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    A Model Study of Viscous FingeringA. L. BENHAM IUNIOR MEMBER AIMER. W. OLSON

    ABSTRACTViscous fingering was studied as it occurred inan open He Ie-Shaw mode I (1 ft x 4 ft x 1/16 in.);

    it was also studied in the same model packed with80-mesh glass beads during miscible displacementsunder unfavorable viscosity ratio conditions. It wasdetermined that the lengths of 20 elements of thefront were distributed normally around the averageposition of the combined elements under al l conditions. Represented by the normal distribution, thelength of the viscous fingers grew linearly with thedistance traveled from the point of finger formation,increased with displacement velocity, and increasedwith increasing mobility ratio. Results obtainedduring the first few inches of displacement were oflittle or no us e in predicting finger growth and/orfinger length throughout th e 4-ft model since thepoint of finger initiation cannot be predicted andwould often occur several inches from th e point ofinjection with either positive or apparent negativecoordinates. Th e fingering occurring during a miscible slug displacement wa s much greater thanwould be predicted based upon the actual mobilityratios between in-place fluid and slug and betweenslug and following fluid, using the results describedabove for miscible displacement in the absence ofa slug. Many of the experiments in the packed modelshowed that the rate of growth of the viscous fingers was diminishing toward the end of the displacement in the 4-ft long model, indicating thatmicroscopic mixing, such as diffusion or dispersion,wa s decreasing th e viscous fingering effect.

    INTRODUCTIONViscous fingering is a manifestation of a finger

    shaped interface between displaced and displacingfluids occurring during typical miscible displacementprojects for oil recovery. It s cause may be tracedto the instability of a viscous fluid being displacedby a more mobile fluid. Viscous fingering takes onimportant significance in the miscible slug process 1, 2 where it may be a dominant factor in determining minimum slug size.

    Experimental 1 -3 an d theoretical4 - 7 studies ofviscous fingering have been made by other investi-

    Original manuscript received in Society of Petroleum Engineersoffice Nov. 1, 1962. Revised manuscript received March 4 , 1963.1References given at end of paper.

    1S 8 4rle

    MARATHON OIL CO.LITTLETON, COLO.

    gators. However, an exhaustive study of the variablesaffecting viscous fingering had not been made. Th epresent study wa s undertaken in an effort to determine the effects of some of th e more obvious variables -such as mobility ratio, displacement velocity, distance displaced, and packing-upon viscousfinger length and growth in a small laboratory model.It does no t necessarily follow that conclusionsreached by studying the results of this model studymay be applied to the field. Future studies wouldhave to evaluate the effect: of model size on extension of these results.EQUIPMENT AND EXPERIMENTAL PROCEDURES

    A flow diagram for th e equipment used in thisstudy is shown in Fig. 1. Basically, it consistedof (1) a constant-rate pump for the injection of displacing fluid into (2) a model made up of two flattransparent plates spaced a small distance apartand (3) provisions for the production of fluid into acalibrated graduate. The displacing fluid containeda dye which allowed visual observation of fingerspacing and length. Th e pump performed at ratesfrom 2.5 cc/hr to 480 cc/hr, while pumping a lightmineral oi l into a bladder in a closed glass containerholding the displacing fluid.

    Th e model wa s made up of two plates of I-in.Plexiglass with dimensions l-ft wide by 4-ft long.These were spaced 1/16 in . apart so that the internalmodel dimensions were I-ft wide x 4-ft long x 1/16-

    FIG. 1 - FLOW SHEET OF EXPERIMENTAL APPARA-TUS.

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    in . thick. Special injection heads were designed togive a planar injection front and a constant velocityacross the model. Two other injection heads, whichgave a square wave of either ~ in. or a modifiedX in., were designed and used for a few experiments.

    Before a run wa s made, th e model was filled withan aqueous glycerine solution of a concentration togive the desired viscosity. This fluid wa s displacedby an aqueous aluminum sulfate solution of concentration to give the same density as th e displacedfluid bu t of a different viscosity. This techniqueavoided gravity effects in the model.

    The displacing fluid was injected into one endof th e model through the injection head describedpreviously. The displaced fluid was forced outthrough the other end of the model, collected, andmeasured in a calibrated burette. The displacingfluid wa s colored with a small amount of red foodcoloring to afford a visual observation of the shapeof th e front; the displaced fluid was colorless. A16-mm. Bolex movie camera was placed over th emodel. A special timing device wa s connected tothe camera which allowed time-lapse filming, withintervals from seconds to 24 hours betweenframes. Th e time lapse used for a particular runwas adjusted to give a showing time of about twominutes for the entire displacement, compared withtotal displacing times from 1 to 40 hours.

    Fo r slug runs, the model was first filled withaqueous glycerine solution, as described previously.Displacement was initiated by th e addition of ared dye until the desired quantity of this fluid(representing the slug) had been injected. At thispoint, a second displacing fluid was employed. Ithad a different viscosity and contained a green dye.The viscosities of th e three fluids were adjustedto give the desired mobility ratios between originalin-place (displaced) fluid, slug fluid, and displacingfluid.

    Some of th e studies utilized the open model, calledth e "H eIe-Shaw Model" by other investigators.However, one series of runs was made using thismodel packed with 80-mesh glass beads (0.0069-in.diameter). Special precautions were taken to assurethat th e model and beads were clean. Th e modelwas then placed on end and filled with a wateralcohol solution. Glass beads were then fed intothe top of th e model, using a vibrating feeder. Th ebeads dropped into the fluid and settled to the bottom.The model wa s tapped to insure uniform packing.When the model wa s completely filled, it wa s placedon the table. Injection and production facilitieswere connected, and the alcohol-water solution wasdisplaced using the desired aqueous glycerinesolution. This technique assured an absence of ai rbubbles in the system.

    EXP ER I MENT AL RESULTSA total of 26 runs were made in the open HeleShaw model with mobility ratios from 1 to 90

    (viscosity displaced/viscosity displacing) an d lineardisplacement velocities from 0.1 to 4.0 fr/hr. Inorder for the Hele-Shaw model to be a satisfactoryJUNE, 1963

    analog of flow in porous media, the fluids must beflowing in th e laminar-flow region. Reynolds numbercalculations showed that laminar flow was alwaysrealized.

    A second series of 12 runs was made in the modelpacked with 80-mesh glass beads. Mobility ratiosfrom 1 to 9.3 and displacement velocities from 0.05to 0. 2 ft/hr were used in these runs. The effect ofmodel packing will be revealed by discussing theeffects of the other variables, such as distance,mobility ratio, and velocity, upon fingering in theopen model, followed immediately by a discussionof the particular variable's effect in the packedmodel.VISCOUS FINGER REPRESENTATION

    The progress of each displacement run was recorded on 16-mm. movie film, as described in theprevious section. In order to determine th e effectof the variables upon the degree of fingering, it isaecessary to characterize fingering by severalparameters. An exact quantitative description ofthe viscous fingering taking place during the runsdescribed is difficult, if not impossible. However,several methods were devised in this study to givea fair representation of the amount of fingeringwhich occurred during these runs in a form whichwould be useful in predicting miscible slug requirements in a miscible slug displacement process.A very rapid, but inadequate method of representingthe degree of fingering is to determine the percentage of the area of th e model swept by displacingfluid behind the point of the longest finger. For acompletely linear front, which would exist in th eabsence of viscous fingering, th e area swept wouldbe 100 per cent. Table 1 reveals that the sweepefficiency behind the longest finger in the openmodel studies varied from 100 to 53 pe r cent as themobility ratio was increased from 1 to 91. Similarly,Table 2 reveals that the sweep efficiency realizedwhen th e model was packed with 80-mesh glassbeads ranged from 100 to 64 per cent while th emobility ratio ranged from 1 to 9.3. It becomesimmediately apparent that the fingering is greaterin the packed model than in the open model for thesame mobility ratio.

    The most useful representation of the viscousfingering was obtained by a statistical procedure.Three or four frames were selected from the 16-mmmovie film for each run which would show th e frontfor three or four different average positions in themodel. These frames were projected on large sheetsof paper, and tracings were made of the outline ofthe front. The mean frontal position for each of thetracings was determined by integrating the area ofthe displacing fluid and converting this area to arectangular shape corresponding to the planar frontalshape. Next, th e model width was divided into 20equal-width slices, or elements. A mean frontalposition wa s determined in each of these elements.Th e distance between the mean frontal position ineach element and the mean frontal position wa smeasured for each element and is referred to here-

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    TABLE 1 - SWEEP EFFICIENCY BEHIND LONGESTFINGER IN OPEN MODEL

    Mobility Ratio(J. Displacedp. Displacing Displacement Vel. Avg. % Sweepft/hr (behind longest fi nger)

    13102091

    0.2 - 3 1000.1- 4 810.1 - 4 750.1 - 2 600.1 - 2 53

    TABLE 2 - SWEEP EFFICIENCY BEHIND LONGESTFINGER IN 80-MESH BEAD PACKED MODEL

    Mobility Ratio(J. Displaced Avg. % Sweepisplacement Vel.p. Displacing ft/hr (behind longest finger)

    1 1002 0.06-0.2 815 0.06-0.24 799.3 0.06-0.10 64

    after as "frontal distortion". The values of thefrontal distortion in each of the 20 elements werearranged in order from the maximum distortion downto the minimum. Distortions could take either positiveor negative values. However, absolute values wereused in this arrangement. Each value of frontaldistortion in this list was then given a numberwhich gave the percentage of the total front havingdistortions equal to or less than the value of thatfrontal distortion. These values would range from5 to 100 per cent. Fig. 2 shows a plot of this information on probability paper for mean frontal positionsof 7.9, 15.0, 22.3, 30.0 and 38.4 inches obtainedfrom a run in which the mobility ratio wa s 3 to 1,th e velocity was 4 ft/hr, and th e model was open.The data give a fair approximation to straight linesat the various frontal positions, lending some support or at least usefulness for this method of fingercharacterization. It wa s found that the frontal distortions were normally distributed for each run,regardless of frontal position, velocity, mobilityratio or packing. .

    ~ , f ' - /...1.1 1RUN NO. 21 __ -

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    3 and 4 it will be noted that the fingers began afterth e front had moved approximately 1 in . from theinjection head. This observation is important inconsidering a method for predicting the finger'slength for any mean frontal position in the modelsince it is necessary to know the point in th e modelat which th e fingers originated to make a prediction.This is a parameter which will never be known in afield miscibledisplacement.A third observation ca n be made from Fig. 4. Th efrontal distortion increased linearly with frontalposition until a mean frontal position of 22.5 in. isreached. Beyond this, the rate of growth appears tobe decreasing. This phenomenon was exhibited inmost of the runs carried out in packed models andmay be explained by retardation of finger growthdue to diffusion or some other microscopic mixingfactor which begins to have a noticeable effectafter the front ha s moved 2 ft.

    The effect of initial finger shape and frequencywas studied by making a series of three runs usinga ~ i n . square wave injection head. A comparisonof these results with those obtained using the planarinjection head is shown in Table 3 and reveals thatth e initial finger shape did not affect finger growthrates.MOBILITY RATIO EFFEC TS

    Th e effect of mobility ratio upon viscous fingerlength or frontal distortion factor is revealed inFigs. 5 and 6 obtained from runs in the open andpacked model. Increased mobility ratio increasesthe rate of growth of the fingers for both cases.However, the effects for th e open and packed modelsdo not take the same mathematical form. Th e frontaldistortion factors for the open model increased asthe log of the mobility ratio; whereas, the frontaldistortion factors obtained in th e packed modeltended to increase proportionally to a number totheM power where M is the mobility ratio.Other data obtained for the two models and variousvelocities show th e same trend. This would indicatethat, at least in th e limit studied, the character ofthe porous media ha s an effect upon the rate ofgrowth of the viscous fingers.

    2.0,.0

    ()ISPLACEMENT VELOCITY I Vtl f t l l l r0I.

    04 --- ::--I-"' 0 ~ 9- ---- ..;.. ---- _ i ~ ~ :..--- .- --------

    ~ ~ ~ ~. ~ __: ~ .-::::::; ~ .0 .09 .. 2 ... '0 10 40 eo .0 1000MOBILITY RATIO 1M" po DISPLACED/", DISPLACING

    FIG. 5 - EFFECT OF MOBILITY RATIO ON FRONTALDISTORTION FACTOR FO R A DISPLACEMENT VELOC

    ITY OF 1 FT HR IN AN OPEN MODEL.JUNE, 1963

    DISPLACEMENT VELOCITY EFFECTIt was found that displacement velocity had very

    little effect upon th e rate of viscous finger growthin the open model when increasing it from 0.5 to4 ft/hr. Although small, the effect of increasedvelocity was to increase the rate of growth of theviscous fingers. Fig. 7 shows the effect of displacement velocity upon frontal distortion factors for thepacked model. The effect of increased displacementvelocity was to increase the viscous finger growthas represented by the frontal distortion factor. Thefrontal distortion factors increased linearly withthe log of the velocity for al l packed model runs.

    Fig. 8 shows a direct comparison of frontal dis-tortion factors (95 per cent parameter) for the openand packed models for mobility ratios from 2 to 9.3and velocities used during th e two different sets

    .S;.....!:.J=:!i'"soJcII:II.

    2.0

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    02

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    .04

    .02 I

    D I I S P L A C ~ M E N i VELbCITY !D. lf t / l r = ft/JI . /"/ "~ r I P ~ ~ 14/"~ I " " " " : ; f/:/"~ Q x / ~ ./

    ~ ~ .. ~ ~ ~ ~ ,h ; ( " ~ ~ ~ t / V t P )i f,..-1'" of ~ V V . . /I ~ ~ ~ v-::'/V ~'- ' .,..,. f--""/ V V / ' V /I'(A-- .....- / / ,.-.... / , /. ,; r V ./~ ~ ,/ /"~ / ' V, /0& ./0& / ' ./,,,! / ' ,.,t ..... /

    2 3 4 5 6 7 B 9 10 IIMOBILITY RATIO, M=u DISPLACED/u DISPLACING

    FIG. 6 -EFFECT OF MOBILITY RATIO UPON FRONTA L DISTORTION FO R A DISPLACEMENT VELOCITYOF 0. 1 FT/HR IN AN 80-MESH BEAD PACK MODEL.0 .6 . - - - , . - - - , - , - - - - - , - - - ,

    '"0.'f

    Ii ' 0c'" - 0 000' 0000

    .06.0 .10 .20DISPLACEMENT VELOCITY, f t /hr

    FIG. 7 - EFFECT OF DISPLACEMENTVELOCITY ON FRONTAL DISTORTION FO RA MOBILITY RATIO OF FIVE IN AN 80-MESH BEAD PACK MODEL.

    141

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    TABLE 3 - FINGER GROWTH IN OPEN MODEL AS AFFECTED BY MODEL FRONTAL INJECTORMobility

    RatioDimensionless Frontal Di stortion Faetor*

    fl DisplacedfJ. Displacing

    10

    DisplacementVel.ft/hr

    Planar(smoothed)0.44

    80%Square(average)0.48

    Planar(smoothed)

    0.40

    90%Square(average)0.35

    Planar(smoothed)

    0.34

    95%.Square(average)

    0.30*D/L Distortion, in./mean frontal distance from point fingering began.

    of runs. These show that the magnitudes of thefrontal distortion factors were approximately equalwhen compared without regard to velocity. However,i f the two sets of data were extrapolated to equalvelocities, the packed model would exhibit higherrates of viscous finger growth than the open model.MISCIBLE SLUG FINGERING

    A series of miscible slug runs was made to determine th e degree and nature of fingers exhibitedduring a miscible slug displacement process. Theseresults were used in an effort to devise a methodof prediction for breakthrough of the displacingfluid through the slug and into the displaced fluid.Fig. 9 shows a series of four frames taken from amiscible slug run in th e open model. These picturesshow that the front between the slug and th e dis-placing fluid assumes essentially the same shapeas that of the interface between the slug and th edisplaced fluid. It ca n be seen that breakthrough ofthe displacing fluid into displaced fluid occurredwhen the longest finger in th e slug-displacing fluidfront reached th e same position as the longest finge r in the slug-displaced fluid front.

    Given th e correlation of the effects of distance,mobility ratio, and displacement velocity discussedabove, one might assume that breakthrough couldbe predicted by applying this correlation to th e dis-placed fluid-slug contact to predict the longestfinger position and the position of the longest fingerin the slug-displacing fluid front; when these havebecome equal breakthrough ha s occurred. Thesepredictions would be based upon the velocity ofdisplacement and upon the mobility ratio betweendisplaced fluid and slug fluid for it s longest finger,and between slug and following fluid for th e pursuing

    1.0

    c.... o.e..

    ! o.s-"z2t-:!i 0. 4t-..is...Ct-i!i o.

    c.i--,- . t : ~ ..-.7'~ .. _9.3, OPE=.-.r -I r-l-. r ~ - I > - ~ ~ T....5" I> ......... T2 _ " ' ~ ~ " ' C ' f . C11i ~ f ' E . " ~ ~ -...;1 I Imo.04 .os.oe J .2 .4 .8.81.0 4

    VELOCITY." H/hrFIG. 8 - COMPARISON OF PACKED AND OPENMODEL RESULTS.

    .142

    finger. These predictions would be based upon the99.9 per cent point in the correlation.At this point, it becomes obvious that as long asthe mobility ratio between the displaced fluid andthe slug is greater than the mobility ratio betweenthe slug and the displacing fluid, breakthroughwould not be anticipated, based upon th e correlationsdescribed. In the slug runs carried out in the openmmel, the mobility ratio between the displacedfluid and the slug was 10, while the mobility ratiobetween the slug and the displacing fluid was 5.Breakthrough did occur in several of these runs.Therefore, the effective or apparent mobility ratiosoperating at the two interfaces must be differentfrom the actual ones, and th e one between the slugand the displacing fluid must be th e largest in thosecases.

    Plots of the frontal distortion for these frontswere made and the rate of growth of the distortion

    RUN No. 26 MI=IO.O-I M2=5.1-1 V= I FtlHr

    RUN No. 26 MI=IO.O-I M2=5.1-1 V=IFtlHr

    RUN No.26 MI=IO.O-I M2=5.1-1 V= 1FtIHr

    RUN No.26 MI=IO.O-I M2=5.1-1 V=IFtlHrFIG. 9 - FINGERING FO R A MISCIBLE SLUG IN ANOPEN MODEL.

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    TABLE 4 - APPARENT MOBILITY RATIOS REQUIREDTO PREDICT FINGERING EXHIBITED DURING MISCIBLESLUG DISPLACEMENT IN OPEN MODELMobi I ty Ratio Mobility Ratia Over-allSlug fLl/fL2 fL2/fLa MobilityRun Length RatioNo. (in.) Act. Apparent Act. Apparent fLl//13

    27 5 10 12 5 20 5028 7 10 35 5 59 5026 10 10 18 5 33 5029 12 10 31 5 17 50

    Average 10 24 5 32 501 = efers to original fluid being displaced2 = efers to slug fluid displacing original fluid3 = efers to fluid displacing slugwith distance was determined. Using this rate ofgrowth (for the 95 per cent point) it was possibleto use the correlation obtained for the open modelto determine the apparent mobility ratio necessaryto cause the frontal distortion actually exhibited.Table 4 contains results of these calculations forslugs of 5-, 7-, 10- and 12-in. in length. It will benoted that in al l of these runs, the apparent mobilityratios between displaced fluid and slug were muchgreater than the actual ratios. The average for al lof th e runs was 24: 1, rather than the 10 whichactually existed. Similarly, the mobility ratios between the slug and displacing fluid gave apparentvalue s ranging from 17 to 59, with an average of32, compared with the actual value of 5. Th e overal l mobility ratio between the displaced fluid andthe displacing fluid was 50: 1. I f the slug size orwidth were zero, the data would represent an actualan d apparent mobility ratio of 50: 1. Th e fact thatapparent mobility ratios were between 5 and 50 forth e front between the slug and displacing fluid, andbetween 10 and 50 for the front between th e displaced fluid and the slug, shows that with the slugsizes used the displaced fluid did have an effect onthe slug-displacing fluid front and that the displacingfluid di d have an effect on the displaced fluid-slugfront. It would be anticipated that the apparentmobility ratio required to predict finger growth in amiscible slug process would be a function of sluglength. However, the data obtained are not sufficiently accurate to show any definite trend.Similar results were obtained for miscible slugruns carried out in the packed model. The evaluationof the packed model miscible slug runs is shown inTable 5. It will be noted that in al l runs exceptRun 46, th e apparent mobility ratios between thedisplaced fluid and the slug fluid were al l greaterthan th e actual ratio of 5. The average ratio for thethree runs was 6.4. Th e apparent mobility ratiosbetween the slug and the displacing fluid rangedfrom 4.5 to 10, whereas the actual value was 2.Th e average for th e four runs was 6.9. The sluglengths for these runs ranged from 4.5 to 29 in .Apparently, these slugs were not long enough toprevent the displacing fluid from having an effectupon the displaced-slug front and the displacedJUNE, 19,63

    TABLE 5 - APPARENT MOBILITY RATIOS REQUIREDTO PREDICT FINGERING EXHIBITED DURING MISCIBLESLUG DISPLACEMENT IN 80-MESH BEAD PACKED MODEL

    Mobility Ratio Mobil ity Ratio Over-allSlug MobilityRun Length fLl/fL2 /12//13 RatioNo. ~ Act. Apparent Act. Apparent J!:J!J!:L45 4.5 5 7.4 2 4. 5 1044 7.0 5 6.0 2 10.0 1043 14.0 5 5.8 2 7.3 1046 29.0 5 1.5 2 5.8 10

    Average 5 6.4* 2 6.9 10* Disregarding Run 46, which is obviously in errorNote: 1 = efers to original fluid being displaced2 = refers to slug fluid displacing original fluid3 = refers to fluid displacing slugfluid from having an effect upon the slug-displacingfluid front. These results do not allow any definiteconclusions with regard to a method for calculatingthe apparent mobility ratio required to predict fingergrowth and breakthrough in the miscible slug process.Considerable additional study to determine such arelationship would be very valuable in predictingth e success of a miscible slug displacement process.

    CONCLUSIONSViscous fingering occurs during a miscible dis

    placement involving unfavorable mobility ratios.The initial formation of th e viscous finger resultsfrom some local heterogeneity in the model. Th efollowing are conclusions reached from a study ofth e results of displacements carried ou t in an openHele-Shaw model using displacement velocities from0.1 to 4 ft/hr and mobility ratios from 1 to 91 andin a packed Hele-Shaw model containing SO-meshglass beads using displacement velocities from0.06 to 0.2 ft/hr and mobility ratios from 1 to 9.3.1. Th e lengths of 20 elements of the front containing viscous fingers are distributed normallyabout th e average position of the combined elements,regardless of front position, displacement velocity,mobility ratio, or packing. This normal distributionserves to characterize th e viscous fingers obtained.

    2_ Th e length of the viscous fingers, as obtainedfrom th e normal distribution of elements, increasedlinearly with the distance displaced relative to thepoint at which th e fingers began.3. Th e point at which fingering began could notbe predicted and was sometimes several inchesfrom th e point of injection with either positive orapparently negative coordinates.

    4. When using a packed model, the rate of growthof th e fingers with distance appeared to decreaseafter 2 to 3 feet. This indicates that microscopicdispersion or diffusion may be becoming effectivein diminishing the viscous fingering effect in thismodel.5. The rate of growth of the fingers was not alteredby altering their initial shape and frequency.

    6. Th e length of the viscous fingers (or frontaldistortion) increases with displacement velocity for

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    both th e open and packed models. However, theeffect of velocity was much greater for the packedmodel studies.

    7. Th e effect of increased mobility ratio is toincrease the length of the viscous finger (or frontaldistortion). Th e form of the dependence of fingergrowth upon mobility ratio was different in the openmodel from that in th e packed model, indicatingthat the packing is an important variable.

    8. Th e apparent mobility ratios exhibited betweenthe displaced fluid and miscible slug and betweenthe miscible slug and displacing fluid during miscible slug displacements are much greater than theactual mobility ratios, regardless of whether thedisplacement is carried out in the onen or packedmodel. In other words, fingering which occurs duringa miscible slug process is much greater than wouldbe anticipated from consideration of mobility ratiosat either of the fronts.

    9. Results obtained during the early stages of amiscible displacement in the model were of littleuse in predicting the fingering which would occurthroughout th e length of the 4-ft model. This is dueto the unpredictability of th e point of finger conception. Cortect scaling of a model for the predictionof viscous fingering in a field would require scalingof th e point of viscous finger initiation. Therefore,results which have been published for fingeringwhich occurs in a 9-in. or 12-in. or any other small

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    model are not reliable in predicting finger size in abigger model. Furthermore, there were indicationsthat the rate of finger growth was diminishing towardthe end of the present model displacement, meaningthat the results obtained in this model should notbe extrapolated beyond present model dimensions.

    REFERENCES1. Habermann, B.: " Th e Efficiencies of Miscible Dis

    placement as a Function of Mobility Ratio", Trans.,AlME (1960) Vol. 219, 264.2. Lacey, J. W., Faris, J. E. and Brinkman, F. H. :"Effect of Bank Size on Oil Recovery in th e HighPressure Gas-Driven LPG-Bank Process" , Jaw.Pet. Tech. (Aug., 1961) 806.3. Weinang, C. F. and Ling, D.: U.S. Patent No.

    2,867,277 (Jan. 6, 1959).4. Chuoke, R. L. : "Character of th e Equilibrium of

    Stratified Viscous Fluids in Slow Flow ThroughUniform Channels" , Bull. Amer. Phys. Soc. (Feb.24, 1956) Sec. n, Vol. 1.

    5. Scheidegger, A. E. : "Fluid Instabili t ies in PorousMedia", Th e Physics of Fluids (1960) Vol. 3, 94.6. Taylor, G. an d Saffman, P. G. : "A Note on th e Motion

    of Bubbles in a Hele-Shaw Cell an d Porous Medium",Quarterly Jour. of Mech. and Appl. Math. (Aug., 1959)xn, Part 3, 265.

    7. Saffman, P. G. an d Taylor, G. : "The Penetration ofa Fluid into a Porous Medium or Hele-Shaw CellContaining a More Viscous Liquid", Proc. Roy. Soc.(1958) A245, 312. ***

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