Mathematical Modelling of Formation Heat TreatmentProcess

A. K. M. JAMALUDDl~ and C. T. BOWEN

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

and

M.HASAN

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

A novel matrix stimulation concept. fonnation beat treatment (FHT), wbicb involves the application of intense beataround die near-wellbore region for the treatment of water blockage and clay related fonnation damage in water sensi-tive formations previously was developed and presented in the literature. The FHT process involves the application ofintense beat around the weUbore using a downbole beater. The beat is conveyed to the near-wellbore region using aninert gas flowing through a downhole beater.

To understand the beat b'anSfer and fluid-flow cbaracteristics of die FHT process, a transient two-dimensional math-ematical model bas been developed and is presented in this paper. The model is based on coupling the momentum andenergy-balance equations for the wellbore gas with the surrounding porous formation. The presence of the heater acrossdie net pay (sandface) is1aken into account in die energy equation as a localized volumetric heat source. A control-volumebased finite-difference scheme is used to solve the model equations on a staggered grid. Parametric studies indicate thatby injecting a suitable quantity of gas through die tube and annulus, and by adjusting the power input to the downholeheater, die temperature near the wellbore can be favourably controlled.

On a mil au point et presenre anrerieurement dans la litterature scientifique un nouveau modele de simulationmatricielle pour Ie traitement de chaleur de fomlation (FHT), base sur I'application d'une chaleur intense autour de laregion du puits de forage pour Ie traitement du blocage des pores et des dornrnages a la formation lies a l'argile dans leiformations sensibles a I' eau. Le procede FHT utilise pour ce faire une chaudiere a orifice descendant. La chaleur est con-duite a la region du puits a I'aide d'un Ccoulement de gaz inerte dans la chaudiere a orifice descendant.

Afin de comprendre lei caractCristiques du transfert de chaleur et de I'ecoulement des fluides du procede FHT, on amis au point un modele mathematique transitoire bidimensionnel. Ce modele s'appuie sur Ie couplage des equationsd'Cquilibre de conservation de la quantitC de mouvement et d'energie pour Ie gaz du puits de forage avec la formationporeuse environnante. La presence de la chaudiere dans Ia production nette (cOte sable) est prise en compte dans I'equa-non d'energie comme source de chaleur volum~ue localisee. On utilise un schema de differences tiDies base sur lesvolmnes de conuole a grille decalee. Les etudes parametriques indiquent qu'en injectant une quantire adequate de gazdans Ie tube et I'espace annulaire, et en ajustant la puissance dans la chaudiere a orifice descendant, on peut contr6lerfavorablement la temperature pres du puits de forage. .

Keywords: wellbore damage. clay swelling. water blocking. fonnation beat treatment. simulation. temperature profile.

P etroleum engineering operations such as drilling, com-pletion, workovers, and stimulation, expose the forma-

tion to a foreign fluid. This exposure results in fluid invasioninto the near wellbore region. The permeability of the fluidinvaded porous zone is reduced because of pore throat con-striction caused by clay swelling, clay migration and waterblocking. This fluid-invaded region with reduced penne-ability is called the "damaged zone," extending roughly 1 minto the reservoir. Clay-related formation damage duringdrilling and completion has long been identified to be amajor problem. Measures to stabilize clay swelling andmigration have been discussed in the literature (Himes et al.,1991; Borchardt et aI., 1984; Theng, 1984; Reed, 1974;CoppeD et aI., 1973; Plummer, 1991). Curative methodshave also been attempted and presented in the literature(Hayatdavoudi et aI., 1992; Lund et aI., 1976; Thomas and

Crowe, 1981; Garst, 1957; Sloat, 1989; Schaible, 1986;Crowe, 1986). The two most popular non-thermal stimula-tion processes are hydraulic fracturing and matrix acidizing.

One of the earliest reports of in situ thermal b:eatment wasthat of Albaugh (1954), on a field test that was carried out inan oil weD in California. Since then, many other curativethermal processes have been described for a variety of pur-poses, including the removal of wax (Nenniger, 1992) orasphaltene (Winckler and McManus, 1990) buildups. ther-mal fracturing of the formation (White and Mass, 1965), andthe consolidation of unconsolidated formations (Friedmanet aI., 1988). More specifically related to clay damage aremethods aimed at evaporating blocked water (Reed.1991a.b), dehydrating bound water from clays (White andMass. 1965; Braun, 1971). or transforming a sensitive typeof clay (e.g. smectite) into a less sensitive type (e.g. illite)(Carroll, 1970; Nooner, 1980).

A new matrix stimulation concept, called formation heattreatment (FHT), was tested in the laboratory and in the field(Jamaluddin and Namrko, 1994; Jamaluddin et aI., 1995,1996a). The FHT process involves die application of beat for

-Author to whom correspondence should be addressed. Present address:Hycai EtIerIY Rearch Laboratooes Ud.. 1338A - 36th Avenue N.E..

Calpry. Alberta, Canada T2E 6T6.

mTHE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997

I-.DER8URDEN

Figure I - Schematic diagram of the fonnation heat treatment(FliT) process logistics.

the treatment of near wellbore damage. The heating aroundthe wellbore is achieved using a downhole heater (Jamaluddinet al., 1996b) located at the sandface. The heat is conveyedfrom the heater to the near wellbore region by an inert gas(e.g. nitrogen) flowing through and around the heater.

To understand the heat transfer and fluid flow character-istics of downhole heating, various modelling efforts havebeen presented in the literature. A one-dimensional mathe-matical model (Shamla et al., 1989) was described for pre-dicting the flowing temperature profile in a well with adownhole heater. The model was identical to Ramey's(1962) model, except that the heater was included throughthe treatment of a source term in the one-dimensional heatbalance equation. Another mathematical model to calculateheat losses to the surrounding formations due to the down-ward injection of hot fluid through the tubing was presentedby Hoang (1980) and referenced by Somerton (1992). Themodel divided the wellbore and its swroundings into tworegions: tubing and formation. Hoang assumed that the heatedfluid flowing through the tubing was losing heat radially tothe surroundings, while in the formation heat was assumedto be conducted both radially and vertically. The analysisdid not take into account any penetration of the hot fluid intothe formation. Hoang's analysis showed that for an injectionrate of 30,000 kg/h of the hot fluid, temperature profileswithin the entire length of the tubing and swrounding for-mations reached a steady state within a few hours. He con-cluded that for a high injection rate of the hot fluid throughthe tubing, a transient analysis of the model equations wasnot necessary.

None of these studies coupled heat transfer and fluid flowphenomena for the injection of a gas in a wellbore where thegas was heated while passing through a downhole heater.All prior studies in this area were variations of Ramey's(1962) original work, where the fluid momentum equationwas completely ignored. Furthermore, almost all studies inthis area have combined the steady-state heat conductionsolution for the wellbore with the approximate and UDSteady-state heat conduction solution for the surrounding rock.

In this study, a two-dimensional mathematical model hasbeen developed for the simulation of localized wellbore heat-ing using nitrogen gas as the injection fluid. The presence ofa downhole heater has been accounted for by incorporatinga volumetric heat source term in the transient energy equation.The model uses two-dimensional axisymmetric turbulentNavier-Stoke's equations and energy equations. Specifically,the model deals with the electrical heating of the nitrogengas near the formation and predicts the flow fields and con-vective and conductive heat transfer characteristics betweenthe heated gas and the surrounding reservoir. The model canbe used to quantify the power requirement of the downholeheater and the heat propagation in the near wellbore regionduring the formation heat treatment process. In addition, themodel allows for the optimization of the operating parameters.

Formation beat treatment (FHT) process logistics

A series of bench scale heating tests was carried out onsandstone cores taken ftom both oil- and gas-bearing for-mations (Jama1uddin et aI., 1995). Sample cores taken ftomactual formation displayed an 84% reduction in penneabilityfollowing water exposure. Heating to a temperature around400°C re-established the baseline penneability of the core.Further heating at 600 and 800°C improved the penneabilityto 500/0 and 7600/0 above the baseline value, respectively.

The physical situation and field logistics of the formationheat treatment process are presented schematically in Figure I.As seen in the figure, a downhole heater is attached to theend of a tubing and placed across the sandface. After lower-ing the tool, nitrogen gas is injected through both the tubingand the casing tubing annulus ftom the surface. The well ispressurized to a pressure higher than the correspondingreservoir pressure forcing the nitrogen into the reservoir.After pressurization, the tool is powered up to heat the injec-tion nitrogen while it is flowing through the downholeheater and convey the heat to the near weUbore region of thereservoir.

The primary objective of the FHT process is the intensebeating of the near weUbore region extending to I m radiallyin the reservoir. The duration of the total heating period isdesigned to be 6 to 8 hours. The heating period starts with aslow power up sequence and continues with a one hour heat-ing period to establish steady state conditions after reachingthe target temperature of the gas exiting the downholeheater.

To validate the field logistics and design, a multi-chamber,multi-pass, 60 kW electrical resistance type heating systemwas designed (Jamaluddin et aI" 1996b), constructed, andtested. Due to voltage losses in the cable and limited spacewithin the wellbore, the heater was restricted to 60 kW. The

The formation heat treatment (PHT) process consists ofexposing the formation to an elevated temperature to cause:

- vaporization of blocked water,- dehydration of the clay structure,- partial destruction of the clay minerals, and- possibly, micro-fracture of the formation in the near-

wellbore area due to thermally induced sttesses.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 7S, AUGUST. 1997118

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 800°C. 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'-*IQ

4. L

~

Model development

1~1oc:dcx1~L Yt: ..Centerlne

Figure 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 in the tubing and tubing-casingannulus for upper segment were estimated using Equation(1) (Beggs, 1984; Carcoana. 1992), which is derived fromthe average pressure and temperature method. This calcula-tion provided the input parameters for the lower segment

p2wf=P}e'+25 Ygq2 TagZfH(e'-l)/sJ5 (1)

In Equation (I), P wfis the pressure in die tubing or annu-lus, P If is the pressure at die inlet of the tube or annulus, His die total height of the tube, Tag is die average geothermaltemperature, q is the total volumetric gas flow rate throughthe tube or annulus, d is the diameter of the tube or theequivalent diameter of the annulus,fis the turbulent frictionfactor and is calculated using Equation (2); parameter S iscalculated using 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 by 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) is injected through the tubingand the remaLT!ing (100/0) is 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 in 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 of the tube or the tub-ing-casing annulus, Re is the Reynolds number for the tubeor annulus.

779THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997

S = 0.0375 Y, HITar Z. (3)

In Equations (1) and (3), Yl.is the specific gravity of thegas, and Z is the gas compressibility factor evaluated at Tag'

loWER SEGMENT OF THE WEU.BORE (DOMAIN MODELLED)

In Equation (5), the second ternl on the left hand side isthe buoyancy term and the quantity Q in Equation (7) is thevolumebic heat source. The value of Q is zero, except in theheater region, where a volumetric fraction of total heat isassigned.

The flow of gas through the perforated casing, perforatedcement, damaged zone and formation is assumed to be gov-erned by the non-Darcy flow equation. Specifically, theBrinkmann extended non-Darcy model (Chan et. al., 1991;Mishima and Szekely, 1989) is used to incorporate the vis-cous effect of the gas in the near wellbore region where fluidvelocities are high. In modelling the flow in this region, thefollowing assumptions are 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 account

through the Boussinesq approximation.With the above assumptions, the general macroscopic

conservation equations for mass, momentum and heattransfer applicable below the overburden and through thecasing, cement, damaged zone and formation 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)-=0Or

Momentum Equations

Axial momentum equation (UD-momenmm equation)

Continuity

~&l=_~_gppf(T-Tr)at axc3(pU) 1 ~-+-

Ox r=0 . (4)

or ~ + ! l. (,. ~ ),l_..e ~& ,.81' Or ~ K

(9)+J1Momentum Equations

Axial momentum equation (UD-momentum equation) Radial momentum equation (V o-momentum equation)

~+~~+!~~= r ~~ = -~at Orat Ox Or

8P [ a2U 1 8 ( aU)~---gPP(T-7:)+J1 -;:r+-- r-Ox ,. & rBr Or (5) +J1 [~+!.!. (r~ ) -~ ] -~. Ox rOr 8r ~ K (lO)

Radial momentum equation (V D-momentum equation)

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) f+(pCp).(l-f) T+i!~~l£~dat Ox

Ilr(pC,),Yorl [ a2T I a ( aT )~+- 1.'\r-'J,'~ J_~ "";:I"+-- r- .

r &- Ox rar ar(11)Energy Equation

where U D and V D are the volume-averaged (Darcian) veloc-ities in the axial and radial directions, respectively; P D is thevolume averaged pressure; (pC,)fand (pCp>,r are 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

a2T 1 a( or,- -;:,-+-- r~ar rbr ar

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997780

is the effective thermal conductivity; ka is the therDlal con-dlM:tivity of the fluid in the porous medium; 41 is the porosityof the porous medium. Because of radial symmetry, only onehalf of the domain shown in Figure 2 has been considered.

The above general equations for a porous medium havebeen modified to account for the flow through the perforatedcasing and cement, damaged and formation zones.

Initial Conditions

The initial conditions are:

(12)u= V= 0 (at t = 0)

. (13)7'(1 = 0, x, r) = 7'(1 = 0, x = 0, r) + a xx

(14)p( t=O x=O r ) =P., , I

where a is the geothermal temperablre gradient; x is theaxial distance from the top of the lower segment; Pi is theaverage inlet pressure in the tubing or annulus.

Boundary Conditions

At T> 0, the boundary conditi"ons are:

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

solved simultaneously as a single domain problem. Thefmite-difference equations were derived by integrating thedifferential equations over an elementary control volumesurrounding a grid node appropriate for each dependentvariable (patankar, 1980). A staggered grid system was usedso that the scalar properties, P and T, were stored midwaybetween the U and V velocity grid nodes (patankar, 1980).A power-law scheme (patankar, 1980) was used for the con-vective tenDS, and the integrated source terms were lin-earized. The pressure-velocity coupling of the momentumequations was resolved using the popular SIMPLER algo-rithm (patankar, 1980). The governing finite-differenceequations were solved iteratively by the tri-diagonal matrixalgorithm (TDMA) and using a bloc-correction scheme withunder-relaxation until the solutions converged.

Simulations were performed using non-uniform grids inboth axial and radial directions. A non-uniform matrix of 80by 80 nodes was used in the simulation. In the axial direc-tion, the first 7 metres from the bottom of the reservoir wasdivided into 0.1 m layers. The next 3 m distance was dividedinto 0.5 m layers and the remaining 10 m distance at the topof the model domain was divided into 2.5 m layers. In theradial direction, the first 0.085 m was divided into 0.005 msteps and the remaining distance was divided into 0.078 msteps. The use of non-uniform grids in the radial directionwas crucial because of relative dimension of the wellbore(casing diameter = 0.11 m), which is extremely small com-pared to the radial extension of the model domain (5 m). Thenon-uniform grid in the axial and radial directions was suit-ably placed to accommodate the various interfacial bound-aries. The axial grid distance was chosen to accommodatethe heater region. The solutions were to be converged, whenthe following criterion was satisfied simultaneously by eachcomputed variable:

(16)

=h,(T-Tz) .

(18)aT

-k.-a;=~(T- Tx>

(19)at x = 0, 0 < r< dll/2, U= Vi' V= 0, T= Ti

(20)at x = 0, d,/l < r < dcj12. U= u" v= 0, T= T,

(21)~+I ",-

,iJ -"'i,J < 0.001 Max _+1

iJ

where ~i. represents any dependent variable and (n + 1)refers to ihe value of QiJ at the (n + 1)1h iteration level. Toreduce computing time, the convergence criterion was mon-itored and it was identified that the relative differencebetween the pammeter values of two consecutive iterationswere within 0.001 after 700 to 800 iterations. As an exam-ple. the relative changes in temperature values are plotted inFigure 3. As seen in the figure. the relative changes in para-meter values start to fluctuate after 800 iterations. The tem-perature values calculated using engineering approximation(mC~7) match very well with the average of the simulatedtemperatures of the exit gas after 800 iterations. Therefore.all simulation nms were tenninated after 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' dlO and 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

781THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997

TABLE IParameters Used in the Simulation Runs

ValuesParametersReservoir. overburden and underburden chIrICteriIticI

Reservoir depth from surface (m) ISOONet pay (m) Spcxosity I SDamaged zone penneability (mD) SReservoir penncability (mD) 2SHeat loss coefficient for over/under bW"dcn (W/m2.K) 2.5Heat loss coefficient for reservoir (W/m2.K) 3.0Reservoir temperature (DC) S2

Relevant dimensions of well, cement and damaged ~Inner radius of the heater (m) 0.04Outer radius of the heater (m) 0.045Inner radius of the casing (m) 0.055Outer radius of the casing (m) 0.065Outer radius of cement (m) 0.085Outer radius of damaged zone (m) 0.961Outer radius of formation (m) 5.0

region was large, resulting in a high rate of conductive andconvective heat transfer. As a result, the temperature adja-cent to the weUbore rose quickly. Because the segments ofthe casing, cement and fonnation adjacent to the heaterlargely controUed the heat transfer rate, the temperaturetherefore became constant when these segments approachedthennaI equilibrium. Convective heat transfer through theseporous segments played a significant role in the attainmentof thennal equilibrium in a short time. Therefore, subse-quent analyses concentrated on the steady-state solution ofthese equations.

30 x 10-"1090

0.0516

Nitrogen characteristicsKinematic viscosity (mo.)Specific beat (J/kgoOC)Thenna1 conductivity (W/moK)

Equivalent Conv~tive Heat Transfer Coefficients (W/m2.K)hi: Heat transfer coeff'1Cient for 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 thermal properties of cbe reservoir, ov~ andunderburden are taken from Butler (1991).

STEADy STATE SOUn'K)NS

The parameters used in these simulation nms are present-ed in Table 1. Simulation conditions, average temperaturesof the exit gas and average temperabUes at a radial distanceof 0.5 m into the reservoir are tabulated in Table 2. Theresults are p~ted graphically in Figwa 4 through 8.

To undelStand the practical feasibility of the FHT processand to identify the critical parameters affecting this down-hole heating process, various simulation nms were carriedout. During the simulation nms, the controllable critical~ such as nitrogel1 flow rate and total power require-ment at the heater were varied. The effects of these changesunder steady state conditions on the temperature of the gasleaving the heater, depth of heat penetration into the reser-voir and the temperablre distribution in the near-wellboreregion were studied and the results are presented in thispaper. Based on these parametric studies, conditions wereselected for field testing of the tool and the FHr process.

An example temperature contow' plot is presented inFigure 4 (Run B, Table 2). As seen in this tig\R, the high-cst temperature is seen to be concentrated around the hotregion of the beater (1 to 4 m). As expected, the temperaturegradually decleases radially to the reservoir temperature.There are no apparent changes in the temperature due to per-meability variation from damaged zone (5 mD ex~ing toI m) to the rest of the reservoir (25 mD). This is possiblybecause of low velocity of nitrogen gas in the porous medi-um. Under pressure the hot nitrogen gas, exiting the heater,enters through the perforated casing and cement and travelsinto die porous reservoir. Gas flow into the region below theheater widlin die wellbore and up the annular space is min-imal as the only exit is through die porous formation.

The model was originally developed for transient solu-tions of the transport ~tions. The transient solution to thegoverning equations uses a fully implicit scheme. Omissionof transient terms from the model equations resulted insteady state solutions. In this paper, results related to steadystate solutions of the modelled equations are presentedalong with a brief discussion of the transient solutions.

Results and discussion

TRANSIENT ~

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 that at every instant the gas, surrounding per-forated casing, cement and porous formation were in a ther-mal equilibrium condition (i.e. there was no thermal disper-sion effect). Also, it was initially assumed that gas in thewellbore and the formation were at a temperature given bythe (constant) ambient surface temperature plus the productof depth and geothermal gradient (assumed to be constant).

Transient calculations at full power input to the toolrevealed that within 30 minutes, the near wellbore regionreached a thennal equilibrium condition. This short timerequired to reach a steady state was due to heating a con-fined region resulting in low heat losses (about S%) to theunproductive strata above and below the fonnation.Initially, when the heater was twned on, the temperature dif-ference between die gas and die surrounding near-wellbore

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 7S, AUGUST, 1997782

TABLE 2Swnmary of Conditions Used in These Simulation Runs and Selective Results

55555555S5804034

2446

1044S.S

9090809090909070

1417735820507325

1045548446

215262261294295346210201

ABCDEFG

H (field test)... Engineering calculations were carried out using mCpAT relations (flow going through the tube is assumed to pick up all heat)... Measured temperature during field test (non steady state): 382°C

0 2 4 6 8 10 12

T0t8 Flow. m3(STP)/min

Figure 7 - Temperature as a function of total flow at 1.1 m fromdie bottom of die model domain (Heater power: 55 kW; 90% flowthrough tubing).

.0

P-

Ii

0 1 2 3 4 5

Radial DIsIance. m

Figure 6 - Temperature profile in dte radial direction as a func-tion of total flow rate at 1.1 m from dte bottom of the modeldomain (Heater power: 55 kW; 900/. flow dtrough tubing).

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, theheater 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 in the figure, temperature increasesand reaches a maxima within the flow rates of about 2 to 10m3(STP)/min. As expected, in all situations, the maximumtempemturc is at the heater openings from where the hot gasexits the heater. The vertical temperature profile is also seen

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 75, AUGUST, 1997 783

To evaluate the efficiency of the stimulation process, anoverall heat balance was carried out for the near wellboreregion. At a steady state, the heat dissipated by the heaterequals the heat losses in the over and underburden and heatgain in the reservoir. The heat loss calculations using 55 kWpower, a total flow of 4 m3(STP)/min, flow through the tub-ing indicated that less than 5% of the 55 kW input powerwas lost in the unproductive strata (overburden and under-burden).

Simulated vs measured temperature from field testingofFHT

To validate the FHT concept in the field, a proprietaryresistance-type electrical 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 382°C, 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 700°C. 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 382°C. The thermocou-ple was located at the bottom end of the heater as shown inFigure 2. An engineering calculation based on 700/0 nitrogenflowing through the heater would correspond to 446°C 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 473°C.Correspondingly, this exit. gas would have resulted in anaverage temperature of 200°C 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 mathematical model presented in this paper demon-strates the feasibility of the formation heat Ueatment processusing a downhole resistance-type electrical heater. The heatis conveyed to the near-wellbore region by niuogen gaspassing through the heater located downhole by means ofconduction and convection. Transient solutions of the mod-eled equations have shown that when initially both the gasin the wellbore and the reservoir are in equilibrium with dIe

to be more rounded for a flow rate of 10 m3(STP)/min indi-cating a greater vertical dispersion of hot gas. Although thehighest temperature at a 0.5 m radial distance is achieved ata flow rate of 6 m3(STP)/min, dle average temperature overthe vertical distance is seen to be almost the same for thesetwo cases (Runs D and E in Table 2).

The temperature profile in dle radial direction at a fIXedvertical location of 1.1 m from the bottom of the modeldomain is presented in Figure 6 as a function of total flowrate. As expected, the increase in flow rate results in a lowertemperature of dle gas exiting the heater because of fIXedheater power (55 kW). It is important to note dIat the higherexit gas temperature will not necessarily provide higher heatpenetration into dle near wellbore region. Fluid velocity inthe porous medium will play an important role in achievinga higher te~ture at various radial distances. At a lowflow rate, 2 m3(STP)/min, the temperature in the near well-bore region (within 0.1 m) is very high (> lOOOOC).However, this temperature quickly drops off to less than200°C at a radial distance of 1 m. This is an indication ofconduction dominated heat transfer mechanism- Low flowrate will result in high temperature, however, the velocityrelated to low flow rate is so small dIat heat penetration inthe porous medium will also be small (Figure 6). On dleother hand, at a high flow rate, 10 m3(STP)/min, the exit gastemperature is low (400°C), but the temperature at a radialdistance of 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 temperature greater than300°C within dle target radial distance of 1 m.

The maximum temperature at a specific radial distance isdependent on dle gas flow rate. If one wanted a high nearwellbore temperature, a low flow rate is recommended.However, if a greater heat penetration into the formation isrequired, then a higher flow rate will be needed. In this case,greater heat penetration will have to be compromised with alower temperature.

Figure 7 presents temperatures as a function of flow rateat a fixed vertical location of 1.1 m from the bottom of themodel domain for various radial locations. For a fIXed heaterpower of 5S kW (Figure 7) and at a 0.5 m radial distance,the ~~ture reaches a maximum of 4QOOC at a flow rateof 6 m (STP)/min. As expected, lower temperatures areachieved at a radial distance of 1 m. Close to the casing wall(O.l m), however, the highest temperature is ~ at the lo~flow rate, 2 m3{STP)/min. This is because the hot gas is stillin the tubular region.

The variation in temperature profile at a fixed radial dis-tance of 0.5 m for three power ratings is presented in Figure8. These three runs were carried out at a 4 m3(STP)/minflow rate and 900/0 flow is flowing through the tubing. Toachieve a maximumte mperature of SOO+°C at 0.5 m into thereservoir, a downhole heater of at least 80 kW is required.As explained earlier, the availability of suitable powercable, voltage losses in the cable (~1500 m long) and inter-nal diameter of the well casing limits dle practicality ofusing higher power in a resistive type heating device.

To study the effect of variation in dle tubing flow fractionon the attainable temperature at a radial distance of 0.5 mfrom dle centre of dle wellbore, two cases were studied.Variations in the tubing flow fraction of 800/0 and 900/0 didnot have significant impact on the average temperature at aradial distance of 0.5 m (Table 2).

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

geothennal temperature profile, the time required for thenear wellbore region (confmed region) to reach thennalequilibrium is less than 30 minutes. Results show that for avolumetric gas flow rate of 6 m3(STP)/min and a heaterpower of 55 kW a temperature of 400°C can be attained at aradial distance of 0.5 m in the reservoir. Simulation resultsindicate that to achieve a higher temperature at 0.5 m in thereservoir, a higher-power heating system is required.Calculations have revealed that the total heat loss to theunproductive strata above and below the fonnation is lessthan 5% of the power rating of the heater.

References

Acknowledgements

The authors wish to thank Noranda Inc. and Noreen EnergyResources Limited for the pennission to publish this work.

Nomenclature

a - geothemlal temperature gradient, °Clmd - diameter of the tube or the equivalent diameter of the

annuIus,mdJi - inner diameter of the tube, m(//0 - outer diameter of the tube, mdci - inner diameter of the casing, mD - diameter of the tube, m .f - turbulent friction factorh - height of the formation (net pay), mhi - equivalent convective heat transfer coefficients at the

formaion, W/m2.Kh2 - equivalent convective heat transfer coefficients at over-

burden, W/m2.Kh3 - equivalent convective heat transfer coefficients at

underburden, W/m2.Kh. - equivalent convective heat transfer coefficients at the

top of the overburden, W/m2.KH - total height of the tube, mk - penneability of the porous medium, mDkb - thermal conductivity of the overburden, W/m.Kke - effective thermal conductivity, W/m.K1a - dlennal conductivity of the fluid in the porous medium,

W/m.KPo - volume averaged pressure, kPaPi - average inlet pressure in the tubing or annulus, kPaP Mf - pressure in the tubing annulus, kPaP d" - pressure at the inlet of the tube or annulus, kPaq - ~ volumetric gas flow rate through the tube or annulus,

m3(STP)/minQ - volumetric beat source, W/m3Re - Reynolds number for the tube or annulus, Re = pvD/~Tag - average geothermal temperature, °Cv - velocity of fluid, m/secUo - volume averaged (Darcian) velocities in the axial

direction, m/secV 0 - volume averaged (Darcian) velocities in the radial

direction, m/secx - axial distance ftom the top of the lower segment, mXL - vertical height of the computational domain, mYL - radial depth of the computational domain, mZ - gas compressibility factor

Greek letters

r, - specific gravity of the gasE - roughness in the tube or the tube casing annulus.u - fluid viscosity, mPa.sp - fluid density, kg/m3(pC,>/- volumebic heat capacity of the fluid. J/m3.oC(pC'",J.r - volumebic heat capacity of the solid. J/m3.oC; - porosity of the porous mediumfJ - represents any dependent variable

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Manuscript ~ivcd May 28, 1996; revised manuscript ~ivcdFebruary 21,1997; accepted for publication March 12, 1997.

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