a: .' - RIEOLOGICAL CHARACTERIZATION OF FRACTURING FLUIDS

Robert K . Prud'homme, A l i c e Chu, and J e f f r e y Kramer*

I. INTRODUCTION

Guar g e l s c r o s s l i n k e d w i t h t r a n s l t l o n - m e t a l i o n s a r e used a s fracturing f l u i d s i n o i l w e l l complet ion. Ge ls are a t t r a c t i v e because c r o s s l i n k i n g c r e a t e s a f l u i d w i t h s u f f i c i e n t v i s c o s i t y t o suspend s o l i d p r o p a n t s w h i l e r e q u i r i n g c n l y s m a l l amounts o f p a r polymer. To s i m u l a t e t h e f r a c t u r i n g o p e r a t i o n and p r e - d l c t f r a c t u r e geometry i t 1s n e c e s s a r y t o u n d e r s t a n d and model t h e rheo logy of g u a r g e l s . To d a t e , t h e r e p r o d u c l b l l i t y of l a b o r a t o r y tests of g u a r g e l rheo logy h a s been poor and models o f g e l rheo logy have i n v o l v e d o n l y e m p i r i c a l modifications o f t h e power-law f l u l d model. These e m p i r i c a l models a r e i n c a - p a b l e of d e s c r i b i n g t h e e f f e c t s o f s h e a r and t i m e h i s t o r y on g e l properties.

The purpose o f t h i s paper i s t o p r o v i d e a working g u l d e t o t h e rheo logy , and c h a r a c t e r i z a t i o n o f g u a r g e l s . F i r s t , i n S e c t l o n 11, we d e s c r i b e the e x p e r i m e n t a l t e c h n i q u e s used t o s t u d y g u a r rheo logy , which i n c l u d e dynamic o s c i l l a t o r y s h e a r measurements and s t e a d y s h e a r measurements. Dynamic o s c l l l a - t o r y s h e a r measurements a r e e s p e c i a l l y l m p o r t a n t i n i n v e s t i g a t i n g g e l s t r u c t u r e because t h e s e measurements can be u s e d t o de te rmine t h e number of network c r o s s - l l n k s . These measurements and t h e i r i n t e r p r e t a t i o n a r e d i s c u s s e d i n d e t a i l , s i n c e t h e y are p r o b a b l y l e s s f a m i l i a r t o r e s e a r c h e r s i n t h e o i l production r e s e a r c h a r e a t h a n a r e s t e a d y s h e a r measurements. I n S e c t i o n I11 we d e s c r i b e t h e r h e o l o g i c a l I n s t r u m e n t s used i n t h i s s t u d y . I n S e c t i o n I V t h e p r e p a r a t l o n o f guar samples i s d e t a i l e d . The compos l t lon o f t h e model g u a r g e l u s e d i n t h l s s t u d y was specified by t h e A P I s t e e r i n g c o m m t t e e . Our o b s e r v a t i o n s on t h e f a c - t o r s c o n t r o l l i n g g e l rheo logy , including chemlca l e f f e c t s , sample p r e p a r a t l o n e f f e c t s , and f l o w h i s t o r y e f f e c t s a r e p r e s e n t e d i n S e c t l o n V. I n S e c t i o n V I a model t h a t describes t h e rheo logy o f g e l l i n g f l u l d s i s described. The model i s based on t h e temporary network t h e o r r e s used t o d e s c r l b e t h e r h e o l o g y of polymer s o l u t i o n s . To t h i s t h e o r y we have i n c o r p o r a t e d t h e chemlca l k l n e t i c s o f m e t a l l o n a d s o r p t i o n o n t o t h e g u a r polymer backbone and s u b s e q u e n t polymer-polymer c r o s s l i n k i n g . I n t h e f i n a l s e c t i o n recommendatlons f o r s t a n d a r d t e s t p rocedures , f o r r h e o l o g i c a l l n s t r u r n e n t a t i o n , and f o r f u t u r e r e s e a r c h a r e p r e s e n t e d .

I I . RHEOLOG ICAL MEASUREMENTS

A. Dynamlc O s c i l l a t o r y Measurements

Dynamic o s c i l l a t o r y s h e a r experiments which measure t h e l i n e a r v l s c o e l a s t l c r e s p o n s e o f materials a r e acknowledged t o be t h e most v a l u a b l e p robes of g e l o r network s t r u c t u r e . Though s t e a d y s h e a r measurements a r e n e c e s s a r y t o d u p l i c a t e p r o c e s s c o n d i t i o n s , t h e o s c i l l a t o r y measurenen ts g i v e more i n s i g h t i n t o t h e p r o p e r t i e s of t h e g e l t h a n d o s t e a d y s h e a r measurenents . When i n t e r p r e t e d u s i n g classical network t h e o r y , l i n e a r v l s c o e l a s t l c measure- ments can be used t o de te rmine t h e k l n e t i c s of g e l f o r m a t i o n , t h e c r o s s l i n k den- s i t y o f a g e l , o r t h e s h e a r d e g r a d a t i o n o f g e l s t r u c t u r e . The g e l a t i o n o f p o l y v i n y l a l c o h o l and g e l a t i n g e l s have been s t u d i e d by a number of r e s e a r c h e r s ( 1 , 2 , 3 ) , and a t P r i n c e t o n we have used t h e s e measurements t o s t u d y p o l y a c r ~ l b - mide g e l s used a s p e r m e a b i l i t y c o n t r o l a g e n t s i n enhanced o i l r e c o v e r y ( 4 , 5 ) .

fc P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , New J e r s e y

I n a l i n e a r v i s c o - e l a s t i c measurement a n o s c i l l a t o r y s t r a i n , ~ ( t ) , i s imposed on a sample,

~ ( t ) = y o s i n wt . ( 1 )

Experimentally t h i s i s accomplished by p l a c i n g a s a a p l e i n a cone and p l a t e geometry , a p a r a l l e l p l a t e geometry, o r between c o n c e n t r i c c y l i n d e r s i n a C o u e t t e geometry, and t h e n imposing a t o r s i o n a l o s c i l l a t i o n on one p l a t e , cone, o r c y l i n d e r . The r e s u l t i n g s t r e s s on t h e s z a t i o n a r y p l a t e , cone, o r c y l i n d e r w i l l o s c i l l a t e w i t h t h e imposed f r e q u e n c y w, b u t w i l l be o u t o f p h a s e n t h t h e f o r c i n g oscillation. The measured stress c a n be f a c t o r e d i n t o two components, one i n phase w i t h t h e d i s p l a c e m e n t and one 90 d e g r e e s o u t o f phase w i t h t h e displacement :

~ ( t ) = yo(G1 s i n w t + G" c o s w t ) . ( 2 )

The ln-phase stress d e f i n e s a s t o r a g e modulus G ' t h a t g i v e s i n f o r m a t i o n a b o u t e l a s t l c r t y and network s t r u c t u r e , whereas t h e out-of -phase component def l n e s a l o s s modulus G" tha t g i v e s i n f o r m a t i o n a b o u t t h e v i s c o u s o r dissipative pro- p e r t i e s o f t h e f l u i i . The f requency and s t r a i n dependence of t h e s t o r a g e and l o s s moduli , G' and G" r e s p e c t i v e l y , p r o v i d e i n f o r m a t i o n a b o u t t h e s t a t e of t h e f l u i d . For a n u n c r o s s l i n k e d guar s o l u t i o n b o t h G ' and G " d e c r e a s e w i t h d e c r e a s i n g f r e q u e n c y , w i t h G" l y i n g above G I . As a g e l c r o s s l i n k s G ' rises u n t i l i t i s h o r i z o n t a l - - i n d e p e n d e n t of f r e q u e n c y . A s a n example, t h i s p r o g r e s s i o n i s shown i n Fig. 1 foz t h e g e l a t i o n o f a p c l y s t y r e n e / c a r b o n d i s u l f i d e s o l u t i o n as t e m p e r a t u r e 1s d e c r e a s e d . As we w i l l show i n S e c t i o n V, G' can be moni tored as t h e a m p l i t u d e of t h e s t r a i n d e f o r m a t i o n i s i n c r e a s e d . I f s t r a i n d e s t r o y s t h e network s t r u c t u r e , t h e n G ' w i l l d e c r e a s e w l t h i n c r e a s i n g s t r a i n .

Classical network t h e o r y ( 7 ) shows t h a t G I , i n t h e low f r e q u e n c y r e g i o n where G ' i s i n d e p e n d e n t of f requency , i s p r o p o r t i o n a l t o t h e number d e n s l t y o f c r o s s l i n k s i n t h e g e l :

where g i s a c o n s t a n t o f o r d e r one, n i s t h e number d e n s l t y o f c r o s s l i n k s , k is Bol tzman 's c o n s t a n t , T i s t h e a b s o l u t e t e m p e r a t u r e , and Ge i s a c o n t r i b u t i o n t o t h e modulus f rorn 'molecular en tang lements . For aqueous g e l s Ge 1s v e r y s m a l l . It i s p o s s i b l e t o f o l l o w t h e k i n e t l c s o f g e l fo rmat ion by measur ing G ' a s a func- t i o n of t ime. The r a t e of t h e c r o s s l i n k i n g r e a c t i o n i s o b t a l n e d by t a k l n g t h e t i m e d e r i v a t i v e o f Eq. 3:

Likewlse t h e d e s t r u c t i o n o f g e l s t r u c t u r e by s h e a r can be moni to red by measuring

G ' a f t e r e x p o s u r e t o s t e a d y s h e a r . The r e s u l t s can be i n t e r p r e t e d I n terms of t h e breakdown i n t h e number of c r o s s l i n k p o i n t s .

111. EOUIPMENT

A. Rheometrics Sys t e m - I V Rheometer

Most of t h e measurements r epo r t ed here w e r e conducted on our Rheometrics Inc. (Piscataway, N J ) System I V rheometer. This s t a t e of t he a r t ins t rument shown i n Fig. 2 has s e v e r a l motor and t r ansduce r op t ions . The i n s t r u - ment i s f u l l y automated and a l l d a t a a c q u i s i t i o n and manipulat ion is under com- p u t e r con t ro l . For measurements w i th t h e F lu ids Transducers a circulating water ba th is a v a i l a b l e w i th a temperature range from -20 C t o 80 C.

For most of t h e guar s o l u t i o n measurements a F lu ids Transducer wi th a 10 g-cm maximum to rqe and 100 g maximum normal f o r c e was used. This F lu ids Transducer a l lows s t eady shea r measuremements of f l u i d v i s c o s i t y , dynamic o s c i l l a t o r y shea r measurements, and, wi th some modi f ica t ion to t h e d r i v e u n i t , s t eady shea r fol lowed by o s c i l l a t o r y shear . The F lu ids Transducers can be run wi th cone-and-plate, p a r a l l e l p l a t e , o r Couet te geometries.

For dynamic o s c i l l a t o r y measurements on guar g e l s t h e 1 0 g-cm t r ans - ducer i s i d e a l ; however, t h e to rque range of t h i s t r ansduce r i s qu ick ly exceeded i f s teady shea r measurements a r e a t tempted on g e l s . Therefore , f o r t h e bulk of t h e g e l measurements a F lu ids Transducer wi th a 100 g-cm torque range was used.

B. Impingement Mixing Device

The homogeneity achieved dur ing t h e rmxing of t he guar and metal i on s o l u t i o n s and s h e a r h i s t o r y of t h e f l u i d a s i t c r o s s l i n k s determines t he g e l p r o p e r t i e s . The recommended procedure of mixing t h e guar s o l u t i o n and metal i on s o l u t i o n i n a b lender and then t r a n s f e r r i n g t h e preformed g e l t o t h e viscometer y i e l d s i r r e p r o d u c i b l e r e s u l t s . This w i l l be d i scussed below. To circumvent t h i s problem, an impingement mixing dev ice was f a b r i c a t e d t h a t i n t i m a t e l y mixes t he two s t reams and i n j e c t s them d i r e c t l y i n t o t h e rheometer t es t ce l l (F ig . 3 ) . The device c o n s i s t s of a s t a l n l e s s s teel double a c t i n g pneumatic c y l i n d e r t h a t is mechanically coupled t o a m i c r o l i t e r g l a s s syr inge . The pneumatic c y l i n d e r is p r e s s u r i z e w i th n i t rogen a t 200 p s i t o f o r c e guar s o l u t i o n i n t h e cy l inde r and metal i on s o l u t i o n i n t h e s y r i n g e through an impingement mixing head and then through a packed bed mixing s e c t i o n . The packed bed c o n s i s t s of t h r e e inches of a 1/4" OD s t a i n l e s s s t e e l t ube packed wi th 200 mesh sand. During i n j e c t i o n through the sand pack t h e Reynolds number i s about one, based on a mean hydrau l i c r a d i u s f o r t h e sand pack and t h e v i s c o s i t y of t h e uncross l inked guar. The connect ions i n t h e device a r e made wi th 1/8" t e f l o n tubing. A three-way va lve i s used to d i v e r t f l u i d e i t h e r t o waste o r t o t h e rheometer c e l l . The f l u i d f lows d i r e c t l y i n t o t h e rheometer c e l l and the dynamic o s c i l l a t o r y measurement can be i n i t i a t e d even before t h e f l u i d f i l l s t h e gap. The t o t a l t i m e between t h e i n i t i a l con tac t ing of t h e p a r and metal i o n s o l u t i o n s and t h e start of an experiment i s on the o r d e r of 5 t o 10 seconds.

IV. MATERIALS AND PREPARATION - The exact formulation of the guar g e l was speci f ied by the API Committee

monitoring t h i s p ro jec t . Specia l l o t s of hydroxypropyl guar and Tyzor AA t i t a - na te were reserved f o r t h i s s tudy by Celanese and DuPont, respect ive ly . The following formulation was used t o produce a 40 lb/bbl ge l :

500 m l d i s t i l l e d water

2.4 g hydroxypropyl guar ((3elanese SCN 9574)

0.6 g sodium d i a c e t a t e buffer ( Celanese SCN 9744 )

10 9 a n a l y t i c a l grade KCL ( f i s h e r l o t 722797)

0.125 m l 25% glutaraldehyde i n water (Eastman Kodak l o t El 1 A )

2 m l of ( 9 : l ) so lu t ion by volume isopropyl a lcohol (JT Baker) and Tyzor AA t i t a n a t e (DuPont)

The base guar (without cross l inking agent) is prepared using an Os te r i ze r blender s e t a t low speed. A t imer and var iac a r e connected with the blender i n s e r i e s t o con t ro l mixing time and speed. The so lu t ion is prepared i n the following way. The blender, with 500 m l of water i n the p i t c h e r , i s s e t a t a low speed t o produce a shallow vortex. The hydroxypropyl guar is sprinked slowly on the f r e e surface t o produce a uniform dispers ion . The potassium chlor ide , sodlum d i a c e t a t e and glutaraldehyde a r e quickly added. The t o t a l mixing time i s three minutes. The so lu t ion i s then t r ans fe r red t o another conta iner and allowed t o mix f o r about 20 hours on a low shear tumbling mixer.

To prepare crossl inked guars f o r rheologica l s t u d i e s two procedures were used. During the f i r s t ha l f year the so lu t ions were mixed by hand, and i n the l a s t ha l f year the impingement mixing device was used. In mixing by hand, 10 m l of base guar so lu t ion i s placed i n a beaker, followed by- a propor t ional amount of Tyzor AA ( d i l u t e d with isopropanol) . The so lu t ion i s s t i r r e d vigorously with a g l a s s s t i r r i n g rod f o r 30 seconds and t r ans fe r red i n t o the rheometer cup. The rheometer s t age is then closed t o s e t the proper gap and the t e s t begins. This technique proved more reproducible than mixing the guar and t i t a n a t e i n a blender. However, some va r i a t ion i n r e s u l t s with hand mixing caused by inade- quate homogenization and time delays l ed t o our development of the impingement mixer.

Between runs the rheometer t o o l s were cleaned with water, followed by alco- hol . Thorough cleaning was required o r the g e l would prematurely s l i p a t the t o o l surfaces during measurement.

V. RESULTS AND DISCUSSION - The p r o p e r t i e s of guar g e l s depend on t h e chemistry choosen, t h e mixing of

t h e components, and s h e a r h i s t o r y of the ge l . I n t h i s s e c t i o n we p r e s e n t t he r e s u l t s of s t u d i e s on t h e s e e f f e c t s .

A. Chemical E f f e c t s

1. Hydration of HP-Guar

HP-guar molecules are n o t p r e s e n t a s i s o l a t e d e n t i t i e s bu t a r e r a t h e r contained i n s i d e an i n t a c t ce l l about t e n microns i n diameter . The guar d e r i v a t i z a t i o n i s done i n such a way t h a t t h e ce l l w a l l i s n o t d i s rup t ed . When t h e HP-guar i s added t o water , water p e n e t r a t e s t h e c e l l w a l l and begins to hydra te the guar. During hydra t ion m i c r o c r y s t a l l i n e c e l l u l o s e domains a r e d i s so lved , t h e c e l l swells, and f i n a l l y the c e l l w a l l r up tu re s r e l e a s i n g the guar . The r a t e a t which t h i s occurs depends on pH, temperature , and the osmotic 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 c e l l wal l . An experiment was conducted t o test how long i t took t o f u l l y hydra te t h e guar. A guar s o l u t i o n was prepared and mixed i n t h e b lender f o r 30 minutes under s t rong a g i t a t i o n . Immediately t h e r e a f t e r dynamic o s c i l l a t o r y measurements were run a s a f u n c t i o n of s t r a i n amplitude. The r e s u l t s i n Fig. 4 show t h a t G' decreases a s s t r a i n amplitude i nc reases . This i n d i c a t e s t h a t t h e r e a r e three dimensional s t r u c t u r e s i n solu- t i o n , probably a r i s i n g from a s s o c i a t i o n of the unhydrated guar domains, t h a t a r e e a s i l y broken down by shear . This i s n o t observed i f the guar is allowed t o age 0 h o u r s a f t e r t h e i n i t i a l mixing. This same phenomenon i s shown i n Fig. 5 where, f o r t h e s o l u t i o n mlxed 30 minutes, t h e ' s t e a d y shea r v i s c o s i t y i nc reases a t low shea r r a t e s , whereas f o r t h e aged s o l u t i o n t h e v i s c o s i t y reaches a --- Newtonian p l a t eau . This i n c r e a s e i n low shea r viscosity a l s o i n d i c a t e s aggrega- t i o n and s L u c t u r e i n so lu t ion . I t is important t o no te t h a t a t h ighe r shear r a t e s t h e v i s c o s i t i e s of t h e two f l u i d s a r e i d e n t i c a l s i n c e moderate shea r f i e l d s can d i s r u p t t he se weak aggrega tes . W e s e e t h a t dynamic o s c i l l a t o r y o r low shear r a t e measurements a r e s e n s i t i v e probes of s t r u c t u r e i n s o l u t i o n .

2. Aging of TyZOr AA So lu t ions

Tyzor AA s o l u t i o n s change co lo r from l i g h t yel low t o orange o r brown over a pe r iod of t i m e . On t h e b a s i s of d i s cus s ions w i t h D r . Donald Futzig o f DuPont, t h r e e exp lana t ions a r e proposed:

a . Photoreduct ion of Titanium. This w i l l g ive a green/blue co lo r . Keeping s o l u t i o n s i n brown b o t t l e s e l i m i n a t e s this problem.

b. Oxidat ion of Acetylacetone by Oxygen. This w i l l g ive an orange co lo r . D r . Pu tz ig d i d n o t t h ink t h i s would a f f e c t t h e c r o s s l i n k i n g r eac t ion .

c . Hydration of Titanium by Water from t h e Air. This w i l l g ive a whi te t i t an ium d iox ide p r e c i p i t a t e .

We found t h a t g e l s made from Tyzor AA s o l u t i o n s t h a t had been opened many times over a s e v e r a l months made g e l s wi th lower va lues of G I . Fig- u r e 8 shows t h e s t o r a g e modulus of g e l s made from newly opened b o t t l e s of Tyzor

AA and from b o t t l e s t h a t had been used f o r s e v e r a l months. To e l i m i n a t e t h e s e problems t h e l a r g e p i n t b o t t l e s of Tyzor AA were divided i n t o s e v e r a l sma l l e r v i a l s and sea l ed . Brown v i a l s were used and molecular s e i v e s (W.R. Grace 3#1' were added t o t h e v i a l t o scavenge water. This worked w e l l and gave reprodu- c i b l e r e s u l t s t h a t d i d n o t show t h e e f f e c t s of aging.

3. Di-ke tone Addition

It has been suggested t h a t d i -ke tones can be used t o slow t h e r a t e o f c ros s l ink ing . Our r e s u l t s show t h a t n o t only does t h e a d d i t i o n of diketone slow t h e r e a c t i o n r a t e , b u t i t a l s o prevents t h e g e l from c r o s s l i n k i n g f u l l y . With the a d d i t i o n of d i -ke tone , a s shown i n Fig. 9, t h e f i n a l value of t he s t o r a g e modulus i s decreased.

4. Isopropanol Addition

S ince i sopropanol is t h e base l i q u i d f o r t h e t i t a n a t e s o l u t i o n , t e s t s were conducted on t h e e f f e c t of added a lcohol . No e f f e c t s were de tec ted .

B. E f f e c t s of Flow on Gel P rope r t i e s ---- I n t h i s s e c t i o n we p r e s e n t t h e main r e s u l t s on t h e rheology of guar

g e l s i n dynamic o s c i l l a t o r y and s t eady shea r flows. The dynamic o s c i l l a t o r y measurements a r e used t o c h a r a c t e r i z e t he network s t r u c t u r e , whereas t h e s t eady shea r measurements a r e meant t o measure g e l v i s c o s i t y under process condi t rons .

1 . Dynamic O s c i l l a t o r y Measurements

a . Mixing

The degree of homogenization a t a microscopic l e v e l is cru- clal i n determining t h e f i n a l g e l s t r u c t u r e . m e problem of mixing r e a c t i v e t i t an ium and guar s o l u t i o n s is i n many ways analogous t o problems encountered I n r e a c t i o n i n j e c t i o n molding of polyurethanes i n t h e p l a s t i c s i ndus t ry . Thorough mixing i n t h e impingement mixing devlce r e s u l t e d i n g e l s with lower va lues of G ' than g e l s made wi th hand mixing. W e be l i eve t h i s i s caused by the inhomogeneous g e l s t r u c t u r e t h a t is produced by incomplete mixing. During hand mixing i n t e r f a c e s a r e developed between s t r i a t i o n s o r l a y e r s of bulk guar solu- t i o n and t h e very concent ra ted t i t a n t a t e s o l u t i o n . The r e a c t i o n r a t e i n t hese i n t e r f a c i a l reg ions i s very h igh wi th t h e r e s u l t t h a t reg ions of dense c r o s s l i n k s t r u c t u r e a r e developed. The f i n a l g e l formed conta ins microscopic threads o r s h e e t s of more h igh ly c r o s s l i k e d guar t h a t a r e e l a s t i c , impar t ing t o t he g e l a h igher l e v e l of G' than would be p red ic t ed i f t h e c r o s s l k i n k s were homogeneously d i s t r i b u t e d . Cha rac t e r i za t ion of t h e s t a t e of mixedness dur ing t h e product ion of g e l s i n t h e l abo ra to ry o r f i e l d i s c r i t i c a l .

b. S torage Modulus Versus S t r a i n Amplitude

F igure 10 shows t h e e f f e c t of s t r a i n on guar g e l s produced i n t h e impingement mixer, in t roduced i n t o t h e gap between p a r a l l e l p l a t e s , and allowed to s i t f o r 1 5 minutes before t h e dynamic o s c i l l a t o r y t e s t was begun. A t low s t r a i n s t he va lue of G' i s above G". A s t h e s t r a i n ampli tude i s increased G I

drops d rama t i ca l l y i n d i c a t i n g t h a t t h e guar g e l s t r u c t u r e i s being d i s rup t ed by s t r a i n . I n t h i s experiment t h e s t r a i n i s f i r s t increased t o 500 % and then decreased. Hys t e r e s i s can be observed i n d i c a t i n g t h a t i n t h e t i m e s c a l e of t he se experiments t h e g e l does n o t rehea l . The n e c e s s i t y of having adequate torque s i g n a l s r e q u i r e s t h a t most of our experiments were run a t 100 % s t r a i n . I t should be k e p t i n mind t h a t a t t h e s e s t r a i n s some d e s t r u c t i o n of g e l network s t r u c t u r e occurs . The s t r a i n s e n s i t i v i t y of guar g e l s i s i n s h a r p c o n t r a s t to t h e p e r f e c t l y e l a s t i c behavior of polyacrylamide g e l s formed wi th chromium c r o s s l i n k s ( 5 ) shown i n Fig. 11. The value of G I is unaf fec ted by s t r a i n s a s l a r g e as 500 % f o r polyacrylamide ge l s .

c. S torage Modulus Versus Frequency

The s t o r a g e and l o s s moduli of guar g e l s produced i n the impingement mixer, a l lowed t o s i t f o r 20 minutes, and measured a t 100 % s t r a l n a r e a s shown i n Fig. 12. The s t o r a g e modulus , G ' , i s above t h e l o s s modulus , Gn, and G I becomes c o n s t a n t a t low f r equenc i e s a s i s i n d i c a t i v e of a c ros s l i nked g e l .

d. S torage ~ d u l u s Versus Time: Chemical Kine t ics

A s descr ibed i n Sec t ion 11, by-measuring t h e t i m e dependence of t he s t o r a g e modulus dur ing c r o s s l i n k i n g it is p o s s i b l e t o fo l l ow the chemical k i n e t i c s of t h e g e l a t i o n r e a c t i o n . From t h e d a t a showing t h e s t r a l n and f r e - quency dependence of t he se guar g e l s i t should be remembered t h a t t h e g e l s t r u c - t u r e has been s l i g h t l y degraded by t h e s t r a i n s imposed dur ing the dynamic o s c i l l a t o r y measurement, and a t 10 rad/s t h e value of G ' is s l i g h t l y h igher than t h e a c t u a l low frequency asymptote. Most measurements were taken a t 10 rad/s and 100% s t r a i n because those va lues y i e l d adequate to rque s i g n a l s and a f a s t d a t a a c q u i s i t i o n t i m e s o t h a t r e a c t i o n dynamics can be followed.

A series of measurements of t h e s t o r a g e modulus versus t i m e were made wh i l e varying guar and t i t a n a t e concen t r a t i ons (F igs . 13 and 14 ) . The g e l s were produced i n t h e impingement mixer and in t roduced d i r e c t l y i n t o t he gap between p a r a l l e l p l a t e s . The t i m e between t h e con tac t ing of t h e guar and t i t a - n a t e and t h e f i r s t d a t a p o l n t is approximately 10 s . Measurements were conducted a t 10 rad/s and 100% s t r a i n . The s u r p r i s i n g th ing about t h e r e s u l t s is t h e speed of t h e r eac t ion . W e were unable t o measure an i nduc t ion pe r lod before t h e ne t - work began t o form. The maximurn r a t e of r e a c t i o n (i.e. maximum s lope i n G I vs t) occurs be fo re t h e f i r s t d a t a po in t . I n t h e f i n a l s e c t i o n of t h e r e p o r t we w i l l show how k i n e t i c parameters could be ob ta ined from t h i s da t a .

Th i s d a t a can be used t o show t h e s e n s i t i v i t y of guar g e l p r o p e r t i e s t o guar and t i t an ium concent ra t ions . I t is p o s s i b l e t o ob t a in t h e same va lues of G I from s o l u t i o n s having d i f f e r e n t compositions.

2. Steady Shear Masurements

a. G e l s Mixed by Hand

Our i n i t i a l s t eady shea r experiments were on guar g e l s mixed by hand and in t roduced t o t h e rheometer tes t c e l l . This process t a k e s on t h e o rde r of 2 minutes t o accomplish. The q u i e s c e n t pe r lod be fo re s t eady s h e a r i s imposed, we now be l i eve , has a dominant e f f e c t on t h e v i s c o s i t y t h a t is measured. The stress versus time behavior f o r g e l s being sheared a t 100, 500, and 1000 s - l , r e s p e c t i v e l y were inves t i ga t ed . Seve ra l series of tests were run where t h e g e l s were allowed t o s i t 2.5, 3.5, and 4.5 minutes before shea r w a s i n i t i a t e d . I n a l l of t h e s t e a d y s h e a r experiments t h e asymptot ic s t r e s s a t long times (1500 s ) is roughly t h e same - independent of shea r r a t e . When t h e s t r e s s d a t a i s d iv ided by shea r r a t e t o o b t a i n a v i s c o s i t y , n,, a log-log p l o t of v i s c o s i t y versus shea r r a t e has a s l o p e of -0.96 shown i n Fig. 15. This s lope i s u n r e a l i s t i c f o r a polymer f l u i d (which normally have s l o p e s between -0.6 and -0.31, and i s much more sugges t ive of w a l l s l i p . A s t anda rd procedure t o tes t f o r t h e presence of w a l l s l i p is t o t ake tw sets of measurements wi th d i f f e r e n t gap s e t t i n g s . I f , a t t h e same shea r stresses, t h e shea r r a t e s a r e t h e same wi th both gaps, then w a l l s l i p i s thought n o t t o occur ; b u t i f , a t t h e same shea r stresses, t h e s h e a r r a t e i n t h e narrower gap i s g r e a t e r , t hen s l i p is occur- r i ng . We performed s t eady shea r measurements a t 25 s'l w i t h gaps of 0.75 mm, 1.5 mm, and 3.0 mm a s shown i n Fig. 16. In a l l c a se s t h e s h e a r s t r e s s e s a t long t i m e s were i d e n t i c a l w i th in exper imenta l unce r t a in ty . This i n d i c a t e s s l i p is n o t occur r ing . The i n t e r p r e t a t i o n of t h e s e observa t ions is unce r t a in . Resolut ion w i l l r e q u i r e d i r e c t measurement of t h e v e l o c i t y f i e l d . W e cons ide r t h i s a major r e sea rch need a t t h i s po in t .

b. Impingement Mixing: E f f e c t of Shear on React ion Rate

By us ing t h e impingment mixer it is p o s s i b l e t o i n t roduce s o l u t i o n s i n t o t h e rheometer and begin measurements before t h e f l u i d completely g e l s . A series of measurements a t 25, 100, and 500 s'l w e r e run and t h e r e s u l t s of s t r e s s versus t i m e a r e shown i n Fig. 17. 19-24. In t h e runs a t 25 and 100 s'l t h e stress i n c r e a s e s l i n e a r l y f o r 100-200 s a s t h e guar network g e l s . A t t he g e l p o i n t t h e f l u i d can no longer f low and t h e stress rises sha rp ly . I t then f a l l s j u s t a s sha rp ly , a s would be observed i f t h e g e l broke away from t h e p a r a l l e l p l a t e s u r f a c e s and began t o s l i p . Af t e r t h e drop i n s t r e s s it remains essen- t i a l l y c o n s t a n t f o r 1500 s. The l o c a t i o n of t h e rise i n stress and a l s o t h e s lope o f t h e s t r e s s versus t i m e p l o t s f o r 25 and 100 s" i n c r e a s e s wi th i nc reas ing s h e a r r a t e . The conc lus ion i s t h a t shea r i n c r e a s e s t h e r a t e of r eac t ion . W e may assume t h a t f o r t h e measurement a t 500 s" t h e jump i n s t r e s s a t very s h o r t times corresponds t o t h e g e l p o i n t which i s seen more c l e a r l y a t t h e lower shea r r a t e s .

This behavior has n o t been r epo r t ed by prev ious i n v e s t i g a t o r s because t he g e l p o i n t i s reached before t h e g e l can be mixed and loaded i n t o a Fann viscome t e r .

c. S teady Shear Followed by O s c i l l a t o r y Shear

Measurements were made bo th on our Sys t e m I V rheome t e r and on t h e Rheometrics P r e s s u r e Rheometer on t h e e f f e c t o f s t e a d y s h e a r fo l lowed by dynamic o s c i l l a t o r y s h e a r . I n t h e s e tests t h e samples were mixed by hand. F i g u r e s 18 and 1 9 show t h e s t r e s s v e r s u s t ime and modulus v e r s u s t i m e d a t a f o r a n exper iment t h a t c o n s i s t e d o f a l t e r n a t i n g p e r i o d s of 1000 s'l s h e a r f o r 60 s and dynamic o s c i l l a t o r y s h e a r a t 100 % s t r a i n and 1 0 r a d / s f o r 60 seconds . The s t e a d y s h e a r p o r t i o n shows a g r a d u a l i n c r e a s e i n stress l e v e l and a f t e r e a c h p e r i o d of dynamic t e s t i n g t h e stress s p i k e s and decays r a p i d l y . The stress o v e r s h o o t when t h e s t e a d y s h e a r i s r e a p p l i e d may be due t o e i t h e r r e h e a l i n g of t h e t h e g e l network o r rebonding o f t h e network t o t h e t o o l s u r f a c e s d u r i n g o s c i l l a t o r y s h e a r . T h i s s u g g e s t s t h a t f low l o o p d e v i c e s w i t h r e g i o n s o f q u i e s c e n t f l u i d may l e a d t o r e s u l t s t h a t are n o t i n d i c a t i v e of f l o w under c o n s t a n t s h e a r .

V I . KINETIC THEORY MODELING

A. Background

The k i n e t i c t h e o r y f o r t h e rheo logy of f l u i d s w i t h temporary network j u n c t i o n s h a s been developed by r e s e a r c h e r s a t t e m p t i n g t o model t h e f l o w of e n t a n g l e d po lymer ic f l u i d s 18). I n t h i s p r e v i o u s work t h e j u n c t i o n s were e n v i - s i o n e d as temporary en tang lement p o i n t s t h a t would form and disengage I n response t o f low. W e have ex tended t h e t h e o r y by i n c o r p o r a t i n g t h e r e a c t i o n o f t h e t i t a n i u m w i t h t h e g u a r polymer and subsequen t c r o s s l i n k i n g of guar molecu- l e s . The temporary en tang lements i n t h e c l a s s i c a l t h e o r y are r e p l a c e d by t h e chemical c r o s s l i n k s i n t h e g u a r g e l system. The t h e o r y a l l o w s one t o c a l c u l a t e t h e c o n s t i t u t i v e e q u a t i o n f o r t h e r e a c t i n g g e l i n e x p l i c i t form. Th is e q u a t i o n can t h e n be used to c a l c u l a t e t h e dynamic o s c i l l a t o r y r e s p o n s e o f a g e l , t h e s h e a r v i s c o s i t y , and t h e s h e a r v i s c o s i t y as a f u n c t i o n o f t ime under s h e a r h i s t o r i e s d u p l i c a t i n g p r o c e s s c o n d i t i o n s .

C r o s s l i n k f o r m a t i o n a p p e a r s t o c o n s i s t o f two s t e p s : f i r s t , t h e a t t a c h m e n t of t h e meta l i o n s t o t h e polymer backbone, fo l lowed by t h e fo rmat ion o f c r o s s l i n k s between a d j a c e n t c h a i n s through metal-lon b r i d g e s . Th i s l a t t e r s t e p i s most p r o b a b l y r e v e r s i b l e and depends on f l o w h i s t o r y . Thus, a network t h e o r y coupled w i t h chemica l r e a c t i o n k i n e t i c s i s e s s e n t i a l t o p r e d i c t t h e rheo logy o f g u a r g e l s . W e p r e s e n t h e r e a model t h a t i n c o r p o r a t e s b o t h o f t h e s e s t e p s -- chemica l k i n e t i c s c o n t r o l s t h e fo rmat ion o f a c t i v e si tes f o r c r o s s l i n k i n g , and t h e r a t e o f j u n c t i o n fo rmat ion i s g i v e n by t h e p r o d u c t of t h e r a t e o f polymer c h a i n c o l l i s i o n and t h e p r o b a b i l i t y t h a t a c o l l i s i o n w i l l l e a d t o a c r o s s l i n k . The model i s based on t h e Bird-Carreau network model ( 8 , 9 ) w i t h t h e a d d l t i o n of chemica l r e a c t i o n k i n e t i c s to d e s c r i b e t h e fo rmat ion o f chemical c r o s s l i n k s among t h e polymer molecules i n s o l u t i o n . Th is model h a s t h e a b i l i t y t o p r e d i c t a n i n c r e a s e i n s t o r a g e modulus w i t h t i m e , s h e a r t h i n n i n g v i s c o s i t y , s t r e s s o v e r s h o o t upon t h e i n c e p t i o n o f s h e a r f low, and v i s c o s i t y changes d u r i n g f l o w h i s t o r y s i m u l a t i o n o f f r a c t u r i n g o p e r a t i o n s .

B. Network T h e o r i e s

As a common s t a r t i n g p o i n t i n network t h e o r i e s f o r macromolecular f l u i d s , t h e e q u a t i o n f o r s t r e s s i n a deformed network i s d e r i v e d from r u b b e r e l a s t i c i t y t h e o r y (10-12). I n what f o l l o w s we a d o p t t h e nomencla ture o f Bi rd , Hassager, Armstrong, and C u r t i s s (91, h e n c e f o r t h deno ted BHAC. The stress ten- s o r is g i v e n by

where g i s a paramete r of o r d e r one, n i s t h e number d e n s i t y o f c r o s s l i n k s , k i s t h e Boltzman c o n s t a n t , T i s t h e a b s o l u t e t empera tu re and Y [ O ] i s t h e F inger t e n - s o r d e s c r i b i n g t h e de format ion o f t h e m a t e r i a l (BHAC, p. C-3, C-4) . To p a s s from t h e t h e o r y f o r a n e l a s t i c s o l i d ( t h a t is , a permanent ly c r o s s l i n k e d network) t o a f l u i d f o r which t h e c r o s s l i n k j u n c t i o n s form and break d u r i n g f low, a model must be developed t o a c c o u n t f o r t h e change i n t h e number d e n s i t y o f c r o s s l i n k s w i t h t i m e . f i e t h e o r y is formula ted n o t i n terms of c r o s s l i n k s , b u t r a t h e r c h a i n segments between c r o s s l i n k s . I t i s , t h e r e f o r e , p o s s i b l e t o d i s t i n g u i s h between c h a i n segments of v a r i o u s l e n g t h s o r k i n d s . A p o p u l a t i o n ba lance on t h e number d e n s i t y of c h a i n segments of k i n d j formed a t t i m e t ' t h a t s t i l l e x i s t a t a la ter t i m e tn i s g iven by,

where n t ~ ~ t t j d t ' is t h e number o f segments of k i n d j p e r u n i t volume i n t h e n e t - work a t t i m e t" c r e a t e d i n t h e t i m e i n t e r v a l from t ' to t ' + d t ' , and X'ldt" i s t h e p r o b a b i l i t y t h a t a segment o f k ind j c r e a t e d a t some p a s t t i m e t ' is l o s t i n t h e t i m e i n t e r v a l from t" t o tn + d t " .

The j u n c t i o n b a l a n c e e q u a t i o n , Eq. 6, is s o l v e d s u b j e c t t o t h e i n i t i a l c o n d i t i o n t h a t a t time t" = t' t h e r e is a n e t r a t e of c r e a t i o n of segments,

A A - where q j ~ j 2dt: i s t h e number o f segments o f k ind 3 t h a t are c r e a t e d p e r u n i t

volume i n t h e t i m e i n t e r v a l t ' t o t ' + d t ' .

The stress i n a n e las t ic network s t r a n d depends on i t s e x t e n s i o n . For t h e temporary j u n c t i o n model t h e e x t e n s i o n depends b o t h on t h e r a t e o f d e f o r - mation and t h e l e n g t h o f t i m e t h e s t r a n d h a s been undergoing deformat ion . The o r i g i n a l e q u a t i o n f o r t h e stress t e n s o r , Eq. 5, must be i n t e g r a t e d o v e r a l l p a s t t ime t o a c c o u n t f o r t h e c r e a t i o n o f network j u n c t i o n s a t p a s t time t h a t s t i l l e x i s t a t t h e p r e s e n t t i m e :

T = - C n - tn tu jKT&[01 d t n - j=1

Thus f o r t h e c l a s s i c a l network t h e o r i e s of polymer f l u i d s Eq. 6 is s o l v e d t o o b t a i n t h e j u n c t i o n k i n e t i c s , and Eq. 8 i s so lved t o o b t a i n t h e s t r e s s t e n s o r . The main modeling problem comes i n t h e s e l e c t i o n of the paramete rs and 1, which r e p r e s e n t t h e r a t e o f j u n c t i o n d e s t r u c t i o n and t h e rate of j u n c t i o n c r e a t i o n , r e s p e c t i v e l y .

A t t h i s p o i n t d i f f e r e n t a u t h o r s propose d i f f e r e n t e m p i r i c a l e x p r e s s i o n f o r t h e k i n e t i c pa ramete rs n and 1. If t h e y are assumed c o n s t a n t , t h e n t h e Lodge ' r u b b e r - l i k e u l i q u i d " is o b t a i n e d . Unfor tuna te ly , t h i s model i s i n c a p a b l e o f d e s c r i b i n g s h e a r t h i n n i n g v i s c o s i t y o r stress overshoo t , as w e l l a s o t h e r i m p o r t a n t phenomena. To c o r r e c t t h i s d e f i c i e n c y Kaye (1966) assumed t h e parame- t e r s are f u n c t i o n s o f s t r e s s . Kay's model l e a d s t o a n i n t e g r o - d i f f e r e n t i a l e q u a t i o n f o r stress f o r which no closed-form s o l u t i o n i s a v a i l a b l e and f o r which even numer ica l e v a l u a t i o n i s d i f f i c u l t . B i r d and Car reau (1968) proposed l e t t i n g n and depend on t h e second i n v a r i a n t of t h e r a t e o f s t r a i n ; t h a t is

[-. P h y s i c a l l y t h i s i m p l i e s t h a t j u n c t i o n c r e a t i o n and d e s t r u c t i o n i s a fun-ction of t h e r a t e o f energy d i s s i p a t i o n i n t h e system. T h i s r e s u l t s i n a c l o s e d form e x p r e s s i o n f o r stress i n terms of t h e v e l o c i t y g r a d i e n t i n s h e a r f low. The model h a s been e x t e n s i v e l y t e s t e d and f i t s bo th l i n e a r v i s c o e l a s t i c and non- l inear m a t e r i a l p r o p e r t i e s w e l l . The model and i t s development a r e o u t l i n e d below because t h e e x t e n s i o n o f t h e model t o i n c l u d e chemical k i n e t i c s fo l lows d i r e c t l y from t h e Blrd-Carreau model.

I n t h e Bird-Carreau model t h e r a t e s of l o s s and c r e a t i o n o f segments i n t h e network a r e assumed to be f u n c t i o n o f s h e a r r a t e . Thus, E q . 2, a ba lance on t h e segments i n t h e network a t t i m e t" i n t h e i n t e r v a l tu<t"<t, g i v e s :

where {" = ; ( tn) . Car reau i n t r o d u c e d e m p i r i c a l e x p r e s s i o n f o r t h e :j and h:

Notice t h a t t h e r e are s i x c o n s t a n t s i n t h i s model: A , a, t l , S,

and R. The f i r s t t h r e e pa ramete rs can be determined from l i n e a r v i s c o e l a s t i c measurements; t h e o t h e r t h r e e may be determined from measurements of t h e n o n l i - n e a r behav ior of t h e m a t e r i a l .

C a r r e a u ' s e x p r e s s i o n f o r j u n c t i o n k i n e t i c s can be coupled w i t h t h e e q u a t i o n of s ta te f o r stress, Eq. 8, t o s o l v e f o r t h e m a t e r i a l f u n c t i o n s :

v i s c o s i t y :

pr imary normal stress 6 = 2 1 nj h . A . g c o e f f i c i e n t : j =1 J I I

complex v i s c o s i t i e s : n f = 1 'J

j = 1 1 + ( A . w ) 2

J

C. ' C o n s t i t u t i v e Equat ion f o r R e a c t i v e Network F l u i d s - I n t h i s modif ied Bi rd -Car reau model i n c o r p o r a t i n g chemical r e a c t i o n

k i n e t i c s , t h e stress i s s t i l l assumed t o be determined by t h e number d e n s i t y o f c r o s s l i n k s . The stress t e n s o r i s t h e n g i v e n by Eq. 8. The same d i f f e r e n t i a l e q u a t i o n govern ing t h e b a l a n c e of segments, Eq. 6, is a l s o used. Note, however, t h a t i n t h e p o p u l a t i o n ba lance e q u a t i o n t h e n a t u r e of t h e chemica l c r o s s l i n k s de te rmines t h e va lue of t h e k i n e t i c pa ramete r Ap, which i s a s s o c i a t e d w i t h t h e rate of c r o s s l i n k d e s t r u c t i o n . No a d d i t i o n a l term is added t o t h e segment b a l a n c e e q u a t i o n t o a c c o u n t f o r chemica l r e a c t i o n , because segments once c r e a t e d a t p a s t t i m e t f can be d e s t r o y e d o n l y a t some later t i m e tn; t h a t is, a t some p r e s e n t t i m e t " segments c a n n o t be c r e a t e d t h a t have a p a s t h i s t o r y o f ( t " - t o . The chemica l r e a c t i o n k i n e t i c s come i n t o t h e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n th rough t h e i n i t i a l c o n d i t i o n t h a t a t t ' = t " , which s p e c i f i e s t h e n e t rate of c r e a t i o n o f segments.

The a d s o r p t i o n o f meta l i o n s o n t o t h e polymer backbone i s assumed t o be t h e p r e c u r s o r of cha in -cha in c r o s s l i n k s format ion. Thus, t h e i n i t i a l con- d i t i o n f o r t h e s o l u t i o n o f Eq. 8 i n c l u d e s t h e chemical k i n e t i c s o f t h i s p rocess . The i n i t i a l c o n d i t i o n i s g i v e n by:

A A -2

a t t i m e t" = t' - ' I t t t t j = , , . ( + ' ) A j ( + ' ) P 3

C o l l i s i o n s between polymer c h a i n s occur a t a r a t e sjlj-2 a s i n t h e Car reau model; however, t h e r e i s p r o b a b i l i t y P t h a t t h e c o l l i s i o n produces a n e f f e c t i v e c r o s s l i n k . The p r o b a b i l i t y can be t h o u g h t of a s t h e " s t i c k i n e s s " o f t h e polymer c h a i n which w i l l depend on t h e f r a c t i o n of a d s o r p t i o n s i t e s on t h e polymer back- bone t h a t have r e a c t e d w i t h metal i o n s .

L e t u s assume t h a t t h e p r o b a b i l i t y of a chain-chain c o l l i s i o n c r e a t i n g a j u c n t i o n i s d i r e c t l y p r o p o r t i o n a l t o t h e f r a c t i o n of sites on t h e polymer c h a i n t h a t a r e occupied by metal i o n s . The f r a c t i o n o f f i l l e d s i tes is S. I f t h e r e a c t i o n f o r t h i s a d s o r p t i o n p r o c e s s is f i r s t o r d e r , then:

where l / k i s t h e t i m e c o n s t a n t f o r t h e a d s o r p t i o n r e a c t i o n . With t h e i n i t i a l c o n d i t i o n t h a t a t t ime e q u a l t o z e r o no si tes are f i l l e d , Eq. 2 1 can be s o l v e d f o r S:

There fore , t h e p r o b a b i l i t y of a cham-cha in c o l l i s i o n c r e a t i n g a j u n c t i o n is 1 1 - e x p ( - k t ) I . The r e a c t i o n r a t e c o n s t a n t k can be o b t a l n e d from t h e t i m e dependence o f t h e s t o r a g e modulus of a c r o s s l i n k e d g e l a s w i l l be shown below. The i n i t i a l c o n d i t i o n f o r t h e network j u n c t l o n e q u a t i o n i s then:

The material f u n c t i o n s f o r a r e a c t l n g network f l u i d are o b t a i n e d from t h e s o l u t i o n of Eqs. 6 and 8, s u b j e c t t o t h e i n i t i a l c o n d i t i o n i n Eq. 23.

D. M a t e r i a l Func t ion i n Shear Flow --- 1 . Dynamic O s c i l l a t o r y Shear

Cons ider a m a t e r i a l s u b j e c t t o t h e o s c i l l a t o r y v e l o c i t y f i e l d g i v e n by

where t h e v e l o c i t y is i n t h e x - d i r e c t i o n and t h e g r a d i e n t of t h e v e l o c i t y is i n t h e y - d i r e c t i o n :

I n t h e l i n e a r v i s c o e l s a t i c regime t h e j u n c t i o n k i n e t i c s and material f u n c t i o n s are independen t o f t h e magnitude of t h e r a t e o f s t r a i n , yo, and depend o n l y on f requency w. There fore , f o l l o w i n g B i r d and Car reau we f a c t o r x ~ ( ~ ) and np(y ) e a c h i n t o two terms, a c o n s t a n t p r e f a c t o r and s t r a i n rate independen t term whlch g o e s t o u n i t y i n t h e l i n e a r v i s c o e l a s t i c l i m i t :

The s h e a r stress can be o b t a i n e d a n a l y t i c a l l y from t h e s o l u t i o n o f E q . 9 f o r t h e j u n c t i o n k i n e t i c s w i t h 4. 23 as the i n i t i a l c o n d i t i o n and Eqs. 10-15 f o r t h e k i n e t i c pa ramete rs . For s m a l l ampl i tude o s c i l l a t o r y motion t h e s h e a r stress, T-, i s g iven by:

'Xj t T X . ( A - T ) e x p ( - t / r 1 - e x p ( - - > +

I j 2 2 ] s i n w t

A j - T T (1) - T) + ( T A ~ W )

where T = l /k . No s i n g u l a r i t y o c c u r s as T + x ~ , which can be proved by u s i n g L ' H o p i t a l ' s r u l e . The u n d e r l i n e d t r a n s i e n t term i s a s s o c i a t e d w i t h t h e start- u p o f f l o w a t t = 0, and l t d i s a p p e a r s q u i c k l y , becuase normal ly A, 1s i n t h e o r d e r of seconds ( C a r r e a u 1972) . The t r a n s i e n t r esponse a s s o c i a t e d wi th t h e chemical r e a c t i o n is of a much l o n g e r t i m e s c a l e . Thus, t h e complex v i s c o s i t i e s are:

TX, ( x ~ - T 1 e x p ( - t / ~ 1 - 2 2 I (29)

( X j - ( T X . U J ) 3

It can be seen t h a t rises t o i t s f i n a l equi l ib r ium va lue w i th a t i m e c o n s t a n t given by t h e r a t e of chemical r e a c t i o n , ~ ( = l / k ) . Figure 20 shows G I

versus t i m e which m i m i c s t h e exper imenta l ly ob t r a ined G' d a t a f o r c r o s s l i n k i n g guar ge l s .

1 . Steady Shear Flow

For a m a t e r i a l i n a v e l o c i t y f i e l d given by

wl th t h e v e l o c i t y g rad i en t :

For t h i s s t eady shea r f low, t h e modifled Car reau ' s model g lves t h e s t eady v i s c o s i t y , n, and t h e primary normal stress coefficient, 0 , a s i n t h e fol lowing:

The shear v i s c o s i t y predicted by Eq. 32 i s shown versus shear r a t e a t a number of times during the ge la t ion process is shown i n Fig. 21. That no s i n g u l a r i t y occurs a s ~ j g j + T can be proved by expanding

A t i n f i n i t e time, which corresponds t o the condit ion of the o r i - g i n a l Bird-Carreau model, Eqs. 28, 29, 32, and 33 s impl i fy t o Eqs. 18, 19, 16, and 17, respect ive ly . This is a cross-check between our r e s u l t s and those of B i rd-Carreau .

Further work i s underway on the v i scos i ty of g e l s under varying shear h i s t o r i e s t o model f i e l d operat ing condit ions.

V. CONCLUSIONS

Dynamic o s c i l l a t o r y and s t e a d y shear measurements have been used t o s t u d y t h e rheology of guar ge ls . The dynamic o s c i l l a t o r y measurements have been used t o study the slow hydration of the guar polymer, and the e f f e c t s of chemical composition and mixing on guar g e l s t r u c t u r e . It is shown t h a t aged t i t a n a t e solutions produce g e l s with poorer p roper t i e s . Adding di-ketones t o modify the r a t e of r eac t ion does no t j u s t slow down the reac t ion , b u t it a l s o prevents the g e l from c ross l ink ing t o the same ex ten t a s g e l s without added di-ketone. Mixing i s shown t o p lay a c r u c i a l r o l e i n t h e develoment of g e l s t r u c t u r e . Poor mixing appears t o produce inhomogeneous g e l networks t h a t have higher l e v e l s of e l a s t i c i t y than homogeneous ge l s . To produce in t imate ly mixed f l u i d s we have developed a novel impingement mixing device.

Measurements of the steady shear v i s c o s i t y of g e l s i n d i c a t e t h a t wal l s l i p i s occuring. However, conventional rheologica l techniques f o r ca lcu la t ing wall s l i p v e l o c i t i e s have given contradic tory r e s u l t s . There is a need f o r d i r e c t measurements of ve loc i ty f i e l d s i n shear flow t o c l a r l f y the mechanism of wall s l i p . In the next year we w i l l be conducting l a s e r doppler mesurements to address t h i s problem.

A novel network theory coupled with chemical r eac t ion k i n e t i c s i s proposed; mater ia l funct ions can be expressed a n a l y t i c a l l y , a s shown i n 4 s . 24, 25, 28, and 29. This model provides a framework f o r predic t ing rheologica l p roper t i e s of

r e a c t i n g molecular networks w i th temporary junc t ions . The a p p l i c a t i o n of t h i s theory t o t h e p r e d i c t i o n of t h e rheology of c r o s s l i n k i n g guar f r a c t u r i n g f l u i d s w i l l be presen ted i n f u t u r e a r t i c l e s .

ACKNOWLEDGMENT

The au tho r s w i t h t o acknowledge t h e f i n a n c i a l a s s i s t a n c e provided by funds from the American Petroleum I n s t i t u t e .

REFERENCES

K. t e J i g e n Nuis, C o l l o i d Polymer Sc ience , - 259, 522 (1 981 ).

J. D. F e r r y , ~ d v . P r o t e i n Chem., Q, 1 (1948).

R. Roscoe, Rheologica Acta, 19, 737 ( 1 980). - J. T. Uhl, R. K. Prud'homme, Macromolecules ( s u b m i t t e d ) .

R. K. Prud'homme, J. T. Uhl, J. P. P o i n s a t t e , and F. Halverson, Soc. Pe t . Engrs. J., 804-808, O c t . 1983.

J. D. F e r r y , V i s c o e l a s t i c P r o p e r t i e s of Polymers, 3 r d ed. , John Wiley and Sons, NY, 1980.

D. S. Pearson and W e W. G r a e s s l e y , Macromolecules, - 13, 1001, ( 1 9 8 0 ) .

R. B. B i rd and P. J. Car reau , Chem. Engr. Sc i . , 23, 427-434 (1968) . - R. B. B i r d , 0. Hassager, R. C. Armstrong and C. F. C u r t i s s , Dynamics of - Polymeric L i q u i d s , Vol. 2, John Wiley: New York ( 1 9771, (3.15.

P. J. Car reau , Trans. Soc. Rheol., 16, 99-120 (1972) . - A. Kaye, B r i t . J. Apl. Phys., - 17, 803-806 (1966) .

A. S. Lodge, Rheol. Acta., 1, 379-392 (1 968) .

R. I. Tanner and J. M. Simmons, Chem. Engr . Sci . , 22, 1803-1 81 5 ( 1 967).

Fig. 1 a. Fig. 1 b.

I

.I .1 10 lo;? w rad/sec

Fig. 1 c. Fig. 1 d.

Fig.1 a-d. Storage (G') and loss (G") moduli as a function of frequency of a - 9.5 wt. % polystyrene (900,000 molecular weight) in carbon disulfide solution

at temperature as ,indicated in each figure and run between parallel plates

at 3% strain.(Clark, et.al., Polym Preprintrs 24,87('1983)). .

4

F i g . 2 . S y s t e m I V R h e o m e t e r .

double - acting pneumatic cylinder

microliter syringe

titanate solution

*

.

Fig.3. Impingement mixing device

0 % strain 300

Fig.4. Storage(G1) and loss (G") moduli as a function of strain of a .48%

HP guar solution (30 minutes agitation) at 10 radlsec : (run 21783 2).

Fig.5. Viscosity (Q) as a function of shear rate of .48% HP guar solustions

J

0 . aa - Q) :

E : 0 . \ . 0) h ,

fi Y . > , : CI - Q ) . 0 - 0 i Q) : - > : r

at different ages (runs 2 1683 and 2 1883 2).

b - 30 min

b

b - 24kr . b b

6

b

1 - 1 . "'..-- .O 1 ra'te (1 lsec) 100

dyn

esls

q c

m

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0

0 - - - -

2

3

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w, - wm

- wm

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w

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wm

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w

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wm

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48

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m

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lm

Y

wm

0

w

l

Y

wm

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a

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0.

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480

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,

Fig.7. Effect of order of addition on G' am and 6". of .48% HP guar mixed

with Tyzor AA by hand. Measurements began immediately after mixing

at 1 redlsed end 100% strain (runs 61583 1. 61583 2, 61583 5 (L 7683 2).

Fig.8. Storage (0') and loss (G"), moduli of HP guar gel (.48% guar

(Of K + N) and .04% Tyzor ma made from a newly opened bottle of Tyzor AA

and from a bottle that had been used for several months. Impingement device

was used and tests began immediately after mixing at 10 radlsec 100% strain

(runs 121583 1 (L 122983 3\

G' 0% diketone

G' . 0 2 5 W e t o n e

Fig.9. Effect of diketone on G' and G" of HP guar gel (.48% guar

(G $ K + N) and .04% Tyzor AA). Tests began immediately after mixing by

hand at 1 radlsec and 100% strain (runs 52483 2, 52483 4 & 61583 1).

-

- 0 6 ' 1 1 I

% strain

Fig.10. Effect of strain on G' and G" of HP guar gel (.48% guar

<G K N) and .04% Tyzor AA) produced in the impingement device.

Mixture was allowed to sit for 15 minutes before tests began at

10 rad/sec (run 1 13083 1).

b I % strain 500

Fig.ll. Effect of strain on G' and G" of polyacrylamide gels

formed with chromium 'crosslinks.(5).

Fig.12. Effect of frequency on dynamic moduli of -48% HP guar (G+ K + N) gel

(-04% Tyzor AA) produced in the impingement device . Mixture was allowed to

sit for 20 minutes before measuring at 100% strain (run 1 13083 4).

Fig. 13. Effect of Tyzor AA concentration on HP guar (G -& K + N) gel

formation at a constant HP guar concentration , .48%. Tests began

immediately after mixing in the impingement device. Only the values of

G" of HP guar (.0.48%) with .04% Tyzor AA is shown because G" are

the same for all runs.(runs 122883 2, 122883 1, 122883 4, 122883 3,

122983 2,& 122983 3).

Fig.14. Effect of HP guar (G+K+N) concentration on gel formation at

a constant Tyzor AA concentration, .08%. Tests began immediately after

mixing in the impingement device. (runs 122883 4, 1484 3, 1484 1,

& 1584 1).

Fig.15. Effect of shear rate on viscosity ( r) ) of .48% HP guar Y

(G + K $ N) gels with .04% Tyzor AA produced by hand mixing.

Tests began after the gels were allowed to sit 2.5 , 3.5 and 4.5

minutes. Solid line is a least squar fit of these points. (runs

81683 1,81683 2, 81683 3, 81683 4, 81683 5. 81683 6, 81783 1.

81783 2, 81783 3, 81983 1 81983 2, & 81983 4).

Fig.16 Steady shear measurements with parallel plates geometry with gaps

of 0.75 mm, 1.5mm & 3.0mm on .48% guar (G +K+ N ) gels (.04% Tyzor AA)

produced in the impingement device. Tests began immediately aftter mixing

. Effect of shear on reaction rate of gel formation. Samples were

-48% guar (G + K-k N) gels (.04% Tyzor AA) produced in the impingement

device. Tests began immediately after mixing at various shear rates with

.75mm gap.(runs 11084 3, 11 184 1 & 11 184 4).

of 10001sec shear for 60 sec. and dynamic oscillatory shear at -100% strain and

500 - I 1 I I 1 I I I

10 rad.lsec. for 60 sec. Sample was .48% HP guar (G+ K+ N) gel with .04%

E 0 0 @a \ @a - Q) C , * 0 ,

,

200

Tyzor AA mixed by hand (run 51 183 5).

-

,

I_LLL,L."-LC ,

- I 1 I I I I I

0.1 I 0 time(min.1 7

Fig.19. Dynamic moduli vs. time for an experiment that consisted of alternating

o 'timb(sec.) 480 Fig.18. Stress vs. time for an experiment that consisted of alternating periods

periods of 10001sec: shear for 60 sec. and dynamic oscillatory shear at 100%

strain and 10 radlsec. for 60 sec. Sample was .47% HP guar (G+ K+ N) gel

with .04% Tyzor AA mixed by hand.(run 5 1183 5).

Figu

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