a: .' - RIEOLOGICAL CHARACTERIZATION OF FRACTURING FLUIDS Robert K. Prud'homme, Alice Chu, and Jeffrey Kramer* I. INTRODUCTION Guar gels crosslinked with transltlon-metal ions are used as fracturing fluids in oil well completion. Gels are attractive because crosslinking creates a fluid with sufficient viscosity to suspend solid propants while requiring cnly small amounts of p a r polymer. To simulate the fracturing operation and pre- dlct fracture geometry it 1s necessary to understand and model the rheology of guar gels. To date, the reproduclbllity of laboratory tests of guar gel rheology has been poor and models of gel rheology have involved only empirical modifications of the power-law fluld model. These empirical models are inca- pable of describing the effects of shear and time history on gel properties. The purpose of this paper is to provide a working gulde to the rheology, and characterization of guar gels. First, in Sectlon 11, we describe the experimental techniques used to study guar rheology, which include dynamic oscillatory shear measurements and steady shear measurements. Dynamic oscllla- tory shear measurements are especially lmportant in investigating gel structure because these measurements can be used to determine the number of network cross- llnks. These measurements and their interpretation are discussed in detail, since they are probably less familiar to researchers in the oil production research area than are steady shear measurements. In Section I11 we describe the rheological Instruments used in this study. In Section IV the preparatlon of guar samples is detailed. The composltlon of the model guar gel used in thls study was specified by the API steering commttee. Our observations on the fac- tors controlling gel rheology, including chemlcal effects, sample preparatlon effects, and flow history effects are presented in Sectlon V. In Section VI a model that describes the rheology of gelling flulds is described. The model is based on the temporary network theorres used to descrlbe the rheology of polymer solutions. To this theory we have incorporated the chemlcal klnetics of metal lon adsorption onto the guar polymer backbone and subsequent polymer-polymer crosslinking. In the final section recommendatlons for standard test procedures, for rheological lnstrurnentation, and for future research are presented. I I. RHEOLOG ICAL MEASUREMENTS A. Dynamlc Oscillatory Measurements Dynamic oscillatory shear experiments which measure the linear vlscoelastlc response of materials are acknowledged to be the most valuable probes of gel or network structure. Though steady shear measurements are necessary to duplicate process conditions, the oscillatory measurenents give more insight into the properties of the gel than do steady shear measurenents. When interpreted using classical network theory, linear vlscoelastlc measure- ments can be used to determine the klnetics of gel formation, the crosslink den- sity of a gel, or the shear degradation of gel structure. The gelation of polyvinyl alcohol and gelatin gels have been studied by a number of researchers (1,2,3), and a t Princeton we have used these measurements to study polyacr~lb- mide gels used as permeability control agents in enhanced oil recovery (4,5). fc Princeton University, Princeton, New Jersey
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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
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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