SPE 119526

Prediction of Wellbore Temperature Profiles During Heavy Oil Production Assisted With Light OilYanmin Yu, Peking U.; Tao Lin and Hongxing Xie, Sinopec; Yanyan Guan, Peking U.; and Kewen Li, SPE, Peking U. and Stanford U.

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Production and Operations Symposium held in Oklahoma City, Oklahoma, USA, 4–8 April 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract In some heavy oilfields, oil can flow in the reservoirs but may not flow through the wellbore without taking measures such as heating or diluting with light oil because of the high viscosity caused by low temperature around the wellbore close to surface. We focused on the latter issue in this study which is being widely used in Tah oilfield and other oilfields in China. Since the viscosity of the crude oil changes with temperature significantly, accurate prediction of wellbore temperature profiles is of great importance to determine the injection rate and temperature of light oil for dilution. An analytical model was derived to predict the wellbore temperature profiles for the heavy oil production assisted with the addition of light oil from the annulus. The model was based on energy balance between the formation and fluids flowing through each conduit. The temperature profiles were calculated using the proposed temperature model and the results were compared with the measured data. Sensitivity analysis was conducted using the proposed model and the main factors that influence the temperature profiles were discussed. Introduction Tah oilfield is one of the ultra-deep and ultra-heavy oil reservoirs in China. Most wells are deeper than 5000 m. Tah oilfield is also featured of high temperature (130oC) and high water salinity (over 100,000 ppm). Although the oil can flow in the reservoir easily (this may be because of the high reservoir temperature), it can not flow to the surface in many wells. The main reason may be because the viscosity of the crude oil increases greatly in the upper part of the wellbore due to the low temperature around the wellbore at the shallow depth as a large amount of heat loses to the formation. In Tah oilfield, one of the solutions to this problem was to inject light oil with much lower viscosity through the annulus to the crude oil in the tubing. This approach has been being extensively applied in Tah oilfield and other oilfields in Xinjiang, China. Accurate prediction of the wellbore temperature profile is important to determining the values of the parameters (for example, the amount of light oil that should be added) in the design of the dilution projects because the oil viscosity varies significantly with temperature. There have been many papers on the calculation of the temperature profiles in the wellbore under different production conditions but few in the cases in which light oil is injected from the annulus to mix with the produced crude oil in the tubing. The physical process and heat-transfer in the case in which heavy oil is produced by adding light oil are similar to those in gas-lift wells. However there are some differences between the two cases. For example, the heat-transfer is of different characteristics for gas and light oil; the viscosity of the crude oil in the tubing is reduced significantly after mixing with the light oil injected from the annulus, which may affect the heat-transfer.

Ramey (1962) might be the first to present a theoretical model to estimate fluid temperature as a function of well depth and production time. The model proposed by Ramey (1962) has been widely used in both geothermal and petroleum industries. However, the effects of kinetic energy and friction were ignored in the model proposed by Ramey (1962) and therefore the model is only applicable to single-phase flow. The approach reported by Satter (1965) improved Ramey's model by honoring the effect of phase change during steam injection. There have been many modified temperature models (Shiu and Beggs, 1980; Sagar, et al., 1991; Alves, et al., 1992; Hasan and Kabir, 1994 and 1996; Jurak and Prnic, 2005) for two-phase flow in wellbore since Ramey (1962) proposed the theoretical temperature model. Sagar et al. (1991) extended Ramey's model to multiphase flow in wellbore by considering kinetic energy and Joule-Thompson expansion effects. Hasan and Kabir (1996) reported a mechanistic model for circulating fluid temperature. Hasan et al. (1996) presented a model for gas-lifting wells.

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In this study, an analytical two-phase model was proposed to estimate the fluid temperature in the wellbore for heavy oil production assisted by the addition of light oil from top to the bottom through the annulus. The heat-transfer coefficient in the proposed model was not given as a constant, instead it was inferred from other parameters using the model proposed by Manabe (2003). Mathematical Models Reverse injection was adopted in the production of heavy oil in Tah oilfield. That is, light oil was injected from the top to the bottom through the annulus and flew back up from the tubing after mixed with the crude oil at the bottom. We followed Ramey’s assumptions: i) heat flows radically in the wellbore; ii) heat travels much faster in the wellbore than that in the formation. So a steady-state flow was assumed in the wellbore, and used the function f (t) introduced by Ramey (1962) to denote the non-steady-state heat-transfer in the formation.

Considering the convective heat-transfer in both the tubing and the annulus as well as the conductive heat-transfer in the cement and formation, the mathematic model for the temperature profiles can be expressed as follows (the detail derivation is presented in Appendix A):

Cpt

attt F

CgTTA

dZdT

+−−−= )( (1)

)()( eaaata

tta TTATTw

wAdZdT

−+−−= (2)

ZgTT Gbhe += (3)

In our calculation, the mixing point is at the bottom of the well, so the boundary conditions are:

bht TTZ == ,0 (4)

ina TTHZ == , (5)

Where H is the well length, Tbh is the fluid temperature at the bottomhole and Tin is the temperature of the injected light oil at the wellhead.

For the situation when mixing point is not at the bottom, one just needs to use Eqs.1-3 above the mixing point and the equations from Hasan (1991) below the mixing point. In this case, the boundary conditions need to be changed accordingly. The calculation procedure is fundamentally the same as shown in this article.

If the wellbore is divided into many elements, and a constant temperature in each element is assumed, the physical properties of the fluid are the same in an individual element. So the analytical solutions of the above equations are expressed as follows:

ta

Gtatta

tZZ

et AA

AgMAMAww

eCeCTT++

++=− 2211λλ

(6)

t

GZZ

ta

Gtta

tZZ

ea AgeCeC

AA

AgMAww

eCeCTT++

++

++=−2221112211

λλλλ λλ (7)

Where, at

a

t AAwwA −⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1 (8)

242

2,1taaa AAAA +±

=λ (9)

Cpt

t FC

gM +−= (10)

Here, C1 and C2 are parameters needed to be decided by the boundary conditions. Theoretically, C1, C2 are not the same in different elements; it is necessary to use an iterative procedure to decide the values of C1 and C2 for each element according to the fluid temperature at the bottom in the annulus (Tba). However, our calculations demonstrated that the values of C1 and C2 did not change significantly in different elements. So it may be reasonable to assume that the values of C1 and C2 are constant for the whole wellbore. It was also assumed that the fluid temperature (Tba) at the bottom of the annulus equal to the

SPE 119526 3

bottomhole formation temperature (Tbh). The physical properties of fluid were calculated using Tbh. The impact of this assumption will be discussed later. Using these physical properties and boundary conditions in Eqs.4 and5, the values of C1 and C2 for the whole wellbore could be inferred.

In the case of Tah oilfield, the Reynold number of fluid flow in the annulus is less than 1000. So it is reasonable to assume a laminar flow in the annulus. The Nusselt number is only a function of the outer and inner radius for laminar flow. Therefore the convective heat-transfer coefficient in the annulus (ha) is constant. For the tubing, we used the two-phase model presented by Manabe (2003) and the flow type was bubble flow.

In the cases studied, it is necessary to calculate the viscosity of the mixture of heavy and light oils in the tubing. The model we used in this study is expressed as follows:

)lg(lg1

1)lg(lg1

)lg(lg hlmix xxx µµµ

++

+= (11)

Where x is the volume ratio of light oil to heavy oil; mixµ , and are viscosities of mixture, light oil and heavy oil respectively. Later, we will discuss how the calculated temperature depends on the viscosity of the mixture.

lµ hµ

Results and Discussion We used the proposed models (Eqs. 6 and 7) to calculate the temperature profiles of a self-flowing well in Tah oilfield, where the mixing point was at the bottom. All the relevant data are shown in table 1. We compared the model results with the temperature data measured in this well. As shown in Fig.1, the calculated temperature data in the tubing match with the measured temperature satisfactorily, except for the data point at the wellhead. Sagar (1991) explained this might be due to the measurement error. This has not been verified yet. We can see that the temperature in the annulus changed so quickly near the wellhead that it almost became equal to the formation temperature at the depth below about 1500 meters, which might be because of the heat loss from the injected light oil to the formation.

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt, calculated

Tt, measured

Ta, calculated

Te

Fig.1: Calculated fluid temperature profiles in the tubing (with a comparison to the measurements) and the annulus

As pointed out previously, we assumed that the fluid temperature in the annulus at the bottom (Tba) was equal to the

temperature in the formation at the same depth (Tbh), which was 123℃ in the cases studied. However this may not be true. Fluid temperature in the annulus near the bottom may be lower than that of the formation, especially when injection rate is large. So we used different temperatures at the bottom to show the impact. The results are shown in Figs. 2 and 3. We set the bottom temperature as 100℃ and 110℃ respectively. Fig.2 shows that the effect of the assumed temperature at the bottom on the fluid temperature in the annulus is small, especially at the top part of the wellbore.

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0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (Tba=123℃),

Ta (Tba=100℃)

Ta (Tba=110℃)

Te

Fig.2: Impact of the fluid temperature at the bottom (Tba) on the temperature profiles in the annulus

Fig.3 shows the effect of the assumed temperature at the bottom (Tba) on the fluid temperature in the tubing. There are almost no differences near the wellhead though the temperature profiles in the tubing are significantly different at deep depth.

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (Tba=123℃)

Tt (Tba=100℃)

Tt (Tba=110℃)

Te

、

Fig.3: Impact of the fluid temperature at the bottom on the temperature profiles in the tubing

SPE 119526 5

As we concern more about temperature at the shallow depth, it is reasonable to take the fluid temperature at the bottom in the annulus as the temperature of the formation at the same depth.

In the above calculations, the values of the overall heat-transfer coefficients Ut and Ua were determined using a two-phase model proposed by Manabe (2003). In the cases of gas-lift wells reported by Hasan et al. (1996), the overall heat-transfer coefficients were given constant values to calculate the temperature profiles. We conducted some calculations with different given overall heat-transfer coefficients. The results are shown in Figs. 4 and 5 respectively. In Figs.4 and5, we first used a given Ut but a calculated Ua, and then a given Ua but a calculated Ut.

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (calculated Ua and Ut)

Ta (calculated Ua,Ut=15W/(m2.℃)Ta (calculated Ut, Ua=1W/(m2.℃)

Fig.4: Impact of overall heat-transfer coefficient on the temperature profiles in the annulus

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0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (Calculated Ua and Ut)

Tt (Calculated Ua,Ut=15W/(m2•℃)

Tt (Calculated Ut,Ua=1W/(m2•℃)

Te

、

Fig.5: Impact of overall heat-transfer coefficient on the temperature profiles in the tubing

When Ut is fixed at 15 W/(m2.oC) (greater than the calculated value), temperatures in both the annulus and tubing are smaller than those determined using the calculated Ut and Ua. This may be because more heat lost to the formation as Ut increases. When Ua is fixed at 1 W/(m2. oC) (less than the calculated value), the fluid temperatures are higher at the shallow depth but little lower at deep depth as shown in Fig.4. Decrease in Ua reduces the heat flux between the annulus and the formation. At shallow depth, where heat flux is from the annulus to the formation, the temperature in the annulus becomes higher when heat flux is smaller. At deeper depth, where the heat flux is from formation to annulus, smaller heat flux causes lower temperature. However, the temperature difference between the annulus and the formation at deep depth is so small that the influence of Ua is not significant. Similar trend happens to temperature profile of tubing in Fig.5 because of the heat exchange between the annulus and the tubing.

To understand how the mixing of the light oil with the heavy oil affects the temperature profiles, we conducted sensitivity analysis and the results are discussed as follows.

In Figs.6 and 7, we set the viscosity of the light oil to 10 times and 50 times as it was, which was 48.1 mPa.s at the injection temperature of 80℃.We can see in Fig.6, no matter how the viscosity changes, temperature profile in the annulus remains almost the same. This is because we assumed a laminar flow in the annulus, the Nusselt number does not change with the fluid property in the annulus; and the heat-transfer coefficient is fixed; therefore the change in viscosity has little impact on temperature in the annulus. But change in viscosity does influence the temperature in the tubing. As shown in Fig.7, temperature in the tubing increases with the viscosity of the light oil. This may be because the viscosity of the mixture becomes greater when the viscosity of the light oil increases; the Reynold number of the fluid in the tubing decreases. So the convective heat-transfer coefficient decreases. As a result, there is less heat loss from the tubing to the annulus. According to the above results, it is essential to calculate the viscosity of the mixture accurately.

SPE 119526 7

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (µ=48.1mPa·s)

Ta (µ*10)

Ta (µ*50)

Te

Fig.6: Temperature profiles in the annulus calculated with different viscosity of the light oil

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (µ=48.1mPa·s)

Tt (µ*10)

Tt (µ*50)

Te

Fig.7: Temperature profiles in the tubing calculated with different viscosity of the light oil

The effects of the temperature of the injected light oil on the temperature profiles in the wellbores are shown in Figs. 8 and 9. As expected, calculated temperatures change significantly near the wellhead.

8 SPE 119526

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (Tin=80℃)

Ta (Tin=20℃)

Ta (Tin=120℃)

Te

Fig.8: Temperature profiles in the annulus calculated with different injection temperatures

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (Tin=80℃)

Tt (Tin=20℃)

Tt (Tin=120℃)

Te

Fig.9: Temperature profiles in the tubing calculated with different injection temperatures

The temperatures in both the tubing and the annulus increase with the injection temperature. But as shown in Fig.8, there is even no difference in temperature profiles in the tubing at the depth below about 500 m. This also happens to the

SPE 119526 9

temperatures in the annulus below about 1000 m. We speculated that the heat loss in the annulus happens so quickly that the temperature in the annulus drops close to the formation temperature about 1000 meters below the wellhead, as shown in Fig.1.

We also investigated the effect of the production rate on temperature profiles. With a fixed ratio of light oil to heavy oil, we changed the production rate to 10 times and 0.1 times as it was, which was 74.7m3/d. In Figs.10 and11, one can see significant changes in temperature profiles. In Fig.10, because the ratio of light oil to heavy oil is fixed, the mass flow rate in the annulus is 10 times greater when the production rate is 10 times greater. The light oil flows too fast to have sufficient heat exchange with the formation. So the temperature in the annulus at shallow depth is much higher than that in the formation. At the deep depth where heat flux is from the formation to the annulus, the temperature in the annulus remains much lower than that in the formation because of the same mechanism. By the contrary, when the mass flow rate is only 0.1 times as it was, fluid flows so slow that it changes heat sufficiently with formation, so even near the wellhead , temperature in the annulus is almost the same as that in the formation. The same mechanism can be used to explain the significant changes in the temperature profiles in the tubing as shown in Fig.11.

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (Qo=74.7m3/d)

Ta (Qo*10)

Ta (Qo/10)

Te

Fig.10: Temperature profiles in the annulus calculated with different production rates

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0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (Qo=74.7m3/d)

Tt (Qo*10)

Tt (Qo/10)

Te

Fig.11: Temperature profiles in the tubing calculated with different production rates

Figs.12 and13 show the influence of the volume ratio of the light oil to heavy oil. We used the ratios of 0.4, 0.8, and 2.0 respectively. As the ratio of the light oil to heavy oil increases, mass flow rates in the annulus increase. As shown in Fig.12, though the change is not significant, temperature profiles in the annulus have the same trend as shown in Fig.10. In the tubing, with fixed production rates of heavy oil, there are two major parameters affected by the change in ratio of the light oil to heavy oil: the viscosity of the mixture and the total mass flow rate in the tubing. As the ratio of the light oil to heavy oil increases, the viscosity of the mixture decreases. As shown in Fig.7, this will lead a decrease in the fluid temperature in the tubing. At the same time, as the ratio of the light oil to heavy oil increases, the total mass flow rates in the tubing will increase. As shown in Fig.11, the temperature in the tubing will increase significantly. Fig.13 shows the latter effect plays a decisive role. Temperature in the tubing increases with the ratio of light oil to heavy oil, especially at shallow depth.

SPE 119526 11

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Ta (X=0.8)

Ta (X=0.4)

Ta (X=2.0)

Te

Fig.12: Temperature profiles in the annulus calculated with different mixing ratio

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140Temperature，°C

Dep

th，

m

Tt (X=0.8)

Tt (X=0.4)

Tt (X=2.0)

Te

Fig.13: Temperature profiles in the tubing calculated with different mixing ratio

Note that all the above calculations and sensitivity analyses were done when the production time was 7 days. Ramey

(1962) reported that the temperature did not change with production time after about 7 days.

12 SPE 119526

Conclusions 1. An analytical temperature model was derived for heavy oil production assisted with the addition of light oil. With

certain reasonable assumptions, the calculation procedure is simple and the model results matched with measured temperature data satisfactorily.

2. The viscosity of the injected light oil, the production rate, and the ratio of the light oil to heavy oil had significant impact on temperature profiles in the tubing, especially near the wellhead. However the influence of the temperature of the injected light oil was limited in the cases studied, especially at the deep depth.

3. The accuracy to calculate the viscosity of the mixture of the light and heavy oil in the tubing was important to predict the temperature profiles.

Nomenclature Cpt = specific heat of fluid in the tubing Cpa = specific heat of fluid in the annulus f(t)= transient heat-conduction time function of formation g = acceleration of gravity gG = geothermal gradient ha = convective heat-transfer coefficient of annulus fluid ht = convective heat-transfer coefficient of tubing fluid

Ht = specific enthalpy of tubing fluid Kc = thermal conductivity of cement Ke = thermal conductivity of formation p = pressure Pwh = wellhead pressure rc = casing radius rh = wellbore radius rt = tubing radius RgL = gas/liquid ratio Q1 = heat flux between tubing and annulus per unit mass Q2 = heat flux between annulus and formation per unit mass t = production time Ta = temperature of annulus fluid Tba = temperature of annulus fluid at the bottom Tbh = bottomhole temperature of formation Tc = temperature of cement Te = temperature of formation Tin = injection temperature of light oil Tt = temperature of tubing fluid Ua = overall heat-transfer coefficient from annulus to formation Ut = overall heat-transfer coefficient from tubing to annulus vt = flow rate in the tubing wa = mass flow rate in the annulus wt = mass flow rate in the tubing x = volume ratio of light oil to heavy oil Z = depth α = thermal diffusivity of formation β = Joule-Thomson coefficient

gγ = gas specific gravity oγ = oil specific gravity hµ = viscosity of heavy oil lµ = viscosity of light oil mixµ = viscosity of the mixture of heavy and light oil References Alves, I.N., Alhanatl, F.J.S., and Shoham, O.: “A Unified Model for Predicting Flowing Temperature Distribution in Wellbores and

Pipelines,” SPEPE (Nov.1992) 363 Hasan, A.R. and Kabir, C.S.: “Heat-transfer during Two-Phase Flow in Wellbores: Part II-Wellbore Fluid Temperature,” paper SPE 22948

presented at the 1991 SPE Annual Technical Conference and Exhibition, Oct. 6-9.

SPE 119526 13

Hasan, A.R. and Kabir, C.S.: “Aspects of Wellbore Heat-transfer during Two-phase flow,” SPEPE (Aug. 1994) 211. Hasan, A.R. and Kabir, C.S. and Ameen, M.M.: “A Mechanistic Model for Circulating Fluid Temperature.” SPE Journal (Jun 1996) 133 Hasan, A.R. and Kabir, C.S.: “A Mechanistic Model for Computing Fluid Temperature Profile in Gas-lift Wells” SPEPF (Aug 1996) Jurak, M. and Prnic, Z.: “Heating of Oil Well by Hot Water Circulation,” Proceedings of the Conference on Applied Mathematics and

Scientific Computing (2005) 235. Manabe, R., Wang, Q. and Zhang, H.Q.: “A Mechanistic Heat-transfer Model for Vertical Two-Phase Flow” paper SPE 84226 presented at

the 2003 SPE Annual Technical Conference and Exhibition, Oct. 5-8. Ramey Jr., H.J: “Wellbore Heat Transmission,” JPT (April 1992) 427; Trans., AIME, 225. Sagar, R.K., Doty, D.R., and Schmidt, Z.: “Predicting Temperature Profiles in a Flowing Well,” SPEPE (Nov. 1991) 441. Satter, A.: "Heat Losses during Flow of Steam Down a Wellbore," JPT (July 1965) 845; Trans., AIME, 234. Shiu, K.C. and Beggs, H.D.: “Predicting Temperature in Flowing Oil Wells,” J.Energy Resources.Tech. (March, 1980); Trans., ASME. Willtite, G.P.: “Overall Heat-transfer Coefficients in Steam and Hot Water Injection Wells,” JPT (May 1967) 607. Appendix A The schematic of the wellbore heat-transfer while mixing light oil is shown in Fig. 14. With the assumptions we described previously, we can write the energy balance equations for fluids in the tubing:

dZdQ

dZdvvg

dZdH ttt 1−=++ (A-1)

( )att

tt TTw

UrdZ

dQ−=

π21 (A-2)

at

att hh

hhU+

= (A-3)

Where the subscripts a and t refer to variables for the annulus and tubing, respectively. H is the specific enthalpy, g is the

ght oil injected is noncompressible; so the energy balance equation can

acceleration of gravity, v is the flow rate, r is the radius, w is the mass flow rate, h is the convective heat-transfer coefficient, T is the temperature, and Q1 is the heat flux from tubing to annulus. Note that the heat flux Q1 is not always positive due to the temperature difference between tubing and annulus.

For the fluids in the annulus, it is assumed that the li be written as:

dZdQ

dZwdQwdTa

dZC

a

tpa

21 +−= (A-4)

Where Cp is the specific heat of fluid and Q2 is the heat flux from the annulus to the formation. Willtite (1967) already gave sufficient discussion about the heat loss from wellbore to formation in self-producing wells. According to Willtite (1967):

( )eaecaa

aec TTkrUtfw

UkrdQ 22 π (A-5) dZ

−+

=))((

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+=c

c

hc

aa krrr

hU

ln11 (A-6)

Where the subscripts c and e refer to the variables for the cement and formation respectively. K is the thermal conductivity. f(t) is the transient heat-conduction time function for the formation introduced by Ramey (1962). In this study, Hasan’s correlation (1994) is used:

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛tα 5.12 ≤hrtα −= 22 3.011.1281

hh rt

rtf α (A-7b)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+=

tr

rttf h

h αα 2

26.01ln5.00463.0 5.12 >

hrtα (A-7b)

Where t is the production time and α is the thermal diffusivity of the formation. Because:

dZdTCdpCdH t

ptPtt +−= β

dZdZ (A-8)

Combining Eqs.A-1, A-2, and A-8, one can get:

)()(2 ttttt dvvdpgUrdT−+−−−= β

ptptat

tpt CdZCTT

wCdZπ (A-9)

The Joule-Thompson effect is estimated using Sagar’s empirical equation (1991):

14 SPE 119526

pt

ttc C

dvvdZdpF −= β (A-10)

If , Fc=0; (A-11a) skgwt /3.2>otherwise:

Ggo

gL

twhc

gR

wpF

1943.010009.45.1315.14110229.310878.5

1020273.41087.14510978.2(5471.0

356

463

−×+⎟⎟⎠

⎞⎜⎜⎝

⎛−×+×−

×+×+×−×=

−−−

−−−

γγ

(A-11b)

Where pwh, RgL, , oγ gγ , gG are the wellhead pressure, gas/liquid ratio, oil specific gravity, gas specific gravity, geothermal gradient respectively. Eq.A-9 can be rewrite as:

Cpt

attt F

CgTTA

dZdT

+−−−= )( (A-12)

Similarly:

)()( eaaata

tta TTATTw

wAdZdT

−+−−= (A-13)

Here:

tpt

ttt wC

UrA

π2= ,

⎥⎦

⎤⎢⎣

⎡+

=eca

aec

apaa krUtf

UkrwC

A)(

2π (A-14)

So the mathematic models (see Eqs. 1 and 2 in the text) for the temperature profiles in the wellbore can be obtained.

Table1: Wellbore, formation and fluid Data

Well depth, m 5500 Wellhead pressure, Mpa 2.98 Bottomhole temperature, oC 123.44 Surface formation temperature, oC 20 Injection temperature of light oil, oC 80 Production time, d 7 Tubing radius, m 0.038 Casing radius, m 0.0889 Wellbore radius, m 0.15 Density of heavy oil, kg/m3 980 Density of light oil, kg/m3 830 Production rate, m3/d 74.7 Volume fraction of gas in oil, m3/m3 25.3 Volume ratio of light to heavy oil, m3/m3 0.8 Thermal diffusivity of formation, m2/h 0.0037 Thermal conductivity of formation, W/(m· oC) 2.4 Thermal conductivity of cement, W/(m· oC) 1.74

SPE 119526 15

Fig. 14: Schematic of the heat-transfer in wellbore