ISociotyof PetfofaumEnglnasmI

SPE 28641

Some Theoretical Aspects of Gravity Drainage in Naturally FracturedReservoirsZhi-An Luan, University of Petroleusn,Dong-Ying, China ESPE Member

Copyright1994,SocietyofPetroleumEngisseemItsc.

lMspaperwaspreparedforpresentationattlse69thAnnealTechnicalConferenceandExhibitionoftheSocietyofPetroleumEngineersheldirrNewOrlesns,Lmrisii,September25-28,1994.

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Abstract

The mathematical model of gravitydrainage in natural ly fracturedreservoirs and singular perturbationanalyses are presented which explorethe effeet of capillary, gravity, andnettability on oil saturationdistribution and oil recovery inreimbibition. This study shows thatwettebility ,capillary end effectand fracture transmissibility appearto be most important factors in oilrecovery. Numerical results are alsopresented which summarize andcompare the effective saturation innonequilibrium state with the actualsaturation in equilibrium state.Based on this work,we believereimbibition or capillary continuityin a stack of matrix blocks can notprovide higher oil recovery than incontinuity core.

Introduction

In recent years, growing attentionhas been focused on reimbibition andcapillary centinuity indrainage.

gravityM Rei~ibition process is

closely related to various forces

(gravity,capillary, viscous,diffusiveand relaxation) . However, theexperimental and theoreticalresearch on gravity drainagemechanism in naturally fracturedreservoirs is limited. Bogomolovaand Glazova(1970)1 have observed theend effects and saturationdiscontinuities on the boundaries ofunconsolidated porous media withsome different permeability.Hagoort(1980)12 used the classicalBuckley-Leverett theory to study thegravity drainage in homogeneousmedia,and indicated that oilrelative permeability is a keyfactor in the gravity-drainageprocess. Barenblatts workl]investigated the end effects inheterogeneous media.Recent experiments and observationsby Firoozabadi and his co-worker(1992)5, Catalan and Dullien(1992)9prompted us to reexamine the gravitydrainage mechanism. This study hastried to examine from both amathematical and a physiicalviewpoint the key features ofgravity drainage in a stack ofmatrix blocks. We present criticalanalytical and numerical solutionsof the reimbibition model anddiscuss the effect of nonequilibriumphenomena upon the saturation

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distribution profile and oilrecovery.

Mathematical Model andSolutions of Gravity Drainage

Gravity drainage in naturallyfractured reservoirs is an importantrecovery mechanism in whichgravity, capillary,viscous anddiffusive are fundamental forces.Relaxation is a more complicatedcombination of the above forces andactions. Gravity drainage can beconsidered as a displacement processin which gas displaces oil. In otherwords,if gravity force and capillaryforce are taken into account,gravitydrainage in a stack of matrix blockswithout forced gas injection can bedescribed as if there is gasinjection at the top of the corestack. Since the gas mobility isrelatively high the potential dropin the gas phase is negligible. Thegas potential can be written as12

o~=p~-p~gz=constant (1)

Darcys flow law of oil phase isgiven by the following form

(2)

The continuity equation of oii phaseis

aso+auo=o$ at az (3)

The capillary pressure is thepressure difference between the oiland gas phases in oil wetting case

p==pg-po=aWJ (%) (4)

Using this notation,the equation ofone-dimensional model ofreimbibition in matrix blocks ofnaturally fractured reservoirs canbe writt& in the following

i?S Ap gkk, L%.o+%+ g x+~;k=o+o

form

(5)

At the entry section in the stack ofmatrix blocks(z=O),we can assume

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that the saturation of oil phase isthe following form

So(z=o, t)=sol (6)

It means that if oil is nonwettingphase,then oil can move out of thematrix block by imbibition into thefree space(zcO) ,then this abovecondition can be satisfied only if a

I,tSo

I

1,1

t1,1

1,4

1,1

I

DMIa!llll

Flgil Saturation Profile

Zd

mixed-wet semi-permeable membrane isinstalled. If oil is wettingphase,then the condition of a zeroflow velocity of oil phase at 2=0 issatisfied

Uo=o (7)

The boundary condition at the exitsection of each matrix block dependson the wetting state of matrixmedium

S(z=li,t)=so. (non-oil-wetting)

S(z=li, t)=s; (oil-wetting)

The nonlinear convection-dominatednroblem(cBq.5-7)~ingeneral,cannot be=solved m a closed form. We shalluse perturbation approach. Using themethod of matched asymptoticexpansion we seek the solution ofoil saturation distribution in astack of matrix blocks . Thisperturbation method breaks theproblem into three parts,for each ofwhich the solution has either aregular perturbation series or asingular perturbation series holdingin some sub-domain. The method

firstly is to find the outersolution in the regzon oiitsid~ erxl.....-effect layer,which is a simpleregular perturbation series.Similarly, an inner solution isfound which holds in end effectlayer. The outer expansion(Buckley-Leverett case) is not suited for thematrix bottom boundary nearest thefracture. On other hand,the fracturezone can be considered as a constantsaturation zone in which oilsaturation only depends on the outersolution. Fig.1 shows the oilsaturation profile in a stack ofmatrix blocks. It clearly shows thelarge gradient of oil saturation inthe boundary layer. A gravitydrainage solution of Buckley-Leverett equation will be used as anouter solution to determinesaturation in the flow zone awayfrom matrix boundaries. Forexample,if the coefficient Ncg ofcapillary item in Eq. (5) can beconsidered as a small parameter,then-.

~ne.-._

equa~loii (5) beccrles thfollowing Buckley-Leverett problemwhich can be solved by the classicalcharacteristics method

/SO+ApWgkk, ds-5E p. .o-&=o (9)

i7siiig some ---special ~~~forms ofrelative permeability and Leverettfunction,Hagoort have presented theanalytical solution of Eq. (9),thenumerical solution and steady statesolution of Eq. (5) in dimensionlessform. But his object mainly was toovercome the end effect by a newcentrifuge technique. Clearly theouter solution only fails to satisfythe boundary condition(8) . Let usnow consider inner solution or innerexpansion. We introduce a capillarylength scale

(lo)

The inner expansion associated withthe end effect boundary layer isexpressed in term of a stretchedvariable

The above conception implies thatthe length of an end effect zone isinversely proportional to thegravity force, and positivelyproportional to the capillary force.Then we can write Eq.5 as thefollowing form

as. as.

7% K;a (12)-$(%J (s.)+=O

The inner and outer expansions arematched over a zone located at theedge of the boundary layer. We shallrestrict our attention to matchingtkAs (zero) .ue.Ieadmg -?+.. expar.smu.of imer and outer solutions. In thezero approximation,we obtainthe inner solution equation

(13)a(KroJf(So)~)=O%%?

Prandtls matching condition avoidsthe need to choose an arbitraryedge of the boundary layer.Applying this condition to ourproblem leads to

The above outer expansion isobtained from Buckley-Leverettsolution at Z=li. The outer solutionslowly varies in the inner solutiontime scale,hence the oil saturationdistribution in the boundary layeris in the quasi-steady state. Onother hand,using Eq.ll,we obtainthat,as the stretched variablebecomes zero ~+O,the inner solutionclearly satisfies the boundarycondition(8) .The question now is how to determinethe unknown constants of integrationof Eq.13 in order to give the innersolution. From the above twoboundary conditions,it is notdifficult to determine theintegration constant,then we havethe implicit solution as thefollowing form

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s0 K=O(SO)J (SO) ~s ~o_oil-wetC=-JK,O(SO)-IC=O(SO,) so

0 K,O(SO)J (S.) ~s oil-wet=/K=O(SO)-K=O(SO,) s:

(15)

We introduce a new variable &i. e.aninfluence degree of end effect,thenthe above solution rewritten as

J~ ~~ 0 K=O(SO)J(SO) ~sApoggli$ Kro(so)-K=o(sol)no:oil-wet

/A.-s rO(sO)J(sO)ds

ApWgli~: Kro(S.)-Kro(Sol) oil-wet

inwhich A=~=tlm

(16)

The inner solution of the above formdescribes the relationship betweenthe influence degree of end effectand the parameters as gravity force,capillary force,wettability. Fromthis soiution,we conclude that(i). When length scales of matrixblocks in reimbibition arecomparable with the capillarycharacteristic scale,then thesaturation profile in the matrixblocks will be greatly distorted bycapillary end effect.(ii). Reducing the surface tensionof the liquid can decrease the endeffect. Surfactant solutions providea means of reducing the energyrequired to overcome oil retention

the boundary layer.;~one,et.al presented works onsurfactant effects on flow throughfractured porous media,which confirmthat surfactant solutions increaseoil recovery and drainage rateh.(iii). The length of an end effectzone,i.e. the length of the oilretention zcine . inverselyproportional to the ~ravity force.& gravity energy during gravitydrainage process decreases,thecapillary retention effect willgradually increase.In Fig.2,the oil saturation profile~s m+vnn f~~ ~~~~~ diffe~~~~=-----specific saturation values whichequate the inner and outer terms.

Computational results in Fig.1 andFig.2 are obtained Qy the explicitdifference stheme and Rombergintegration, respectively. Note thatthere is saturation discontinuityacross any fracture. But thissaturation discontinuity exist onlyon the matrix bottom boundaries. (seeFig.1) To prove that this conclusionis valid,we can think of gravitydrainage across the boundary betweenmatrix and fracture as adisplacement process in which thepermeability of matrix medium isk.,of fracture medium is kf.Using theinner solution model (Eq.12) tinner

I Fig.2 Inner Solutionsi QC-8 Integrll

;[&,w-

when the permeability of thedownstream medium approaches aninfinite permeability:the capillaryend effect in the (fracture) mediumwill be neglected(see Fig.1) .

Nonequilibrium Phenomena inGravity Drainage

In some gravity drainage processes,nonequilibrium phenomena caused by*:-- J1-l_..LA1lle ue.~ay ,-.----LJ--\[re~axac~onj !cfestablishing a new equilibrium stateare most important. This is partlydue to that the characteristicimbibition time of low-permeabilitymatrix blocks in gravity drainagecan be comparable to the timerequired to establish capillarypressure eouilibriurn.and it is

=- ,----

partly a reflection of a greatdifference in the permeabilitybetween fracture and matrix systems.Barenblatt constructs a generaltheoretical framework fornonequilibrium phenomena inwaterflooding processes anddiscusses specific technique forcounterblow capillary imbibition.It is apparent that for gravitydrainagertwo phases move alongdifferent channels,i.e.the stronglywetting phase (oil) along narrowerchannels and nonwetting phase(gas) along wider channels. Then thedisplaced oil will come to the widerchannels. Also,a typical case occurswhen oil and gas flow through thefracture between matrix blocks. Forthese conditions the phasepermeability for the wetting phaseis temporarily higher, incontrast,for non-wetting phase islower than that in the equilibriumstate. It implies that there is anincreased saturation of the wettingphase. It will be assumed that thedifference between the effectivesaturation and the actual saturationonly depends on the local rate ofvariation of the actual saturationand the relaxation time for thenonequilibrium

Generalized Darcys law can bewritten as

--~krj(30)VPJi- lJ:2__ (20)

(l=Q,g;

The pressure difference of betweenthe two phases is given by thefollowincJ

pg-p.=&(K)=a~J(%) (21)

With equations 17,18,19rthe---LJ-..JA.- _.L,-cunLMmLLy equa~~on 3 ?hefi can be-express as

(22)The simplest algorithm for thisnonlinear convection-dominatedequation is the FTCS scheme. Thecorresponding forward time,centredspace finite differencerepresentation for Eq.22 isconstructed by locally freezing thenonlinear coefficients. Let usassume the nonlinear coefficientshave the following form

and constants

w=O.25+$; nO=0.5Lz (24)

,- At-Aza; 6=A:2

Then we have the following implicitfinite difference expression

--ii?r-_d-ajS~~!l+bjS~~l+cjSOj+l-j (25)

where

ajo=(~+nO) a(S$)+%P(SOj-l/2)bjo=l+t[p(s$+,/J+p(sojJP6)Cjo=(m+no) a(S$)+n6P(SOj+112)

and the right item in the implicitscheme is

361

d,=[1-(%-n~)(13j+l/2+i3j-l/2)1Z

+ [ (no-w) a+ (m--m) 13j+112)1so~+l- [ (no-%) a- (w-nj ) Pj-,/2) 1Sj-I

(27)It is clear that the coefficients al,bj, Cj, are function for So and must

Flg,3 Nonequillbrium Effect

Nco=D. 03, Td=D,2

so

1,tI 1

.

=-:zz:-.=-=i;-::-l

4E---------------------------.........1.1 ---------------- -:1X ----------- xf+.=-~------~---=------------./---------------------------------------1-Jt -7

initial cfravity-capillary potentialprofile is a smooth line becausecapillary continuity is establishedfor oil phase. Even in no-oil-wetting case,gravity profile is asmooth 1ine. The capillarydiscontinuity can only influence theparts near fractures(see Fig.4(a)).

2.Contact property and ContactArea

some simulation researches indicatethat the key of effects onreimbibition is the with-contact orthe without-contact, and that thereis dramatically higher recovery forthe with-contact case than for thewithout contact case. The presentedsimulation conclusions are seemlylacking in support of fundamentallaboratory studies. Though only afew experimental researches havetouched with the physical mechanismof gravity drainage in fracturedreservoirs,it is enough to improveour understanding of reimbibition.Firoozabadi et.al.s experimentsdemonstrate that the rate ofdrainage across a stack of matrixblocks is governed by the fractureliquid transmissibility which isvery sensitive to the fractureaperture. They have indicated thattransmissibility may not besensitive to the number of contactpoints or the contact area. Catalanand Dulliens experiments also showsthat permeable contacts and evenclosed contacts can not ensurecanillarv.-.=-.4 continuity . It impliesthat contact and contact area arenot a main governed factor inreimbibition. In fact,there alwaysexist some contacts between matrixblocks. AS many researchersrecognize that matrix blocks can notjust remain suspended in the fluidsexisting in the fractures and that acertain degree of continuity mustexist across the fractures,at leastmechanically.In order to study this mechanicalcontinuity,let us consider a stackof matrix blocks as showed in Fig.5.Without loss of generality,wesuppose that the stack includes twotypes of mechanical medium betweentwo matrix blocks:(i) an impermeable rock bridge withthe permeability value 1$ (near zerovalue) and its average contact widthbi,

(ii) a permeable rock bridge withthe permeability value ~ and itsaverage contact width bP.Let & be the matrix permeability ,&be the permeability of a purefracture with width b~ ,as showed inFig.5. Using width as the weightparameter,we can give an averagepermeability kf,w,of this fracturedsystem:

~~dver=b-l[K~b~+K#P+~#fl(28)

For practical fracturedreservoirs,it generally is true that& >> %, Ki->O and Kic~; I$>=&.In these conditions ,the averagewidth b~ of a pure fracture can takesame order of magnitude as theaverage width bi of impermeable rockbridge,and then we obtain thefollowing result:

(29)

The above simple equation shows thatthe conception of fracture systempermeability in double porosity andpermeability theoryll generally iscorrect . Hence tin naturallyfractured media,the contact area ofan impermeable or permeable rockbridge between two matrix blocks cannot strongly influence thetransmissibility of the fracturesystem.

3.Nettability

The ...capll~ary continuity or thecapillary discontinuity has a farmore narrowly specified physicalmeaning. Capillary discontinuitymeans that capillary pressuredisappears at some sections ofmedium. Conversely, permeable matrixcontacts do not ensure completecapillary continuity since in astack of matrix blocks theestablishment of capillarycontinuity strongly is governed bycapillary end effect in matrixblocks . In heterogeneous media,thecapillary pressure curves depend onthe permeability,the wetting phaseis concentrated in a less permeablezone and nonwetting phase in a morepermeable zone!when EWQ phases flowcrosses the boundary as showed inFig.1.In Catalan and Dulliens presentedexperiments ,the connate brine

363

saturations after gravity drainagewere very close in their Exp.3 andExp.4. But in Exp.3,the faces of thecore plugs were polished andcontacted directly;and in Exp.4,alayer of mixed-wet paste wasinserted between the two plugs. Itshows that capillary continuity wasestablished for the wettingphase(water) . In Exp.3,the gassaturation was relatively high at

FIJ. } LIumld lrlmmloslblllty

I bloct! IId II

the fracture .Decauge capillarycontinuity was not established forno-wetting oil phase. When the oilflow is directed from the lesspermeable matrix into a morepermeable f r a c tur eregion,specially, in the limitingcase when the permeability of thedownstream fracture region mayincrease infinitely(fracture orfreedom space),the oil saturationdiscontinuity in nonwetting mediamay appear on the boundaries. Thecapillary pressure in the fractureregion may become a little or zerovaiue,wnicn generally is called asthe capillary discontinuity. Infracture reservoirs,if the length ofmatrix is comparable with thecharacteristic length scale ofcapillary(see Eq.10) , then thesaturation distribution will beconsiderably distorted by thiscapillary discontinuity effect,andnonequilibrium effect can not beneglected. 1

4.Recovery Efficiency

As we have indicated,some simulationworks considered that reimbibitionor called block to block processin fractured reservoirs can greatlyincrease oil recovery efficiency. Itis not an accepted fact. Infact,gravity drainage in a stack of

matrix blocks,even with somecapillary continuity,can not give ahigher oil recovery than that in thecontinuity core. It means thatreimbibition mechanism can notprovide same oil recovery efficiencyof waterflooding in naturallyfractured reservoirs as inhomogeneous porous media. In amixed/wet paste experiment, Catalanand Dullien9.found that though thecontact faces of the matrix blockswere polished and contacteddirectlytafter 52 days of gravitydrainage,the waterflooding residualoil recovered of a stack of matrixblocks was only- a~+ C2 7s.>0.La!AL is VA.*.for continuous core) . Howeverjinsame case if a layer of mixed-wetpaste was inserted between the twomatrix blocks one can reproduce theresults obtained with continuouscores because the paste eliminatedcapillary end effects. Suffridge andRenners experiments show anotherinteresting fact that addition of6895 Kpa net over-burden pressure tothe fractured core appeared toIlhealll fractures,and resulting~r=~~-age r~~e and recovery wereessentially that of the continuouscore. But in the absence ofoverburden pressure the core stackgiven lower recovery factor since asignificant capillary end effect ineach of matrix blocks exists.7We consider that,for the coreanalysis of fractured reservoirs,ifthe capillary end effect can beeliminated or largely reduced usingsome continuity agent, forexampie,mixed-wet semi-permeablemembranes,then called reimbibitionphenomenon itself will have not anynDw..- .. .Waterflooding recovery in practicalfractured reservoirs also shows thatthe presence of natural fractureswhich become potential avenues forthe water to bypass large volumes ofthe oil contained in the matrixoften causes total failures inconventional waterflooding. In somefractured reservoirs ,its primaryrecovery appears to be aided by thepresence of natural fractures whichact as important drainagechannels,however, secondary recoveryoperations are not effective becausefrackwes frequently act as channelsfor the early breakthrough ofinjected water. Our point of view isthat reimbibition phenomena can notincrease oil recovery in stack ofmatrix blocks,and that recoveryefficiency of waterflooding innaturally fractured reservoirs is

364

lower than that in homogeneousreservoirs.

Conclusions

1. The new model of reimbibitionand singular perturbation analysesare presented. Then it becomespossible to determine the oilsaturation profile in a stack ofmatrix blocks.2. The numerical results in thispaper confirms that when thecharacteristic time of thesaturation variation in matrixL. -,--D1OCJCS is ~~iilj?~~~bk -2LWALU tk tiRE

required to establish capillaryequilibrium the nonequilibriumphenomena in gravity drainage mustbe considered.3. Some important mechanisms inreimbibition processes fully isrevealed. The nettability, capillaryend effect,and fracture liquidtransmissibility govern flow in astack of matrix blocks.4. The more number of contactpoints,contact area or capillarycontinuity can not give a higher Oil-.---_-

recovery than that in the continuitycore. It is not true thatreimbibition may be greatly increaseoil recovery efficiency ofwaterflooding in naturally fractured

reservoirs.

Nomenclature

ajo= coefficientb. total contactb~ contact width

of j-lth space itemwidthof Dure fracture

b~= contact width of i-~ermeable rockbjo = coefficient of jth space itemb. = contact width of matrix blockbP = contact width of permeable rockCjo =coefficient of j+lth space itemdj= right item of difference schemefO= fractional flow for oilg= acceleration of gravityJ= Leveretts capillary functionk= absolute permeabilitykp permeability of pure fracture%er= ,average permeability of system

-. J..-&l - --a: -ki=p2i31WFd3ilit-y- UL J.[llpe?I1lleaLJAe IllCUAa~= permeability of permeable mediak.= oil relative permeabilityli= length of i-th matrix blocklP= capillary length scale~,~= constantNC~= capillary number~,~= constantPC= capillary pressureP~= Pressure of gas phasep.= pressure of oil phase

SO= actual oil saturations. . effective oil saturationSO*

= high oil saturations& = low oil saturationSfl= inner solution saturationSoalt

= outer solution saturationt = timetd = dimensionless timeAt= time step~= oil flow velocityz= vertical distanceAz= space stepzd= dimensionless distance

a = nonlinear coefficient functionf3= nonlinear coefficient functionA%= density difference of oil-gas6= small parameter~= influence degree of end effectp~= gas viscositypO. oil viscosity~ = inner variable(a . oil density~. gas densityo . interracial tension~ = relaxation parameter*O = oil potentialz~ . gas notential= _ -.@ = porosity

References

1. Saidi,A.M. : Simulation ofNaturally Fractured Reservoirs, SPE 12270 presented the 1983 SPEReservoir Simulation Symposium,SanFrancisco,Nov. 16-18.2. Van Golf-Rachat,T.D. :Fundamental of Fractured ReservoirEngineering, Elsevier ScientificPublishing Co. ,Amsterdam(1982).3. Saidi.A.M. :Reservoir Engineering of FracturedResenoirs,Total, Paris (1988).4. Larry S.K.Fung:Simulation ofBlock-to-Block Processes inNaturally Fractured Reservoirs,,SPE RE Nov.1991 477-84.5. Firoozabadi,A.and Markeset,T. :f!~ Experimental Study of the Gas-LiquidTransmissibility in FracturedPorous Media, SPE 24919 presentedthe 1992 SPE Annual TechnicalConference & Exhibition,wasningtonDC,October 4-7.6.Stones,E.J. ,Zimmerman,S.A. ,Chien,C.V. and Marsden, S.S. :TheEffeCt ofCapillary Connectivity AcrossHorizontal Fractures on GravityDrainage From Fractured PorousMedia, t SPE 24920 presented the1992 SPE Annual Technical Conference& Exhibition,Washington DC,October4-7.

*

7. Suffridge, F.E. ,Renner, T.A. :llDiff~~i~n and Gravity drainageTests to support the Development ofa Dual Porosity Simulation,the 6th. European IOR-Symposium inStavanger,Norway 21-23,1991.8. F.V.da Silva andB.Meyer: Improved Formulation fOrGravity Segregation in NaturallyFractured Reservoirs,the 6th. European IOR-Symposium inStavanger,Norway 21-23,1991.9. Catalan,L.and Dullien,F.A.L:Applications of Mixed-Wet Pastes inGravity Drainage Experiments, JCPT, May 1992.28-30.10. Bogomolova,A.F.and Glazova,V.M. :tThe influence of oil and gasreservoirs heterogeneity on residualwater distribution, (in Russian), NeftepromyslovOye Delo,1970,N0.9,5-9.11. Barenblatt,G. I.,Entov,V.M. andRyzhik,V.M. :Theory of Fluid Flows Trough NaturalRocks,Kluwer Academic Publishers,1990.12. Hagoort,J. : Oil Recovery byGravity Drainage, Soc.Pet.Eng.J.(June.1980) 139-150.

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