Correlation of Drag Reduction With Modified Deborah Number For Dilute Polymer Solutions J. M. RODRIGUEZ U. OF MISSOURI A T ROLLA J. L. ZAKIN ROLLA, MO. G. K. PATTERSON Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 ABSTRACT factor ratio, defined as the ratio of the observed pressure drop to that predicted for a solution of Correlation has been obtained between drag­ the same viscosity characteristics and density_ reducing characteristics for turbulent flow in a at equal flow rates using the Dodge-Metzner pipe and measurable properties of several polymer friction factor equation 4 solutions. Several concentrations of high molecular weight polymethyl methacrylate in toluene, high _I = molecular weight polyisobutylene in both toluene and cyclohexane, medium molecular weight poly­ .fi isobutylene in cyclohexane and benzene and low molecular weight polystyrene in toluene were OAO (1) studied. Data obtained in these nonpolar solvents and literature data for more polar solvents were (n ') I 2 successfully correlated as the ratio of measured friction factor to purely viscous friction factor vs Viscous solutions with drag ratios greater than the modified Deborah number VT1/DO.2, where Tl 1.0 can have friction factor ratios less than 1.0. is the first-mode relaxation time of the solution F or practical applications, it is drag reduction estimated by the Zimm theory. A shift factor which which is of interest. However, for correlation the is a function of intrinsic viscosity 11(4[7]] - 1) fundamental ratio is the friction factor ratio. allowed all the data obtained with nonpolar solvents In recen t years, drag reduction has been studied to be correlated as a single function. For these extensively. Recent studies have shown that systems, most of the data fit a single curve to reasonable predictions of the incipience of drag within ± 5 percent of the average friction factor reduction in polymer solutions can be made from ratio. The shift factor did not give a unique the properties of the solutions and the flow function of the data for the more polar systems. variables.S However, it has not been possible to predict accurately the amount of drag reduction to INTRODUCTION be expected for a given polymer solution without The phenomenon of drag reduction in polymer any drag-reducing turbulent flow data on the same solutions was first studied by Toms 1 in dilute solution. 6 solutions of polymethyl methacrylate in mono­ For flow through circular tubes it is well chlorobenzene. The drag ratio for flow through established that there is a concentration effect circular tubes has been defined2 as the ratio of and a diameter effect on drag reduction. Toms 1 the pressure _drop of the solution to the pressure observed that drag reduction increases with an drop of the solvent at the same flow rate. The increase in the concentration of the solution up drag ratio is less than 1.0 for a drag-reducing to an optimum concentration beyond which, due fluid. Practical use of drag reduction is being to the increased viscosity of the solutions, the made in fracturing operations in the petroleum drag ratio increased with an increase in concen­ industry.3 tration. Hershey 7 observed that for dilute Newtonian A more fundamental quantity is the friction polymer solutions in nonpolar hydrocarbon solvents, a normal transition region may be obtained followed by a departure from the von Karman equation Original manuscript received in Society of Petroleum Engineers office Oct. 19, 1966. Revised manuscript received (on a friction factor-generalized Reynolds number June 11, 1967. Paper (SPE 1678) was presented at SPE Symposium on Mechanics of Rheologically Complex Fluids chart). The point of departure is a function of held in Houston, Tex., Dec. 15-16, 1966. © Copyright 1967 diameter: the smaller tube s depart at the lower American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Reynolds numbers. Concentrated solutions may lReferences given at end of paper. show no abrupt transition reglOn, but merely an S EPTEM RER, 1967 325 extension of the laminar regime followed by of the ratio of specific viscosity to concentration gradual departure from the laminar flow line, again favor drag reduction. This was observed experi­ showing a diameter effect. 4 mentally by Hershey .5, 7 Thus, intrinsic viscosity Several different theories have been offered to [TJ] defined as [TJJ = lim (TJsplc) and related to explain drag reduction such as the slip at the c->o wall theory,8 the anisotropic viscosity theory9 molecular we ight by the Mark-Houwink equation 17 and the various viscoelasticity theories.5 ,10-12 [TJ] = KMva , can be used as an indication of the Presently, viscoelasticity is widely accepted as tendency of a polymer-solvent system to be drag the cause of friction factor reduction, but the reducing. mechanism is not completely understood and it Park 18 was the first experimenter to try to is possible to explain some of the experimentally correlate drag-reducing data with the viscoelastic observed facts about friction factor reduction in properties of the fluids studied. The viscoelasticity terms of the other theories. term appeared in the Weissenberg number, a In a viscoelastic fluid, the stress is dependent dimensionless group relating the first normal stress on both the amount of strain (elastic response) difference to the shear stress, both evaluated at and the rate of strain (viscous response). A measure the wall shear rate. Park correlated data for only of the relative amounts of viscous and elastic one system. Since normal stress data on dilute Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 response is the relaxation time. If the time scale polymer solutions in which turbulent pressure drop of the experiment is of the order of or shorter measurements can be made are difficult to obtain, than its relaxation time, any fluid will exhibit his correlation has not been tested on other systems. elastic as well a's viscous properties.13 Hershey,S The correlation presented in this paper relates the Astarita,12 Fabula, Lumley and Taylorll and amount of friction factor reduction for six different Elata, Lehrer and Kahanovitz 10 have suggested polymer solvent systems to variables which can be that friction reduction occurs when the relaxation obtained from simple measurements of fluid time of the polymer molecules in solution is equal properties and flow conditions. to or larger than a certain "characteristic flow time". In general, the characteristic flow time EXPERIMENT AL has been taken as the reciprocal of the shear rate Pumping experiments were carried out in two at the wall (evaluated at the wall using the 'power recirculating systems'? The large system consisted law or the Newtonian viscosity equation relating of a 100-gal reservoir, two pumps (0 to 35 and 0 to shear stress and shear rate), and the relaxation 200 gal/min) in parallel (each equipped with a time has been estimated using the Zimm14 or variable-speed drive), two turbine meters (1.5 to 15 Rouse 15 theory. and 10 to 140 gal/min) in paralle I and Yz-, 1- and The model for the Zimm theory assumes 2-in. ID smooth bore steel tubing test sections in conditions such as monodispersity (molecules parallel. Appropriate manometer systems permitted having all the same length) and Gaussian distribu­ pressure drops from 0.01 to 50 psi to be measured. tion of the segments of the polymer molecules which Test section and entrance region dimensions are are generally not met by a real polymer. However, listed in Table 1. Friction factor data could be the predicted values of relaxation times are useful obtained in the Reynolds number range of 15,000 for comparisons. The following equation is used to 207,000 with toluene. for predicting relaxation times. 16 The small system consisted of a reservoir, one of two gear pumps (0 to 50 or 0 to 550 ml/min) equipped with a variable-speed drive and stainless steel capillary tubing test sections. The capillary . (2) = stream was discharged horizontally into a specially 0586 RT Ak designed funnel and refed to the reservoir. Pressure drops across the capillaries were measured by manometers or Bourdon gauges and covered the where M v is the viscosity average molecular range of 0.005 to 300 psi. Capillary tubing dimen­ weight, TJ sp is the specific viscosity defined as sions are listed in Table 1. The minimum length in 11- s - 11-0 11- ' 11-0 is solvent viscosity, 11- s is solution 1 0 diameters was 385, and appropriate entrance viscosity, c is concentration, T is absolute corrections were applied to all data. Friction temperature, R is the gas constant and Ak is the factors could be obtained in the laminar region and eigenvalue corresponding to the kth mode of relaxation. The eigenvalues for the different modes have been calculated, the smallest eigen­ TABLE 1 - TEST SECTION DIMENSIONS Tube Diameter Entrance Length Test Section Length values (longest T corresponding to the first W (in.) in Diameters in Diameters mode. The Zimm theory predicts only discrete .0325 values of relaxation times, while in actual polymer 744 .0463 524 solutions a continuous spectrum of relaxation .0629 385 times is observed. .509 100 300 Si.nce drag reduction is favored by long relaxation .999 75 200 times, increased molecular weight and high values 1.998 50 100 326 SOCIETY OF PETROLEUM ENGINEERS JOURNAL up to Reynolds numbers of 15,000 for toluene. analysis may be performed upon the equation of Temperature control to ± O.IC was maintained on motion using a viscoelastic model as the equation both systems.