SPE 10510

A Three-Phase Model For Fractured ReservoFluid Segregation

by Abraham de Swaan. Mex/can Pefro/eum hsthife

SPEsocwrJJtlfR?~Err@w5% . of AIME

rs Presenting

Copyright 1982. Sociely of Petroleum Engineers

Th(s paper was presented at the Sixth SPE Symposium on Reservoir Simulation of the :ociely of Pelroleum Engineers of AIME held m NewOrleans, LA January 31February 3.1982. The material is subject to correction by the aulhor. Permission 10 COPYis restricted 10an abstract ofnot more than 300 words. Write SPE, 6200 North Central Expressway. Dallas. TX, 75206 USA. Telex 730989

ABSTRACT A common approach of models for fracturedreservoirs is the one introducedby Barenblattand

A model has been developed.to describe the Zheltov2 for the transient flow of one phase. Itbehavior of naturally fracturedreservoirswith considers that the flow occurs only in the fracturesblack oil in which high transmissibilityin the and that it is affected by the behavior of the fluidfracturesand low oil productionrates allow the contained in the matrix rock through wh~i has beengravitationalsegregationof the gas, oil and water called the “interactionterm”, which is a source orphases. The influxesof these fluids from the sink term \iitha continuousdistributionwhitin thematrix rock to the fracturesare described by func- fracturesmedium.tions dependent on the propertiesof the reservoirfluids and the matrix rock blocks. These properties The formulationsused for the interactiontermare measurableby laboratorytests on cores and have originatedtwo tendencies:formationstudies in the reservoir. Thisapproachcontrastswith a common trend in fracturedreservoir 11 The interactionterm is a parameter,an interacsimulation favoring the use of parameters to account tion factor,which influences the flow of flui~sfor the matrix-frac~ureinteraction;these parameters due to the differencesbetween the pressurecannot be explicitly linked to the reservoir charac- distributionin the fractures and an averageteristics. pressurewithin the matrix rock blocks.

The presented formulationresults in a system 2) The interactionbetween matrix rocks and frac-of three simultaneousintegro-differentialequations tures is a function that depends explicitlyonwhere the dependent variables are: the dephta of the blocks characteristicsand on the past andthe gas-oil and water-oil contacts, and the pressure present conditionsin the fractures.at the datum. Observed and calculatedvariables,obtained applying the model, are in good agreement The first approachwas presented in Ref. 2 andthroughoutthe.forty-fiveyears long production was solved in the innovativepaper of Warren andhistory of an actual reservoir that includes a period Root3 studying the pressure transientsof well tests.with null production,while all the wells were This solution requires three parameters:closed,

a parame-that resulted in an unusual fluctuationof ter, w, which implies the storage radio between frac-

the variables. tures and matrix and a second parameter,A, whichthrough a third parameter,a, determines the matrix-

INTRODUCTION fracture interaction.

T!levery involvednature of fractured reser- The approachbased on interaction functionswasvoirs, usually an aggregateof vugs, cavities, suggested for the first time by Aronofsky,Masse andfracturesand matrix rock blocks, is generally Natanson4 in a numerical scheme accounting for theassumed to be schematicallydescribed by a connect- flow of oil when displacedby water in the pressenceed fracturemedium in which matrix rock blocks are of imbibitionof water by the matrix rock blocks.buried, each medium having definite porosities and The same approachwas used by the author of thepermeabilities. The studies on naturally fractured present paper for the transient flow due to wellreservoirsassume a locally, i. e., in every space tests5—= in a theory furtherly improvedby Najurieta6.elementaryvolume, homogeneousdistributionof It predicts the results of numerical simulationsoffracturesand matrix blocks since, further than a well tests’ using the data of the simulationwhereascert.zindeterminationof aniaotropictrendsl,adetail- the Warren and Root3 formulation,mentioned above,ed descriptionof fracturesand blocks in an actual finds, after matches to the simulationsresults’,reservoiris impracticable. parameters that are not explicitly related to the

data. The previouslymentioned theory5’6withReferencesand illustrationsat end of paper.

.

A THREE-P}{ASEMODEL FOR FRACTUREDRESERVOIRSPRESENTING FLUID SEGREGATION SPK 10510

interactionfunctionsallows to asses analytically venience of the later.the actual share of every property of the reservoiron the particular pressure response of well tests or The frecuentpublicationof models with interac-interferencetests in fracturedreservoirswithout tion parametershas produced in many managers thethe need of adjusting param~ters. For two-phase flow impressionthat satisfactorymodels, needing tiomoreanother paper by the author shows the suitability informationthan the one usually obtained in homoge-of the interactionfunctionapproach comparing com - neous reservoirs,are available for the descriptionputations of the ensuing theory with the results of of fracturedreservoirswith its connectedbenefits.physical and numerical experiments of oil displaced The extended uae of interactionparametershasby water in a fracturedporous medium. stemmed from the attractivenesspresented by para -

The first numerical simulation of f$ow in frac-metric simulationmodels when opposed to mathematical

tured reservoirswas performed by Kazemlmodels, as stated in a paper by Schmidt17 “,.. many

for one- systems are sufficientlycomplex to defy a completephaae transient flow, Two-phase flow numerical sim- mathematicaldescriptionwhile being amenable to reg-ulations were presentedby Kleppe and Morseg. These resentationthrough a [parametric.1simulationmodel,numerical simulationsare fully causal, that ia, In other cases, the system studied may be amenable toevery mechaniam present in the flow of the fluids mathematicalanalysis but the level of mathematicalthrough the porous materials is accounted for spe - sophisticationrequired is beyond the background orcifically in discrete grids representingthe frac-tures and the blocks,

capabilityof the analyst while he may possess theIn this aspect, the numerical capability to develop a valid simulationmodel.

simulationof Peaceman10 includes such an involved That is, for reasonably complex systems the level ofprocess as fluid convection in the fractures. Since mathematicalsophisticationrequired for developmentnumerical simulationscan, in principle, account for of a valid mathematicalmodel ia usually more exten-all the phenomena involved in the flow of fluids in sive than that required for developmentof the ccr-fracturedreservoirs the only reasms to attempt oth- responding [parametric]simulationmodel”. But,er type of approaches are economic competitiveness quoting the same source: “,.. if a system can be mod-in field case applicationsand, not less important, eled mathematicallyit should be. A mathematicalgetting a better insight of the reservoirmechanics. model can usually be evaluated more quickly than a

Another paper bv Kazemill presents a numerical[~arametricjsimulationmodel since replicationof agiven system condition is normally required for re-

stimulationof the fracturesmedium with an interac- liable estimationof the medsure of system perfora-tion parameter tvpe term for the matrix contributionto the two-phase flow.

ante when the system is modeled through [parametric]The formulationof Braester12 simulation. Thus the cost of the analysis is likely

assumes a modality different from the ones previously to be greater when the system is modeled throughmentioned using propertiesof the compositemediumintermediatebetween the fracture and matrix prop -

$~~~tric] simulation than through a mathematical. (Words in brackets were inserted by the

erties; the characteristicsthus treated are the rel- author of this paper).ative permeabilitieato water and oil. Both paperahave been commentedelaewheree Most of the approaches outlined in previous

Notwithstandingthat since the first applica -paragraphsare intermediatebetween the two extreme

tions”outlooks in the quotation.

of the Warren and Root mode13 to actualfield cases it was found that the determinationof The model presented here uses interactionfunc-tiand A, by matches to observed well test pressure tions in a theory on a producing fracturedreservoirresponses,has no direct link to the reservoir char- where gas, oil and water segregatedue to gravity.acteristics the use of this approach has ~roliferated Saidi and van Golfracht180utlineda model for thatin the literaturedealing with fractured reservoirs. type of reservoirwith a conventionalmaterial bal-Some papers, for instanceRef. 14, consider, contra-ry to Ref. 13, that once u and A are adjusted the

ante of the fracturesand matrix rock fluid contribu-tions, The model presented here introducesthe volu-

reservoir descriptionis consequent although no a?gu- metric contributionsfrom the matrix rock blocks inments neither proofs are rendered to support this their dynamic, time-dependentforms.thinking.

Far from being complete, the bibliographycitedThomas, Dixon and Pierson15 present numerical Z!lJOVemerely refers to the papers representingthe

simulationsof a one-block-fracturesystem and com- outlined approaches to the problems of fluid flow inputationswith equivalentmodels based on interac - fracturedreservoira;a comprehensivebibliographytion parameters. The adjusted A’s are different for on the subject,with “a gravitationaldrainage process and for a water im-

,judicious comments, is found inthe book bjjAguilera .

bibition process; both values are different from avalue of A ccmmuted after a method presented by THEORYKazemi** based on geometricalconsiderations. ‘hereis no way to relate the adjusted parametersand the THE INTERACTIONFUNCTION FORMULATIONassumed data used in the realiatic numerical simula-tions; it is not possible to foresee how a change in This approach uses the functionof response ofthe conditions in the fracturesmay affect the valuesof the parameters.

a matrix rock block to a unit step or Heaviside func-tion change of the variable under consideration,

Duguid and Leel.6pressure or saturation, in the boundary of the block,

propose an interactionvariable i.e., in the fracture surroundingthe block. Thisfactor as complicatedto compute as an interactionfunctionbut without the meaningfulnessnor the con-

~onse is a rate of fluid flow to or from the block.In that way, the formulationdescribing the varia -

!

SPE 10510 A. DE SWAM

tions of the consideredvariable in the fractures incrementused in the numerical solution of themedium incorporatesthe source or sink block term. resultingequations. In that way, definite contacts

Letof oil and gas, and water and oil, are formed in thefractures, Owing to the productionandfor injectionof fluids the dephts of the gas-oil and water-oil

y[(t)=H(t) , . . . . , . . . . . . . . .(1) contactsvary. These contacts dephts and the pres-sure at a datum depht are the dependentvariables

be the variable under considerationat the block in the formulationand define three distinct zonessurface;H(t) is the Heaviside or unit step func- in the fracturemedium: the zone where water hastion (being zero for L = O and having unit valuefor t > O), and let

advanced, the oil column and the gas zone. No capil-lary effects are considered in the fracturesmedium;a negligible capillary pressure between the phases

fmi,,,=fma,,(t,x~ ,X2,....XJ . . . , . . + .(2) is ‘ie result of high porosity and permeabilityina porous medium. General conditions favoring segre-

be the functionof block response to that assumed gation are: high transmissibilityin the fracturesunit step disturbance. Xj, j = l,n are the val- and low production or injectionrates of the involv-ues of the n propertiesof the matrix rock block ed phases. These considerationsimply also theand of the reservoir fluids which are involved in assumptionof horizontal contacts of the fluidsthe mechanism under study. The blocksrespmse af- regardlessof the areal distributionof productionter an arbitrarychange of yf, covering the time and injectionwells. Schemes of the initial stateinterval [O,t],is the accumulationof the product of the reservoir and of a later stage, after someof :he differentialchanges of yf by the unit oil has been produced, as well as diagrams of thestep responsescorrespondingto these differential flow mechanisms present in the matrix blocks arechanges, i.e., a convolution, depicted in Fig, 1.

fro,,s~’ 2Yr(~) g (Since the only medi!lmwhere the formulation

— .md{ t-T,xl, x2,...,xn)dT . . (3)~T

is made is the fracturesspace the outflow from,or inflow to, the matrix rock blocks are consider-ed as influxes to this fracturesmedium. It mustbe kept in mind that, except an actual externalwater encroachment,all the influxes to the frac-

Notice that this response function does not imply tures originate in the reservoir itself.assumptionson the value of the variable inside theblock: yma. Water-oil Contact

In the formulationof the variation of yf in The depht of the water-oil contact, Dwoc, varies

the fractures it influencesand is reciprocallyin- according to the rates at which volumes of waterfluencedby fmq. For the point-wise or elementary enter or leave the fracturesporous volume. These

volume formulationsused in mathematicalmodeling volumetric rates are determinedby:

the source or sink strength is the volumetric rate,fma, divided by the correspondinglocal fracture 1) Encroachment from the acquifer.

volume, 2) Water production.

The matrix-fract~ireresponse or interaction 3) Injectionof water.

functionsmay be 4) Imbibitionby the matrix rock.

a) Analyticalb) Empirical

The correspondingequation is,

c) Numerical simulations aD~oc

Analytical functionsare used in Refs. 5 and—= (eWa -qw+ iW-ewW)/A~(DWOc) . . . . (4)at

6 fo~ transientone-phase flow problems;empiricalfunctionsare used to account for water imbibitionin the blocks in Ref. 8; functionsderived from The first subscripts of the influxes,e’s,

numerical simulationsof a single matrix block have denote the influx phase,tllesecond subscriptsmark

been suggestedby Yamamoto et al,20 the zone whzre the influx originates. Thus, ewa lathe water influx from the acquifer and eW is the

The three types of functions,representingthe water lost, in this case, in the water zone, I.e.,

same mechanism,may be interaubatitutedin a model the zone where the water-oil contact has advanced,

to obtain greater accuracy in the repreaentationofthe phenomenonas the knowledge of the implied vari- All the volumes are at reservoir conditions.

ables progreases or for computatiowl.convenience. The influxzs comprise the interactionfunctions,for instance, the water encroachmentis dependent

FRACTUREDRESERVOIRWITH GRAVITATIONALSEGREGATION on a function,an acquifer response function. This

OF THE FLUIDS response is localized and influencedby the localpressure at the original water-oil contact, Dwoci,

To assume instantaneousfluid segregationin in the manner outlined in the previous paragraph.

the fracturesis not a severe limitationof themodel; it means, practically,that a high percentage e~a =gt-fw(t-’) d~ . . ...”””(s)of the different phases segregateawithin every time

*

A THREE-PHASEMODEL FOR FRACTURED RESERVOIRSPRESENTING FLUID SEGREGATION SPE 10510

.,

faH is the acquifer response to a unit step is given by the pressure at the top of the cap pluschange of pressure at the boundary of interest, that the hydrostaticpressure of the fluids column. Ifis, the common boundary between acquifer and reser- there is no gas cap a similar formulationcan be .Sta~

voir, Dwoci. ed, In the most general case, for saturatedoil, Ehefactors that influence the pressure at the datum are:

The imbibitionof water by the blocks actsalong the zone invaded by the water, from &oc to 1) Total gas cap volume, determinedby the original

Dwoci! gas cap plus the secondary gas cap expanded inthe fracturesand in the matrix porous volume void

eWW = p’” fll{(t-T(D),D)dD. . . . . . . (6) by the drained oil.woc 2) Free gas in the reservoir,being the sum of:

where fIH is the imbibition function of the blocksinitial gas in the reservoir;gas liberated from

at depht D for a Heaviside disturbanceof the waterthe oil in fracturesand matrix down to the depht

saturationon their surface. Since it is consider-where the bubble-pointpressure is rea~hed~ Dbp;

ed that the advance of water in the fractures isinjection and production of gas.

piston-like this is equivalent to local unit step 3) The hydrostaticpressure of the fluids columnwater saturationchanges at varying dephts, effec- over the datum.tive at different times T(D) determined by themovement of the water-oil contact. According to Thus,its definition,it is the imbibitionresponse funstion itself, acting for [t-T(D)], which describes

J= B“”l(VPgci + V~gc~)/Gl,,c]

the influx. That makes unnecessarythe use of the pd

convolutionintegral,Eq. 3, for the imbibition D ‘dinfluy, The dependence of the imbibition function + [ ‘0’pgdll+ ~ f)OdD.. . . . . . . . . (8)on depht has been written directly in Eq. 6; the D

function is influenced through the fluids, rockgoc

-1

and fluids-rocksystem properties. ‘~ is the inverse functionof Bg; it is the Pressurecorrespondingto the ratio of the volumes of gas at

Other water influxes may be included in Eq. 4, reservoir conditionsand at standard conditionsandfor instance the water expanding out of the blocks is determined by the same measurementsused for Bg.in any of the three zones in the reservoir. Thewater saturation in blocks in the original oil The secondary gas cap volume iszone may reach significantvalues in some reser-voirs, see Ref. 18.

‘,gcs ‘/j’c {Af+( fgc~H[t-T(D),D]dT}dD. (9)Gas-oil Contact goc i

The top of the oil coiumn, Dgoc, is influencedby:

‘gcsH is the response functiondescribingthe rate ofsecondary gas cap formation in the blocks due todrainage and shrinkageof the oil.

1) The rate of change of the water-oil contacton which the oil column floats. The total free gas is

2) Rate of oil production.

3) Oil coming from the water ’zonedue to water GFS. = [GFi +Ge +Gi ‘GFp]~Cimbibitionin the blocks.

4) Oil that expands or shrinks in the blocks and ~ ,: ~@~ Af dDd~ . . . . . . . . (10)in the fracturesof the oil zone. BoaT d?

goc

5) Amounts of oil that drain from the blocks inthe secondarygas cap.

all the terms being at standard conditions. The gas

These effects are stated in cumulative influx from the matrix blocks is

tG, = J (Yg + ego& = aDwoc +eRW)dT . . . . ...(n)

+ (-qO + eow + eoo oat at

where the e’s depend on specific functionsfor the““c1 38.3

+ (goc~~ Af dD + eOs)/Af(DEOC) . . . . (7)gas liberation in the blocks.

Equations 4, 7 and 8 constitute the system ofsimultaneousequations of the problem. The pressureequation, Eq. 8, involves cumulativevariables

The influxesdepend on correspondingspecific contrastingwith Eqs. 4 and 7 in which rates of chang(blocks functions in the way described above. are considered. Equivalent formulationscan be set

Pressureup using combinationsof cumulativevalues or ratesof change of the involved dependentvariables.

The pressure at the reference depht or datum

k

SPE 10510 A. DE SWAAN

APPLICATIONTO A FIELD CASE.—— —

Saidi22 presents data of a fracturedreservoir fluid interchangemechanisms.with a productionhistory of forty-fiveyears. Theoil production is shown in Fig. 2 no gas nor water Althoughj due to lack of certain data for inputwere injected during that interval. into the interactionfunctions,some characteristics

were found fitting the calculationsto the obscrva-Since not all the characteristicsneeded in the tions, the values thus found arc amenable to be vali-

interactionfunctionsused in the formulationare dated experimentally. For instance, the imbibitionavailable some of them were determinedmatching functionuscsamean timeof inbibitionfithat depends onthe calculatedto the observed history of the the blocks dimensions,matrix permeabilityand porosi-.variables. The fit can be done in a piecewise ty, intcrfacialtension between water and oil in themanner; one or two dependent variables of the systcm matrix rock and water viscosity. Knowledge of theseof equations are treated as unknowns whereas the properties separatelyo evaluation, through labora-remaining dependentvariable(s) are read from the tory tests on matrix-rock cores, of a dimensionlessobserved values. For instance,to find the acquifer group relating these properties allows to cross-checkand imbibitionfunctionsdata ~oc is left as an the adjusted and measured mean imbibitiontimes.unknown, i,c., calculatedby Eq. 4,while Dgoc andpq are su~tuted by their known values at every AS compared to numerical simulations,interactiontime step of the numerical solution of the equations function formulationscan provide computingeconomyuntil a good match is obtained with the observedDwoc.

without over-simplifyingthe mathematicalstatementA similar partial model, limited to the pres- of the processes that occur in the reservoir. That

sure equation, Eq. 8, is used to find the volume of type of formulationsoffer adaptable and progressivelythe origin~l gas cap. The match of calculatedand improvablemathematicalmodels and correspondingobserved vsriables for the published exploitation reservoir descriptions.period is shown in Fig. 2.

The use of interactionfunctionsdocc not meanThe functionsused in the model accounted for the immediate determinationof all the variables

reversals in the fracturesconditions;such reversals implied in tllcmechanisms present in the fracturedoccur in the twenty-thirdyear of the production reservoir,but it can unite and synthcsizetheknowl-history, when production rates fell sharply. (Nega- edge gathered by a synergic group applied to thetive production rates in years twenty-fiveand description of the reservoir. Thus, as in the casetwenty-sixare the results of oil encroachment from of homogeneous reservoirs,the characteristicsofa near reservoir, see Ref. 21). the fluids, the rock and the fluids-rocksystem can

stand as the basis of fractured reservoirmodelling.An iterativeprocedure and the Newton method

were used to solve the system of equations 4, 7, CONCLVST.’3NS8 ; no significantdifferenceswere found between

—- -

their computationalneeds. Commerciallyavailable The present study leads to the followingcon-softwarewas used for the optimizationof the ad- Clusio[ls:justed variables.

1) A mathematicalmodel has been developed describingThe best fit was obtained using a lineal varia- the behavior of a naturally fracturedreservoir

tion of the fractures-totalporous volume ratio, with black saturated oil and presenting fastthat is 18,6 percent at the original gas-oil contact gravitationalsegregationof the gas, oil andand 3,6 percent at the original water-oil contact; water phases. The model includeswater encroach-Chc value of that ratio presented by Saidi22 is an ment from an acquifer. It reproduces fairlywellalmost constant 4. percent throughout the produc- the forty-fiveyears long history of an actualtive pay. ‘l’heoriginal oil in place was fitted to producing reservoir.1.10 Gm3, not far from the 1,18 Gm~ estimated bySaidi. 2) The presentedmodel illustratesthe convenience

and advantagesof the use of interactionfunctionsAs acquifer response functionwas used the to describe the interchangeof fluids between

one correspondingto an infinite lineal acquifer. fracturesand matrix-blocksin the reservoir.

The published PVT propertiesof the fluids Nf)}lENCLATURE

were used throughout the oil bearing formation.Af = horizontal area correspondingto the frac-

CO}blENTs tures porous volume

The field case studied in this work presents B = gas formationvolume factor,volume at res-g

particularcharacteristics. The uneven productionervoir conditions divided by volume at

~at~scausedan unusual fluctuationof the variables standard conditions.-1

and is a good proof of the validity of the model. B . inverse functionof Bg(p), pressure, PaMhile the wells were shut down water encroachment, g

water imbibitionof the blocks and gravitational B. = oil formationvolume factordrainageof oil from the blocks uncovered by thegas-oil contact continued to be active processes D = depht, mdependenton the past history of the variables.At the same time, the reversalof the gas-oil con-

influx of phase j originating in zGne k,‘jk = m3,s

tact advance and of the pressure decllne set new

f = function REFERENCES

G= = cumulativegas influx, m31. Janet, P.: “Determiningthe ElementaryMatrix

GFi =

initial reservoir free-gasvolume, 3n3 310ck in a Fissured Reservoir - Eschau Field,

cumulative free gas produced,m3France,” J. Pet. Tech. (May, 1973) 523 - 530.

‘Fp =———

Gi = cumulativegas injected,m3 2. Barenblatt,G.I. and Zheltov, Yu.P.: “Fundamental

H = Heaviside or unit step functionEquations of Filtrationof HomogeneousLiquidsin Fissured Rocks,” Soviet Physics, Doklady

i= injectionrate at reservoir cond:Ltions, (1960) Vol. 5, 522.m3/s

p = pressure,Pa3. Warren, J. E. and Root, P.J.: “The Behavior

of Naturally Fractured Reservoirs,”Sot. Pet.q = ;;p;uction rate at reservoir conditions, *J. (Sept. 1963) 245 - 255; Trans., ,—f!=

Vol. ~8.

Rs = solution gas oil-ratio, gas volume at4.

standard conditionsper unit oil volumeAronoCsky,J.S., Masse, L. and Natanson,“S.G.:

at standard conditions“A Model for the kfechanismof Oil Recovery fromthe Porous Matrix Due to Water Invasion in Fra~

t = time, s tured Reservoirs,”Trans., AIME (1958) Vol. 213,—.

v = porous volume in the original gas cap, m~17.

pgci

v = porous volume of the secondary gas cap, m3 5. de Swaan, A.: “Analytic Solutions for deter-pgcs mining Naturally Fractured Reservoir Properties

x = fluids or matrix-rockblock propertyby Well Testing,” Sot. Pet. Eng. J. (.June,1976)117 - 122; Trans.,~E=. 261.

y = variable

~ = parameter in Ref. 36. Najurieta,H.L.: “A Theory for the Pressure

Transient Analysis in Naturally Fractured Rese~A = parameter in Ref. 3 voirs,” J. Pet. Tech. (July, 1980) 1241 - 1250.—— —~= time of arrival of a disturbanceand

integrationparameter, s7. Kazemi, H. “PressureTransient Analysis of

Naturally Fractured Reservoirswith Uniformp = density, Mg/m3 Fracture Distributions,”Sot. Pet. Eng. J.

W = parameter in Ref. 3(Dec. 1969) 451 - 462; Tr~,— AIMR, Vol. 246

8, de Swaan, A.: “Theory of Waterfloodingin Frac-tured Reservoirs,”sec.Pet.~. JA(April, 1978)117 -

.—122; Trans., ~, Vol. 265.

SUBSCRIPTS 90 Kleppe, J. and Morse, R.A.: “i’v Production fromFracturedReservoirs by Water Displacement”,

a . acquifer paper SPE 5084 presented at the SPE-AIME 49thbp = bubble-point Annual Fall Technical Conference and Exhibition,d = datum Houston, Oct. 6-9, 1974.f = fracturesg = gas phase 10. Peaceman, D,W.: “Convectionin Fractured Reser-

gci = original gas cap voirs-NumericalCalculationof Convection in agcs = secondary gas cap Vertical Fissure Including the Effect of Matrix-goc = gas-oil contact Fissure Transfer,” Sot. Pet. ~ J. (Oct. 1976)H = related to a Heaviside or unit step

——281 - 301,

—

functionI . imbibition 11. Kazemi, H., Merrill, L.S., Jr., Porterfield,i= initial or original K.L. and Zeman, P.R.: “NumericalSimulationof

ma = matrix-rock Water-Oil Flow in Naturally Fractured Reser-0 . oil phase voirs,” ~ Pet, Eng. J. (Dec. 1976) 317 - 326;Sc = standard conditions Trans., AIME, Vol. 261.w=

—.water phase

woc = water-oil contact 12. Braester, C.: “SimultaneousFlow of InmiacibleLiquids Through Porous Media,” Sot. Pet. Eng. J.

AKNOWLEDGEXENTS (Augs 1972) 297 - 305. ‘—

The author thanks the Directive of the Mexican 13. Crawford,G.E.,Hagedorn,A.R. and Pierce> A.E.:Petroleum Institute for granting permission to “Analysiaof Pressure Buildup Tests in a Natu-publish this paper. The help of Enrique Aguilar and rally Fractured Reservoirs,”J. Pet. Tech.Ren6 Hidalgo in various computationalaspects is (NOV. 1976), 1295 - 1300.

—— —

gratefullyacknowledged.

—

●

=PE 10510 A, DE SWAM

4,

5,

—6,

=7.

=8.

=9.

20.

21.

Da Prat, G.C., Cinco-Lcy,H. and Ramey, H.J.,Jr.: “Decline Curve Analysis Using Type-Curvesfor Two-PorositySystems,” SOC. Pet. Eng. J.(June 1981) 354 - 362. ‘—

Thomas, L.K., Dixon, T.N.,and Pierson, R.G.:“FracturedReservoir Simulation,”paper SpE9305 presented at the SPE 55th Annual FallTechnical Conference and Exhibition,Dallas,Sept. 21-24, 1980.

Duguid, J.P. and Lee, P. C,Y.: “Flow in Frac-tured Porous Media,” Water Resour. Res.(June, 1977) Vol. 13, No. 3, 558 - ~

Schmidt, J. W,: “Introductionto SimulationModeling,~tproceedingsof the 1978 Winter

.— —Simulation Conference,Vol. 1, IEEE, (1978),3 - 12,

Saidi, A.M. and van Golfracht, T.:“Consid6ratiorsur les M6canismes de Base Dan les ReservoirsFractures,”Rev. Inst. Francais Pet. (Dec. 1971)1167 - 1180,——

Aguilcra, R.: Naturally FracturedReservoirs,PennWell Books, Tulsa (1980).

Yamamoto, R.H., Padgett, J.D., Ford, W.T., andBoubeguira,A.: “ComputationalReservoirSimulator for Fissured Systems - The Single-Block Nodel,” Sot. Pet. Eng. J. (June, 1571)—— ——113,

Saidi, A.M.: “MathematicalSimulationModelDescribing Iranian Fractured Reservoirsandits Application to Haft- Kel Field,” Proceed-ings 9th World Petroleum Congress, Tokyo,(1975) 209 - 219.

,-----

Dgoc

Dbp

Dwoc

-----

INITIAL CONDITIONS

~-j GAS -OIL 1- WATER

Fig. 1—Sclwr,. v; o! !rac. t.r(;u’ fcseruoir aIId wa!:ix I,lucks lE!MV,OIS

-..——-----”-. — —.

u OBSERVED —THIS THEORY o A

E

nId

= 400 - “ “-” ..——_ —.——.—-_—._— —

500 “--

600 -- -.

1 1=1 10s,5 10 15 20 25 30 35 40 45

TIM E, YEARS

Fig. 2-Observe@ ancl calculated oehav,ors. and yearly producl,on rates of actual resewot: m Rel21.

, .