The Development, Testing, and Application Of a Numerical Simulator for Predicting Miscible Flood...

8/13/2019 The Development, Testing, and Application Of a Numerical Simulator for Predicting Miscible Flood Performance

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3yA7~The Development, Testing, and ApplicationOf a Numerical Simulator for PredictingMiscible Flood PerformanceM. R. Todd, SPE-AIME, Shell Development Co.. ----W. J. Longstafi, SPE-AIME, Sheij canada Ltd.

IntroductionWhen designing or evaluating a miscible displace-ment project, one would like to be capable of accu-rately forecasting reservoir performance for a varietyof operating conditions. In the past, physical model *studies have provided the buiicof the data used in *these evaluations. Unfortunately, proper scaling ofmiscible displacement is difficult to obtain and modelconstruction is time-consuming and expensive. Con-sequently, considerable effort has been devoted inrecent years to the development of numerical reser-voir simulators capable of predicting miscible floodperformance. In this paper, a method is ciescfiDecifor modifying an existing three-phase simulator sothat it may be used to simulate miscible flooding. Themodel employed is unique in that the essential char-acteristics of miscible displacement are describedwithout reproducing the fine structure of unstablefrontal advance. This feature is of particular im-portance as it allows the reservoir to be representedby a fairly coar e numerical grid.We shall describe here three- and four-componentversions of the miscible flood simulator. In the three-component version, wetting (water) and nonwetting(hydrocarbon) phases are considered with two-component miscible flow of oil and solvent in thehydrocarbon phase. With the four-component ver-sion, slug as well as continuous solvent injection maybe examined. The validity of the miscible simulatoris demonstrated by comparing predicted performancewith experimental results from both linear and areal

displacement in laboratory models. Predictions arealso compared with reported results of miscible dis-placements carried out in a shallow, water-bearingreservoir. Finally, possible application of the simu-,. . . 3..-.-. .,-J L.. :.. ---- :- AL- -. ..-1..-.:-- -47Ia[or ]s aemonsmdteu Dy IL> usc m LIIC cviiIua LIuII UIalternative flooding configurations and operationalpolicies for a field example.Simulator RequirementsA common characteristic of miscible recovery proc-esses is unstable frontal advance, in the form of eitherviscous fingering or gravity tonguing. Tlnese itistabiii-ties are the natural result of the highly adverse vis-cosity ratio and large density difference that generallyexist between an oil and the displacing solvent. Fig. 1,for example, depicts the swept zone at solvent break-through for a five-spot pattern flood as actually ob-served in a Hele-Shaw (parallel plate) model. Themodel is simulating the miscible displacement of oilby solvent at an adverse mobility ratio of about 15.The effect that an unstable frontal advance exertsupon oil recovery might be considerably worse thanactually observed if it were not for the fact that sol-vent disperses in the oil, which renders the effectiveviscosity and density differences less extreme thanthose of the pure components. Thus, a successfulmiscible-flood simulator should allow for the possi-bility of unstable frontal advance and must describethe dispersion phenomenon, at least as related to thedetermination of effective fluid properties.

Here is a method for modifying an existing three-phase simulator so that it may be usedto forecast miscible flood performance. The simulator is capable of modeling theessential features of miscible displacement while leaving the fine structure oj unstablemiscible flow unresolved, making it possible to represent the reservoir by a jairlycoarse numerical grid.


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Past simulation effort has been directed toward thenumerical solution of those transport equations thatspecifically include the mutual dispersion of misciblefluids during flow in porous media. -9 Unfortu-nately, attempts at solutions based on these formula-tions have met with only limited success. This is partlybecause standard finite-difference techniques, such asthose used in immiscible reservoir simulators, intro-duce a numerical dispersion into the solution, whichtends to mask or distort the effects of the true physicaldispersion of the solvent in the oil. 3A more formi-dable obstacle, however, is that rigorous formulationof the problem requires that the fine structure of theunstable frontal advance be reproduced in detail bythe simulation; otherwise, mixing cell effects will con-siderably dampen the simulated flow instabilities, re-sulting in optimistic forecasts of oil recovery per-formance. This requirement necessitates the use ofimpractically large numerical grids for simulating allbut the simplest of reservoir systems. For example,the grid structure overlying the displacement shownin Fig. 1 contains 121 grid blocks. Since this is onlyone five-spot, this grid would be considered fairlylarge with respect to the requirements of a full-scalereservoir simulation; however, the grid is far toocoarse to reproduce in detail the viscous fingeringshown in the figure. Since the required size of a gridblock is on the order of that of a finger width orsmaller, about 10 to 50 times this number of blocksmight be required for accurate numerical simulationof th~~ ~~~ob fivr==nmt Aicl. nr.rn-+b. - ...- Oyw. u&oylcL&ti,l, tiUL.We have found that the effects of dispersion oneffective fluid properties can be adequately repre-sented by means of an empirical model. Conse-quently, we find that accurate simulation of miscibledisplacement performance can be obtained withoutreproducing the fine structure of the flow. This, then,~z~.~c f,=ncihlta +h- mothc.mot;n.ml .: . . ...1-+.-- -c t..llu .wbAaAuAe .Lti .L&a L1ltillla L1bal mL1lula L1uIl U1 lull-

Fig. lSwept zone at solvent breakthrough for a five-spotpattern flood as observed in a Hele-Shaw (parallel plate)model. M ~ 15.

scale miscible flooding projects.Model FormulationIn this section we develop a model for two-componentmiscible flow for use in a three-component reservoirsimulator. A four-component model, developed sothat solvent slugs as well as continuous solvent in-jection might be investigated, is described in theAppendix. As the miscible formulations are merelyvariants of immiscible models, the actual solution ofthe conservation equations will not be discussed.Eqs. 1a through lC are the familiar expressionsfor the superficial velocities of water, oil, and gasused in three-pha~e re~en~ir ~i,mu ater~ for the cal-culation of component fluxes.

kk,,cu,~ = (VP,C flogVD), . . (la)/4. kk,oMO= (VP. pogVD) , . . (lb)

PO

Ug = - (VP, f%gv~) . . . (lC)1%

Here, the relative permeability, k,,,, is a single-valuedfunction of the water saturation, S,,, as is the differencein phase pressures, P,.t~O= P. P,,,. Similarly, the gasrelative permeability and the capilla~ pressure, P~,,~= pg PO,are single-valued functions of the total li-quid saturation, (S,c + SJ. The oil relative permeabil-ity is normally modeled as a function of both the watersaturation and the total liquid saturation. These func-tions of saturation are assumed to be known and aregenerally entered into a three-phase immiscible simu-lator in tabular form.For our purposes, gas is identified with a selectedmiscible drive agent (or solvent). Hence, oil and gasare considered miscible components of the non-wetting phase. Consequently, the capillary pressureE...+ha.-vCOOs zerc. J w L1lGiliGi~, Ske the i lKN~in~ilt Of Ori~component through a porous medium is not impededby the presence of other components in the samephase except for a reduction in the area available forflQw,each of the rnkcih e cmnponents asswnes a frrx=tion of the nonwetting phase relative permeabilityequal to its volume fraction in that phase. Thus, therelative permeabilities of the miscible oil and gas maybe determined as follows:k,o+. k,n, . . . . . .n

(2a)1 s,firg=-= km,. . . . . . . .s. @bj

where in this case S. = S,, + S~ is the nonwettingphase saturation and k, = k,. (SJ is the imbibitionr~ ati~e ni=rmt=ahilitv nf t~e nnnmmttinm nh. s=.~-.. ----... -J . . . ...* ..w. bu,5 ~,,= b.For the development of the viscosity model, weconsider the miscible displacement of oil by gas (sol-vent) in a two-diinefi~icmal system. It is assumed thata rectangular grid such as that overlying the displace-ment shown in Fig. 1 has been prescribed for nu-merical simulation. Fig. 2 depicts possible fluid dis-tributions in an arbitrary grid block during frontalpassage. The clear areas represent zones were the oilJULY: 19?2


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and solvent have been mixed by dispersion. Whetherthe displacement is stable or unstable, we must choosefor the viscosity and density of each component asingle value that reflects the effective fluid propertyaveraged over the entire grid block. These values arerequired for the numerical calculation of componentfluxes with Eq. 1.If the rate of dispersion of solvent and oil weresuch that the size of the dispersed zone is large withrespect to the size of the grid block, the misciblecomponents could be considered completely mixedin the block. Both components would have the sameeffective density and viscosity, as determined by ap-propriate mixing rules. However, if the dispersionrate were so low that the mixed zone is negligiblewith respect to the size of the block, the effectivefluid properties of solvent and oil would be those ofthe pure components. It is reasonable to expect thatthe actual effective fluid properties will fall some-where between these mixing limits. Eqs. 3a and 3bdescribe the effect of partial mixing on effective vis-

cosities in terms of an empirical model suggested byK. S. Lee and E. L. Claridge of Shell Develop-ment Co.POe = /LO*w/bLu , . . . . . . . (3a)I.% = /%=pn,w. . . . . . . (3b)

A value of the mixing parameter, O, of zero corre-sponds to the case of a negligible dispersion rate.Note that the effective viscosities equal those of thepure components. A value of u = 1 corresponds tocomplete mixing within the grid block. Here, theeffective viscosity of each component equals the mix-ture viscosity, which in our model is determined bythe %-power fluidity-mixing rule, Eq. 4 or Eq. 4a.(i) WY (WY 9 ~ 4)/( s, sl-k = l-w% _Opo +JLOS,j s. U)4. (4a)

An independent model has not as yet been devel-oped for evaluating the effective densities of themiscible components. However, a density model canbe derived that is at least consistent with the vis-cosity model. The only assumption is that mixingis ideal.If the gas and oil were completely mixed in a two-component miscible system, they would have the--- A-.; +., o. rlo+-.-~--~ h., Gn UkLWLInLLLL. U Uy J+. d.

F%= PO*$+P9 ;... . . . 4 (5)n uHowever, as with the component viscosities, we willassume that the fluids are only partially mixed or,alternatively, that the fluids are completely mixed,but at the effective saturation fractions that wouldresult in the effective viscosities derived by Eq. 5.To determine the necessam effective saturationsin the miscible tluids, thepressed alternatively as

Pm = P../[

M%

mixing rule (Eq. 4) is ex-

1MM1 ; , . j

n

where M = mobility ratio p.~/;Lg,and Sg/sn = 1 S0/S,,. Rearranging, we have

As Eq. 7 is valid for any mixture viscosity, it is validfor mixing at the effective fractional saturations; thus(-)o M% (po/poJ$~s,, g. = M?A_l Y . . . . (8a)()

so _ M% (p./poJ~zoe M~_~Y.. (8b)

where the subscripts ge and oe are used to denoteeffective fractional saturations in the gas and oil, re-spectively. The effective densities may now be ex-pressed by

oe=O i_)Oe Pg[1 (*)., y ga[s. )Pw=P.~~) Pv [l-(*) ]. @Jn ge II ,e

Note that Eq. 8 is indeterminate for a unit gas/oilmobility ratio. In other words, no information can beobtained from the mixing parameter model for unitviscosity displacements. For these cases, we suggestthe following alternative mixing model for the effec-tive densities:

FLOW DIRECTION-

a. STABLEDISPLACEMENT

~DISPERSED ZONEFlnW nlRFf TinN------- .. . . . . . . .

b. UNSTABLEDISPLACEMENT

~ DISPERSED ZONESFig. 2Schematic representation of solvent displacing oilin a grid block.


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poe=(l *J)po+~p,,i , . . . . . (lOa)pve=(l o))pg+copm . . . . . . (lOb)In an immiscible simulator, values of water, oil,and gas relative permeabilities, viscosities, and den-sities, together with water/oil and oil/gas capillarypressures are normally taken from tables of inputdata as mentioned previously. For miscible floodsimulation, these values are modified by Eq. 2 foroil and gas relative permeabilities, Eq. 3 for viscosities,and either Eq. 9 or Eq. 10 for densities.The only additional input variable required is thevalue of the mixing parameter, o). In addition, theinput values of the capillary pressure between the oiland gas, PCOO,must be set equal to zero.This completes the derivation of the three-com-ponent model. An analogous derivation for a four-component model, intended for the study of miscibleslug flooding, is presented in the Appendix.

Program Testing Simulation ofExperimental Datal-h. -.,4,.11llG llluuCL di%~tibd iii tk ~~~@di~fj ~i?di~rl forevaluating apparent miscible fluid viscosities, den-sities, and relative permeabilities has been incor-porated into Sheii Development Co.s compressible,three-phase, three-dimensional, black-oil reservoirsimulator. To test the efficacy of the resultant misciblesimulator and to determine the appropriate magnitudeof the mixing parameter, ~),simulator predictions werecompared with results from physical model experi-ments reported in the literature. -I~In addition, pre-dictions were compared with reported results ofmiscible floods carried out in a shallow, water-saturated reservoir.] All results are for two-compon-ent, miscible displacements in horizontal systems

I .7=2//QJ , 1 2

EXPERIMENTALCOMPUTED

LINEARM=B6

u0 1 10 1.0 2.0PORE VOLUMES OF SOLVENT INJECTED

Fig. 3Linear miscible displacement performancefor M = 86.JULY, 1972

using fluids of equal density.Simulation of Physical Model PerformanceFig. 3 depicts the recovery performance for a lineardisplacement of oil by solvent at an oil:solvent vis-cosity ratio of 86. The experimental data are thereported results of Blackwell et al.fi for a horizontal,two-dimensional linear displacement in a sand pack.The computed results were determined using one-dimensional simulations. Grid requirements will bediscussed in a later section. Note that the computedresults are quite sensitive to the value of the mixingparameter, and that good correspondence betweenthe experimental and computed recovery perform-ance is obtained with ., = 2/3.The mixing parameter model would be of limitedutility if it had to be calibrated for each displacement;however, as shown in Fig. 4, we have found that onevalue of w can be used to predict oil recovery per-formance for this system over a wide range of un-favorable mobility ratios.

To evaluate the areal performance of the simulator,computed resuits were compared with experimentaldata for displacements carried out in a quarter five-spot symmetry element. As shown in Fig. 5, thepredicted oil recovery performance is in good agree-ment with the reported experimental data of Lacey[1 ail; for mobility ratios of 10. 40, and 85. Thecomputed results were obtained using a two-dimen-sional grid and a mixing parameter of 2/3.Fig. 6 also depicts the recovery performance fora five-spot displacement, but all data shown here arefor a mobility ratio near 10. Note that the recoverycurves determined from the three experimental in-vestigations are considerably different. Lowest pro-duction performance reported is for a Hele-Shaw

T 1 -Y M=86/M=150r

EXPERIMENTALCOMPUTED

{,

LINEAR(lJ= 2/3

111.0 2.0PORE VOLUMES OF SOLVENT INJECTED

Fig. 4-Linear miscible displacement performance forvarious unfavorable mobility ratios.877


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(parallel plate) model. Performance is better for theunconsolidated bead-pack model,l; and is best foran artificially consolidated sand pack.1~ The variationin performance is thought to result from differingrates of transverse dispersion of the solvent in the oilfor the three experimental models. Dispersion is held~l~Q~tfi th~ rn.ojecu]ar eve in the He e-Shaw model..., .- . ..-With increasing pore complexity, convective disper-sion increases. This increase results in wider mixed~--.~~rles ~ild h~ii~~ idid~d t?4LGW U V & V I~&&.yT- f.+1..l ..; CP* it., TQt;rl, C&U,bringing about improved recovery performance. Inthe numerical simulator, the influence of transversedispersion on effective fluid properties is specified bymeans of the mixing parameter. Thus, it is in accordwith the reported experimental results that the com-puted oil recovery performance improves with in-creasing values of wNote that all of the experimental data shown inFig. 6 are bracketed by the computed recovery curvesfor the mixing limits of ~ = O and ,,, = 1. Furthernote that if we do not take into account the mutualdispersion of solvent and oil as, for example, whenattempting to use an immiscible reservoir simulatordirectly to forecast miscible flood performance thepredicted oil recovery would probably be pessimistic,as indicated by the computed curve for ~,,= O. Onthe other hand, if we use a simulator that explicitlyincludes diffusive mass transfer and do not reproduceunstable frontal advance in the simulation, the pre-dicted oil recovery will probably be optimistic, asindicated by the computed curve for o,= 1.We he iev~ the Hele-Shaw model to be most closelyscaled to reservoir conditions for miscible displace-

For excellent descriptions of the requirements and tiiiiicu itiesassociated with the construction of scale models for studyingmiscible displacements. JIease refer to tJa Ders by Pozzi and Black.well and by Heller. ]g

i

. EXPERIMENTAL

COMPUTED7 5-SPOT~ . 2/3r1 I

1.0 2.0E VOLUMES OF SOLVENT INJECTED

Fig. &Miscible displacement performance in a confined,five-spot pattern for various unfavorable mobili~ ratios.878

ment;* consequently, we believe that the appropriatevalue of OJ to be used for miscible flood simulation ison the order of 1/3.Simulation of Reservoir Performance The Chandler ExperimentsAs a further test of the miscible simulator, the Chand-ler experiments were simulated. These experimentswere miscible floods carried out in a confined five-spot~Q@lJ~~~~Q~ ~~ ~ Sha]]ow water-bearing reservoir.Adjustment of the brine viscosity with Dextran al-lowed the investigation of two-component misciblefloods having mobility ratios of 0.1, 1.0, and 10.0.The experiments were conducted by Esso ProductionResearch Co. and are fully described in Ref. 15. TheIithology of the Chandler sand is described in Refs.16 and 17. Note that although thickened brines wereactually used for the Chandler experiments, forclarity the resident fluid will be referred to here asoil and the invading fluid as solvent.An initial investigation indicated that introducinginto the simulator the measured areal variation oftransmissibility at Chandler would have little effecton the computed flood performance. Consequently, auniform quarter five-spot symmetry element was usedfor the remainder of the study.Fig. 7 shows the oil recovery performance for theunit mobility ratio displacement in the Chandler reser-voir. Note that because of the vertical reservoirheterogeneity, the observed recovery performance isconsiderably poorer than that predicted for a homo-geneous five-spot. Our computed recovery curve fora homogeneous reservoir. by the way, is in goodagreement with the analytical solution of Morel-13e~tow,19To include reservoir heterogeneity in the simula-tions, the method of Hearn was used to generate

5-SPOT U=l.ofAJ %3~:1/ 3

QJ, ()

COMPUTEDI A HELE-SHAW MODEL BEAD PACK= CO NS.0Lii2ATEi2 5AiND PACKI I

1.0 2.0PORE VOLUMES OF SOLVENT INJECTEDFig. 6--Miscible displacement performance in a confined,

fivespot pattern for M ~ 10.JOURNAL OF PETROLEUM TECHNOLOGY


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pseudo oil/solvent relative permeability curves (Fig.8) for the permeability distribution of the Chandlersand.; Use of those pseudo relative permeabilitycurves describing Lithotype A, thought to be thedominant lithotype at Chandler, resulted in the bestagreement between the computed and the observedreservoir performance as shown in Fig. 7. The mix-ing parameter model, of course, has no influence onthese results because the viscosities of the oil andsolvent are equal.Fig. 9 shows the experimental and computed oilrecovery performance for the unfavorable mobilityratio displacement at Chandier. H-ere, the pseudorelative permeability curves that describe LithotypeA were used for all three simulations. Use of ,, =1/2 yields the best agreement between computed and-,--- --Jomerveu reservoir perfOMiatlW. lk3G3:i tii~i gdcorrespondence between experimental and computedresults is obtained with (,J= 2/3 when forecastingmiscible flood performance in a laboratory sand packor bead pack model, and that we believe 6) = 1/3is the approximate value to be used for simulatingfull-scale displacements (say 10 to 40 acres/well).Because of the smaii pattern size (0.03 acre/weiij andrelatively low injection rate, scaling considerationsdictate that the performance of the Chandler flood liebetween that of a sand pack model and that of afull-scale reservoir flood. Thus. the value of ,,)= 1/2found here to give the best correspondence betweencomputed and observed Chandler performance seemsreasonable. It should be noted, however, that Hearnsmodel for pseudo relative perrneabilities is strictlyvalid for unit mobility ratio displacements only. It ispossible, therefore, that use of the Hearn method insimulating the unfavorable Chandler flood is in-appropriate.

/HOMO.

A

A+B

EXPERIMENTAL COMPUTED

~HANDl&RM=l

o 1.0 2.0PORE VOLUMES OF SOLVENT INJECTED

Fig. 7Recovery performance for the unit mobility ratiodisplacement in the Chandler reservoir.

Z?:al,.1 A--l:nn+:n -F tha M:. n;hla c;m* l 3+n*A lGIU Appllwluull WI Lll& LrmLci&lul& uxllluLu LwaTo demonstrate possible applications of the misciblesimulator, we examine briefly the prospects for high--pressure gas miscible flooding of a watered-out oilr,, n.nela. rrlmary depietiori, together with a iiigi ii~ siiC-cessful peripheral line drive waterflooding operation,is assumed to have reduced the oil saturation to water-flood residual.Laboratory testing has confirmed that the high--pressure gas will become sufficiently enriched atreservoir temperature and pressure to miscibly dis-piace practically aii of the oii contacted. The gas:oiimobility ratio is 16 and the immiscible gas:watermobility ratio is 70. The viscous: gravity-force ratiofor the displacement is large enough so that gravity-... -.:.J - -c -- d--- --- . . . ..-. -s..+. ..a L1...T.. .-.AA;uvcllluc ul d> uum uut plcxu L d pIUUIGIIi. 111 auul-tion, the reservoir is practically uniform and isotropic.Consequently, simulations were carried out using atwo-dimensional areal grid. A mixing parameter of1/3 was employed.

A peripheral line drive was considered the primecandidate for the flooding configuration because ofthe success of the writer_i300dittgOperadOii. lHOW~V~i,laboratory study has indicated that pattern floodingcould be more efficient for highly unfavorable dis-placements. Consequently, an inverted, confinednine-spot pattern was also selected for study.For the line drive, gas is injected into wells adjacentto the reservoir boundary. and two or three down-stream wells are put on production. The injection ofhigh-pressure gas effects an immiscible displacementof the water injected during the prior watetioodingoperation while reconnecting and banking up thewaterflood residual oil. A large portion of this oil isrecovered as the oil bank is swept downstream past

1.0

LITHOTYPES A680.8 -

k: rs~m$ 0.6 -s HOMOGENEOUSaw RESERVOIRo.u: 0.4 -au~

//0.2 - /

() I I I I Io 0.2 0.4 0.6 0.8 to

OIL SATURATION, soFig. 8-Pseudo relative permeability curves derived fromthe vertical permeabili ty distribut ion ofthe Chandler reservoir.


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the production wells. When production from a givenproducer reaches an economic GOR limit, the wellis shut in and the next well downstream is added tothe line of producers. For an inverted, confined nine-spot flooding pattern there are effectively three pro-ducers surrounding each injector. As these are nearlyequidistant from the injector, all may be expected tosimultaneously produce displaced tertiary oil eventhough their behavior will not be identical.Some results of the numerical evaluation of theseflooding schemes are shown in Figs. 10 and 11. Acomparison of the predicted recovery efficiencies forthe line drive and the confined nine-spot is given inFig. 10. These results show that the nine-spot patternflood is clearly superior, recovering almost 70 per-cent of the tertiary oil before reaching the economiclimit. Predicted recovery for the line drive, on theother hand, is less than 45 percent.Once a preferred flooding pattern has been de-termined, alternative operational schemes may beevaluated using the numerical simulator. For example,continued gas injection in the confined nine-spot untilthe economic limit is reached results in a recoveryof 1.3 million STB of oil in 2,000 days under the fieldconditions as shown by Curve a. Fig. 11. (The pat-tern area originally contained almost 1.9 million STBof tertiary oil.) Blowdown of the reservoir to recovera portion of the gas remaining after the miscible floodmight recover an additional 160.000 STB of oil. ifblowdown is effected through the comer productionwell, Curve c. Alternatively, water injected after1,200 days of gas injection results in only 1 millionSTB of oil recovered, Curve d. The results of varioussimulator predictions can now be evaluated economi-cally to aid in selecting the optimum flood design.

In addition to assisting in the selection of a pre-ferred flooding scheme and operational policy, thesimulator can be further employed for the designand subsequent surveillance of both pilot operations

CHANDLERM=IO1 1

1.0 2.0PORE VOLUMES OF SOLVENT INJECTED

Fig. 9-Recovery performance for the unfavorable mobilityratio displacement in the Chandler reservoir.8R0

and fieldwide expansions.Program LimitationsIn evaluating the applicability of the miscible dis-placement model presented here for a specific reser-voir problem, it is important to recognize that thismodel does not represent an attempt to describerigorously the thermodynamic and transport phe-nomena that determine the details of the local fluidcomposition and flow characteristics. Instead. it is acomputational expedient that allows for the engineer-ing evaluation of miscible flood performance. Evenso, the explicit and implicit assumptions inherent inthe model do restrict to a certain degree the practicalapplication of the miscible simulator. We shall de-scribe those limitations with which we are familiar.The model does not include diffusive flow betweengrid blocks due to spatial gradients of concentration.Consequently, it is not applicable to systems charac-terized by dispersed zones perpendicular to the localflow vector, which are large with respect to the sizeof the grid blocks used for the simulation. This draw-back might be particularly important in modelinghighly stratified reservoirs (with communicatinglayers) using multilayer grid systems. A combinationof the miscible and immiscible simulator analogy de-scribed by Lantzt and the model presented in thispaper might result in a miscible simulator without thislimitation.

-1

GAS lN,C 140N lHVD @0C AR80N PO tf VO LUM EFig. 10-Comparison of predicted tertiary oil recoveryperformances for peripheral line drive and invertednine-spot flooding patterns.

1. I C ONTNUOUS G AS INJC C11ON[b] ~LOWDO WN THRO UG H O RG INAL INJEC TO R[ . BLO WD OWN lM RO UG H C ORN E@ PRO DUC ER-1600 [d) WAIEt INJECTIONFROM1200DAYS: l.) WATtt INJEC1OONRO M 700 DAYS +.o~00 Km IIW Ibm mm 2400lo w, DAM

Fig. 1 lTertiary oil reeovery from an invertedninespot pattern.

00

JOURNAL OF PETROLEUM TECHNOLOGY


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1

The model does not include mass transfer effects;oil and solvent are assumed to be completely miscibleon first contact. This treatment might seem to p,re-clude the use of the miscible simulator for forecastingdisplacements where multiple contacts are requiredbefore miscibility is obtained; for example, in con-densing or vaporizing gas drives. However, so longas the distance required to obtain miscibility is shortwith respect to the well spacing, the simulator shouldbe adequate for determining oil recovery performance.The model inherently assumes that the invadingsolvent has equal chance of contacting all of the oilin a grid block. Consequently, the model may not beapplicable to systems exhibiting strong gravity segre-gation where an attempt is made to represent thesystem with a single layer of grid blocks by partialanalytical integration of the flow equations. 2 If, forthis system. the oil saturation is at waterflood resi-dual when the miscible displacement is started, Eqs.2 through 4 will exaggerate the influence of theoil properties in attenuating solvent mobility. Thiswill result in optimistic forecasts of oil recoveryperformance.Although this paper does not treat the actualnumerical techniques used in a simulator for solvingthe conservation equations. some observations per-taining to the selection of the numerical grid arewarranted. It is well known that the normal nlanifesta-tion of truncation error in finite-difference-basednumerical simulators is a nonphysical dispersion atthe flood front that results in an apparent smearingor degradation of the saturation profiles. For aspecified time-step size the magnitude of the trunca-tion error decreases with decreasing grid spacing andincreases with increasing frontal velocity and in-creasing saturation gradient at the front. Thus, itrequires fewer grid blocks to simulate accurately ahiphlv ~nfav~ra~ e n~~~ilitx, ratin Aic nlac tmm-nt .hnl... C...= . . . . . .-... . ..oy.utiti fifi. b. t {DA,CI,low saturation gradient) than to simulate one whosemobility ratio is near unity (steep saturation gradient).The computed results shown for a linear system inFigs. 3 and 4 were obtained using a one-dimensionalgrid. The results for mobility ratios of 86 and 150exhibited no sensitivity to the grid spacing for gridscomprising 10 elements or more. However, it took40 grid elements to obtain a convergent solution forthe M = 5 displacement. This grid requirement stillcompares favorably with the two-dimensional, 800-point mesh used by Peaceman and Rachford toobtain detailed simulations of linear sand-pack dis-placements. For our quarter five-spot simulations. atwo-dimensional, 5 X 5 point grid was adequate forthe AI = 40 and M = 85 displacements; however, a7 < 7 point grid was required for the M = 10 dis-placements. A 10 X 10 point grid was required forsimulating the unit mobility ratio displacements shownin Fig. 7.Nomenclature

D = depth, measured positive downward fromsea levelk = rock permeabilityk, = relative permeability-component speci-

fied with additional subscriptsJULY. 1972

M.~=

P=roPCKO=s=u=/.t =P =1,) =

v=

Subscriptse=g=

yn =n=o =s =w=

References

mobility ratio (viscosity ratio for misciblefluids)pressuregas-oil capillary pressurewater-oil capillary pressuresaturationsuperficial fluid velocityviscositydensitymixing parametergradient operator

effectivegas in four-component simulator, solventin three-component simulatormixturenonwetting (hydrocarbon)oilsolventwetting (water)

1. Pozzi. A. L. and Blackwell, R. J.: Design of Labora-tory Models for Study of Miscible Displacement, SOc. F , (J. j. (Marc h, ]963 ) 2 3-40..- ..-. .2. Lantz. R. B.: -Rigorous Calculation of Miscible Dis-placement Using immiscible Reservoir Simulators. Sot.Per, ,51AI. J. (June, 1970) 192-202.3. Lantz, R. B.: Quantitative Evaluation of NumericalDiffusion (Truncation Error) , SOC. Pet. Eng. J. (Sept.,1971 ) 315-320.4. Peaceman, D. W. and Rachford. H. H., Jr.: NumericalCalculation of Multidimensional Miscible Displacement,.Soc. Per. Eng. 1. (Dec.. 1962) 327-339.5. Blackwell, R. J., Rayne, J. R. and Terry, W. M.: Fac-tors Influencing the Efficiency of Miscible Displacement.Trans., AIME ( 1 959) 216, 1-8.6. Garder. A. O.. Peaceman. D. W. and Pozzi, A. L.:

Numerical Calculation of Multidimensional MiscibleDisplacement by Melhod of Characteristics, Sot. Per.F , r> 1 [M. --h 10 CA\ ?L lz= .-rr A. . . , 1.1-IL . ,, -?, A .JU.7. Perrine, R. L. and Gay, G. M.: Unstable Miscible Flowin Heterogeneous Systems, SOc. Pet. Eng. J. (Sept.,1966) 228-238.8. Price, H. S. and Donahue, D. A. T.: Isothermal Dis-placement Processes with interphase Mass Transfer,Sot. Pet. Eng. J. (June, 1967) 205-220.9. Chaudhari, N. M.: An Improved Numerical Techniquefor Solving Multidimensional Miscible DisplacementEquations, Sot. Pet. ErIR. J. (Sept., 1971) 277-284.

10. Koval. E. J.: A Method ~or Predicting the Performanceof Unstable Miscible Displacement ;n HeterogeneousMedia, SOC. Pef. Eng. J. (June, 1963) 145-154.1. Brigham, W. E., Reed, P. W. and Dew, J. N.: Experi-ments on Mixing During Miscible Displacement inPorous Media, SOc. Pd. Eng. J. (March, 1961) 1-8.2. Mahaffey, J. L., Rutherford, W. M. and Matthews, C. S.:Sweep Efficiency by Miscible Displacement in a Five-Spot, .SOC.Pet. Ew. J. (March, 1966) 73-80.3. Lacey. J. W., Faris, J. E. and Brinkman, F. H.: Effectof Bank Size on Oil Recovery in the High Pressure Gas-Driven LPG-Bank Process. J. Pet. Tech. (Aug., 1961)806-816.4. Habermann, B.: The Efficiency of Miscible Displace-ment as a Function of Mobility Ratio, Trans., AIME( 1960) 219, 264-272.5. Greenkorn, R. A.. Johnson, C. R. and Haring, R. E.:Miscible Displacement in a Controlled Natural System,j. Per. Tech. (Nov.. 1965) 1329-1335.

L6. Greenkorn, R. A., Johnson. C. R. and Shallenberger,L. K.: Directional Permeability of Heterogeneous Anl-sotropic Porous Media, Sot. Per. Eng. J. (June. 1964)115-123.17. Johnson, C. R. and Greenkorn, R. A.: Description of

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Gross Reservoir Heterogeneity by Correlation of Litho-logic and Fluid Properties from Core %mpie, Buii.,Intl. Assoc. Scientific Hydrology ( 1963).18. Heller, J. P.: The Interpretation of Model Experimentsfor the Displacement of Fluids Through Porous Media,AIChE Jour. (July, 1963).19. Morel-Seytoux: H. J.: Unit Mobility Ratio Displace-ment Calculations for Pattern Floods in HomogeneousMedium, Sot. Pet. Eng. J. (Sept., 1966) 217-227.20. Hearn, C. L.: Simulation of Stratified Watertlooding

by Pseudo Relat ive Permeabil ity Curves, J. Pet. Tceh.Jdy, 1971) 805-813.21. Coats, K. H., Nielsen, R. L., Terhune, Mary H. andWeber, A. G.: Simulation of Three-Dimensional , Two-Phase Flow in Oil and Gas Reservoirs, Sot. Pet Eng.J. (Dec., 1967) 377-388.APPENDIXFour-Component ModelIn order that miscible slug displacements as well ascontinuous solvent injection may be examined, theequations developed for two-component miscible flowmay be extended to allow for three-component mis-cible flow in the nonwetting phase. Here it is assumedthat a slug of solvent (denoted by subscript s) ismiscible with the oil and with the gas driving thes yg. The drive gas, however, is not miscible directlywith the oil. Water is again considered the wettingphase. The four-component model requires the addi-tion of a fourth conservation equation to an existingthree-phase simulator. However, if no mobile wateris present in the system being simulated, a three-phasesimulator could be modified so that the conservationequation for water is used instead for the solvent.So long as miscibility is maintained, oil, solvent,and gas are all considered miscible components ofthe nonwetting phase. The capillary pressure Pc~ois thus zero as for the three-component simulator.However, when the oil-solvent and solvent-gas dis-persed zones contact (note Fig. 12) miscibility is lost.This occurs when the local solvent saturation dropsbelow a value sufficient to maintain both dispersedzones. This critical value will vary with both timeand space in a manner consistent with the growth of

F1OW DIRECTION

......e~. ..c- lmmlaL, eLKOISSIACEMENTOF OIL BY GAS

the dispersed zones. As the dispersed zones will gen-erally be small relative to the grid biock size, an ap-proximate IOW value (say, S,/S,, = 0.01) shouldsuffice for simulation purposes. Thus, when the sol-vent saturation at a grid point falls below this criticalvalue, the miscible model discussed below is bypassed.The remaining solvent is assumed immobile, and theoil and gas are treated as immiscible phases. Thus,the capillary pressure and the component viscosities,relative permeabilities, and densities are taken fromtables of input data in the normal fashion.While miscibility is maintained, the relative per-meabilities for miscible oil, solvent, and gas are givenas follows:

k,. = *k,%, . . . . . . . (A-la)n

krg=~% , . . .n. . . . (A-lb)

k,, = 5* k,,,, . . . . . . . (A-lc)n

where, now, S. = So + S. + So and the nonwetting-phase reiative permeabiiii~, k,,,, k again assumed %be a known, single-valued function of the watersaturation.For the application of the mixing parameter model,we assume that oil viscosity and density are modifiedonly by the presence of the solvent, as are the viscosityand density of the gas. The viscosity and density ofthe solvent, however, are modified by both the oiland the gas since the solvent shares dispersed zoneswith both of these components. Thus, in analogy withEqs. 3 and 4, we have:Poe = pol-u ,L1.,,,08~ . , . . . . (A-2a)P.e = P8-J/1,,,~ , . , . . . . (A-2b)pg. = l.(c-wp,,lv , . . . . . . (A-2c)

. . . . . . . . . (A-3a)/( s s )pmsg = hpg 7j--- v +-. s p. ,89 89. . . . . . . . . . (A-3b)

/( sPm= P0P8P9 ~ ~~ pgti + n s.

Eq. A-3c is the M-power fluidity-mixing rule for threemiscible hydrocarbon components.The effective densities for the three miscible com-ponents may be calculated in a manner analogous tothat described in the body of the paper. -Tf-GAS/WLVENT \ so~E~T/o,~

DISPERSED ZONE DISPERSED ZONEFig. 12Schematic representation of a three-component,miscible displacement in a grid block.

Original manuscript received in Society of Petroleum Engineersoffice July 23, 1971. Revised manuscript received Feb. 29, 1972.Paper (SPE 3484) waa presented at SPE 46th Annual Fall Meeting,held in New Orleans, Oct. 3-6, 1971. 0 Copyright 1972 AmericanInstitute of Mining, Metallurgical, end Petroleum Engineers, Inc.

This paper will be printed in Tranaactiona volume 253, which willcover 1972.

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