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Mathematical Modelling of Formation Heat TreatmentProcessA. K. M. JAMALUDDl~ and C. T. BOWEN

Noranda Technology Centre, 240 Hymus Blvd., Pointe Claire, QC H9R IGS, Canadaand

M.HASANDepartment of Mining and Metallurgical Engineering, University of McGill, 3450 University Street,Montreal, QCH3A 2A 7, Canada

A novel matrix stimulation concept. onnation beat reatment FHT), wbicb involves the applicationof intensebeatarounddie near-wellbore egion for the treatmentof water blockageand clay related onnation damage n water sensi-tive formations previously was developedand presentedn the literature.The FHT process nvolves the application ofintensebeat around he weUboreusing a downbolebeater.The beat s conveyed o the near-wellbore egion using aninert gas lowing through a downhole beater.To understandhe beat b'anSfer nd fluid-flow cbaracteristics f die FHT process,a transient wo-dimensionalmath-ematicalmodel bas been developedand s presentedn this paper.The model is basedon coupling the momentumandenergy-balance quations or the wellbore gaswith the surroundingporous ormation. The presence f the heateracrossdie net pay (sandface)s1aken nto account n die energyequation s a localizedvolumetricheatsource.A control-volumebased inite-difference scheme s used o solve he model equationson a staggered rid. Parametric tudies ndicate hatby injecting a suitable quantity of gas hrough die tube and annulus,and by adjusting he power input to the downholeheater,die temperature ear he wellbore can be favourablycontrolled.

On a mil au point et presenrenrerieurement ans la litterature scientifique un nouveau modele de simulationmatricielle pour Ie traitement de chaleur de fomlation (FHT), basesur I'application d'une chaleur ntenseautour de laregion du puits de forage pour Ie traitementdu blocagedesporeset desdornrnages la formation ies a l'argile dans eiformationssensibles I' eau.Le procedeFHT utilise pour ce faire une chaudiere orifice descendant. a chaleurest con-duite a la region du puits a I'aide d'un Ccoulement e gaz nerte dans a chaudierea orifice descendant.Afin de comprendre ei caractCristiques u transfertde chaleuret de I'ecoulementdes luides du procedeFHT, on amis au point un modele mathematique ransitoire bidimensionnel.Ce modele s'appuie sur Ie couplagedes equationsd'Cquilibre de conservationde la quantitCde mouvement t d'energie pour Ie gaz du puits de forage avec a formationporeuse nvironnante. a presence e a chaudiere ans a productionnette cOtesable)est prise en comptedans 'equa-non d'energiecomme sourcede chaleurvolum~ue localisee.On utilise un schemade differences iDiesbasesur lesvolmnesde conuole a grille decalee.Les etudesparametriquesndiquent qu'en injectant une quantireadequate e gazdans e tube et I'espaceannulaire, et en ajustant a puissance ans a chaudierea orifice descendant, n peut contr6lerfavorablementa temperature res du puits de forage. .Keywords:wellbore damage. lay swelling. water blocking. fonnation beat reatment.simulation. emperature rofile.

P etroleum engineering operations such as drilling, com-pletion, workovers, and stimulation, expose the forma-tion to a foreign fluid. This exposure esults n fluid invasioninto the near wellbore region. The permeability of the fluidinvadedporouszone s reducedbecause f pore throat con-striction causedby clay swelling, clay migration and waterblocking. This fluid-invaded region with reduced penne-ability is called he "damagedzone," extending oughly 1 minto the reservoir. Clay-related formation damageduringdrilling and completion has long been identified to be amajor problem. Measures o stabilize clay swelling andmigration havebeendiscussedn the iterature Himeset al.,1991; Borchardt et aI., 1984; Theng, 1984; Reed, 1974;CoppeDet aI., 1973; Plummer, 1991). Curative methodshave also been attempted and presented n the literature(Hayatdavoudiet aI., 1992; Lund et aI., 1976; Thomasand

Crowe, 1981; Garst, 1957; Sloat, 1989; Schaible, 1986;Crowe, 1986).The two most popular non-thermalstimula-tion processes re hydraulic fracturing and matrix acidizing.One of the earliest eportsof in situ thermalb:eatment asthat of Albaugh (1954), on a field test hat was carriedout inan oil weD in California. Since then, many other curativethermal processes ave been described or a variety of pur-poses, ncluding the removal of wax (Nenniger, 1992) orasphaltene Winckler and McManus, 1990)buildups. ther-mal fracturing of the formation (White andMass, 1965),andthe consolidation of unconsolidated ormations (Friedmanet aI., 1988). More specifically related o clay damagearemethods aimed at evaporating blocked water (Reed.1991a.b),dehydratingbound water from clays (White andMass. 1965;Braun, 1971).or transforminga sensitive ypeof clay (e.g. smectite) nto a less sensitive ype (e.g. illite)(Carroll, 1970;Nooner, 1980).A new matrix stimulation concept,called formation heattreatment FHT), was ested n the aboratoryand n the field(Jamaluddinand Namrko, 1994; Jamaluddinet aI., 1995,1996a). he FHT processnvolvesdie applicationof beat or

-Author to whom correspondence should be addressed. Present address:Hycai EtIerIY Rearch Laboratooes Ud.. 1338A - 36th Avenue N.E..Calpry. Alberta, Canada T2E 6T6.

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I-.DER8URDENFigure - Schematic iagramof the fonnationheat reatment(FliT) processogistics.

the treatmentof near wellbore damage.The heatingaroundthewellbore s achieved singa downholeheater Jamaluddinet al., 1996b) ocatedat the sandface. he heat s conveyedfrom the heater o the near wellbore region by an inert gas(e.g. nitrogen) lowing through and around he heater.To understandhe heat ransferand fluid flow character-istics of downhole heating, various modelling efforts havebeenpresentedn the literature. A one-dimensionalmathe-matical model (Shamla et al., 1989)was described or pre-dicting the flowing temperatureprofile in a well with adownhole heater. The model was identical to Ramey's(1962) model, except hat the heaterwas included throughthe treatmentof a source erm in the one-dimensional eatbalanceequation.Another mathematicalmodel to calculateheat osses o the surrounding ormationsdue to the down-ward njection of hot fluid through he tubing was presentedby Hoang (1980) and referenced y Somerton 1992). Themodel divided the wellbore and its swroundings nto tworegions: ubing and ormation.Hoangassumedhat heheatedfluid flowing through he tubing was osing heat radially tothe surroundings,while in the formation heat was assumedto be conductedboth radially and vertically. The analysisdid not take nto accountany penetration f the hot fluid intothe formation.Hoang's analysisshowed hat for an njectionrate of 30,000 kg/h of the hot fluid, temperatureprofileswithin the entire length of the tubing and swrounding for-mations eacheda steadystatewithin a few hours. He con-cluded hat for a high injection rate of the hot fluid throughthe tubing, a transientanalysisof the model equationswasnot necessary.None of thesestudiescoupledheat ransferand luid flowphenomenaor the njection of a gas n a wellbore where hegas was heatedwhile passing hrough a downhole heater.All prior studies n this area were variations of Ramey's(1962) original work, where the fluid momentumequationwas completely gnored. Furthermore,almost all studies nthis area have combined the steady-state eat conductionsolution or the wellborewith the approximate ndUDSteady-stateheatconductionsolution for the surrounding ock.

In this study, a two-dimensionalmathematicalmodel hasbeendevelopedor the simulationof localizedwellboreheat-ing using nitrogengasas he njection fluid. The presence fa downholeheaterhas been accounted or by incorporatinga volumetricheatsource erm n the ransient nergyequation.The model uses two-dimensional axisymmetric turbulentNavier-Stoke'sequations nd energyequations.Specifically,the model dealswith the electrical heating of the nitrogengasnear he formation and predicts he flow fields and con-vective and conductiveheat ransfercharacteristics etweenthe heatedgasand he surrounding eservoir.The model canbe used o quantify the power requirementof the downholeheaterand the heat propagation n the near wellbore regionduring the formation heat reatmentprocess. n addition, hemodelallows or the optimizationof the operating arameters.Formation beat treatment (FHT) process ogistics

A seriesof bench scale heating ests was carried out onsandstone ores aken ftom both oil- and gas-bearing or-mations Jama1uddin t aI., 1995).Samplecores aken tomactual ormationdisplayedan 84% reduction n penneabilityfollowing water exposure.Heating to a temperature round400C re-establishedhe baselinepenneability of the core.Furtherheatingat 600 and 800C mproved he penneabilityto 500/0 nd 7600/0 bove he baselinevalue, respectively.The physical situation and field logistics of the formationheat reatment rocess represented chematicallyn Figure I.As seen n the figure, a downhole heater s attached o theend of a tubing and placed across he sandface.After lower-ing the tool, nitrogen gas s injected hrough both the tubingand he casing ubing annulus tom the surface.The well ispressurized o a pressure higher than the correspondingreservoir pressure orcing the nitrogen into the reservoir.After pressurization,he tool is poweredup to heat he njec-tion nitrogen while it is flowing through the downholeheaterandconvey he heat o the nearweUbore egion of thereservoir.The primary objective of the FHT process s the intensebeatingof the nearweUbore egion extending o I m radiallyin the reservoir.The duration of the total heating period isdesigned o be 6 to 8 hours. The heatingperiod startswith aslow powerup sequence nd continueswith a one hour heat-ing period o establish teadystateconditionsafter reachingthe target temperature of the gas exiting the downholeheater.To validate he field logisticsanddesign,a multi-chamber,multi-pass,60 kW electrical resistance ype heating systemwas designed Jamaluddinet aI" 1996b),constructed,andtested.Due to voltage osses n the cable and imited spacewithin the wellbore, he heaterwas restricted o 60 kW. The

The formation heat treatment PHT) processconsistsofexposing he formation to an elevated emperatureo cause:- vaporizationof blocked water,- dehydrationof the clay structure,- partial destructionof the clay minerals, and- possibly, micro-fracture of the formation in the near-wellbore areadue o thermally inducedsttesses.THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 7S, AUGUST. 1997118


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current rating limited the operating power to 55 kW. Sincethe power was limited, the desired temperature had to beachieved by varying the total nitrogen flow rate. Based onthe bench-scale results, the preferred temperature in thenear-wellbore region was considered to be 800C. Toachieve this temperature in the formation, a higher tempera-ture was required at the ex.it of the downhole heater. Due topower limitations and practical concerns of the effect ofhigh temperature on casing and cement, the target tempera-ture of the exit gas was set at around 700 to 8000C. The pur-pose of this simulation study was to identify the depth ofheat penetration and to determine the temperature profile inthe near wellbore region given the fIXed power constraints.

r.-Tl u - u..- (Nitrogen)~ I

~c.8Ig..j;J- TIMIg~AM,*-T'-*IQ4. L

Model development

1~1oc:dcx1~L Yt: ..CenterlneFigure 2- Schematic diagram of the model domain.damaged formation region. and part of the formation. Thefour concentric zones in the overburden are: tubing, casing-tubing annulus with unperforated casing. unperforated cementregion. and impervious overburden. The near-wellboreregion and the formation are bounded at the bottom by animpervious underburden.Velocity, pressure, density of the injection gas and thegeothermal temperature at the lowest end of the upper seg-ment make up the input of the lower segment of the modelregion. In view of the complexity of the computationaldomain, the turbulent conditions (due to high injection rates)in the energy and momentum equations is modelled usingthe ad-hoc viscosity approach (ttr"jsbima and Szekely, 1989;Chao et aI., 1991), where the thermal conductivity and vis-cosity of the gas is increased by factors of 100.UPPER SEOMENT OF mE WELLBORE

The flowing pressures n the tubing and tubing-casingannulus or upper segmentwere estimatedusing Equation(1) (Beggs, 1984; Carcoana.1992), which is derived fromthe averagepressure nd temperaturemethod.This calcula-tion provided he input parametersor the lower segmentp2wf=P}e'+25 Ygq2 TagZfH(e'-l)/sJ5 (1)In Equation I), P wfis the pressuren die tubing or annu-lus, P f is the pressureat die inlet of the tube or annulus,His die total height of the tube, Tag s die averagegeothermal

temperature, is the total volumetric gas low rate throughthe tube or annulus, d is the diameter of the tube or theequivalentdiameterof the annulus,fis the turbulent rictionfactor and is calculatedusing Equation (2); parameterS iscalculatedusing Equation (3).

f= 1.01[1.14 2.0 x log (eld + 21.2SIReO.~f ... (2)

In this model, the vertical height of the well is dividedinto two segments: an upper segment and a lower segmentIn the upper segment, the injected gas is assumed to enterthe top of the well at a fixed volumetric flow at an atmos-pheric temperature and at a fIXed injection pressure. A pre-set ftaction of the injection volume is assumed to flow downthe tubing and the remaining fraction of the total volume ofthe gas is assumed to flow through the tubing-casing annu-lus. Typically, 90% by volume is pumped through the tub-ing and 100/0 y volume through the casing-tubing annulus.The temperature of the gas in this upper vertical segment ofthe well is assumed to be in thermal equilibrium with thegeothermal temperature. No account is made of the heat lossor heat gain in this region from the surroundings. The pres-sure profiles for the tubing and tubing-casing annulus forthis upper vertical segment of the well are obtained afterintegrating the differential mechanical-energy balance equa-tion and assuming that an average geothermal temperatureprevails in this section. Since this study is concerned withthe mass, momentum and heat transfer in the near wellborewith a heater near the bottomhole, the detailed mass,momentum and heat transfer analyses for the upper segmenthave not been carried out It was verified through the pre-liminary analysis that the downstream results did not haveany impact on the upstream calculations.The target reservoir and the associated overburden andunderburden regions constitute the domain of the FHTmodel (Figure 2). The FHT model domain is set at 20 mhigh (XL =20 m) and 10 m in diameter (YL =5 m). Out ofthe 20 In, the bottom 5 m is the net pay (h =5 m) and theremAining 15 m is overburden. A 4 m long downhole heateris positioned across the net pay (Figure 2). The first 1 m ofthe heater is a cold section where junction box and coolingchamber are housed (Jamaluddin et al., 1996b). The subse-quent 3 m length is the hot region and the hot gas exit at thebottom of the heater (0.3 m opening). In the model, the outerdiameter of the tool and the internal diameter of the casingis set to be 0.09 and 0.11 m (dci =0.11 m), respectively. Thecasing and cement across the net pay are perforated. Themajority of the nitrogen (900/0) s injected through the tubingand the remaLT ing100/0) s injected through the casing-tubingannulus. The purpose of this annular injection is to reduceheat propagation upwards through the annular space.The model domain (Figure 2) consists of five concentriczones in the radial direction across the net pay and four con-centric zones n the overburden region. The five radial zonesat the sandface are: tubing, casing-tubing annulus with per-forated casing, perforated cement region outside the casing,

In Equation 2), E is the roughness f the tube or the tub-ing-casingannulus,Re is the Reynoldsnumber or the tubeor annulus.

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S=0.0375Y, HITarZ. (3)In Equations 1) and (3), Yl.is the specific gravity of thegas,and Z is the gascompressibility actor evaluatedat Tag'

loWER SEGMENT OF THE WEU.BORE (DOMAIN MODELLED)

In Equation 5), the second ernl on the left hand side isthe buoyancy erm and he quantity Q in Equation (7) is thevolumebic heatsource.The value of Q is zero, except n theheater region, where a volumetric fraction of total heat isassigned.The flow of gas hrough he perforatedcasing,perforatedcement,damaged oneand ormation s assumed o be gov-erned by the non-Darcy flow equation. Specifically, theBrinkmann extendednon-Darcymodel (Chan et. al., 1991;Mishima and Szekely, 1989) s used o incorporate he vis-couseffect of the gas n the nearwellbore region where luidvelocities are high. In modelling he flow in this region, thefollowing assumptions re made:- the porous medium is considered as a continuum,- the gas and the porous matrix are in local themla1 equi-

librium,- the effect of natural convection is taken into accountthrough the Boussinesq approximation.With the above assumptions, he general macroscopicconservation equations for mass, momentum and heattransfer applicable below the overburden and through thecasing,cement,damaged oneand ormation can be writtenas follows:

Continuity

The lower segment of the wellbore is considered to be theFlIT model domain. Model regions are considered to be aspresented in Figure 2. The coordinate system as well as var-ious geometrical parameters also are presented in Figure 2.The nitrogen gas with constant physical properties enters thetube and annulus at a unifonn (but not necessarily the same)velocity. Prior to the start of heating, the fluid is assumed tobe stationary and in thennal equilibrium with the surround-ings. The transient process starts by switching the heater on(I> 0). The gas is assumed to be incompressible, viscous,heat conducting and obedient to the ideal gas laws. The rele-vant physical properties of the gas are thermal conductivity(k), dynaDlic viscosity 0, and specific heat capacity (Cp).Due to the complex nature of the model domain, the actualdesign of the heater is not taken into account in this model'sequations. The heater aspect of this simulation was simpli-fied by considering a volwnetric heat source in the regionwhere the downhole heater is located.With the Boussinesq approximation assumed, the fluidmotion and energy ~port in the tube and annulus aregoverned by the axisymmetric, time-dependent turbulentNavier-Stokes equations and energy equation, respectively.Referring to a cylindrical coordinate frame (x,r) with corre-sponding velocity components (U,J'>, these equations are asfollows, using standard notation:

~&1 + 5l -2ax r (8)=0OrMomentum EquationsAxial momentumequation UD-momenmm quation)

Continuity~&l=_~_gppf(T-Tr)at ax3(pU) 1~-+-Ox r =0 . (4)or ~ + l. (.~ )l_..e& ,.81' Or ~ K (9)J1MomentumEquations

Axial momentumequation UD-momentum quation) Radial momentumequation Vo-momentumequation)~+~~+ ~~= r ~~ = -~at Ort Ox Or8P [ a2U 1 8 ( aU~--gPP(T-7:)+J1 -;:r+-- r-Ox ,. & rBr Or (5) +J1 [~+ . . (r~ ) -~ ] -~. Ox rOr 8r ~ K (lO)

Radial momentumequation VD-momentum quation)cXpV) tJ(pUV) 1 cXrpVV)-+ +-=at Ox r or

o2V 1 0 ( oV\ V-;::1""+-- r 1--::1'"Ox: rOr r

Energy Equation

8Pc3r (6)Ji "8;")(pCp) +(pCp).(l-f) T+i ~~l~d

at OxIlr(pC,),Yorl [a2T I a ( aT )~- 1.'\r-'J,'~ J_~ "";:I"+-- r- .r &- Ox rar ar (11)Energy Equation

where UD and VD are the volume-averaged (Darcian) veloc-ities in the axial and radial directions, respectively; P D is thevolume averaged pressure; (pC,)fand (pCp>,rare the volu-metric heat capacities of the fluio"and solid, respectively; k isthe permeability of the porous medium; k~ fkaf + k" (1- f)]7)k +Q~ +~~ ~ + ~rpC,mat ax. r ar

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is the effective thermal conductivity; ka s the therDlalcon-dlM:tivity of the fluid in the porous medium; 41 is the porosityof the porousmedium.Because f radial symmetry,only onehalf of the domainshown n Figure2 hasbeenconsidered.The above generalequations or a porousmedium havebeenmodified to account or the low through he perforatedcasingand cement,damaged nd ormation zones.Initial ConditionsThe initial conditions are:

(12)= V= 0 (at t = 0). (13)'(1= 0, x, r) = 7'(1= 0, x = 0, r) + a xx

(14)( t=O x=O r ) =P, , Iwhere a is the geothermal emperablregradient; x is theaxial distance rom the top of the lower segment;Pi is theaverage nlet pressure n the tubing or annulus.Boundary ConditionsAt T> 0, the boundary conditi"ons are:

au aTat r=O.-= Y=-=OOr Or (IS)

solved simultaneouslyas a single domain problem. Thefmite-differenceequationswere derived by integrating thedifferential equationsover an elementary control volumesurrounding a grid node appropriate for each dependentvariable patankar,1980).A staggered rid systemwas usedso that the scalarproperties,P and T, were stored midwaybetween he U and V velocity grid nodes (patankar, 1980).A power-lawscheme patankar,1980)was used or the con-vective tenDS,and the integrated source terms were lin-earized.The pressure-velocity oupling of the momentumequationswas resolvedusing the popular SIMPLER algo-rithm (patankar, 1980). The governing finite-differenceequationswere solved teratively by the tri-diagonal matrixalgorithm TDMA) andusing a bloc-correctionschemewithunder-relaxation ntil the solutionsconverged.Simulationswere performedusing non-uniform grids inboth axial and adial directions.A non-uniform matrix of 80by 80 nodeswas used n the simulation. In the axial direc-tion, the first 7 metres rom the bottom of the reservoir wasdivided nto 0.1 m layers.The next 3 m distancewas dividedinto 0.5 m layersand he remaining 10 m distanceat the topof the model domain was divided into 2.5 m layers. In theradial direction, he first 0.085 m was divided into 0.005 mstepsand the remainingdistancewas divided into 0.078 msteps.The use of non-uniform grids in the radial directionwas crucial because f relative dimension of the wellbore(casing diameter =0.11 m), which is extremely small com-pared o the radial extension f the model domain (5 m). Thenon-uniformgrid in the axial and radial directions was suit-ably placed o accommodatehe various interfacial bound-aries. The axial grid distancewas chosen o accommodatethe heater egion.The solutionswere o be converged,whenthe following criterion was satisfiedsimultaneouslyby eachcomputed ariable:

(16)

=h,(T-Tz) .

(18)T-k.-a;=~(T- Tx>(19)t x = 0, 0 < r< dll/2, U= Vi' V= 0, T= Ti(20)t x = 0, d,/l < r < dcj12.U= u" v= 0, T= T,

(21)~I ",-

,J -"'iJ < 0.001ax _+1iJ

where ~i. represents ny dependentvariable and (n + 1)refers o ihe value of QiJ at the (n + 1)1hteration level. Toreduce omputing ime, the convergence riterion was mon-itored and it was identified that the relative differencebetween he pammeter aluesof two consecutive terationswere within 0.001 after 700 to 800 iterations. As an exam-ple. the relativechangesn temperature alues are plotted nFigure 3. As seen n the figure. the relative changes n para-metervaluesstart o fluctuateafter 800 iterations. The tem-perature aluescalculatedusing engineeringapproximation(mC~7) matchvery well with the averageof the simulatedtemperatures f the exit gas after 800 iterations. Therefore.all simulationnms were enninatedafter 800 iterations.

where hI, h~_h), and h~ are the equivalent convective heattransfer coefficients at the fonnation. overburden, underbur-den and top of the overburden, respectively; h is the heightof the fonnation (net pay); XL and YL are the vertical heightand radial depth of the computational domain; dti' dlOand dciare inner diameter of the tube, outer diameter of the tube anainner diameter of the casing, respectively; kb is the thermalconductivity of the overburden.Numerical procedure

The dimensional form of the above sets of elliptic partialdifferential equations was solved numerically by a controlvolume fmite difference scheme. The lower segment, incor-porating a part of the overburden and the formation. consti-tutes the full computational domain. The governing trans-port equations for the fluid, solid and porous regions were

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TABLEParameters sed n the Simulation RunsValuesarameters

Reservoir.overburdenand underburden hIrICteriIticIReservoirdepth rom surface m) ISOONet pay (m) Spcxosity I SDamaged one penneability (mD) SReservoirpenncability (mD) 2SHeat osscoefficient for over/underbW"dcnW/m2.K) 2.5Heat osscoefficient for reservoir W/m2.K) 3.0Reservoir emperature DC) S2Relevantdimensionsof well, cementand damagedInner radiusof the heater m) 0.04Outer radius of the heater m) 0.045Inner radiusof the casing m) 0.055Outer radiusof the casing m) 0.065Outer radius of cement m) 0.085Outer radius of damaged one m) 0.961Outer radiusof formation (m) 5.0 region was arge, resulting n a high rate of conductiveandconvectiveheat transfer.As a result, the temperatureadja-cent to the weUbore ose quickly. Because he segments fthe casing, cement and fonnation adjacent o the heaterlargely controUed he heat transfer rate, the temperaturethereforebecame onstantwhen hesesegments pproachedthennaI equilibrium. Convectiveheat ransfer hrough heseporoussegments layed a significant role in the attainmentof thennal equilibrium in a short time. Therefore, subse-quent analyses oncentrated n the steady-state olution oftheseequations.

30 x 10-"10900.0516

Nitrogen characteristicsKinematicviscosity (mo.)Specific beat J/kgoOC)Thenna1 onductivity (W/moK)Equivalent Conv~tive Heat Transfer Coefficients (W/m2.K)hi: Heat transfer coeff'1Cient or formation 3.0~: Heat transfer coefficient for overburden 2.5h): Hcat transfer coefficient for underburden 2.5h.: Heat transfer coefficient for top of domain surface 2.0

Note:Other hermalproperties f cbe eservoir, v~ andunderburdenre aken rom Butler 1991). STEADy STATE SOUn'K)NSThe parameters sed n thesesimulationnms are present-ed in Table 1. Simulation conditions,average emperaturesof the exit gasand average emperabUest a radial distanceof 0.5 m into the reservoir are tabulated n Table 2. Theresultsare p~ted graphically n Figwa 4 through 8.To undelStandhe practical easibility of the FHT process

and to identify the critical parameters ffecting this down-hole heating process,various simulation nms were carriedout. During the simulation nms, the controllable critical~ suchasnitrogel1low rateand otal power equire-ment at the heaterwere varied. The effectsof thesechangesunder steadystateconditions on the temperature f the gasleaving the heater,depth of heat penetration nto the reser-voir and the temperablredistribution in the near-wellboreregion were studied and the results are presented n thispaper. Basedon theseparametric studies,conditions wereselected or field testing of the tool and he FHr process.An example temperaturecontow' plot is presented nFigure 4 (Run B, Table 2). As seen n this tig\R, the high-cst temperature s seen o be concentrated round the hotregion of the beater 1 to 4 m). As expected,he temperaturegradually decleases adially to the reservoir temperature.Thereare no apparent hangesn the temperature ue o per-meability variation from damaged one 5 mD ex~ing toI m) to the rest of the reservoir (25 mD). This is possiblybecause f low velocity of nitrogen gas n the porousmedi-um. Under pressure he hot nitrogen gas,exiting the heater,enters hrough he perforatedcasingand cementand ravelsinto die porous eservoir.Gas low into the region below heheaterwidlin die wellbore and up the annularspace s min-imal as he only exit is through die porous ormation.

The model was originally developed or transient solu-tions of the transport~tions. The transientsolution o thegoverningequationsusesa fully implicit scheme.Omissionof transient terms from the model equations resulted insteadystatesolutions. n this paper, esults elated o steadystate solutions of the modelled equations are presentedalong with a brief discussionof the transientsolutions.Results and discussionTRANSIENT ~

Transient solutions of the partial differential equationsdescribing the mass, momentum and energy of the gasinjected down the tubing and annulus were carried out. Itwas assumed hat at every instant he gas,surroundingper-foratedcasing,cementand porous ormation were n a ther-mal equilibrium condition (i.e. therewas no thermal disper-sion effect). Also, it was initially assumed hat gas n thewellbore and the formation were at a temperature iven bythe (constant)ambient surface emperature lus the productof depthand geothermalgradient assumed o be constant).Transient calculations at full power input to the toolrevealed hat within 30 minutes, the near wellbore regionreacheda thennal equilibrium condition. This short timerequired o reach a steadystate was due to heating a con-fined region resulting in low heat osses about S%) to theunproductive strata above and below the fonnation.Initially, when he heaterwas wned on, the emperature if-ferencebetweendie gas and die surroundingnear-wellbore

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TABLE2Swnmary of Conditions Used in These Simulation Runs and Selective Results

55555555S5804034

24461044S.S

9090809090909070

14177358205073251045548446

215262261294295346210201

ABCDEFGH (field est)... Engineering alculationswere carried out using mCpAT elations flow going through the tube is assumedo pick up all heat)... Measuredemperature uring field test (non steadystate): 382C

0 2 4 6 8 10 12T0t8 Flow. m3(STP)/min

Figure 7- Temperatureas a function of total flow at 1.1 m fromdie bottom of die model domain (Heaterpower: 55 kW; 90% flowthrough ubing).

.0P-Ii

0 1 2 3 4 5RadialDIsIance.m

Figure 6 - Temperature rofile in dte radial direction as a func-tion of total flow rate at 1.1 m from dte bottom of the modeldomain Heaterpower: 55 kW; 900/. low dtrough ubing).The effect of the total volumetric flow rate, m3(STP)/min,on the vertical temperature distribution at a fixed radial dis-tance of 0.5 m is presented in Figure 5. In these cases, the

heater power was 55 kW with a tubing flow fraction of 900/0and the heater was placed at I m from the bottom of the

model domain. As seen n the figure, temperaturencreasesand eaches maxima within the flow ratesof about2 to 10m3(STP)/min.As expected, n all situations, he maximumtempemturc s at the heateropenings rom where he hot gasexits the heater.The vertical temperature rofile is also seenTHE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997 783


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To evaluate he efficiency of the stimulation process,anoverall heat balancewas carried out for the near wellboreregion. At a steadystate, he heat dissipatedby the heaterequals he heat osses n the over and underburden nd heatgain n the reservoir.The heat osscalculationsusing 55 kWpower, a total flow of 4 m3(STP)/min, low through he tub-ing indicated hat less than 5% of the 55 kW input powerwas lost in the unproductivestrata (overburdenand under-burden).Simulated vs measured emperature from field testingofFHTTo validate the FHT concept in the field, a proprietaryresistance-typeelectrical heater was developed to conduct theheating process downhole. The heater was successfully test-ed at the surface several times and subsequently, the heaterwas tested in the field (Jamaluddin et aI., 1996a). To avoidthe risks of damaging the wellbore casing due to thermalshock in a producing wen, a depleted well slated for aban-donment was chosen. During the field test, the heater waslowered into the target reservoir 1.5 km downhole, heated upto a temperature of 382C, and retrieved from the wellbore.The total nitrogen flow was maintained at 5.5 m3(STP)/min, 4 m3(STP)/min in the tubing and 1.5 m3(STP)/min inthe annulus. The injection pressure was stabilized at 3.7 MPa.The heater was slowly powered up to achieve a target exitgas temperature of 700C. This target temperature wasdesigned corresponding to 5.5 m3(STP)/min and a powerinput of 55 kW. However, the heater failed within 1 minuteof achieving an input power of 34 kW at the heater. Theheater failure occurred due to an electrical short circuitcaused by water leakage into the junction box. At the instantof failure, the measured temperature of the gas exiting theheater at downhole conditions was 382C. The thermocou-ple was located at the bottom end of the heater as shown inFigure 2. An engineering calculation based on 700/0nitrogenflowing through the heater would correspond to 446C ofthe exit gas at steady state conditions. Simulation resultsindicate that at a steady state condition corresponding to atotal nitrogen flow of 5.5 m3(STP)/min and at a power inputof 34 kW, the exit gas temperature would be 473C.Correspondingly, this exit. gas would have resulted in anaverage temperature of 200C at a radial distance of 0.5 minto the reservoir.As presented in the earlier paper (Jamaluddin et aI.,1996a), the target field test objective of raising the down-hole temperature to +700oC was not satisfied due to theheater failure, but the most important aspect of the effect ofheat on the reservoir characteristics was estimated usingtype curve matching technique. The permeability of the nearwellbore reservoir was improved six fold, which has enor-mous potential benefits for hydrocarbon producing wells.Concluding remarks

The mathematicalmodel presented n this paperdemon-strates he easibility of the formation heatUeatment rocessusing a downhole esistance-type lectrical heater.The heatis conveyed o the near-wellbore region by niuogen gaspassing hrough the heater ocated downhole by meansofconductionand convection.Transientsolutions of the mod-eled equationshave shown hat when initially both the gasin the wellbore and he reservoirare n equilibrium with dIe

to be more rounded or a flow rate of 10 m3(STP)/min ndi-cating a greatervertical dispersionof hot gas.Although thehighest emperature t a 0.5 m radial distance s achievedata flow rate of 6 m3(STP)/min, le average emperature verthe vertical distance s seen o be almost he same or thesetwo cases Runs D and E in Table 2).The temperature rofile in dle radial direction at a fIXedvertical location of 1.1 m from the bottom of the modeldomain s presented n Figure 6 as a function of total flowrate. As expected, he ncreasen flow rate results n a lowertemperatureof dle gas exiting the heater becauseof fIXedheaterpower (55 kW). It is important o note dIat the higherexit gas emperaturewill not necessarily rovide higher heatpenetration nto dle near wellbore region. Fluid velocity inthe porousmedium will play an important role in achievinga higher te~ture at various radial distances.At a lowflow rate, 2 m3(STP)/min, he temperaturen the near well-bore region (within 0.1 m) is very high (> lOOOOC).However, this temperaturequickly drops off to less than200C at a radial distanceof 1 m. This is an indication ofconduction dominatedheat ransfer mechanism-Low flowrate will result in high temperature, owever, the velocityrelated o low flow rate is so small dIat heat penetration nthe porous medium will also be small (Figure 6). On dleother hand,at a high flow rate, 10 m3(STP)/min, he exit gastemperatures low (400C), but the temperatureat a radialdistanceof 1 m is around 3000C.This is an indication ofconvection dominated heat transfer mechanism.Since dlepower is limited to 55 kW, a flow rate in the range of 4 to10 m3(STP)/min will provide a temperaturegreater than300C within dle target radial distanceof 1 m.The maximum emperature t a specific radial distance sdependent n dle gas flow rate. If one wanted a high nearwellbore temperature,a low flow rate is recommended.However, f a greaterheatpenetration nto the formation isrequired, hen a higher low ratewill be needed. n this case,greaterheatpenetrationwill have o be compromisedwith alower temperature.Figure 7 presents emperatures s a function of flow rateat a fixed vertical location of 1.1 m from the bottom of themodel domain or various adial ocations.For a fIXedheaterpower of 5S kW (Figure 7) and at a 0.5 m radial distance,the ~~ture reaches maximum of 4QOOCt a flow rateof 6 m (STP)/min. As expected, ower temperaturesareachievedat a radial distance f 1 m. Close o the casingwall(O.l m), however, he highest emperatures ~ at the o~flow rate,2 m3{STP)/min. his is becausehe hot gas s stillin the tubular region.The variation in temperature rofile at a fixed radial dis-tanceof 0.5 m for threepower ratings s presented n Figure8. These hree runs were carried out at a 4 m3(STP)/minflow rate and 900/0 low is flowing through the tubing. Toachievea maximumtemperature f SOO+C t 0.5 m into thereservoir,a downhole heaterof at least 80 kW is required.As explained earlier, the availability of suitable powercable, voltage osses n the cable ~1500 m long) and inter-nal diameter of the well casing limits dle practicality ofusing higher power in a resistive ype heating device.To study he effect of variation n dle tubing flow fractionon the attainable emperature t a radial distance of 0.5 mfrom dle centre of dle wellbore, two caseswere studied.Variations n the tubing flow fraction of 800/0 nd 900/0 idnot have significant impact on the average emperature t aradial distanceof 0.5 m (Table 2).

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997784


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geothennal emperatureprofile, the time required for thenear wellbore region (confmed region) to reach thennalequilibrium is less han 30 minutes.Resultsshow that for avolumetric gas flow rate of 6 m3(STP)/minand a heaterpower of 55 kW a temperature f 400C can be attainedat aradial distanceof 0.5 m in the reservoir. Simulation resultsindicate hat to achievea higher temperature t 0.5 m in thereservoir, a higher-power heating system is required.Calculations have revealed hat the total heat loss to theunproductivestrata above and below the fonnation is lessthan 5% of the power rating of the heater.

References

AcknowledgementsThe authorswish to thankNorandanc. and NoreenEnergyResourcesimited or thepennissiono publish his work.

Nomenclaturea - geothemlalemperatureradient,Clmd - diameter f the ubeor theequivalent iameter f the

annuIus,mdJi - innerdiameter f the ube,m(//0 - outerdiameter f the ube,mdci - inner diameterof the casing,mD - diameterof the tube, m .f - turbulent riction factorh - height of the formation (net pay), mhi - equivalentconvectiveheat ransfercoefficients at the

formaion, W/m2.Kh2 - equivalentconvectiveheat ransfercoefficients at over-burden,W/m2.Kh3 - equivalentconvectiveheat ransfercoefficients atunderburden,W/m2.Kh. - equivalent convective heat transfer coefficients at thetop of the overburden,W/m2.KH - total height of the tube, mk - penneability of the porousmedium, mDkb - thermalonductivityf theoverburden,/m.Kke - effective thermalconductivity, W/m.K1a - dlennal onductivity f the luid in theporousmedium,W/m.KPo - volume averaged ressure, PaPi - average nlet pressuren the tubing or annulus,kPaPMf - pressuren the tubing annulus,kPaPd" - pressure t the inlet of the tube or annulus,kPaq - ~ volumetricgas low rate hrough he tube or annulus,m3(STP)/minQ - volumetric eat ource,W/m3Re - Reynoldsnumber or the tube or annulus,Re= vD/~Tag - averageeothermalemperature,Cv - velocity of fluid, m/secUo - volume averaged Darcian) velocities n the axialdirection, m/secV0 - volume averaged Darcian) velocities n the radialdirection, m/secx - axial distance tom the top of the lower segment,mXL - vertical height of the computationaldomain, m

YL - radial depthof the computationaldomain, mZ - gas compressibility actorGreek lettersr, - specific gravity of the gasE - roughnessn the tube or the tube casingannulus.u - fluid viscosity, mPa.sp - fluid density,kg/m3(pC,>/- volumebic eat apacity f the luid. J/m3.oC(pC'",J.r volumebic heat capacity of the solid. J/m3.oC; - porosity f theporousmediumfJ - representsnydependentariable

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Sloat. B. F., "Nitrogen Stimulation of a Potassium HydroxidWellbore T~ent", U.S. Patent4,844,169 1989).ScxneI1oo.W. H., "ThennaI Propertiesand Temperature RelatBehaviour of Rock/FIuidSystems,Elsevier, New York (1992pp.215-222.Theog, B. K. G., "The OIemiltl'y of Clay-Organic ReactionsHalstedPressDiv., John Wiley &:.Sons,New York (1984).Thomas,R. L. andC. W. Crowe, Matrix TreatmentEmploys NeAcid System or Stimulationand Control of Fines Mi~tion Sandstone mations", J. Petrol.Techno ., 1491-1SOOAugus1981 .White, P. D. and J. T. Mass, "High Temperature ThermT~hniques for Stimulating Oil R;overy", J. Petrol. T~hnol1007-1015 September, 965).Winckler, E. and J. W. McManus, "Method and Apparatus fRemovalof Oil Well Puaffin", U.S. Patent4,911,239 1990)

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Manuscript ivcd May28, 1996;evisedmanuscript ivcdFebruary 1,1997;acceptedor publicationMarch 12, 1997.

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