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SPE 30127Evaluating the Performance of Open-Hole, Perforated and FracturedWater Injection WellsShutongPang andMukulM. SharmaUniversityof Texas at AustinCapyI@ 1ss5, SCCiety of Petroleum Engineers, Inc.This paper was prepared for preeenlatlon at the Europ%an Formalwn Damage Conference held inThe Hague, The Nethertends, 15-16 May 1995.This paper was selected forpreeentstian by an SPE Program Committee following reviawofinformation contained inan abstract submitted by the author(a). Contents ofthe paper, as presented,have not baan reviewed by the Society of Petroleum Engineers and are subpcted to corractM by the author(s). The mater ia l, as preeentad, does not nsceeserify reflect any position of theSociity of Petroleum Enginaers, * offiirs , or members. Papers presented at SPE meetings are subject topubfiiation review by Editorial Commitfeea ofthe Bc-Sity of Petrolaum Engineers.o.-L.A. .. . . d ~.,++~ t. n c,hc+~,.r .f ~Qt ~ re ~han w ~r& . l l lu~t


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.2 EvaluatinghePerformancef Open-Hole, Perforated and Fractured Water Injection Wells SPE 30127transition time. The model can be used effectively if core flowtest data is available.The objective of this work is to provide a reliable anduser friendlysimulatorfor predictinginjectivitydeclinein waterinjection wells. The simulator considers both external andinternalfaltercake formationfor variouswell completions,suchas openhole, cased/perforat~ ad hy~ulic~ly ~ct~ we~sThe model equations used in the simulator are formulated.Methodsforcalculatingflltmtionparametersamdevelopedwhencore flow test data is not available. Examples are given toillustmte how the simulator can be run to predict the injectivitydecline. These results can be used in conjunction with a costanalysiswhichis presentedin AppendixA.

All of the model equations used in the simulator arelistedinTable 1.There are eight sets of equationsthat havebeenderived for the various types of well completions and filtrationmechanisms.Equations for linear geometryare usually used forparameter estimation using data from core flow experiments.Modelequationsfor internaland externalfaltercakeformationaregiven separately. They can be used separately to compute theinjectivity decline for each impairment mechanism.They canalso be used together to calculate the overall decrease ininjectivityby using the transition time concept (l).In all equations, a is well injectivityand t is injectiontime.A derivationof themodel equationsfor lineargeometryandopenholewells can be found in Ref. (l). Derivationsof modelequations for perforated and hydraulically fractured wells areprovidedbelow.Models for Perforated WellsA perforation can be considered to be one half of anellipse that is rotated about its primary axis, Fig. 1. shows thise~ipsoi~ geometrywhererp is the radiusof entryholeandLPisr...l:~m...:~m.. ~mk hm . f& \ An f l r i .x r rnt o in 9h ~neht ioi i (k@h i%)l lUW1l l& Odbl lLGL \ UJ,u- ----- w..-perforat ion canb e expnxsedas,27rkoHpAp g +1@= ~ /M~lgp-1 (1)W* ~p= exp(wp) (2)tanh wp = rp/Lp (3)Hp = r@nh Wp (4)We use 13q.(1) to derive the injectivity equations forbothinternalandexternalfdtrationinperforatedwells.Internal Filtration. If we view the fluid flow in aperforation as radial flow around the perforation axis, then thepermeabilitydistribution at a distanceof r horn the axis can bederivedfromthe fdtrationequationsin radialgeometry(1),k(r,t)= ~1+ ~ n(r,t) (5)

n(r, t) = kC t exp(-kn(r2-$)Wqp) (6)&P= NPh m

Having-assumed an ellipsoidal sham for the damagedzone and discretizing the permeability distribution around the

perforation into several layers, it is possible to develop anequationfor the flowratevs.pressuredrop.(8)where R is the total flow resistance which can be computedasthe summationof the flow resistance in each layer as shown inFig. 1.The resistancein one layer@i) can be writtenas

Ri = (ln~-ln~i+l+l)k(ri~t)Hi &i-1 ~i+l -1(9)

-Wke gi ~ ~xp(.w l) [10~h wi = ri/Li (11)Hi= ri/sinh wi (12ri = rp + (i-l)Ar (13aLi = Lp + (i-l)& (13bthus the total resistanceis 1 ~ r.R= $.Ri+_ (14

i= ] k(rp + nAr) Lp + nArwherethenumberof layersaccountedfor in the summation(n)isdeterminedsuch that the radius of the n-th layer doesnot exceedhalf the distancebetweentwoadjacentperforations. The secondtermon the RHS of the above equation is the flow resistanceinthe radial flow region which can be computed from the radiaflowequations.Forthesameinjectionpressurein both thedamagedandundamagedcases, the injectivitydecline (a), definedas %1%, isgivenby

1 ~q[~P+ 1].=&lG@ (15whtxeR is defti byR~s.(14)and(9).External Filtration. In the case of externalfdtration,we can assume that the external filter cakejust is lik-c -=.:... A .-~ -m.=mk;l~tu k- l%l lnw ino Ffi (14).a kyer ul pulurw y ~c -,U p....-u.-.J ~. - ----- ---e -l ., - .,.the flow resistanceswith and without external cake formation

ale,%=

i 1~ +11 ln[~ + l].&llQJ +--hn[~l

W% k-l ~p-] ~ ~P-l (16g +1R = ~n[~]* &p-l (17Becausethecake thicknessis very small, we can assumethat thcake is concentricwith theellipsoidal perforation(H= I-Q. Thinjectivitydeclinecanbe definedas R& thus~ ln(~

a= P-

h an approximationwhen ~ >>rpand ~ >>rc,~p=r*p+l~=r~p+l

Ifr~= rp- k, rearrangingEq. (18)yields,

(18(19a(19b

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SPE 30127 Shutong Pang and Mukul M. Shanna 3

:=l+*(ln(l+*)/ln(l +y.l) (20)c cNext,we need to derivea relationshipbetweeninjectiontime and the thickness of the external filter cake (hc). Thecumulativeinjected solid volume (Vs) at injectiontime t isV%=C\qdt (21)

Ontheother han~,the volumeof solids in theexternalfaltercakeof thickness~ in each perforationisVps=(l-Oc) [~~Lp-~~(rp -hc)2(Lp-hc)l(z)

u ~ 100), it is likely that particles willflow from the fracture into the formation and damage theformation. In this case, it can be assumed that the injectivitydecline is caused by near-fracture formation permeability---- .reduction,sowe can use equations for lineargeome~ to predictinjectivitydecline.Following Appendix A in Ref. (l), if the flow area isgivenby A = 2 h Lf and Lf = &, V = q/A, then the injectivitydeclinewillbea(t) = r~~ + (re-xf)]

(27)Xf=vt, forVt < re (28a)Xf= re for Vt > re* (28b)

1_=l+Mt (29k,(t)M=6f (306= pvclxf (31f= 1- exp(-kxf/V) (32External Damage on Rock Face. If the fracture

ennAls~t iv i t v /9w ti11re ic re ln t ive lv !~rg~ ~n~ ~h~ fQrmat iOI l .1 - .. . . . J ,-y - ---- - ------- --,permeability is relatively small, it is possible for particles toaccumulateat the rcwkfacearound the fracture. In this case, tcmwe can use the equations for linear geometry to calculateinjectivitydecline,

~=l+Mt (33a ~vM=~ (34kC l-$C LwhereL=&, V=q/A, A=2h Lf.

Internal Damage in Fracture. The flow rate in anundamagedfracturedwell(6) is~= 2rclq_)hAp/pRf+ ln(rJLf) (35Rf = ~(~)rw/Lf + CR (36wkfCR=zL& (37If the injected particles plug the proppant pack itself, theconductivityof the fracture (CR) will decrease. Therefore,theinjectivitydeclinedueto*hue damagecanbeexpressedasink)+ ~ h( l+ai )1. L i= 1 ri.l/ri + CRia l@) +~(JALCL)L rw/Lf + CR (38CRi = *7cr& (39where kfi is the damaged fracture permeability. kfi can beobtained by using the internal filtration equations for packeproppmt in a Iin= geomeby. kfi can also be experimentallobtained b.y injecting particulate suspensions into a poroumediumformedbyproppantandmeasuringthepmsure dropasfunctionof injectiontime.

he model presented here needs input data on the wegeometry, the type of impairment, the external filter caktifiwrt;pc mA tw n in t erna l f ihm t inn nnramet ers The We!!yavyv. .. mom .- . . . .. ...... . ..---=.. ~.--...... -.w .geometryis commonlyavailable, while other parameters musbe determinedexplicitlybeforecomputingthe injectivitydeclineType of Well Impairment and Transition TimeThe typeof well impairment internal filtration vexternalfiltrationhas traditionallybeen done usingthe 1/3to 1/rule. This rule of thumb states that an external filter cake iformedif the averageparticle/ Pre sizeratio is greater than 1/and an internalfflter cake is formed if the ratio is less than 1/7In general,however, it is likely that an internal filter cake wi

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A Evaluating the Performance of Open-Hole, Perforated and Fractured Water Injection Wells SPE 30127be hnku ill ally ,,,11= W,nlv al, WA.U*.CU . . w. w.2 ..---1 . .:-a . ..h.b .-Am awf,an.ml f i lk=r r& e w i l l f@m at .. . . . . . .. . -.large time. This transition from intemrd to external filtmtion isobserved in flow experiments and has led to the concept ofTmnsitionTime (t*) (Ref, 1).Internalfiltrationmodelsapplyfort < t* and external filtration models apply for t > P. t* can bedetermined experimentally when core flow test data is notavailable.t* can also be determined using StokesianDynamicssimulations(l).External Filter Cake PropertiesWhen an extemai fiiter cake is formed, die injectivitydecline is dependent on the cake properties (permeability andporosity). Experimentaldata on cake permeabilityand porosityfor some types of particles are available in the literature.Permeabilityand porositydata for latex and silicaparticleswerereportedbyEylartder(3) andme listed inTable2. Thesedatacanbe useddirectlyif the exWrimentalconditionsaresimilar.Extemai fiitercakepropertiescanah w&aimd hx ndirec t measurementor empiricalcorrelations.The cakeporositycan be obtained if the particle volume in the filter cake isgravimetricallymeasured.The cakepermeabilitycanbeobtainedfrom flow tests if the cake thickness is measured. Empiricalco~!ations such ~ the Bla.ke-Cozenyequationcan be used toobtain a permeability-porosity correlation for packed beds oftmticles,

dz 43kp = &.-1501-$ (40)

WI. - --.-:... J. - . . ..+ ,.I -A. -o ;. n1 In - n m An* n& fig ~~I IE PUIUMLY UI lhu~au=t%FaIuu Q=Is U. v -V.JWWP..the type of particle, size distribution and applied pressure.Exampledata derived fromEq. (40) are shown inTable 2. Theresultsamveryclose to theexperimentaldata.Internal Filtration ParametersTwo parameters, the filtration coefficient (k) and thedamagefactor(fI),are introducedin the internalfiltrationmodel.They can be determined from either experimental data orempirical correlations.When core flow test data is available,kand ~ can be obtained from history matching the pressuredropdata. The filtration coefficient is a local and timedependentparameter.Any factors affecting the particledepositionprocesscan affect it. It is believed that the following factors areimptanti (1) Pore structure(posit y,pore thrw@ody size anddistribution).(2) Particle sizeand distribution.(3) Fluid velocity.(4)Interactions(variousforces)amongthe pmticle,fluidandpmsurface.Variousmodelswereproposedby earlierinvestigatorstosimulatethe process and to obtain the filtrationcoefficient.Payatakes at al. (7) obtained the values of filtrationcoefficient from trajectory analysis based on various flowmodels. These values along with some experimental data areshownin Table4.3. Both Payatakesat al. (8)andRajagopalanetal. (9) also reported some experimental data for filtrationcoefficients which are shown in Table 4.4. These data arereplottedin Fig. 3. The followingempiricalconflation betweenfiltrationcoefficientandparticle/grainsizemtiocanbe obtained,

k= f(v)* 14.481[dP/dg]0.587 (41)where f(v) is the coefficient accounting for the effect of fluidvelocity on the filtration coefficient. The form of f(v) can be

obttinti from theliterature(10).Additionalresearchis currentlybeingconductedto betterquan~y the dependenceof 1 on v.Thedefaultvaluefor f(v)is 1(novelocityeffects).The retained particles (n(x,t)) can be retained at porethroats or in pore bodies. To quantify the effect of the retainedparticles on the permeability it is important to distinguishbetweenthese tworetentionmechanismssince particlesretainedin pore throats have a much larger impact on permeability.Wedefine ~ to be the fraction of retained particles trapped at porethroats.It is evident that as the injectedparticle size increasesa1--.,.- ~..:,.. ,.#9.,a+.+,-1a+iebc w i l l & !O@ @ rf)-re $h_~_@w BGI UdWJ I1 U1 ~VWnw~ .1 .. . -andwill contribute to permeability impairment. Particles muchsmallerthan the pore throat size are likely to be retained in porebodiesand therefore,contributeless topermeabilityimpairmentA typical shapeof the functiont is shown in Figure4.If we assume that the permeability reduction is causedby pore throat biocking, the correlation between permeability-A... :Am-A *&.&nmr&.lsvm rpn~t inn P21) h? w ~~en & $lc uudluu anu RUUIIWU yu -.. . .. ... - - .. .- .,- . .1 _ z n(x,t) + (1- z n(x, t)) (42)k(x,t) kp kc)where kp is the permeabilityof the plugged pore throats and isrelatedto theptmneabilityof theretainedparticlesthatarepackedin thepore throats.RearrangingEq. (42),weobtain

b=l+(k- 1)Tn(x,t)k(x,t) (43)kPWe can express the above equation as the followingequationwhere~ is caiieddamagefactor.&Jl=~b 1+ ~n(x, t) (44)~= c(ko/kP - 1) (45)

Ifwe apply the Blake-Cozenyequation (40) to both the rock andthepackedparticleswiththe sameporosity,we obtain,$=~d~~ (46)wheredp is the averageparticle size and dg is the average grainsize of the rock. Inserting Eq. (46) into Eq. (45) gives us acondation for$,~= ~ (d~/d;-1) (47)

where ~ is determined from Fig. 4. Eq. (47) can be used toestimate~ if no coreflowdata is available.If the injected particles are much smaller than theformationgrain size, ~ + Oand most particles will be retainedin pore bodies. The permeability impairment can then becomputedusingEquation(40),~= $S(x,t) 1-+2 (48)k. l-#(x,t)Twhere $(x,t)=@ - (1~) n(x,t) (49)

A simulatorhas been written to compute the injectivitydecline in water injection wells. The model equations listed inTable 1are usedin the simulator.The simulator is available on PCs and Unixworkstations.It is windowbasedandmenu-drivenprogram.Help

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SPE 30127 Shutong Pang and Mukul M. Shanna 5informationforeach commandis provided.Theinputdatacanberead from the file or keyboard. All input data and parametervaluescanbemodifiedby theuser at anytime.Defaultparametervalues are provided if there is no user input available. All ofthesefeaturesmakethe simulatoreasy-to-useanduser-friendly.Input/output of the SimulatorThe input @ta forthesimu!ator are divided into twocategories required and optional. The following data is

Formationpermeability~ (red)Formation porosity $ (fraction)Formation height h (m)Wellboreradiusrw (m)Drainageradiusre (m)Waterinjectionrateq (m3/s)AvemgeinjectedparticlesizedP@m)SolidsconcentrationC (volumefraction)Them@red data also includesomespecificdata foreach typeof

w el l cnmnle~inn Q.17. . .. w .,...v .v ..,.. , rw rfnrat inn c Jensi [y and geometry for., ~-. -----..perforated completions, and fracture length, width andconductivityforhydraulicallyfracturedwells.The optional input data for the simulator are: (1) Thecore-flow test data. This includes the pressure drop and/or theeffluent particle concentration. (2) External filter cake. .. ..permeatnnty and porosity. (3) Internai fiittatioii piIriUIICLGI-.-. . . .values. If values are not provided for the optional data, thesimulator estimates these values based on the empiricalcorrelationsdiscussedearlier.

Results from SimulationsFig. 5 is a flow chart showing the steps involved inpredictinginjectivitydeclineusedby the simulator.The requireddata needed to be input depends on the type of completion.Theoptionatdata need notbe input but valuesfor parameterssuch asfiltration coefficients and damage fac!or are best obtained from---- n-... a-.- m,. .:-..l..*A. . . . m . ...4 +- hid S.S, -.m tf ih en.mcum l luw Uak i. 1 UCM luul ia l vl Ca l ik UWAJLu lllmolyIllabw,,Wulwflow data to obtain these parameters. If no optional data isprovided, reasonable estimates of these parameters can beobtainedfromempiricaicorrelations,The output of the simulator will be the injectivitydecline,the skin factor, thedepthandextentof solidsinvasioninthenearwellboreregion and thepressuredropdue to theexternalfiltercake. The simulator results provide not only the decline ininjectivity with time (hdf-iife of the injector) but also moredetailed information about the location of the damage. Thisinformation is usefui in making stimulation decisions andA.a.%wh%m ..1.3*JMradmpntcuw l~l lnl~ .dnu la.w l. UWCA* ... WO..O.The simulator can be used in a histo~ matchingmodeto matchinjectivitydecline for existing injectionweIls.it canbeused to evaluate the impact of water treatment facilities on theeconomicsof a waterinjectionproject.

In this section, we provide examples to showhow thesimulator can be used to predict well injectivity decline forvtious types of completions.

Example 1This example illustratesa case where external filtratiodominates.Thedata for this example is taken fromRef. (11):Averageinjectedparticlesizedp = 5 pm,ParticleconcentrationC = 0.25 ppm,Injectionrateq = 6 cc/rein.,Corepermeability~ = 501md,core porosity@= 0.22,Core lengthL = 2.54cm.Experimentalobservationsshow thatexternalfiltercakformation is the primary damaging mechanism. The simulatowas run in a history matchingmode to match the experimentadata.Goodagreementis obtainedfor a valueof ~ = 1.03mdand$.= 0.18. These two parametersare then used inwell injectivitcalculationsforopen-holeandperforatedwells.Allotherrequireparametersare listed in Table 5. The final results are shown inFig.6. Half-livesof411 and 267 days areobtainedfor openholeandperforatedwellsrespectively.Example 2This exampledata set is taken from van Oort et ai. (5)Their lab experiments indicated that internal filtration is thedominantmechanism.Data listed in Table 5 and the followin&tame lequiredFormationpermeability& = 1260md,Forma~Qnpor@!y $ = ~.~~,Avemgeinjectedparticlesizedp= 6.4Vm,Injectedsolids concentrationC = 0.25ppm.Basedon the aboveparametervalues, internal filtratioparametersare determinedconsistentlyby the simulator (defauvalues).A damage factor of417 and a filtration coefficient o0.025 were obtained. The simulator was then run to give thinjectivitydeclinefor open-holeand perforatedwells.A half-lifof 136 days was obtained for an openhole completion. Usingcore flowdata van Gort et. al. obtained a half-life of 175daysThis indicates that our empirical correlations for the filtratio~%%ei_~O~1 ~~~~hie. .A-~orddr & rna@d_e f iMt er ddine kinjectivitywasnoted for perforatedwells.This suggests that theffectof high fluid flow rates near the perforationsneeds to b

C. I-..,... ..,.,44%.: --+ #.n-lm :*m ( ff .rh + 1 inaccoimkd for in tiie UU4UII GUG1l lULG1lL ~ul lvmk l l ~ . i ., -. . .equation41). By setting f(v)= 0.1 for perforated completionsaIudf-lifeof 68 dayswas obtained.Example 3Thisexamplewas selectedto illustratecaseswherebotinternaland externalfaltercakeformationare important.Thedatis fromEyiander{Y):

-A veraQe in ie c t e d particlesizedp = 12pm,.. JParticleconcentrationC = 0.18 ppm,Injectionrate q = 6 cc/rein.,Corepermeability~ = 992 md,Coreporosity$ = 0.20,Core lengthL = 5.08 cm.,Corediameterd= 2.54cm.Basedon thegrain size of the formationand the injecteparticle size, the transition time is calculated to be 33 minutefor the coreexperiments.Theexperimentaldata wasmatched toobtain& =4.5 md,$== 0.18, f3= 200, I = 0.035. These valuecan be used to calculate the injectivity decline in open-holeand

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6 Evaluating the Performance of Open-Hole, Perforated and Fractured Water Injection Wells.

SPE 30127perforatedwells.The final results are shownin Fig. 8 predictinga half-lifeof 630 and280 days foropenholeandperfomtedwellsrespectively.

-- .,-. . - .L---I ne amve mree ex~pies show-injectivity decline fi3iopen-holeand perforatedwells.The injectivityfor hydraulicallyfracturedwells in each example has been calculatedand it wasfound that (a) the injectivity decline in fractured weiis isprimarily due to internal damage to the fracture, other damagemechanisms are minimal, (b) if the default values for filtrationparametersare used, the well injectivity decreases fast at first,then tends to stabilize (see Fig. 9), and (c) if smaller filtrationparametervalues are used, the injectivity will decreasesslowly(seeFig. 10).l%ese findings are reasonabledue to the fact thattheinjectivityfor hydraulicallyfracturedwelldecmses slowlyinpracticeand flowvelocityin fractureis high.Our results show that the type and geometry of wellcompletionshave a large effect on the well injectivitydecline ifother conditions remain the same. Our results also show thatinternalfracturedamage is the primary reason for the injectivitydecline in hydraulically fractured wells. This is consistentwiththecommonpractice.In addition to predicting well injectivity decline, thesimulator we present here can also be used to conduct a costbenefit analysis (water treatment vs. stimulation) and/or asensitivity analysis.

Equationsare presentedfor pm.dietinginjectivitydeclineandnearwellboreformationdamagein waterinjectionwells.Themodel is incorporated into a simulator that is easy-to-use anduser friendly.Both internal and external filtercakes are allowedfor in the simulator for various types of well completions.Thesimulator can be used whether or not core flow test data isavailable. All model parameters can be obtained either byexperimental core flow data or from empirical correlations.Examplesareprovided to illustrate theuse of the simulator.

The authors wish to acknowledgethe financial supportprovidedby the companies sponsoringthe Stimulation,Loggingand Formation DamageResearchProgram at the UniversityofTexasat Austin.

c = solidsconcentrationCR = fractureconductivity% = themean grain sizeofthemck$ = @tie -m= formationheighthe = the thicknessof the externalfaltercakek = permeabilityk = rockpermeabilitykc = the permeabilityof the externalfiltercake

kf =%=Lf =Lp .=:=re =rf,xfsr- s-Yrw =t =t* =v=Wf =;, df:Ap =o =+C =$m a, fc=v=Np =T =

6actumpemabilitythepermeabilityof thepackedparticlesfracturelengthperforationlengththeconcentrationof particlestrappedin the formationinjectionratedminageradiustheradial/lineardistanceof injectionfrontentryradiuswellradiusinjectiontimethe transitiontimefluidvelocitytime widththedegreeofwell impairmentdamagefactorplwsuredropporositythecakeporositythe porosityof the rockvolumefiltrationcoeftlcientfluidviscosityshotdensityvol.frac.of retainedparticleswhichcontributetothepore threat bloc~ng (permeabilityirnpainnent)

1.

2.

3.4.

5.

6.7.

8.9.

Pang, S. and Sharma, M. M.: A Model for predictingInjectivity Decline in Water Injection Wells, SPE 28489,presentedat the SPE 69th AnnurdTechnical Conference&Exhibitionheld in NewOrleans, Louisiana September 25-28, 1994.Barkman, J. H, and Davidson, D. H., Measuring WaterQuality and Predicting Well Impairment, J. Pet. Tech.(July 1972),865-873.Eylander, J. G. R., Suspended Solids Specifications forWater Injection from Coreflooding Tests, SPE [email protected]).Vitthal, S., Sharma, M. M. and Sepehrnoori, K A OneDimensionalFormationDamageSimulator for Damagedueto Fines Migration, Proceedings of SPE Symposium onFormationDamageControl,Bakersfield,29-42(1988).van Oort, E., van Velzen, J.F.G., AND Leerlooijer, K.,Impairment by Suspended Solids Invasion: Testing andPredictionn,SPE IYoduction.&Facilities (Aug. 1993) 178-184.Schechter, R, S.: Oil Well Stimulation, Prentice-Hall,EnglewoodCliffs,NewJersey(1992).Payatakes, A. C., Rajagopalan, R., and Tien, C.:Applicationof PorousMediaModelsto the Study of DeepBed Filtration, Tine Canadian Joumai of ChemicaiEna in tww ina [Vnl 5)1 TMr 1Q7A-..e.-.=e=e 1 , -. -] ---- - ~ . .Payatakes,A. C., Tien, C., and Turian, R. M.: TrajectoryCalculationof Particle Deposition in Deep Bed Filtration,AIChEJournal (Vol. 20, No. 5) Sept. 1974Rajagopalan,R. andTien,C.: TrajectoryAnalysis of Deep-BedFiltrationwith theSphere-incell PorousMediaModelAIChEJournal (Vol. 22, No. 3)May. 1976

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SPE 30127 Shutong Pang and Mukul M. Sharma 710,11.

Gruesbeck, C. and Collins, R. E.: Entrainment andDepositionof Fine Particles in Porous Media,SPEJ (Dec.1982)847-856.Todd. A. C. . et al., Review of Permeability DamageStudies and Related North Sea Water Injection, (1979),SPE 7883, SPE International Symposium on Oilfield andGeothermalChemistry, Dallas, ~, Jaft.22-% 1979.

In the design and selectionof water treatmentfacilities,the two important parameters for water quality are the particlesize and solids concentration. Clean injectionwater (low solidscontentand small particle size) tends to stabilizewell injectivitybut requires additional water treatment costs. In the followingdiscussion,we present a method to systematicallycomparewelltreatment costs with water treatment costs anti obttn optiniaiwnt t = r -Wi rn t innc. .. . =-. -. .- . . .The water treatment cost depends on the water qualityrequirements. If Cu is the water treatment cost per unit volumeof wa~r, c is the solids concentrationand, dp is the particlesizein the treated water, then after injection time t, the watertreatmentcost Ct will beC t=Q CU(C, dg) (A-1)whereQ is the injectedvoiumeand Cuis a functionof c and dg.

4 Cleanwater iniection1.0 Unfilteredwaterinjection>.ga8.-C

td tfInjectionTime

Fig. Al. Schematicof water injectionscheme

As shown in Fig. Al, if tf is the time required for theinjectivity to math as for clean water injection, then the totalcleanwater injectioncost (Cf)will kCf=Ct+QCP+Cs (A-2)whereCp is the water pumping cost Pr unit volume of water?CSis the stimulationcost for recoveringwell injectivity.For unfilteredwater injection, there is no cost forwatertreatment.If td is the life time for injectivity being as then thetotal injectioncost (Cd) for injection time tf will beCd=Q~+tf/tdC~ (A-3)Therefore,the cost ratio for clean and unfilteredwater injectioncartbeObtfdM!dh ~S. (A-1) to (A-3),

G=l+QcdC@ -(tdu-l)c ,cd Q CP+ tdtdC, (AA)Ifwe havetwodifferentwater quality requirements,thecost ratio for these two typesof water injectionwiii mcf_l+Q(cu- cJ-otif-l)cs6- QC. +QCP+ C, (A-5)Eq. (A4) is a special case of Equation (A-5)whereCu

is equal to O(no water treatment). The net cost increasedue tothe injection of cleaner water (ACf = Cf - Cf) can be derivedtiomEq. (A-5),AC. s=O (C.., . Q - (~~~f~ . ~) ~a--, y ,.- ()@where the first term in the RHS is the water treatment costincreasedue to thebetterwater quality requirements,the secondterm is the decrease in stimulation cost due to the longerinjectionlife time.Eqs. (A-5) and (A-6) show us how the total cost ofwater injectionis relatedto thewater treatmentcost for specifiedwater quality requirements, the pumping cost, the stimulationcostand the life timeof the injectionproject.Inmany instancesthedecreasein injectivityis offsetby

increasingthe pumpingpressure. Eventuallypumping pressureswillexceedtiture gradientsand the injectorwillbe fractured.Insuch cases the injectivity will increasewithout any stimulationcost associated with it. This may be desimble in many cases(low cost) and very undesirable in other cases for reservoirengineeringreasons (neighboring water zones, well placemenetc. ) or forenvironmentalreasons.

Table 1, ModelEquationsFlowgeometry Internalfdtmtionmodel Externalfdtrationmodel

LinearGeometry1=l+Mt ~=l+Mta aM=6f6= pvt/L M=~~~kc i-$C Lf= 1- exp(-1.IJV)

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.8 Evaluating the Performance of Open-Hole, Perforated and Fractured Water Injection Wells SPE 30127

Table 1. ModelEquations(continued)

1_=l+Mtk,(t)M=8f6= ~cq2~~h$m 1$Openhole wellf= exp(a)Ei(a) - Ei(arf/r#)l

n-1~ .~(l~ -lnLi+l + 1) ~ln~n + 1 ln(l + a)1 i=okr(riJ)Hi Li -1 Li+l-1 Hn ~rt-1 ~ rp - hc= a-l+~(- - 1)a -&& ln(l + ~.r -rk(r,t) = 1Perforatedwell 1+ ~ n(r,t) hC= ~~qdtP

n(r, t)= LCt exp(-ln(r2-#)L@n/qP)Clp=$ G=(l~c)~rpLp(2 +rp/LP)&= exp(wi)tih wi = ri/LiHi= ri / sinh wi

ln~)+ ~ ln(il+CRi \

1. f i=l ri.l/ri + CRi1= L+ (l-x:) a ~(!!) + ln( 1 + CR )a ~xf) rw/Lf + CR*-CR=~Hydraulically Xf=vt for VtC re xL~CRi - wkfi!Yactunxlwell xf=re for Vt > re

~iko~=~1_=l+Mt 1+ ~ n(x, t)k,(t)

M=8f n(x, t) = O for x >Vt8= ~VC/re n(x, t) = LCt exp(-kx/V) for x


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.,SPE 30127 Shutong Pang and Mukul M. Shanna 9

Table 2. External Filter Cake Properties

Latex 12 0.18 5.8 6.2Latex 12 0.31 32 38Latex 12 0.12 1.7 1.7Silica 5 0.2 0.95Silica 5 0.15 ;:;8 0.45Silica 5 0.13 0.37 0.26

Table 3. Filtration CoefficientsL (Ref.7)alp@)= 3.5 9.5 21

Expt1data 2.20 5.10 17.00Comptddata(capillaricmodel) 0.46 3.41 17.06Comptddata(Brinkmanmodel) 2.09 13.93 61.20Note (1) data is given in L1O%V(cm-l)(2) dg = 720 pm, $ = 0.39. V = 0.136 Cln/St!C

Table4. Filtration Coefficients(Refs. 8 and 9)Run %v ExperimentalDataNo. (w) (Cln/S) dp = 2.75 4.5 9.012345678

460 .159.143460 .127.175z .191548 .159658 .159777 .159

6.0 2.48.1 3.111.0 4.13.1 1.83.1 1.24.5 2.13.9 1.92.7 1.7

2.42.62.72.21.81.51.91.7Note: his given in ~l@/V (cm-l)> $ = 0.38.

Table 5. Data Set for Examples

Fig. 1. Internalfiltrationin perforations

Injectionrate (m3/day)Formationheight (m)Wellbore/Drainagemdius (m)Shotdensity (shots/m)Entryradius(m)Perforationlength (m)FmctmePermeability(d)Fracturewidth(m)Fracturehalf-length(m)

1360%00::30.212.60.00566

Fig. 2. Verticalhydraulicfracture3) 1

n0.00 0.01 0.02 0.03Particle/GrainSizeRatioFig. 3. Filtration coefficient vs. size ratio

10.80.60.40.20 0

Fig. 4.

0.1 0.2 0.3 0.4 0.5Particle/PoreSizeRatioCoefficient t vs. size ratio

497


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..,.10 Evaluating the Performance of Open-Hole, Perforated and Fractured Water Injection Wells SPE 30127

I Reauired Data ~ OptionalData I. II I[ Core Flow TestData? ~

Yes ~ NO

~ , ~ = R: J (Fiy=..,I intemai - ExtendFiltrationModeling IA

q!:f

AOpenhole ~orated Fractured

~ ResultsAnalysis ~

I output(Dataand Plots) I

Fig. 5. Schematicof proceduresforpredictinginjectivitydecline

~ :~ open-hole- peafomted

___0 100 200 300 400 500InjectionTime(days)Fig. 6. Injectivitydecline (Example 1)

--- 0 100 200 3m 400 500 600InjectionTime(days)

Fig. 7. Injectivitydecline (Example2)

open-hole

i :~. 0 200 400 600 800 1000InjectionTime (days)Fig. 8. InjectivityDecline (Example3)

.2 T*-.,* A*.. km avmnnlt= 9Lllpub Uala. Lsuml Waum..p -0.6-- 0 5 10 15 20Injectiontime(days)Fig. 9. Injectivitydeclinedue to frachuedamage

L df=400, fc==.m125

.g ---- df4000, fc=o.oo1259.

~:s ---- 0 100 200 300 400 500Injectiontime(days)Fig. 10. Effectof filtrationparametersoninjectivitydue to fracturedamage

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