Coupled calculation model for transient temperature and pressure of carbon dioxide injection well

International Journal of Heat and Mass Transfer 121 (2018) 680–690

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Coupled calculation model for transient temperature and pressure of carbon dioxide injection well Liang-Ping Yi a,⇑, Xiao-Gang Li a, Zhao-Zhong Yang a,⇑⇑, Jun Sun b a b

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Xindu Avenue 8, Chengdu 610500, China Down-Hole Operating Company, Chuanqing Drilling and Exploration Engineering Co., Ltd., Chengdu 610051, China

a r t i c l e

i n f o

Article history: Received 30 August 2017 Received in revised form 29 December 2017 Accepted 9 January 2018

Keywords: Supercritical carbon dioxide Temperature Pressure Transient Wellbore

a b s t r a c t Based on the law of thermodynamics, this study establishes a numerical calculation model for the temperature and pressure of a carbon dioxide (CO2) injection well. Our model assumes that the well structure has a three-layer casing completion, and that the heat generated by the fluid friction losses is absorbed by the CO2 and tubing according to a contact coefficient. The heat transfer in the wellbore and formation is considered to be transient, and the model is solved using a fully implicit finite difference method. The results of numerical experiments show that the CO2 pressure and temperature are significantly affected by the injection rate, injection temperature, and tubing roughness. Therefore, it should be possible for engineers to adjust the injection temperature or injection rate to ensure that the CO2 reaches the supercritical state at the downhole. Although the Joule–Thomson effect and frictional heat proportional absorption effect also influence the CO2 pressure and temperature, they can be neglected at the engineering scale. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Carbon dioxide (CO2) is widely used in the field of fossil energy exploitation for enhanced oil recovery [1] and enhanced coalbed methane recovery [2–4], and supercritical carbon dioxide is a promising operating fluid in drilling and fracturing technologies [5–13]. Many such applications involve CO2 flowing through the wellbore and heat transfer in and around the wellbore. Therefore, there is a significant benefit to establishing a temperature and pressure prediction model for CO2 injection wells. At present, the methods used to study the heat transfer of CO2 flow in a tube are generally experimental [14–18] or mathematical [5,6,12,19,20]. Although experimental methods can directly measure many heat transfer parameters (CO2 and tube wall temperature, heat transfer coefficient, etc.), they still have many limitations, such as the dimensional effect, and experimental equipment cannot fully reflect the formation within a wellbore. Fortunately, mathematical methods can compensate for these limitations. The first temperature prediction model for drilling processes was established by Raymond [21]. Inspired by this model, researchers have successively developed wellbore temperature ⇑ Corresponding author. ⇑⇑ Corresponding author. E-mail addresses: [email protected] (L.-P. Yi), yangzhaozhong@ swpu.edu.cn (Z.-Z. Yang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.036 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

and pressure prediction models [22–25]. Although these models consider the fluid to be water rather than CO2, they provide the basis for a temperature and pressure field prediction model for CO2 injection wells. Wang et al. [6] developed a coupled model for calculating the wellbore temperature and pressure during drilling with supercritical CO2. The physical CO2 parameters were calculated using the Span–Wagner equation, and the heat transfer in the wellbore was assumed to be steady while that around the wellbore was unsteady. This assumption is reasonable for long-term injection wells, particularly as the thermal resistance of the casing and tubing was neglected. Dou et al. [19] established a simple wellbore pressure and temperature prediction model for CO2 injection wells using the model assumptions and basic equations from Wang’s model. Motivated by Wang’s model, Song et al. [5] established a wellbore pressure and temperature calculation model for managed pressure drilling with supercritical CO2. Unlike Wang’s model, they also considered the influence of the casing and tubing thermal resistance on heat transfer. The models described above are suitable for simulating the long-term injection of CO2. However, for short-term injection conditions such as fracturing, the unsteady heat transfer process in the wellbore is not negligible. Therefore, Guo and Zeng [12] established a transient temperature and pressure coupling model for supercritical CO2 fracturing, and predicted the physical CO2

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parameters using the REFROP software. However, their model neglects axial heat conduction in the annulus, and is only suitable for single-casing well completion conditions. In real-life situations, multi-layer casing completion is applied to ensure the security of the wells. Additionally, current models assume that the heat generated by fluid friction is only absorbed by the fluid; however, it is more reasonable to assume that the friction heat is absorbed by both the fluid and the tubing. Inspired by previous studies and aware of their limitations, this research develops a coupled model for the transient temperature and pressure of CO2 injection wells. Our model considers a threelayer casing completion structure, and calculates the physical CO2 parameters using the Span–Wagner and Vesovic equations. The heat generated by fluid friction is absorbed by both the CO2 and tubing according to a contact coefficient.

681

In Eqs. (1) and (2), r1 is the inner radius of the tubing; q1 is the CO2 density; v1 is the CO2 volume flow velocity; Re is the Reynolds number; and D is the absolute roughness of the tubing. 3.2. Heat transfer model (1) Transient heat transfer inside the tubing According to the first law of thermodynamics, and considering the Joule–Thomson effect, the equation of heat transfer within tubing can be described as [12]

Q1

pr21

 q1 v 1 c 1

¼ q1 c1

@T 1 @p 2h1 ðT 2  T 1 Þ þ aJ q1 v 1 c1 1 þ r1 @z @z

@T 1 @p  aJ v 1 c 1 1 ; @t @t

ð3Þ

where 2. Physical model The physical model of a CO2 injection well is illustrated in Fig. 1. The process of CO2 injection is as follows: (1) Before injection, the wellbore is full of CO2 and reaches a temperature balance with the formation; (2) CO2 is injected to the downhole through the tubing, and the fluid in the annulus remains stationary; (3) The heat transfers in the wellbore and formation are both transient; (4) The CO2 temperature, pressure, and flow velocity remain constant in the same section of tubing.

3.1. Pressure calculation model Based on the continuity equation and flow equation, the pressure control equation for CO2 flowing in the tubing can be written as follows:

ð1Þ

where [26] 8 " #3=2 91=12
16 < = 0:27D 12 0:9 16 f ¼ 8 ð8=ReÞ þ 2:457 ln ð7=ReÞ þ þ ð37530=ReÞ : ; 2r 1

ð2Þ

0:027k1 Re0:8 Prm ; 2r1

Q1 ¼ Q

b ; 1þb

Q ¼ qDpf ;

ð4Þ ð5Þ ð6Þ

and [27]

b¼

3. Mathematical model

dp1 q v2 dv 1 ¼ q1 g sin h  f 1 1  q1 v 1 ; dz 4r 1 dz

h1 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 q1 c 1 ; k2 q2 c 2

ð7Þ

where c1 and c2 are the heat capacities of CO2 and the tubing, respectively; T1 and T2 are the temperatures of CO2 and the tubing, respectively; k1 and k2 are the thermal conductivities of CO2 and the tubing, respectively; aJ is the Joule–Thomson coefficient; h1 is the convection coefficient inside the tubing; Pr is the Prandtl number; q2 is the density of tubing; Q is the total energy produced by the fluid friction losses of a unit length of tubing; Q1 is the proportion of Q absorbed by the CO2; q is the flow rate inside the tubing; and Dpf is the gradient of the friction pressure drop. (2) Heat transfer in the tubing According to the law of conservation of energy, the energy balance in the tubing can be described as follows:

Fig. 1. Physical model of CO2 injection well.

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  2k2 k3 ðT 3  T 2 Þ 2r 1 h1 ðT 2  T 1 Þ @ @T 2   þ k 2 @z @z ðr 22  r 21 Þ k3 ln r22rþr2 3 þ k2 ln r22rþr3 3 ðr22  r21 Þ þ

Q2

pðr22  r21 Þ

¼ q2 c2

@T 2 ; @t

ð8Þ

where

Q2 ¼ Q

1 : 1þb

ð9Þ

Here, r2 is the outer radius of the tubing; r3 is the inner radius of the casing; k3 and T3 are the thermal conductivity and temperature of the fluid in the annulus, respectively; and Q2 is the proportion of Q absorbed by the tubing. (3) Heat transfer in the annulus, casing, cement sheath, and formation The temperature of the annulus, casing, cement sheath, and formation are mainly determined by heat conduction in the radial and axial directions: 2ki kiþ1 ðT iþ1  T i Þ 2ki ki1 ðT i  T i1 Þ     r þr 2r þr i i þ ki1 ln r 2rþr kiþ1 ln i 2riþ1 þ ki ln r þriþ1 ðr 2i  r 2i1 Þ ki ln ri1 ðr 2i  r 2i1 Þ 2ri1 i i iþ1 i1 i   @ @T i @T i þ ð10Þ ¼ qi c i ði P 3Þ; ki @z @t @z

For brevity, the parameters of Eqs. (14) and (15) are not detailed in this article, but can be found in [29]. In this study, we adopt Vesovic’s model to calculate the viscosity and thermal conductivity coefficient of CO2. These can be written as [30,31]

lðq; TÞ ¼ l0 ðTÞ þ Dlðq; TÞ þ Dc lðq; TÞ and

ð16Þ

kðq; TÞ ¼ k0 ðTÞ þ Dkðq; TÞ þ Dc kðq; TÞ;

ð17Þ

0

0

where l and k are the viscosity and thermal conductivity at the zero-density limit, respectively; Dl and Dk are the excess viscosity and thermal conductivity of CO2 due to elevated density, respectively; Dcl and Dck are the enhancements of viscosity and thermal conductivity near the critical point, respectively. 3.4. Initial and boundary conditions To solve the coupled pressure and temperature equations described above, the corresponding boundary and initial conditions must be specified. The initial temperature for each profile of the computational domain can be given as

T i;j ðr; z; t ¼ 0Þ ¼ T s þ Gf z;

ð18Þ

where ri is the distance from the i-th element in the radial direction to the center of the wellbore; ki, Ti, qi, and ci are the thermal conductivity, temperature, density, and heat capacity of the i-th element in the radial direction, respectively.

where Ts is the surface temperature and Gf is the geothermal gradient. As the inlet fluid temperature is equal to the injection temperature, the injection temperature boundary conditions can be written as

3.3. Calculation of CO2 physicochemical properties

T 1;1 ðr ¼ 0; z ¼ 0; tÞ ¼ T in :

The CO2 is compressible, and its physicochemical properties change greatly with changing temperature and pressure. The Span–Wagner (S-W) model can be used to calculate the density, heat capacity, and Joule–Thomson coefficient of CO2 (216 K  T  1100 K, 0 MPa  p  800 MPa) [28]. According to the S-W model, the density, heat capacity, and Joule–Thomson coefficient of CO2 can be expressed as follows [29]:

p ; q¼ Rc Tð1 þ d/rd Þ 2

ð11Þ

 3 r 2 r 2 !  1 þ d @/  ds @@d@/s 7 @d @ 2 /o @ 2 /r þ þ r 2 r 5 @ s2 @ s2 1 þ 2d @/ þ d2 @@d/2 @d

6 cp ¼ Rc 4s2

r

aJ ¼ 




Rc q

r

1 þ d @/ @d

2 r

ð12Þ

2 r

ð13Þ where d = q/qc and s s = Tc/T are the reduced density and inverse reduced temperature, respectively; cp is the heat capacity at a constant pressure; and Rc is the specific gas constant. /o and /r are the ideal and remaining parts of the Helmholtz free energy, which can be expressed as 8 X aoi ln½1  expðshoi Þ i¼4

ð14Þ /r ðd; sÞ ¼

7 34 39 X X X 2 2 c n i d di st i þ ni ddi sti ed i þ ni ddi sti eai ðdei Þ bi ðsci Þ i¼1

i¼8

42 X 2 2 þ ni Dbi deCi ðd1Þ Di ðs1Þ i¼40

As the temperature at the outer radial boundary is constant, the temperature boundary condition can be described as

Tðr ! 1; z; tÞ ¼ T s þ Gf z:

ð20Þ

The temperature boundary condition at the surface and bottom boundary is

@Tðr; z; tÞ ¼ 0: @z z¼0;z¼H

ð21Þ

Besides, the initial and boundary pressure conditions can be written as

d @/ þ d2 @@d/2 þ ds @@d@/s @d  2 o   2 2 r 2 r r 2 @ 2 /r  ds @@d@/s  s2 @@ s/2 þ @@ s/2 1 þ 2d @/ þ d 2 @d @d

/o ðd; sÞ ¼ lnðdÞ þ ao1 þ ao2 s þ ao3 lnðsÞ þ

ð19Þ

i¼35

ð15Þ

p1;j ðz; t ¼ 0Þ ¼ pin þ qg sin hz

ð22Þ

p1;1 ðz ¼ 0; tÞ ¼ pin :

ð23Þ

The numerical discretization and solution procedure for the above equations are elaborated in the Appendix A. 4. Results and discussion 4.1. Model comparison and verification To verify the validity of our models, field data from the Cao 8 well in Jiangsu oilfield are compared with the output from our model and the calculation data given by Dou’s model. The basic parameters of the Cao 8 well are presented in Table 1 [19]. Fig. 2 displays the field-logged wellbore temperature and temperature predicted by the proposed model and Dou’s model. It is clear that both our model and Dou’s model predict temperatures that are in agreement with the field-logged data after injection for 26.58 days. However, we can also see that there are some differences between the output from our model and Dou’s model. Dou’s model predicts slightly lower temperatures than the fieldlogged temperature, whereas our model predicts a value that is slightly higher than the actual temperature. This is because Dou’s

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model considers the heat transfer in the wellbore to be steady (wellbore temperature does not vary with injection time), and ignores the heat generated by fluid friction. We can also see that, after an injection time of 30 min, the difference in the predicted temperature between our model and Dou’s model is approximately 4 °C at depths of more than 800 m. That is, for long-term injection conditions, both our model and Dou’s model can accurately predict the wellbore temperature when CO2 is injected, but in the initial stages of injection, Dou’s model will produce significant errors whereas our model accurately predicts the wellbore temperature. 4.2. Example analysis with different injection temperatures The wellbore combination and casing program are illustrated in Fig. 1, and the tubing and casing parameters are listed in Table 2. The thermodynamic parameters of the completion material (casing, cement sheath, and tubing) and formation are presented in Table 1, the well depth and geothermal gradient are 1257.4 m and 2.7 °C/100 m, respectively. In this section, we analyze the influence of the injection temperature on the temperature and pressure distribution in the wellbore. Injection temperatures of 20 °C, 0 °C, and 20 °C were simulated at an injection rate and pressure of 2.5 m3/min and 70 MPa, respectively. Fig. 3 displays the CO2 temperature distribution under different injection temperatures. Fig. 3a shows that the CO2 temperature in the wellbore decreases as the injection time increases under the 20 °C condition. The temperature drops rapidly in the initial stage, becoming a more gradual decrease after about 200 s. At an injection temperature of 0 °C (Fig. 3b), the CO2 temperature variation in the wellbore is similar to that in the 20 °C case, although the rate of decrease in the CO2 temperature is lower. For an injection temperature of 20 °C (Fig. 3c), the CO2 temperature first decreases, then increases before becoming stable. This behavior occurs because the primary cold fluid at the top of the wellbore will cool the fluid in lower parts of the wellbore as it flows down in the primary stage. As the primary fluid is gradually replaced, the CO2 temperature increases through the frictional heating of the fluid. Fig. 4 shows the impact of injection temperature on CO2 temperature at the downhole. It can be seen that higher injection tem-

Table 1 Basic parameters of Cao 8 well. Diameter of wellbore (mm) Inner diameter of casing (mm) Inner diameter of tubing (mm) Mud line depth (m) Injection pressure (MPa) Surface temperature (°C)

215.9

Thermal conductivity of tubing (W/(mk)) Thermal conductivity of casing (W/(mk)) Thermal conductivity of formation (W/(mk)) Density of fluid in the annulus (kg/m3) Density of cement sheath (kg/m3) Heat capacity of tubing (J/(kgk)) Heat capacity of casing (J/(kg. k)) Heat capacity of formation (J/(kg. k))

43.5

124.37 62 2074 30 15

43.5 2.09

Outer diameter of casing (mm) Outer diameter of tubing (mm) Well depth (m)

137

Injection rate (t/d) Injection temperature (°C) Geothermal gradient (°C/100 m) Thermal conductivity of fluid in the annulus (W/(mk)) Thermal conductivity of cement sheath (W/(mk)) Density of tubing (kg/m3)

21.17 20 3

3

73 3100

0.8 0.52 7800

1100

Density of casing (kg/m )

7800

2200

Density of formation (kg/m3)

2600

460

Heat capacity of fluid in the annulus (J/(kg. k)) Heat capacity of cement sheath (J/(kg. k))

4180

460 837.279

1880

683

peratures result in higher CO2 temperatures at the downhole. When the injection temperature is 20 °C, the CO2 temperature in the downhole can exceed its critical temperature (31.1 °C). A common phenomenon in all three cases is that the CO2 temperature at the downhole is higher than the injection temperature, because CO2 is heated by geothermal and frictional heat. We also plot the CO2 pressure distribution under different injection temperatures (Fig. 5). The smaller the difference between the injection temperature and the wellhead temperature, the smaller the variation of pressure in the wellbore. In general, however, the pressure in the wellbore exhibits little variation with time, and the pressure reaches the steady state before the temperature. Another common phenomenon is that the CO2 pressure in the wellbore decreases as the well depth increases, this is because the frictional pressure drop is greater than the CO2 weight pressure gradient. 4.3. Example analysis with different injection rates Because the wellbore temperature and pressure determine the state of the CO2, it is crucial to analyze the pressure and temperature in this region are affected by various factors. Therefore, we now analyze the influence of the injection rate on the wellbore CO2 temperature and pressure. Injection rates of 2, 2.5, and 3 m3/ min were simulated under an injection temperature and pressure of 0 °C and 70 MPa, respectively. The other basic parameters are same as the input data in ‘‘4.2Example analysis with different injection temperatures” part. The effect of the injection rate on CO2 temperature at the downhole is shown in Fig. 6. It can be seen that the rate of decrease in the downhole temperature increases as the injection rate decreases, and that higher injection rates result in higher CO2 temperatures at the downhole. This is because higher injection rates generate more frictional heat. In the case of a 3 m3/min injection rate, the downhole CO2 temperature in the stable stage is higher than its critical temperature; if the pressure at the downhole is also higher than its critical pressure (7.38 MPa), the CO2 will enter the formation as a supercritical fluid, which is the goal of many field techniques. A common phenomenon in all three cases is that the CO2 temperature at the downhole drops rapidly in the initial stage, and then tends to stabilize after injecting about 5 min. Fig. 7 displays the influence of injection rate on CO2 pressure at the downhole. The pressure at the downhole decreases as the injection rate increases. For a given injection rate, the pressure at the downhole slowly increases in the initial stage and then tends to become stable. This is because, as the injection time increases, the temperature in the wellbore decreases and the CO2 density increases, which will lead to an increase in wellbore fluid gravity and a reduction in volume flow rate (the frictional pressure drop decreases as the volume flow rate becomes smaller). Hence, the pressure at the downhole increases. In addition, the pressure at the downhole is above the critical pressure (7.38 MPa) in all three examples. For a 2 m3/min injection rate, the pressure drop from the wellhead to the downhole is only 7.4 MPa; when the injection rate increases to 3 m3/min, the pressure drop rises to 36.7 MPa. Therefore, it is necessary to study drag reducers that are suitable for CO2. In order to further investigate the difference between our model and other model, we plotted wellbore temperature and pressure profiles predicted by different models under different injection rates in Figs. 8 and 9. It can be seen that the wellbore temperature predicted by two different models are both approximately linearly increases with the increase of the wellbore depth (Fig. 8). But there are also some difference, the wellbore temperature predicted by the Dou’s model[19] decreases with the increase of injection rates, while wellbore temperature predicted by the our model increases with the increase of injection rates, this is because Dou’s model

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Fig. 2. Comparison between the wellbore temperature obtained using our new model and that obtained using Dou’s model.

Table 2 Basic parameters of tubing and casing program. Parameter

First casing

Second casing

Third casing

Tubing

Inner diameter (mm) Outer diameter (mm)

322.9 339.7

228.7 244.5

127.3 139.7

62 73

ignores the heat generated by fluid friction, and the injection temperature is lower than the geothermal. However, the wellbore pressure predicted by different models almost have the same result (Fig. 9), the slight pressure difference is caused by the difference of wellbore temperature. 4.4. Example analysis with different tubing roughness We now study the influence of the tubing roughness on the wellbore CO2 temperature and pressure. Absolute tubing roughness values of 0.015, 0.03, and 0.045 mm were considered with an injection rate, injection pressure and injection temperature of 2.5 m3/min, 70 MPa and 0 °C, respectively. Fig. 10 shows the impact of the tubing roughness on the CO2 temperature at the downhole. It can be seen that, as the tubing roughness increases, the CO2 temperature at the downhole increases and the rate of decrease in the downhole temperature decreases. When the absolute tubing roughness is 0.045 mm, the downhole CO2 temperature in the stable stage is slightly higher than its critical temperature. This is because the higher tubing roughness generates more frictional heat. Fig. 11 displays the effect of tubing roughness on CO2 pressure at the downhole. The three curves in this figure have the same shape, whereby the pressure slowly increases in the initial stage before maintaining a constant value approximately 300 s after injection. The pressure at the downhole decreases as the tubing roughness increases, with pressure differences between the downhole and wellhead of 20.1 MPa, 25.8 MPa, and 29.9 MPa, respectively. The wellbore temperature and pressure profiles predicted by different models under different tubing roughness values are presented in Figs. 12 and 13, respectively. From Fig. 12 we can see that the wellbore temperature predicted by Dou’s model change little with the change of tubing roughness, but the wellbore temperature calculated by our model increase rapidly with increase of tubing roughness. This is attributed to the fact that the frictional heat

increase with the increase of tubing roughness, but Dou’s model did not consider it. Therefore, the wellbore temperature calculated by our model is higher than that of Dou’s model. From Fig. 13 we can see that the wellbore pressure calculated by Dou’s model and our model are both decrease with the increase of tubing roughness, and there are almost no differences in the wellbore pressure profiles calculated by different models. 4.5. Example analysis considering different factors This section analyzes the influence of the Joule–Thomson effect and frictional heat proportional absorption effect (frictional heat is absorbed by the CO2 and tubing according to a contact coefficient) on the downhole temperature and pressure. We define the temperature difference and pressure difference as the difference between our model output and the value calculated when the Joule–Thomson effect and frictional heat proportional absorption effect are neglected. The injection pressure and temperature are 70 MPa and 0 °C, respectively. The other basic parameters are listed in Tables 1 and 2. Fig. 14 shows the influence of the Joule–Thomson effect on the CO2 temperature difference at the downhole under different injection rates. As can be seen, the variation in temperature difference with time caused by the Joule–Thomson effect is different for each injection rate. For the 3 m3/min injection rate, the temperature difference is negative in the initial stages, and later becomes positive. For the 2.5 and 2 m3/min injection rates, the temperature difference is positive for the whole injection process, and increases with time in the initial stages of injection before attaining a constant value after around 300 s. That is, the downhole temperature is higher when the Joule–Thomson effect is included in the model. The temperature difference in the stable stage increases as the injection rate increases. Fig. 15 displays the impact of the Joule–Thomson effect on downhole pressure difference under different injection rates. It can be seen that, with a 3 m3/min injection rate, the pressure

L.-P. Yi et al. / International Journal of Heat and Mass Transfer 121 (2018) 680–690

(a) Injection temperature of -20°C

(a) Injection temperature of -20°C

(b) Injection temperature of 0°C

(b) Injection temperature of 0°C

(c) Injection temperature of 20°C

(c) Injection temperature of 20°C

Fig. 3. CO2 temperature distribution under different injection temperatures.

Fig. 5. CO2 pressure distribution under different injection temperatures.

Fig. 4. CO2 temperature at the downhole under different injection temperatures.

Fig. 6. CO2 temperature at the downhole under different injection rates.

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Fig. 7. CO2 pressure at the downhole under different injection rates.

Fig. 11. CO2 pressure at the downhole under different tubing roughness values.

Fig. 8. Wellbore temperature profiles predicted by different models under different injection rates (injection time is 5 min).

Fig. 12. Wellbore temperature profiles predicted by different models under different tubing roughness values (injection time is 5 min).

Fig. 9. Wellbore pressure profiles predicted by different models under different injection rates (injection time is 5 min).

Fig. 13. Wellbore pressure profiles predicted by different models under different tubing roughness values (injection time is 5 min).

Fig. 10. CO2 temperature at the downhole under different tubing roughness values.

difference is only positive in the initial stages, before becoming negative and fluctuating around a value of 0.51 MPa after approximately 300 s. For the 2.5 and 2 m3/min injection rates, the pressure difference increases with time in the initial stages, and then tends toward values of 0.2 MPa and 0.05 MPa, respectively. In other words, the downhole pressure is lower when the Joule– Thomson effect is included in the model, and the greater the injection rate, the greater the pressure difference. Fig. 16 shows the influence of the frictional heat proportional absorption effect on the CO2 temperature difference at the downhole under different values of the tubing roughness. The temperature difference is negative in all three cases, that is, the downhole temperature is lower when the frictional heat proportional absorption effect is included in the model. The temperature difference decreases over time, with higher value of tubing roughness leading to greater differences in temperature. The maximum temperature

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687

Fig. 14. CO2 temperature difference at the downhole caused by Joule–Thomson effect under different injection rates. Fig. 17. Downhole pressure difference caused by frictional heat proportional absorption effect under different tubing roughness values.

5. Conclusions Based on the numerical results reported in the previous section, the following conclusions can be drawn:

Fig. 15. Downhole pressure difference caused by Joule–Thomson effect under different injection rates.

(1) The CO2 pressure at the downhole is generally higher than its critical pressure. Thus, engineers can theoretically adjust the injection temperature or injection rate to control the downhole temperature, ensuring that the CO2 reaches the supercritical state at the downhole. (2) The CO2 temperature at the downhole increases as the tubing becomes rougher, whereas the downhole pressure decreases with the increase in tubing roughness. Therefore, it is not reasonable to ignore the influence of tubing roughness on the temperature and pressure at the downhole, unless the tubing is new. (3) In the early stages of injection, the pressure and temperature in the wellbore are unsteady and the trend is complicated. Therefore, it is not reasonable to assume that the heat transfer process in the wellbore is in the steady state, unless the injection process continues over the longer term. (4) The downhole temperature is higher when the Joule–Thomson effect is considered, whereas the downhole pressure is lower in the presence of the Joule–Thomson effect. (5) The frictional heat proportional absorption effect has a slight influence on the pressure and temperature, but this can be neglected at the engineering scale. Conflict of interest The authors declared that there is no conflict of interest.

Fig. 16. CO2 temperature difference at the downhole caused by frictional heat proportional absorption effect under different tubing roughness values.

difference in each case is 0.195 °C, 0.188 °C, and 0.169 °C. However, differences of this magnitude can be neglected at the engineering scale. Fig. 17 displays the impact of the frictional heat proportional absorption effect on the downhole pressure difference under different values of the tubing roughness. The pressure difference is positive in all three cases, that is, the downhole pressure is higher when the frictional heat proportional absorption effect is included in the model, although the maximum pressure difference is less than 0.02 MPa.

Acknowledgments This work was financially supported by the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (PLN1208), Qihang Foundation of Southwest Petroleum University (No. 431), and the National Science and Technology Major Project of China (2016ZX05044-004-002). Appendix A. Equation discretization and numerical solutions As the pressure and temperature are coupled, we adopt a fully implicit finite difference method to discretize the governing equations. The corresponding program was written and executed using the MATLAB software. We assume that the heat transfer in the wellbore and formation is axisymmetric, so the schematic diagram

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of mesh grids in the computational domain can be represented by the two-dimensional form in Fig. A1. Therefore, the heat transfer equation within the tubing can be discretized as follows:

Dt a Dt ¼ 2 Q nþ1 þ pr1 1;j 

q

nþ1 1;j

v

nþ1 nþ1 1;j c 1;j

Dzj

"

!

nþ1 nþ1 pnþ1 þ anþ1 J;j v 1;j c 1;j 1;j

nþ1 nþ1 Dtqnþ1 1;j v 1;j c 1;j T nþ1 1;j1 Dz j

ðA-1Þ

We adopt the fast successive over-relaxation (SOR) iteration method to solve Eq. (A-1). The iterative form is expressed as follows: ðkÞ

¼ T nþ1 þ 1;j

x a1j

ðkþ1Þ

ðb1j þ c1j T nþ1 1;j1



r 2 þr 3 2;j c 2;j k3;j ln 2r 2

q

ðkÞ

¼

nþ1 nþ1 nþ1 Dt qnþ1 2Dth1;j 1;j v 1;j c 1;j nþ1 þ þ qnþ1 1;j c 1;j ; Dzj r1

þ

c1j ¼



ðr 22

 r 21 Þ

5T nþ1 2;j

DtQ nþ1 2r1 Dth1;j T nþ1 Dtk2;j0:5 T nþ1 2;j 1;j 2;j1 þ T n2;j þ þ 2 2 2 2 pq2;j c2;j ðr2  r1 Þ q2;j c2;j ðr2  r1 Þ q2;j c2;j Dzj Dzj0:5

þ

Dtk2;jþ0:5 T nþ1 2Dtk2;j k3;j T nþ1 2;jþ1 3;j   þ and q2;j c2;j Dzj Dzjþ0:5 q c2;j k3;j ln r2 þr3 þ k2;j ln 2r3 ðr2  r2 Þ 2;j 2 1 2r 2 r 2 þr 3 ðA-4Þ

ðkÞ

T nþ1 2;j

ðkþ1Þ

ðkÞ

¼ T nþ1 þ 2;j

where x is the iteration factor; a1j, b1j, c1j, and d1j are the coefficients, which can be written as

Dt nþ1 n þ qnþ1 b1j ¼ 2 Q nþ1 1;j c 1;j T 1;j pr1 1;j

þ k2;j ln r22rþr3 3

 a1j T nþ1 þ d1j T nþ1 Þ; 1;j 2;j ðA-2Þ

a1j ¼

3

2Dtk2;j k3;j

nþ1

nþ1

ðkþ1Þ

2r 1 Dth1;j k2;jþ0:5 Dt k2;j0:5 Dt þ þ Dzj Dzjþ0:5 q2;j c2;j Dzj Dzj0:5 q2;j c2;j q2;j c2;j ðr 22  r 21 Þ

þ

2Dth1;j T nþ1 2;j þ r1

T nþ1 1;j

nþ1

1þ

nþ1 nþ1 nþ1 Dt anþ1 J;j q1;j v 1;j c1;j nþ1 nþ1 n pnþ1 1;j1 þ q1;j c1;j T 1;j Dz j

nþ1 nþ1 n  anþ1 J;j v 1;j c 1;j p1;j þ

ðA-3dÞ

Similarly, the discretization and iterative form of the heat transfer equation in the tubing can be expressed as follows:

! nþ1 nþ1 nþ1 Dtqnþ1 2Dth1;j 1;j v 1;j c 1;j nþ1 nþ1 þ þ q1;j c1;j T nþ1 1;j Dz j r1 nþ1 J;j

nþ1

2Dth1;j : r1

d1j ¼

x

ðkþ1Þ

a2j

ðb2j þ c2j T nþ1 1;j

ðkþ1Þ

þ d2j T nþ1 2;j1

 a2j T nþ1 2;j

ðkÞ

ðkÞ

ðkÞ

nþ1 þ e2j T nþ1 Þ; 2;jþ1 þ f 2j T 3;j

ðA-3aÞ

ðA-5Þ

where

a2j ¼ 1 þ

!

nþ1 nþ1 nþ1 Dtanþ1 J;j q1;j v 1;j c 1;j nþ1 nþ1 þ anþ1 pnþ1 J;j v 1;j c 1;j 1;j Dzj

k2;jþ0:5 Dt k2;j0:5 Dt þ Dzj Dzjþ0:5 q2;j c2;j Dzj Dzj0:5 q2;j c2;j nþ1

þ

nþ1 nþ1 nþ1 Dtanþ1 J;j q1;j v 1;j c 1;j nþ1 nþ1 nþ1 n  pnþ1 1;j1  aJ;j v 1;j c1;j p1;j ; Dz j

ðA-3bÞ

nþ1 nþ1 Dt qnþ1 1;j v 1;j c 1;j ; and Dz j

ðA-3cÞ

þ

b2j ¼

2r1 Dth1;j

q2;j c2;j ðr22  r21 Þ 2Dtk2;j k3;j   ; r 2 þr 3 q2;j c2;j k3;j ln 2r2 þ k2;j ln r22rþr3 3 ðr22  r21 Þ

DtQ nþ1 2;j þ Tn ; pq2;j c2;j ðr22  r21 Þ 2;j

ðA-6aÞ

ðA-6bÞ

nþ1

c2j ¼

d2j ¼

e2j ¼

f 2j ¼

Fig. A1. Schematic diagram of mesh grids in the computational domain.

2r 1 Dth1;j

q2;j c2;j ðr22  r21 Þ

;

Dtk2;j0:5

q2;j c2;j Dzj Dzj0:5 Dtk2;jþ0:5

q2;j c2;j Dzj Dzjþ0:5

ðA-6cÞ

;

ðA-6dÞ

; and

ðA-6eÞ

2Dtk2;j k3;j   : r 2 þr 3 q2;j c2;j k3;j ln 2r2 þ k2;j ln r22rþr3 3 ðr22  r21 Þ

ðA-6fÞ

Similarly, the discretization and iterative form of the heat transfer equation in the annulus, casing, cement sheath, and formation can be written as follows:

689

L.-P. Yi et al. / International Journal of Heat and Mass Transfer 121 (2018) 680–690

Fig. A2. Flowchart of the calculation procedure.

2 41 þ

2Dtki;j ki1;j  r i1 þri qi;j ci;j ki;j ln 2r þ ki1;j ln r i1

þ

i1



dij ¼

2r ðr2i  r 2i1 Þ qi;j ci;j kiþ1;j ln ri þr2riþ1 þ ki;j ln ri þriþ1 i iþ1

#

þ þ

þ

ki;jþ0:5 Dt ki;j0:5 Dt T nþ1 ¼ T ni;j þ Dzj Dzjþ0:5 q2;j c2;j Dzj Dzj0:5 q2;j c2;j i;j 2Dtki;j ki1;j T nþ1 i1;j





þri i ðr2i  r 2i1 Þ qi;j ci;j ki;j ln ri1 þ ki1;j ln r 2rþr 2r i1 i1 i

Dtki;jþ0:5 T nþ1 i;jþ1

qi;j ci;j Dzj Dzjþ0:5

þ

þ

eij ¼

Dtki;j0:5 T nþ1 i;j1

qi;j ci;j Dzj Dzj0:5

2Dtki;j kiþ1;j T nþ1 iþ1;j



ðkþ1Þ

ðkÞ

¼ T nþ1 þ i;j þ

x aij

ðkÞ eij T nþ1 i;jþ1

ðkþ1Þ

ðbij þ cij T nþ1 i1;j

ðkþ1Þ

þ dij T nþ1 i;j1

ðkÞ

 aij T nþ1 i;j

ðkÞ

þ f ij T nþ1 iþ1;j Þ;

f ij ¼



2r qi;j ci;j kiþ1;j ln ri þr2riþ1 þ ki;j ln r þriþ1 ðr 2i  r 2i1 Þ i i iþ1

ðA-7Þ T nþ1 i;j

þ

ðA-9cÞ

ki;jþ0:5 Dt

qi;j ci;j Dzj Dzjþ0:5

;

ðA-9dÞ

; and

ðA-9eÞ

2Dtki;j kiþ1;j r i þr iþ1 i;j c i;j ðkiþ1;j ln 2ri

q

2r

þ ki;j ln ri þriþ1 Þðr 2i  r 2i1 Þ iþ1

nþ1

pnþ1 1;j

þr i i qi;j ci;j ðki;j ln ri1 þ ki1;j ln r 2rþr Þðr 2i  r 2i1 Þ 2ri1 i1 i

ðkþ1Þ

ðkþ1Þ

¼ pnþ1 1;j1 

ðA-9aÞ ðA-9bÞ

ðA-9fÞ

ðA-10Þ nþ1ðkÞ

ðkÞ

qnþ1 ðv nþ1 Þ 1;j 1;j 4r1

ðkÞ

2

4r 1

þ qnþ1 g sin hDz  f 1;j 1;j

ðkÞ

2r qi;j ci;j ðkiþ1;j ln ri þr2riþ1 þ ki;j ln ri þriþ1 Þðr 2i  r 2i1 Þ i iþ1

:

nþ1 qnþ1 1;j ðv 1;j Þ

nþ1 nþ1 nþ1  qnþ1 1;j v 1;j ðv 1;j  v 1;j1 Þ

ðA-8Þ

2Dtki;j ki1;j

ki;jþ0:5 Dt ki;j0:5 Dt þ þ Dzj Dzjþ0:5 q2;j c2;j Dzj Dzj0:5 q2;j c2;j

ki;j0:5 Dt

qi;j ci;j Dzj Dzj0:5

nþ1 nþ1 pnþ1 1;j ¼ p1;j1 þ q1;j g Dz sin h  f 1;j Dz

2Dtki;j kiþ1;j

bij ¼ T ni;j ;

;

Finally, the discretization and iterative form of the pressure equation can be described as follows:

where

aij ¼ 1 þ

2ki;j ki1;j Dt þr i i qi;j ci;j ðki;j ln ri1 þ ki1;j ln r 2rþr Þðr2i  r 2i1 Þ 2ri1 i1 i

i

2Dtki;j kiþ1;j



cij ¼

 2ri ðr2i  r 2i1 Þ þr

Dz

2 ðkÞ

 qnþ1 1;j

v nþ1 ðv nþ1 1;j 1;j ðkÞ

ðkÞ

 v nþ1 ðA-11Þ 1;j1 Þ ðkÞ

Combined with the above discretized equations and specific conditions, the whole coupled system of equations can be solved by the following calculation procedures. The iteration continues until both the pressure and temperature meet the following convergence conditions. A flowchart of the calculation procedure is shown in Fig. A2.

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L.-P. Yi et al. / International Journal of Heat and Mass Transfer 121 (2018) 680–690

8   ðkþ1Þ ðkÞ > < max jpnþ1  pnþ1 j 6 ep 1;j 1;j   ðkþ1Þ ðkÞ > : max jT nþ1  T nþ1 j 6 eT i;j i;j

ðA-12Þ

(1) Firstly, input the basic data including the thermophysical properties of the tubing, fluid in the annulus, casing, cement sheath, and formation. Supply suitable initial and boundary conditions. (2) Using pressure and temperature trial solutions p1,jn+1(k) and Ti,jn+1(k), the CO2 density is calculated using Eqs. (11) and (15), the CO2 viscosity is given by Eq. (16), and the wellbore pressure p1,jn+1(k+1) is obtained from Eq. (A-11). The total heat generated by the fluid friction losses Q can be obtained from Eq. (6). (3) Calculate the convection coefficient through Eq. (4), recalculate the CO2 density by substituting the CO2 pressure p1,jn+1 (k+1) and temperature T1,jn+1(k) into Eqs. (11) and (15), and compute the thermal conductivity coefficient of CO2 by substituting the CO2 density and temperature T1,jn+1(k) into Eq. (17). The temperature Ti,jn+1(k+1) can then be obtained combining Eqs. (A-1)–(A-9). (4) Repeat steps (2) and (3) until the pressure and temperature meet the convergence conditions. Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.01.036. References [1] M.L. Godec, V.A. Kuuskraa, P. Dipietro, Opportunities for using anthropogenic CO2 for enhanced oil recovery and CO2 storage, Energy Fuels 27 (8) (2013) 4183–4189. [2] H. Kumar, D. Elsworth, J. Liu, et al., Optimizing enhanced coalbed methane recovery for unhindered production and CO2 injectivity, Int. J. Greenhouse Gas Control 11 (2012) 86–97. [3] J.E. Fitzgerald, Z. Pan, M. Sudibandriyo, et al., Adsorption of methane, nitrogen, carbon dioxide and their mixtures on wet Tiffany coal, Fuel 84 (2005) 2351– 2363. [4] H.E. Ross, P. Hagin, M.D. Zoback, CO2 storage and enhanced coalbed methane recovery: reservoir characterization and fluid flow simulations of the Big George coal, Powder River Basin, Wyoming, USA, Int. J. Greenhouse Gas Control 3 (2009) 773–786. [5] W. Song, H. Ni, R. Wang, et al., Pressure controlling method for managed pressure drilling with supercritical carbon dioxide as the circulation fluid, Pet. Explor. Dev. 43 (5) (2016) 787–792. [6] H. Wang, Z. Shen, G. Li, Wellbore temperature and pressure coupling calculation of drilling with supercritical carbon dioxide, Pet. Explor. Dev. 38 (1) (2011) 97–102. [7] Z. He, S. Tian, G. Li, et al., The pressurization effect of jet fracturing using supercritical carbon dioxide, J. Nat. Gas Sci. Eng. 27 (2015) 842–851. [8] B.R. Scanlon, R.C. Reedy, J.-P. Nicot, Comparison of water use for hydraulic fracturing for unconventional oil and gas versus conventional oil, Environ. Sci. Technol. 48 (2014) 12386–12393.

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