ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 19, Studies of the Rheological Properties of Drilling Fluids

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1 ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 9, 0 Studies of the Rheological Properties of Drillig Fluids Iurii Salyzhy, Mykhaylo Myslyuk Ivao-Frakivsk Natioal Techical Uiversity of Oil ad Gas, Ivao-Frakivsk, Ukraie ABSTRACT The methodology ad results of ivestigatios of rheological properties of drillig fluids o the basis of rotatioal viscosimetry data have bee described. Special attetio has bee paid to rheological properties of biviscosity fluids ad applied aspects of determiatio of state equatios. INTRODUCTION Techological processes i oil ad gas idustry are closely coected with the use of o-newtoia fluids. Efficiet maagemet of these processes requires precise determiatio of rheological parameters ad ifluece of ay factors upo them. For this purpose the rotatioal viscosimeters are most widely used but the simplified methodologies for viscosimetry data treatmet do ot allow usig all possibilities of the measurig equipmet. The Couette flow i a gap betwee two coaxial cyliders of rotatioal viscosimeters is described by equatio ω = τ τout γ ξ dξ ɺ () ξ where ω is agular speed of rotatio of outer cylider, τ ad τ out are shear stresses o ier ad outer cyliders, γɺ ( τ) is fluid rheological model, γɺ is shear rate gradiet. The depedece betwee stresses o the outer ad ier cyliders is determied by the followig relatio: α τ τ τ α, if 0 ; τ out = τ 0 τ 0 τ < τ 0 α, if, () where τ 0 is yield stress, α = R R, ad R ad R are the radii of the ier ad outer cyliders. I Eq. the rheologically statioary models are used which permits clear γ ɺ = γɺ τ. aalytical solutio of METHODOLOGY The proposed methodology for rotatioal viscosimetry data treatmet is based o exact solutio of the Eq. ad cosiders iformatioal relevace of the experimets. The aim of rotatioal viscosimetry data processig is formalized as the task of search of ˆ idex of rheological model ad its parameters â i some a priori kow class of ϑ models ( ω ) either A,a + ε;... ; τ = or A ( ω,a ) + ε, ϑ;..., (3)

2 where τ is vector of measured shear stresses at agular rates ω, A ( ω,a ) is operatio of direct task of rotatioal viscosimetry Eqs. ad, a is vector of rheological properties, is idex of rheological model; ε is vector of cetered ormal radom compoet, caused by measuremet iaccuracies. The solutio of the task Eq. 3 is built usig the priciple of maximum of likelihood fuctio ad is realized with the help of the followig procedures ( ( ω ) τ) ( ) mi C A,a a ˆ, ˆ ; (4) mi σ ˆ = N r N ω τ i= ( A ( i,aˆ ) i ) ˆ, (5) where N is umber of rates of cylider rotatio, r is umber of rheological parameters beig evaluated, C is matrix of covariatios of radom compoet i Eq. 3. Accordig to Eq. 4 at first for each model ϑ evaluatios â of rheological properties are built, ad the accordig to Eq. 5 rheological model ˆ is evaluated. Cotrary to other existig methods, methodology allows more exact selectio of the proper rheological model ad precise estimatio of its parameters. Methodology allows to select the most appropriate model i the class of the rheologicaly statioary models: Newto, Bigham, Ostwald, Herschel-Bulkley, Schulma-Casso ad biviscosity models, estimate their rheological properties ad build the estimatio error covariace matrix O. ( T ˆ ) ˆ ˆ ˆ O = A ω,a C A ω,a, (6) ˆ ˆ ω,a is matrix of derivatives of rheological parameters ad trasposed to it T is matrix A ( ω,aˆ ˆ ). where A The biviscosity model 3 is cosidered as ay combiatio of rheological models, which describes rheological properties of fluid i differet rages of shear rates. ( γ ) * ɺ ( τ,a ), τ τ ; γ ɺ = γ ( ) * ɺ ( τ,a ), τ > τ, (7) where a, a are rheological properties accordigly for the low ad high gradiets * of shear rates, τ is boudary shear stress, calculated from the equatio * * γɺ τ,a = γɺ τ,a. For the selectio of the most appropriate rheological model, which describes fluid i the etire techological process (uder the ifluece of parameters of state or ay additives) the procedure of batch processig is proposed 4. I this case Eqs. 4 ad 5 are geeralized as follows: ( ) mi A ω,a τ a ˆ, =, z ; (8) mi σ ˆ c = z( N r ) ω τ = i= ( A ( i,aˆ ) i ) ˆ z N ˆ, (9) where σ c is dispersio of adequacy for the whole umber of experimets z with differet values of ifluecig factors.

3 The polyomial ad splie models are used for the describig of ay factors ifluece o the rheological properties. The splie fuctios are represeted accordig to V.A. Vasyleko 5 z 3 α p( x) = b Gm,z x x + b z+ ( x ), = = where b, b z + are parameters of aalytical splie represetatio, z is umber of experimetal poits, G ( x x ) = m,z m z x x l x x, if z is eve; = m z x x, if z is odd, q z x x = ( xi x i ), i= q = z + m! m!z!, x 3 = ( x, x,, x ) T z is vector of ifluece factors i the experimet, α = α, α,, α is multiidex, z α α α αz = z ( x ) (x ) (x ) (x ) ad m is parameter of variatioal fuctioal. THE STUDY As it was metioed i methodology descriptio, biviscosity models have bee icluded ito the class of rheological models, amog which the most adequate oe is beig selected. First-priority task was to fid out whether fluids beig described by these models do exist. The search of such Table. Results of rotatioal viscometry data treatmet where most appropriate is biviscosity rhelogical model Studied fluid Device gap Most appropriate biviscosity rhelogical model Model parameters Boudary shear stress Dispersio σ ˆ, Pa Herschel- Dispersio for other models σ ˆ, Pa shear τ0, k, η, rates Pa Pa s Pa s τ, Pa Bulkley Ostwald Bigham Bigham low & Ostwald high low Ostwald & Newto high low Ostwald & Bigham high low Ostwald & Ostwald high Note. τ 0 is yield poit, k is cosistecy idex, η is plastic viscosity ad is flow-behavior idex. Figure. Rheogram for studied fluid Figure. Rheogram for studied fluid

4 Figure 3. Rheogram for studied fluid 3 Figure 4. Rheogram for studied fluid 4 fluids was carried out by treatmet of rheometry data, obtaied by our colleagues ad us ad take from literature. I table ad o figures 4 the most vivid results for differet biviscosity models, which are as follows: studied fluid is CMC water solutio (Sample 33 6 ), studied fluid is cemet slurry with alumia-silicate microspheres, studied fluid 3 ad 4 are betoite ligite suspesios (Sample 5 ad Sample 7 respectively 6 ). It should be metioed that biviscosity fluids amog drillig fluids occur rather frequetly, but the fractio of such fluids is difficult to evaluate while it depeds o the type of fluid. Aother importat issue for the evaluatio of parameters of biviscosity fluids is the umber of rates of viscosimeter, used to make measuremets. The rheological properties of drillig fluids are more sesitive to ifluece of temperature, pressure ad additives the other parameters. So, while selectig drillig fluids formulatio, the attetio must be previously paid o rheological properties especially whe the aim is to select the most thermostable prescriptio. I the case of selectio drillig mud formulatio the thermostability meas: mud properties before ad after heatig must be i specified boudaries ad the parameters chages must be miimal. I table there are rotatioal viscosimetry data, obtaied accordig to the pla of experimet, the aim of which was the selectio of optimal formulatio of humate-biopolimer drillig mud. Table. Rotatioal viscometry data for humate-biopolimer drillig mud Values of factors Agles of tur (degree) at rotatig rates (rpm) Experimet Polypac Duo- before heatig after heatig % UL, % vis, % Note. CAR is coal-alkali reaget. Device costat is Pa/degree, device gap is

5 Thermostatig of drillig fluid i each experimet has bee carried out for 4 hours i autoclave (with the capacity of 400 cm 3 ) i roller ove at the temperature of 0 С. As a result of treatmet of preseted data, parameters of rheological models i each poit of experimet pla have bee evaluated ad the most adequate amog them have bee selected (Table 3). Exp. Table 3. Most appropriate rheological models at experimetal poits Most appropriate models ad dispersio σ ˆ, Pa before heatig after heatig Biviscosity Bigham & Ostwald Ostwald 0.65 Herschel-Bulkley Ostwald Schulma- Casso Ostwald Herschel-Bulkley.850 Herschel-Bulkley Herschel-Bulkley Ostwald Herschel-Bulkley 0.37 Biviscosity Bigham & Ostwald Herschel-Bulkley 0.6 Herschel-Bulkley Herschel-Bulkley 0.08 Herschel-Bulkley Herschel-Bulkley Herschel-Bulkley Biviscosity 0 Herschel-Bulkley Ostwald & Ostwald Herschel-Bulkley 0.64 Herschel-Bulkley Herschel-Bulkley Ostwald Herschel-Bulkley.040 Herschel-Bulkley Ostwald Ostwald Herschel-Bulkley Ostwald 0.36 But it is clear that it s practically impossible to use several rheological models for the aalysis of ifluece of reagets upo rheological properties of drillig fluid. Thus, accordig to Eqs. 8 ad 9 dispersio of adequacy for the whole umber of experimets was estimated (Table 4). Table 4 cotais estimatios of dispersio of adequacy ust for five rheological models, because for other models it was t possible to make such estimatio while data could ot be described with these models i some experimetal poits. From Table 4 oe ca make a coclusio that i geeral Herschel- Bulkley model is the most adequate for the descriptio of results of experimets, the evaluatios of parameters of which are give i Table 5. Table 4. Geeralized fuctioal for model selectio Dispersio of adequacy σ ˆ c, Pa Rheological model before heatig after heatig Herschel-Bulkley Schulma-Casso.696 -* Ostwald Bigham Newto Note. * - data could ot be described with this model i some experimetal poits. Table 5. Results for Herschel-Bulkley model Model parameters before heatig after heatig Exp. τ0, k, τ0, k, a Pa Pa s Pa Pa s Rheological properties refer to vector quatities, specified chage of which ca be estimated o the basis of multidimesioal fuctio of probability desity 7 T a = a a O a a 0 0, (0) where a, a are vectors of rheological 0 properties of drillig fluid correspodigly before ad after thermal ifluece.

6 Figure 5. Ifluece of reagets o the rheological properties of humate-biopolimer drillig mud Accordig to Eq. 0 estimatios of chage of rheological properties i the course of thermostatig have bee built, which are give i Table 5. restrictios ad which allowed estimatig regio of acceptable formulatios. Regio of acceptable formulatios is quite broad, that s why selectio of optimal While experimet was carried out formulatio of drillig fluid requires accordig to orthogoal cetral compositio pla, for the descriptio of its parameters formalizatio of optimizatio criterio. I this case, the aim is to select the most stable secod-degree polyoms have bee used. formulatio i terms of rheological I figure 5 depedecies of rheological parameters chagig, which is esured by parameters upo cocetratios of reagets miimizatio of criterio of Eq. 0. i drillig fluid at the specified CAR value of 3.5 % are represeted by isolies. Aalogical depedecies were built for other properties of drillig fluid, upo Figure 5 represets depedece of criterio of Eq. 0 upo cocetratios of reagets ad optimal drillig fluid formulatio, obtaied due to its use. values of which were imposed iterval

7 ACKNOWLEDGMENTS A lot of thaks to Dr. Sami Hietala ad his colleagues for the chace to participate i the coferece. REFERENCES. Myslyuk, M.A. (988), "Determiig rheological parameters for a dispersio system by rotatioal viscometry", Izheero-Fizicheskii Zhural, Vol. 54, No. 6, pp Myslyuk, M.A., Vasylcheko, A.A., Salyzhy, Yu.M., ad Kusturova, E.V. (006), Evaluatio of rheological properties of drillig muds from the rotatioal viscometry data, Stroit. Neft. Gaz. Skvazhi Sushe More, No., pp Myslyuk, M.A., Salyzhy, Yu.M. (008), The evaluatio of biviscosity fluids rheological properties o the basis of rotatioal viscometry data, Neft. Khoz., No., pp. 40-4,. 4. Myslyuk, M.A., Salyzhy, Yu.M. (007), Rotatioal viscometry: ew approaches to data processig, Naft. Gaz. Prom., No. 6, pp Vasyleko, V.A. (983) Splie fuctios: theory, algorithms, programs, Nauka, Novosibirsk. 6. Kelessidis, V.C., Maglioe, R. (008) Shear rate correctios for Herschel-Bulkley fluids i Couette geometry, Appl. Rheol., Vol. 8, Myslyuk, M.A., Vasylcheko, A.A., Salyzhy, Yu.M., Kusturova, E.V. (006), To the selectio of drillig mud coditioig prescriptio cosiderig the heat resistace, Stroit. Neft. Gaz. Skvazhi Sushe More, No. 8, pp