... ~ This pap<!r was presented at the Symposimn "Multiphase fllow in the Plv,xI,uction a,nd Drilling of Oil Wells", A.I.Cb . B. - S.P.B . Joint Symposium. September 25 ~ 28~ 1960, in Tulsa, Oklahoma. , , ABSTRACT The addition of water to crude oil pipelines appears to be an b,npoltlnt motb,od tor din, th, preS8QJ'e Fadtent for a Slv.n 011 tm'ouabP\J~ .Althov.gb the .Q«lCfttrie otJmwat». tlow IHlttCrB pMVtdeIJ the ateat. PQHibIe reduction, the general case, in which the liquids are stratified as a result of the 011 and water having different densities, is also characterized by reductions in pressure gradient,. to be expected in stratified flow systems, the Navier-Stokesequations were solved by a numerical procedureut~ina a digltal computer, for the laminar etretit1ed.fiow of two Newtonian liquids - oil and water - in a Circular pipe. Liquid flow rates were obtained for the flow of five hypothetical oils ranging in viscosity from 4 to 1500 centipoises in the presence of water, by computing the oil and water velocity-profiles for a .series of-arbitrary on .. water in~ce " positions. It was found that the maxim wn values for the pressure gradient re- duction factor ranged from 1.12 to 1.31 for the five oils and occurred at water contents ranging from 12 to 93 per cent respectively. The computed reduction factors were considerably lower than experimental values and this appears to indicate that wave motion and miXing at the oil-water interface produces a further and very Significant reduction in the pressure gradient. '" ~ 1 ;. INTROQUctlON The most convenient and usually the moSt econoillical me~od of transporting petroleum overland is by pipeline 40 In this mode of transportation ~e P9W~t requir~d to mQve the oil is a function t;Jf the oil viscosity which, for different crUde oils, may vary Over a wide range. The flow of mm'e viscQUs oils under pipeline conditions tends to be laminar and in this state the pressure gradient neCessary to move oil at a giV'en throughput is proportional to the viscosity of the oil. With highly viscous crude oils several methods are available for reducing the pressure gradient necessary to maintain a given -:t thrOUghput. The oil viscosity may be reduced by raising the temperature of the' oil or by adding a dUuent,or the resistance to the flow of the oil may be reduced by the addition of a less viscous immiscible liquid. A discussion of the relative mente of these methods in a particular case has been given by Leach (1). The resistance to flow in either laminar or turbulent flow results for the most part from the friction at the pipe wall and if the viscous Uquid at the pipe wall is replaced by a liquid of much lower viscosity the resistance to flow is considerably reduced. The addition of water to crude oil pipelines can therefore be an attractive method of reducing pressure gradient. Maximmn pressure gradient reduction is obtained with a con- centric oil-in-water flow pattern. This condition is possible if the densities of the oil and water are approximately equal. The cOncentric flow of oil and water has been analysed theoretically by RUssell and Charles (2) for the case .. 2 - whc;m both the ott Alld die watet ate in laminar motion, by Cbemi.kl~ (3) for tile ease When the oil is in 18lllu.ar motion aDd the water is tutbUient. Yuster (4). Odeb (5) and Baker (6) have considered the c.oncentric flow mOdel in lde_l ed e"plilary systems. C()neentrtc flow was shown to be stable lil\der certaln CGa- ditl ef new by the ~rimentai work of Charlea, 'Govier alld Hodgson (7) in Wbich the densitles of the water and oit were equal. The patents by Clark and Sh"piro (8) and ClUlton aDd Handley (9); Were als. concerned with the appUootlonof the concentric flowaystem. Isaacs and Speed (10) and Cher- nildn (3) Indicated that when the density differential is sufficient to produce stratification of the oil and water it is possible that concentric flow may be established if a rotational motion is imparted to the flowing liquids by meaDS ~ a rifle on the insi4e of the pipe. . , If no attempt is made to equalize the density differential or produce a rifling effect, the oil and water will tend to stratify and the water will be in contact with the oil on a horizontal interface. This type of flow is probably the most important to be considered because of its simple and w1de- spread application. Looman (11) was probably the first to suggest the use of water as a bottom layer in oil pipelines to reduce the pressure gradient necessary to convey oil at a given rate. The effect of the oil viscosity in stratified flow between parallel plates has been predicted by Russell and Charles (2) in a theoretical study. The analysis dealt with Newtonian fluids in laminar motion and the maximum - 3- pressure gradient reduction factor, i.e. the ratio of the pressure gradient for the oil flowing alone to the pressure gradient for the same oil throughput in the presence of water, was predicted to approach a constant value of approximately 4 at high oil viscosities. Dumitrescu and Stanescu (12) presented a general analytical method for the treatment of stratified laminar flow in a conduit of any shape and have indicated solutions for the special cases of a circular pipe and a rectangular conduit half full of each liquid. Teletov (13) has also indicated a highly com- plex analytical method for obtaining velocity profiles for stratified flow in a circular conduit. Neither Dumitrescu and Stanescu nor Teletov obtained flow rate equations or pressure gradient reduction factors. Russell, Hodgson and Govier (14) investigated the general flow characteristics of the simultaneou~ horizontal flow of a light oil and water and noted small reductions in pressure gradient for flow in the stratified regime. Charles (15) has reported pressure gradients for the stratified flow of a viscous crude oil and water in a I-inch laboratory pipeline as well as in a 2.45-inch experimental field pipeline. The results obtained with the 2 45-inch line are reproduced in Figure 1 where the pressure gradient reduction factors are plotted as a function of the percentage of water in the flowing stream. ReductiOns fn pressure gradient by factors of more than 10 were recorded for water percentages ranging from 30 to 50 per cent. Oil viscosities ranged from 124 to 910 centipoise. The object of the present investigation was to evaluate the effect on the velocity profile of the introduction of water as a lower layer in oil .. 4 .. p!pew 4mt *~ t the nHlltli~¢ ~ ~~ pte ~~ _adlt t.clueq~ bt tetltls bf the ctUde ijti ViS¢08itY ind the W,ter now tate. The analysis Is ri!stHcted to Newtmtiaii fiilidj in iamittlf mttltii tb .·Qjn·· .. " beNb tat inteHac ' it i"umed that tit' . et it .' . . " .' n .. e, 8 141(1 .' e v ~ Je8 'iftK" hlil~iil .tii~ tf'tial at l't'etfaee tI ~et6 ii tit . N.' M. 'wilti i"'d thAi ~ it q Hie ,n. ., ,~ ,e .' ; H, ,. , the sHelf fottes eXertEkl Chi each Ph_e. at ihe intetl.~e are ~Wti. THEOR.Y " The Navlet .. Stokes equathms iOvem the three .. dlmensionai flow of flUid ~ When the flow is undil'ectional and laminar, and the fluid. tn- compressible, under steady state conditions the Navier-Stokes equations reduce to the single equation: (1) which is an elliptic differential equation for Which it is possible to specify the conditions at all points along a closed boundary. There is a simple analytical .olution of this equation for the flow of a single fluid in a circular pipe. How- ever in the case of stratified flow, although Oumitrescu and Stanescu (12) and Teletov (13) have indicated general analytical epproaches, and the fonner have giVq the solution for the special case when the interface passes through the axis of the pipe, the general analytical solution for any interface position il ~eedtngly complex. Iterative and analosue procedures are 8V'ai1able as sub-- stitutes for analytical methods and a numerical techirlqt1e was therefore used to .. 5- give an approximate solution to the differential equation (1). Usually in such a process the nmnber of repetitive calculations is very large and several investi- gators, including Frankel (16) and Radd and Tek (17), have suggested and demonstrated the use of digital computers to shorten the computation time. The general approach of Radd and Tek was modified to suit the present problem and entailed the replacement of the differential equation (1) by the finite difference equation Ui,J = ! rl+l,J + UI-l,J + UI,J+l + U1,J-l + and the superimposition of a grid With a square mesh on the pipe cross-section. A grid which is too coarse to obtain a reasonably accurate solution of the present problem but which will be useful in explaining the method, is shown super- imposed on the pipe cross-section in Figure 2 where the relationship between adjacent grid points is illustrated. The finite difference equation is applied to each point in tum and the new velocity, Ui, j' calculated from the four neigh- boring points, is a better approximation to the true velocity at that point. Only one-half.of the pipe cross-section is illustrated because the system is symmetrical about. the vertical diameter. The position of the pipe wall is approximated by the grid points labelled zero. The finer the mesh, the more accurate is the approximation to the solution of equation (1). The use of equation (2) gives slow convergence to the final velocities and the extrapolated Liebmann method (18) which utilizes and over- relaxation factor ~ , was used to produce faster convergence. In the extra- polated Liebmann method the finite difference equation (2) becomes: - 6 - Ui,l ~ Ul,l ~[1-<Ui+l'i + UI_1,! + UI,j+l + UI,l-1 + C) - UI where dp dz (3) The over-relaxation factor, ~ ; was calculated from the equation given by Frankel (15): ~ = 2 - 211" m2 + ~ . [2 ( 1 1 )] 1/2 (4) The relaxatiOn constant, C, in Equation (3) contains the fluid viscosity and hence has different values when EqUation (3) is applied to the oil and water layers. Equation (3) governs the flow within the oil and water layers but does not apply at the oil-water interface. At the interface the shear forces exerted on each liquid are equal and for Newtonian liquids it follows that: = (5) in which the velocity gradients are evaluated at the interface. Equation (5) may be introduced into the iteration procedure in at least two ways: (i) The interface is coincident with a row of grid points and the differential equation (5) is rep~aced by the finite difference equation: .. U' =fh )u .fA )u i,j fo+,fw i,j-l ~o + iN i,j+l (6) which is used instead of Equation (3) at the grid points which lie at the interface. The relationship of the adjacent grid points and the interface position is illus- trated in Figure 3. .. 7 - This method suffers from the obvioUS limitation that iterations are perfonned in one dimension only for the interface grid points. (it) The interface lies equidistant between two adjacent J;'OWS Of grid points. The method is illustrated in Figure 4. Temporary interface points are assigned and the finite difference equation: U' = ( to ) u + (- tw ) i,jfr, + fw i,j-1 \fo + fw U. j 1, (7) which again corresponds to the differential equation (5) is used to calculate the interface velocity values which are subsequently used in the finite difference equation (3) modified, as suggested by Round, Newton and Redberger (19), to take into account the variation in grid size brought about by placing the inter- face between two grid rows. This modified finite difference equation, written for a point i, j in the grid row below the interface is Ui,J = UI,J UI+l'J + UI-1,J + UI,J+l + ~ (Ui,J+l + 8'l,j)+ c]- UI,j] (8) This method has the advantage over method (i) in that iterations are perfonned in two-dimensions closer to the interface and hence should yield more accurate results. Both methods (i) and (ii) were investigated and the relative accuracy of the two methods will be indicated. In the iterative procedure the grid points were treated suc- cessively and new velocity values were calculated. nweeps through the grid were continued until the differences between the old and the new velocity values at each point were within a predetennlned arbitrary tolerance, I. e ./ Uj, J - U I, J 1< e. - 8 - COMPUf ATION In general, in an iterative procedure of the type used in the present study, the smaller the mesh size is, the greater is the accuracy of the computed result and the longer is the comli\ltation time. The mesh size used in this case represented a compromise between the accuracy of the result and the time required for computation. Initial test runs were made on a Royal McBee LGP-30 comt:uter and three mesh size ratios, namely 1/16, 1/32, and 1/48, were investigated. (The mesh SiZe- ratio. of the grid shown in Figure 2 is 1/8.) Accuracy was gauged by comparing the computed velocity profile for a pipe flowtug full of a sillgle' liquid with the theoretical profile, and the computed cross-sectional area of the approximate pipe boundary with the theoretical area of the pipe. The mesh , size ratio of 1/16 resulted in errors of 3 per cent for the maximum velocity and 4 per cent for the pipe area compared with approximately one per cent errors given by the mtio of 1/32. The ratio of 1/48 gave results very little better than that of 1/32 but required a much longer computation time and thus a mesh size ratio of 1/32 was chosen for the main investigation. Approximately 100 sweeps through the grid were necessary to achieve the above accuracies in the velocity values. It was found that the computation time was 4.5 min. per sweep on the LGP-30 computer and consequently an IBM 704 was used for the complete solutions because it gave the shorter computational time of 0.6 sec. per sweep_ -9- Flow systems were studied for oils of viscosity 4, 20, 150, 450, and 1500 centipoise floWing above water With a Viscosity of 0.896 centipoise. For each oil viscosity six interface positions were investigated, more iIlterface positions being in the lower hall of the pipe than in the upper so that the effect of combining relatively small flow rates Of water with the oil could be evaluated. For each interface position the computer printed out the velocity values at each grid point as well as the oil and water flow rates which were calculated by nwneri- cal integration. RESULTS The analytical and nwnerical values of the velocities on the axiS of the pipe were compared for the special case when the interface coincided With the horizontal diameter. The equations of Dumitrescu and Staniscu (12) are easily solved for this particular point and the analytical velocity value is given by: U = n2gc 8 ~ dz (9) Numerical velocity values were calculated using both of the methods outlined for incorporating the interface condition into the iterative method. The corresponding velocities given by the analytical and numerical methods are given in Table I for the five oil viscosities. It was apparent that method (ii), in which the interface was placed between two grid rows, gave the more accurate results and this method was subsequently used throughout the computation. - 10- The effect on the oil phase of the addition of wa~er to fonn a lower layer in oil pipelines is illustrated py the fonn of the velocity profiles on the vertical pipe diameter. The profiles obtained at each interface position for each .oil viscosity are basically Similar, and as examples, profiles are given in Figure 5 for.an oil viscosity of ISO centipoise. The pressure gradient is the same for each profile, and the interface positions and percent- ages of water in the flowing stream a~e indicated. The parabolic profile for the oil flowing alone lm,der the same pressure gradient is included for com- parison. The effect of the.addition of the water is to sharply extend the pro- file in the oil phase for a given pressure gradient. The general shape of the velocity profile obtained by the addition of water to the floWing stream is further illlustrated in Figure 6 where horizontal profi~e planes are superimposed in the plan view and vertical profile planes are superimposed in the elevation to give a three-dimensional representation of the profile. The oil viscosity is 150 centipoise and the water flow rate constitutes 45.8 per cent of the total flow for the profile shown. The effect of the oil viscosity on the shape of the velocity profile for a fixed interface pOSition is illuotrated in F!zure 7. The profiles are awin drawn for a constant pressure gradient. Although the interface position is the same for each oil viscosity the relative flow rates of the two 'liquids differ considerably. For an oil viscosity of 4 centipoise, water con- stitutes a9.8 per cent of the total flow, while for an oil viscosity of 1500 centi- poise water constitutes 99.0 per cent of the total flow. - 11 - In pipeline design, interest may focus on the oil flow rate which may be obtained for a given pressure gradient, or aitematively, the pressure gradient which is necessary to provide a given oil throughput. Hence the re- sults of the present investigation could be reported either in terms of the in- crease in oil flow rate for a given pressure gradient obtabied by addition of water, or alternatively, in tenns of the redUCtion in pressure gradient for a given oil flow rate obtained by the addition of water. In practice the oil throughput is usually known and accordingly the present results are reported in tenns of the pressure gradient reduction factor which is defined as the ratio of the pressure gradient for the oil flowing alone to the pressure gradient for the same oil throughput in the presence of water. The pressure gradient reduction factors calculated from the results of the numerical analysis are plotted against the interface pOSition with the oil viscosity as parameter in Figure 8. The curves all have the same general shape; with the interface at the bottom of the pipe, i.e. the pipe flowing full of oil, the pressure gradient reduction factor is unity for each oil viscosity and increases to a maximum on the addition of water and then falls off to zero as the interface rises to the top of the pipe on the further addition of water. The greater the oil viscosity, t.~e greater is the rate of increase of the reduction factor with increasing water percentage, although for the higher oil viscosities the curves are very cloce together. The pressure gradient reduction factors are also plotted against the percentage of water in the flOwing stream in Figure 9. Curves are shown only for oil viscosities of 4, 150 and 1,500 centipoise to avoid confuSion. The per- - 12 centage of water needed to bdng about the maximtim beneficiation varies from 12 per cent for a 4-centipoise oil to 93 per cent for a 150centipoise oil. For all viscosities the reduction factor decreases vety rapidly as the water percentage approaches 100. The maximum pres$ure gradient reduction factors are shown in Figure 10 as a function of oil viscosity and range up to abOut 1.31 for the viscosity values investigated. It is evident that for oil vismsities greater than about 100 centipoise the reduction factor is approximately constant. Also in Figure 10 the maximum pressure gradient reduction factors are com- pared with those predicted by Russell and Charles (2) for concentric flow in a circular pipe and stratified flow between parallel plates. For concentric flow the maximum pressure gradient reduction factor is directly proportional to the oil viscosity whereas for stratified flow between parallel plates the maximum reduction factor approaches a value of approximately 4. It is apparent, there- fore, that concentric flow, provided it may be established, is very much more successful in reducing pressure gradients. A further analysis of the experimental pressure drop data ob- tained by Russell et al (14) for the stratified flow of an 18 centipoise mineral oil with water indicates that for the regimes of flow in which both phases were apparently in laminar motion, pressure gradient reduction factors of approximately 1.2 were noted when water constituted approximately 10 per cent of the total flow. These values compare very closely with the value of 1.21 obtained for the 20 centipoise oil anr. 10 per cent water by the numerical analysis. - 13 However, the experiniental prcessure gradient reduction factors reported by Charles (15) for the stratified flow of a crude oil and water in a 2.45-inch diameter pipeline and reproduced in Figure 1 diffet considera.fjly t,rom the results obtained in the numen,cal analysis. Up to a water percentage of about 20 the experimental reduction factors are Ilttle different from unity but at water percentages in excess of 20, pressure gradient reduction factors greater than 10 were recorded over a wide range of oil flow rates. In the experimental tests the oil was almost certainly in laminar flow and the water in turbulent flow . The discrepancy between the experimental reduction factors and the results of the numerical analysis is probably accounted for by the turbulent water layer producing wave action and mixing of the oil and water at the interface which would effectively give a gradual change in viscosity across the boundary between the phases and reduce the sharpness of the discontinuity in the velocity ~rofile. Part of the oil phase would then be in the relatively fast moving water phase with a consequent very substantial increase in the pressure gradient reduction factor. CONCLUSIONS 1. The use of a digital computer has been demonstrated for obtaining velocity profiles and flow rates of two liquids flowing laminarly and stratified within a Circular conduit. The method may be applied to any number of liquid layers flOWing in a conduit having any geometry. 14 - 2 A reduction is predicted in the pressure gradient necessary to maintain a given oil throughput by inttoduclng water as a bottom layer into a pipeline carrying oil. The range of oil viscosity investigated was from 4 to 1500 centipoise and the maximum pressure gradient reduction factors ranged from 1.12 to 1.31 respectively. 3. The pressure gradient reduction factors of the present analysiS are considerably less than certain experimental values for stratified flow. This appears to indicate that wave motion and mixing of the two phases at the inter- fdee due to turbulence in the water layer appears to give a further and very significant reduction in pressure gradient. ACKNOWLEDGEMENT The authors acknowledge the helpful suggestions and criticisms of R. Newton 2nd G. W. Hodgson. C = D = e = &: = h = m = n = dp - = (~Z S = t = Ui . = ,J UI = I,J ~ = f- = Subscripts i j f 0 w - 15 - NOMENCLATURE h2 d relaxation constant = gc ~ , ft. per sec. - dz inside diameter of PiPe tolerance, ft. per sec. dimensional conversion factor, lb. M ft. per lb. F sec.2 length of side of grid square, ft. number of columns in grid. number of rows of grid. pressure gradient, lb. F per ft.2 per ft. ratio of distance from top of pipe/diameter of pipe; dimensionless ratio. of distance from centre of pipe/radius of pipe, dimensionless. general point velocity, ft. per sec. newly calculated general point velocity, ft. per sec. over-relaxation factor, dimensionless. liquid viscosity, lb. M per ft. sec. = grid column number (1 i !fm) = grid row number (1 ~ j $.n) = temporary grid row number defined by Equation (6) = oil = water - 16- REFERENCES 1. Leach, R. W., tlPipeline Designed for Viscous Crude", Pipeline Engineer (Nov., 1957 D-25. 2. Russell, T. W. F ., and Charles, M. E ., "The Effect of the Less Viscous Liquid in the Laminar Flow of Two Immiscible Liquids", Can. jour. Chem. Eng. (1959), 37, 18. 3. Chemikin, V.I., "Combined Pumping of Petroleum and Water in Pipelines", Trudi Mock. Neft. in-ta, (1956I, 101. 4. Yuster, S. T ., "Theoretical Considerations of Multlphase Flow in Idealized Capillary Systems" Proc. Third World Petroleum Congress (1951), ~, 437. 5. Odeh, A.S., "Effect of Viscosity Ratio on Relative Permeability", Trans. A.I.M.E., (1959), 216, 346. 6. Baker, P.E., "Discussion of Effect of Viscosity Ratio on Relative Permeability", Jour. Pet. Tech. (1960 12, 65. 7. Charles, M. E ., Govier, G. W., and Hodgson, G. W., "The Horizontal Pipeline Flow of Equal Density Oil- Water Mixtures", Can. Jour. Chem. Eh'g. (in press). 8. Clark, A. F ., and Shapiro, A., "Method of Pumping Viscous Petroleum", (1950) U.S. Patent 2,533,878. 9. Chilton, E. G., and Handley, L. R., "Method and Apparatus for Lubricating Pipe1in3s", (1958), U.S. Patent 2,821,205. - 17 - 10.. Isaacs, J. D." and Speed, J .B., "Method of Piping Fluids". (1904), U.S~ Patent 759.374. 11. Looman, M~. "Method of Conveying Oil". (1916). U.S. Patent 1.192, 438~ 12. Dumitrescu, L., and ~taniscu, C., "A Boundary Value Problem with Application to the Flow of Two Viscous Fluids in Contact and me Stress in a Non-homogenous Rod". Acad. Repub. Pop. Romane, Rev. Mecan~ Appl. (1957 2. 13. Teletov, S.G., "The Separated Laminar Flow of Gas-Liquid MiXtures", Compt. Rend. Acad. Sci. U.R.S.S. (1946g, 179. 14. Russell, T. W.F., Hodgson, G. W., and Govier, G. W., "Horizontal Pipeline Flow of Mixtures of Oil and Water", (1959). Can. Jour. Chern. Eng. 37, 9. 15. Charles, M. E ., "The Reduction of Pressure Gradients in Oil Pipelines: Experimental Results for the Stratified Pipeline Flow of a Heavy Crude Oil and Water". (1960). Trans. Can. !nst. Min. Met. 63, 305. 16. Frankel, S .P .. , "Convergence Rates of Iterative Treatment of Partla,l Differential Equations". Math. Tables and Other Aids to Computation (19503. 65. 17. Radd, M.E .. , and Tek, M.R. , "Engineering Application of Relaxation Procedures by Digital Computation." A.I.Ch.E. jour. (1959). ~, 111. - 18 - 18. Liebmann, H., "The Approximate Solution of Harmonic FunCtions and Conformal Representations", Bayer Akad. Wiss., Math. Phys. K~asse, Sitz., (1918), p. 385. 19. Round, G.F., Newton, R., and Redberger, P.J., "Variable Mesh Size in Iteration Methods of Solving Partial Differential Equations and Application to Heat Transfer", (1960). Presented at the Annual Meeting, A.I.Ch.E., Washington, D.C. Figti.re 1. Pigute 2, Figure 3. Figure 4. Figure 5. Figure 6. .. 19 .. CAP'TION$ FOR FIGt)RBS Experimental results for pressure gradient redlictlb11 factbt' as a function of percentage of water in the flowing stream. as reponed by Charles (15) fot a heavy crude oil. Illustrative grid on half of cross-sectional area of a circular pipe carrying stratified oil and wate The position of the in ter!ece in rl3lations to the grid points when the interface lies in a row of grid points. The position of the interface in relation to the grid points when the interface lies between two adjacent rows of grid points. Typical velocity profiles on the vertical pipe diameter for the 150 centipoise oil flowing above water compared with the parabolic profile for the oil f!oVling alone. The pressure gradient is the sam e I)file 2l1d tile lower parts of the three profiles are identical ~thin the limits of the figure. Plan view and elevation of superimposed profile planes to illustrate a three-dimensionp.1 proile for an oil viSCOSity of 150 centipoise when weter constitutes ~5. 8 per cent of the tot21 foVl Figttte 1. Figure 8. Figure 9~ Figure 10. - 20 ~ The eff~ct of the oil viscosity ori the shape of the velocity profiles for a fixed interface position. The pressure gradient is the same for each profile and the lower pints of the profiles are identical" within the limits of the figure. Pressure gradient reduction factors calculated by the numerical procedure as a function of interface position. Pressure gradient reduction factors calculated by the numerical procedure as a function of percentage of water in the flowing stream. Comparison of maximum pressure gradient reduction factors for stratified flow in a circular pipe, stratified flow between ,parallel plates and concentric flow in a circular pipe. Viscosity of oil Centipoise 4 20 150 450 1500 - 21 - TABLE I COMPARISON OF ANALYTICAL AND NUMERICAL VELOCITY VALUES I Velocity Computed velocity values calculated for Interface method (i) I lntedace inr;thod jiO centre point i I Per cent from analytical Velooity Velocity Per cent equation predicted I error predicted error Ft. per sec. Ft. per I - Ft. per - sec. sec. 5.10 4.77 6.5 5.08 0.39 1.19 I 1.11 I 6.7 1.18 0.84 0.166 ! 0.154 7.2 0.164 1.2 0.0557 0.0513 7.9 0.0547 1.8 1 0.0163 2.4 0.0167 I 0.0152 9.0 i a o r- o « 50 LL 20 z o r- 10 o ::J o W a 5.0 r- z w - 2.0 0 « a 9 Figure 1 . / , , 245 / , -, , 176 213 194~J OIL VELOCITY, ft. per sec . 1.0 ~ ,200 L LI66 ~ 124 ~ 182 1.20 2.18 2.94 3.88 5.08 6.87 W a ::J (.f) 0.5 (.f) W a CL o 153 10 PRESSURE GRAD:~NT REDUCTION FACTORS FOR 2.45" 1.0. PIPELINE NUMBERS ATTACH~D TO POINTS INDIC/~TE OIL VISCOSITY IN CENTIPOISE 20 30 PERCENTAGE IN 60 FLOWING STREAM 90 100 n rows ! i ,j-I I, J eO i-I,j i+l,j i, j + I interior point /boundary / point ~ ex fer i 0 r point o I I I I I I I I I o o o m columns i ,j-I 01 L 1 N T E R F A C E I, J WAT ER i ,j + 1 01 L i,j -I I N T E R F A C E - - . . / I, J i-I,j .. i + 1 ,j I,J WAT ER i ,j + I W 0.. o 0.. LL W o 0..0.25 0.. 1 0.. OLL f-O 2: 1 0 O!~ 0.5 OI W LL2: w« u- ZO « f-: 0.7Sf-U): ! 0' I o <t 1 Oil viscosity: 150 centipoise I \ \ Oi I alone Water flow rate I I Is-~alues at (percentage of I i li0J~_fus_~ I total flow) I II -- c I 0.297 99.7 i 0.641 90.9 0.828 45.8 CL I. 0 iL ~= 1 1 o 1.0 2.0 3.0 VELOCITY arbitrary units r-I {lu Qc s- Figure 6 1.0 W 0 0 LL, 0: cr: w W 0 f Z 0 w LL U 0 L 0 cr: 0 LL « w, cr: U z «, fi ':01 0 0 f « 0 1.0 W 0 0 LL' OW oe: o 0,0 fILL LO 0'00.5 OiW LLIf WW UL z<c <cO tfj: 0.7 0 1 o I r- 1 ! S volls for horizontal j l profile pi ones _ 0.843 . 0.969 - PLAN VIEN 011 viscos ty: ,SU Cef1tlpol~e s valve at in'. ,fa, ~ = CJ.828 Water flow ratc:L; .8% of total flow ,- I. r oi' _r~/7~1 GA water below 011 J t values for~et~~ [profile planes J ELEVATION I :- ~ 10LIJLj L "- 1.0 2.0 3.0 er:: VELOC lTV, arbitrary units tJ) .. w 0.. 0. uW 00. 04~ 39.8 20 L~ 63.2 0.0. o ~ ~ 0.25 20:: Ow o uw w 2 U« Zo OIL VISCOSITIES centipoise / WATER FLOW RATE AS PERCENTAGE OF TOTAL FLOW « 0.5 ~ I 1500 o o ~ <{ 0.75 0 99.0 s value at interface =0.641 L 1.0 1.0 2.0 3.0 VELOCITY, arbitrary units hf.l)Qtl 0:: 1.4 o r- U ~ 1.2 z o r- 1.0 U :) o ~ 0.8 r- z wO.6 o « 0:: lJ 0.4 w 0:: :) ~0.2 w 0:: Cl. I Oil viscosity, centipoise I 1500 and 450 ~ 4 OLL ." cO" t:: CD co o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE FROM TOP OF PI PE INTERFACE POSITION: RATIO DIAMETER OF PIPE ~ 1.4 I- U <t LL z o I- U o w cr 0.8~ I- z w o I L 0.6~ i '-I i-- cr ' I \J 0.4f- ! W i I cr - I ~ O.2~ W i JOil viscosity ,centipoisel ~ CD 1500 co cr ~ CL 0 L l lJ 1 ~ __ L _ __L o 10 20 30 40 50 60 70 80 90 100 PERCENTAGE OF WATER IN TOTAL FLOW Figure 10 1000 cr:: i o I f- I U ! <t LL z o f- U => o 100~ w cr:: f- Z w o <t cr:: U w cr:: => 10 oL rJ) . if) w cr:: Cl. 2 => 2 X <t --r -- lAMINAR CONCENTRIC FlOvV IN A CiRCULAR PI~E (refere nce 2) lAMINAR STRATIFIED FlOVV BETWEEN PARALLEL PLATES (reference 2) LAMINAR STRATIFIED FLOW IN A CIRCULAR PIPE (present investigation) 2 I O Lr . I 10 100...