Coupling model for calculation of transient temperature and pressure during coiled tubing drilling with supercritical carbon dioxide

International Journal of Heat and Mass Transfer 125 (2018) 400–412

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

Coupling model for calculation of transient temperature and pressure during coiled tubing drilling with supercritical carbon dioxide Xiao-Gang Li a, Liang-Ping Yi a,⇑, Zhao-Zhong Yang a,⇑⇑, Yu-Ting Chen b, Jun Sun c a

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Xindu Avenue 8, Chengdu 610500, China CNPC Offshore Engineering Co., Ltd., Beijing 100028, China c Down-Hole Operating Company, Chuanqing Drilling and Exploration Engineering Co., Ltd., Chengdu 610051, China b

a r t i c l e

i n f o

Article history: Received 26 October 2017 Received in revised form 16 March 2018 Accepted 19 April 2018

Keywords: Supercritical carbon dioxide Coiled tubing drilling Transient Wellbore Temperature Pressure

a b s t r a c t Based on thermodynamics, heat transfer mechanisms, and fluid mechanics, a transient temperature and pressure coupling calculation model for supercritical carbon dioxide coiled tubing drilling is established in this study. In this model, the Joule-Thomson effect is considered, the CO2 physical properties are varied with the temperature and pressure, and the heat transfer in the wellbore and formation are both considered unsteady. The model is solved using the fully implicit finite difference method. The results show that the wellbore CO2 temperatures predicted by Gupta’s model and Wang’s model are both lower than that of the proposed model. The primary reason for this discrepancy is that the previous models considered the heat transfer in the wellbore as a steady state and ignored friction heat. In the shallower and deeper sections of the well, the wellbore temperature changes rapidly with the depth, whereas in the middle section of well, the wellbore temperature increases linearly with increasing depth. The wellbore temperature changes with circulation time, but the wellbore pressure is unaffected by the circulation time. The injection rate and nozzle diameter both have a significant effect on the downhole temperature and tubing pressure, and an injection rate that is too larger or a nozzle diameter that is too small may lead to CO2 that cannot exist in a supercritical state. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Supercritical carbon dioxide (SC-CO2) drilling is a novel drilling technique that uses the SC-CO2 as a drilling fluid [1]. SC-CO2 has many unique physicochemical properties; its very low viscosity and diffusivity are close to those of the gas, and it has a high density that is close to that of the liquid [2]. Compared to conventional drilling technology, using SC-CO2 as a drilling fluid provides the following advantages: (1) SC-CO2 cuts rock at a much lower pressure than water [3–6], which can improve drilling speed; (2) it does not cause damage to the reservoir even if CO2 incursion occurs; (3) using CO2 as a drilling fluid complies with global environmental policy and can reduce greenhouse gas emissions [7]. CO2 can only become a supercritical fluid when the temperature and pressure are both higher than their critical values (Tc = 30.98 °C and Pc = 7.38 MPa) [8–10]. However, the heat transfer process and pressure variations during SC-CO2 drilling are complex. As CO2 flows through the nozzle, it causes a significant drop in ⇑ Corresponding author. ⇑⇑ Corresponding author. E-mail addresses: [email protected] (L.-P. Yi), [email protected] (Z.-Z. Yang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.04.095 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

pressure and temperature [11], and thus, the CO2 may not reach a supercritical state at the downhole when the pressure drop and temperature are too high. Therefore, it is important to accurately predict the temperature and pressure of the CO2 during SC-CO2 drilling. Some temperature and pressure prediction models for the drilling process have been developed [12–14], but the drilling fluid in these models is water rather than CO2. Because CO2 is a compressible fluid, its physical properties are greatly affected by temperature and pressure. Therefore, it is necessary to establish a temperature and pressure computational model for SC-CO2 drilling. At present, there have been few studies of temperature prediction models for SC-CO2 drilling. Gupta [5] established a temperature and pressure calculation model to study the feasibility of SC-CO2 drilling. However, that model assumed that the CO2 physical parameters (thermal conductivity, viscosity, etc.) were constant and ignored the Joule-Thomson effect. Wang et al. [15] developed a wellbore temperature and pressure calculation model for SC-CO2 drilling that took the Joule-Thomson effect into account. In their model, the CO2 physical parameters are varied with temperature and pressure. Subsequently [1], consideration of the formation water influx into the annulus led to the establishment of

X.-G. Li et al. / International Journal of Heat and Mass Transfer 125 (2018) 400–412

401

Nomenclature A c1 c2 c3 c4 cp cv Gf h1 h2 h3 K Nu pu pd Pr q1 qm Q1 Q2 r1 r2 r3 r4 r5 Rc Re T1

nozzle cross-section area, m2 heat capacity of CO2 inside the coiled tubing, J/(kg K) heat capacity of coiled tubing, J/(kg K) heat capacity of CO2 in the annulus, J/(kg K) casing heat capacity, J/(kg K) isobaric heat capacity, J/(kg K) volumetric heat capacity, J/(kg K) geothermal gradient, K/m convection coefficient inside the coiled tubing, W/(m2 K) convection coefficient outside the coiled tubing, W/(m2 K) convection coefficient inside the casing, W/(m2 K) isentropic index, dimensionless Nusselt number, dimensionless CO2 pressure at nozzle upstream, Pa CO2 pressure at nozzle downstream, Pa Prandtl number, dimensionless CO2 flow rate inside the coiled tubing, m3/s CO2 mass flow rate, kg/s energy produced by the fluid friction losses of unit length coiled tubing, W/m energy produced by the fluid friction losses per unit length of annulus, W/m coiled tubing inner radius, m coiled tubing outer radius, m casing inner radius, m casing outer radius, m cement sheath outer radius, m gas constant, 0.1889 kJ/(kg K) Reynolds number, dimensionless temperature of CO2 inside the coiled tubing, K

a wellbore flow model for coiled tubing drilling. However, the effects of the casing and drill pipe thermal resistance on heat transfer were not considered [1,15]. Ni et al. [16] built on this by considering the effect of the casing and tubing thermal resistance on heat transfer to establish a coupling model to investigate the fluid flow and heat transfer during SC-CO2 coiled tubing drilling. Song et al. [17] also accounted for the effect of the casing and tubing thermal resistance on heat transfer and developed a temperature and pressure coupling calculation model for SC-CO2 pressure controlling drilling. Although previous studies [1,15–17] have preliminarily determined the variation in CO2 temperature and pressure from the construction parameters, all of these studies have assumed that the heat transfer in both the wellbore and the formation are steady. In addition, the current models all ignore the heat generated by fluid friction. Building on these previous studies and aware of the possible limitations, a full transient pressure and temperature coupling calculation model for SC-CO2 coiled tubing drilling is proposed in this study. In the proposed model, the influence of the casing and tubing thermal resistance on heat transfer is considered, and the CO2 physical properties are varied with temperature and pressure. In addition, the heat transfer in the wellbore and formation are both treated as unsteady. Finally, the model simulation results of a case study were analyzed and compared with results of previous studies to verify the reliability of the model.

T2 T3 T4 Ts Tu Td

v1 v3

x q1 q2 q3 q4 d

s aJ uo ur

l0

D1 D3 k0 k1 k2 k3 k4 k5 Dl Dk Dcl Dck

coiled tubing temperature, K temperature of CO2 in the annulus, K casing temperature, K surface temperature, K CO2 temperature at nozzle upstream, K CO2 temperature at nozzle downstream, K CO2 flow velocity in the coiled tubing, m/s CO2 flow velocity in the annulus, m/s iteration factor, dimensionless CO2 density in the coiled tubing, kg/m3 density of coiled tubing, kg/m3 CO2 density in the annulus, kg/m3 casing density, kg/m3 reduced density, dimensionless inverse reduced temperature, dimensionless Joule-Thomson coefficient, K/Pa ideal part of Helmholtz energy, dimensionless residual part of Helmholtz energy, dimensionless viscosity at the zero-density limit, Pa s coiled tubing absolute roughness, m casing absolute roughness, m thermal conductivity at the zero-density limit, W/(m K) thermal conductivity of CO2 inside the coiled tubing, W/(m K) thermal conductivity of coiled tubing, W/(m K) thermal conductivity of CO2 in the annulus, W/(m K) thermal conductivity of casing, W/(m K) thermal conductivity of cement sheath, W/(m K) excess viscosity, Pa s excess thermal conductivity, W/(m K) enhancements of viscosity, Pa s enhancements of thermal conductivity, W/(m K)

CO2 flows out of the tanker, pressurized with a triple plunger pump, and then flows into the coiled tubing; (2) high pressure CO2 flows from the wellhead to the downhole through the coiled tubing; (3) CO2 is ejected from the nozzle on a drilling bit, which improves drilling speed; and (4) CO2 flowing through the nozzle results in a drop in pressure and temperature, and the CO2 flows back to the ground along the annulus.

2. Physical model A schematic of coiled tubing drilling with SC-CO2 is shown in Fig. 1. The specific operation process is as follows: (1) the liquid

Fig. 1. Schematic of coiled tubing drilling with SC-CO2 [18].

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3. Mathematical model

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

8 X  aoi ln 1  expðshoi Þ

A schematic of the fluid flow and heat transfer of SC-CO2 during coiled tubing drilling is shown in Fig. 2. It is clear that the flow and heat transfer process during SC-CO2 drilling is complex. Therefore, to solve this problem, the following assumptions are made: (1) the CO2 flow pattern in both the drilling pipe and the annulus is onedimensional; (2) the change in internal energy induced by the CO2 phase change is ignored; (3) heat transfer is ignored for the CO2 flow in the ground pipeline; and (4) before injection, the wellbore is filled with CO2 and reaches a temperature equilibrium with the formation. 3.2. Calculation of CO2 physical properties Changes in temperature and pressure greatly affect the physical properties of CO2. Previous studies have indicated that the SpanWagner (S-W) model can accurately calculate the density, heat capacity, and Joule-Thomson coefficient of CO2 for temperatures between 216 K and 1100 K, and pressures between 0 MPa and 800 MPa [16,19]. The relevant equations can be expressed as follows [19]:

q¼

ð4Þ

i¼4

3.1. Basic hypothesis

p
 r Rc T 1 þ d @/ @d

ð1Þ

8 2 2 3 2 !  > r > 2 2 1 þ d @/  ds @@d@/sr 7 > @d @ / @ / > 6 o r > > þ c ¼ Rc 4s2 þ 5 > 2 < p @ s2 @ s2 1 þ 2d @/r þ d2 @ /2r @d

! > > > > @ 2 /o @ 2 /r > 2 > > : cv ¼ s Rc @ s2 þ @ s2

@d

/r ðd; sÞ ¼

7 34 X X c ni ddi sti þ ni ddi sti ed i i¼1

þ

39 X

i¼8

ni ddi sti eai ðdei Þ

2

bi ðsci Þ2

i¼35 42 X 2 2 þ ni Dbi deC i ðd1Þ Di ðs1Þ

Previous research has indicated that the model proposed by Vesovic et al. can be used to calculate the viscosity and thermal conductivity of CO2. The viscosity and thermal conductivity can both be described with the following expression [20,21]:

Xðq; TÞ ¼ X 0 ðTÞ þ DXðq; TÞ þ Dc Xðq; TÞ

ð6Þ

where X0(T) is the value in the zero-density limit; DX(q, T) is the excess value due to elevated density; and DcX(q, T) is the increase in value near the critical point. Therefore, the CO2 viscosity can be expressed as:

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

ð7Þ

Similarly, the CO2 thermal conductivity can be calculated with the following:

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

ð2Þ

ð5Þ

i¼40

ð8Þ

The details of the fluid viscosity and thermal conductivity calculations have been described in previous publications [20,21]. 3.3. Pressure calculation model

2

2

r d @/ þ d2 @@d/2r þ ds @@d@/sr @d aJ ¼    2   2 2 2 2 r r Rc q 1 þ d @/  ds @@d@/sr  s2 @@ s/2o þ @@ s/2r 1 þ 2d @/ þ d2 @@d/2r @d @d

ð3Þ where d = q/qc and s = Tc/T; cp and cv are the heat capacity at a constant pressure and constant volume, respectively; Rc is the gas constant; and /o and /r are the ideal and residual parts of the Helmholtz energy, respectively, which can be calculated with Eqs. (4) and (5) below.

Based on the law of conservation of mass and momentum theory, the pressure drop for CO2 flowing down inside the coiled tubing is obtained as follows:

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

ð9Þ

Similarly, the pressure drop for CO2 flowing up in the annulus can be expressed as:

dp3 q3 v 23 dv 3 ¼ q3 g sin h  f 3  q3 v 3 dz 4ðr 3  r 2 Þ dz

ð10Þ

where f1 and f3 are friction factors. Churchill’s model is able to predict the friction factor for all Reynolds numbers and pipe-roughness ratios [22–24], and thus it is used to calculate the friction factor in this study. The friction factor expressions are given in Eqs. (11) and (12) below: 8 "

#3=2 91=12  16 = < 0:27D1 f 1 ¼ 8 ð8=ReÞ12 þ 2:457 ln ð7=ReÞ0:9 þ þ ð37530=ReÞ16 ; : 2r 1

ð11Þ 8 "

#3=2 91=12  16 = < 0:27D3 12 0:9 16 f 3 ¼ 8 ð8=ReÞ þ 2:457 ln ð7=ReÞ þ þ ð37530=ReÞ ; : 2ðr 3  r 2 Þ

ð12Þ

Fig. 2. Schematic of fluid flow and heat transfer during SC-CO2 coiled tubing drilling.

where r1 and r2 are the inner and outer radius of the coiled tubing, respectively; r3 is the inner radius of the casing; q1 and q3 are the CO2 density in the coiled tubing and the annulus, respectively; v1 and v3 are the CO2 flow velocities in the coiled tubing and the annulus, respectively; Re is the Reynolds number; and D1 and D3 are the absolute roughness of the coiled tubing and the casing, respectively.

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CO2 flowing through the nozzle will produce an additional pressure and temperature drop, and an excessive temperature drop will cause the CO2 to become solid and plug the wellbore. To prevent this phenomenon, the flow rate in the nozzle should remain subsonic, and the pressure drop at the nozzle can then be calculated with the following implicit equation [15]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2 kþ1 #ffi u u 2k pd k p k p q q m ¼ At  d k  1 u u pu pu

ð13Þ

where

k¼

cp cv

ð14Þ

where qm is the CO2 mass flow rate, A is the cross-sectional area of the nozzle, k is the isentropic index of CO2, pu is the CO2 pressure upstream of the nozzle, and pd is the CO2 pressure downstream of the nozzle. 3.4. Heat transfer model 3.4.1. Heat transfer inside the coiled tubing Based on the first law of thermodynamics, and taking the JouleThomson effect into account, the equation to describe the heat transfer inside the coiled tubing can be written as [2]:

Q1

pr21

 q1 v 1 c1

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

@T 1 @p ¼ q1 c 1  aJ v 1 c1 1 @t @t

ð15Þ

where k2 is the thermal conductivity of the coiled tubing; T3 is the temperature of the CO2 in the annulus; q2 and c2 are the density and heat capacity of the coiled tubing, respectively; and h2 is the convection coefficient outside the coiled tubing. 3.4.3. Heat transfer in the annulus The annulus CO2 temperature is affected by the forced convection heat transfer. Considering the Joule-Thomson effect, the equation to describe the heat transfer in the annulus can be written as:

2r 2 h2 ðT 2  T 3 Þ 2r 3 h3 ðT 4  T 3 Þ @T 3 @p
2  þ þ q3 v 3 c3 þ aJ q3 v 3 c3 3 @z @z ðr 23  r 22 Þ r3  r22 Q @T 3 @p þ
2 2 2  ¼ q3 c3  aJ v 3 c 3 3 ð21Þ @t @t p r3  r2 where T4 is the temperature of the casing, c3 is the heat capacity of CO2 in the annulus, h3 is the convection coefficient inside the casing, and Q2 is the energy produced by the fluid friction loss per unit length of the annulus. 3.4.4. Heat transfer in the casing Three factors influence the casing temperature: heat conduction in the axial direction, convection heat transfer between the casing inner wall and the annulus CO2, and heat conduction between the casing outer wall and the cement sheath in the radial direction. Therefore, the heat transfer in the casing can be expressed as:

 @ @T 4 2k4 k5 ðT 5  T 4 Þ k4 þ
 @z @z 2 2 r  r k5 ln r4 þr5 þ k4 ln 4

k1 Nu h1 ¼ 2r 1

ð16Þ

Q 1 ¼ q 1 Dp f

ð17Þ

where Q1 is the energy produced by the fluid friction loss per unit length of coiled tubing; c1 is the CO2 heat capacity inside the coiled tubing; T1 and T2 are the temperatures of the CO2 and coiled tubing, respectively; aJ is the Joule-Thomson coefficient; h1 is the convection coefficient inside the coiled tubing; q1 is the CO2 flow rate inside the coiled tubing; Dpf is the gradient of the friction pressure drop inside the coiled tubing; and Nu is the Nusselt number, which can be calculated with Eq. (18) [25].

ðf 1 =8ÞRePr  pffiffiffiffiffiffiffiffiffiffi 1:07 þ 12:7 f 1 =8 Pr2=3  1

ð18Þ

Considering the Joule-Thomson effect, the temperature at the downstream of the nozzle can be calculated with the following expression [1,15,26]:

Td ¼ Tu

k1 pd k pu

ð19Þ

where Tu is the CO2 temperature upstream of the nozzle, and Td is the CO2 temperature downstream of the nozzle. 3.4.2. Heat transfer in the coiled tubing The coiled tubing temperature is primarily determined by the CO2 temperature and flow velocity inside the coiled tubing and the annulus. Based on the first law of thermodynamics, the expression to describe the heat transfer in the coiled tubing can be written as:

 @ @T 2 2r1 h1 ðT 2  T 1 Þ 2r 2 h2 ðT 2  T 3 Þ @T 2
2 
2   ¼ q2 c2 k2  @z @z @t r 2  r 21 r 2  r 21

2r 4



2r 3 h3 ðT 4  T 3 Þ
2  r4  r23

@T 4 ¼ q4 c4 @t

where

Nu ¼

3

2r 5 r 4 þr 5

ð20Þ

ð22Þ

where k4 and k5 are the thermal conductivity of the casing and cement sheath, respectively; r4 and r5 are the outer radius of the casing and cement sheath, respectively; and q4 and c4 are the density and heat capacity of the casing, respectively. 3.4.5. Heat transfer in the cement sheath and formation Heat transfer in the cement sheath and formation consists of heat conduction in both the radial and axial directions. Therefore, based on the first law of thermodynamics, the heat transfer equation for the cement sheath and formation can be written as:



2ki kiþ1 ðT iþ1  T i Þ 
 2r þ ki ln ri þriþ1 r2i  r2i1 iþ1

r þr kiþ1 ln i 2riþ1 i

 2ki ki1 ðT i  T i1 Þ @ @T i 
þ k  i  @z þri @z i ki ln ri1 þ ki1 ln ri12rþr r 2i  r 2i1 2r i1 i

¼ qi ci

@T i @t

ði P 5Þ

ð23Þ

3.5. Initial and boundary conditions To solve the above equations, the corresponding boundary and initial conditions need to be defined. The initial temperature for the profile of each region of calculation is a function of depth, z, and can be expressed as:

T i;j ðr; z; t ¼ 0Þ ¼ T s þ Gf z

ð24Þ

where Ts is the surface temperature, and Gf is the geothermal gradient. The CO2 temperature at the injection point is equal to the injection temperature, and can be defined as:

T 1;1 ðr ¼ 0; z ¼ 0; tÞ ¼ T in

ð25Þ

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It is assumed that the temperature at the outer radial boundary does not change, and thus the outer radial temperature boundary condition can be expressed as follows:

Tðr ! 1; z; tÞ ¼ T s þ Gf z

ð26Þ

In addition, it is assumed that the surface and bottom boundary are both insulated, and can be expressed as:

@Tðr; z; tÞ

¼0

@z z¼0;z¼H

ð27Þ

The initial fluid pressure at the wellbore is also assumed to be a function of depth, z, and is expressed as follows:

p1;j ðz; t ¼ 0Þ ¼ pin þ qg sin hz

ð28Þ

The pressure at the annulus outlet can be obtained from onsite well log data. Therefore, the pressure boundary condition can be described as follows:

p3;1 ðz ¼ 0; tÞ ¼ pout

ð29Þ

The numerical discretization and calculation steps for these equations are described in Appendix A.

4. Results and discussion

4.2. Wellbore temperatures and pressures at different circulation times Figs. 4 and 5 show the temperature and pressure profiles, respectively, at different circulation times. Fig. 4 shows that in the shallowest and deepest parts of the wellbore, the wellbore temperature decreases with increasing circulation time. However, in the middle section, the circulation time has little impact on the wellbore temperature, which is similar to the results reported by Yang et al. [13]. Similar to the trend in Fig. 3, the wellbore temperature increases rapidly with increasing depth in the shallower part of wellbore. At depths of 400–1700 m, the wellbore temperature increases linearly with increasing depth. However, near the downhole the temperature decreases abruptly. At depths greater than 600 m, the wellbore CO2 temperature exceeds its critical value (304.19 K). If the pressure also exceeds the critical value (7.38 MPa) in this region, the CO2 will enter the supercritical state. Fig. 5 shows that the circulation time has little effect on the pressure inside the annulus and coiled tubing, i.e., the pressure distribution in the wellbore can be considered steady. The pressure inside the annulus and coiled tubing both vary linearly with depth. When CO2 flows through the nozzle, a pressure drop of approximately 6.3 MPa occurs, which is consistent with the results reported by Ni et al. [16]. At depths of greater than 235 m, the CO2 pressure in the annulus is greater than its critical value (7.38 MPa). Therefore, in combination with the results in Fig. 4, it can be concluded that for depths between 600 m and the downhole, the CO2 is in a supercritical state, whereas the CO2 is in a liquid state in the other sections of the wellbore.

4.1. Model validation 4.3. Impact of the injection rate on wellbore temperature and pressure Because SC-CO2 drilling is still in the basic research stage, there is a lack of field data for wellbore temperatures during SC-CO2 drilling, and thus the validity of the proposed model cannot be verified by comparing the model calculation results to field logging data. In this section, the wellbore temperature data obtained from the proposed model is compared to that from Gupta’s model [5] and Wang’s model [15], as shown in Fig. 3. This can be considered a verification of the proposed model. The basic calculation parameters for the model are summarized in Table 1 [15]. Fig. 3 shows that the wellbore temperature profile curves obtained from the different models have similar transformation trends. The wellbore temperature profile curve can be divided into three sections: in the first section, the wellbore temperature increases rapidly with increasing well depth because the injected cryogenic CO2 is rapidly heated by the formation; in the second section, the wellbore temperature increases almost linearly with increasing well depth; and in the third section near the bottom of the well, the wellbore temperature decreases with increasing well depth. This is because as the CO2 flows through the nozzle, the temperature drops abruptly, and the cryogenic CO2 cools the coiled tubing and the CO2 inside the tubing as it flows upward along the annulus. However, there are some differences among the wellbore temperature profile curves obtained from the different models. In most parts of the wellbore, the CO2 temperature calculated by the proposed model is higher than that from Gupta’s model [5] and Wang’s model [15]. Two factors contribute to this difference. One is that the proposed model treats the heat transfer in the wellbore as transient, while Gupta and Wang both assume the heat transfer in the wellbore is steady. The other factor is that the proposed model takes into account the heat generated by the fluid friction losses, while Gupta and Wang both ignored this effect. It is clear that the model proposed in this study is more similar to the actual heat transfer process, and assuming that the heat transfer in the wellbore is steady state will lead to very large calculation errors.

The injection rate is an important optimization parameter for SC-CO2 drilling, because this parameter determines the ability of the process to efficiently cut rock and return the debris to the wellhead. Therefore, the influence of the injection rate on the wellbore temperature and pressure is analyzed in this section. For comparison, injection rates of 0.5 kg/s, 0.625 kg/s, and 0.75 kg/s are investigated. The other input parameters are as listed in Table 1. Fig. 6 shows the effect of the injection rate on the wellbore temperature profile. It illustrates that the temperature profile can be divided into three parts: (1) from the wellhead to a depth of 300 m, the wellbore temperature decreases slightly with increasing injection rate; (2) between well depths of 300 and 1600 m, the injection rate has little effect on the wellbore temperature; and (3) near the downhole, the wellbore temperature decreases with increasing injection rate. This is because a higher injection rate results in a larger pressure drop when the CO2 flows through the nozzle, which results in a greater temperature difference between the upstream and downstream of the nozzle, in accordance with Eq. (19). In addition, for injection rates of 0.625 kg/s and 0.75 kg/ s, the temperature downstream of the nozzle is lower than the CO2 critical temperature after 10 h of circulation, which is not conducive for cutting rock [27]. Therefore, in order to effectively break rock, the injection rate must be optimized to ensure the CO2 flow out of the nozzle remains in a supercritical state. Fig. 7 shows the effect of the injection rate on the wellbore pressure profile. These results illustrate that the pressure in the annulus increases slightly with increasing injection rate, and the annulus pressure difference at different injection rates increases with increasing well depth. This is because higher injection rates cause greater friction pressure drops. In these three cases, the pressure difference between the upstream and downstream of the nozzle are 6.3 MPa, 9.8 MPa, and 12.6 MPa, respectively. Therefore, the pressure inside the coiled tubing increases with increasing injection rate under conditions in which the outlet pressure is fixed.

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Fig. 3. Wellbore temperature profiles obtained from the different models.

Table 1 Basic model calculation parameters. Parameter name

Value

Parameter name

Value

Outer diameter of coiled tubing Outer diameter of casing Thermal conductivity of coiled tubing Thermal conductivity of casing Thermal conductivity of cement sheath Thermal conductivity of formation Heat capacity of coiled tubing Heat capacity of cement sheath Injection temperature Outlet pressure Surface temperature Injection rate

31.8 mm 73 mm 43.5 W/(m K) 43.5 W/(m K) 0.52 W/(m K) 2.3 W/(m K) 460 J/(kg K) 1880 J/(kg K) 253.15 K 5 MPa 288.71 K 0.5 kg/s

Inner diameter of coiled tubing Inner diameter of casing Density of tubing Density of casing Density of cement sheath Density of formation Heat capacity of casing Heat capacity of formation Well depth Circulation time Geothermal gradient Diameter of nozzle

22.9 mm 57 mm 7800 kg/m3 7800 kg/m3 2200 kg/m3 2600 kg/m3 460 J/(kg K) 837 J/(kg K) 2000 m 10 h 0.02723 K/m 4.7 mm

4.4. Impact of the injection temperature on wellbore temperature and pressure

4.5. Impact of nozzle diameter on the wellbore temperature and pressure

The influence of the injection temperature on the wellbore temperature and pressure is analyzed in this section. Injection temperatures of 253.15 K, 263.15 K, and 273.15 K were investigated. The other basic parameters are as listed in Table 1. Fig. 8 shows the effect of the injection temperature on the wellbore temperature profile. From the injection point to a well depth of approximately 300 m, the temperature in the annulus and inside the coiled tubing both decrease with decreasing injection temperature, because in this section of the well, the wellbore temperature is primarily affected by the injection temperature. For depths of greater than 300 m, the injection rate has little influence on the wellbore temperature. This is because the injection rate is small, and thus, the low temperature CO2 is rapidly heated by the high temperature formation as it flows down through the coiled tubing. At depths of greater than 300 m, the wellbore reaches a temperature equilibrium with the formation. In addition, at depths of greater than 600 m, the wellbore CO2 temperature is higher than its critical value. Fig. 9 shows the impact of the injection temperature on the wellbore pressure profile. As Fig. 9 illustrates, the injection temperature does not affect the pressure profile inside either the tubing or the annulus. Therefore, it can be concluded that injection temperature is not a key factor affecting SC-CO2 coiled tubing drilling.

The nozzle diameter is also an important parameter that needs to be optimized during SC-CO2 drilling. If the nozzle diameter is too large, it cannot play the role of a jet in assisting in the breaking of rock, whereas if the nozzle is too small, it will produce a large pressure and temperature drop when the CO2 flows through the nozzle, which may lead to freezing of the downhole. Therefore, the impact of the nozzle diameter on the wellbore temperature and pressure is analyzed in this section. Nozzle diameters of 3.7 mm, 4.7 mm, and 5.7 mm are investigated. The other calculation parameters are as listed in Table 1. Fig. 10 illustrates the influence of nozzle diameter on the wellbore temperature profile. It can be seen that the temperature profiles in Fig. 10 are similar to the temperature profiles in Fig. 6. Fig. 10 shows that the nozzle diameter does not affect the wellbore temperature at depths from the wellhead to 1700 m, but it has a significant impact on the wellbore temperature near the downhole section. The smaller the nozzle diameter, the lower the temperature is near the downhole. This is because a smaller nozzle diameter results in a larger pressure drop when CO2 flows through the nozzle, which results in a larger temperature difference between the upstream and downstream of the nozzle, in accordance with Eq. (19). Upon closer examination, it can be observed that for a nozzle diameter of

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Fig. 4. Wellbore temperature profiles at different circulation times.

Fig. 5. Wellbore pressure profiles at different circulation times.

Fig. 6. Wellbore temperature profiles at different injection rates.

X.-G. Li et al. / International Journal of Heat and Mass Transfer 125 (2018) 400–412

Fig. 7. Wellbore pressure profiles at different injection rates.

Fig. 8. Wellbore temperature profiles at different injection temperatures.

Fig. 9. Wellbore pressure profiles at different injection temperatures.

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X.-G. Li et al. / International Journal of Heat and Mass Transfer 125 (2018) 400–412

Fig. 10. Wellbore temperature profiles for different nozzle diameters.

Fig. 11. Wellbore pressure profiles for different nozzle diameters.

3.7 mm, the temperature at the downhole is lower than the CO2 critical temperature after 10 h of circulation. Although the smaller nozzle diameter can make the CO2 jet exit the nozzle faster, which is beneficial for cutting rock, a nozzle diameter that is too small may cause the CO2 to no longer be in a supercritical state at the downhole, which is not conducive for cutting rock. Therefore, rock breaking should be considered in the optimization of the nozzle diameter and injection rate in future research. Fig. 11 illustrates the effect of nozzle diameter on the wellbore pressure profile. It can be seen that the nozzle diameter has little impact on the annulus pressure, but has a significant influence on the pressure inside the coiled tubing. In these three cases, the pressure drops when CO2 flows through the nozzle are 15.1 MPa, 6.3 MPa, and 2.9 MPa, respectively. Because the outlet pressure is fixed, the pressure inside the coiled tubing increases with decreasing nozzle diameter.

5. Conclusions (1) Although the wellbore temperature profile curves calculated with the proposed model are consistent with the wellbore temperature curves calculated with previous models, there are some important differences between the results from different models. The wellbore CO2 temperature predicted by Gupta’s model and Wang’s model are lower than that of the model proposed in this study. This is attributed to the previous models treating the heat transfer in the wellbore as steady state, and ignoring friction heat. (2) In the shallower part of the well, the wellbore temperature increases rapidly with increasing well depth, whereas in the middle section of the well, the wellbore temperature increases linearly with increasing depth. However, near the downhole, the temperature decreases abruptly with increasing depth.

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(3) Near the injection point and the downhole, the wellbore temperature decreases with increasing circulation time, whereas in the middle part of the well, the circulation time has only a very small influence on the wellbore temperature. In addition, the circulation time has little effect on the wellbore pressure. (4) In the shallower and deeper parts of the well, the wellbore temperature decreases with increasing injection rate, and an injection rate that is too large may lead to CO2 that does not remain in a supercritical state. The pressure in the annulus and the tubing both increase with increasing injection rate under conditions in which the outlet pressure is fixed. (5) The wellbore temperature near the downhole decreases significantly with decreasing nozzle diameter. However, in other parts of the well, the temperature of the wellbore is not affected by the nozzle diameter. The pressure in the tubing increases as the nozzle diameter decreases, whereas the pressure in the annulus is not affected by the nozzle diameter. Conflict of interest The authors declared that there is no conflict of interest.

!

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

¼

! nþ1 nþ1 nþ1 Dt anþ1 Dt nþ1 J;j q1;j v 1;j c1;j nþ1 nþ1 nþ1 Q þ þ a v c pnþ1 J;j 1;j 1;j 1;j Dz j pr21 1;j 

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

þ

nþ1 nþ1 nþ1 Dtqnþ1 2Dth1;j T nþ1 1;j v 1;j c 1;j 2;j T nþ1 1;j1 þ Dz j r1

The iteration form is written as follows:

T nþ1 1;j

ðkþ1Þ

ðkÞ

¼ T nþ1 þ 1;j

x

a1j

ðkþ1Þ

b1j þ c1j T nþ1 1;j1

ðkÞ

ðkÞ

 a1j T nþ1 þ d1j T nþ1 1;j 2;j

 ðA:2Þ

where x is the iteration factor; a1j, b1j, c1j, and d1j are the corresponding coefficients. These coefficients can be expressed as follows:

a1j ¼

nþ1 nþ1 nþ1 Dt qnþ1 2Dth1;j 1;j v 1;j c1;j nþ1 þ þ qnþ1 1;j c 1;j Dzj r1

Acknowledgments

b1j ¼ This work was financially supported by the Open Fund (PLN1208) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) and the National Science and Technology Major Project of China (2016ZX05044-004-002).

Dt

pr21 þ

ðA:3aÞ

nþ1 nþ1 n Q nþ1 1;j þ q1;j c 1;j T 1;j

! nþ1 nþ1 nþ1 Dt anþ1 J;j q1;j v 1;j c 1;j nþ1 nþ1 pnþ1 þ anþ1 v c J;j 1;j 1;j 1;j Dz j

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 c 1;j p1;j Dzj



Appendix A. Equation discretization and numerical solutions To ensure the stability of the numerical solution, the fully implicit finite difference method is used to discretize the coupling equations in this study, and the fast successive over-relaxation (SOR) iteration method is used to solve the discrete equations. Because the heat transfer surrounding the wellbore is axisymmetric, the calculation nodes can be defined by a two-dimensional mesh grid, as shown in Fig. A.1. Therefore, the discrete equation for the heat transfer equation inside the coiled tubing can be expressed as:

ðA:1Þ

c1j ¼

nþ1 nþ1 Dt qnþ1 1;j v 1;j c 1;j Dz j

d1j ¼

2Dth1;j r1

ðA:3bÞ

ðA:3cÞ

nþ1

ðA:3dÞ

Similarly, the discretization and iterative equation of the coiled tubing heat transfer equation can be given as follows:

q2;j c2;j Dt ¼

nþ1

2r1 h1;j 2r 2 h2;j k2;jþ0:5 k2;j0:5 þ þ þ 2 Dzj Dzjþ0:5 Dzj Dzj0:5 r 22  r 21 r2  r21

q2;j c2;j T n2;j Dt þ

T nþ1 2;j

nþ1

þ

nþ1

þ

2r1 h1;j T nþ1 1;j r22  r21

þ

! T nþ1 2;j

k2;j0:5 T nþ1 k2;jþ0:5 T nþ1 2;j1 2;jþ1 þ Dzj Dzj0:5 Dzj Dzjþ0:5

nþ1 2r 2 h2;j T nþ1 3;j r 22  r21

ðkþ1Þ

ðkÞ

¼ T nþ1 þ 2;j

ð4Þ

x

b2j þ c2j T nþ1 1;j  ðkÞ þ f 2j T nþ1 3;j

a2j ðkÞ

þ e2j T nþ1 2;jþ1

ðkþ1Þ

ðkþ1Þ

þ d2j T nþ1 2;j1

 a2j T nþ1 2;j

ðkÞ

ð5Þ

where

a2j ¼

b2j ¼

q2;j c2;j Dt

nþ1

þ

q2;j c2;j T n2;j Dt

nþ1

2r 1 h1;j 2r 2 h2;j k2;jþ0:5 k2;j0:5 þ þ þ 2 Dzj Dzjþ0:5 Dzj Dzj0:5 r 22  r 21 r 2  r 21

ðA:6aÞ

ðA:6bÞ

nþ1

c2j ¼ Fig. A.1. Schematic diagram of the mesh grids of the computational domain.

2r1 h1;j r22

 r21

ðA:6cÞ

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X.-G. Li et al. / International Journal of Heat and Mass Transfer 125 (2018) 400–412

d2j ¼

k2;j0:5 Dzj Dzj0:5

e2j ¼

k2;jþ0:5 Dzj Dzjþ0:5

f 2j ¼

nþ1 2r 2 h2;j r 22  r21

ðA:6dÞ

The corresponding coefficients can be expressed as follows: nþ1 qnþ1 3;j c 3;j

a3j ¼

Dt

ðA:6eÞ

Q nþ1 2;j

b3j ¼

pðr23  r22 Þ

ðA:6fÞ

nþ1 qnþ1 3;j c 3;j

Dt

nþ1

þ

ðr 23

Q nþ1 2;j

¼



þ

r 22 Þ

2r3 h3;j ðr23

þ

nþ1 nþ1 nþ1 þ anþ1 J;j q3;j v 3;j c 3;j

pðr23  r22 Þ nþ1 n pnþ1 qnþ1 3;j 3;j c3;j T 3;j 



r22 Þ

þ

nþ1 qnþ1 3;j v 3;j c 3;j

pnþ1 3;jþ1

Dz

nþ1 nþ1 nþ1  anþ1 J;j q3;j v 3;j c 3;j

ðkþ1Þ

T nþ1 3;j

ðkÞ

¼ T nþ1 þ 3;j

x a3j

ðkþ1Þ

b3j þ c3j T nþ1 2;j

ðkþ1Þ

þ d3j T nþ1 3;jþ1

4

q4;j c4;j

¼

Dt

ðkÞ

ðkÞ

 a3j T nþ1 þ e3j T nþ1 3;j 4;j

k4;jþ0:5 k4;j0:5 2k4;j k5;j 
þ þ þ Dzj Dzjþ0:5 Dzj Dzj0:5 k5;j ln r42rþr4 5 þ k4;j ln r42rþr5 5 r24  r23

q4;j c4;j T n4;j Dt

nþ1

þ

2r3 h3;j T nþ1 3;j ðr 24  r 23 Þ

þ

T nþ1 4;j

ðkþ1Þ

2Dtki;j ki1;j






þri i r2i  r2i1 qi;j ci;j ki;j ln ri1 þ ki1;j ln r 2rþr 2r i1 i1 i



q4;j c4;j

a4j ¼

Dt

T nþ1 i;j




ðkþ1Þ

k4;jþ0:5 k4;j0:5 þ Dzj Dzjþ0:5 Dzj Dzj0:5

3 nþ1 2r3 h3;j 
5 þ
2 r 4  r 23 r2  r2 4

ðA:14aÞ

3

q4;j c4;j T n4;j

b4j ¼

ðA:14bÞ

Dt

ðA:14cÞ

d4j ¼

k4;j0:5 Dzj Dzj0:5

ðA:14dÞ

e4j ¼

k4;jþ0:5 Dzj Dzjþ0:5

ðA:14eÞ

þ

ðA:14fÞ

2Dtki;j ki1;j  
 r i1 þr i i r 2i  r 2i1 qi;j ci;j ki;j ln 2ri1 þ ki1;j ln ri12rþr i 2Dtki;j kiþ1;j






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

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

ðkþ1Þ ðkþ1Þ ðkÞ x ¼ T nþ1 þ bij þ cij T nþ1 þ dij T nþ1  aij T nþ1 i;j i1;j i;j1 i;j aij  ðkÞ nþ1ðkÞ ðA:12Þ þ eij T nþ1 i;jþ1 þ f ij T iþ1;j

ðA:15bÞ

2ki;j ki1;j Dt  r i1 þr i qi;j ci;j ki;j ln 2r þ ki1;j ln r i1

ðA:11Þ dij ¼

ðkÞ

eij ¼

 ðA:15aÞ

bij ¼ T ni;j cij ¼



þ

þ k5;j ln r42rþr4 5 þ k4;j ln r42rþr5 5

þ

 
 þri i qi;j ci;j ki;j ln ri1 þ ki1;j ln ri12rþr r2i  r2i1 2r i1 i



ðA:13eÞ

2k4;j k5;j

2Dtki;j ki1;j T nþ1 i1;j

2r qi;j ci;j kiþ1;j ln ri þr2riþ1 þ ki;j ln r þriþ1 r2i  r2i1 i i iþ1

ðA:13dÞ

Dz

aij ¼ 1 þ

Dtki;j0:5 T nþ1 Dtki;jþ0:5 T nþ1 i;j1 i;jþ1 þ þ qi;j ci;j Dzj Dzj0:5 qi;j ci;j Dzj Dzjþ0:5 þ

ðA:13cÞ

2k4;j k5;j 
f 4j ¼   k5;j ln r42rþr4 5 þ k4;j ln r42rþr5 5 r 24  r23

2Dtki;j kiþ1;j  
 ri þr iþ1 qi;j ci;j kiþ1;j ln 2ri þ ki;j ln ri2rþriþ1iþ1 r2i  r2i1 # ki;jþ0:5 Dt ki;j0:5 Dt T nþ1 ¼ T ni;j þ þ Dzj Dzjþ0:5 q2;j c2;j Dzj Dzj0:5 q2;j c2;j i;j

2Dtki;j kiþ1;j T nþ1 iþ1;j

ðA:13bÞ

Dt

nþ1

þ

þ

Dt

nþ1 nþ1 n anþ1 J;j v 3;j c 3;j p3;j

2r 3 h3;j  c4j ¼
2 r4  r23

ðkÞ ðkþ1Þ ðkþ1Þ ðkÞ x ¼ T nþ1 þ b4j þ c4j T nþ1 þ d4j T nþ1  a4j T nþ1 4;j 3;j 4;j1 4;j a4j  ðkÞ nþ1ðkÞ þ e4j T nþ1 ðA:10Þ 4;jþ1 þ f 4j T 5;j

2 41 þ

nþ1 2r 3 h
2 3;j2 5T nþ1 4;j r4  r3

ðA:9Þ


þ  k5;j ln r42rþr4 5 þ k4;j ln r42rþr5 5 r 24  r 23

Dz

nþ1 nþ1 n 3;j c 3;j T 3;j

ðr 23  r 22 Þ



k4;j0:5 T nþ1 k4;jþ0:5 T nþ1 4;j1 4;jþ1 þ Dzj Dzj0:5 Dzj Dzjþ0:5

2k4;j k5;j T nþ1 5;j

q

pnþ1 3;jþ1

nþ1 nþ1 qnþ1 3;j v 3;j c 3;j

"

3

þ



þ

ðA:13aÞ

nþ1

ðA:8Þ 2

Dz

Dz

2r 3 h3;j

e3j ¼

ðA:7Þ 

pnþ1 nþ1 nþ1 3;j 3;j c 3;j

nþ1 nþ1 qnþ1 3;j v 3;j c 3;j

ðr23  r 22 Þ

d3j ¼

nþ1 nþ1 nþ1 anþ1 J;j v 3;j c 3;j p3;j

nþ1 2r 3 h3;j T nþ1 4;j
2  r 3  r 22

þ

ðr23  r22 Þ

nþ1 nþ1 nþ1 þ anþ1 J;j q3;j v 3;j c 3;j

Dt

þ

2r2 h2;j

c3j ¼

þ Dt Dt nþ1 nþ1 nþ1 nþ1 n anþ1 v c p 2r h T T nþ1 2 2;j 3;j J;j 3;j 3;j 2;j nþ1 nþ1 nþ1 3;jþ1   q v c þ
2 þ 3;j 3;j 3;j Dt Dz r 3  r 22

Dz

nþ1

2r3 h3;j

þ

nþ1

T nþ1 3;j

Dz

ðr 23  r 22 Þ

nþ1 nþ1 nþ1 anþ1 J;j v 3;j c 3;j p3;j

þ

# nþ1

nþ1

2r 2 h2;j

2r 2 h2;j

nþ1  anþ1 J;j q3;j v

Similarly, the discretization and iterative form of the heat transfer equation for the annulus, casing, and formation can be expressed as given in Eqs. (A.7)–(A.12).

"

nþ1

þ

ki;j0:5 Dt

qi;j ci;j Dzj Dzj0:5 ki;jþ0:5 Dt

qi;j ci;j Dzj Dzjþ0:5

2ri i1 þri




r 2i  r2i1



ðA:15cÞ

ðA:15dÞ

ðA:15eÞ

X.-G. Li et al. / International Journal of Heat and Mass Transfer 125 (2018) 400–412

411

Fig. A.2. Flow chart of the calculations.

f ij ¼

2Dtki;j kiþ1;j






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

ðA:15fÞ



The discretization and iterative form of the pressure equation in the coiled tubing can be written as Eqs. (A.16) and (A.17), respectively.



nþ1 f 1;j Dz

nþ1 nþ1 pnþ1 1;j1 ¼ p1;j  q1;j g Dz sin h þ   nþ1 nþ1 þ qnþ1 v nþ1 1;j v 1;j 1;j  v 1;j1

qnþ1 v nþ1 1;j 1;j

2

¼

ðkþ1Þ pnþ1 1;j

þ qnþ1 1;j

ðkÞ

q

nþ1ðkÞ g 1;j

4r 1 ðA:16Þ

v nþ1 1;j



ðkÞ

sin hDz þ

v nþ1 1;j

ðkÞ

nþ1ðkÞ f 1;j Dz

 v nþ1 1;j1

ðkÞ

qnþ1 1;j



v nþ1 1;j

ðkÞ

2

ðA:17Þ

The discretization and iterative form of the pressure equation in the annulus can be written as Eqs. (A.18) and (A.19), respectively.



nþ1 f 3;j Dz

nþ1 nþ1 pnþ1 3;j ¼ p3;j1  q3;j g Dz sin h    nþ1 nþ1  qnþ1 v nþ1 3;j v 3;j 3;j  v 3;j1

qnþ1 v nþ1 3;j 3;j

2

4ðr 3  r 2 Þ ðA:18Þ ðkþ1Þ

ðkþ1Þ pnþ1 3;j

¼

ðkþ1Þ pnþ1 3;j1

 qnþ1 3;j

q

ðkþ1Þ

nþ1ðkþ1Þ g 3;j

 ðkþ1Þ

v nþ1 3;j

sin hDz 

v nþ1 3;j

ðkþ1Þ

nþ1ðkþ1Þ f 3;j Dz

 v nþ1 3;j1

ðkÞ



qnþ1 3;j



Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.04.095. References

4r1



ðA:20Þ

Appendix B. Supplementary material

ðkÞ

ðkþ1Þ pnþ1 1;j1

  8

nþ1ðkþ1Þ nþ1ðkÞ > max p  p

6 ep1 > 1;j 1;j > >

 <  ðkþ1Þ ðkÞ

max pnþ1  pnþ1

6 ep3 3;j 3;j > >

  > > : max

T nþ1ðkþ1Þ  T nþ1ðkÞ

6 e i;j i;j T

v nþ1 3;j

ðkþ1Þ

2

4ðr 3  r 2 Þ ðA:19Þ

The iteration continues until both the pressure and temperature meet the convergence conditions given in Eq. (A.20). The flow chart of the calculation process is shown in Fig. A.2.

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