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Energy 75 (2014) 419e429

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Energy

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Prediction of thermophysical properties of saturated steam andwellbore heat losses in concentric dual-tubing steam injection wells

Hao Gu a, *, Linsong Cheng a, Shijun Huang a, Baojian Du a, Changhao Hu b

a Department of Petroleum Engineering, China University of Petroleum, Beijing, 18 Fuxue Road, Changping 102249, Chinab Research Institute of Petroleum Exploration & Development, Liaohe Oilfield Company, PetroChina, Panjin 124000, China

a r t i c l e i n f o

Article history:Received 16 April 2014Received in revised form27 July 2014Accepted 30 July 2014Available online 28 August 2014

Keywords:Concentric dual-tubing steam injection wellThermophysical propertiesWellbore heat lossesPressure gradient in annuliInsulation materials

* Corresponding author. Tel./fax: þ86 10 89733726E-mail address: guhao110110@163.com (H. Gu).

http://dx.doi.org/10.1016/j.energy.2014.07.0910360-5442/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Concentric dual-tubing steam injection is important in the process of thermal recovery for heavy oils.This paper firstly presented a mathematical model to predict thermophysical properties of saturatedsteam (i.e. steam pressure, temperature and quality) and wellbore heat losses in CDTSIW (concentricdual-tubing steam injection wells). More importantly, a semi-analytical model for estimating pressuregradient for steam/water flow in annuli was developed. Then the mathematical model is solved using aniterative technique. Predicted results were compared with measured field data to verify the accuracy ofthe model. The results indicate that the direction of heat transfer between fluids in the integral jointtubing and in the annulus depends not only on wellhead injection conditions but on temperature drop ineach tubing. In addition, the steam qualities in CDTSIW are significantly influenced by heat exchangebetween fluids in dual tubing, which can cause steam boiling or condensation. Moreover, the papershows that to effectively reduce the wellbore heat losses and to ensure high bottomhole steam qualitiesin Well Xing 67 of Liaohe Oilfield, the thermal conductivity of insulation materials should be less than0.7 W/(m K).

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Steam injection techniques are widely used in the process ofthermal recovery for heavy oils, such as in steam stimulation,steamflooding and steam-assisted gravity drainage [1,2]. One of themost important reasons is that high-temperature steam carriesmuch heat, and injecting the heat into oil layers can reduce theviscosity of heavy oil whose mobility is relatively low under initialformation temperature. However, traditional single-tubing steaminjection technique is not perfect. For instance, in Liaohe Oilfield,Panjin, single-point steam injection method applied in horizontalwells always leads to obvious steam fingering phenomena anduneven exploitation of oil layers [3], especially in seriously het-erogeneous reservoirs. In addition, single-tubing steam injection isnot the best choice for multiple-oil-layer steamflooding when lowcost and easy control are taken into account [4,5]. In these cases,concentric dual-tubing steam injection may be one of the mosteffective measures to alleviate these problems. As steam flows in aCDTSIW (concentric dual-tubing steam injection well), the

.

thermophysical properties of saturated steam (i.e. steam pressure,temperature and quality) always changewith well depth, therefore,the first task in the design of steam injection projects is to predictthese properties before steam enters the oil layers [6]. Also, not allheat carried by steam injected from wellhead can enter the oillayers, there are still some heat losing from wellbore to the sur-rounding formation, so the second task is to predict wellbore heatlosses.

For the above two tasks in the design of steam injection projects,some classic researches have been conducted. Ramey [7] firstlypresented an approximate method for predicting fluid temperaturein wellbores on the assumption that heat transfer inside the well-bore is steady-state, while heat transfer in the formation is un-steady radial conduction. His work laid a foundation for subse-quent researchers, although he only considered single phase (idealgas and incompressible liquid) flow in the wellbore. Satter [8] tookinto account the effect of phase change and suggested a method forestimating steam quality, but he ignored kinetic energy changewhen modelling steam quality based on energy conservationprinciple. In addition, his assumption that pressure drops due topotential energy change and friction loss can cancel each other maynot be true in the case of a deep well or a high injection rate. Hasanand Kabir [9], whose work is very important in determining the

Nomenclature

a geothermal gradient, K/mCJ JouleeThompson coefficient, K/PaCp heat capacity at constant pressure, J/(kg K)De equivalent hydraulic diameter, mDii inside diameter of integral joint tubing, mdQan/dz wellbore heat losses or rate of heat flow from annulus

to the surrounding formation, W/mdQij/dz rate of heat flow from fluid in the integral joint tubing

to the annulus, W/mftp two-phase friction factor, dimensionlessf(t) transient heat-conduction time function,

dimensionlessg gravitational acceleration, m/s2

han specific enthalpy of mixture fluid in the integral jointtubing, J/kg

hc convective heat transfer coefficient, W/(m2 K)hfii forced-convection heat transfer coefficient on inside of

integral joint tubing, W/(m2 K)hfio forced-convection heat transfer coefficient on outside

of integral joint tubing, W/(m2 K)hf1i forced-convection heat transfer coefficient on inside of

tubing 1, W/(m2 K)hr radiative heat transfer coefficient, W/(m2 K)hs specific enthalpy of dry steam, J/kghw specific enthalpy of saturated water, J/kgJ0 first kind Bessel functions of zero orderJ1 first kind Bessel functions of first orderLv latent heat of vaporization of steam, J/kgN segment numbers or data numbersp pressure, Par radius distance from the center of the wellbore, mr1i inside radius of tubing 1, mr1o outside radius of tubing 1, mr2i inside radius of tubing 2, mr2o outside radius of tubing 2, mrci inside radius of casing, mrco outside radius of casing, mrh outside radius of the wellbore, mrii inside radius of integral joint tubing, mrio outside radius of integral joint tubing, mt injection time, hT0 surface temperature of the formation

T temperature, KTei initial temperature of the formation, KTh wellbore/formation interface temperature, Ku dummy variable for integration, dimensionlessU2o over-all heat transfer coefficient between the annulus

and the cement/formation interface, W/(m2 K)Uio over-all heat transfer coefficient between inside and

outside of integral joint tubing, W/(m2 K)n velocity, m/snsgan superficial gas velocity in the annulus, m/snsgij superficial gas velocity in the integral joint tubing, m/sW mass flow rate, kg/sx steam quality, dimensionlessxan(0) wellhead steam quality in the annulus, dimensionlessxij(0) wellhead steam quality in the integral joint tubing,

dimensionlessy dependent variablesY0 the second kind Bessel functions of zero orderY1 the second kind Bessel functions of first orderz variable well depth from surface, m

Greek lettersa thermal diffusivity of the formation (m2/h)lcas thermal conductivity of casing, W/(m K)lcem thermal conductivity of cement sheath, W/(m K)le thermal conductivity of formation, W/(m K)lins thermal conductivity of insulation materials, W/(m K)ltub thermal conductivity of tubing wall, W/(m K)r density, kg/m3

rns no-slip density of mixture fluid, kg/m3

u ratio of the formation heat capacity to the wellboreheat capacity, dimensionless

tD dimensionless timeq well angle from horizontal

Subscriptsij integral joint tubingan annulusm mixturemea measured valuepre predicted values dry steamw saturated water

H. Gu et al. / Energy 75 (2014) 419e429420

wellbore heat losses, established a formation heat-transfer modeland derived an expression for formation temperature distributionas a function of radial distance and injection time, although theeffect of wellbore heat capacity was not included in their study. Inrecent years, Cheng et al. [10,11] improved the formation heat-transfer model by considering the wellbore heat capacity andproposed a novel transient heat-conduction time function that willbe adopted to calculate the wellbore heat losses in this paper.

The above classic studies are significant bases of predictingthermophysical properties of saturated steam and wellbore heatlosses in CDTSIW. However, in order to successfully accomplish thetwo tasks, we must also overcome a critical bottleneck: how toaccurately estimate pressure gradient for steam/water flow indownward annuli. In fact, it is not always easy to solve this problemand this difficulty can further influence thewhole predicted results.Caetano [12], Hasan and Kabir [13], Antonio et al. [14,15] and Yuet al. [16] presented different mechanistic models to estimatepressure gradient for two-phase flow in annuli. In their models, the

flow mechanism and the transition criterion for each flow patternwere researched independently, and the governing equations forpressure drop and flow parameters for a given flow pattern werealso suggested. However, the calculation methods for intermediatevariables were very complicated and time-consuming. Moreimportantly, what they studied was upflow, which differs fromdownward steam/water flow, and the difference in flow directioncan affect buoyancy effect of gas bubbles, bubble distribution acrossthe channel, flow patterns and final computational model [17].Besides mechanistic models, empirical correlations were alsoadopted in previous works. Griston et al. [5] and Wu et al. [18]treated the annuli as pipes based on equivalent hydraulic diam-eter concept and calculated the pressure drop for two-phase flow inannuli with the methods that had been extensively employed inpipe systems. While for downward or upward gas/liquid flow inpipes, the calculation methods for pressure drop are relativelysimple and have been well verified and improved in practice[19e21]. Orkiszewski [22], Beggs and Brill [23] and Hasan et al.

H. Gu et al. / Energy 75 (2014) 419e429 421

[17,24] are the representatives of these methods. However, thehydraulic diameter is not always the most suitable characteristicdimension for two-phase flow in annuli [12]. Moreover, the flowpatterns in annuli have been proved to be different from pipe-flowpatterns [16], in other words, their method for pressure gradient indownward annuli may be a rough approximation.

The authors and our team have done a series of researches onprediction of thermophysical properties in the cases of saturatedsteam injection [20] and superheated steam injection [25], and onestimation of wellbore heat losses under unsteady wellhead in-jection conditions [26]. Based on previous studies, our team beginsto focus on the concentric dual-tubing steam injection techniquethat is applied in Liaohe Oilfield. In this paper, to accomplish theabove two tasks, we firstly establish a mathematical model topredict thermophysical properties of saturated steam and wellboreheat losses in CDTSIW. In this section, a semi-analytical model forestimating pressure gradient for steam/water flow in annuli wasproposed based on thermodynamic principles, mass and energybalances and limit and derivative theories in mathematics. Sec-ondly, the mathematical model is solved using an iterative tech-nique. Next, the accuracy of the theoretical model is verified bycomparisons of simulated results with measured field data. Finally,based on the validated model, the predicted results and the effectsof thermal conductivity of insulation materials are analyzed indetail.

2. Mathematical model

A simplified schematic of a concentric dual-tubing steam in-jection well is shown in Fig. 1.The concentric dual tubing mainlyincludes a 1.9-in. integral-joint tubing and a 41/2-in. insulatedtubing, and the latter consists of three components: tubing 1,insulationmaterials and tubing 2. In this article, in order to simplifythe model calculation, several major assumptions are made asfollows:

(1). The wellhead injection conditions (i.e. injection pressure,temperature, mass flow rate and steam quality) do notchange with injection time, which guarantees relativelyconstant profiles of thermophysical properties of saturatedsteam.

(2). Heat transfer inside the wellbore, including heat exchangebetween fluids in the integral joint tubing and in the annulusand heat transfer from fluid in the annulus to the cement

r

zdzd

z

ijir ijor 1ir 1or 2ir 2or cir cor hr

Insulation materials Tubing 1 Tubing 2

Integral joint tubing

Ann

ulus zzzzzzzzzzzzzzzddddddddddddddzzzzzzzz

Formation Cement Insulated tubing Casing

Cas

ing

annu

lus

Fig. 1. Schematic of concentric dual-tubing steam injection well.

sheath, is assumed to be steady-state, while heat transfer inthe formation is assumed to be transient radial conduction.Holst and Flock [27] and Wang [28] discussed the assump-tion in detail.

(3). The physical and thermal properties of the formation areindependent of the well depth and temperature [27].

(4). The casing annulus is filled with air at a constant pressure.

The mathematical model is mainly composed of three parts:steam pressure, wellbore/formation heat transfer and steam qual-ity, which will be introduced in detail.

2.1. Steam pressure model

2.1.1. Steam pressure in the integral joint tubingFor steam/water two-phase flow in the integral joint tubing, the

pressure drop can be calculated by using the classic method pro-posed by Beggs and Brill [23]. In their work, the flow patterns inpipes were divided into segregated flow, intermittent flow anddistributed flow. Moreover, for a given flow pattern, the liquidholdup was firstly estimated, then other key parameters used forpredicting the pressure drop were determined, such as two-phasemixture density and two-phase friction factor.

The mass and momentum balance equations for the fluid in theintegral joint tubing are given as follows:

Mass balance equation:

vWij

vz¼ pD2

ii4

v�rijnij

�vz

¼ 0 (1)

Momentum balance equation:

dpijdz

¼ rijg sin q� ftprnsDii

n2ij

2� rijnij

dnijdz

(2)

where Wij, rij, nij and pij are the mass flow rate, the density, thevelocity and the pressure of two-phase mixture in the integral jointtubing, respectively; Dii is the inside diameter of integral jointtubing; rns is the no-slip density of mixture fluid; ftp is the two-phase friction factor; g is the gravitational acceleration; q is thewell angle from horizontal.

The pressure drop due to kinetic energy change denoted by thethird term of the right side in Eq. (2) can be rewritten as Eq. (3), andthe detailed derivation can be found in Ref. [23].

rijnijdnijdz

¼ �rijnijnsgij

pij

dpijdz

(3)

where nsgij is the superficial gas velocity in the integral joint tubing.Substituting Eq. (3) into Eq. (2) yields the final expression for

pressure gradient in the integral joint tubing.

dpijdz

¼rijg sin q� ftprnsn2ij

2Dii

1� rijnijnsgij

.pij

(4)

2.1.2. A semi-analytical model for estimating pressure gradient forsteam/water flow in annuli

For pressure drop in the annulus, Griston et al. [5] and Wu et al.[18] treated the annuli as pipes, namely, Dii in Eq. (4) was replacedby equivalent hydraulic diameter, De ¼ 2(r1i � rio), where r1i is theinside radius of tubing 1 and rio is the outside radius of integral jointtubing. As stated above, this may be a rough approximation. Here, amore rigorousmethod is presented to predict steam pressure in theannulus.

H. Gu et al. / Energy 75 (2014) 419e429422

If dQij/dz and dQan/dz are defined as the rate of heat flow fromthe fluid in the integral joint tubing to the annulus and from theannulus to the surrounding formation, respectively, then a generalenergy balance on the annulus fluid can be written as.

1Wan

�dQan

dz� dQij

dz

�¼ �dhan

dz� ddz

�v2an2

�þ g sin q (5)

where dQan/dz � dQij/dz denotes the rate of net heat losses ofannulus fluid; Wan, han and nan are the mass flow rate, the specificenthalpy and the velocity of two-phase mixture in the annulus,respectively.

The first term of the right side in Eq. (5) represents the enthalpygradient, which can be written in terms of temperature and pres-sure gradients based on thermodynamic principles [29e31]:

dhandz

¼ CpmdTandz

� CJmCpmdpandz

(6)

where Tan and pan are the temperature and pressure of annulusfluid, respectively; Cpm and CJm are the heat capacity at constantpressure and the JouleeThompson coefficient of mixture fluid,respectively, the calculation methods for which are presented indetail in Appendix A.

Similarly, according to the derivationmethod proposed by Beggsand Brill [23], the kinetic energy change per unit mass of annulusfluid per unit length of the wellbore represented by the secondterm of the right side in Eq. (5) can also be rewritten as

ddz

�v2an2

�¼ nandnan

dz¼ �nannsgan

pan

dpandz

(7)

where nsgan is the superficial gas velocity in the annulus.Incorporating Eqs. (6) and (7) into Eq. (5) results in

1Wan

�dQan

dz� dQij

dz

�¼ CJmCpm

dpandz

� CpmdTandz

þ nannsgan

pan

dpandz

þ g sin q

(8)

A general relationship between temperature gradient andpressure gradient for saturated steam can be further created basedon limit and derivative theories in mathematics. Fig. 2 shows asegment of an annulus. Assuming that the steam temperaturevaries from Ti toTiþ1 and the steam pressure varies from pi to piþ1

Inlet: iT , ip

zΔ

θ

Outlet: 1i+T , 1i+p

θ

Integral joint tubing

Tubing 1

zΔzz

Fig. 2. A segment of an annulus.

when steam flows from the inlet to the outlet, then the temperatureand the pressure changes over Dz can be expressed as DT¼ Tiþ1� Tiand Dp ¼ piþ1 � pi, respectively. Consequently, the temperaturegradient can be obtained from the definition of derivative and thealgorithm of limit:

dTdz

¼ limDz/0

DTDz

¼ limDz/0

�DTDp

DpDz

�¼ lim

Dz/0

DTDp

$ limDz/0

DpDz

¼ limDz/0

DTDp

$dpdz

(9)

In Eq. (9), there are two methods to estimate the value oflimDz/0

ðDT=DpÞ.The first one is interpolation of steam tables [32],

however, not only is interpolation itself inconvenient in computerprocedure, but also this method for solving Eq. (9) is time andmemory consuming. Because in order to ensure that the interpo-lated results are precise enough, the wellbore must be divided intoa large number of segments so that the length of each segment (Dz)is very small and lim

Dz/0ðDT=DpÞ is approximately equal to DT/Dp.

Here, a more practical approach is recommended. According tosteam tables, there is one-to-one correspondence between satu-rated temperature and pressure. By regression analysis and poly-nomial interpolation, Ejiogu et al. [33] and Tortike et al. [34]proposed different empirical correlations to describe this rela-tionship. In this paper, the empirical correlation suggested by Tor-tike et al. [34] is adopted due to its high accuracy, and theexpression is given by

T ¼ f ðpÞ¼280:034þ14:0856 lnp

1000

þ1:38075�ln

p1000

�2�0:101806

�ln

p1000

�3þ0:019017

�ln

p1000

�4;611 Pa�p�2:212�107 Pa

(10)

Fig. 3 shows a comparison of the results from Eq. (10) and thosefrom steam tables. It is obviously found that the empirical corre-lation agrees very well with steam-table data, also, the maximumand the mean absolute residuals are only 0.11% and 0.03%, respec-tively [34]. Therefore, Eq. (10) can be used to simulate the rela-tionship between saturated temperature and pressure in thecomputer procedure. More importantly, the proposed empiricalfunction is differentiable, which lays a foundation for furthersimplification of Eq. (9).

200

300

400

500

600

700

0 3 6 9 12 15 18 21 24

Steam-table data

Empirical correlation

Tem

pera

ture

(K)

Pressure (MPa)

Water

Superheated steam

Critical pressure:22.12MPa

Critical temperature:647.30K

Fig. 3. Comparison of empirical correlation with steam-table data.

H. Gu et al. / Energy 75 (2014) 419e429 423

In Fig. 2, Dz/0 is equivalent to piþ1/pi, so limDz/0

ðDT=DpÞ can bedeveloped into another form:

limDz/0

DTDp

¼ limDz/0

Tiþ1 � Tipiþ1 � pi

¼ limDz/0

f ðpiþ1Þ � f ðpiÞpiþ1 � pi

¼ limpiþ1/pi

f ðpiþ1Þ � f ðpiÞpiþ1 � pi

¼ df ðpÞdp

(11)

where df(p)/dp can be obtained from Eq. (10):

df ðpÞdp

¼ 1p

�14:0856þ 2:7615

�ln

p1000

�� 0:305418

��ln

p1000

�2þ 0:076068

�ln

p1000

�3(12)

Substituting Eq. (11) into Eq. (9) yields the general relationshipbetween temperature gradient and pressure gradient for saturatedsteam:

dTdz

¼ df ðpÞdp

$dpdz

(13)

Incorporating Eq. (13) into Eq. (8) reduces to the governingdifferential equation for steam pressure in the annulus.

dpandz

¼1

Wan

�dQandz � dQij

dz

�� g sin q

CJmCpm � Cpmdf ðpÞdp

p¼pan

þ nannsganpan

(14)

2.2. Wellbore/formation heat transfer model

Steam/water two-phase flow in CDTSIW involves heat transferinside the wellbore and heat conduction in the formation. Barua[35] and Hight et al. [4] mentioned concentric dual-tubing steaminjection, but neither set up a concrete wellbore/formation heattransfer model. In this part, based on the above assumptions, wecompleted this work.

2.2.1. Heat exchange between fluids in the integral joint tubing andin the annulus

For concentric dual-tubing steam injection, there is heat ex-change between fluids in the integral joint tubing and in theannulus if the fluid temperatures are not equal. Based on Fourierlaw, the rate of heat flow per unit length of the wellbore can begiven by,

dQij

dz¼ 2prioUio

�Tij � Tan

�(15)

where Uio is the over-all heat transfer coefficient between insideand outside of integral joint tubing, representing the net resistanceto heat flow offered by films on inside and outside of integral jointtubing and the tubing wall. The detailed expression for Uio is givenas follows:

Uio ¼"

riohfiirii

þ rioltub

lnriorii

þ 1hfio

#�1

(16)

where rii is the inside radius of integral joint tubing; ltub is thethermal conductivity of tubing wall; hfii and hfio are the forced-convection heat transfer coefficient on inside and outside of inte-gral joint tubing, respectively.

In Eq. (15), the temperature of fluids in the integral joint tubingand in the annulus can be determined from Eq. (10):

Tk ¼ f ðpkÞ; k ¼ ij or an (17)

2.2.2. Heat transfer from fluid in the annulus to the cement sheathSimilarly, for steady-state heat transfer from fluid in the annulus

to the cement sheath, the heat flow rate can be expressed as

dQan

dz¼ 2pr2oU2oðTan � ThÞ (18)

where r2o is the outside radius of tubing 2; Th is the cement/for-mation interface temperature; U2o is the over-all heat transfer co-efficient between the annulus and the cement/formation interface,representing the net resistance to heat flow offered by films oninside of tubing 1, tubing 1 wall, insulation materials, tubing 2 wall,casing annulus, casing wall and cement sheath. The completeexpression for U2o is given by,

U2o ¼"

r2ohf1ir1i

þ r2oltub

lnr1or1i

þ r2olins

lnr2ir1o

þ r2oltub

�lnr2or2i

þ 1hc þ hr

þ r2olcas

lnrcorci

þ r2olcem

lnrhrco

#�1

(19)

where r1o, r2i, rci, rco and rh are the outside radius of tubing 1, insideradius of tubing 2, inside and outside radius of casing and outsideradius of the wellbore, respectively; hf1i is the forced-convectionheat transfer coefficient on inside of tubing 1; lins, lcas and lcemare the thermal conductivity of insulation materials, casing walland cement sheath, respectively; hc and hr are the convective heattransfer coefficient and the radiative heat transfer coefficient,respectively, which can be estimated by using the method pre-sented in Refs. [6,11,36].

Since hf1i, ltub and lcas are generally much higher than lcem, hcand hr, the resistances offered by films on inside of tubing 1, tubing1 wall, tubing 2 wall and casing wall are negligible. Consequently,Eq. (19) can be reduced to:

U2o ¼�r2olins

lnr2ir1o

þ 1hc þ hr

þ r2olcem

lnrhrco

�1(20)

2.2.3. Heat transfer in the formationUsually, the formation heat-transfer model is set up according to

energy balance on the formation. If vertical heat diffusion is nottaken into account due to relatively small vertical temperaturedifference [9,11,37], the rate of transient heat transfer in the for-mation can be give as Eq. (21), and the detailed derivation waspresented by Hasan et al. [9] and Cheng et al. [10,11].

dQan

dz¼ 2pleðTh � TeiÞ

f ðtÞ (21)

where le is the thermal conductivity of formation; Tei is the initialtemperature of the formation at any given depth, which is generallyassumed to vary linearly with well depth, Tei ¼ T0 þ azsinq, T0 is thesurface temperature of the formation, a is the geothermal gradient;f(t) is the transient heat-conduction time function.

In Eq. (21), in order to estimate the rate of heat flow in theformation, it is necessary to determine the expression for f(t).Ramey [7], Chiu et al. [38], Hasan et al. [9] and Bahonar et al. [39]proposed different empirical expressions, but in their studies,they ignored the wellbore heat capacity, which in fact has

H. Gu et al. / Energy 75 (2014) 419e429424

significant effects on heat transfer in the wellbore and the finalexpression for f(t), especially in short injection time. Recently,Cheng et al. [10,11] improved the formation heat-transfer model byconsidering the wellbore heat capacity and suggested a newexpression for f(t), which is given by Refs. [10,40]

f ðtÞ ¼ 16u2

p2

Z∞0

1� exp��tDu2

�u3Dðu;uÞ du (22)

where u is the ratio of the formation heat capacity to the wellboreheat capacity, which can be calculated by using the method pro-posed by Cheng et al. [10]; tD represents the dimensionless time,tD ¼ at=r2h, a is the thermal diffusivity of the formation, t is theinjection time; u is the dummy variable for integration and D(u,u) is[10,11,40]

Dðu;uÞ ¼ ½uJ0ðuÞ � uJ1ðuÞ�2 þ ½uY0ðuÞ � uY1ðuÞ�2 (23)

where J0 and J1 are the first kind Bessel functions of zero and firstorders, respectively; Y0 and Y1 are the second kind Bessel functionsof zero and first orders, respectively.

Combining Eqs. (18) and (21) to eliminate Th, we have

dQan

dz¼ 2pr2oU2ole

r2oU2of ðtÞ þ leðTan � TeiÞ (24)

2.3. Steam quality model

The methods for predicting steam qualities in the integral jointtubing and in the annulus are almost the same. Here, we take themodel of steam quality in the annulus as an example.

The specific enthalpy of wet steam can be represented byfunctions of steam quality and pressure [41]:

han ¼ hanðx; pÞ ¼ ð1� xÞhw þ xhs ¼ xLv þ hw (25)

where hw and hs are the specific enthalpies of saturated water anddry steam, respectively; Lv is the latent heat of vaporization ofsteam.

So the enthalpy gradient of mixture fluid in the annulus can alsobe written as

dhandz

¼ dxdz

Lv þ xdLvdp

p¼pan

$dpandz

þ dhwdp

p¼pan

$dpandz

(26)

where dLv/dp and dhw/dp can be obtained from interpolation ofsteam tables [32] or empirical correlations proposed by Tortikeet al. [34].

Substituting Eqs. (26) and (7) into Eq. (5) gives the governingequation for steam quality in the annulus:

C1dxdz

þ C2xþ C3 ¼ 0 (27)

where C1 ¼ Lv; C2 ¼ dLv=dpjp¼pandpan=dz; C3 ¼ 1=Wan

ðdQan=dz� dQij=dzÞ þ ðdhw=dpjp¼pan� nannsgan=panÞ

�dpan=dz� g sin q:

Usually, the steam quality at the wellhead is known, so theboundary condition for Eq. (27) can be given by

xjz¼0 ¼ xanð0Þ (28)

where xan(0) is the wellhead steam quality in the annulus.Solving Eqs. (27) and (28) yields the final mathematical

expression for steam quality in the annulus,

xanðzÞ ¼ e�C2C1

z�� C3C2

eC2C1z þ xanð0Þ þ C3

C2

�(29)

Similarly, for the steam quality in the integral joint tubing, wehave

xijðzÞ ¼ e�C0

2C01z

� C03

C02eC02

C01z þ xijð0Þ þ

C03

C02

!(30)

where xij(0) is the wellhead steam quality in the integraljoint tubing; C0

1 ¼ Lv; C02 ¼ dLv=dpjp¼pij

dpij=dz;C03 ¼ 1=WijdQij=dzþ ðdhw=dpjp¼pij

� nijnsgij=pijÞdpij=dz� g sin q:

3. Calculation flowchart for the mathematical model

Because the steam pressure, temperature, quality and wellboreheat losses all change with well depth, the whole wellbore is firstlydivided into many segments, and then Eqs. (4), (14), (15), (17), (24),(29) and (30) are coupled and solved iteratively for each segment.In addition, the steam quality in each tubing is estimated by simpleupwind approximation. A calculation flowchart for the mathe-matical model is illustrated in Fig. 4.

4. Results and discussion

4.1. Validation of the model with measured field data

To validate the above model, we compared theoretical pre-dictions with measured field data. The field test was performed atWell Xing 67 in Block Du 84 of Liaohe Oilfield, Northeast of China.The well trajectory data can be found in Ref. [3], and the total depthof the insulated tubing string is 905.62 m. In the field test, 59%-quality steamwas injected into the 1.9-in. integral-joint tubing at arate of 8388 kg/h with a wellhead pressure of 11.55 MPa and thesame quality steam was injected into the annulus at a rate of5364 kg/h with a wellhead pressure of 8.62 MPa, and the injectiontime was 17.8 d. Other related parameters of the wellbore and theformation used for the field test are listed in Table 1.

Figs. 5 and 6 show comparisons of calculated steam pressureand temperature from the theoretical model and those from themeasured field data, respectively. It is observed that the simulatedresults agree very well with the measured values. In addition, wefurther analyzed the errors in the prediction of steam pressure andtemperature in each tubing by choosingMax RE (maximum relativeerror) and MRE (mean relative error) as the evaluation criteria, andthey are defined as

Max RE ¼ Max�ypre � ymea

ymea� 100%

�(31)

MRE ¼ 1N

Xypre � ymea

ymea� 100%

(32)

where ypre and ymea are the predicted and measured values,respectively; N is the data number.

Table 2 shows the maximum and mean relative errors in theprediction of steam pressure and temperature in each tubing forthe field test. It is found that these deviations are all acceptable inengineering calculation, which should support the reliability andaccuracy of the theoretical model.

However, it should be noted that the empirical correlation in Eq.(10) proposed by Tortike et al. [34] is valid only for saturated steam.In other words, Eqs. (14) and (17) would be invalid if saturatedsteam was completely condensed into saturated water or heated

No

Yes

Yes

Yes

Start

Divide the wellbore intoN segments

Input given parameters

Guess kpijΔ

0=k

Calculate kTan , using Eq. (17); calculate kQijd , using Eq. (15);

calculate kU o2 , using Eq. (20) and the method presented in

Refs. [6,11,36]; calculate kQand , using Eq. (24); calculate

kpand , using Eq. (14)

<Δ− kk pp ijijd

acceptable value

Calculate kTij , using Eq. (17);

calculate ijρ , tpf et al., using the method

proposed by Beggs and Brill [23];

calculate kpijd , using Eq. (4)

<Δ− kan

kand pp

acceptable value

Output kpij , kTij , kxij , kpan , kTan , kxan , kQijd , kQand et al. No

Iteration

No

Iteration

Calculate kxij , using Eq. (30); calculate kxan , using Eq. (29)

Guess kpanΔ

1−≤ Nk

Stop

1+= kk

Fig. 4. Calculation flowchart for the mathematical model.

H. Gu et al. / Energy 75 (2014) 419e429 425

into superheated steam. In these cases, we can use the methodsproposed by Ramey [7] and Zhou et al. [25] to solve these problems,although these investigations are not included in this paper.

4.2. Analyses of the predicted results

In this section, based on the validated model, the above well-head injection conditions and the basic data provided in Table 1, we

Table 1Wellbore and the formation parameters used for field test at Well Xing 67 in Block Du 8

Parameter Value, Unit Parameter

Inside radius of integral joint tubing (rii) 0.01905 m Forced-conveOutside radius of integral joint tubing (rio) 0.02415 m Thermal condInside radius of tubing 1 (r1i) 0.0380 m Thermal condOutside radius of tubing 1 (r1o) 0.0445 m Thermal condInside radius of tubing 2 (r2i) 0.0509 m Thermal condOutside radius of tubing 2 (r2o) 0.0572 m Thermal diffuInside radius of casing (rci) 0.0807 m Surface tempOutside radius of casing (rco) 0.0889 m Geothermal gOutside radius of wellbore (rh) 0.1236 m Ratio of the fo

analyzed the profiles of thermophysical properties of saturatedsteam and the characteristics of heat transfer in Well Xing 67.However, it should be stressed that these predicted results mainlydepend on the wellbore structure and the wellhead injectionconditions.

From Fig. 5, we can also see that the saturated pressures both inthe integral joint tubing and in the annulus decrease with welldepth. Since there is one-to-one correspondence between

4 of Liaohe Oilfield.

Value, Unit

ction heat transfer coefficient (hfii, hfio and hf1i) 3255 W/(m2 K)uctivity of the tubing and casing wall (ltub and lcas) 57 W/(m K)uctivity of the insulation materials (lins) 0.07 W/(m K)uctivity of the cement (lcem) 0.933 W/(m K)uctivity of the formation (le) 1.73 W/(m K)sivity of the formation (a) 0.0037 m2/herature of the formation (T0) 303.15 Kradient (a) 0.029 K/mrmation heat capacity to the wellbore heat capacity (u) 0.7273

2.0

4.0

6.0

8.0

10.0

12.0

0 100 200 300 400 500 600 700 800 900

Measured integral joint tubing

Measured annulus

Simulated integral joint tubing

Simulated annulus

Satu

rate

d pr

essu

re (M

Pa)

Well depth (m)

Fig. 5. Comparison of simulated pressure with measured field data.

Table 2Maximum and mean relative errors in the prediction of steam pressure and tem-perature in each tubing for the field test.

Parameter Integral joint tubing Annulus

Max RE (%) MRE (%) Max RE (%) MRE (%)

Steam pressure 4.71 1.99 3.97 2.23Steam temperature 0.56 0.24 0.47 0.27

300

310

320

330

340

350

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

0 100 200 300 400 500 600 700 800 900

From integral joint tubing to annulus

Net heat losses in the annulus

From annulus to cement sheathH

eat f

low

rat

e (W

/m)

Well depth(m)

Wel

lbor

e he

at lo

sses

(W/m

)

Fig. 7. Profiles of rate of heat transfer in Well Xing 67.

H. Gu et al. / Energy 75 (2014) 419e429426

saturated temperature and pressure, the variation trend of satu-rated temperature in each tubing is the same with that of saturatedpressure, as illustrated in Fig. 6.

Fig. 7 shows profiles of rate of heat transfer in Well Xing 67.Firstly, it is observed that from the wellhead to a depth of 777 m,heat is transferred from the fluid in the integral joint tubing to theannulus and the heat flow rate decreases withwell depth, but whenthe well depth is larger than 777 m, the direction of heat transferchanges and the annulus fluid begins to release heat to the integraljoint tubing. This is because the temperature of fluid in the integraljoint tubing drops faster than the annulus fluid temperature, asshown in Fig. 6, although at the wellhead, the former is muchhigher than the latter. In addition, it should be noted that the over-all heat transfer coefficient between inside and outside of integraljoint tubing (Uio) is always very high and here the value of Uio is1254.402 W/(m2 K), so the rate of heat exchange between fluids inthe dual tubing is relatively large even when the temperature dif-ference is small. Secondly, from Figs. 6 and 8, we can see that theannulus fluid temperature is much higher than the cement/for-mation interface temperature, which is also higher than the initialformation temperature, consequently, heat is always transferredfrom the annulus to the surrounding formation through the cementsheath and the wellbore heat losses cannot be avoided. Moreover,Fig. 7 indicates that the rate of heat flow from the annulus to thecement sheath, namely, the rate of wellbore heat losses, decreaseswith well depth due to the diminishing temperature difference,

520

540

560

580

600

0 100 200 300 400 500 600 700 800 900

Measured integral joint tubing

Measured annulus

Simulated integral joint tubing

Simulated annulus

Satu

rate

d te

mpe

ratu

re(K

)

Well depth (m)

Fig. 6. Comparison of simulated temperature with measured field data.

which can be explained according to Eq. (18) or (21). Lastly, the netheat losses in the annulus are negative when the heat absorbedfrom the fluid in the integral joint tubing is much enough to offsetthe wellbore heat losses. But after the annulus fluid releases heat tothe integral joint tubing, the net heat losses in the annulus in-creases sharply with well depth.

Fig. 9 shows steam quality profiles inWell Xing 67. It is found thatthe annulus steam quality increases firstly and drops afterwards,which is opposite to the variation trend of steam quality in the

Fig. 8. Profiles of cement/formation interface temperature, initial formation temper-ature, Tan � Th and Th � Tei.

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500 600 700 800 900

Integral joint tubing

Annulus

Stea

m q

ualit

y

Well depth (m)

Fig. 9. Profiles of steam quality in Well Xing 67.

300

350

400

450

500

550

600

650

0 1 2 3 4 5 6 7 8Thermal conductitivity of insulation matereials (W ·m-1·K-1)

Ave

rage

wel

lbor

e he

at lo

sses

(W/m

)

Fig. 11. Effect of lcem on average wellbore heat losses.

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800

0.07 W/(m·K) 0.1 W/(m·K)

0.3 W/(m·K) 0.5 W/(m·K)

0.7 W/(m·K)

Well depth(m)

Stea

mqu

ality

in th

e in

tegr

al jo

int t

ubin

g

Fig. 12. Profiles of steam quality in the integral joint tubing for different lcem.

H. Gu et al. / Energy 75 (2014) 419e429 427

integral joint tubing. In fact, in the case of concentric dual-tubingsteam injection, the steam quality in each tubing is influenced notonly by the wellhead injection conditions, but also by the heat ex-change between fluids in the dual tubing. Take the Well Xing 67 asan example, when the annulus fluid absorbs much heat from thefluid in the integral joint tubing, the latter is just like a heat sourcethat can heat the steam in the annulus, causing the annulus steamquality to increase with well depth, even may reach 1 to becomesuperheated steam. In fact, field experience has also shown thatboiling does occur in CDTSIW [4]. On the contrary, steam quality inthe integral joint tubing drops with well depth due to a lot of heatlosses, even may reach 0 to become condensed water. Similarly, theopposite results may occur if the fluid in the integral joint tubingabsorbs much heat from the annulus fluid, as shown in Fig. 9.

4.3. Effects of the thermal conductivity of insulation materials

In Liaohe Oilfield, steam injection wells are relatively deep, soinsulation materials are often utilized to reduce the wellbore heatlosses and to ensure high bottomhole steam qualities. In thefollowing, we studied the effects of thermal conductivity of insu-lation materials (lcem) on wellbore heat losses and steam qualitiesin CDTSIW. The results are presented in Figs. 10e13.

Fig. 10 shows profiles of wellbore heat losses for different lcem. Itis obviously found that the wellbore heat losses decline almostlinearly with well depth for a given lcem, but they all increase with

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 800

0.07 W/(m·K) 0.1 W/(m·K)

0.3 W/(m·K) 0.5 W/(m·K)

0.7 W/(m·K)

Well depth(m)

Wel

lbor

e he

at lo

sses

(W/m

)

Fig. 10. Profiles of wellbore heat losses for different lcem.

lcem at the same well depth. Fig. 11 shows a relationship betweenaverage wellbore heat losses and lcem, we can see that when lcem isless than 0.7 W/(m K), the average wellbore heat losses increasesharply with lcem, but when lcem is greater than 0.7 W/(m K), theaverage wellbore heat losses rise very slowly with lcem. Therefore,in order to effectively reduce the wellbore heat losses, lcem shouldbe smaller than 0.7 W/(m K).

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800

0.07 W/(m·K) 0.1 W/(m·K)

0.3 W/(m·K) 0.5 W/(m·K)

0.7 W/(m·K)

Well depth(m)

Stea

mqu

ality

in th

e an

nulu

s

Fig. 13. Profiles of steam quality in the annulus for different lcem.

H. Gu et al. / Energy 75 (2014) 419e429428

Figs. 12 and 13 show profiles of steam qualities in the integraljoint tubing and in the annulus for different lcem, respectively. It canbe seen that lcem has little effects on steam qualities when the wellis shallow, but for deep wells, the steam quality in each tubing issignificantly influenced by lcem. Also, from Fig. 13, we can find thatthe annulus steam quality is near 0 at a depth of 710 m when lcemequals to 0.7 W/(m K). Consequently, in order to ensure high bot-tomhole steam qualities, especially in deep wells of Liaohe Oilfield,lower thermal-conductivity insulation materials are required.

5. Conclusions

In this study, a mathematical model for predicting thermo-physical properties of saturated steam and wellbore heat losses inCDTSIW was established. More importantly, a semi-analyticalmodel for estimating pressure gradient for steam/water flow inannuli was proposed. After the theoretical model was verified usingmeasured field data from Well Xing 67 of Liaohe Oilfield, weanalyzed the profiles of thermophysical properties of saturatedsteam and the characteristics of heat transfer in Well Xing 67. Also,based on the theoretical model, the effects of the thermal con-ductivity of insulation materials on wellbore heat losses and steamqualities were further studied. The findings can be summarized asfollows:

(1). Compared with measured field data, the proposed mathe-matical model was proved to be reliable in engineeringcalculation.

(2). The direction of heat transfer between fluids in the integraljoint tubing and in the annulus depends not only on well-head injection conditions but on temperature drop in eachtubing, however, heat is always transferred from the annulusto the surrounding formation and the wellbore heat lossescannot be avoided.

(3). The heat exchange between fluids in dual-tubing has sig-nificant effect on steam quality in each tubing, and steamboiling and condensation can occur during steam injection.

(4). In order to effectively reduce the wellbore heat losses and toensure high bottomhole steam qualities in Well Xing 67 ofLiaohe Oilfield, the thermal conductivity of insulation ma-terials should be less than 0.7 W/(m K).

Acknowledgements

The authors wish to thank the Research Institute of PetroleumExploration & Development, Liaohe Oilfield Company, PetroChina(2013-JS-9399). This work was also supported in part by a grantfrom National Science and Technology Major Projects of China(2011ZX05024-005-006 and 2011ZX05012-004).

Appendix A. Heat capacity and JouleeThompson coefficient

For steam/water two-phase flow system, the total enthalpy ofmixture fluid is the sum of the enthalpy of each phase [30].Therefore, the enthalpy gradient in the annulus can be given by

dhandz

¼ Ws

Wan

dhsdz

þ Ww

Wan

dhwdz

(A-1)

where Ws and Ww are the mass flow rates of steam and water,respectively; and

dhsdz

¼ CpsdTandz

� CJsCpsdpandz

(A-2)

dhwdz

¼ CpwdTandz

� CJwCpwdpandz

(A-3)

where Cps and Cpw are the heat capacities of steam and water atconstant pressure, respectively; CJs and CJw are the JouleeThomp-son coefficients of steam and water, respectively. Cps, Cpw, CJs andCJw are given by

Cps ¼ ðvhs=vTÞp (A-4)

Cpw ¼ ðvhw=vTÞp (A-5)

CJs ¼1Cps

(T�v

vT

�1rs

�p� 1rs

)(A-6)

CJw ¼ 1Cpw

(T�v

vT

�1rw

�p� 1rw

)(A-7)

If the compressibility of water is ignored, Eq. (A-7) can bereduced to.

CJw ¼ � 1Cpwrw

(A-8)

Substituting Eqs. (A-2) and (A-3) into Eq. (A-1) gives

dhandz

¼WsCpsþWwCpwWan

dTandz

��WsCJsCps

WanþWwCJwCpw

Wan

�dpandz(A-9)

Therefore, the heat capacity at constant pressure and the Jou-leeThompson coefficient of mixture fluid are

Cpm ¼ WsCps þWwCpwWan

(A-10)

CJm ¼ 1Cpm

�WsCJsCps

WanþWwCJwCpw

Wan

(A-11)

Substituting Eqs. (A-6) and (A-8) into Eq. (A-11), we have

CJm ¼ 1CpmWan

(WsT

�v

vT

�1rs

�p�Ws

rs�Ww

rw

)(A-12)

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