 -1 8239 4 .0899 4 21 83 3 .04 CRUDE26 1 4 .0 -1 1 380 3 .4701 3 5325 1 .77 CRUDE27 14 .0 -0 6936 3 .0980 . 3 0881 -0 .32 CRUDE28 14 .0 -0 4798 2 .9492 2 8742 -2 .61 -210-T a b l e D - 7 R e s u l t s o f L e a s t S q u a r e s F i t f o r a l l C l a r u s - B a n d C l a r u s - C D a t a , ( L o g f * a s a F u n c t i o n o f L o g R* ) _ - ( U B C - O L S ) N u m b e r o f D a t a P o i n t s I n t e r c e p t S l o p e LOG ( f * ) S T . D E V . V A R I A N C E MEAN RANGE 4.844 235 2 . 6 6 7 - 0 . 9 3 3 0 . 9 6 4 0 . 9 3 0 6 . 6 0 0 9 . 4 5 7 L 0 G ( R * ) e S T . D E V . V A R I A N C E MEAN RANGE - 7 . 4 7 1 1 . 0 1 3 1 . 0 2 6 - 4 . 2 1 4 - 2 . 3 6 2 C o r r e l a t i o n C o e f f i c i e n t 0 . 9 8 0 -211-APPENDIX E Sample C a l c u l a t i o n F o r 2.5% wt. i n C l a r u s - B o i l f o r t h e l a r g e c olumn a t T = 16°C Runs No. B151 t o B157 i n t h e A p p e n d i x . 1) C a l c u l a t i o n o f t h e r h e o l o g i c a l p a r a m e t e r s , T , H, and n. S h e a r s t r e s s e s and s h e a r r a t e s were c a l c u l a t e d by u s i n g t h e f o l l o w i n g e q u a t i o n s : 2 x = 5.336 S dyne/cm E - l y = 5.41 n 1 s e c 1 E-2 where n' and S a r e t h e v i s c o m e t e r r e a d i n g s . The f o l l o w i n g t a b l e was o b t a i n e d ., Log 2 T (dyne/cm <*•) Log n' Y ( s ~ 1 ) <Y) • S . T — T O 0.01 0.054 0 .2 1.07* 0 2 10.82 1.034 5 .5 29.35 28 .28 1.451 4 21.64 1. 335 10 .0 53.36 52 .29 1.718 6.4 34.62 1.539 14 .9 79.51 78 .44 1.895 8 43.28 1.636 18 .3 97.65 96 .58 1.985 12.8 69 .25 1.840 27 .7 147.81 146 .74 2.167 16 86.56 1.937 34 .6 184.63 183 .56 2.264 25.6 138.50 2.141 53 .9 287.61 286 .54 2.457 32 173.12 2.238 66 .9 356.98 355 .91 2.551 51.2 276.99 2.442 103 .2 550.68 249 .61 2.740 T q = 1.07 t a k e n a t l o w e s t s p e e d n = 0.01 min -212-By t a k i n g t h e l o g a r i t h m o f t h e H e r s c h e l - B u l k l e y model ( E q u a t i o n 2-6) Log (x - x ) = l o g H + n l o g (y) E-3 A f t e r l e a s t s q u a r e f i t o f t h e d a t a H and n were d e t e r m i n e d f r o m t h e i n t e r c e p t and t h e s l o p e o f E q u a t i o n E-3, i . e . H = 3.085 (dyne/cm 2) s e c 1 1 n = 0.92 2) C a l c u l a t i o n o f t h e m o d i f i e d D a r c y ' s e q u a t i o n p a r a m e t e r s (K, O" q, V J=£" n and AP) by u s i n g E q u a t i o n 3-18. AP = Q n + a L K A n E-4 u - ,v- fL Log (AP - a L) = l o g ( ) + n l o g Q ° K A n E-5 By l e a s t s q u a r e f i t o f t h e e x p e r i m e n t a l d a t a t h e f o l l o w i n g t a b l e was c o n s t r u c t e d . Q l o g Q APexp l o g ( A P e x p - A P e a l ( c . c . / s ) — (kPa) a Q L ) (kPa) 0.0 - 1.84* 0 . 354 -0.451 9 .67 0.89 4 9.90 0.663 -0.178 16.03 1.152 16.20 0.983 -0.0074 23.36 1.333 22.47 1.433 0.156 32.13 1.481 31.02 1.833 0.263 39.39 1.575 38.44 2.417 0.383 47.13 1.656 49.05 2.875 0.459 55.97 1.733 57.22 T h i s v a l u e ( a 0 L ) was d e t e r m i n e d f r o m t h e l e a s t s q u a r e f i t o f t h e l o w e s t t h r e e f l o w r a t e s . — 2 1 3 -From t h e f i t t i n g we g e t t h e s l o p e n = 0.92 t h e i n t e r c e p t e = 20.955 KP s n era" K A n where U e f f . , n c r i n 5 ,, -n-3, K = 1.1956 x 10 (dyne.cm ) E-6 S i n c e L = 100 cm, e = 0.44, A = 81.1 cm 2 2 n = 0.92 , H = 3.085, x 0 = 1-07 dyne/cm t h e n e q u a t i o n (3-21) c a n be a p p l i e d t o d e t e r m i n e t h e p e r m e a b i l i t y , K 1-n -1-n = I (H" + 3 ) 1 1 ( 8 ^ K E-7 s u b s t i t u t e E-6 i n t o E-7 1-0.92 -1-0.92 1.1956 x 10 5 = 3 , ° 8 5 ( o - % 2 + ° ^ 9 2 (8 x 0.44) 2 K K = 15.92 x 1 0 ~ 6 cm 2 _ 3 a Q = 184 dyne/cm d e t e r m i n e d f r o m t h e i n t e r c e p t o f t h e AP v e r s u s Q c u r v e . From E q u a t i o n 3-20 -214-1.07 _ Q 1 8 4 = 3 •15.92 x 10-6 3 = 0.686 w h i l e y e £ f was d e t e r m i n e d f r o m E q u a t i o n 3-19 1-n ^ e f f = ! (n" + 3 ) 1 1 ( 8 £ K ) 2 1-0.92 0 . 9 2 2 ^ e f f = 3 ' 4 8 5 (0T9 '2 + 3 ) ( 8 x ° ' 4 4 X 1 5 ' 9 2 x 1 0 _ 6 ) = 1 . 9 0 3 8 E - 9 by s u b s t i t u t i n g t h e s e v a l u e s i n t o E q u a t i o n E - 4 A T ) _ 1 . 9 0 3 8 X 100 n 0 - 9 2 ^ IQ/I i n n ^ / 2 AP = — T ; Q + 184 x 100 dyne/cm 15.92 x 10 x ( 8 1 . 1 ) ' y A P c a l = 2 0 . 9 5 9 Q ° " 9 2 + 1 . 8 4 KPa A P c a l = 3 8 . 4 4 K P a f o r Q = 1 . 8 3 3 c m 3 / s c (Run NO .B155) 3) C a l c u l a t i o n o f f r i c t i o n f a c t o r and R y n o l d s number, by u s i n g t h e f o l l o w i n g e q u a t i o n s * AP D_ 3 f* = e x P _Z y-^— E-10 exp L T72 1 - £ * 248 f c a l - E - " K -215-where R* was c a l c u l a t e d u s i n g E q u a t i o n 3-29 e> 12p C M o 2H D e 3 V N + e 2 x K p o o ^ E-12 and t, = D £ p 3 ( l - e ) n+1 (1-e) E-13 S u b s t i t u t i n g t h e f o l l o w i n g v a l u e s f o r r u n No. B155 2 L = 100 cm n = 0.92 2 T 0 = 1.07 dyne/cm D = 0.128 cm P 3 Q = 1 . 8 3 3 cm / s e c 3.085 (dyne/cm 2) s e c 1 1 e = 0.44 H p = 0.889 gm/cm AP A = 81.1 cm exp = 3.939 x 1 05 dyne/cm 2 T = 16°C V q = Q/A = 0.02 26 cm/sec f r o m E q u a t i o n E-13 £ = 6.375 x 10 f r o m E q u a t i o n E-12 R|= 1.586 x 10" -4 f r o m E q u a t i o n E - l l f * , 1.564 x 10-f r o m E q u a t i o n E-10 f * = 1.689 x 10 -^ -exp The r e s t o f t h e c a l c u l a t e d r e s u l t s a r e shown i n T a b l e D-3 page (197) . -216-T a b l e F - l L u b r i c a t i n g O i l S p e c i f i c a t i o n s B r a n d Name : H V I - 2 5 0 N e u t r a l " C l a r u s - B " ( S h e l l O i l o f C a n a d a ) C H A R A C T E R I S T I C S E n g l i s h S y s t e m M e t r i c S y s t e m G r a v i t y at 15°C 3 0 . 3 A P | 0 . 8 7 4 1 ; C o l o u r , ASTM D 1 5 0 0 L 0 . 5 L 0 . 5 P o u r P o i n t +5 ° F - 1 5 °C F l a s h , COC 4 5 0 ° F -V i s c o s i t y ASTM D 2 6 0 2 SSU c S t . 2 6 3 . 5 6 1 0 0 ° F 5 1 . 2 @ 4 0 ° C> 5 0 . 2 @ 2 1 0 ° F 7 . 0 8 @ 1 0 0 ° C V i s c o s i t y I n d e x 94 94 C a r b o n R e s i d u e , %W 0 . 0 7 0 . 0 7 N e u t r a l i z a t i o n V a l u e , T A N - C 0 . 0 1 0 . 0 1 S u l p h u r , %W 0 . 4 6 0 . 4 6 C o r r o s i o n , C u . s t r i p N o . 1 @ 2 1 2 ° F N o . 1 @ 1 0 0 ° C A c t i v e S u l p h u r N i l @ 4 0 0 ° F N i l @ 2 0 4 ° C A n i l i n e P o i n t 2 2 1 ° F 1 0 5 ° C A p p e a r a n c e c l e a r c l e a r H e r s c h e l d e m u l s i b i l i t y m l o i l s e p a r a t e d p e r h o u r 1 4 7 0 @ 1 3 0 ° F 1 4 7 0 @ 5 4 ° C -217-T a b l e F - 2 L u b r i c a t i n g O i l S p e c i f i c a t i o n s B r a n d Name : H V I -5 8 0 N e u t r a l " C l a r u s -C " C H A R A C T E R I S T I C S E n g l i s h S y s t e m M e t r i c S y s t e m G r a v i t y a t 1 5 ° C 2 9 . 3 ° A . P . I . 0 . 8 7 9 5 C o l o u r , ASTM D 1 5 0 0 L 1 . 0 L 1 . 0 P o u r P o i n t +10 ° F - 1 2 °C F l a s h , COC 5 2 0 ° F -V i s c o s i t y , A S T M D 2 6 0 2 1 2 3 . 7 c s @ 1 0 0 ° F 1 1 0 . 7 e s t @ 4 0 ° C 1 2 . 8 c s @ 2 1 0 ° F 1 1 . 8 6 e s t @ 1 0 0 ° C V i s c o s i t y I n d e x 9 5 9 5 C a r b o n R e s i d u e , %wt 0 . 0 7 0 . 0 7 N e u t r a l i z a t i o n V a l u e , T A N - C 0 . 0 1 0 . 0 1 S u l p h u r , %wt 0 . 4 8 0 . 4 8 C o r r o s i o n , C u . s t r i p N o . 1 @ 2 1 2 ° F N o . 1 @ 1 0 0 ° C A c t i v e S u l p h u r N i l @ 4 0 0 ° F N i l @ 2 0 4 ° C A n i l i n e P o i n t 239 ° F 1 1 5 ° C A p p e a r a n c e C l e a r C l e a r D e m u l s i b i l i t y A STM D 1 4 0 1 m i s . 4 0 - 4 0 - 0 ( 1 0 ) 4 0 - 4 0 - 0 ( 1 0 ) @ 1 8 0 ° F @ 8 2 ° C -218-Table K-3 Paraffin Wax Characteristics PARVAN 55 (Esso Petroleum Co) CHARACTERISTICS Test Method ASTM Melting Point - ASTM, °C D87 55 Odour D1833 0 Colour, Saybolt (min.) D156 +30 Oil Content, wt% D721 0.4 Blocking Point, °C D1465 35 Flash Point, COC, °C Viscosity, cSt @ 100°C D92 220 D445 3.4 cSt @ 80°C 4.7 SSU 6 210°F 37.6 SSU @ 180°F 41.2 Needle Penetration D1321 mm/10 @ 25°C 12 35 °C 48 45°C 118 Density D1465 25°C Solid 0.90 80°C 0.77 100°C 0.759 -219-T a b l e F - 4 P h y s i c a l P r o p e r t i e s o f P e a c e R i v e r C r u d e O i l o f A l b e r t a ( C h e v r o n C a n a d a L i m i t e d ) G r a v i t y , A P I 3 9 . 5 P o u r P o i n t , ° F 0 . 0 V i s c o s i t y , c S t . @ 1 0 0 ° F 2 . 9 4 @ 1 3 0 ° F 2 . 3 5 S u l p h u r , % w t . 0 . 3 3 F l a s h P o i n t , ° F 1 0 0 H S , ppm 4 0 C h r o m a t o g r a p h o f S T - 3 0 0 ° F C u t C a r b o n P a r a f f i n s N a p h t h e n e s A r o m a t i c s T o t a l s N u m b e r N o r m a l I s o C 4 C 5 ' S C 2 0 . 0 6 - - - - 0 . 0 6 C 3 1 . 0 7 - - - - 1 . 0 7 C 4 4 . 9 0 1 . 3 3 - - - 6 . 2 3 C 5 5 . 8 2 4 . 2 1 0 . 5 9 - - 1 0 . 6 2 C 6 5 . 1 6 5 . 3 2 3 . 1 9 2 . 2 0 0 . 9 1 1 6 . 7 9 C 7 5 . 9 3 4 . 7 4 8 . 2 3 4 . 8 9 2 . 4 5 2 6 . 2 3 C 8 4 . 9 7 6 . 2 8 5 . 5 7 4 . 8 0 4 . 2 6 2 5 . 8 8 C 9 2 . 2 6 4 . 0 7 0 . 9 5 3 . 7 1 0 . 0 5 1 1 . 0 4 C 1 0 — 0 . 9 1 0 . 5 9 0 . 5 8 — 2 . 0 8 T o t a l 3 0 . 1 7 2 6 . 8 6 1 9 . 1 2 1 6 . 1 8 7 . 6 7 1 0 0 . 0 0 T o t a l 5 7 . 0 3 3 5 . 30 7 . 6 7 1 0 0 . 0 0 -220-T a b l e F - 5 S p e c i f i c a t i o n s o f t h e P r e s s u r e T r a n s d u c e r DPT 3 6 2 J - 5 Q f r o m D y n i s c o PERFORMANCE C H A R A C T E R I S T I C S A c c u r a c y W i t h i n ±0 .5% F u l l S c a l e o u t p u t ( t e r m i n a l l i n e a r i t y ) i n c l u d i n g n o n l i n e a r i t y , h y s t e r e s i s a n d r e p e a t a b i l i t y . R e p e a t a b i l i t y W i t h i n 0 .2% F u l l S c a l e o u t p u t R e s o l u t i o n I n f i n i t e S t o r a g e - 5 0 ° F t o + 2 5 0 ° F t e m p . r a n g e ( - 4 5 ° C t o + 1 2 0 ° C ) C o m p e n s a t e d t e m p . + 3 0 ° F t o + 1 3 0 ° F r a n g e ( 0 ° C t o + 5 5 ° C ) T h e r m a l e f f e c t s o n z e r o L e s s t h a n ±0 . 01% F u l l S c a l e / ° F ( 0 . 0 2 % F . S . / ° C ) : T h e r m a l e f f e c t s o n L e s s t h a n ±0 .02% R e a d i n g / ° F s e n s i t i v i t y ( 0 . 0 4 % R e a d i n g / ° C ) E L E C T R I C A L C H A R A C T E R I S T I C S F u l l S c a l e o u t p u t ± 5 . 0 VDC O u t p u t i s a d j u s t a b l e ± 0 . 5 VDC w i t h a n i n t e r n a l p o t e n t i o m e t e r M i n i m u m l o a d 2 5 0 0 ohms E x c i t a t i o n ±12 VDC t o ±16 V D C ; o r 28 VDC ±4 VDC Z e r o b a l a n c e F a c t o r y s e t w i t h i n ±1 .0% o f F u l l S c a l e o u t p u t . A d j u s t a b l e ±5% o f F u l l S c a l e w i t h i n t e r n a l p o t e n t i o m e t e r . I n p u t r e s i s t a n c e 5 0 0 ohms a p p r o x i m a t e l y O u t p u t r e s i s t a n c e L e s s t h a n 1 ohm C o n n e c t o r B e n d i x P T 0 2 A - 1 0 - 6 P ; M a t i n g c o n n e c t o r s u p p l i e d MECHAN ICAL C H A R A C T E R I S T I C S R a n g e s ± 5 0 , P S I D L i n e p r e s s u r e r a t i n g s 1 5 0 0 p s i m a x . f o r ±50 P S I D r a n g e S a f e o v e r p r e s s u r e T w i c e r a t e d d i f f e r e n t i a l p r e s s u r e r a n g e w i t h o u t d amage B u r s t p r e s s u r e 3 0 0 0 p s i f o r ±50 P S I D r a n g e M a t e r i a l A M 3 5 0 , 1 7 - 4 P H , a n d 3 0 3 S t a i n l e s s S t e e l s B u n a - N 0 - r i n g s s t a n d a r d ; O t h e r 0 - r i n g c o m p o u n d s a v a i l a b l e o n r e q u e s t W e i g h t 24 o u n c e s = 221-T a b l e F-6 D e n s i t i e s o f t h e Waxy O i l s a t D i f f e r e n t T e m p e r a t u r e s T e m p a r a t u r e D e n s i t y D e n s i t y * * (gm/c.c.) ( g m / c c . ) (°C) 2.5% wax 4% wax i n i n C l a r u s - C C l a r u s - B 8* 0.8900 0.8918 10 0.8884 0.8904 12* 0.8870 0.8904 14* 0.8856 0.8897 15 0.8850 0.8896 16* 0.8841 0.8890 18* 0.8826 0.8883 20 0.8811 0.8875 T e m p e r a t u r e °C D e n s i t y (gm/c.c.) P e a c e R i v e r C r u d e 2 0.8736 4* 0.8719 6* 0.8703 7 0.8693 8* 0.8688 10* 0.8672 12* 0.8656 14 0.8642 D e n s i t y was d e t e r m i n e d by i n t e r p o l a t i o n . ** The same d e n s i t i e s were u s e d f o r 2.5 and 5% wax. 

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